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Solo Hermelin
Solo Hermelin

Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements. Estimation is a prerequisite. The Probability Theory summary is included. The presentation is at graduate level in math and engineering. For comments please connect me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

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1 Recursive Bayesian Estimation SOLO HERMELIN Updated: 22.02.09 11.01.14 http://www.solohermelin.com
2 SOLO Table of Content Recursive Bayesian Estimation Review of Probability Conditional Probability Total Probability Theorem Conditional Probability - Bayes Formula Statistical Independent Events Expected Value or Mathematical Expectation Variance and Central Moments Characteristic Function and Moment-Generating Function Probability Distribution and Probability Density Functions (Examples) Normal (Gaussian) Distribution Existence Theorems 1 & 2 Monte Carlo Method Estimation of the Mean and Variance of a Random Variable Generating Discrete Random Variables Existence Theorem 3 Markov Processes Functions of one Random Variable The Laws of Large Numbers Central Limit Theorem Problem Definition Stochastic Processes
3 SOLO Table of Content (continue -1) Recursive Bayesian Estimation Bayesian Estimation Introduction Linear Gaussian Markov Systems Closed-Form Solutions of Estimation Kalman Filter Extended Kalman Filter General Bayesian Nonlinear Filters Additive Gaussian Nonlinear Filter Gauss – Hermite Quadrature Approximation Unscented Kalman Filter Monte Carlo Kalman Filter (MCKF) Non-Additive Non-Gaussian Nonlinear Filter Nonlinear Estimation Using Particle Filters Importance Sampling (IS) Sequential Importance Sampling (SIS) Sequential Importance Resampling (SIR) Monte Carlo Particle Filter (MCPF) Bayesian Maximum Likelihood Estimate (Maximum Aposteriori – MAP Estimate)
4 SOLO Table of Content (continue -2) Recursive Bayesian Estimation References Nonlinear Filters based on the Fokker-Planck Equation
5 SOLO Recursive Bayesian Estimation kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Since this is a probabilistic problem, we start with a remainder of Probability Theory A discrete nonlinear system is defined by ( ) ( )kkk kkk vxkhz wxkfx ,, ,,1 11 = −= −− State vector dynamics Measurements kk vw ,1− State and Measurement Noise Vectors, respectively Problem Definition: Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements . kx kz Table of Content
6 SOLO Pr (A) is the probability of the event A if S nAAAA ∪∪∪= 21 1A 2A nA jiOAA ji ≠∀/=∩ ( ) 0Pr ≥A(1) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 ( ) 1Pr =S(2) then ( ) ( ) ( ) ( )nAAAA PrPrPrPr 21 +++=  Probability Axiomatic Definition Probability Geometric Definition Assume that the probability of an event in a geometric region A is defined as the ratio between A surface to surface of S. ( ) ( ) ( )SSurface ASurface A =Pr ( ) 0Pr ≥A(1) ( ) 1Pr =S(2) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 then ( ) ( ) ( ) ( )nAAAA PrPrPrPr 21 +++=  S A Review of Probability A more detailed explanation of the subject is given in the “Probability” Presentation
7 SOLO From those definition we can prove the following:( ) 0=/OP(1’) Proof: OOSandOSS /=/∩/∪= ( ) ( ) ( ) ( ) ( ) 0PrPrPrPr 3 =/⇒/+=⇒ OOSS ( ) ( )APAP −= 1(2’) Proof: OAAandAAS /=∩∪= ( ) ( ) ( ) ( ) ( ) ( ) ( )AAAAS Pr1PrPrPr1Pr 32 −=⇒+==⇒ ( ) 1Pr0 ≤≤ A(3’) Proof: ( ) ( ) ( ) ( ) ( ) 1Pr0Pr1Pr 1'2 ≤⇒≥−= AAA ( ) ( )APr0 1 ≤ ( ) 0Pr ≥A(1) ( ) 1Pr =S(2) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 then ( ) ( ) ( ) ( )n AAAA PrPrPrPr 21 +++=  ( ) ( )AABAIf PrPr ≤⇒⊂(4’) Proof: ( ) ( ) ( ) ( ) ( ) ( )BAAABB PrPr0PrPrPr 00 3 ≤⇒≥+−= ≥≥  ( ) ( ) OAABandAABB /=∩−∪−= ( ) ( ) ( ) ( )BABABA ∩−+=∪ PrPrPrPr(5’) Proof: ( ) ( ) ( ) ( ) ( ) ( ) OABBAandABBAB OABAandABABA /=−∩∩−∪∩= /=−∩−∪=∪ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )BABABA ABBAB ABABA ∩−+=∪⇒     −+∩= −+=∪ PrPrPrPr PrPrPr PrPrPr 3 3 Table of Content Review of Probability
8 SOLO Conditional Probability S nAAAA ααα ∪∪∪= 21  1αA jiOAA ji ≠∀/=∩ 1αβA mAAAB βββ ∪∪∪= 212αA 2αβA 1βA 2βA  Given two events A and B decomposed in elementary events jiOAAandAAAAA ji n i in ≠∀/=∩=∪∪∪= = αααααα  1 21 lkOAAandAAAAB lk m k km ≠∀/=∩=∪∪∪= = ββββββ  1 21 jiOAAandAAABA jir ≠∀/=∩∪∪∪=∩ αβαβαβαβαβ 21 ( ) ( ) ( ) ( )n AAAA ααα PrPrPrPr 21 +++=  ( ) ( ) ( ) ( )mAAAB βββ PrPrPrPr 21 +++=  ( ) ( ) ( ) ( ) nmrAAABA r ,PrPrPrPr 21 ≤+++=∩ βαβαβα  We want to find the probability of A event under the condition that the event B had occurred designed as P (A|B) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )B BA AAA AAA BA m r Pr Pr PrPrPr PrPrPr |Pr 21 21 ∩ = +++ +++ = βββ βαβαβα   Review of Probability
9 SOLO Conditional Probability S nAAAA ααα ∪∪∪= 21  1αA jiOAA ji ≠∀/=∩ 1αβA mAAAB βββ ∪∪∪= 212αA 2αβA 1βA 2βA  If the events A and B are statistical independent, that the fact that B occurred will not affect the probability of A to occur. ( ) ( ) ( )B BA BA Pr Pr |Pr ∩ = ( ) ( ) ( )A BA AB Pr Pr |Pr ∩ = ( ) ( )ABA Pr|Pr = ( ) ( ) ( ) ( ) ( ) ( ) ( )BAAABBBABA PrPrPr|PrPr|PrPr ⋅=⋅=⋅=∩ Definition: n events Ai i = 1,2,…n are statistical independent if: ( ) nrAA r i i r i i ,,2PrPr 11  =∀=      ∏== Table of Content Review of Probability
10 SOLO Conditional Probability - Bayes Formula Using the relation: ( ) ( ) ( ) ( ) ( )llll AABBBABA ββββ Pr|PrPr|PrPr ⋅=⋅=∩ ( ) ( ) ( ) klOBABABAB lk m k k , 1 ∀/=∩∩∩∩= = βββ ( ) ( )∑ = ∩= m k k BAB 1 PrPr β we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑= ⋅ ⋅ = ⋅ = m k kk llll l AAB AAB B AAB BA 1 Pr|Pr Pr|Pr Pr Pr|Pr |Pr ββ ββββ β Bayes Formula Thomas Bayes 1702 - 1761 Table of Content Review of Probability
11 SOLO Total Probability Theorem Table of Content jiOAAandSAAA jin ≠∀/=∩=∪∪∪ 21If we say that the set space S is decomposed in exhaustive and incompatible (exclusive) sets. The Total Probability Theorem states that for any event B, its probability can be decomposed in terms of conditional probability as follows: ( ) ( ) ( ) ( )∑∑ == == n i i n i i BPBABAB 11 |Pr,PrPr Using the relation: ( ) ( ) ( ) ( ) ( )llll AABBBABA Pr|PrPr|PrPr ⋅=⋅=∩ ( ) ( ) ( ) klOBABABAB lk n k k , 1 ∀/=∩∩∩∩= =  ( ) ( )∑= ∩= n k k BAB 1 PrPr For any event B we obtain: Review of Probability
12 SOLO Statistical Independent Events ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∏∑∏∑∏∑ ∑∑∑ = −       ≠≠ =       ≠ =       = = −       ≠≠       ≠       == −+−+−=       −+−+−=      n i i n n kji kji i i n ji ji i i n i i tIndependen lStatisticaA n i i n n kji kji kji n ji ji ji n i i n i i AAAA AAAAAAAA i 1 1 3 ,. 3 1 2 . 2 1 1 1 1 1 3 ,. 2 . 1 11 Pr1PrPrPr Pr1PrPrPrPr    From Theorem of Addition Therefore ( )[ ]∏== −=      − n i i tIndependen lStatisticaA n i i AA i 11 Pr1Pr1  ( )[ ]∏== −−=      n i i tIndependen lStatisticaA n i i AA i 11 Pr11Pr  Since OAASAA n i i n i i n i i n i i /=               =               ====   1111 &         =      − ==  n i i n i i AA 11 PrPr1 ( )∏== =      n i i tIndependen lStatisticaA n i i AA i 11 PrPr  If the n events Ai i = 1,2,…n are statistical independent than are also statistical independentiA ( )∏= = n i iA 1 Pr      = =  n i i MorganDe A 1 Pr ( )[ ]∏= −= n i i tIndependen lStatisticaA A i 1 Pr1 ( ) nrAA r i i r i i ,,2PrPr 11  =∀=      ∏== Table of Content Review of Probability
13 SOLO Review of Probability Expected Value or Mathematical Expectation Given a Probability Density Function p (x) we define the Expected Value For a Continuous Random Variable: ( ) ( )∫ +∞ ∞− = dxxpxxE X: For a Discrete Random Variable: ( ) ( )∑= k kXk xpxxE : For a general function g (x) of the Random Variable x: ( )[ ] ( ) ( )∫ +∞ ∞− = dxxpxgxgE X: ( )xp x 0 ∞+∞− 0.1 ( )xE ( ) ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = dxxp dxxpx xE X X : The Expected Value is the center of surface enclosed between the Probability Density Function and x axis. Table of Content
14 SOLO Review of Probability Variance Given a Probability Density Functions p (x) we define the Variance ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( )22222 2: xExExExExxExExExVar −=+−=−= Central Moment ( ) { }k k xEx =:'µ Given a Probability Density Functions p (x) we define the Central Moment of order k about the origin ( ) ( )[ ]{ } ( ) ( )∑= −− −      =−= k j jk j jkk k xE j k xExEx 0 '1: µµ Given a Probability Density Functions p (x) we define the Central Moment of order k about the Mean E (x) Table of Content
15 SOLO Review of Probability Moments Normal Distribution ( ) ( ) ( )[ ] σπ σ σ 2 2/exp ; 22 x xpX − = [ ] ( )    −⋅ = oddnfor evennforn xE n n 0 131 σ [ ] ( )      += =−⋅ = + 12!2 2 2131 12 knfork knforn xE kk n n σ π σ Proof: Start from: and differentiate k time with respect to a( ) 0exp 2 >=−∫ ∞ ∞− a a dxxa π Substitute a = 1/(2σ2 ) to obtain E [xn ] ( ) ( ) 0 2 1231 exp 12 22 > −⋅ =− + ∞ ∞− ∫ a a k dxxax kk k π [ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 12 ! 0 122/ 0 222221212 !2 2 exp 2 22 2/exp 2 2 2/exp 2 1 2 + ∞+ = ∞∞ ∞− ++ =−= −=−= ∫ ∫∫ kk k k k xy kkk kdyyy xdxxxdxxxxE σ πσ σ π σ σπ σ σπ σ    Now let compute: [ ] [ ]( )2244 33 xExE == σ Chi-square
16 SOLO Review of Probability Functions of one Random Variable Let y = g (x) a given function of the random variable x defined o the domain Ω, with probability distribution pX (x). We want to find pY (y). Fundamental Theorem Assume x1, x2, …, xn all the solutions of the equation ( ) ( ) ( )n xgxgxgy ==== 21 ( ) ( ) ( ) ( ) ( ) ( ) ( )n nXXX Y xg xp xg xp xg xp yp ''' 2 2 1 1 +++=  ( ) ( ) xd xgd xg =:' Proof ( ) ( ) ( ) ( ) ( ) ( )∑∑∑ === ==±≤≤=+≤≤= n i i iX n i iiX n i iiiY yd xg xp xdxpxdxxxydyYyydyp 111 ' PrPr: q.e.d.
17 SOLO Review of Probability Functions of one Random Variable (continue – 1) Example 1 bxay += ( )       − = a by p a yp XY 1 Example 2 x a y = ( )       = y a p y a yp XY 2 Example 3 2 xay = ( ) ( )yU a y p a y p ya yp XXY                 −+         = 2 1 Example 4 xy = ( ) ( ) ( )[ ] ( )yUypypyp XXY −+= Table of Content
18 SOLO Review of Probability Characteristic Function and Moment-Generating Function Given a Probability Density Functions pX (x) we define the Characteristic Function or Moment Generating Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )     = ==Φ ∑ ∫∫ +∞ ∞− +∞ ∞− x X XX X discretexxpxj continuousxxPdxjdxxpxj xjE ω ωω ωω exp expexp exp: This is in fact the complex conjugate of the Fourier Transfer of the Probability Density Function. This function is always defined since the sufficient condition of the existence of a Fourier Transfer : Given the Characteristic Function we can find the Probability Density Functions pX (x) using the Inverse Fourier Transfer: ( ) ( ) ( ) ∞<== ∫∫ +∞ ∞− ≥+∞ ∞− 1 0 dxxpdxxp X xp X ( ) ( ) ( )∫ +∞ ∞− Φ−= ωωω π dxjxp XX exp 2 1 is always fulfilled.
19 SOLO Review of Probability Properties of Moment-Generating Function ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X X ω ω ω exp ( ) ( ) 10 ==Φ ∫ +∞ ∞− = dxxpXX ω ω ( ) ( ) ( )xEjdxxpxj d d X X == Φ ∫ +∞ ∞−=0ω ω ω ( ) ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X X 22 2 2 exp ω ω ω ( ) ( ) ( ) ( ) ( )2222 0 2 2 xEjdxxpxj d d X X == Φ ∫ +∞ ∞−=ω ω ω ( ) ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X nn n X n ω ω ω exp ( ) ( ) ( ) ( ) ( )nn X nn n X n xEjdxxpxj d d == Φ ∫ +∞ ∞−=0ω ω ω   ( ) ( ) ( )∫ +∞ ∞− =Φ dxxpxj XX ωω exp This is the reason why ΦX (ω) is also called the Moment-Generation Function.
20 SOLO Review of Probability Properties of Moment-Generating Function ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   +++++= + Φ ++ Φ + Φ +Φ=Φ === = n n n n X n XX XX xE n j xE j xE j d d nd d d d !!2!1 1 ! 1 !2 1 2 2 0 2 0 2 2 0 0 ωωω ω ω ω ω ω ω ω ω ω ωω ωωω ω Develop ΦX (ω) in a Taylor series ( ) ( ) ( )∫ +∞ ∞− =Φ dxxpxj XX ωω exp
21 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (2) Poisson’s Distribution ( ) ( )0 0 exp ! , k k k nkp k −≈ (1) Binomial (Bernoulli) ( ) ( ) ( ) ( ) knkknk pp k n pp knk n nkp −− −      =− − = 11 !! ! , 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k ( )nkP , (3) Normal (Gaussian) ( ) ( ) ( )[ ] σπ σµ σµ 2 2/exp ,; 22 −− = x xp (4) Laplacian Distribution ( )         − −= b x b bxp µ µ exp 2 1 ,;
22 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (5) Gama Distribution ( ) ( ) ( )      < ≥ Γ − = − 00 0 /exp ,; 1 x xx k x kxp k k θ θ θ (6) Beta Distribution ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 1 0 11 11 1 1 1 ,; −− −− −− − ΓΓ +Γ = − − = ∫ βα βα βα βα βα βα xx duuu xx xp (7) Cauchy Distribution ( ) ( )       +− = 22 0 0 1 ,; γ γ π γ xx xxp
23 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) SOLO (8) Exponential Distribution ( ) ( )    < ≥− = 00 0exp ; x xx xp λλ λ (9) Chi-square Distribution ( ) ( ) ( ) ( )      < ≥− Γ= − 00 02/exp 2/ 2/1 ; 12/ 2/ x xxx kkxp k k Γ is the gamma function ( ) ( )∫ ∞ − −=Γ 0 1 exp dttta a (10) Student’s t-Distribution ( ) ( )[ ] ( ) ( )( ) 2/12 /12/ 2/1 ; + +Γ +Γ = ν ννπν ν ν x xp
24 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) SOLO (11) Uniform Distribution (Continuous) ( )      >> ≤≤ −= bxxa bxa abbaxp 0 1 ,; (12) Rayleigh Distribution ( ) 2 2 2 2 exp ; σ σ σ       − = x x xp (13) Rice Distribution ( )             + − = 202 2 22 2 exp ,; σσ σ σ vx I vx x vxp
25 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (14) Weibull Distribution SOLO ( )      < >≥               − −      − = − 00 0,,exp ,,; 1 x x xx xp αγµ α µ α µ α γ αµγ γγ Table of Content
26 SOLO Review of Probability Normal (Gaussian) Distribution Karl Friederich Gauss 1777-1855 ( ) ( ) ( )σµ σπ σ µ σµ ,;: 2 2 exp ,; 2 2 x x xp N=       − − = ( ) ( ) ∫ ∞−       − −= x du u xP 2 2 2 exp 2 1 ,; σ µ σπ σµ ( ) µ=xE ( ) σ=xVar ( ) ( )[ ] ( ) ( )       −=       − −= =Φ ∫ ∞+ ∞− 2 exp exp 2 exp 2 1 exp 22 2 2 σω µω ω σ µ σπ ωω j duuj u xjE Probability Density Functions Cumulative Distribution Function Mean Value Variance Moment Generating Function
27 SOLO Review of Probability Moments Normal Distribution ( ) ( ) ( )[ ] ( )σ σπ σ σ ,0;: 2 2/exp ,0; 22 x x xpX N= − = [ ] ( )    −⋅ = oddnfor evennforn xE n n 0 131 σ [ ] ( )      += =−⋅ = + 12!2 2 2131 12 knfork knforn xE kk n n σ π σ Proof: Start from: and differentiate k time with respect to a( ) 0exp 2 >=−∫ ∞ ∞− a a dxxa π Substitute a = 1/(2σ2 ) to obtain E [xn ] ( ) ( ) 0 2 1231 exp 12 22 > −⋅ =− + ∞ ∞− ∫ a a k dxxax kk k π [ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 12 ! 0 122/ 0 222221212 !2 2 exp 2 22 2/exp 2 2 2/exp 2 1 2 + ∞+ = ∞∞ ∞− ++ =−= −=−= ∫ ∫∫ kk k k k xy kkk kdyyy xdxxxdxxxxE σ πσ σ π σ σπ σ σπ σ    Now let compute: [ ] [ ]( )2244 33 xExE == σ Chi-square
28 SOLO Review of Probability Normal (Gaussian) Distribution (continue – 1) Karl Friederich Gauss 1777-1855 ( ) ( ) ( ) ( )PxxxxPxxPPxxp T ,;: 2 1 exp2,; 12/1  N=    −−−= −− π A Vector – Valued Gaussian Random Variable has the Probability Density Functions where { }xEx  = Mean Value ( )( ){ }T xxxxEP  −−= Covariance Matrix If P is diagonal P = diag [σ1 2 σ2 2 … σk 2 ] then the components of the random vector are uncorrelated, and x  ( ) ( ) ( ) ( ) ( ) ∏= − −       − − =       − −      − −      − − =                               − − −                             − − − −= k i i i ii k k kk kk k T kk xxxxxxxx xx xx xx xx xx xx PPxxp 1 2 2 2 2 2 2 2 2 22 1 2 1 2 11 22 11 1 2 2 2 2 1 22 11 2/1 2 2 exp 2 2 exp 2 2 exp 2 2 exp 0 0 2 1 exp2,; σπ σ σπ σ σπ σ σπ σ σ σ σ π    therefore the components of the random vector are also independent
29 SOLO Review of Probability The Laws of Large Numbers The Law of Large Numbers is a fundamental concept in statistics and probability that describes how the average of randomly selected sample of a large population is likely to be close to the average of the whole population. There are two laws of large numbers the Weak Law and the Strong Law. The Weak Law of Large Numbers The Weak Law of Large Numbers states that if X1,X2,…,Xn,… is an infinite sequence of random variables that have the same expected value μ and variance σ2 , and are uncorrelated (i.e., the correlation between any two of them is zero), then ( ) nXXX nn /: 1 ++=  converges in probability (a weak convergence sense) to μ . We have { } ∞→=<− nforXn 1Pr εµ converges in probability The Strong Law of Large Numbers The Strong Law of Large Numbers states that if X1,X2,…,Xn,… is an infinite sequence of random variables that have the same expected value μ and variance σ2 , and are uncorrelated (i.e., the correlation between any two of them is zero), and E (|Xi|) < ∞ then ,i.e. converges almost surely to μ.{ } ∞→== nforXn 1Pr µ converges almost surely
3030 SOLO Review of Probability The Law of Large Numbers Differences between the Weak Law and the Strong Law The Weak Law states that, for a specified large n, (X1 + ... + Xn) / n is likely to be near μ. Thus, it leaves open the possibility that | (X1 + ... + Xn) / n − μ | > ε happens an infinite number of times, although it happens at infrequent intervals. The Strong Law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have for any positive value ε, the inequality | (X1 + ... + Xn) / n − μ | > ε is true only a finite number of times (as opposed to an infinite, but infrequent, number of times). Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law.
3131 SOLO Review of Probability The Law of Large Numbers Proof of the Weak Law of Large Numbers ( ) iXE i ∀= µ ( ) iXVar i ∀= 2 σ ( )( )[ ] jiXXE ji ≠∀=−− 0µµ ( ) ( ) ( )[ ] µµ ==++= nnnXEXEXE nn //1  ( ) ( )[ ]{ } ( ) ( ) ( )( )[ ] ( )[ ] ( )[ ] nn n n XEXE n XX E n XX EXEXEXVar n jiXXE nn nnn ji 2 2 2 2 22 1 0 2 1 2 12 σσµµ µµ µ µµ == −++− =               −++− =               − ++ =−= ≠∀=−−   Given we have: Using Chebyshev’s inequality on we obtain:nX ( ) 2 2 / Pr ε σ εµ n Xn ≤≥− Using this equation we obtain: ( ) ( ) ( ) n XXX nnn 2 2 1Pr1Pr1Pr ε σ εµεµεµ −≥≥−−≥>−−=≤− As n approaches infinity, the expression approaches 1. Chebyshev’s inequality q.e.d. Monte Carlo Integration Monte Carlo Integration Table of Content
3232 SOLO Review of Probability Central Limit Theorem The first version of this theorem was first postulated by the French-born English mathematician Abraham de Moivre in 1733, using the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This was published in1756 in “The Doctrine of Chance” 3th Ed. Pierre-Simon Laplace (1749-1827) Abraham de Moivre (1667-1754) This finding was forgotten until 1812 when the French mathematician Pierre-Simon Laplace recovered it in his work “Théory Analytique des Probabilités”, in which he approximate the binomial distribution with the normal distribution. This is known as the De Moivre – Laplace Theorem. De Moivre – Laplace Theorem The present form of the Central Limit Theorem was given by the Russian mathematician Alexandr Lyapunov in 1901. Alexandr Mikhailovich Lyapunov (1857-1918)
3333 SOLO Review of Probability Central Limit Theorem (continue – 1) Let X1, X2, …, Xm be a sequence of independent random variables with the same probability distribution function pX (x). Define the statistical mean: m XXX X m m +++ = 21 ( ) ( ) ( ) ( ) µ= +++ = m XEXEXE XE m m 21 ( ) ( )[ ]{ } ( ) ( ) ( ) mm m m XXX EXEXEXVar m mmmXm 2 2 22 21 22 σσµµµ σ ==               −++−+− =−==  Define also the new random variable ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : We have: The probability distribution of Y tends to become gaussian (normal) as m tends to infinity, regardless of the probability distribution of the random variable, as long as the mean μ and the variance σ2 are finite.
