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# "Modern Tracking" Short Course Taught at University of Hawaii

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A short course I taught in 2002 at the University of Hawaii's Kauai Community College. This course was offered to professionals working on radar tracking systems for air, missile, surface, and subsurface vehicle tracking. The material is a decade old, so it does not cover the latest technology. However, it is an excellent primer for those just starting on the subject.

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### "Modern Tracking" Short Course Taught at University of Hawaii

1. 1. August 6-8, 2002
2. 2. Topics Overview
3. 3. Overview Mathematics Overview – Linear Algebra and Linear Systems – Probability and Hypothesis Testing – State Estimation Filtering Fundamentals – Linear and Non-linear Filtering – Multiple Model Filtering Tracking Basics – Track Maintenance – Data Association Techniques – Activity Control
4. 4. Mathematics Review
5. 5. Mathematics Review Linear Algebra and Linear Systems – Definitions, Notations, Jacobians and Matrix Inversion Lemma – State-Space Representation (Continuous and Discrete) and Observability Probability Basics – Probability, Conditional Probability, Baye’s and Total Probability Theorem – Random Variables, Gaussian Mixture, and Covariance Matrices Bayesian Hypothesis Testing – Neyman-Pearson Lemma and Wald’s Theorem – Chi-Square Distribution Estimation Basics – Maximum Likelihood (ML) and Maximum A Posteriori (MAP) Estimators – Least Squares (LS) and Minimum Mean Square Error (MMSE) Estimators – Cramer-Rao Lower Bound, Fisher Information, Consistency and Efficiency
6. 6. Vector and Matrix Basics
7. 7. Definitions and Notations  a1  a  T  a = [ ai ] =  2  a = [ a1 a2  an ]    an   a11 a12  a1m   a11 a21  an1  a  a2 m  a  an 2  [ ]A = aij =  21   a22      [ ] AT = a ji =  12   a22          an1 an 2  anm  a1m a2 m  anm 
8. 8. Basic Matrix and Vector Properties Symmetric and Skew Symmetric Matrix A = AT A = − AT Matrix Product (NxS = [NxM] [MxS]): [ ] m C = cij = AB = ∑ aik bkj k =1 Transpose of Matrix Product [ ] m C = c ji = ( AB ) = B A = ∑ b jk aki T T T T k =1 Matrix Inverse AA−1 = I
9. 9. Basic Matrix and Vector Properties  Inner Product (Vectors must have equal length) n 〈 a, b〉 = a b = ∑ ai bi T i =1  Outer Product (NxM = [N] [M]) [ ] [ abT = C = cij = ai b j ]  Matrix Trace n Tr ( A) = ∑ aii = Tr ( AT ) i =1  Trace of Matrix Product ∂ ∂ ( AB )Tr ( AB) = Tr ( BA) ∂A (Tr ( ABAT ) ) = A( B + BT ) = BT ∂A
10. 10. Matrix Inversion Lemma In Estimation Theory, the following complicated inverse appears: (P −1 +H R H T −1 ) −1 The Matrix Inversion Lemma yields an alternative expression which does not depend on the inverses of the matrices in the above expression: ( P − PH T HPH T + R ) −1 HP An alternative form of the Matrix Inversion Lemma is: ( A + BCB )T −1 −1 = A − A B B A B+C−1 ( T −1 ) −1 −1 B T A−1
11. 11. The Gradient The Gradient operator with respect to an n-dimensional vector “x” is: T  ∂ ∂  ∇x =      ∂x1 ∂xn  Thus the gradient of a scalar function “f” is: T  ∂f ∂f  ∇x f =      ∂x1 ∂xn  The gradient of an m-dimensional vector-valued function is: T T  ∂ ∂    ∇x f =     [ f1 ( x )  f m ( x )] = NxM  ∂x1 ∂xn 
12. 12. The Jacobian Matrix The Jacobian Matrix is a matrix of derivatives describing a linear mapping from one set of coordinates to another. This is the transpose of the gradient of a vector-valued function (p. 24):  ∂x1 ∂x1   ∂x′   ∂xn  ′   ∂x ∂ ( x1 , x2 ,..., xm )  1  J ( x , x ′) =  = =     ∂x ′ ∂ ( x1 , x′ ,..., xn ) ′ 2 ′  ∂xm  ∂xm   ∂x1 ′ ∂xn  ′   This is typically used as part of a Vector Taylor Expansion for approximating a transformation.     ∂x   x = x ( xo ) +  ⋅ ( x ′ − xo ) +  ′ ′ ∂x ′ xo′ 
13. 13. The Jacobian Matrix: An Example  The conversion from Spherical to Cartesian coordinates yields:  x = [ r Sin(b) Cos (e) r Cos (b) Cos (e) r Sin(e)]  x ′ = [ r b e]  Sin(b) Cos (e) r Cos (b) Cos (e) − r Sin(b) Sin(e) J ( x , x ′) = Cos (b) Cos (e) − r Sin(b) Cos (e) − r Cos (b) Sin(e)       Sin(e) 0 r Cos (e)         ∂x ∂x ′ J ( x , x ′) J ( x ′, x ) =   = I ∂x ′ ∂x
14. 14. Linear Systems Basics
15. 15. Dirac Delta Function The Dirac Delta Function is defined by: δ (t − τ ) = 0 ∀t ≠ τ This function is defined by its behavior under integration: τ ∈ [ a, b ] b ∫ δ (t − τ )dt = 1 a In general, the Dirac Delta Function has the following “sifting” behavior: f (t ) δ (t − τ )dt = f (τ ) τ ∈ [ a, b] b ∫a The discrete version of this is called the Kronnecker Delta: 0 ∀i ≠ j δ ij =   1 i= j
16. 16. State-Space Representation (Continuous) A Dynamic Equation is typically expressed in the standard form (p. 27):     (t ) = A(t ) x (t ) + B(t )u (t ) x  x (t ) is the state vector of dimension “nx”  u (t ) is the control input vector of dimension “ny” A(t ) is the system matrix of dimension “nx x nx” B (t ) is the input gain matrix of dimension “nx x ny” While the Measurement Equation is expressed in the standard form:   z (t ) = C (t ) x (t )  z (t ) is the measurement vector of dimension “nz” C (t ) is the observation matrix of dimension “nz x nx”
17. 17. Example State-Space System A typical (simple) example is the constant velocity system: ξ(t ) = 0  This system is not yet in state-space form: ξ  0 1 ξ  0 0  u1     =   ξ  + 0 0 u  ξ  0 0       2      (t ) = A(t ) x (t ) + B(t )u (t ) x And suppose that we only have position measurements available: ξ  ξ meas = [1 0]   ξ    z (t ) = C (t ) x (t )
18. 18. State-Space Representation (Discrete) A continuous state-space system can also be written in discrete form (p. 29):    xk = Fk −1 xk −1 + Gk −1 u k −1  xk is the state vector of dimension “nx” at time “k”  uk is the control input vector of dimension “ny” at time “k” Fk is the transition matrix of dimension “nx x nx” at time “k” Gk is the input gain matrix of dimension “nx x ny” at time “k” While the Measurement Equation is expressed in the discrete form:   z k = H k xk  zk is the measurement vector of dimension “nz” at time “k” Hk is the observation matrix of dimension “nz x nx” at time “k”
19. 19. Example Revisited in Discrete Time The constant velocity discrete time model is given by: ξ k  1 t k − t k −1  ξ k −1  0 0  u1k −1  ξ  = 0   ξ  + 0 0 u  1   k −1    k    2 k −1     xk = Fk −1 xk −1 + Gk −1 u k −1 Since there is no time-dependence in the measurement equation, it is a trivial extension to the continuous example: ξ k  ξ meas = [1 0]    ξ k  k   z k = H k xk
20. 20. State Transition Matrix  We wish to be able to convert a continuous linear system to a discrete time linear system. Most physical problems are easily expressible in the continuous form while most measurements are discrete. Consider the following time-invariant homogeneous linear system (pp. 180-182):    (t ) = A(t ) x (t ) where x A(t ) = A for t ∈ [ t k −1 , t k ]  We have the solution:  { } x (t ) = Fk −1 ( t , t k −1 ) x ( t k −1 ) = L−1 ( sI − A) = e A( t −tk −1 ) x ( t k −1 ) −1 for t ∈ [ t k −1 , t k ]  If we add a term, making an inhomogeneous linear system, we obtain:     x (t ) = A(t ) x (t ) + B(t )u (t ) where B(t ) = B for t ∈ [ t k −1 , t k ]