3434 SOLO Review of Probability Central Limit Theorem (continue – 2) ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : Proof The Characteristic Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) m X m i m i i m Y m X m j E m X jE m XXX jEYjE i               Φ=                     − =               − =               −++−+− ==Φ − = ∏ ω σ µω σ µ ω σ µµµ ωωω σ µexpexp expexp 1 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0/lim 2 1 !3 / !2 / !1 / 1 2222 33 1 22 0 =      Ο/      Ο/+−= +               − +               − +      − +=      Φ ∞→ − mmmm X E mjX E mjX E mj m m iii Xi ωωωω σ µω σ µω σ µωω σ µ     Develop in a Taylor series( )       Φ − miX ω σ µ
35 SOLO Review of Probability Central Limit Theorem (continue – 3) Proof (continue – 1) The Characteristic Function ( ) ( ) m XY m E i               Φ=Φ − ω ω σ µ ( ) 0/lim 2 1 2222 =      Ο/      Ο/+−=      Φ ∞→ − mmmmm m Xi ωωωωω σ µ ( ) ( )2/exp 2 1 2 22 ω ωω ω −→            Ο/+−=Φ ∞→m m Y mm Therefore ( ) ( ) ( ) ( ) ( )2/exp 2 1 2/exp 2 1 exp 2 1 22 ydyjdyjyp m YY −=−−→Φ−= ∫∫ +∞ ∞− ∞→+∞ ∞− π ωωω π ωωω π The probability distribution of Y tends to become gaussian (normal) as m tends to infinity (Convergence in Distribution). Characteristic Function of Normal Distribution Convergence Concepts Monte Carlo Integration Table of Content
36 SOLO Review of Probability Central Limit Theorem (continue – 2) ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : Proof The Characteristic Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) m X m i m i i m Y m X m j E m X jE m XXX jEYjE i               Φ=                     − =               − =               −++−+− ==Φ − = ∏ ω σ µω σ µ ω σ µµµ ωωω σ µexpexp expexp 1 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0/lim 2 1 !3 / !2 / !1 / 1 2222 33 1 22 0 =      Ο/      Ο/+−= +               − +               − +      − +=      Φ ∞→ − mmmm X E mjX E mjX E mj m m iii Xi ωωωω σ µω σ µω σ µωω σ µ     Develop in a Taylor series( )       Φ − miX ω σ µ
37 SOLO Review of Probability Central Limit Theorem (continue – 3) Proof (continue – 1) The Characteristic Function ( ) ( ) m XY m E i               Φ=Φ − ω ω σ µ ( ) 0/lim 2 1 2222 =      Ο/      Ο/+−=      Φ ∞→ − mmmmm m Xi ωωωωω σ µ ( ) ( )2/exp 2 1 2 22 ω ωω ω −→            Ο/+−=Φ ∞→m m Y mm Therefore ( ) ( ) ( ) ( ) ( )2/exp 2 1 2/exp 2 1 exp 2 1 22 ydyjdyjyp m YY −=−−→Φ−= ∫∫ +∞ ∞− ∞→+∞ ∞− π ωωω π ωωω π The probability distribution of Y tends to become gaussian (normal) as m tends to infinity (Convergence in Distribution). Characteristic Function of Normal Distribution Convergence Concepts Table of Content
38 SOLO Review of Probability Existence Theorems Existence Theorem 1 Given a function G (x) such that ( ) ( ) ( ) 1lim,1,0 ==∞+=∞− ∞→ xGGG x ( ) ( ) 2121 0 xxifxGxG <=≤ ( G (x) is monotonic non-decreasing) ( ) ( ) ( )xGxGxG n xx xx n n == ≥ → + lim We can find an experiment X and a random variable x, defined on X, such that its distribution function P (x) equals the given function G (x). Proof of Existence Theorem 1 Assume that the outcome of the experiment X is any real number -∞ <x < +∞. We consider as events all intervals, the intersection or union of intervals on the real axis. 5x 1x 2x 3x 4x 6x 7x 8x ∞− ∞+ To specify the probability of those events we define P (x)=Prob { x ≤ x1}= G (x1). From our definition of G (x) it follows that P (x) is a distribution function. Existence Theorem 2 Existence Theorem 3
39 SOLO Review of Probability Existence Theorems Existence Theorem 2 If a function F (x,y) is such that ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, 1,,0,, 11122122 ≥+−− =+∞∞+=−∞=∞− yxFyxFyxFyxF FxFyF for every x1 < x2, y1 < y2, then two random variables x and y can be found such that F (x,y) is their joint distribution function. Proof of Existence Theorem 2 Assume that the outcome of the experiment X is any real number -∞ <x < +∞. Assume that the outcome of the experiment Y is any real number -∞ <y < +∞. We consider as events all intervals, the intersection or union of intervals on the real axes x and y. To specify the probability of those events we define P (x,y)=Prob { x ≤ x1, y ≤ y1, }= F (x1,y1). From our definition of F (x,y) it follows that P (x,y) is a joint distribution function. The proof is similar to that in the Existence Theorem 1
40 SOLO Review of Probability Monte Carlo Method Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm. The term Monte Carlo method was coined in the 1940s by physicists Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, working on nuclear weapon projects in the Los Alamos National Laboratory (reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble) Stanislaw Ulam 1909 - 1984 Enrico - Fermi 1901 - 1954 John von Neumann 1903 - 1957 Monte Carlo Casino Nicholas Constantine Metropolis (1915 –1999)
41 SOLO Review of Probability Monte Carlo Approximation Monte Carlo runs, generate a set of random samples that approximate the distribution p (x). So, with P samples, expectations with respect to the filtering distribution are approximated by ( ) ( ) ( ) ( )∑∫ = ≈ P L L xf P dxxpxf 1 1 and , in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some degree of approximation. { } ( ) ( ) ∑∫ = ≈== P L L x P dxxpxxE 1 1 1 µ ( ){ } ( ) ( ) ( ) ( )∑∫ = −≈−=−= P L nLnn n x P dxxpxxE 1 111 1 µµµµ  Table of Content x(L) are generated (draw) samples from distribution p (x) ( ) ( )xpx L ~
42 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (Unknown Statistics) { } { } jimxExE ji ,∀== Define Estimation of the Population mean ∑= = k i ik x k m 1 1 :ˆ A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ ( ) { } ( ) ( ) ( )[ ] ( ) ( )[ ] 2 1 2 1 222 2 22222 1 11 2 1 2 2 11 2 1 2 11 1 1 1 1 1 21 11 2 1 ˆˆ2 1 ˆ 1 σσ σσσ k k kk mkmkk k mmk k m k xx k Ex k xExE k mxmxE k mx k E k i k i k i k l l k j j k j jii k k i ik k i i k i ki − =      −=       ++−+++−−+=               +       −=       +−=       − ∑ ∑ ∑ ∑∑∑ ∑∑∑ = = = === === { } { } jimxExE ji ,2222 ∀+== σ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } { } { } jimxExExxE ji tindependenxx ji ji ,2 , ∀== Compute Biased Unbiased Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
43 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 1) { } { } jimxExE ji ,∀== Define Estimation of the Population mean ∑= = k i ik x k m 1 1 :ˆ A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ ( ) 2 1 2 1 ˆ 1 σ k k mx k E k i ki − =       −∑= { } { } jimxExE ji ,2222 ∀+== σ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } { } { } jimxExExxE ji tindependenxx ji ji ,2 , ∀== Biased Unbiased Therefore, the unbiased estimation of the sample variance of the population is defined as: ( )∑= − − = k i kik mx k 1 22 ˆ 1 1 :ˆσ since { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE Unbiased Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
44 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 2) A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
45 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 3) { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EEWe found: Let Compute: ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ){ } ( ){ } k mxEmxEmxE k mxmxEmxE k mx k Emx k EmmE k i k ij j ji k i i k i k ij j ji k i i k i i k i ikmk 2 1 1 00 1 2 2 1 11 2 2 2 1 2 1 22 ˆ 2 1 1 11 ˆ: σ σ σ =           −−+−=           −−+−=               −=               −=−= ∑ ∑∑ ∑∑∑ ∑∑ = ≠ == = ≠ == ==  ( ){ } k mmE kmk 2 22 ˆ ˆ: σ σ =−=
46 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 4) Let Compute: ( ){ } ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )               −− − +− − − +− − =               −−+−−+− − =               −−+− − =               −− − =−= ∑∑ ∑ ∑∑ == = == 2 22 11 2 2 2 1 22 2 2 1 2 2 2 1 22222 ˆ ˆ 11 ˆ2 1 1 ˆˆ2 1 1 ˆ 1 1 ˆ 1 1 ˆ:2 σ σ σσσσσσ k k i i k k i i k i kkii k i ki k i kik mm k k mx k mm mx k E mmmmmxmx k E mmmx k Emx k EE k ( ) ( ){ } ( ){ } ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ }                 k k k i i k k i i k k k i i k k i i k k k i i k k k k i i k k k i k ij j ji k k i i mmE k k mxE k mmE mxE k mmEk mxE k mxE k mmEk mxE k mmE mmE k k mxE k mmE mxEmxEmxE kk / 2 2 1 0 2 0 1 0 2 3 1 2 2 1 2 2 / 2 1 3 2 0 44 2 2 1 2 2 / 2 1 1 22 1 4 2 2 ˆ 2 222 22 22 4 2 ˆ 1 2 1 ˆ4 1 ˆ4 1 2 1 ˆ2 1 ˆ4 ˆ 11 ˆ4 1 1 σ σσσ σσ σσ µ σ σσ σ σσ − − −− − − −− − − + − − −− − − +− − − + +− − +− − − +             −−+− − ≈ ∑∑ ∑∑∑ ∑∑ ∑∑ == === == ≠ == Since (xi – m), (xj - m) and are all independent for i ≠ j:( )kmm ˆ−
47 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 4) Since (xi – m), (xj - m) and are all independent for i ≠ j:( )kmm ˆ− ( ) ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }4 2 2 4 22 4 44 2 4 44 2 2 2 4 2 4 2 42 ˆ ˆ 11 7 11 2 1 2 1 2 ˆ 11 4 1 1 1 2 k k mmE k k k k k k kk k k k mmE k k kk kk k k k − − + − +− + − = − − − − − + +− − + − + − − + − ≈ σ µσσσ σ σσµ σσ kk 4 42 ˆ 2 σµ σσ − ≈ ( ){ }4 4 : mxE i −=µ ( ) ( ){ } ( ){ } ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ }                 k k k i i k k i i k k k i i k k i i k k k i i k k k k i i k k k i k ij j ji k k i i mmE k k mxE k mmE mxE k mmEk mxE k mxE k mmEk mxE k mmE mmE k k mxE k mmE mxEmxEmxE kk / 2 2 1 0 2 0 1 0 2 3 1 2 2 1 2 2 / 2 1 3 2 0 44 2 2 1 2 2 / 2 1 1 22 1 4 2 2 ˆ 2 222 22 22 4 2 ˆ 1 2 1 ˆ4 1 ˆ4 1 2 1 ˆ2 1 ˆ4 ˆ 11 ˆ4 1 1 σ σσσ σσ σσ µ σ σσ σ σσ − − −− − − −− − − + − − −− − − +− − − + +− − +− − − +             −−+− − ≈ ∑∑ ∑∑∑ ∑∑ ∑∑ == === == ≠ ==
48 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 5) { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE We found: ( ){ } k mmE kmk 2 22 ˆ ˆ: σ σ =−= ( ){ } ( ) k mx k EE k i kik k 4 4 2 2 1 22222 ˆ ˆ 1 1 ˆ:2 σµ σσσσσ − ≈               −− − =−= ∑= ( ){ }4 4 : mxE i −=µ Kurtosis of random variable xi Define 4 4 : σ µ λ = ( ){ } ( ) ( ) k mx k EE k i kik k 42 2 1 22222 ˆ 1 ˆ 1 1 ˆ:2 σλ σσσσσ − ≈               −− − =−= ∑=
49 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 6) [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n For high values of k, according to the Central Limit Theorem the estimations of mean and of variance are approximately Gaussian Random Variables. kmˆ 2 ˆkσ We want to find a region around that will contain σ2 with a predefined probability φ as function of the number of iterations k. 2 ˆkσ Since are approximately Gaussian Random Variables nσ is given by solving: 2 ˆkσ ϕζζ π σ σ =      −∫ + − n n d2 2 1 exp 2 1 nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 Cumulative Probability within nσ Standard Deviation of the Mean for a Gaussian Random Variable 22 k 22 1 ˆ- 1 σ λ σσσ λ σσ k n k n − ≤≤ − − 22 k 2 1 1 ˆ-1 1 σ λ σσ λ σσ         − − ≤≤        + − − k n k n ( ) ( ) ( ) ( )( )42222 1,0;ˆ~ˆ&,0;ˆ~ˆ σλσσσσ −−− kkkk kmmmk NN
50 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 7) [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n 22 k 22 1 ˆ- 1 σ λ σσσ λ σσ k n k n − ≤≤ − − 22 k 2 1 1 ˆ-1 1 σ λ σσ λ σσ         − − ≤≤        + − − k n k n 22 ˆ 1 2 k σ λ σσ k − = 22 k 2 1 1ˆ 1 1 σ λ σσ λ σσ         − −≥≥        − + k n k n         − − ≥≥         − + k n k n 1 1 ˆ 1 1 2 2 k 2 λ σ σ λ σ σσ k n k n 1 1 :ˆ: 1 1 k − − =≥≥= − + λ σ σσσ λ σ σσ
51 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 8)
52 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 9)
53 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 10) k n k n kk 1ˆ 1 :& 1ˆ 1 : 00 − − = − + = λ σ σ λ σ σ σσ Monte-Carlo Procedure Choose the Confidence Level φ and find the corresponding nσ using the normal (Gaussian) distribution. nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 1 Run a few sample k0 > 20 and estimate λ according to2 ( ) ( ) 2 1 2 0 1 4 0 0 0 0 0 0 ˆ 1 ˆ 1 :ˆ       − − = ∑ ∑ = = k i ki k i ki k mx k mx k λ∑= = 0 0 10 1 :ˆ k i ik x k m 3 Compute and as function of kσ σ 4 Find k for which [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n 5 Run k-k0 simulations
54 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue – 11) Monte-Carlo Procedure Choose the Confidence Level φ = 95% that gives the corresponding nσ=1.96. nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 1 The kurtosis λ = 32 3 Find k for which ϕσ λ σσ σ σ =             − ≤≤  2 kˆ 22 k 2 1 ˆ-0Prob k n 4 Run k>800 simulations Example: Assume a Gaussian distribution λ = 3 95.0 2 96.1ˆ-0Prob 2 kˆ 22 k 2 =             ≤≤  σ σσσ k Assume also that we require also that with probability φ = 95 %22 k 2 1.0ˆ- σσσ ≤ 1.0 2 96.1 = k 800≈k
55 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators • First attempts to generate “random numbers”: - Draw balls out of a stirred urn - Roll dice • 1927: L.H.C. Tippett published a table of 40,000 digits taken “at random” from census reports. • 1939: M.G. Kendall and B. Babington-Smith create a mechanical machine to generate random numbers. They published a table of 100,000 digits. • 1946: J. Von Neumann proposed the “middle square method”. • 1948: D.H. Lehmer introduced the “linear congruential method”. • 1955: RAND Corporation published a table of 1,000,000 random digits obtained from electronic noise. • 1965: M.D. MacLaren and G. Marsaglia proposed to combine two congruential generators. • 1989: R.S. Wikramaratna proposed the additive congruential method.
56 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators A Random Number represents the value of a random variable uniform distributed on (0,1). Pseudo-Random Numbers constitute a sequence of values, which although are deterministically generated, have all the appearances of being independent uniform distributed on (0,1). One approach 1. Define x0 = integer initial condition or seed 2. Using integers a and m recursively compute mxax nn modulo1−= mxIntegerxkmaxmkxa nnn <∈+⋅=− ,,,1 Therefore xn takes the values 0,1,…,m-1 and the quantity un=xn/m , called a pseudo-random number is an approximation to the value of uniform (0,1) random variable. In general the integers a and m should be chose to satisfy three criteria: 1. For any initial seed, the resultant sequence has the “appearance” of being a sequence of independent (0,1) random variables. For any initial seed, the number of variables that can be generated before repetition begins is large. The values can be computed efficiently on a digital computer. Multiplicative congruential method Return to Monte Carlo Approximation
57 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators (continue – 1) A guideline is to choose m to be a large prime number compared to the computer word size. Examples: 32 bits word computer: 807,16712 531 ==−= am 125,35312 535 ==−= am36 bits word computer: Another generator of pseudo-random numbers uses recursions of the type: ( ) mcxax nn modulo1 += − mxIntegerxkmcaxmkcxa nnn <∈+⋅=+− ,,,,1 Mixed congruential method
58 SOLO Review of Probability Generating Discrete Random Variables Histograms Return to Table of Content A histogram is a graphical display of tabulated frequencies, shown as bars. It shows what proportion of cases fall into each of several categories: it is a form of data binning. The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent. The intervals (or bands, or bins) are generally of the same size. Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram always equals 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot. A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram Mi of a histogram mi is defined as: An ordinary and a cumulative histogram of the same data. The data shown is a random sample of 10,000 points from a normal distribution with a mean of 0 and a standard deviation of 1. Mathematical Definition ∑= = k i imn 1 In a more general mathematical sense, a histogram is a mapping mi that counts the number of observations that fall into various disjoint categories (known as bins), whereas the graph of a histogram is merely one way to represent a histogram. Thus, if we let n be the total number of observations and k be the total number of bins, the histogram mi meets the following conditions: ∑= = i j ji mM 1
59 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method Suppose we want to generate a discrete random variable X having probability density function: ( ) 1,1,0)( ==−= ∑j jjj pjxxpxp δ To accomplish this, let generate a random number U that is uniformly distributed over (0,1) and set:            <≤ +<≤ < = ∑∑ = − =   j i i j i ij pUpifx ppUpifx pUifx X 1 1 1 1001 00 j j i i j i ij ppUpPxXP =       <<== ∑∑ = − = 1 1 1 )( Since , for any a and b such that 0 < a < b < 1, and U is uniformly distributed P (a ≤ U < b) = b-a, we have: and so X has the desired distribution.
60 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 1) Suppose we want to generate a discrete random variable X having probability density function: ( ) 1,1,0)( ==−= ∑j jjj pjxxpxp δ Draw X, N times, from p (x) Histogram of the Results
61 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 2) Generating a Poisson Random Variable: 1,1,0 ! )( ===== ∑− i i i i pi i eiXPp  λλ ( ) 1 ! !1 1 1 + = + = − + − + i i e i e p p i i i i λ λ λ λ λ Draw X , N times, from Poisson Distribution Histogram of the Results
62 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 3) Generating Binominal Random Variable: ( ) ( ) 1,1,01 !! ! )( ==− − === ∑− i i ini i pipp ini n iXPp  ( ) ( ) ( ) ( ) ( ) p p i in pp ini n pp ini n p p ini ini i i −+ − = − − − −−+ = − −−+ + 111 !! ! 1 !1!1 ! 11 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k ( )nkP , Histogram of the Results
63 SOLO Review of Probability Generating Discrete Random Variables The Accaptance-Rejection Technique Suppose we have an efficient method for simulating a random variable having a probability density function { qj, j ≥0 }. We want to use this to obtain a random variable that has the probability density function { pj, j ≥0 }. Let c be a constant such that: 0.. ≠∀≤ j j j qtsjc q p If such a c exists, it must satisfy: cqcp j j j j ≤⇒≤ ∑∑ 1 11  Rejection Method Step 1: Simulate the value of Y, having probability density function qj. Step 2: Generate a random number U (that is uniformly distributed over (0,1) ). Step 3: If U < pY/c qY, set X = Y and stop. Otherwise return to Step 1.
64 SOLO Review of Probability Generating Discrete Random Variables The Acceptance-Rejection Technique (continue – 1) Theorem The random variable X obtained by the rejection method has probability density function P { X=i } = pi. Proof { } { } { } { } { }Acceptance , Acceptance Acceptance, Acceptance Method Acceptance Method Acceptance P qc p UiYP P iYP iYPiXP i i Bayes       ≤= = = ==== { } { } { } { }AcceptanceAcceptanceAcceptance (0,1)ddistribute uniformlyU ceindependen by Pc p P qc p q P qc p UPiYP ii i i i i qi ==      ≤= =  Summing over all i, yields { } { }Acceptance 1 1 Pc p iXP i i i   ∑ ∑ == { } 1Acceptance =Pc { } ipiXP == { } 1 1 Acceptance ≤= c P q.e.d.
65 SOLO Review of Probability Generating Discrete Random Variables The Acceptance-Rejection Technique (continue – 2) Example Generate a truncated Gaussian using the Accept-Reject method. Consider the case with ( ) [ ]     −∈ ≈ − otherwise xe xp x 0 4,42/2/2 π Consider the Uniform proposal function ( ) [ ]    −∈ ≈ otherwise x xq 0 4,48/1 In Figure we can see the results of the Accept-Reject method using N=10,000 samples.
66 SOLO Review of Probability Generating Continuous Random Variables The Inverse Transform Algorithm Let U be a uniform (0,1) random variable. For any continuous distribution function F the random variable X defined by ( )UFX 1− = has distribution F. [ F-1 (u) is defined to be that value of x such that F (x) = u ] Proof Let Px(x) denote the Probability Distribution Function X=F-1 (U) ( ) { } ( ){ }xUFPxXPxPx ≤=≤= −1 Since F is a distribution function, it means that F (x) is a monotonic increasing function of x and so the inequality “a ≤ b” is equivalent to the inequality “F (a) ≤ F (b)”, therefore ( ) ( )[ ] ( ){ } ( )[ ] ( ){ } ( ) ( ) ( )xFxFUP xFUFFPxP uniformU xF UUFF x 1,0 10 1 1 ≤≤ = − =≤= ≤= −
67 SOLO Review of Probability Importance Sampling Let Y = (Y1,…,Ym) a vector of random variables having a joint probability density function f (y1,…,ym), and suppose that we are interested in estimating ( )[ ] ( ) ( )∫== mmmmf dydyyyfyyhYYhE  1111 ,,,,,,θ Suppose that a direct generation of the random vector Y so as to compute h (Y) is inefficient possible because (a) is difficult to generate the random vector Y, or (b) the variance of h (Y) is large, or (c) both of the above Suppose that W=(W1,…,Wm) is another random vector, which takes values in the same domain as Y, and has a joint density function g(w1,…,wm) that can be easily generated. The estimation θ can be expressed as: ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )       === ∫ Wg WfWh Edwdwwwg wwg wwfwwh YYhE gmm m mm mf     11 1 11 1 ,, ,, ,,,, ,,θ Therefore, we can estimate θ by generating values of random vector W, and then using as the estimator the resulting average of the values h (W) f (W)/ g (W). Return to Particle Filters
68 SOLO Review of Probability Monte Carlo Integration Monte Carlo Method can be used to numerically evaluate multidimensional integrals ( ) ( )∫∫ == xdxgdxdxxxgI mm  11 ,, To use Monte Carlo we factorize ( ) ( ) ( )xpxfxg ⋅= ( ) ( ) 1&0 =≥ ∫ xdxpxp in such a way that is interpreted as a Probability Density Function such that( )xp We assume that we can draw NS samples from ( )xp( )S i Nix ,,1, = ( ) S i Nixpx ,,1~ = Using Monte Carlo we can approximate ( ) ( )∑= −≈ SN i S i Nxxxp 1 /δ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑∫ ∫ ∑∫ == = =−⋅= −⋅=≈⋅= SS S S N i i S N i i S N i S i N xf N xdxxxf N xdNxxxfIxdxpxfI 11 1 11 / δ δ
69 SOLO Review of Probability Monte Carlo Integration we draw NS samples from ( )xp( )S i Nix ,,1, = ( ) S i Nixpx ,,1~ = ( ) ( ) ( )∑∫ = =≈⋅= S S N i i S N xf N IxdxpxfI 1 1 If the samples are independent, then INS is an unbiased estimate of I. i x According to the Law of Large Numbers INS will almost surely converge to I: II sa N N S S .. ∞→ → ( )[ ] ( ) ∞<−= ∫ xdxpIxff 22 :σIf the variance of is finite; i.e.:( )xf then the Central Limit Theorem holds and the estimation error converges in distribution to a Normal Distribution: ( ) ( )2 ,0~lim fNS N IIN S S σN− ∞→ The error of the MC estimate, e = INS – I, is of the order of O (NS -1/2 ), meaning that the rate of convergence of the estimate is independent of the dimension of the integrand. Numerical Integration of and ( )kk xzp |( )1| −kk xxp Return to Particle Filters
70 SOLO Review of Probability Existence Theorems Existence Theorem 3 Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation. Proof of Existence Theorem 3 Define ( ) ( ) ( ) ( ) ( )ω π ω π ω ωωω π −= − === ∫ +∞ ∞− f a S a S fdSa 22 2 :& 1 : Since , according to Existence Theorem 1, we can find a random variable ω with the even density function f (ω), and probability density function ( ) ( ) 1&0 =≥ ∫ +∞ ∞− ωωω dff ( ) ( )∫∞− = ω ττω dfP : We now form the process , where is a random variable uniform distributed in the interval (-π,+π) and independent of ω. ( ) ( )ϑω += tatx cos: ϑ
71 SOLO Review of Probability Existence Theorems Existence Theorem 3 Proof of Existence Theorem 3 (continue – 1) Since is uniform distributed in the interval (-π,+π) and independent of ω, its spectrum is ( ){ } ( ){ } ( ){ } ( ){ } ( ){ } 0sinsincoscos 00 , =−=  ϑωϑω ϑωϑω ϑω EtEaEtEatxE tindependen ϑ ( ) { } ( ) ϖπ ϖπ ϖπϖπ ϑ π ϖ πϖπϖπ π ϑϖπ π ϑϖϑϖ ϑϑ sin 2 1 2 1 2 1 = − ==== −+ − + − ∫ j ee j e deeES jjj jj or { } ( ){ } ( ){ } ( ) ϖπ ϖπ ϑϖϑϖ ϑϑ ϑϖ ϑ sin sincos =+= EjEeE j 1=ϖ 1=ϖ ( ) ( ){ } ( ) ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ }  0 2 0 22, 22 2 2sin2sin 2 2cos2cos 2 cos 2 22cos 2 cos 2 coscos ϑτωϑτωτω ϑτωτω ϑτωϑωτ ϑωϑωω ϑω EtE a EtE a E a tE a E a ttEatxtxE tindependen +−++= +++= +++=+ 2=ϖ 2=ϖ Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation.
72 SOLO Review of Probability Existence Theorems Existence Theorem 3 Proof of Existence Theorem 3 (continue – 2) ( ){ } 0=txE ( ) ( ){ } ( ){ } ( ) ( ) ( )τωωτωτωτ ω xRdf a E a txtxE ===+ ∫ +∞ ∞− cos 2 cos 2 22 ( ) ( )ϑω += tatx cos:We have Because of those two properties x (t) is wide-sense stationary with a power spectrum given by: ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫ +∞ ∞− −=+∞ ∞− =−= ττωτττωτωτω ττ dRdjRS x RR xx xx cossincos ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫ +∞ ∞− −=+∞ ∞− =+= ωτωω π ωτωτωω π τ ωω dSdjSR x SS xx xx cos 2 1 sincos 2 1 Therefore ( ) ( )ωπω faSx 2 = q.e.d. Fourier Inverse Fourier ( ) ( )∫ +∞ ∞− = ωωτω df a cos 2 2 f (ω) definition ( )ωS= Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation.
73 SOLO Markov Processes A Markov Process is defined by: Andrei Andreevich Markov 1856 - 1922 ( ) ( )( ) ( ) ( )( ) 111 ,|,,,|, tttxtxptxtxp >∀ΩΩ=≤ΩΩ ττ i.e. the Random Process, the past up to any time t1 is fully defined by the process at t1. Examples of Markov Processes: 1. Continuous Dynamic System ( ) ( ) ( ) ( )vuxthtz wuxtftx ,,, ,,, = = 2. Discrete Dynamic System ( ) ( ) ( ) ( )kkkkk kkkkk vuxthtz wuxtftx ,,, ,,, 1111 = = −−−− x - state space vector (n x 1) u - input vector (m x 1) - measurement vector (p x 1)z v - white measurement noise vector (p x 1) - white input noise vector (n x 1)w Recursive Bayesian Estimation
74 Recursive Bayesian EstimationSOLO Using this property we obtain: ( ) ( )1021 |,,,| −−− = kkkkk xxpxxxxp  Markov Processes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∏= − −−−− −−−−−− = = = −− − k i ii k xxp kkkk kk xxp kkkkkk xxpxp xxpxxxpxxp xxxpxxxxpxxxxp kk kk 1 10 02 | 0211 021 | 021021 | ,,,,|| ,,,,,,|,,,, 21 1           Markov Process: Table of Content the present discrete state probability depends only on the previous state. The Markov Process is defined if we know p (x0) and p(xi|xi-1) for each i.
75 Recursive Bayesian EstimationSOLO In a Markovian system the probability of the current true state depends only on the previous state, and is independent of the other earlier states ( ) ( )1021 |,,,| −−− = kkkkk xxpxxxxp  Similarly the measurements at the k-th time- step is dependent upon the current true state, so is conditionally independent of all other earlier states, given the current state ( ) ( )kkkkk xzpxxxzp |,,,| 01 =−  ( ) ( ) ( ) ( ) ( )kkkkkkkk zpzxpxpxzpxzp ||, == From the definition of the Markovian system (see Figure) p (xk|xk-1) is defined by f and the statistics of x and w and p (zk|xk) is defined by h and statistics of x and v. kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )111 ,, −−− kkk wuxf ( )kk vxh , Markov Processes ( )000 ,, wuxf ( )11,vxh ( )111 ,, wuxf ( )22 ,vxh Hidden States Measurements
76 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx :, ,,:,, 011111 0 = = −−−−− Markov Processes ( ) ( )j kkkkxkkkw j k wuxfxtsNjuxxfw k 11111 1 1 ,,..,..,1,, −−−−− − − === Suppose that we can obtain all for which j kw 1− ( ) ( ) ( )∑= − −−−−− ∇= kxN j j kkkw j kwkk wuxfwpxxp 1 1 11111 ,,|then ( ) ( ) ( )∑= − ∇= kx k N j j kkv j kvkk vxhvpxzp 1 1 ,| ( ) ( )j kkkzkkv j k vxhztsNjxzhv k ,..,..,1,1 === − In the same way, suppose that we can obtain all for whichj kv then ( ) ( ) ( ) ( ) ( )∑ ∑ = − −−−− = −−−− ∇= =+≤≤= kx kx N j k j kkkw j kw N j j k j kwkkkkkkkk xdwuxfwp wdwpxxdxXxxdxxp 1 1 1111 1 1111 ,, |Pr| This is a Conceptual Not a Practical Procedure Analytic Computations of and .( )kk xzp |( )1| −kk xxp
77 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx : ,,:, 011111 0 += += −−−−− kx1−kx kz1−kz ( ) 111, −−− + kkk wuxf ( ) kk vxh + Markov Processes ( ) ( )[ ]111 ,| −−− −= kkkwkk uxfxpxxptherefore ( ) ( )[ ]kkvkk xhzpxzp −=|and For additive noise we have ( ) ( )kkk kkkk xhzv uxfxw −= −= −−− 111 , Analytic Computations of and (continue – 1)( )kk xzp |( )1| −kk xxp
78 SOLO ( ) ( )kkk kkk vxhz wxfx , , 11 = = −− kk vw &1− are system and measurement white-noise sequences independent of past and current states and on each other and having known P.D.F.s ( ) ( )kk vpwp &1− We want to compute p (xk|Z1:k) recursively, assuming knowledge of p(xk-1|Z1:k-1) in two stages, prediction (before) and update (after measurement) ( ) ( )( ) ( )∫ −−−−− −= 11111 ,| kkkkkkk wdwpwxfxxxp δ We need to evaluate the following integrals: ( ) ( )( ) ( )∫ −= kkkkkkk vdvpvxhzxzp ,| δ We use the numeric Monte Carlo Method to evaluate the integrals: Generate (Draw): ( ) ( ) Sk i kk i k Nivpvwpw ,,1~&~ 11 =−− ( ) ( )( ) S N i i k i k i kkk Nwxfxxxp S ∑= −−− −≈ 1 111 /,| δ ( ) ( )( ) S N i i k i k i kkk Nvxhzxzp S ∑= −≈ 1 /,| δ or ( ) ( ) ( ) S N i i kkkk i k i k i k Nxxxxpwxfx S ∑= −−− −≈→= 1 111 /|, δ ( ) ( ) ( ) S N i i kkkk i k i k i k Nzzxzpvxhz S ∑= −≈→= 1 /|, δ Analytic solutions for those integral equations do not exist in the general case. Recursive Bayesian Estimation Numerical Computations of and .( )kk xzp |( )1| −kk xxp Markov Processes Prediction (before measurement) ( ) ( ) ( )∫ −−−−− = 11:1111:1 ||| kkkkkkk xdZxpxxpZxp1 Update (after measurement) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ − − − − = − === kkkkk kkkk kk kkkk Bayes bp apabp bap kkkkk xdZxpxzp Zxpxzp Zzp Zxpxzp ZzxpZxp 1:1 1:1 1:1 1:1 | | 1:1:1 || || | || ,|| 2
79 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx :, ,,:,, 011111 0 = = −−−−− Markov Processes Monte Carlo Computations of and .( )kk xzp |( )1| −kk xxp Generate (Draw) ( ) Sx i Nixpx ,,1~ 00 0 = For { }∞∈ ,,1 k Initialization0 1 At stage k-1 Generate (Draw) NS samples ( ) Skw i k Niwpw ,,1~ 11 =−− 2 State Update ( ) S i kk i k i k Niwuxfx ,,1,, 111 == −−− 3 Generate (Draw) Measurement Noise ( ) Skv i k Nivpv ,,1~ = k:=k+1 & return to 1 Compute Histograms of to obtain ( )kk xzp | kk xz | ( ) ( )∑= − −≈ SN i S i kkkk Nxxxxp 1 1 /| δ ( ) ( )∑= −≈ SN i S i kkkk Nzzxzp 1 /| δ Compute Histograms of to obtain 1| −kk xx ( )1| −kk xxp 4 Measurement , Update ( ) S i k i k i k Nivxhz ,,1, ==kz
SOLO Stochastic Processes deal with systems corrupted by noise. A description of those processes is given in “Stochastic Processes” Presentation. Here we give only one aspect of those processes. ( ) ( ) ( ) [ ]fttttwddttxftxd ,, 0∈+= A continuous dynamic system is described by: Stochastic Processes ( )tx - n- dimensional state vector ( )twd - n- dimensional process noise vector Assuming system measurements at discrete time tk given by: ( ) ( )( ) [ ]fkkkkk tttvttxhtz ,,, 0∈= kv - m- dimensional measurement noise vector at tk We are interested in the probability of the state at time t given the set of discrete measurements until (included) time tk < t. x ( )kZtxp |, { }kk zzzZ ,,, 21 = - set of all measurements up to and including time tk. The time evolution of the probability density function is described by the Fokker–Planck equation.
A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. The initial condition is a Dirac delta function in x = 1, and the distribution drifts towards x = 0. The Fokker–Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation. The first use of the Fokker– Planck equation was the statistical description of Brownian motion of a particle in a fluid. In one spatial dimension x, the Fokker–Planck equation for a process with drift D1(x,t) and diffusion D2(x,t) is More generally, the time-dependent probability distribution may depend on a set of N macrovariables xi. The general form of the Fokker–Planck equation is then where D1 is the drift vector and D2 the diffusion tensor; the latter results from the presence of the stochastic force. Fokker – Planck Equation Adriaan Fokker 1887 - 1972 Max Planck 1858 - 1947 SOLO Adriaan Fokker „Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld" Annalen der Physik 43, (1914) 810- 820 Max Plank, „Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie“, Sitzungberichte der Preussischen Akadademie der Wissenschaften (1917) p. 324-341 Stochastic Processes ( ) ( ) ( )[ ] ( ) ( )[ ]txftxD x txftxD x txf t ,,,,, 22 2 1 ∂ ∂ + ∂ ∂ −= ∂ ∂ ( )[ ] ( )[ ]∑∑∑ = == ∂∂ ∂ + ∂ ∂ −= ∂ ∂ N i N j Nji ji N i Ni i ftxxD xx ftxxD x f t 1 1 1 2 2 1 1 1 ,,,,,, 
Fokker – Planck Equation (continue – 1) The Fokker–Planck equation can be used for computing the probability densities of stochastic differential equations. where is the state and is a standard M-dimensional Wiener process. If the initial probability distribution is , then the probability distribution of the state is given by the Fokker – Planck Equation with the drift and diffusion terms: Similarly, a Fokker–Planck equation can be derived for Stratonovich stochastic differential equations. In this case, noise-induced drift terms appear if the noise strength is state-dependent. SOLO Consider the Itô stochastic differential equation: ( ) ( ) ( )[ ] ( ) ( )[ ]txftxD x txftxD x txf t ,,,,, 22 2 1 ∂ ∂ + ∂ ∂ −= ∂ ∂
Fokker – Planck Equation (continue – 2) Derivation of the Fokker–Planck Equation SOLO Start with ( ) ( ) ( )11|1, 111 |, −−− −−− = kxkkxxkkxx xpxxpxxp kkkkk and ( ) ( ) ( ) ( )∫∫ +∞ ∞− −−− +∞ ∞− −− −−− == 111|11, 111 |, kkxkkxxkkkxxkx xdxpxxpxdxxpxp kkkkkk define ( ) ( )ttxxtxxttttt kkkk ∆−==∆−== −− 11 ,,, ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )∫ +∞ ∞− ∆−∆− ∆−∆−∆−= ttxdttxpttxtxptxp ttxttxtxtx || Let use the Characteristic Function of ( ) ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )ttxtxtxtxdttxtxpttxtxss ttxtxttxtx ∆−−=∆∆−∆−−−=Φ ∫ +∞ ∞− ∆−∆−∆ |exp: || ( ) ( ) ( ) ( )[ ]ttxtxp ttxtx ∆−∆− || The inverse transform is ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( )∫ ∞+ ∞− ∆−∆∆− Φ∆−−=∆− j j ttxtxttxtx sdsttxtxs j ttxtxp || exp 2 1 | π Using Chapman-Kolmogorov Equation we obtain: ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( )ttxdsdttxpsttxtxs j ttxdttxpsdsttxtxs j txp j j ttxttxtx ttx ttxtxp j j ttxtxtx ttxtx ∆−∆−Φ∆−−= ∆−∆−Φ∆−−= ∫ ∫ ∫ ∫ ∞+ ∞− ∞+ ∞− ∆−∆−∆ +∞ ∞− ∆− ∆− ∞+ ∞− ∆−∆ ∆− | | | exp 2 1 exp 2 1 | π π    Stochastic Processes
Fokker – Planck Equation (continue – 3) Derivation of the Fokker–Planck Equation (continue – 1) SOLO The Characteristic Function can be expressed in terms of the moments about x (t-Δt) as: ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( )ttxdsdttxpsttxtxs j txp j j ttxttxtxtx ∆−∆−Φ∆−−= ∫ ∫ +∞ ∞− ∞+ ∞− ∆−∆−∆ |exp 2 1 π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ }∑ ∞ = ∆−∆∆−∆ ∆−∆−− − +=Φ 1 || | ! 1 i i ttxtx i ttxtx ttxttxtxE i s s Therefore ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )ttxdsdttxpttxttxtxE i s ttxtxs j txp j j ttx i i ttxtx i tx ∆−∆−       ∆−∆−− − +∆−−= ∫ ∫ ∑ +∞ ∞− ∞+ ∞− ∆− ∞ = ∆− 1 | | ! 1exp 2 1 π Use the fact that ( ) ( ) ( )[ ]{ } ( ) ( ) ( )[ ] ( )[ ] ,2,1,01exp 2 1 = ∂ ∆−−∂ −=∆−−−∫ ∞+ ∞− i tx ttxtx sdttxtxss j i i i j j i δ π ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )∫∑ ∫ ∫ ∞+ ∞− ∞ = ∆− +∞ ∞− ∆− ∞+ ∞− ∆−∆−∆−∆−− ∂ ∆−−∂− + ∆−∆−∆−−= 1 | ! 1 exp 2 1 i ttx i i ii ttx j j tx ttxdttxpttxttxtxE tx ttxtx i ttxdttxpsdttxtxs j txp δ π where δ [u] is the Dirac delta function: [ ] { } ( ) [ ] ( ) ( ) ( ) ( ) ( )000..0exp 2 1 FFFtsuFFduuuFsdus j u j j ==∀== −+ +∞ ∞− ∞+ ∞− ∫∫ δ π δ Stochastic Processes
Fokker – Planck Equation (continue – 4) Derivation of the Fokker–Planck Equation (continue – 2) SOLO [ ] ( ){ } ( ) [ ] ( ) ( ) ( ) ( ) ( )afafaftsufufduuaufsduas j ua au j j ==∀=−−=− −+= +∞ ∞− ∞+ ∞− ∫∫ ..exp 2 1 δ π δ [ ] ( ){ } ( ) ( ) { } ( ) ( ) { }∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =→=− − =− j j j j j j sdussFs j uf du d sdussF j ufsduass j ua ud d exp 2 1 exp 2 1 exp 2 1 πππ δ ( ) [ ] ( ) ( ){ } ( ) ( ){ } { } ( ) { } { } ( ) ( ) au j j j j j j j j ud ufd sdsFass j sdduusufass j sdduuasufs j dusduass j ufduua ud d uf = ∞+ ∞− ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− −= − =− − = − − =− − =− ∫∫ ∫ ∫ ∫∫ ∫∫ exp 2 1 expexp 2 1 exp 2 1 exp 2 1 ππ ππ δ [ ] ( ) ( ){ } ( ) ( ) { } ( ) ( ) { }∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =→=− − =− j j i i ij j j j i i i i sdussFs j uf du d sdussF j ufsduass j ua ud d exp 2 1 exp 2 1 exp 2 1 πππ δ ( ) [ ] ( ) ( ) ( ){ } ( ) ( ) ( ){ } ( ) { } ( ) { } ( ) ( ) { } ( ) ( ) au i i i j j i ij j i i j j i ij j i i i i ud ufd sdassFs j sdduusufass j sdduuasufs j dusduass j ufduua ud d uf = −= − =− − = − − =− − =− ∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− 1exp 2 1 expexp 2 1 exp 2 1 exp 2 1 ππ ππ δ Useful results related to integrals involving Delta (Dirac) function Stochastic Processes
Fokker – Planck Equation (continue – 5) Derivation of the Fokker–Planck Equation (continue – 3) SOLO ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ]txpttxdttxpttxtxttxdttxpsdttxtxs j ttxttxttx ttxtx j j ∆− +∞ ∞− ∆− +∞ ∞− ∆− ∆−− ∞+ ∞− =∆−∆−∆−−=∆−∆−∆−− ∫∫ ∫ δ π δ    exp 2 1 ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∑ ∫ ∫∑ ∞ = =∆ ∆−∆− ∞ = ∞+ ∞− ∆−∆− +∞ ∞− ∞ = ∆−∆− ∂ ∆−∆−−∂− = ∆−∆−∆−∆−− ∂ ∆−−∂− = ∆−∆−∆−∆−− ∂ ∆−−∂− 1 0 | 1 | 1 | | ! 1 | ! 1 | ! 1 i t i ttx i ttxtx ii i ttx i ttxtxi ii i ttx i ttxtxi ii tx txpttxttxtxE i ttxdttxpttxttxtxE tx ttxtx i ttxdttxpttxttxtxE tx ttxtx i δ δ ( ) [ ] ( ) ( ) ( ) [ ] [ ] ( ) auau i i i i i i i i i ud ufd duua uad d uf ud ufd duua ud d uf == =− − →−=− ∫∫ +∞ ∞− +∞ ∞− δδ 1We found ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = =∆ ∆−∆− ∆− ∂ ∆−∆−−∂− += 1 0 | | ! 1 i t i ttx i ttxtx ii ttxtx tx txpttxttxtxE i txptxp ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = ∆− →∆ ∆− →∆ ∂ ∆−∆−−∂ ∆ − = ∆ − 1 00 |1 lim ! 1 lim i i ttx ii t i ttxtx t tx txpttxttxtxE tit txptxp Therefore Rearranging, dividing by Δt, and tacking the limit Δt→0, we obtain: Stochastic Processes
Fokker – Planck Equation (continue – 6) Derivation of the Fokker–Planck Equation (continue – 4) SOLO We found ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = ∆−∆− →∆ ∆− →∆ ∂ ∆−∆−−∂ ∆ − = ∆ − 1 | 00 |1 lim ! 1 lim i i ttx i ttxtx i t i ttxtx t tx txpttxttxtxE tit txptxp Define: ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } t ttxttxtxE txtxm i ttxtx t i ∆ ∆−∆−− =− ∆− →∆ − | lim: | 0 Therefore ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]∑ ∞ = − ∂ −∂− = ∂ ∂ 1 ! 1 i i tx iii tx tx txptxtxm it txp ( ) ( )ttxtx t ∆−= →∆ − 0 lim: and: This equation is called the Stochastic Equation or Kinetic Equation. It is a partial differential equation that we must solve, with the initial condition: ( ) ( )[ ] ( )[ ]000 0 txptxp tx === Stochastic Processes
Fokker – Planck Equation (continue – 7) Derivation of the Fokker–Planck Equation (continue – 5) SOLO We want to find px(t) [x(t)] where x(t) is the solution of ( ) ( ) ( ) [ ]fg ttttntxf dt txd ,, 0∈+= ( ){ } 0: == tnEn gg  ( )tng ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( )τδττ −=−− ttQnntntnE gggg ˆˆ Wiener (Gauss) Process ( ) ( )[ ] ( ) ( )[ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tQnEtxnE t ttxttxtxE txtxm gg t === ∆ ∆−∆−− =− →∆ − 22 2 2 0 2 | | lim: ( ) ( )[ ] ( ) ( )[ ] ( ){ } ( ) ( ) ( ) ( ) ( )txfnEtxftx td txd E t ttxttxtxE txtxm g t ,,| | lim: 0 0 1 =+=             = ∆ ∆−∆−− =− →∆ −  ( ) ( )[ ] ( ) ( )[ ] ( ){ } 20 | lim: 0 >= ∆ ∆−∆−− =− →∆ − i t ttxttxtxE txtxm i t i Therefore we obtain: ( ) ( )[ ] ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ] ( )[ ]2 2 2 1, tx txp tQ tx txpttxf t txp txtxtx ∂ ∂ + ∂ ∂ −= ∂ ∂ Stochastic Processes Fokker–Planck Equation Return to Daum
89 Recursive Bayesian EstimationSOLO Given a nonlinear discrete stochastic Markovian system we want to use k discrete measurements Z1:k={z1,z2,…,zk} to estimate the hidden state xk. For this we want to compute the probability of xk given all the measurements Z1:k={z1,z2,…,zk} . If we know p ( xk| Z1:k ) then xk is estimated using: { } ( )∫== kkkkkkkk xdZxpxZxEx :1:1| ||:ˆ ( )( ){ } ( )( ) ( )∫ −−=−−= kkk T kkkkk T kkkkkk xdZxpxxxxZxxxxEP :1:1| |ˆˆ|ˆˆ or more general we can compute all moments of the probability distribution p ( xk| Z1:k ): ( ){ } ( ) ( )∫= kkkkkk xdZxpxgZxgE :1:1 || Bayesian Estimation Introduction Problem: Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements. kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh The knowledge of p ( xk| Z1:k ) allows also the computation of Maximum a Posteriori (MAP) estimate using: ( )kk x MAP kk Zxpx k :1| |maxargˆ =
90 Recursive Bayesian EstimationSOLO To find the expression for p ( xk| Z1:k ) we use the theorem of joint probability (Bayes Rule): ( ) ( ) ( )k kk RuleBayes kk Zp Zxp Zxp :1 :1 :1 , | = Since Z1:k ={ zk, Z1:k-1 }: ( ) ( ) ( )1:1 1:1 :1 , ,, | − − = kk kkk kk Zzp Zzxp Zxp The denominator of this expression is ( ) ( ) ( )1:11:11:1 ,,|,, −−− = kkkkk RuleBayes kkk ZxpZxzpZzxp ( ) ( ) ( )    1:11:11:1 |,| −−−= kkkkkk ZpZxpZxzp Since the knowledge of xk supersedes the need for Z1:k-1 = {z1, z2,…,zk-1} ( ) ( )kkkkk xzpZxzp |,| 1:1 ≡− ( ) ( ) ( ) ( ) ( ) ( )1:11:1 1:11:1 :1 | || | −− −− = kkk kkkkk kk ZpZzp ZpZxpxzp ZxpTherefore: ( ) ( ) ( )1:11:11:1 |, −−− = kkk RuleBayes kk ZpZzpZzp and the nominator is Bayesian Estimation Introduction
91 Recursive Bayesian EstimationSOLO The final result is: ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp Therefore: Since p ( xk| Z1:k ) is a probability distribution it must satisfy: ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ − − − − === 1:1 1:1 1:1 1:1 :1 | || | || |1 kk kkkkk k kk kkkk kkk Zzp xdZxpxzp xd Zzp Zxpxzp xdZxp ( ) 1| :1 =∫ kkk xdZxp ( ) ( ) ( )∫ −− = kkkkkkk xdZxpxzpZzp 1:11:1 ||| ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | and: This is a recursive relation that needs the value of p (xk|Z1:k-1), assuming that p (zk|xk) is obtained from the Markovian system definition ( zk = h (xk,vk) ). Bayesian Estimation Introduction kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Hidden States Measurements
92 Recursive Bayesian EstimationSOLO The Correction Step is: ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp Bayesian Estimation Introduction evidence priorlikeliood posterior ⋅ = or: prior: given by prediction equation ( )kk xzp | likelihood: given by observation model ( )1:1| −kk Zxp evidence: the normalized constant on the denominator ( ) ( ) ( )∫ −− = kkkkkkk xdZxpxzpZzp 1:11:1 |||
93 Recursive Bayesian EstimationSOLO ( ) ( ) ( )1:111:111:11 |,||, −−−−−− = kkkkk Bayes kkk ZxpZxxpZxxp ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp Using: We obtain: Since for a Markov Process the knowledge of xk-1 supersedes the need for Z1:k-1 = {z1, z2,…,zk-1} ( ) ( )11:11 |,| −−− = kkkkk xxpZxxp Chapman – Kolmogorov Equation Sydney Chapman 1888 - 1970 Andrey Nikolaevich Kolmogorov 1903-1987 Bayesian Estimation Introduction kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Hidden States Measurements
94 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp Using p (xk-1|Z1:k-1) from time-step k-1 and p (xk|xk-1) of the Markov system, compute: Initialize with p (x0) ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | Using p (xk|Z1:k-1) from Prediction phase and p (zk|xk) of the Markov system, compute: { } ( )∫== kkkkkkkk xdZxpxZxEx :1:1| ||ˆ ( )( ){ } ( )( ) ( )∫ −−=−−= kkk T kkkkk T kkkkkk xdZxpxxxxZxxxxEP :1:1| |ˆˆ|ˆˆ At stage k k:=k+1 ( )1|11| ˆˆ −−− = kkkk xfx 0 Prediction phase (before zk measurement) 1 Correction Step (after zk measurement)2 Filtering3 kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Bayesian Estimation Introduction - Summary
95 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | Prediction phase (before zk measurement) 1 Correction Step (after zk measurement)2 kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Bayesian Estimation Introduction - Summary This is a Conceptual Solution because the Integrals are Often Not Tractable. An optimal solution is possible for some restricted cases: • Linear Systems with Gaussian Noises (system and measurements) • Grid-Based Filters Table of Content
96 SOLO Linear Gaussian Systems A Linear Combination of Independent Gaussian random vectors is also a Gaussian random vector mmm XaXaXaS +++= 2211: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    +++++++−=     +−    +−    +−= ΦΦ⋅Φ==Φ ∫ ∫ +∞ ∞− +∞ ∞− mmmm mmmm YYYm YpYp mYYmS aaajaaa ajaajaaja YdYdYYpSj m mmYY mm µµµωσσσω µωσωµωσωµωσω ωωωωω          2211 222 2 2 2 2 1 2 1 2 222 22 2 2 2 2 2 11 2 1 2 1 2 11,, 2 1 exp 2 1 exp 2 1 exp 2 1 exp ,,exp 21 11 1 ( ) ( )       − −= 2 2 2 exp 2 1 ,; i ii i iiiX X Xp i σ µ σπ σµ ( ) ( ) ( )     +−==Φ ∫ +∞ ∞− iiiiXiX jXdXpXj ii µωσωωω 22 2 1 expexp: Moment- Generating Function Gaussian distribution Define Proof: ( ) ( )iX ii i X i iYiii Xp aa Y p a YpXaY iii 11 : =      =→= ( ) ( ) ( ) ( ) ( ) ( )       +−=Φ===Φ ∫∫ +∞ ∞− +∞ ∞− iiiiiiX asign asign ii i iX iiiiYiY ajaXaXda a Xp XajYdYpYj i i ii µωσωωωω 222 2 1 expexpexp: 1 1 Review of Probability
97 SOLO Linear Gaussian Systems (continue – 1) A Linear Combination of Independent Gaussian random vectors is also a Gaussian random vector mmm XaXaXaS +++= 2211: Therefore the Linear Combination of Independent Gaussian Random Variables is a Gaussian Random Variable with mmS mmS aaa aaa m m µµµµ σσσσ +++= +++=   2211 222 2 2 2 2 1 2 1 2 Therefore the Sm probability distribution is: ( ) ( )         − −= 2 2 2 exp 2 1 ,; m m m mm S S S SSm x Sp σ µ σπ σµ Proof (continue – 1): ( ) ( ) ( )      +++++++−=Φ mmmmS aaajaaam µµµωσσσωω  2211 222 2 2 2 2 1 2 1 2 2 1 exp We found: Review of Probability q.e.d.