21. 21. Matrix Superposition Integral Then, the state transition matrix is applied to the additive term and integration is performed to obtain the generalized solution:  x (t ) = Fk −1 ( t , t k −1 ) x ( t k −1 ) + ∫ Fk −1 ( t ,τ ) B(τ )u (τ ) dτ for t ∈ [ t k −1 , t k ] t t k −1 Consider the following example: u(t)=(t) 2σ 2 β x2 1 x1 s+β s  x1  0 1   x1   0    x  = 0 − β   x  +  2σ 2 β u (t )  2    2   
22. 22. Observability Criteria A system is categorized as observable if the state can be determined from a finite number of observations, assuming that the state-space model is correct. For a time-invariant linear system, the observability matrix is given by:  H   HF  Ω=      n x −1  H F  Thus, the system is observable if this matrix has a rank equal to “nx” (pp. 25,28,30).
23. 23. Observability Criteria: An Example For the nearly constant velocity model described above, we have:  [1 0]  1 0   Ω= 1 ∆t   =  [1 0]     1 ∆t    0 1   The rank of this matrix is “2” only if the delta time interval is non-zero. Thus, we can only estimate position and velocity both (using only position measurements) if these position measurements are separated in time. The actual calculation of rank is a subject for a linear algebra course and leads to ideas such as linear independence and singularity (p. 25)
24. 24. Probability Basics
25. 25. Axioms of Probability Suppose that “A” and “B” denote random events, then the following axioms hold true for probabilities: – Probabilities are non-negative: P{ A} ≥ 0 ∀A – The probability of a certain event is unity: P{ S } = 1 – Additive for mutually exclusive events: If P{ A ∩ B} = 0 then P{ A ∪ B} = P{ A} + P{ B} Mutually Exclusive
26. 26. Conditional Probability The conditional probability of an event “A” given the event “B” is: P{ A ∩ B} P{ A | B} = P{ B} For example, we might ask the following tracking related questions: – Probability of observing the current measurement given the previous estimate of the track state – Probability of observing a target detection within a certain surveillance region given that a true target is present Formulating these conditional probabilities is the foundation of track initiation, deletion, data association, SNR detection schemes…
27. 27. Total Probability Theorem Assume that we have a set of events “Bi” which are mutually exclusive: P{ Bi ∩ B j } = 0 ∀ i ≠ j And exhaustive: n ∑ P{ B } = 1 i =1 i Then the Total Probability Theorem states: n n P{ A} = ∑ P{ A ∩ Bi } = ∑ P{ A | Bi } P{ Bi } i =1 i =1
28. 28. Baye’s Theorem  We can work the conditional probability definition in order to obtain the reverse conditional probability: P{ Bi ∩ A} P{ A | Bi } P{ Bi } P{ Bi | A} = = P{ A} P{ A}  This conditional probability “Bi” is called the Posterior Probability while the unconditional probability of “Bi” is called the Prior Probability.  In the case of “Bi” being mutually exclusive and exhaustive, we have (p. 47): P{ A | Bi } P{ Bi }Posterior Probability P{ Bi | A} = Prior Probability ∑ P{ A | B } P{ B } n j j j =1 Likelihood Function
29. 29. Gaussian (Normal) Random Variables The Gaussian Random Variable is the most well-known, well- investigated type because of its wide application in the real world and its tractable mathematics. A Gaussian Random Variable is one which has the following probability density function (PDF) : ( x−µ ) 2 1 − p ( x) = N ( x; µ , σ ) = 2 e 2σ 2 2πσ 2 and is denoted: x ~ N (µ ,σ 2 )
30. 30. Gaussian (Normal) Random Variables The Expectation and Second Central Moment of this distribution are: ( x−µ ) 2 ∞ x − E[ x ] = ∫ e 2σ 2 dx = µ Mean −∞ 2π σ  ∞ x2 ( x−µ )  2 − E[( x − E[ x]) ] = E[ x ] − E[ x] =  ∫ 2 2 2 e 2σ dx  − µ 2 = σ 2 2  −∞ 2π σ    Mean Square Variance These are only with respect to scalar random variables…what about vector random variables?