98 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 2) ( ) ( )kkkk kkkk vuxkhz wuxkfx ,,, ,,,1 111 = −= −−− kkkk kkkkkkk vxHz wuGxx += Γ++Φ= −−−−−− 111111 wk-1 and vk, white noises, zero mean, Gaussian, independent ( ) ( ) ( ){ } ( ) ( ){ } ( )kPkekeEkxEkxke x T xxx =−= &: ( ) ( ) ( ){ } ( ) ( ){ } ( ) lk T www kQlekeEkwEkwke , 0 &: δ=−=  ( ) ( ) ( ){ } ( ) ( ){ } ( ) lk T vvv kRlekeEkvEkvke , 0 &: δ=−=  ( ) ( ){ } { }0=lekeE T vw    = ≠ = lk lk lk 1 0 ,δ ( ) ( )Qwwpw ,0;N= ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − wQw Q wp T nw 1 2/12/ 2 1 exp 2 1 π ( ) ( )     −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π A Linear Gaussian Markov Systems is defined as ( ) ( )0|0000 ,;0 Pxxxp ttx == = N ( ) ( ) ( ) ( )    −−−= = − == 00 1 0|0002/1 0|0 2/0 2 1 exp 2 1 0 xxPxx P xp t T tntx π
99 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 3) 111111 −−−−−− Γ++Φ= kkkkkkk wuGxx Prediction phase (before zk measurement) { } { } { }   0 1:111111:1111:11| |||:ˆ −−−−−−−−−− Γ++Φ== kkkkkkkkkkkk ZwEuGZxEZxEx or 111|111| ˆˆ −−−−−− +Φ= kkkkkkk uGxx The expectation is { }[ ] { }[ ]{ } ( )[ ] ( )[ ]{ }1:1111|111111|111 1:11|1|1| |ˆˆ |ˆˆ: −−−−−−−−−−−−− −−−− Γ+−ΦΓ+−Φ= −−= k T kkkkkkkkkkkk k T kkkkkkkk ZwxxwxxE ZxExxExEP ( ) ( ){ } ( ){ } ( ){ } { } T k Q T kkk T k T kkkkk T k T kkkkk T k P T kkkkkkk wwExxwE wxxExxxxE kk 11111 0 1|1111 1 0 11|11111|111|111 ˆ ˆˆˆ 1|1 −−−−−−−−−− −−−−−−−−−−−−−− ΓΓ+Φ−Γ+ Γ−Φ+Φ−−Φ= −−         T kk T kkkkkk QPP 1111|111| −−−−−−− ΓΓ+ΦΦ= { } ( )1|1|1:1 ,ˆ;| −−− = kkkkkkk PxxZxP N Since is a Linear Combination of Independent Gaussian Random Variables: 111111 −−−−−− Γ++Φ= kkkkkkk wuGxx
100 SOLO For the particular vector measurement equation where the measurement noise, is Gaussian (normal), with zero mean: ( ) ( )kkkv Rvvp ,0;N= ( ) ( ) ( )xp zxp xzp x zx xz , | , | = and independent of , the conditional probability can be written, using Bayes rule as: kx ( )xzp xz || ( )           − − ==−= 1 111 1111 1 1 , nxpp nx pxnxpxnpxpx xHz xHz zxfxHzv xn xn  ( ) ( ) 2/1 ,, /,, T vxzx JJvxpzxp = The measurement noise can be related to and by the function:v zx pxp p pp p I z f z f z f z f z f J =                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =      ∂ ∂ =    1 1 1 1 ( ) ( ) ( ) ( )vpxpvxpzxp vxvxzx ⋅== ,, ,, kv Since the measurement noise is independent of :xv zThe joint probability of and is given by:x Recursive Bayesian Estimation Linear Gaussian Markov Systems (continue – 4) kkkk vxHz += Correction Step (after zk measurement) - 1st Way ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp
101 ( ) ( )kkkv Rvvp ,0;N= kkkk vxHz += Consider a Gaussian vector , where , measurement, , where the Gaussian noise is independent of and . v kx ( ) [ ]1|1| ,; −−= kkkkkkx Pxxxp  N kx ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− == kkxkkxzkkkzxkz xdxpxzpxdzxpzp |, |, is Gaussian with( )kz zp ( ) ( ) ( ) ( ) 1| 0 −=+=+= kkkkkkkkk xHvExEHvxHEzE   ( ) ( )[ ] ( )[ ]{ } [ ][ ]{ } ( )[ ] ( )[ ]{ } [ ]{ } [ ]{ } [ ]{ } { } k T kkkk T kk T k T kkkk T kkkkk T k T kkkkkkk T kkkkkkkkkk T kkkkkkkkkkkk T kkkkk RHPHvvEHxxvEvxxEH HxxxxEHvxxHvxxHE xHvxHxHvxHEzEzzEzEz +=+−−−− −−=+−+−= −+−+=−−= −−− −−−− −− 1| 0 1| 0 1| 1|1|1|1| 1|1|cov           ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]       −−+−−−− +− = − xHzRHPHxHz RHPH zp TT Tpz ˆˆ 2 1 exp 2 1 1 2/12/ π ( ) ( ) ( ) ( )      −−−= − − −− − −− 1| 1 1|1|2/1 1| 2/1:1| 2 1 exp 2 1 |1:1 kkkkk T kkk kk nkkZx xxPxx P Zxp kk  π ( ) ( ) ( ) ( ) ( )    −−−=−= − kkk T kkkpkkkvkkxz xHzRxHz R xHzpxzp 1 2/12/| 2 1 exp 2 1 | π Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 5) Correction Step (after zk measurement) 1st Way (continue – 1)
102 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 6) kkkk vxHz += ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π Correction Step (after zk measurement) 1st Way (continue – 2) ( ) ( ) ( ) ( )    −−−= − − −− − −− 1| 1 1|1|2/1 1| 2/1:1| 2 1 exp 2 1 |1:1 kkkkk T kkk kk nkkZx xxPxx P Zxp kk  π ( ) ( ) ( ) ( ) ( )    −−−=−= − kkk T kkkpkkkvkkxz xHzRxHz R xHzpxzp 1 2/12/| 2 1 exp 2 1 | π ( ) ( ) [ ] [ ] [ ]       −+−− + = − − −− − 1| 1 1|1|2/1 1| 2/ ˆˆ 2 1 exp 2 1 kkkk T kkkk T kkk k T kkkk p kz xHzRHPHxHz RHPH zp π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ]      −+−+−−−−−−⋅ + == − − −−− − −− − − −− − 1| 1 1|1|1| 1 1|1| 1 2/1 1| 2/12/1 1|2/1:1 1:1 :1 ˆˆ 2 1 2 1 2 1 exp 2 1 | || | kkkkk T kkkk T kkkkkkkkk T kkkkkkk T kkk k T kkkk kkknkk kkkk kk xHzRHPHxHzxxPxxxHzRxHz RHPH RPZzp Zxpxzp Zxp  π from which
103 ( ) ( ) ( ) ( ) ( ) [ ] ( )1| 1 1|1|1| 1 1|1| 1 − − −−− − −− − −+−−−−+−− kkkk T kkkkk T kkkkkkkkk T kkkkkkk T kkk xHzHPHRxHzxxPxxxHzRxHz  ( )[ ] ( )[ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ]{ }( ) ( ) ( ) ( ) ( ) ( ) [ ]( )1| 11 1|1|1| 1 1|1| 1 1| 1| 1 1| 1 1|1| 1 1|1| 1| 1 1|1|1|1| 1 1|1| − −− −−− − −− − − − − − − −− − −− − − −−−− − −− −+−+−−−−−− −+−−=−+−− −−+−−−−−−= kkkkk T kkk T kkkkkkkk T kkkkkkkkk T k T kkk kkkk T kkkkkk T kkkkkkkk T kkkkk T kkkk kkkkk T kkkkkkkkkkkk T kkkkkkkk xxHRHPxxxxHRxHzxHzRHxx xHzHPHRRxHzxHzHPHRxHz xxPxxxxHxHzRxxHxHz    [ ] [ ] 1111 1| 1111 1| 1 −−−− − −−−− − − ++/−/=+− k T kkk T kkkkkkk LemmaMatrixInverse T kkkkkk RHHRHPHRRRHPHRRwe have Define: [ ] [ ] 1 1| 1 1| 1 1| 1 1| 111 1|| : − − − − − − − − −−− − +−=+= kk T k T kkkkkkkkkk LemmaMatrixInverse kk T kkkkk PHHPHRHPPHRHPP ( )[ ] ( )[ ]1| 1 |1| 1 |1| 1 |1| − − − − − − − −+−−+−= kkkkk T kkkkkkkk T kkkkk T kkkkkk xHzRHPxxPxHzRHPxx  ( ) ( ) ( )[ ] ( )[ ]      −+−−+−−⋅= − − − − − − − 1| 1 |1| 1 |1| 1 |1|2/1 | 2/:1| 2 1 exp 2 1 | kkkkk T kkkkkkkk T kkkkk T kkkkkk kk nkkzx xHzRHPxxPxHzRHPxx P Zxp  π Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 7) Correction Step (after zk measurement) 1st Way (continue – 3) then ( ) ( ) ( ) ( ) ( ) [ ] ( )1| 1 1|1|1| 1 1|1| 1 − − −−− − −− − −+−−−−+−− kkkkk T kkkk T kkkkkkkkk T kkkkkkk T kkk xHzRHPHxHzxxPxxxHzRxHz  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1| 1 |1|1| 1 || 1 1| 1| 1 | 1 |1|1| 1 | 1 || 1 1| − − −− −− − − −− −− −−− − −−+−−− −−−−−= kkkkk T kkkkkkkkkkkk T kkkk kkkkk T kkkkk T kkkkkkkk T kkkkkkkkk T kkkk xxPxxxxPPHRxHz xHzRHPPxxxHzRHPPPHRxHz  
104 then ( )kkzx x Zxp k :1| |max ( ) { }kk kkkkk T kkkkkkkk ZxE xHzRHPxxx :1 1| 1 |1| * | | ˆˆ:ˆ = −+== − − − Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 8) Correction Step (after zk measurement) 1st Way (continue – 4) ( ) ( ) ( )[ ] ( )[ ]      −+−−+−−⋅= − − − − − − − 1| 1 1| 1 |1| 1 1|2/1 | 2/:1| 2 1 exp 2 1 | kkkkk T kkkkkk T kkkkk T kkkk kk nkkzx xHzRHxxPxHzRHxx P Zxp  π where:[ ] ( )( ){ }k T kkkkkkkk T kkkkk ZxxxxEHRHPP :1|| 111 1|| ˆˆ: −−=+= −−− −
105 { } ( ) ( )  ki kkkkkkkkkkkkkkkkk zzKxxHzKxZxEx 1|1|1|1|:1| ˆ| −−−− −+=−+== Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 9) Summary 1st Way – Kalman Filter Initial Conditions: [ ] 111 1|| : −−− − += kk T kkkkk HRHPP Prediction phase (before zk measurement) 111|111| ˆˆ −−−−−− +Φ= kkkkkkk uGxx Correction Step (after zk measurement) T kk T kkkkkk QPP 1111|111| −−−−−−− ΓΓ+ΦΦ= 1 |: − = k T kkkk RHPK { }00|0 ˆ xEx = ( ) ( ){ }T xxxxEP 0|000|000|0 ˆˆ: −−= kkkk wxHz += { } { } { } 0 1:11|1:11:11| |ˆ||ˆ −−−−− +=+== kkkkkkkkkkkkk ZwExHZwxHEZzEz 1|1| ˆˆ −− = kkkkk xHz
106 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 10) kkkk vxHz += ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π ( ) ( ) [ ] [ ] [ ]       −+−− + = − − −− − 1| 1 1|1|2/1 1| 2/ ˆˆ 2 1 exp 2 1 kkkkk T kkkk T kkkk k T kkkk p kz xHzRHPHxHz RHPH zp π from which { } 1|1:11| ˆ|ˆ −−− == kkkkkkk xHZzEz ( ) ( ){ } kk T kkkkk T kkkkkk zz kk SRHPHZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } [ ] ( )[ ]{ } T kkkk T kkkkkkkk k T kkkkkk xz kk HPZvxxHxxE ZzzxxEP 1|1:11|1| 1:11|1|1| ˆˆ ˆˆ −−−− −−−− =+−−= −−= We also have Correction Step (after zk measurement) 2nd Way Define the innovation: 1|1| ˆˆ: −− −=−= kkkkkk xHzzzi
107 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables       = k k k z x yDefine: assumed that they are Gaussian distributed Prediction phase (before zk measurement) 2nd way (continue -1) { }         =             = − − − − − 1| 1| 1:1 1:1 1:1 ˆ ˆ | | | kk kk kk kk kk z x Zz Zx EZyE         =                 − −         − − = −− −− − − − − − − zz kk zx kk xz kk xx kk k T kkk kkk kkk kkkyy kk PP PP Z zz xx zz xx EP 1|1| 1|1| 1:1 1| 1| 1| 1| 1| ˆ ˆ ˆ ˆ where: [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } kk T kkkkk T kkkkkk zz kk SRHPHZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−= Linear Gaussian Markov Systems (continue – 11)
108 ( ) ( ) ( )    −−−= − − −− − − 1| 1 1|1|2/1 1| 1:1, ˆˆ 2 1 exp 2 1 |, kkk yy kk T kkk yy kk kkkzx yyPyy P Zzxp π Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables The conditional probability distribution function (pdf) of xk given zk is given by: Prediction phase (before zk measurement) 2nd Way (continue – 2) ( ) ( ) ( )      −−−= − − −− − − 1| 1 1|1|2/1 1| 1:1 ˆˆ 2 1 exp 2 1 | kkk zz kk T kkk zz kk kkz zzPzz P Zzp π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    −−−     −−− === − − −− − − −− − − − − − 1| 1 1|1| 1| 1 1|1| 2/1 1| 2/1 1| 1:1 1:1, |1:1| ˆˆ 2 1 exp ˆˆ 2 1 exp 2 2 | |, |,| kkk zz kk T kkk kkk yy kk T kkk yy kk zz kk kkz kkkzx kkzxkkkzx zzPzz yyPyy P P Zzp Zzxp zxpZzxp π π ( ) ( ) ( ) ( )    −−+−−−= − − −−− − −− − − 1| 1 1|1|1| 1 1|1|2/1 1| 2/1 1| ˆˆ 2 1 ˆˆ 2 1 exp 2 2 kkk zz kk T kkkkkk yy kk T kkk yy kk zz kk zzPzzyyPyy P P π π Linear Gaussian Markov Systems (continue – 12) We assumed that is Gaussian distributed:      = k k k z x y
109 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 3) ( ) ( ) ( ) ( ) ( )    −−+−−−= − − −−− − −− − − 1| 1 1|1|1| 1 1|1|2/1 1| 2/1 1| | ˆˆ 2 1 ˆˆ 2 1 exp 2 2 | kkk zz kk T kkkkkk zz kk T kkk yy kk zz kk kkzx zzPzzyyPyy P P zxp π π Define: 1|1| ˆ:&ˆ: −− −=−= kkkkkkkk zzxx ςξ ( ) ( ) ( ) ( ) k zz kk T kk zz kk T kk zx kk T kk xz kk T kk xx kk T k kkkzz T k k k zz kk zx kk xz kk xx kk T k k k zz kk T k k k zz kk zx kk xz kk xx kk T k k kkk zz kk T kkkkkk yy kk T kkk PTTTT P TT TT P PP PP zzPzzyyPyyq ςςςςξςςξξξ ςς ς ξ ς ξ ςς ς ξ ς ξ 1 1|1|1|1|1| 1 1| 1|1| 1|1| 1 1| 1 1|1| 1|1| 1| 1 1|1|1| 1 1|1| ˆˆˆˆ: − −−−−− − − −− −− − − − −− −− − − −−− − −− −+++= −                    = −                    = −−−−−= Linear Gaussian Markov Systems (continue – 13)
110 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd way (continue – 4) Using Inverse Matrix Lemma: ( ) ( ) ( ) ( )         −−− −−− =      −−−−− −−−−−− 11111 111111 nxmnxnmxnmxmmxnmxmnxmnxnmxnmxm mxmnxmmxnmxmnxmnxnmxnmxmnxmnxn mxmmxn nxmnxn BADCDCBADC CBDCBADCBA CD BA         =         −− −− − −− −− zz kk zx kk xz kk xx kk zz kk zx kk xz kk xx kk TT TT PP PP 1|1| 1|1| 1 1|1| 1|1| in 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| − −−− − − − − −−− − − − − −−− − − −= −= −= zz kk xz kk xz kk xx kk xz kk xx kk zx kk zz kk zz kk kkzxkkzzkkxzkkxxkkxx PPTT TTTTP PPPPT k zz kk T kk zz kk T kk zx kk T kk xz kk T kk xx kk T k PTTTTq ςςςςξςςξξξ 1 1|1|1|1|1| − −−−−− −+++= ( ) k zz kk T kk zz kk T k k xz kk xx kk zx kk T kk xz kk xx kk zx kk T kk xz kk T kk xx kk xx kk zx kk T k T k PT TTTTTTTTTT ςςςς ςςςςςξξςξ 1 1|1| 1| 1 1|1|1| 1 1|1|1|1| 1 1|1| − −− − − −−− − −−−− − −− −+ −+++= ( ) ( ) ( ) ( ) ( )k xz kk xx kkk xx kk T k xz kk xx kkkk zz kk xz kk xx kkkkzx zz kk T k k xz kk xx kk xx kk T k xz kk xx kkkk xx kk T k xz kk xx kkk TT TTTTTPTTTT TTTTTTTT zx kk Txz kk ςξςξςς ςςξξςξ 1| 1 1|1|1| 1 1| 0 1|1| 1 1|1|1| 1| 1 1|1|1| 1 1|1|1| 1 1| 1|1| − − −−− − −−− − −−− − − −−− − −−− − − = ++=−−+ +++= −−    Linear Gaussian Markov Systems (continue – 14)
111 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd way (continue – 5)         =         −− −− − −− −− zz kk zx kk xz kk xx kk zz kk zx kk xz kk xx kk TT TT PP PP 1|1| 1|1| 1 1|1| 1|1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| − −−− − − − − −−− − − − − −−− − − −= −= −= zz kk xz kk xz kk xx kk xz kk xx kk zx kk zz kk zz kk kkzxkkzzkkxzkkxxkkxx PPTT TTTTP PPPPT ( ) ( )k xz kk xx kkk xx kk T k xz kk xx kkk TTTTTq ςξςξ 1| 1 1|1|1| 1 1| − − −−− − − ++= 1|1| ˆ:&ˆ: −− −=−= kkkkkkkk zzxx ςξ ( ) ( )[ ] ( )[ ]       −−−−−−−=       −= −−−−− − − − − 1|1|1|1|1|2/1 1| 2/1 1| 2/1 1| 2/1 1| | ˆˆˆˆ 2 1 exp 2 2 2 1 exp 2 2 | kkkkkkk xx kk T kkkkkkk yy kk zz kk yy kk zz kk kkzx zzKxxTzzKxx P P q P P zxp π π π π ( )1| 1 1|1|1| 1 1|1| ˆˆ − − −−− − −− −−−=+ kkk K zz kk xz kkkkkk xx kk xz kkk zzPPxxTT k   ςξ Linear Gaussian Markov Systems (continue – 15)
112 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 6) ( ) ( )[ ] ( )[ ]      −−−−−−−= − − −−−−− − −−− 1| 1 1|1|1|1|1| 1 1|1|1|| ˆˆˆˆ 2 1 exp| kkk xx kk xz kkkkk xx kk T kkk xx kk xz kkkkkkkzx zzPPxxTzzPPxxczxp From this we can see that { } ( )1| 1 1|1|1|| ˆˆˆ| − − −−− −+== kkk K zz kk xz kkkkkkkk zzPPxxzxE k    ( )( ){ } T k zz kkk xx kk zx kk zz kk xz kk xx kk xx kkk T kkkkkk xx kk KPKP PPPPTZxxxxEP 1|1| 1| 1 1|1|1| 1 1|:1||| ˆˆ −− − − −−− − − −= −==−−= [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } k T kkkkkk T kkkkkk zz kk SHPHRZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−= Linear Gaussian Markov Systems (continue – 16)
113 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 7) From this we can see that ( ) [ ] 111 1|1| 1 1|1|1|| −−− −− − −−− +=+−= kk T kkkkkk T kkkkk T kkkkkkk HRHPPHHPHRHPPP ( ) 1 1| 1 1|1| 1 1|1| − − − −− − −− =+== k T kkk T kkkkk T kkk zz kk xz kkk SHPHPHRHPPPK Linear Gaussian Markov Systems (continue – 17) kk T kkkkk KSKPP −= −1|| or [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } k T kkkkkk T kkkkkk zz kk SHPHRZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−=
114 We found that the optimal Kk is [ ] 1 1|1| − −− += T kkkkk T kkkk HPHRHPK [ ] [ ] 1111 |1 11 & 1 |1 1 1| 1 −−−− + −−− + +−=+ − − − k T kkk T kkkkkk LemmaMatrixInverse existPR T kkkkk RHHRHPHRRHPHR kkk [ ] 1111 1| 1 1| 1 1| −−−− − − − − − +−= k T kkk T kkkkk T kkkk T kkkk RHHRHPHRHPRHPK [ ]{ } [ ] 1111 |1 111 |1|1 −−−− + −−− ++ +−+= k T kkk T kkkkk T kkk T kkkkk RHHRHPHRHHRHPP [ ] 1 | 1111 |1 −−−−− + =+= RHPRHHRHPK T kkk T kkk T kkkk If Rk -1 and Pk|k-1 -1 exist: Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 18) Relation Between 1st and 2nd ways 2nd Way 1st Way = 2nd Way
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3 recursive bayesian estimation

  • 1. 1 Recursive Bayesian Estimation SOLO HERMELIN Updated: 22.02.09 11.01.14 http://www.solohermelin.com
  • 2. 2 SOLO Table of Content Recursive Bayesian Estimation Review of Probability Conditional Probability Total Probability Theorem Conditional Probability - Bayes Formula Statistical Independent Events Expected Value or Mathematical Expectation Variance and Central Moments Characteristic Function and Moment-Generating Function Probability Distribution and Probability Density Functions (Examples) Normal (Gaussian) Distribution Existence Theorems 1 & 2 Monte Carlo Method Estimation of the Mean and Variance of a Random Variable Generating Discrete Random Variables Existence Theorem 3 Markov Processes Functions of one Random Variable The Laws of Large Numbers Central Limit Theorem Problem Definition Stochastic Processes
  • 3. 3 SOLO Table of Content (continue -1) Recursive Bayesian Estimation Bayesian Estimation Introduction Linear Gaussian Markov Systems Closed-Form Solutions of Estimation Kalman Filter Extended Kalman Filter General Bayesian Nonlinear Filters Additive Gaussian Nonlinear Filter Gauss – Hermite Quadrature Approximation Unscented Kalman Filter Monte Carlo Kalman Filter (MCKF) Non-Additive Non-Gaussian Nonlinear Filter Nonlinear Estimation Using Particle Filters Importance Sampling (IS) Sequential Importance Sampling (SIS) Sequential Importance Resampling (SIR) Monte Carlo Particle Filter (MCPF) Bayesian Maximum Likelihood Estimate (Maximum Aposteriori – MAP Estimate)
  • 4. 4 SOLO Table of Content (continue -2) Recursive Bayesian Estimation References Nonlinear Filters based on the Fokker-Planck Equation
  • 5. 5 SOLO Recursive Bayesian Estimation kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Since this is a probabilistic problem, we start with a remainder of Probability Theory A discrete nonlinear system is defined by ( ) ( )kkk kkk vxkhz wxkfx ,, ,,1 11 = −= −− State vector dynamics Measurements kk vw ,1− State and Measurement Noise Vectors, respectively Problem Definition: Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements . kx kz Table of Content
  • 6. 6 SOLO Pr (A) is the probability of the event A if S nAAAA ∪∪∪= 21 1A 2A nA jiOAA ji ≠∀/=∩ ( ) 0Pr ≥A(1) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 ( ) 1Pr =S(2) then ( ) ( ) ( ) ( )nAAAA PrPrPrPr 21 +++=  Probability Axiomatic Definition Probability Geometric Definition Assume that the probability of an event in a geometric region A is defined as the ratio between A surface to surface of S. ( ) ( ) ( )SSurface ASurface A =Pr ( ) 0Pr ≥A(1) ( ) 1Pr =S(2) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 then ( ) ( ) ( ) ( )nAAAA PrPrPrPr 21 +++=  S A Review of Probability A more detailed explanation of the subject is given in the “Probability” Presentation
  • 7. 7 SOLO From those definition we can prove the following:( ) 0=/OP(1’) Proof: OOSandOSS /=/∩/∪= ( ) ( ) ( ) ( ) ( ) 0PrPrPrPr 3 =/⇒/+=⇒ OOSS ( ) ( )APAP −= 1(2’) Proof: OAAandAAS /=∩∪= ( ) ( ) ( ) ( ) ( ) ( ) ( )AAAAS Pr1PrPrPr1Pr 32 −=⇒+==⇒ ( ) 1Pr0 ≤≤ A(3’) Proof: ( ) ( ) ( ) ( ) ( ) 1Pr0Pr1Pr 1'2 ≤⇒≥−= AAA ( ) ( )APr0 1 ≤ ( ) 0Pr ≥A(1) ( ) 1Pr =S(2) (3) If jiOAAandAAAA jin ≠∀/=∩∪∪∪= 21 then ( ) ( ) ( ) ( )n AAAA PrPrPrPr 21 +++=  ( ) ( )AABAIf PrPr ≤⇒⊂(4’) Proof: ( ) ( ) ( ) ( ) ( ) ( )BAAABB PrPr0PrPrPr 00 3 ≤⇒≥+−= ≥≥  ( ) ( ) OAABandAABB /=∩−∪−= ( ) ( ) ( ) ( )BABABA ∩−+=∪ PrPrPrPr(5’) Proof: ( ) ( ) ( ) ( ) ( ) ( ) OABBAandABBAB OABAandABABA /=−∩∩−∪∩= /=−∩−∪=∪ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )BABABA ABBAB ABABA ∩−+=∪⇒     −+∩= −+=∪ PrPrPrPr PrPrPr PrPrPr 3 3 Table of Content Review of Probability
  • 8. 8 SOLO Conditional Probability S nAAAA ααα ∪∪∪= 21  1αA jiOAA ji ≠∀/=∩ 1αβA mAAAB βββ ∪∪∪= 212αA 2αβA 1βA 2βA  Given two events A and B decomposed in elementary events jiOAAandAAAAA ji n i in ≠∀/=∩=∪∪∪= = αααααα  1 21 lkOAAandAAAAB lk m k km ≠∀/=∩=∪∪∪= = ββββββ  1 21 jiOAAandAAABA jir ≠∀/=∩∪∪∪=∩ αβαβαβαβαβ 21 ( ) ( ) ( ) ( )n AAAA ααα PrPrPrPr 21 +++=  ( ) ( ) ( ) ( )mAAAB βββ PrPrPrPr 21 +++=  ( ) ( ) ( ) ( ) nmrAAABA r ,PrPrPrPr 21 ≤+++=∩ βαβαβα  We want to find the probability of A event under the condition that the event B had occurred designed as P (A|B) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )B BA AAA AAA BA m r Pr Pr PrPrPr PrPrPr |Pr 21 21 ∩ = +++ +++ = βββ βαβαβα   Review of Probability
  • 9. 9 SOLO Conditional Probability S nAAAA ααα ∪∪∪= 21  1αA jiOAA ji ≠∀/=∩ 1αβA mAAAB βββ ∪∪∪= 212αA 2αβA 1βA 2βA  If the events A and B are statistical independent, that the fact that B occurred will not affect the probability of A to occur. ( ) ( ) ( )B BA BA Pr Pr |Pr ∩ = ( ) ( ) ( )A BA AB Pr Pr |Pr ∩ = ( ) ( )ABA Pr|Pr = ( ) ( ) ( ) ( ) ( ) ( ) ( )BAAABBBABA PrPrPr|PrPr|PrPr ⋅=⋅=⋅=∩ Definition: n events Ai i = 1,2,…n are statistical independent if: ( ) nrAA r i i r i i ,,2PrPr 11  =∀=      ∏== Table of Content Review of Probability
  • 10. 10 SOLO Conditional Probability - Bayes Formula Using the relation: ( ) ( ) ( ) ( ) ( )llll AABBBABA ββββ Pr|PrPr|PrPr ⋅=⋅=∩ ( ) ( ) ( ) klOBABABAB lk m k k , 1 ∀/=∩∩∩∩= = βββ ( ) ( )∑ = ∩= m k k BAB 1 PrPr β we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑= ⋅ ⋅ = ⋅ = m k kk llll l AAB AAB B AAB BA 1 Pr|Pr Pr|Pr Pr Pr|Pr |Pr ββ ββββ β Bayes Formula Thomas Bayes 1702 - 1761 Table of Content Review of Probability
  • 11. 11 SOLO Total Probability Theorem Table of Content jiOAAandSAAA jin ≠∀/=∩=∪∪∪ 21If we say that the set space S is decomposed in exhaustive and incompatible (exclusive) sets. The Total Probability Theorem states that for any event B, its probability can be decomposed in terms of conditional probability as follows: ( ) ( ) ( ) ( )∑∑ == == n i i n i i BPBABAB 11 |Pr,PrPr Using the relation: ( ) ( ) ( ) ( ) ( )llll AABBBABA Pr|PrPr|PrPr ⋅=⋅=∩ ( ) ( ) ( ) klOBABABAB lk n k k , 1 ∀/=∩∩∩∩= =  ( ) ( )∑= ∩= n k k BAB 1 PrPr For any event B we obtain: Review of Probability
  • 12. 12 SOLO Statistical Independent Events ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∏∑∏∑∏∑ ∑∑∑ = −       ≠≠ =       ≠ =       = = −       ≠≠       ≠       == −+−+−=       −+−+−=      n i i n n kji kji i i n ji ji i i n i i tIndependen lStatisticaA n i i n n kji kji kji n ji ji ji n i i n i i AAAA AAAAAAAA i 1 1 3 ,. 3 1 2 . 2 1 1 1 1 1 3 ,. 2 . 1 11 Pr1PrPrPr Pr1PrPrPrPr    From Theorem of Addition Therefore ( )[ ]∏== −=      − n i i tIndependen lStatisticaA n i i AA i 11 Pr1Pr1  ( )[ ]∏== −−=      n i i tIndependen lStatisticaA n i i AA i 11 Pr11Pr  Since OAASAA n i i n i i n i i n i i /=               =               ====   1111 &         =      − ==  n i i n i i AA 11 PrPr1 ( )∏== =      n i i tIndependen lStatisticaA n i i AA i 11 PrPr  If the n events Ai i = 1,2,…n are statistical independent than are also statistical independentiA ( )∏= = n i iA 1 Pr      = =  n i i MorganDe A 1 Pr ( )[ ]∏= −= n i i tIndependen lStatisticaA A i 1 Pr1 ( ) nrAA r i i r i i ,,2PrPr 11  =∀=      ∏== Table of Content Review of Probability
  • 13. 13 SOLO Review of Probability Expected Value or Mathematical Expectation Given a Probability Density Function p (x) we define the Expected Value For a Continuous Random Variable: ( ) ( )∫ +∞ ∞− = dxxpxxE X: For a Discrete Random Variable: ( ) ( )∑= k kXk xpxxE : For a general function g (x) of the Random Variable x: ( )[ ] ( ) ( )∫ +∞ ∞− = dxxpxgxgE X: ( )xp x 0 ∞+∞− 0.1 ( )xE ( ) ( ) ( )∫ ∫ ∞+ ∞− +∞ ∞− = dxxp dxxpx xE X X : The Expected Value is the center of surface enclosed between the Probability Density Function and x axis. Table of Content
  • 14. 14 SOLO Review of Probability Variance Given a Probability Density Functions p (x) we define the Variance ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( )22222 2: xExExExExxExExExVar −=+−=−= Central Moment ( ) { }k k xEx =:'µ Given a Probability Density Functions p (x) we define the Central Moment of order k about the origin ( ) ( )[ ]{ } ( ) ( )∑= −− −      =−= k j jk j jkk k xE j k xExEx 0 '1: µµ Given a Probability Density Functions p (x) we define the Central Moment of order k about the Mean E (x) Table of Content
  • 15. 15 SOLO Review of Probability Moments Normal Distribution ( ) ( ) ( )[ ] σπ σ σ 2 2/exp ; 22 x xpX − = [ ] ( )    −⋅ = oddnfor evennforn xE n n 0 131 σ [ ] ( )      += =−⋅ = + 12!2 2 2131 12 knfork knforn xE kk n n σ π σ Proof: Start from: and differentiate k time with respect to a( ) 0exp 2 >=−∫ ∞ ∞− a a dxxa π Substitute a = 1/(2σ2 ) to obtain E [xn ] ( ) ( ) 0 2 1231 exp 12 22 > −⋅ =− + ∞ ∞− ∫ a a k dxxax kk k π [ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 12 ! 0 122/ 0 222221212 !2 2 exp 2 22 2/exp 2 2 2/exp 2 1 2 + ∞+ = ∞∞ ∞− ++ =−= −=−= ∫ ∫∫ kk k k k xy kkk kdyyy xdxxxdxxxxE σ πσ σ π σ σπ σ σπ σ    Now let compute: [ ] [ ]( )2244 33 xExE == σ Chi-square
  • 16. 16 SOLO Review of Probability Functions of one Random Variable Let y = g (x) a given function of the random variable x defined o the domain Ω, with probability distribution pX (x). We want to find pY (y). Fundamental Theorem Assume x1, x2, …, xn all the solutions of the equation ( ) ( ) ( )n xgxgxgy ==== 21 ( ) ( ) ( ) ( ) ( ) ( ) ( )n nXXX Y xg xp xg xp xg xp yp ''' 2 2 1 1 +++=  ( ) ( ) xd xgd xg =:' Proof ( ) ( ) ( ) ( ) ( ) ( )∑∑∑ === ==±≤≤=+≤≤= n i i iX n i iiX n i iiiY yd xg xp xdxpxdxxxydyYyydyp 111 ' PrPr: q.e.d.
  • 17. 17 SOLO Review of Probability Functions of one Random Variable (continue – 1) Example 1 bxay += ( )       − = a by p a yp XY 1 Example 2 x a y = ( )       = y a p y a yp XY 2 Example 3 2 xay = ( ) ( )yU a y p a y p ya yp XXY                 −+         = 2 1 Example 4 xy = ( ) ( ) ( )[ ] ( )yUypypyp XXY −+= Table of Content
  • 18. 18 SOLO Review of Probability Characteristic Function and Moment-Generating Function Given a Probability Density Functions pX (x) we define the Characteristic Function or Moment Generating Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )     = ==Φ ∑ ∫∫ +∞ ∞− +∞ ∞− x X XX X discretexxpxj continuousxxPdxjdxxpxj xjE ω ωω ωω exp expexp exp: This is in fact the complex conjugate of the Fourier Transfer of the Probability Density Function. This function is always defined since the sufficient condition of the existence of a Fourier Transfer : Given the Characteristic Function we can find the Probability Density Functions pX (x) using the Inverse Fourier Transfer: ( ) ( ) ( ) ∞<== ∫∫ +∞ ∞− ≥+∞ ∞− 1 0 dxxpdxxp X xp X ( ) ( ) ( )∫ +∞ ∞− Φ−= ωωω π dxjxp XX exp 2 1 is always fulfilled.
  • 19. 19 SOLO Review of Probability Properties of Moment-Generating Function ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X X ω ω ω exp ( ) ( ) 10 ==Φ ∫ +∞ ∞− = dxxpXX ω ω ( ) ( ) ( )xEjdxxpxj d d X X == Φ ∫ +∞ ∞−=0ω ω ω ( ) ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X X 22 2 2 exp ω ω ω ( ) ( ) ( ) ( ) ( )2222 0 2 2 xEjdxxpxj d d X X == Φ ∫ +∞ ∞−=ω ω ω ( ) ( ) ( ) ( )∫ +∞ ∞− = Φ dxxpxxjj d d X nn n X n ω ω ω exp ( ) ( ) ( ) ( ) ( )nn X nn n X n xEjdxxpxj d d == Φ ∫ +∞ ∞−=0ω ω ω   ( ) ( ) ( )∫ +∞ ∞− =Φ dxxpxj XX ωω exp This is the reason why ΦX (ω) is also called the Moment-Generation Function.
  • 20. 20 SOLO Review of Probability Properties of Moment-Generating Function ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   +++++= + Φ ++ Φ + Φ +Φ=Φ === = n n n n X n XX XX xE n j xE j xE j d d nd d d d !!2!1 1 ! 1 !2 1 2 2 0 2 0 2 2 0 0 ωωω ω ω ω ω ω ω ω ω ω ωω ωωω ω Develop ΦX (ω) in a Taylor series ( ) ( ) ( )∫ +∞ ∞− =Φ dxxpxj XX ωω exp
  • 21. 21 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (2) Poisson’s Distribution ( ) ( )0 0 exp ! , k k k nkp k −≈ (1) Binomial (Bernoulli) ( ) ( ) ( ) ( ) knkknk pp k n pp knk n nkp −− −      =− − = 11 !! ! , 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k ( )nkP , (3) Normal (Gaussian) ( ) ( ) ( )[ ] σπ σµ σµ 2 2/exp ,; 22 −− = x xp (4) Laplacian Distribution ( )         − −= b x b bxp µ µ exp 2 1 ,;
  • 22. 22 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (5) Gama Distribution ( ) ( ) ( )      < ≥ Γ − = − 00 0 /exp ,; 1 x xx k x kxp k k θ θ θ (6) Beta Distribution ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 1 0 11 11 1 1 1 ,; −− −− −− − ΓΓ +Γ = − − = ∫ βα βα βα βα βα βα xx duuu xx xp (7) Cauchy Distribution ( ) ( )       +− = 22 0 0 1 ,; γ γ π γ xx xxp
  • 23. 23 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) SOLO (8) Exponential Distribution ( ) ( )    < ≥− = 00 0exp ; x xx xp λλ λ (9) Chi-square Distribution ( ) ( ) ( ) ( )      < ≥− Γ= − 00 02/exp 2/ 2/1 ; 12/ 2/ x xxx kkxp k k Γ is the gamma function ( ) ( )∫ ∞ − −=Γ 0 1 exp dttta a (10) Student’s t-Distribution ( ) ( )[ ] ( ) ( )( ) 2/12 /12/ 2/1 ; + +Γ +Γ = ν ννπν ν ν x xp
  • 24. 24 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) SOLO (11) Uniform Distribution (Continuous) ( )      >> ≤≤ −= bxxa bxa abbaxp 0 1 ,; (12) Rayleigh Distribution ( ) 2 2 2 2 exp ; σ σ σ       − = x x xp (13) Rice Distribution ( )             + − = 202 2 22 2 exp ,; σσ σ σ vx I vx x vxp
  • 25. 25 SOLO Review of Probability Probability Distribution and Probability Density Functions (Examples) (14) Weibull Distribution SOLO ( )      < >≥               − −      − = − 00 0,,exp ,,; 1 x x xx xp αγµ α µ α µ α γ αµγ γγ Table of Content
  • 26. 26 SOLO Review of Probability Normal (Gaussian) Distribution Karl Friederich Gauss 1777-1855 ( ) ( ) ( )σµ σπ σ µ σµ ,;: 2 2 exp ,; 2 2 x x xp N=       − − = ( ) ( ) ∫ ∞−       − −= x du u xP 2 2 2 exp 2 1 ,; σ µ σπ σµ ( ) µ=xE ( ) σ=xVar ( ) ( )[ ] ( ) ( )       −=       − −= =Φ ∫ ∞+ ∞− 2 exp exp 2 exp 2 1 exp 22 2 2 σω µω ω σ µ σπ ωω j duuj u xjE Probability Density Functions Cumulative Distribution Function Mean Value Variance Moment Generating Function
  • 27. 27 SOLO Review of Probability Moments Normal Distribution ( ) ( ) ( )[ ] ( )σ σπ σ σ ,0;: 2 2/exp ,0; 22 x x xpX N= − = [ ] ( )    −⋅ = oddnfor evennforn xE n n 0 131 σ [ ] ( )      += =−⋅ = + 12!2 2 2131 12 knfork knforn xE kk n n σ π σ Proof: Start from: and differentiate k time with respect to a( ) 0exp 2 >=−∫ ∞ ∞− a a dxxa π Substitute a = 1/(2σ2 ) to obtain E [xn ] ( ) ( ) 0 2 1231 exp 12 22 > −⋅ =− + ∞ ∞− ∫ a a k dxxax kk k π [ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 12 ! 0 122/ 0 222221212 !2 2 exp 2 22 2/exp 2 2 2/exp 2 1 2 + ∞+ = ∞∞ ∞− ++ =−= −=−= ∫ ∫∫ kk k k k xy kkk kdyyy xdxxxdxxxxE σ πσ σ π σ σπ σ σπ σ    Now let compute: [ ] [ ]( )2244 33 xExE == σ Chi-square
  • 28. 28 SOLO Review of Probability Normal (Gaussian) Distribution (continue – 1) Karl Friederich Gauss 1777-1855 ( ) ( ) ( ) ( )PxxxxPxxPPxxp T ,;: 2 1 exp2,; 12/1  N=    −−−= −− π A Vector – Valued Gaussian Random Variable has the Probability Density Functions where { }xEx  = Mean Value ( )( ){ }T xxxxEP  −−= Covariance Matrix If P is diagonal P = diag [σ1 2 σ2 2 … σk 2 ] then the components of the random vector are uncorrelated, and x  ( ) ( ) ( ) ( ) ( ) ∏= − −       − − =       − −      − −      − − =                               − − −                             − − − −= k i i i ii k k kk kk k T kk xxxxxxxx xx xx xx xx xx xx PPxxp 1 2 2 2 2 2 2 2 2 22 1 2 1 2 11 22 11 1 2 2 2 2 1 22 11 2/1 2 2 exp 2 2 exp 2 2 exp 2 2 exp 0 0 2 1 exp2,; σπ σ σπ σ σπ σ σπ σ σ σ σ π    therefore the components of the random vector are also independent
  • 29. 29 SOLO Review of Probability The Laws of Large Numbers The Law of Large Numbers is a fundamental concept in statistics and probability that describes how the average of randomly selected sample of a large population is likely to be close to the average of the whole population. There are two laws of large numbers the Weak Law and the Strong Law. The Weak Law of Large Numbers The Weak Law of Large Numbers states that if X1,X2,…,Xn,… is an infinite sequence of random variables that have the same expected value μ and variance σ2 , and are uncorrelated (i.e., the correlation between any two of them is zero), then ( ) nXXX nn /: 1 ++=  converges in probability (a weak convergence sense) to μ . We have { } ∞→=<− nforXn 1Pr εµ converges in probability The Strong Law of Large Numbers The Strong Law of Large Numbers states that if X1,X2,…,Xn,… is an infinite sequence of random variables that have the same expected value μ and variance σ2 , and are uncorrelated (i.e., the correlation between any two of them is zero), and E (|Xi|) < ∞ then ,i.e. converges almost surely to μ.{ } ∞→== nforXn 1Pr µ converges almost surely
  • 30. 3030 SOLO Review of Probability The Law of Large Numbers Differences between the Weak Law and the Strong Law The Weak Law states that, for a specified large n, (X1 + ... + Xn) / n is likely to be near μ. Thus, it leaves open the possibility that | (X1 + ... + Xn) / n − μ | > ε happens an infinite number of times, although it happens at infrequent intervals. The Strong Law shows that this almost surely will not occur. In particular, it implies that with probability 1, we have for any positive value ε, the inequality | (X1 + ... + Xn) / n − μ | > ε is true only a finite number of times (as opposed to an infinite, but infrequent, number of times). Almost sure convergence is also called strong convergence of random variables. This version is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). The strong law implies the weak law.
  • 31. 3131 SOLO Review of Probability The Law of Large Numbers Proof of the Weak Law of Large Numbers ( ) iXE i ∀= µ ( ) iXVar i ∀= 2 σ ( )( )[ ] jiXXE ji ≠∀=−− 0µµ ( ) ( ) ( )[ ] µµ ==++= nnnXEXEXE nn //1  ( ) ( )[ ]{ } ( ) ( ) ( )( )[ ] ( )[ ] ( )[ ] nn n n XEXE n XX E n XX EXEXEXVar n jiXXE nn nnn ji 2 2 2 2 22 1 0 2 1 2 12 σσµµ µµ µ µµ == −++− =               −++− =               − ++ =−= ≠∀=−−   Given we have: Using Chebyshev’s inequality on we obtain:nX ( ) 2 2 / Pr ε σ εµ n Xn ≤≥− Using this equation we obtain: ( ) ( ) ( ) n XXX nnn 2 2 1Pr1Pr1Pr ε σ εµεµεµ −≥≥−−≥>−−=≤− As n approaches infinity, the expression approaches 1. Chebyshev’s inequality q.e.d. Monte Carlo Integration Monte Carlo Integration Table of Content
  • 32. 3232 SOLO Review of Probability Central Limit Theorem The first version of this theorem was first postulated by the French-born English mathematician Abraham de Moivre in 1733, using the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This was published in1756 in “The Doctrine of Chance” 3th Ed. Pierre-Simon Laplace (1749-1827) Abraham de Moivre (1667-1754) This finding was forgotten until 1812 when the French mathematician Pierre-Simon Laplace recovered it in his work “Théory Analytique des Probabilités”, in which he approximate the binomial distribution with the normal distribution. This is known as the De Moivre – Laplace Theorem. De Moivre – Laplace Theorem The present form of the Central Limit Theorem was given by the Russian mathematician Alexandr Lyapunov in 1901. Alexandr Mikhailovich Lyapunov (1857-1918)
  • 33. 3333 SOLO Review of Probability Central Limit Theorem (continue – 1) Let X1, X2, …, Xm be a sequence of independent random variables with the same probability distribution function pX (x). Define the statistical mean: m XXX X m m +++ = 21 ( ) ( ) ( ) ( ) µ= +++ = m XEXEXE XE m m 21 ( ) ( )[ ]{ } ( ) ( ) ( ) mm m m XXX EXEXEXVar m mmmXm 2 2 22 21 22 σσµµµ σ ==               −++−+− =−==  Define also the new random variable ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : We have: The probability distribution of Y tends to become gaussian (normal) as m tends to infinity, regardless of the probability distribution of the random variable, as long as the mean μ and the variance σ2 are finite.
  • 34. 3434 SOLO Review of Probability Central Limit Theorem (continue – 2) ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : Proof The Characteristic Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) m X m i m i i m Y m X m j E m X jE m XXX jEYjE i               Φ=                     − =               − =               −++−+− ==Φ − = ∏ ω σ µω σ µ ω σ µµµ ωωω σ µexpexp expexp 1 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0/lim 2 1 !3 / !2 / !1 / 1 2222 33 1 22 0 =      Ο/      Ο/+−= +               − +               − +      − +=      Φ ∞→ − mmmm X E mjX E mjX E mj m m iii Xi ωωωω σ µω σ µω σ µωω σ µ     Develop in a Taylor series( )       Φ − miX ω σ µ
  • 35. 35 SOLO Review of Probability Central Limit Theorem (continue – 3) Proof (continue – 1) The Characteristic Function ( ) ( ) m XY m E i               Φ=Φ − ω ω σ µ ( ) 0/lim 2 1 2222 =      Ο/      Ο/+−=      Φ ∞→ − mmmmm m Xi ωωωωω σ µ ( ) ( )2/exp 2 1 2 22 ω ωω ω −→            Ο/+−=Φ ∞→m m Y mm Therefore ( ) ( ) ( ) ( ) ( )2/exp 2 1 2/exp 2 1 exp 2 1 22 ydyjdyjyp m YY −=−−→Φ−= ∫∫ +∞ ∞− ∞→+∞ ∞− π ωωω π ωωω π The probability distribution of Y tends to become gaussian (normal) as m tends to infinity (Convergence in Distribution). Characteristic Function of Normal Distribution Convergence Concepts Monte Carlo Integration Table of Content
  • 36. 36 SOLO Review of Probability Central Limit Theorem (continue – 2) ( ) ( ) ( ) ( ) m XXXXEX Y m X mm m σ µµµ σ −++−+− = − = 21 : Proof The Characteristic Function ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) m X m i m i i m Y m X m j E m X jE m XXX jEYjE i               Φ=                     − =               − =               −++−+− ==Φ − = ∏ ω σ µω σ µ ω σ µµµ ωωω σ µexpexp expexp 1 21  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0/lim 2 1 !3 / !2 / !1 / 1 2222 33 1 22 0 =      Ο/      Ο/+−= +               − +               − +      − +=      Φ ∞→ − mmmm X E mjX E mjX E mj m m iii Xi ωωωω σ µω σ µω σ µωω σ µ     Develop in a Taylor series( )       Φ − miX ω σ µ
  • 37. 37 SOLO Review of Probability Central Limit Theorem (continue – 3) Proof (continue – 1) The Characteristic Function ( ) ( ) m XY m E i               Φ=Φ − ω ω σ µ ( ) 0/lim 2 1 2222 =      Ο/      Ο/+−=      Φ ∞→ − mmmmm m Xi ωωωωω σ µ ( ) ( )2/exp 2 1 2 22 ω ωω ω −→            Ο/+−=Φ ∞→m m Y mm Therefore ( ) ( ) ( ) ( ) ( )2/exp 2 1 2/exp 2 1 exp 2 1 22 ydyjdyjyp m YY −=−−→Φ−= ∫∫ +∞ ∞− ∞→+∞ ∞− π ωωω π ωωω π The probability distribution of Y tends to become gaussian (normal) as m tends to infinity (Convergence in Distribution). Characteristic Function of Normal Distribution Convergence Concepts Table of Content
  • 38. 38 SOLO Review of Probability Existence Theorems Existence Theorem 1 Given a function G (x) such that ( ) ( ) ( ) 1lim,1,0 ==∞+=∞− ∞→ xGGG x ( ) ( ) 2121 0 xxifxGxG <=≤ ( G (x) is monotonic non-decreasing) ( ) ( ) ( )xGxGxG n xx xx n n == ≥ → + lim We can find an experiment X and a random variable x, defined on X, such that its distribution function P (x) equals the given function G (x). Proof of Existence Theorem 1 Assume that the outcome of the experiment X is any real number -∞ <x < +∞. We consider as events all intervals, the intersection or union of intervals on the real axis. 5x 1x 2x 3x 4x 6x 7x 8x ∞− ∞+ To specify the probability of those events we define P (x)=Prob { x ≤ x1}= G (x1). From our definition of G (x) it follows that P (x) is a distribution function. Existence Theorem 2 Existence Theorem 3
  • 39. 39 SOLO Review of Probability Existence Theorems Existence Theorem 2 If a function F (x,y) is such that ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,,,, 1,,0,, 11122122 ≥+−− =+∞∞+=−∞=∞− yxFyxFyxFyxF FxFyF for every x1 < x2, y1 < y2, then two random variables x and y can be found such that F (x,y) is their joint distribution function. Proof of Existence Theorem 2 Assume that the outcome of the experiment X is any real number -∞ <x < +∞. Assume that the outcome of the experiment Y is any real number -∞ <y < +∞. We consider as events all intervals, the intersection or union of intervals on the real axes x and y. To specify the probability of those events we define P (x,y)=Prob { x ≤ x1, y ≤ y1, }= F (x1,y1). From our definition of F (x,y) it follows that P (x,y) is a joint distribution function. The proof is similar to that in the Existence Theorem 1
  • 40. 40 SOLO Review of Probability Monte Carlo Method Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm. The term Monte Carlo method was coined in the 1940s by physicists Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, working on nuclear weapon projects in the Los Alamos National Laboratory (reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble) Stanislaw Ulam 1909 - 1984 Enrico - Fermi 1901 - 1954 John von Neumann 1903 - 1957 Monte Carlo Casino Nicholas Constantine Metropolis (1915 –1999)
  • 41. 41 SOLO Review of Probability Monte Carlo Approximation Monte Carlo runs, generate a set of random samples that approximate the distribution p (x). So, with P samples, expectations with respect to the filtering distribution are approximated by ( ) ( ) ( ) ( )∑∫ = ≈ P L L xf P dxxpxf 1 1 and , in the usual way for Monte Carlo, can give all the moments etc. of the distribution up to some degree of approximation. { } ( ) ( ) ∑∫ = ≈== P L L x P dxxpxxE 1 1 1 µ ( ){ } ( ) ( ) ( ) ( )∑∫ = −≈−=−= P L nLnn n x P dxxpxxE 1 111 1 µµµµ  Table of Content x(L) are generated (draw) samples from distribution p (x) ( ) ( )xpx L ~
  • 42. 42 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (Unknown Statistics) { } { } jimxExE ji ,∀== Define Estimation of the Population mean ∑= = k i ik x k m 1 1 :ˆ A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ ( ) { } ( ) ( ) ( )[ ] ( ) ( )[ ] 2 1 2 1 222 2 22222 1 11 2 1 2 2 11 2 1 2 11 1 1 1 1 1 21 11 2 1 ˆˆ2 1 ˆ 1 σσ σσσ k k kk mkmkk k mmk k m k xx k Ex k xExE k mxmxE k mx k E k i k i k i k l l k j j k j jii k k i ik k i i k i ki − =      −=       ++−+++−−+=               +       −=       +−=       − ∑ ∑ ∑ ∑∑∑ ∑∑∑ = = = === === { } { } jimxExE ji ,2222 ∀+== σ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } { } { } jimxExExxE ji tindependenxx ji ji ,2 , ∀== Compute Biased Unbiased Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
  • 43. 43 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 1) { } { } jimxExE ji ,∀== Define Estimation of the Population mean ∑= = k i ik x k m 1 1 :ˆ A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ ( ) 2 1 2 1 ˆ 1 σ k k mx k E k i ki − =       −∑= { } { } jimxExE ji ,2222 ∀+== σ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } { } { } jimxExExxE ji tindependenxx ji ji ,2 , ∀== Biased Unbiased Therefore, the unbiased estimation of the sample variance of the population is defined as: ( )∑= − − = k i kik mx k 1 22 ˆ 1 1 :ˆσ since { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE Unbiased Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
  • 44. 44 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 2) A random variable, x, may take on any values in the range - ∞ to + ∞. Based on a sample of k values, xi, i = 1,2,…,k, we wish to compute the sample mean, , and sample variance, , as estimates of the population mean, m, and variance, σ2 . 2 ˆkσ kmˆ { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE Monte Carlo simulations assume independent and identical distributed (i.i.d.) samples.