31. 31. Vector Gaussian Random Variables The vector generalization is straight forward:     ( x − µ ) T P −1 ( x − µ )    1 − p( x ) = N ( x; µ , P) = e 2 2π P The Expectation and Second Central Moment of this distribution are:   E[ x ] = µ     E[( x − E[ x ])( x − E[ x ])T ] = P Notice that the Variance is now replaced with a matrix called a Covariance Matrix. If the vector “x” is a zero-mean error vector than the covariance matrix is called the Mean Square Error.
32. 32. Baye’s Theorem: Gaussian Case The “noise” of a device, denoted “x”, is observed. Normal functionality is denoted by event “B1” while a defective device is denoted by event “B2”: B1 = N ( x;0, σ 12 ) B2 = N ( x;0, σ 2 ) 2 The conditional probability of defect is (using Baye’s Theorem): P{ x | B2 } P{ B2 } 1P{ B2 | x} = = P{ x | B1} P{ B1} + P{ x | B2 } P{ B2 } 1 + P{ x | B1} P{ B1 } P{ x | B2 } P{ B2 } Using the two distributions, we have: 1 P{ B2 | x} = x2 x2 σ 2 P{ B1} − 2+ 2 2σ 1 2σ 2 1+ e σ 1 P{ B2 }
33. 33. Baye’s Theorem: Gaussian Case If we assume the diffuse prior, that the probability of each event is equal, then we have a simplified formula: 1 P{ B2 | x} = x2 x2 σ2 − 2+ 2 2σ 1 2σ 2 1+ e σ1 If we further assume that  2 = 4  1 and that x =  2, then we have: P{ B2 | x} ≈ 0.998 Note that the likelihood ratio largely dominates the result of this calculation. This quantity is crucial in inference and statistical decision theory and often called “evidence from the data”. P{ x | B1} Λ( B1 , B2 ) = P{ x | B2 }
34. 34. Gaussian Mixture Suppose we have “n” possible events “Aj” which are mutually exclusive and exhaustive. And further suppose that each event has a Gaussian PDF as follows (pp. 55-56): A j ={ x ~ N ( x j , Pj )} and P{ A j } = p j ∆ Then, the total PDF is given by the Total Probability Theorem: n p ( x) = ∑ p ( x | A j ) P{ Ai } j =1 This mixture can be approximated as another Gaussian once the mixed moments are computed.