  • 45. 45 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 3) { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EEWe found: Let Compute: ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ){ } ( ){ } k mxEmxEmxE k mxmxEmxE k mx k Emx k EmmE k i k ij j ji k i i k i k ij j ji k i i k i i k i ikmk 2 1 1 00 1 2 2 1 11 2 2 2 1 2 1 22 ˆ 2 1 1 11 ˆ: σ σ σ =           −−+−=           −−+−=               −=               −=−= ∑ ∑∑ ∑∑∑ ∑∑ = ≠ == = ≠ == ==  ( ){ } k mmE kmk 2 22 ˆ ˆ: σ σ =−=
  • 46. 46 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 4) Let Compute: ( ){ } ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )               −− − +− − − +− − =               −−+−−+− − =               −−+− − =               −− − =−= ∑∑ ∑ ∑∑ == = == 2 22 11 2 2 2 1 22 2 2 1 2 2 2 1 22222 ˆ ˆ 11 ˆ2 1 1 ˆˆ2 1 1 ˆ 1 1 ˆ 1 1 ˆ:2 σ σ σσσσσσ k k i i k k i i k i kkii k i ki k i kik mm k k mx k mm mx k E mmmmmxmx k E mmmx k Emx k EE k ( ) ( ){ } ( ){ } ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ }                 k k k i i k k i i k k k i i k k i i k k k i i k k k k i i k k k i k ij j ji k k i i mmE k k mxE k mmE mxE k mmEk mxE k mxE k mmEk mxE k mmE mmE k k mxE k mmE mxEmxEmxE kk / 2 2 1 0 2 0 1 0 2 3 1 2 2 1 2 2 / 2 1 3 2 0 44 2 2 1 2 2 / 2 1 1 22 1 4 2 2 ˆ 2 222 22 22 4 2 ˆ 1 2 1 ˆ4 1 ˆ4 1 2 1 ˆ2 1 ˆ4 ˆ 11 ˆ4 1 1 σ σσσ σσ σσ µ σ σσ σ σσ − − −− − − −− − − + − − −− − − +− − − + +− − +− − − +             −−+− − ≈ ∑∑ ∑∑∑ ∑∑ ∑∑ == === == ≠ == Since (xi – m), (xj - m) and are all independent for i ≠ j:( )kmm ˆ−
  • 47. 47 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 4) Since (xi – m), (xj - m) and are all independent for i ≠ j:( )kmm ˆ− ( ) ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }4 2 2 4 22 4 44 2 4 44 2 2 2 4 2 4 2 42 ˆ ˆ 11 7 11 2 1 2 1 2 ˆ 11 4 1 1 1 2 k k mmE k k k k k k kk k k k mmE k k kk kk k k k − − + − +− + − = − − − − − + +− − + − + − − + − ≈ σ µσσσ σ σσµ σσ kk 4 42 ˆ 2 σµ σσ − ≈ ( ){ }4 4 : mxE i −=µ ( ) ( ){ } ( ){ } ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ){ } ( ) ( ){ } ( ) ( ){ }                 k k k i i k k i i k k k i i k k i i k k k i i k k k k i i k k k i k ij j ji k k i i mmE k k mxE k mmE mxE k mmEk mxE k mxE k mmEk mxE k mmE mmE k k mxE k mmE mxEmxEmxE kk / 2 2 1 0 2 0 1 0 2 3 1 2 2 1 2 2 / 2 1 3 2 0 44 2 2 1 2 2 / 2 1 1 22 1 4 2 2 ˆ 2 222 22 22 4 2 ˆ 1 2 1 ˆ4 1 ˆ4 1 2 1 ˆ2 1 ˆ4 ˆ 11 ˆ4 1 1 σ σσσ σσ σσ µ σ σσ σ σσ − − −− − − −− − − + − − −− − − +− − − + +− − +− − − +             −−+− − ≈ ∑∑ ∑∑∑ ∑∑ ∑∑ == === == ≠ ==
  • 48. 48 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 5) { } { } mxE k mE k i ik == ∑=1 1 ˆ { } ( ) 2 1 22 ˆ 1 1 :ˆ σσ =       − − = ∑= k i kik mx k EE We found: ( ){ } k mmE kmk 2 22 ˆ ˆ: σ σ =−= ( ){ } ( ) k mx k EE k i kik k 4 4 2 2 1 22222 ˆ ˆ 1 1 ˆ:2 σµ σσσσσ − ≈               −− − =−= ∑= ( ){ }4 4 : mxE i −=µ Kurtosis of random variable xi Define 4 4 : σ µ λ = ( ){ } ( ) ( ) k mx k EE k i kik k 42 2 1 22222 ˆ 1 ˆ 1 1 ˆ:2 σλ σσσσσ − ≈               −− − =−= ∑=
  • 49. 49 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 6) [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n For high values of k, according to the Central Limit Theorem the estimations of mean and of variance are approximately Gaussian Random Variables. kmˆ 2 ˆkσ We want to find a region around that will contain σ2 with a predefined probability φ as function of the number of iterations k. 2 ˆkσ Since are approximately Gaussian Random Variables nσ is given by solving: 2 ˆkσ ϕζζ π σ σ =      −∫ + − n n d2 2 1 exp 2 1 nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 Cumulative Probability within nσ Standard Deviation of the Mean for a Gaussian Random Variable 22 k 22 1 ˆ- 1 σ λ σσσ λ σσ k n k n − ≤≤ − − 22 k 2 1 1 ˆ-1 1 σ λ σσ λ σσ         − − ≤≤        + − − k n k n ( ) ( ) ( ) ( )( )42222 1,0;ˆ~ˆ&,0;ˆ~ˆ σλσσσσ −−− kkkk kmmmk NN
  • 50. 50 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 7) [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n 22 k 22 1 ˆ- 1 σ λ σσσ λ σσ k n k n − ≤≤ − − 22 k 2 1 1 ˆ-1 1 σ λ σσ λ σσ         − − ≤≤        + − − k n k n 22 ˆ 1 2 k σ λ σσ k − = 22 k 2 1 1ˆ 1 1 σ λ σσ λ σσ         − −≥≥        − + k n k n         − − ≥≥         − + k n k n 1 1 ˆ 1 1 2 2 k 2 λ σ σ λ σ σσ k n k n 1 1 :ˆ: 1 1 k − − =≥≥= − + λ σ σσσ λ σ σσ
  • 51. 51 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 8)
  • 52. 52 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 9)
  • 53. 53 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue - 10) k n k n kk 1ˆ 1 :& 1ˆ 1 : 00 − − = − + = λ σ σ λ σ σ σσ Monte-Carlo Procedure Choose the Confidence Level φ and find the corresponding nσ using the normal (Gaussian) distribution. nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 1 Run a few sample k0 > 20 and estimate λ according to2 ( ) ( ) 2 1 2 0 1 4 0 0 0 0 0 0 ˆ 1 ˆ 1 :ˆ       − − = ∑ ∑ = = k i ki k i ki k mx k mx k λ∑= = 0 0 10 1 :ˆ k i ik x k m 3 Compute and as function of kσ σ 4 Find k for which [ ] ϕσσσ σσ =≤≤ 2 ˆ 2 k 2 k ˆ-0Prob n 5 Run k-k0 simulations
  • 54. 54 SOLO Review of Probability Estimation of the Mean and Variance of a Random Variable (continue – 11) Monte-Carlo Procedure Choose the Confidence Level φ = 95% that gives the corresponding nσ=1.96. nσ φ 1.000 0.6827 1.645 0.9000 1.960 0.9500 2.576 0.9900 1 The kurtosis λ = 32 3 Find k for which ϕσ λ σσ σ σ =             − ≤≤  2 kˆ 22 k 2 1 ˆ-0Prob k n 4 Run k>800 simulations Example: Assume a Gaussian distribution λ = 3 95.0 2 96.1ˆ-0Prob 2 kˆ 22 k 2 =             ≤≤  σ σσσ k Assume also that we require also that with probability φ = 95 %22 k 2 1.0ˆ- σσσ ≤ 1.0 2 96.1 = k 800≈k
  • 55. 55 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators • First attempts to generate “random numbers”: - Draw balls out of a stirred urn - Roll dice • 1927: L.H.C. Tippett published a table of 40,000 digits taken “at random” from census reports. • 1939: M.G. Kendall and B. Babington-Smith create a mechanical machine to generate random numbers. They published a table of 100,000 digits. • 1946: J. Von Neumann proposed the “middle square method”. • 1948: D.H. Lehmer introduced the “linear congruential method”. • 1955: RAND Corporation published a table of 1,000,000 random digits obtained from electronic noise. • 1965: M.D. MacLaren and G. Marsaglia proposed to combine two congruential generators. • 1989: R.S. Wikramaratna proposed the additive congruential method.
  • 56. 56 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators A Random Number represents the value of a random variable uniform distributed on (0,1). Pseudo-Random Numbers constitute a sequence of values, which although are deterministically generated, have all the appearances of being independent uniform distributed on (0,1). One approach 1. Define x0 = integer initial condition or seed 2. Using integers a and m recursively compute mxax nn modulo1−= mxIntegerxkmaxmkxa nnn <∈+⋅=− ,,,1 Therefore xn takes the values 0,1,…,m-1 and the quantity un=xn/m , called a pseudo-random number is an approximation to the value of uniform (0,1) random variable. In general the integers a and m should be chose to satisfy three criteria: 1. For any initial seed, the resultant sequence has the “appearance” of being a sequence of independent (0,1) random variables. For any initial seed, the number of variables that can be generated before repetition begins is large. The values can be computed efficiently on a digital computer. Multiplicative congruential method Return to Monte Carlo Approximation
  • 57. 57 SOLO Review of Probability Generating Discrete Random Variables Pseudo-Random Number Generators (continue – 1) A guideline is to choose m to be a large prime number compared to the computer word size. Examples: 32 bits word computer: 807,16712 531 ==−= am 125,35312 535 ==−= am36 bits word computer: Another generator of pseudo-random numbers uses recursions of the type: ( ) mcxax nn modulo1 += − mxIntegerxkmcaxmkcxa nnn <∈+⋅=+− ,,,,1 Mixed congruential method
  • 58. 58 SOLO Review of Probability Generating Discrete Random Variables Histograms Return to Table of Content A histogram is a graphical display of tabulated frequencies, shown as bars. It shows what proportion of cases fall into each of several categories: it is a form of data binning. The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent. The intervals (or bands, or bins) are generally of the same size. Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram always equals 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot. A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram Mi of a histogram mi is defined as: An ordinary and a cumulative histogram of the same data. The data shown is a random sample of 10,000 points from a normal distribution with a mean of 0 and a standard deviation of 1. Mathematical Definition ∑= = k i imn 1 In a more general mathematical sense, a histogram is a mapping mi that counts the number of observations that fall into various disjoint categories (known as bins), whereas the graph of a histogram is merely one way to represent a histogram. Thus, if we let n be the total number of observations and k be the total number of bins, the histogram mi meets the following conditions: ∑= = i j ji mM 1
  • 59. 59 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method Suppose we want to generate a discrete random variable X having probability density function: ( ) 1,1,0)( ==−= ∑j jjj pjxxpxp δ To accomplish this, let generate a random number U that is uniformly distributed over (0,1) and set:            <≤ +<≤ < = ∑∑ = − =   j i i j i ij pUpifx ppUpifx pUifx X 1 1 1 1001 00 j j i i j i ij ppUpPxXP =       <<== ∑∑ = − = 1 1 1 )( Since , for any a and b such that 0 < a < b < 1, and U is uniformly distributed P (a ≤ U < b) = b-a, we have: and so X has the desired distribution.
  • 60. 60 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 1) Suppose we want to generate a discrete random variable X having probability density function: ( ) 1,1,0)( ==−= ∑j jjj pjxxpxp δ Draw X, N times, from p (x) Histogram of the Results
  • 61. 61 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 2) Generating a Poisson Random Variable: 1,1,0 ! )( ===== ∑− i i i i pi i eiXPp  λλ ( ) 1 ! !1 1 1 + = + = − + − + i i e i e p p i i i i λ λ λ λ λ Draw X , N times, from Poisson Distribution Histogram of the Results
  • 62. 62 SOLO Review of Probability Generating Discrete Random Variables The Inverse Transform Method (continue – 3) Generating Binominal Random Variable: ( ) ( ) 1,1,01 !! ! )( ==− − === ∑− i i ini i pipp ini n iXPp  ( ) ( ) ( ) ( ) ( ) p p i in pp ini n pp ini n p p ini ini i i −+ − = − − − −−+ = − −−+ + 111 !! ! 1 !1!1 ! 11 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k ( )nkP , Histogram of the Results
  • 63. 63 SOLO Review of Probability Generating Discrete Random Variables The Accaptance-Rejection Technique Suppose we have an efficient method for simulating a random variable having a probability density function { qj, j ≥0 }. We want to use this to obtain a random variable that has the probability density function { pj, j ≥0 }. Let c be a constant such that: 0.. ≠∀≤ j j j qtsjc q p If such a c exists, it must satisfy: cqcp j j j j ≤⇒≤ ∑∑ 1 11  Rejection Method Step 1: Simulate the value of Y, having probability density function qj. Step 2: Generate a random number U (that is uniformly distributed over (0,1) ). Step 3: If U < pY/c qY, set X = Y and stop. Otherwise return to Step 1.
  • 64. 64 SOLO Review of Probability Generating Discrete Random Variables The Acceptance-Rejection Technique (continue – 1) Theorem The random variable X obtained by the rejection method has probability density function P { X=i } = pi. Proof { } { } { } { } { }Acceptance , Acceptance Acceptance, Acceptance Method Acceptance Method Acceptance P qc p UiYP P iYP iYPiXP i i Bayes       ≤= = = ==== { } { } { } { }AcceptanceAcceptanceAcceptance (0,1)ddistribute uniformlyU ceindependen by Pc p P qc p q P qc p UPiYP ii i i i i qi ==      ≤= =  Summing over all i, yields { } { }Acceptance 1 1 Pc p iXP i i i   ∑ ∑ == { } 1Acceptance =Pc { } ipiXP == { } 1 1 Acceptance ≤= c P q.e.d.
  • 65. 65 SOLO Review of Probability Generating Discrete Random Variables The Acceptance-Rejection Technique (continue – 2) Example Generate a truncated Gaussian using the Accept-Reject method. Consider the case with ( ) [ ]     −∈ ≈ − otherwise xe xp x 0 4,42/2/2 π Consider the Uniform proposal function ( ) [ ]    −∈ ≈ otherwise x xq 0 4,48/1 In Figure we can see the results of the Accept-Reject method using N=10,000 samples.
  • 66. 66 SOLO Review of Probability Generating Continuous Random Variables The Inverse Transform Algorithm Let U be a uniform (0,1) random variable. For any continuous distribution function F the random variable X defined by ( )UFX 1− = has distribution F. [ F-1 (u) is defined to be that value of x such that F (x) = u ] Proof Let Px(x) denote the Probability Distribution Function X=F-1 (U) ( ) { } ( ){ }xUFPxXPxPx ≤=≤= −1 Since F is a distribution function, it means that F (x) is a monotonic increasing function of x and so the inequality “a ≤ b” is equivalent to the inequality “F (a) ≤ F (b)”, therefore ( ) ( )[ ] ( ){ } ( )[ ] ( ){ } ( ) ( ) ( )xFxFUP xFUFFPxP uniformU xF UUFF x 1,0 10 1 1 ≤≤ = − =≤= ≤= −
  • 67. 67 SOLO Review of Probability Importance Sampling Let Y = (Y1,…,Ym) a vector of random variables having a joint probability density function f (y1,…,ym), and suppose that we are interested in estimating ( )[ ] ( ) ( )∫== mmmmf dydyyyfyyhYYhE  1111 ,,,,,,θ Suppose that a direct generation of the random vector Y so as to compute h (Y) is inefficient possible because (a) is difficult to generate the random vector Y, or (b) the variance of h (Y) is large, or (c) both of the above Suppose that W=(W1,…,Wm) is another random vector, which takes values in the same domain as Y, and has a joint density function g(w1,…,wm) that can be easily generated. The estimation θ can be expressed as: ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )       === ∫ Wg WfWh Edwdwwwg wwg wwfwwh YYhE gmm m mm mf     11 1 11 1 ,, ,, ,,,, ,,θ Therefore, we can estimate θ by generating values of random vector W, and then using as the estimator the resulting average of the values h (W) f (W)/ g (W). Return to Particle Filters
  • 68. 68 SOLO Review of Probability Monte Carlo Integration Monte Carlo Method can be used to numerically evaluate multidimensional integrals ( ) ( )∫∫ == xdxgdxdxxxgI mm  11 ,, To use Monte Carlo we factorize ( ) ( ) ( )xpxfxg ⋅= ( ) ( ) 1&0 =≥ ∫ xdxpxp in such a way that is interpreted as a Probability Density Function such that( )xp We assume that we can draw NS samples from ( )xp( )S i Nix ,,1, = ( ) S i Nixpx ,,1~ = Using Monte Carlo we can approximate ( ) ( )∑= −≈ SN i S i Nxxxp 1 /δ ( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑∫ ∫ ∑∫ == = =−⋅= −⋅=≈⋅= SS S S N i i S N i i S N i S i N xf N xdxxxf N xdNxxxfIxdxpxfI 11 1 11 / δ δ
  • 69. 69 SOLO Review of Probability Monte Carlo Integration we draw NS samples from ( )xp( )S i Nix ,,1, = ( ) S i Nixpx ,,1~ = ( ) ( ) ( )∑∫ = =≈⋅= S S N i i S N xf N IxdxpxfI 1 1 If the samples are independent, then INS is an unbiased estimate of I. i x According to the Law of Large Numbers INS will almost surely converge to I: II sa N N S S .. ∞→ → ( )[ ] ( ) ∞<−= ∫ xdxpIxff 22 :σIf the variance of is finite; i.e.:( )xf then the Central Limit Theorem holds and the estimation error converges in distribution to a Normal Distribution: ( ) ( )2 ,0~lim fNS N IIN S S σN− ∞→ The error of the MC estimate, e = INS – I, is of the order of O (NS -1/2 ), meaning that the rate of convergence of the estimate is independent of the dimension of the integrand. Numerical Integration of and ( )kk xzp |( )1| −kk xxp Return to Particle Filters
  • 70. 70 SOLO Review of Probability Existence Theorems Existence Theorem 3 Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation. Proof of Existence Theorem 3 Define ( ) ( ) ( ) ( ) ( )ω π ω π ω ωωω π −= − === ∫ +∞ ∞− f a S a S fdSa 22 2 :& 1 : Since , according to Existence Theorem 1, we can find a random variable ω with the even density function f (ω), and probability density function ( ) ( ) 1&0 =≥ ∫ +∞ ∞− ωωω dff ( ) ( )∫∞− = ω ττω dfP : We now form the process , where is a random variable uniform distributed in the interval (-π,+π) and independent of ω. ( ) ( )ϑω += tatx cos: ϑ
  • 71. 71 SOLO Review of Probability Existence Theorems Existence Theorem 3 Proof of Existence Theorem 3 (continue – 1) Since is uniform distributed in the interval (-π,+π) and independent of ω, its spectrum is ( ){ } ( ){ } ( ){ } ( ){ } ( ){ } 0sinsincoscos 00 , =−=  ϑωϑω ϑωϑω ϑω EtEaEtEatxE tindependen ϑ ( ) { } ( ) ϖπ ϖπ ϖπϖπ ϑ π ϖ πϖπϖπ π ϑϖπ π ϑϖϑϖ ϑϑ sin 2 1 2 1 2 1 = − ==== −+ − + − ∫ j ee j e deeES jjj jj or { } ( ){ } ( ){ } ( ) ϖπ ϖπ ϑϖϑϖ ϑϑ ϑϖ ϑ sin sincos =+= EjEeE j 1=ϖ 1=ϖ ( ) ( ){ } ( ) ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ } ( )[ ]{ } ( ){ }  0 2 0 22, 22 2 2sin2sin 2 2cos2cos 2 cos 2 22cos 2 cos 2 coscos ϑτωϑτωτω ϑτωτω ϑτωϑωτ ϑωϑωω ϑω EtE a EtE a E a tE a E a ttEatxtxE tindependen +−++= +++= +++=+ 2=ϖ 2=ϖ Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation.
  • 72. 72 SOLO Review of Probability Existence Theorems Existence Theorem 3 Proof of Existence Theorem 3 (continue – 2) ( ){ } 0=txE ( ) ( ){ } ( ){ } ( ) ( ) ( )τωωτωτωτ ω xRdf a E a txtxE ===+ ∫ +∞ ∞− cos 2 cos 2 22 ( ) ( )ϑω += tatx cos:We have Because of those two properties x (t) is wide-sense stationary with a power spectrum given by: ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫ +∞ ∞− −=+∞ ∞− =−= ττωτττωτωτω ττ dRdjRS x RR xx xx cossincos ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )∫∫ +∞ ∞− −=+∞ ∞− =+= ωτωω π ωτωτωω π τ ωω dSdjSR x SS xx xx cos 2 1 sincos 2 1 Therefore ( ) ( )ωπω faSx 2 = q.e.d. Fourier Inverse Fourier ( ) ( )∫ +∞ ∞− = ωωτω df a cos 2 2 f (ω) definition ( )ωS= Given a function S (ω)= S (-ω) or, equivalently, a positive-defined function R (τ), (R (τ) = R (-τ), and R (0)=max R (τ), for all τ ), we can find a stochastic process x (t) having S (ω) as its power spectrum or R (τ) as its autocorrelation.
  • 73. 73 SOLO Markov Processes A Markov Process is defined by: Andrei Andreevich Markov 1856 - 1922 ( ) ( )( ) ( ) ( )( ) 111 ,|,,,|, tttxtxptxtxp >∀ΩΩ=≤ΩΩ ττ i.e. the Random Process, the past up to any time t1 is fully defined by the process at t1. Examples of Markov Processes: 1. Continuous Dynamic System ( ) ( ) ( ) ( )vuxthtz wuxtftx ,,, ,,, = = 2. Discrete Dynamic System ( ) ( ) ( ) ( )kkkkk kkkkk vuxthtz wuxtftx ,,, ,,, 1111 = = −−−− x - state space vector (n x 1) u - input vector (m x 1) - measurement vector (p x 1)z v - white measurement noise vector (p x 1) - white input noise vector (n x 1)w Recursive Bayesian Estimation
  • 74. 74 Recursive Bayesian EstimationSOLO Using this property we obtain: ( ) ( )1021 |,,,| −−− = kkkkk xxpxxxxp  Markov Processes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∏= − −−−− −−−−−− = = = −− − k i ii k xxp kkkk kk xxp kkkkkk xxpxp xxpxxxpxxp xxxpxxxxpxxxxp kk kk 1 10 02 | 0211 021 | 021021 | ,,,,|| ,,,,,,|,,,, 21 1           Markov Process: Table of Content the present discrete state probability depends only on the previous state. The Markov Process is defined if we know p (x0) and p(xi|xi-1) for each i.
  • 75. 75 Recursive Bayesian EstimationSOLO In a Markovian system the probability of the current true state depends only on the previous state, and is independent of the other earlier states ( ) ( )1021 |,,,| −−− = kkkkk xxpxxxxp  Similarly the measurements at the k-th time- step is dependent upon the current true state, so is conditionally independent of all other earlier states, given the current state ( ) ( )kkkkk xzpxxxzp |,,,| 01 =−  ( ) ( ) ( ) ( ) ( )kkkkkkkk zpzxpxpxzpxzp ||, == From the definition of the Markovian system (see Figure) p (xk|xk-1) is defined by f and the statistics of x and w and p (zk|xk) is defined by h and statistics of x and v. kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )111 ,, −−− kkk wuxf ( )kk vxh , Markov Processes ( )000 ,, wuxf ( )11,vxh ( )111 ,, wuxf ( )22 ,vxh Hidden States Measurements
  • 76. 76 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx :, ,,:,, 011111 0 = = −−−−− Markov Processes ( ) ( )j kkkkxkkkw j k wuxfxtsNjuxxfw k 11111 1 1 ,,..,..,1,, −−−−− − − === Suppose that we can obtain all for which j kw 1− ( ) ( ) ( )∑= − −−−−− ∇= kxN j j kkkw j kwkk wuxfwpxxp 1 1 11111 ,,|then ( ) ( ) ( )∑= − ∇= kx k N j j kkv j kvkk vxhvpxzp 1 1 ,| ( ) ( )j kkkzkkv j k vxhztsNjxzhv k ,..,..,1,1 === − In the same way, suppose that we can obtain all for whichj kv then ( ) ( ) ( ) ( ) ( )∑ ∑ = − −−−− = −−−− ∇= =+≤≤= kx kx N j k j kkkw j kw N j j k j kwkkkkkkkk xdwuxfwp wdwpxxdxXxxdxxp 1 1 1111 1 1111 ,, |Pr| This is a Conceptual Not a Practical Procedure Analytic Computations of and .( )kk xzp |( )1| −kk xxp
  • 77. 77 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx : ,,:, 011111 0 += += −−−−− kx1−kx kz1−kz ( ) 111, −−− + kkk wuxf ( ) kk vxh + Markov Processes ( ) ( )[ ]111 ,| −−− −= kkkwkk uxfxpxxptherefore ( ) ( )[ ]kkvkk xhzpxzp −=|and For additive noise we have ( ) ( )kkk kkkk xhzv uxfxw −= −= −−− 111 , Analytic Computations of and (continue – 1)( )kk xzp |( )1| −kk xxp
  • 78. 78 SOLO ( ) ( )kkk kkk vxhz wxfx , , 11 = = −− kk vw &1− are system and measurement white-noise sequences independent of past and current states and on each other and having known P.D.F.s ( ) ( )kk vpwp &1− We want to compute p (xk|Z1:k) recursively, assuming knowledge of p(xk-1|Z1:k-1) in two stages, prediction (before) and update (after measurement) ( ) ( )( ) ( )∫ −−−−− −= 11111 ,| kkkkkkk wdwpwxfxxxp δ We need to evaluate the following integrals: ( ) ( )( ) ( )∫ −= kkkkkkk vdvpvxhzxzp ,| δ We use the numeric Monte Carlo Method to evaluate the integrals: Generate (Draw): ( ) ( ) Sk i kk i k Nivpvwpw ,,1~&~ 11 =−− ( ) ( )( ) S N i i k i k i kkk Nwxfxxxp S ∑= −−− −≈ 1 111 /,| δ ( ) ( )( ) S N i i k i k i kkk Nvxhzxzp S ∑= −≈ 1 /,| δ or ( ) ( ) ( ) S N i i kkkk i k i k i k Nxxxxpwxfx S ∑= −−− −≈→= 1 111 /|, δ ( ) ( ) ( ) S N i i kkkk i k i k i k Nzzxzpvxhz S ∑= −≈→= 1 /|, δ Analytic solutions for those integral equations do not exist in the general case. Recursive Bayesian Estimation Numerical Computations of and .( )kk xzp |( )1| −kk xxp Markov Processes Prediction (before measurement) ( ) ( ) ( )∫ −−−−− = 11:1111:1 ||| kkkkkkk xdZxpxxpZxp1 Update (after measurement) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ − − − − = − === kkkkk kkkk kk kkkk Bayes bp apabp bap kkkkk xdZxpxzp Zxpxzp Zzp Zxpxzp ZzxpZxp 1:1 1:1 1:1 1:1 | | 1:1:1 || || | || ,|| 2
  • 79. 79 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( ) ( )kvkkk xkkwkkkk vpgivenvxhz xpuwpgivenwuxfx :, ,,:,, 011111 0 = = −−−−− Markov Processes Monte Carlo Computations of and .( )kk xzp |( )1| −kk xxp Generate (Draw) ( ) Sx i Nixpx ,,1~ 00 0 = For { }∞∈ ,,1 k Initialization0 1 At stage k-1 Generate (Draw) NS samples ( ) Skw i k Niwpw ,,1~ 11 =−− 2 State Update ( ) S i kk i k i k Niwuxfx ,,1,, 111 == −−− 3 Generate (Draw) Measurement Noise ( ) Skv i k Nivpv ,,1~ = k:=k+1 & return to 1 Compute Histograms of to obtain ( )kk xzp | kk xz | ( ) ( )∑= − −≈ SN i S i kkkk Nxxxxp 1 1 /| δ ( ) ( )∑= −≈ SN i S i kkkk Nzzxzp 1 /| δ Compute Histograms of to obtain 1| −kk xx ( )1| −kk xxp 4 Measurement , Update ( ) S i k i k i k Nivxhz ,,1, ==kz
  • 80. SOLO Stochastic Processes deal with systems corrupted by noise. A description of those processes is given in “Stochastic Processes” Presentation. Here we give only one aspect of those processes. ( ) ( ) ( ) [ ]fttttwddttxftxd ,, 0∈+= A continuous dynamic system is described by: Stochastic Processes ( )tx - n- dimensional state vector ( )twd - n- dimensional process noise vector Assuming system measurements at discrete time tk given by: ( ) ( )( ) [ ]fkkkkk tttvttxhtz ,,, 0∈= kv - m- dimensional measurement noise vector at tk We are interested in the probability of the state at time t given the set of discrete measurements until (included) time tk < t. x ( )kZtxp |, { }kk zzzZ ,,, 21 = - set of all measurements up to and including time tk. The time evolution of the probability density function is described by the Fokker–Planck equation.
  • 81. A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. The initial condition is a Dirac delta function in x = 1, and the distribution drifts towards x = 0. The Fokker–Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation. The first use of the Fokker– Planck equation was the statistical description of Brownian motion of a particle in a fluid. In one spatial dimension x, the Fokker–Planck equation for a process with drift D1(x,t) and diffusion D2(x,t) is More generally, the time-dependent probability distribution may depend on a set of N macrovariables xi. The general form of the Fokker–Planck equation is then where D1 is the drift vector and D2 the diffusion tensor; the latter results from the presence of the stochastic force. Fokker – Planck Equation Adriaan Fokker 1887 - 1972 Max Planck 1858 - 1947 SOLO Adriaan Fokker „Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld" Annalen der Physik 43, (1914) 810- 820 Max Plank, „Ueber einen Satz der statistichen Dynamik und eine Erweiterung in der Quantumtheorie“, Sitzungberichte der Preussischen Akadademie der Wissenschaften (1917) p. 324-341 Stochastic Processes ( ) ( ) ( )[ ] ( ) ( )[ ]txftxD x txftxD x txf t ,,,,, 22 2 1 ∂ ∂ + ∂ ∂ −= ∂ ∂ ( )[ ] ( )[ ]∑∑∑ = == ∂∂ ∂ + ∂ ∂ −= ∂ ∂ N i N j Nji ji N i Ni i ftxxD xx ftxxD x f t 1 1 1 2 2 1 1 1 ,,,,,, 
  • 82. Fokker – Planck Equation (continue – 1) The Fokker–Planck equation can be used for computing the probability densities of stochastic differential equations. where is the state and is a standard M-dimensional Wiener process. If the initial probability distribution is , then the probability distribution of the state is given by the Fokker – Planck Equation with the drift and diffusion terms: Similarly, a Fokker–Planck equation can be derived for Stratonovich stochastic differential equations. In this case, noise-induced drift terms appear if the noise strength is state-dependent. SOLO Consider the Itô stochastic differential equation: ( ) ( ) ( )[ ] ( ) ( )[ ]txftxD x txftxD x txf t ,,,,, 22 2 1 ∂ ∂ + ∂ ∂ −= ∂ ∂
  • 83. Fokker – Planck Equation (continue – 2) Derivation of the Fokker–Planck Equation SOLO Start with ( ) ( ) ( )11|1, 111 |, −−− −−− = kxkkxxkkxx xpxxpxxp kkkkk and ( ) ( ) ( ) ( )∫∫ +∞ ∞− −−− +∞ ∞− −− −−− == 111|11, 111 |, kkxkkxxkkkxxkx xdxpxxpxdxxpxp kkkkkk define ( ) ( )ttxxtxxttttt kkkk ∆−==∆−== −− 11 ,,, ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )∫ +∞ ∞− ∆−∆− ∆−∆−∆−= ttxdttxpttxtxptxp ttxttxtxtx || Let use the Characteristic Function of ( ) ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )ttxtxtxtxdttxtxpttxtxss ttxtxttxtx ∆−−=∆∆−∆−−−=Φ ∫ +∞ ∞− ∆−∆−∆ |exp: || ( ) ( ) ( ) ( )[ ]ttxtxp ttxtx ∆−∆− || The inverse transform is ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( )∫ ∞+ ∞− ∆−∆∆− Φ∆−−=∆− j j ttxtxttxtx sdsttxtxs j ttxtxp || exp 2 1 | π Using Chapman-Kolmogorov Equation we obtain: ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( )ttxdsdttxpsttxtxs j ttxdttxpsdsttxtxs j txp j j ttxttxtx ttx ttxtxp j j ttxtxtx ttxtx ∆−∆−Φ∆−−= ∆−∆−Φ∆−−= ∫ ∫ ∫ ∫ ∞+ ∞− ∞+ ∞− ∆−∆−∆ +∞ ∞− ∆− ∆− ∞+ ∞− ∆−∆ ∆− | | | exp 2 1 exp 2 1 | π π    Stochastic Processes
  • 84. Fokker – Planck Equation (continue – 3) Derivation of the Fokker–Planck Equation (continue – 1) SOLO The Characteristic Function can be expressed in terms of the moments about x (t-Δt) as: ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( )ttxdsdttxpsttxtxs j txp j j ttxttxtxtx ∆−∆−Φ∆−−= ∫ ∫ +∞ ∞− ∞+ ∞− ∆−∆−∆ |exp 2 1 π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ }∑ ∞ = ∆−∆∆−∆ ∆−∆−− − +=Φ 1 || | ! 1 i i ttxtx i ttxtx ttxttxtxE i s s Therefore ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )ttxdsdttxpttxttxtxE i s ttxtxs j txp j j ttx i i ttxtx i tx ∆−∆−       ∆−∆−− − +∆−−= ∫ ∫ ∑ +∞ ∞− ∞+ ∞− ∆− ∞ = ∆− 1 | | ! 1exp 2 1 π Use the fact that ( ) ( ) ( )[ ]{ } ( ) ( ) ( )[ ] ( )[ ] ,2,1,01exp 2 1 = ∂ ∆−−∂ −=∆−−−∫ ∞+ ∞− i tx ttxtx sdttxtxss j i i i j j i δ π ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( )∫∑ ∫ ∫ ∞+ ∞− ∞ = ∆− +∞ ∞− ∆− ∞+ ∞− ∆−∆−∆−∆−− ∂ ∆−−∂− + ∆−∆−∆−−= 1 | ! 1 exp 2 1 i ttx i i ii ttx j j tx ttxdttxpttxttxtxE tx ttxtx i ttxdttxpsdttxtxs j txp δ π where δ [u] is the Dirac delta function: [ ] { } ( ) [ ] ( ) ( ) ( ) ( ) ( )000..0exp 2 1 FFFtsuFFduuuFsdus j u j j ==∀== −+ +∞ ∞− ∞+ ∞− ∫∫ δ π δ Stochastic Processes
  • 85. Fokker – Planck Equation (continue – 4) Derivation of the Fokker–Planck Equation (continue – 2) SOLO [ ] ( ){ } ( ) [ ] ( ) ( ) ( ) ( ) ( )afafaftsufufduuaufsduas j ua au j j ==∀=−−=− −+= +∞ ∞− ∞+ ∞− ∫∫ ..exp 2 1 δ π δ [ ] ( ){ } ( ) ( ) { } ( ) ( ) { }∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =→=− − =− j j j j j j sdussFs j uf du d sdussF j ufsduass j ua ud d exp 2 1 exp 2 1 exp 2 1 πππ δ ( ) [ ] ( ) ( ){ } ( ) ( ){ } { } ( ) { } { } ( ) ( ) au j j j j j j j j ud ufd sdsFass j sdduusufass j sdduuasufs j dusduass j ufduua ud d uf = ∞+ ∞− ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− −= − =− − = − − =− − =− ∫∫ ∫ ∫ ∫∫ ∫∫ exp 2 1 expexp 2 1 exp 2 1 exp 2 1 ππ ππ δ [ ] ( ) ( ){ } ( ) ( ) { } ( ) ( ) { }∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− =→=− − =− j j i i ij j j j i i i i sdussFs j uf du d sdussF j ufsduass j ua ud d exp 2 1 exp 2 1 exp 2 1 πππ δ ( ) [ ] ( ) ( ) ( ){ } ( ) ( ) ( ){ } ( ) { } ( ) { } ( ) ( ) { } ( ) ( ) au i i i j j i ij j i i j j i ij j i i i i ud ufd sdassFs j sdduusufass j sdduuasufs j dusduass j ufduua ud d uf = −= − =− − = − − =− − =− ∫∫ ∫ ∫ ∫∫ ∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∞+ ∞− +∞ ∞− +∞ ∞− ∞+ ∞− +∞ ∞− 1exp 2 1 expexp 2 1 exp 2 1 exp 2 1 ππ ππ δ Useful results related to integrals involving Delta (Dirac) function Stochastic Processes
  • 86. Fokker – Planck Equation (continue – 5) Derivation of the Fokker–Planck Equation (continue – 3) SOLO ( ) ( )[ ]{ } ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ]txpttxdttxpttxtxttxdttxpsdttxtxs j ttxttxttx ttxtx j j ∆− +∞ ∞− ∆− +∞ ∞− ∆− ∆−− ∞+ ∞− =∆−∆−∆−−=∆−∆−∆−− ∫∫ ∫ δ π δ    exp 2 1 ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∑ ∫ ∫∑ ∞ = =∆ ∆−∆− ∞ = ∞+ ∞− ∆−∆− +∞ ∞− ∞ = ∆−∆− ∂ ∆−∆−−∂− = ∆−∆−∆−∆−− ∂ ∆−−∂− = ∆−∆−∆−∆−− ∂ ∆−−∂− 1 0 | 1 | 1 | | ! 1 | ! 1 | ! 1 i t i ttx i ttxtx ii i ttx i ttxtxi ii i ttx i ttxtxi ii tx txpttxttxtxE i ttxdttxpttxttxtxE tx ttxtx i ttxdttxpttxttxtxE tx ttxtx i δ δ ( ) [ ] ( ) ( ) ( ) [ ] [ ] ( ) auau i i i i i i i i i ud ufd duua uad d uf ud ufd duua ud d uf == =− − →−=− ∫∫ +∞ ∞− +∞ ∞− δδ 1We found ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = =∆ ∆−∆− ∆− ∂ ∆−∆−−∂− += 1 0 | | ! 1 i t i ttx i ttxtx ii ttxtx tx txpttxttxtxE i txptxp ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = ∆− →∆ ∆− →∆ ∂ ∆−∆−−∂ ∆ − = ∆ − 1 00 |1 lim ! 1 lim i i ttx ii t i ttxtx t tx txpttxttxtxE tit txptxp Therefore Rearranging, dividing by Δt, and tacking the limit Δt→0, we obtain: Stochastic Processes
  • 87. Fokker – Planck Equation (continue – 6) Derivation of the Fokker–Planck Equation (continue – 4) SOLO We found ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ){ } ( ) ( )[ ]( ) ( )[ ]∑ ∞ = ∆−∆− →∆ ∆− →∆ ∂ ∆−∆−−∂ ∆ − = ∆ − 1 | 00 |1 lim ! 1 lim i i ttx i ttxtx i t i ttxtx t tx txpttxttxtxE tit txptxp Define: ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } t ttxttxtxE txtxm i ttxtx t i ∆ ∆−∆−− =− ∆− →∆ − | lim: | 0 Therefore ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]∑ ∞ = − ∂ −∂− = ∂ ∂ 1 ! 1 i i tx iii tx tx txptxtxm it txp ( ) ( )ttxtx t ∆−= →∆ − 0 lim: and: This equation is called the Stochastic Equation or Kinetic Equation. It is a partial differential equation that we must solve, with the initial condition: ( ) ( )[ ] ( )[ ]000 0 txptxp tx === Stochastic Processes
  • 88. Fokker – Planck Equation (continue – 7) Derivation of the Fokker–Planck Equation (continue – 5) SOLO We want to find px(t) [x(t)] where x(t) is the solution of ( ) ( ) ( ) [ ]fg ttttntxf dt txd ,, 0∈+= ( ){ } 0: == tnEn gg  ( )tng ( ) ( )[ ] ( ) ( )[ ]{ } ( ) ( )τδττ −=−− ttQnntntnE gggg ˆˆ Wiener (Gauss) Process ( ) ( )[ ] ( ) ( )[ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tQnEtxnE t ttxttxtxE txtxm gg t === ∆ ∆−∆−− =− →∆ − 22 2 2 0 2 | | lim: ( ) ( )[ ] ( ) ( )[ ] ( ){ } ( ) ( ) ( ) ( ) ( )txfnEtxftx td txd E t ttxttxtxE txtxm g t ,,| | lim: 0 0 1 =+=             = ∆ ∆−∆−− =− →∆ −  ( ) ( )[ ] ( ) ( )[ ] ( ){ } 20 | lim: 0 >= ∆ ∆−∆−− =− →∆ − i t ttxttxtxE txtxm i t i Therefore we obtain: ( ) ( )[ ] ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ] ( )[ ]2 2 2 1, tx txp tQ tx txpttxf t txp txtxtx ∂ ∂ + ∂ ∂ −= ∂ ∂ Stochastic Processes Fokker–Planck Equation Return to Daum
  • 89. 89 Recursive Bayesian EstimationSOLO Given a nonlinear discrete stochastic Markovian system we want to use k discrete measurements Z1:k={z1,z2,…,zk} to estimate the hidden state xk. For this we want to compute the probability of xk given all the measurements Z1:k={z1,z2,…,zk} . If we know p ( xk| Z1:k ) then xk is estimated using: { } ( )∫== kkkkkkkk xdZxpxZxEx :1:1| ||:ˆ ( )( ){ } ( )( ) ( )∫ −−=−−= kkk T kkkkk T kkkkkk xdZxpxxxxZxxxxEP :1:1| |ˆˆ|ˆˆ or more general we can compute all moments of the probability distribution p ( xk| Z1:k ): ( ){ } ( ) ( )∫= kkkkkk xdZxpxgZxgE :1:1 || Bayesian Estimation Introduction Problem: Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements. kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh The knowledge of p ( xk| Z1:k ) allows also the computation of Maximum a Posteriori (MAP) estimate using: ( )kk x MAP kk Zxpx k :1| |maxargˆ =
  • 90. 90 Recursive Bayesian EstimationSOLO To find the expression for p ( xk| Z1:k ) we use the theorem of joint probability (Bayes Rule): ( ) ( ) ( )k kk RuleBayes kk Zp Zxp Zxp :1 :1 :1 , | = Since Z1:k ={ zk, Z1:k-1 }: ( ) ( ) ( )1:1 1:1 :1 , ,, | − − = kk kkk kk Zzp Zzxp Zxp The denominator of this expression is ( ) ( ) ( )1:11:11:1 ,,|,, −−− = kkkkk RuleBayes kkk ZxpZxzpZzxp ( ) ( ) ( )    1:11:11:1 |,| −−−= kkkkkk ZpZxpZxzp Since the knowledge of xk supersedes the need for Z1:k-1 = {z1, z2,…,zk-1} ( ) ( )kkkkk xzpZxzp |,| 1:1 ≡− ( ) ( ) ( ) ( ) ( ) ( )1:11:1 1:11:1 :1 | || | −− −− = kkk kkkkk kk ZpZzp ZpZxpxzp ZxpTherefore: ( ) ( ) ( )1:11:11:1 |, −−− = kkk RuleBayes kk ZpZzpZzp and the nominator is Bayesian Estimation Introduction
  • 91. 91 Recursive Bayesian EstimationSOLO The final result is: ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp Therefore: Since p ( xk| Z1:k ) is a probability distribution it must satisfy: ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫ ∫ − − − − === 1:1 1:1 1:1 1:1 :1 | || | || |1 kk kkkkk k kk kkkk kkk Zzp xdZxpxzp xd Zzp Zxpxzp xdZxp ( ) 1| :1 =∫ kkk xdZxp ( ) ( ) ( )∫ −− = kkkkkkk xdZxpxzpZzp 1:11:1 ||| ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | and: This is a recursive relation that needs the value of p (xk|Z1:k-1), assuming that p (zk|xk) is obtained from the Markovian system definition ( zk = h (xk,vk) ). Bayesian Estimation Introduction kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Hidden States Measurements
  • 92. 92 Recursive Bayesian EstimationSOLO The Correction Step is: ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp Bayesian Estimation Introduction evidence priorlikeliood posterior ⋅ = or: prior: given by prediction equation ( )kk xzp | likelihood: given by observation model ( )1:1| −kk Zxp evidence: the normalized constant on the denominator ( ) ( ) ( )∫ −− = kkkkkkk xdZxpxzpZzp 1:11:1 |||
  • 93. 93 Recursive Bayesian EstimationSOLO ( ) ( ) ( )1:111:111:11 |,||, −−−−−− = kkkkk Bayes kkk ZxpZxxpZxxp ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp Using: We obtain: Since for a Markov Process the knowledge of xk-1 supersedes the need for Z1:k-1 = {z1, z2,…,zk-1} ( ) ( )11:11 |,| −−− = kkkkk xxpZxxp Chapman – Kolmogorov Equation Sydney Chapman 1888 - 1970 Andrey Nikolaevich Kolmogorov 1903-1987 Bayesian Estimation Introduction kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Hidden States Measurements
  • 94. 94 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp Using p (xk-1|Z1:k-1) from time-step k-1 and p (xk|xk-1) of the Markov system, compute: Initialize with p (x0) ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | Using p (xk|Z1:k-1) from Prediction phase and p (zk|xk) of the Markov system, compute: { } ( )∫== kkkkkkkk xdZxpxZxEx :1:1| ||ˆ ( )( ){ } ( )( ) ( )∫ −−=−−= kkk T kkkkk T kkkkkk xdZxpxxxxZxxxxEP :1:1| |ˆˆ|ˆˆ At stage k k:=k+1 ( )1|11| ˆˆ −−− = kkkk xfx 0 Prediction phase (before zk measurement) 1 Correction Step (after zk measurement)2 Filtering3 kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Bayesian Estimation Introduction - Summary
  • 95. 95 Recursive Bayesian EstimationSOLO ( ) ( ) ( ) ( )∫∫ −−−−−−−− == 11:11111:111:1 |||,| kkkkkkkkkkk xdZxpxxpxdZxxpZxp ( ) ( ) ( ) ( ) ( )∫ − − = kkkkk kkkk kk xdZxpxzp Zxpxzp Zxp 1:1 1:1 :1 || || | Prediction phase (before zk measurement) 1 Correction Step (after zk measurement)2 kx1−kx kz1−kz 0x 1x 2x 1z 2z kZ :11:1 −kZ ( )11, −− kk wxf ( )kk vxh , ( )00 ,wxf ( )11,vxh ( )11,wxf ( )22 ,vxh Bayesian Estimation Introduction - Summary This is a Conceptual Solution because the Integrals are Often Not Tractable. An optimal solution is possible for some restricted cases: • Linear Systems with Gaussian Noises (system and measurements) • Grid-Based Filters Table of Content
  • 96. 96 SOLO Linear Gaussian Systems A Linear Combination of Independent Gaussian random vectors is also a Gaussian random vector mmm XaXaXaS +++= 2211: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    +++++++−=     +−    +−    +−= ΦΦ⋅Φ==Φ ∫ ∫ +∞ ∞− +∞ ∞− mmmm mmmm YYYm YpYp mYYmS aaajaaa ajaajaaja YdYdYYpSj m mmYY mm µµµωσσσω µωσωµωσωµωσω ωωωωω          2211 222 2 2 2 2 1 2 1 2 222 22 2 2 2 2 2 11 2 1 2 1 2 11,, 2 1 exp 2 1 exp 2 1 exp 2 1 exp ,,exp 21 11 1 ( ) ( )       − −= 2 2 2 exp 2 1 ,; i ii i iiiX X Xp i σ µ σπ σµ ( ) ( ) ( )     +−==Φ ∫ +∞ ∞− iiiiXiX jXdXpXj ii µωσωωω 22 2 1 expexp: Moment- Generating Function Gaussian distribution Define Proof: ( ) ( )iX ii i X i iYiii Xp aa Y p a YpXaY iii 11 : =      =→= ( ) ( ) ( ) ( ) ( ) ( )       +−=Φ===Φ ∫∫ +∞ ∞− +∞ ∞− iiiiiiX asign asign ii i iX iiiiYiY ajaXaXda a Xp XajYdYpYj i i ii µωσωωωω 222 2 1 expexpexp: 1 1 Review of Probability
  • 97. 97 SOLO Linear Gaussian Systems (continue – 1) A Linear Combination of Independent Gaussian random vectors is also a Gaussian random vector mmm XaXaXaS +++= 2211: Therefore the Linear Combination of Independent Gaussian Random Variables is a Gaussian Random Variable with mmS mmS aaa aaa m m µµµµ σσσσ +++= +++=   2211 222 2 2 2 2 1 2 1 2 Therefore the Sm probability distribution is: ( ) ( )         − −= 2 2 2 exp 2 1 ,; m m m mm S S S SSm x Sp σ µ σπ σµ Proof (continue – 1): ( ) ( ) ( )      +++++++−=Φ mmmmS aaajaaam µµµωσσσωω  2211 222 2 2 2 2 1 2 1 2 2 1 exp We found: Review of Probability q.e.d.