35. 35. Gaussian Mixture The first moment (mean) is easily derived as: n  n [x = E [ p ( x)] = E ∑ p ( x | A j ) P{ Ai }  = ∑ E p ( x | A j ) p j]  j =1  j =1 n x = ∑ pj xj j =1 The covariance matrix is more complicated, but we simply apply the definition: [ ] [ ] n P = E ( x − x )( x − x ) = ∑ E ( x − x )( x − x ) | A j p j T T j =1 [ ] = ∑ E ( x − x + x j − x j )( x − x + x j − x j ) | A j p j n T j =1
36. 36. Gaussian Mixture Continuing the insanity: [ ]P = ∑ E ( x − x j )( x − x j ) | A j p j + ∑ ( x j − x )( x j − x ) p j n n T T j =1 j =1 = ∑ Pj p j + ∑ ( x j − x )( x j − x ) p j n n T j =1 j =1 Spread of the Means The spread of the means term inflates the covariance of the final mixed random variable to account for the differences between each individual mean and the mixed mean.
37. 37. Bayesian Hypothesis Testing
38. 38. Bayesian Hypothesis Testing  We consider two competing hypotheses about a parameter “” defined as: Null Hypothesis H0 : θ = θ0 Alternate Hypothesis H 1 : θ = θ1  We also define standard definitions concerning the decision errors: ∆Type I Error (False Alarm) PeI = P{ accept H1 | H 0 true} = α ∆ Type II Error (Miss) PeII = P{ accept H 0 | H1 true} = β
39. 39. Neyman-Pearson Lemma  The power of the hypothesis test is defined as: ∆Test Power (Detection) π = P{ accept H1 | H1 true} = 1 − β  The Neyman-Pearson Lemma states that the optimal decision (most powerful test) rule subject to a fixed Type I Error () is the Likelihood Ratio Test (pp.72-73): P{ z | H1}  H1 ; > Λ 0 Λ( H1 , H 0 ) ⇒ = P{ z | H 0 }  H 0 ; < Λ 0 Likelihood Functions P{ Λ ( H1 , H 0 ) > Λ 0 | H 0 } = PeI = α
40. 40. Sequential Probability Ratio Test  Suppose, we have a sequence of independent identically distributed (i.i.d.) measurements “Z={zi}” and we wish to perform a hypothesis test. We can formulate this in a recursive form as follows: P{ H1 ∩ Z } P{ Z | H1} P0 { H1} PR( H1 , H 0 ) = = P{ H 0 ∩ Z } P{ Z | H 0 } P0 { H 0 } Likelihood Functions a priori Probabilities P0 { H1} n P{ zi | H1} nPRn ( H1 , H 0 ) = ∏ P{ z | H } = PR0 ( H1 , H 0 ) ∏ Λ i ( H1 , H 0 ) P0 { H 0 } i =1 i 0 i =1 n ln ( PRn ( H1 , H 0 ) ) = ln ( PR0 ( H1 , H 0 ) ) + ∑ ln ( Λ i ( H1 , H 0 ) ) i =1
41. 41. Sequential Probability Ratio Test So, the recursive for of the SPRT is: ln( PRk ( H1 , H 0 ) ) = ln( PRk −1 ( H1 , H 0 ) ) + ln( Λ k ( H1 , H 0 ) ) Using Wald’s Theorem, we continue to test this quantity against two thresholds until a decision is made:  H1 ; > T2  ln ( PRk ( H1 , H 0 ) ) ⇒ continue ; > T1 and < T2  H 0 ; < T1  1 − β   β  T2 = ln   and T1 = ln 1 − α   α    Wald’s Theorem applies when the observations are an i.i.d. sequence.
42. 42. Chi-Square Distribution The chi-square distribution with “n” degrees of freedom has the following functional form: n−2 x 1 − χ n ( x) = 2 n x 2 e 2 n 2 Γ  2  2 It is related to an “n” dimensional vector Gaussian distribution as follows: ( x − x ) T P −1 ( x − x ) ~ χ n2 More generally, the sum of squares of “n” independent zero-mean, unity variance random variables is distributed as a chi-square with “n” degrees of freedom (pp.58-60).