  • 98. 98 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 2) ( ) ( )kkkk kkkk vuxkhz wuxkfx ,,, ,,,1 111 = −= −−− kkkk kkkkkkk vxHz wuGxx += Γ++Φ= −−−−−− 111111 wk-1 and vk, white noises, zero mean, Gaussian, independent ( ) ( ) ( ){ } ( ) ( ){ } ( )kPkekeEkxEkxke x T xxx =−= &: ( ) ( ) ( ){ } ( ) ( ){ } ( ) lk T www kQlekeEkwEkwke , 0 &: δ=−=  ( ) ( ) ( ){ } ( ) ( ){ } ( ) lk T vvv kRlekeEkvEkvke , 0 &: δ=−=  ( ) ( ){ } { }0=lekeE T vw    = ≠ = lk lk lk 1 0 ,δ ( ) ( )Qwwpw ,0;N= ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − wQw Q wp T nw 1 2/12/ 2 1 exp 2 1 π ( ) ( )     −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π A Linear Gaussian Markov Systems is defined as ( ) ( )0|0000 ,;0 Pxxxp ttx == = N ( ) ( ) ( ) ( )    −−−= = − == 00 1 0|0002/1 0|0 2/0 2 1 exp 2 1 0 xxPxx P xp t T tntx π
  • 99. 99 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 3) 111111 −−−−−− Γ++Φ= kkkkkkk wuGxx Prediction phase (before zk measurement) { } { } { }   0 1:111111:1111:11| |||:ˆ −−−−−−−−−− Γ++Φ== kkkkkkkkkkkk ZwEuGZxEZxEx or 111|111| ˆˆ −−−−−− +Φ= kkkkkkk uGxx The expectation is { }[ ] { }[ ]{ } ( )[ ] ( )[ ]{ }1:1111|111111|111 1:11|1|1| |ˆˆ |ˆˆ: −−−−−−−−−−−−− −−−− Γ+−ΦΓ+−Φ= −−= k T kkkkkkkkkkkk k T kkkkkkkk ZwxxwxxE ZxExxExEP ( ) ( ){ } ( ){ } ( ){ } { } T k Q T kkk T k T kkkkk T k T kkkkk T k P T kkkkkkk wwExxwE wxxExxxxE kk 11111 0 1|1111 1 0 11|11111|111|111 ˆ ˆˆˆ 1|1 −−−−−−−−−− −−−−−−−−−−−−−− ΓΓ+Φ−Γ+ Γ−Φ+Φ−−Φ= −−         T kk T kkkkkk QPP 1111|111| −−−−−−− ΓΓ+ΦΦ= { } ( )1|1|1:1 ,ˆ;| −−− = kkkkkkk PxxZxP N Since is a Linear Combination of Independent Gaussian Random Variables: 111111 −−−−−− Γ++Φ= kkkkkkk wuGxx
  • 100. 100 SOLO For the particular vector measurement equation where the measurement noise, is Gaussian (normal), with zero mean: ( ) ( )kkkv Rvvp ,0;N= ( ) ( ) ( )xp zxp xzp x zx xz , | , | = and independent of , the conditional probability can be written, using Bayes rule as: kx ( )xzp xz || ( )           − − ==−= 1 111 1111 1 1 , nxpp nx pxnxpxnpxpx xHz xHz zxfxHzv xn xn  ( ) ( ) 2/1 ,, /,, T vxzx JJvxpzxp = The measurement noise can be related to and by the function:v zx pxp p pp p I z f z f z f z f z f J =                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =      ∂ ∂ =    1 1 1 1 ( ) ( ) ( ) ( )vpxpvxpzxp vxvxzx ⋅== ,, ,, kv Since the measurement noise is independent of :xv zThe joint probability of and is given by:x Recursive Bayesian Estimation Linear Gaussian Markov Systems (continue – 4) kkkk vxHz += Correction Step (after zk measurement) - 1st Way ( ) ( ) ( ) ( )1:1 1:1 :1 | || | − − = kk kkkk kk Zzp Zxpxzp Zxp
  • 101. 101 ( ) ( )kkkv Rvvp ,0;N= kkkk vxHz += Consider a Gaussian vector , where , measurement, , where the Gaussian noise is independent of and . v kx ( ) [ ]1|1| ,; −−= kkkkkkx Pxxxp  N kx ( ) ( ) ( ) ( )∫∫ +∞ ∞− +∞ ∞− == kkxkkxzkkkzxkz xdxpxzpxdzxpzp |, |, is Gaussian with( )kz zp ( ) ( ) ( ) ( ) 1| 0 −=+=+= kkkkkkkkk xHvExEHvxHEzE   ( ) ( )[ ] ( )[ ]{ } [ ][ ]{ } ( )[ ] ( )[ ]{ } [ ]{ } [ ]{ } [ ]{ } { } k T kkkk T kk T k T kkkk T kkkkk T k T kkkkkkk T kkkkkkkkkk T kkkkkkkkkkkk T kkkkk RHPHvvEHxxvEvxxEH HxxxxEHvxxHvxxHE xHvxHxHvxHEzEzzEzEz +=+−−−− −−=+−+−= −+−+=−−= −−− −−−− −− 1| 0 1| 0 1| 1|1|1|1| 1|1|cov           ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]       −−+−−−− +− = − xHzRHPHxHz RHPH zp TT Tpz ˆˆ 2 1 exp 2 1 1 2/12/ π ( ) ( ) ( ) ( )      −−−= − − −− − −− 1| 1 1|1|2/1 1| 2/1:1| 2 1 exp 2 1 |1:1 kkkkk T kkk kk nkkZx xxPxx P Zxp kk  π ( ) ( ) ( ) ( ) ( )    −−−=−= − kkk T kkkpkkkvkkxz xHzRxHz R xHzpxzp 1 2/12/| 2 1 exp 2 1 | π Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 5) Correction Step (after zk measurement) 1st Way (continue – 1)
  • 102. 102 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 6) kkkk vxHz += ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π Correction Step (after zk measurement) 1st Way (continue – 2) ( ) ( ) ( ) ( )    −−−= − − −− − −− 1| 1 1|1|2/1 1| 2/1:1| 2 1 exp 2 1 |1:1 kkkkk T kkk kk nkkZx xxPxx P Zxp kk  π ( ) ( ) ( ) ( ) ( )    −−−=−= − kkk T kkkpkkkvkkxz xHzRxHz R xHzpxzp 1 2/12/| 2 1 exp 2 1 | π ( ) ( ) [ ] [ ] [ ]       −+−− + = − − −− − 1| 1 1|1|2/1 1| 2/ ˆˆ 2 1 exp 2 1 kkkk T kkkk T kkk k T kkkk p kz xHzRHPHxHz RHPH zp π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ]      −+−+−−−−−−⋅ + == − − −−− − −− − − −− − 1| 1 1|1|1| 1 1|1| 1 2/1 1| 2/12/1 1|2/1:1 1:1 :1 ˆˆ 2 1 2 1 2 1 exp 2 1 | || | kkkkk T kkkk T kkkkkkkkk T kkkkkkk T kkk k T kkkk kkknkk kkkk kk xHzRHPHxHzxxPxxxHzRxHz RHPH RPZzp Zxpxzp Zxp  π from which
  • 103. 103 ( ) ( ) ( ) ( ) ( ) [ ] ( )1| 1 1|1|1| 1 1|1| 1 − − −−− − −− − −+−−−−+−− kkkk T kkkkk T kkkkkkkkk T kkkkkkk T kkk xHzHPHRxHzxxPxxxHzRxHz  ( )[ ] ( )[ ] ( ) ( ) ( ) [ ] ( ) ( ) [ ]{ }( ) ( ) ( ) ( ) ( ) ( ) [ ]( )1| 11 1|1|1| 1 1|1| 1 1| 1| 1 1| 1 1|1| 1 1|1| 1| 1 1|1|1|1| 1 1|1| − −− −−− − −− − − − − − − −− − −− − − −−−− − −− −+−+−−−−−− −+−−=−+−− −−+−−−−−−= kkkkk T kkk T kkkkkkkk T kkkkkkkkk T k T kkk kkkk T kkkkkk T kkkkkkkk T kkkkk T kkkk kkkkk T kkkkkkkkkkkk T kkkkkkkk xxHRHPxxxxHRxHzxHzRHxx xHzHPHRRxHzxHzHPHRxHz xxPxxxxHxHzRxxHxHz    [ ] [ ] 1111 1| 1111 1| 1 −−−− − −−−− − − ++/−/=+− k T kkk T kkkkkkk LemmaMatrixInverse T kkkkkk RHHRHPHRRRHPHRRwe have Define: [ ] [ ] 1 1| 1 1| 1 1| 1 1| 111 1|| : − − − − − − − − −−− − +−=+= kk T k T kkkkkkkkkk LemmaMatrixInverse kk T kkkkk PHHPHRHPPHRHPP ( )[ ] ( )[ ]1| 1 |1| 1 |1| 1 |1| − − − − − − − −+−−+−= kkkkk T kkkkkkkk T kkkkk T kkkkkk xHzRHPxxPxHzRHPxx  ( ) ( ) ( )[ ] ( )[ ]      −+−−+−−⋅= − − − − − − − 1| 1 |1| 1 |1| 1 |1|2/1 | 2/:1| 2 1 exp 2 1 | kkkkk T kkkkkkkk T kkkkk T kkkkkk kk nkkzx xHzRHPxxPxHzRHPxx P Zxp  π Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 7) Correction Step (after zk measurement) 1st Way (continue – 3) then ( ) ( ) ( ) ( ) ( ) [ ] ( )1| 1 1|1|1| 1 1|1| 1 − − −−− − −− − −+−−−−+−− kkkkk T kkkk T kkkkkkkkk T kkkkkkk T kkk xHzRHPHxHzxxPxxxHzRxHz  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )1| 1 |1|1| 1 || 1 1| 1| 1 | 1 |1|1| 1 | 1 || 1 1| − − −− −− − − −− −− −−− − −−+−−− −−−−−= kkkkk T kkkkkkkkkkkk T kkkk kkkkk T kkkkk T kkkkkkkk T kkkkkkkkk T kkkk xxPxxxxPPHRxHz xHzRHPPxxxHzRHPPPHRxHz  
  • 104. 104 then ( )kkzx x Zxp k :1| |max ( ) { }kk kkkkk T kkkkkkkk ZxE xHzRHPxxx :1 1| 1 |1| * | | ˆˆ:ˆ = −+== − − − Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 8) Correction Step (after zk measurement) 1st Way (continue – 4) ( ) ( ) ( )[ ] ( )[ ]      −+−−+−−⋅= − − − − − − − 1| 1 1| 1 |1| 1 1|2/1 | 2/:1| 2 1 exp 2 1 | kkkkk T kkkkkk T kkkkk T kkkk kk nkkzx xHzRHxxPxHzRHxx P Zxp  π where:[ ] ( )( ){ }k T kkkkkkkk T kkkkk ZxxxxEHRHPP :1|| 111 1|| ˆˆ: −−=+= −−− −
  • 105. 105 { } ( ) ( )  ki kkkkkkkkkkkkkkkkk zzKxxHzKxZxEx 1|1|1|1|:1| ˆ| −−−− −+=−+== Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 9) Summary 1st Way – Kalman Filter Initial Conditions: [ ] 111 1|| : −−− − += kk T kkkkk HRHPP Prediction phase (before zk measurement) 111|111| ˆˆ −−−−−− +Φ= kkkkkkk uGxx Correction Step (after zk measurement) T kk T kkkkkk QPP 1111|111| −−−−−−− ΓΓ+ΦΦ= 1 |: − = k T kkkk RHPK { }00|0 ˆ xEx = ( ) ( ){ }T xxxxEP 0|000|000|0 ˆˆ: −−= kkkk wxHz += { } { } { } 0 1:11|1:11:11| |ˆ||ˆ −−−−− +=+== kkkkkkkkkkkkk ZwExHZwxHEZzEz 1|1| ˆˆ −− = kkkkk xHz
  • 106. 106 Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 10) kkkk vxHz += ( ) ( )Rvvpv ,0;N= ( ) ( )       −= − vRv R vp T pv 1 2/12/ 2 1 exp 2 1 π ( ) ( ) [ ] [ ] [ ]       −+−− + = − − −− − 1| 1 1|1|2/1 1| 2/ ˆˆ 2 1 exp 2 1 kkkkk T kkkk T kkkk k T kkkk p kz xHzRHPHxHz RHPH zp π from which { } 1|1:11| ˆ|ˆ −−− == kkkkkkk xHZzEz ( ) ( ){ } kk T kkkkk T kkkkkk zz kk SRHPHZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } [ ] ( )[ ]{ } T kkkk T kkkkkkkk k T kkkkkk xz kk HPZvxxHxxE ZzzxxEP 1|1:11|1| 1:11|1|1| ˆˆ ˆˆ −−−− −−−− =+−−= −−= We also have Correction Step (after zk measurement) 2nd Way Define the innovation: 1|1| ˆˆ: −− −=−= kkkkkk xHzzzi
  • 107. 107 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables       = k k k z x yDefine: assumed that they are Gaussian distributed Prediction phase (before zk measurement) 2nd way (continue -1) { }         =             = − − − − − 1| 1| 1:1 1:1 1:1 ˆ ˆ | | | kk kk kk kk kk z x Zz Zx EZyE         =                 − −         − − = −− −− − − − − − − zz kk zx kk xz kk xx kk k T kkk kkk kkk kkkyy kk PP PP Z zz xx zz xx EP 1|1| 1|1| 1:1 1| 1| 1| 1| 1| ˆ ˆ ˆ ˆ where: [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } kk T kkkkk T kkkkkk zz kk SRHPHZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−= Linear Gaussian Markov Systems (continue – 11)
  • 108. 108 ( ) ( ) ( )    −−−= − − −− − − 1| 1 1|1|2/1 1| 1:1, ˆˆ 2 1 exp 2 1 |, kkk yy kk T kkk yy kk kkkzx yyPyy P Zzxp π Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables The conditional probability distribution function (pdf) of xk given zk is given by: Prediction phase (before zk measurement) 2nd Way (continue – 2) ( ) ( ) ( )      −−−= − − −− − − 1| 1 1|1|2/1 1| 1:1 ˆˆ 2 1 exp 2 1 | kkk zz kk T kkk zz kk kkz zzPzz P Zzp π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    −−−     −−− === − − −− − − −− − − − − − 1| 1 1|1| 1| 1 1|1| 2/1 1| 2/1 1| 1:1 1:1, |1:1| ˆˆ 2 1 exp ˆˆ 2 1 exp 2 2 | |, |,| kkk zz kk T kkk kkk yy kk T kkk yy kk zz kk kkz kkkzx kkzxkkkzx zzPzz yyPyy P P Zzp Zzxp zxpZzxp π π ( ) ( ) ( ) ( )    −−+−−−= − − −−− − −− − − 1| 1 1|1|1| 1 1|1|2/1 1| 2/1 1| ˆˆ 2 1 ˆˆ 2 1 exp 2 2 kkk zz kk T kkkkkk yy kk T kkk yy kk zz kk zzPzzyyPyy P P π π Linear Gaussian Markov Systems (continue – 12) We assumed that is Gaussian distributed:      = k k k z x y
  • 109. 109 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 3) ( ) ( ) ( ) ( ) ( )    −−+−−−= − − −−− − −− − − 1| 1 1|1|1| 1 1|1|2/1 1| 2/1 1| | ˆˆ 2 1 ˆˆ 2 1 exp 2 2 | kkk zz kk T kkkkkk zz kk T kkk yy kk zz kk kkzx zzPzzyyPyy P P zxp π π Define: 1|1| ˆ:&ˆ: −− −=−= kkkkkkkk zzxx ςξ ( ) ( ) ( ) ( ) k zz kk T kk zz kk T kk zx kk T kk xz kk T kk xx kk T k kkkzz T k k k zz kk zx kk xz kk xx kk T k k k zz kk T k k k zz kk zx kk xz kk xx kk T k k kkk zz kk T kkkkkk yy kk T kkk PTTTT P TT TT P PP PP zzPzzyyPyyq ςςςςξςςξξξ ςς ς ξ ς ξ ςς ς ξ ς ξ 1 1|1|1|1|1| 1 1| 1|1| 1|1| 1 1| 1 1|1| 1|1| 1| 1 1|1|1| 1 1|1| ˆˆˆˆ: − −−−−− − − −− −− − − − −− −− − − −−− − −− −+++= −                    = −                    = −−−−−= Linear Gaussian Markov Systems (continue – 13)
  • 110. 110 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd way (continue – 4) Using Inverse Matrix Lemma: ( ) ( ) ( ) ( )         −−− −−− =      −−−−− −−−−−− 11111 111111 nxmnxnmxnmxmmxnmxmnxmnxnmxnmxm mxmnxmmxnmxmnxmnxnmxnmxmnxmnxn mxmmxn nxmnxn BADCDCBADC CBDCBADCBA CD BA         =         −− −− − −− −− zz kk zx kk xz kk xx kk zz kk zx kk xz kk xx kk TT TT PP PP 1|1| 1|1| 1 1|1| 1|1| in 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| − −−− − − − − −−− − − − − −−− − − −= −= −= zz kk xz kk xz kk xx kk xz kk xx kk zx kk zz kk zz kk kkzxkkzzkkxzkkxxkkxx PPTT TTTTP PPPPT k zz kk T kk zz kk T kk zx kk T kk xz kk T kk xx kk T k PTTTTq ςςςςξςςξξξ 1 1|1|1|1|1| − −−−−− −+++= ( ) k zz kk T kk zz kk T k k xz kk xx kk zx kk T kk xz kk xx kk zx kk T kk xz kk T kk xx kk xx kk zx kk T k T k PT TTTTTTTTTT ςςςς ςςςςςξξςξ 1 1|1| 1| 1 1|1|1| 1 1|1|1|1| 1 1|1| − −− − − −−− − −−−− − −− −+ −+++= ( ) ( ) ( ) ( ) ( )k xz kk xx kkk xx kk T k xz kk xx kkkk zz kk xz kk xx kkkkzx zz kk T k k xz kk xx kk xx kk T k xz kk xx kkkk xx kk T k xz kk xx kkk TT TTTTTPTTTT TTTTTTTT zx kk Txz kk ςξςξςς ςςξξςξ 1| 1 1|1|1| 1 1| 0 1|1| 1 1|1|1| 1| 1 1|1|1| 1 1|1|1| 1 1| 1|1| − − −−− − −−− − −−− − − −−− − −−− − − = ++=−−+ +++= −−    Linear Gaussian Markov Systems (continue – 14)
  • 111. 111 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd way (continue – 5)         =         −− −− − −− −− zz kk zx kk xz kk xx kk zz kk zx kk xz kk xx kk TT TT PP PP 1|1| 1|1| 1 1|1| 1|1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| 1| 1 1|1|1| 1 1| − −−− − − − − −−− − − − − −−− − − −= −= −= zz kk xz kk xz kk xx kk xz kk xx kk zx kk zz kk zz kk kkzxkkzzkkxzkkxxkkxx PPTT TTTTP PPPPT ( ) ( )k xz kk xx kkk xx kk T k xz kk xx kkk TTTTTq ςξςξ 1| 1 1|1|1| 1 1| − − −−− − − ++= 1|1| ˆ:&ˆ: −− −=−= kkkkkkkk zzxx ςξ ( ) ( )[ ] ( )[ ]       −−−−−−−=       −= −−−−− − − − − 1|1|1|1|1|2/1 1| 2/1 1| 2/1 1| 2/1 1| | ˆˆˆˆ 2 1 exp 2 2 2 1 exp 2 2 | kkkkkkk xx kk T kkkkkkk yy kk zz kk yy kk zz kk kkzx zzKxxTzzKxx P P q P P zxp π π π π ( )1| 1 1|1|1| 1 1|1| ˆˆ − − −−− − −− −−−=+ kkk K zz kk xz kkkkkk xx kk xz kkk zzPPxxTT k   ςξ Linear Gaussian Markov Systems (continue – 15)
  • 112. 112 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 6) ( ) ( )[ ] ( )[ ]      −−−−−−−= − − −−−−− − −−− 1| 1 1|1|1|1|1| 1 1|1|1|| ˆˆˆˆ 2 1 exp| kkk xx kk xz kkkkk xx kk T kkk xx kk xz kkkkkkkzx zzPPxxTzzPPxxczxp From this we can see that { } ( )1| 1 1|1|1|| ˆˆˆ| − − −−− −+== kkk K zz kk xz kkkkkkkk zzPPxxzxE k    ( )( ){ } T k zz kkk xx kk zx kk zz kk xz kk xx kk xx kkk T kkkkkk xx kk KPKP PPPPTZxxxxEP 1|1| 1| 1 1|1|1| 1 1|:1||| ˆˆ −− − − −−− − − −= −==−−= [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } k T kkkkkk T kkkkkk zz kk SHPHRZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−= Linear Gaussian Markov Systems (continue – 16)
  • 113. 113 Recursive Bayesian EstimationSOLO Joint and Conditional Gaussian Random Variables Prediction phase (before zk measurement) 2nd Way (continue – 7) From this we can see that ( ) [ ] 111 1|1| 1 1|1|1|| −−− −− − −−− +=+−= kk T kkkkkk T kkkkk T kkkkkkk HRHPPHHPHRHPPP ( ) 1 1| 1 1|1| 1 1|1| − − − −− − −− =+== k T kkk T kkkkk T kkk zz kk xz kkk SHPHPHRHPPPK Linear Gaussian Markov Systems (continue – 17) kk T kkkkk KSKPP −= −1|| or [ ][ ]{ } 1|1:11|1|1| ˆˆ −−−−− =−−= kkk T kkkkkk xx kk PZxxxxEP [ ][ ]{ } k T kkkkkk T kkkkkk zz kk SHPHRZzzzzEP =+=−−= −−−−− :ˆˆ 1|1:11|1|1| [ ][ ]{ } T kkkk T kkkkkk xz kk HPZzzxxEP 1|1:11|1|1| ˆˆ −−−−− =−−=
  • 114. 114 We found that the optimal Kk is [ ] 1 1|1| − −− += T kkkkk T kkkk HPHRHPK [ ] [ ] 1111 |1 11 & 1 |1 1 1| 1 −−−− + −−− + +−=+ − − − k T kkk T kkkkkk LemmaMatrixInverse existPR T kkkkk RHHRHPHRRHPHR kkk [ ] 1111 1| 1 1| 1 1| −−−− − − − − − +−= k T kkk T kkkkk T kkkk T kkkk RHHRHPHRHPRHPK [ ]{ } [ ] 1111 |1 111 |1|1 −−−− + −−− ++ +−+= k T kkk T kkkkk T kkk T kkkkk RHHRHPHRHHRHPP [ ] 1 | 1111 |1 −−−−− + =+= RHPRHHRHPK T kkk T kkk T kkkk If Rk -1 and Pk|k-1 -1 exist: Recursive Bayesian EstimationSOLO Linear Gaussian Markov Systems (continue – 18) Relation Between 1st and 2nd ways 2nd Way 1st Way = 2nd Way

Editor's Notes

  1. John Minkoff, “Signals, Noise, and Active Sensors - Radar, Sonar, Laser Radar”
  2. A. Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.147-148
  3. Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.126-132
  4. Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.126-132
  5. John Minkoff, “Signals, Noise, and Active Sensors - Radar, Sonar, Laser Radar”
  6. A. Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.147-148
  7. A.Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.263-266 http://en.wikipedia.org/wiki/Law_of_large_numbers
  8. Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.263-266 http://en.wikipedia.org/wiki/Law_of_large_numbers
  9. A.Papoulis, “Probability, Random Variables and Stochastic Processes”,McGraw-Hill, 1965, pp.260-263 http://en.wikipedia.org/wiki/Law_of_large_numbers
  10. http://en.wikipedia.org/wiki/Central_limit_theorem
  11. A. Papoulis, “ Probability, Random Variables and StochasticProcesses”, McGraw-Hill, 1965, pp.99-100
  12. A. Papoulis, “ Probability, Random Variables and StochasticProcesses”, McGraw-Hill, 1965, pp.169
  13. http://en.wikipedia.org/wiki/Monte_Carlo_sampling http://www.lanl.gov/news/pdf/Metropolis_bio.pdf
  14. A. Gelb, Ed., “Applied Optimal Estimation”,MIT Press, 1974, pg.147, Prob. 4-10
  15. A. Gelb, Ed., “Applied Optimal Estimation”,MIT Press, 1974, pg.147, Problem 4-10
  16. A. Gelb, Ed., “Applied Optimal Estimation”,MIT Press, 1974, pg.147, Problem 4-10
  17. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  18. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  19. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  20. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  21. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  22. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  23. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  24. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
  25. Taylor, J., H., “Handbook of the Direct Statistical Analysis of Missile Guidance Systems via CADET”,“ The Analytic Sciences Corporation”, NTIS, AD-A013 397, 31 May 1975, Appendix C, “The Monte-Carlo Method: Application and Reliability”
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  29. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 36 - 37
  30. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 36 – 37 Coddington, P., “Monte Carlo Simulation for Statistical Physics”, CPS 713, Northest Parallel Architectures Center, January 1996 http://en.wikipedia.org/wiki/Histogram
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  32. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 44 - 45
  33. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 49 - 50
  34. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 50 - 51
  35. S.M. Ross, “ A Course in Simulation”, Macmillan &amp; Collier MacmillanPublishers, 1990, pp. 51 - 52
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  43. A. Papoulis, “ Probability, Random Variables and StochasticProcesses”, McGraw-Hill, 1965, pp.350
  44. A. Papoulis, “ Probability, Random Variables and StochasticProcesses”, McGraw-Hill, 1965, pp.303, 350
  45. Sage, A.P., &amp; Melsa, J.L., “Estimation Theory with Applications to Communications and Control”, McGraw Hill, 1971, pp. 272 - 283
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