43. 43. Chi-Square Distribution The chi-square distribution with “n” degrees of freedom has the following statistical moments: E[ x] = n E[( x − E[ x]) 2 ] = 2n The sum of two independent random variables which are chi-square are also chi-square: q1 ~ χ 2 n1 q2 ~ χ 2 n2 q1 + q2 ~ χ 2 n1 + n2
44. 44. Estimation Basics
45. 45. Parameter Estimator A parameter estimator is a function of the observations (measurements) that yields an estimate of a time-invariant quantity (parameter). This estimator is typically denoted as: [ ] where Z ={ z j } j =1 ∆ ∆ k xk = x k , Z ˆ ˆ k k Estimate Estimator Observations We also denote the error in the estimate as: ∆ ~ =x−x xk ˆk True Estimate
46. 46. Estimation Paradigms Non-Bayesian (Non-Random): – There is no prior PDF incorporated – The Likelihood Function PDF is formed – This Likelihood Function PDF is used to estimate the parameter ∆ Λ Z ( x) = p( Z | x ) Bayesian (Random): – Start with a prior PDF of the parameter – Use Baye’s Theorem to find the posterior PDF – This posterior PDF is used to estimate the parameter p( Z | x ) p( x ) 1 p( x | Z ) = = p( Z | x ) p( x ) p( Z ) c Posterior Likelihood Prior
47. 47. Estimation Methods Maximum Likelihood Estimator (Non-Random): x ML ( Z ) = arg max[ p( Z | x ) ] ˆ x dp( Z | x ) =0 dx x ML ˆ Maximum A Posteriori Estimator (Random): x MAP ( Z ) = arg max[ p( Z | x ) p( x ) ] ˆ x
48. 48. Unbiased Estimators Non-Bayesian (Non-Random): E[ xk ( Z k )] p ( Z k | x = x ) = x0 ˆ 0 Bayesian (Random): [ ( )] ( E xk Z k ˆ p x∩Z k ) = E[ x] p ( x ) General Case: [ ( )] E ~k Z k = 0 x
49. 49. Estimation Comparison Example Consider a single measurement of an unknown parameter “x” which is susceptible to additive noise “w” that is zero-mean Gaussian: z = x + w w ~ N 0, σ 2 ( ) The ML approach yields: ( z−x ) 2 1 − Λ ( x ) = p ( z | x ) = N ( z; x, σ ) = 2 e 2σ 2 2πσ 2 x ML ˆ = arg max[ Λ ( x)] = z x Thus, the MLE is the measurement itself because there is no prior knowledge.
50. 50. Estimation Comparison Example The MAP, with a Gaussian prior, approach yields: p( x) = N ( x; x , σ 0 ) 2 ( z − x)2 ( x− x )2 ( x −ξ ( z )) 2 − − − 2 2σ 2 2σ 02 p( z | x) p( x) e e 2σ 1 p( x | z ) = = = p( z ) 2πσσ 0 p( z ) 2πσ 12  x z  1 1 1 ξ ( z ) = σ 12  2 + 2  and σ  = 2+ 2  0 σ  σ 12 σ σ0 Prior Information x MAP = arg max[ p( x | z )] = ξ ( z ) ˆ x Measurement Information Thus, the MAPE is a linear combination of the prior information and the observation and it is weighted based upon the variance of each. NOTE: The MLE and MAPE are equivalent for a diffuse prior !
51. 51. Batch Estimation Paradigms Consider that we now have a set of observations available for estimating a parameter and that in general these observations are corrupted by measurement noise: Z k = { z j = h j ( x) + w j } j =1,,k Least Squares (Non-Random) k 2 ˆ x LS k [ = arg min ∑ z j − h j ( x)  ] x  j =1  Minimum Mean Square Error (Random): xkMMSE = arg min E ( x − x ) | Z k ˆ ˆ x ˆ 2 [ ] [ ] ( ) ∆ ∞ ˆ x MMSE k = E x | Z = ∫ x p x | Z k dx k −∞
52. 52. Unbiasedness of ML and MAP Estimators Maximum Likelihood Estimate: E[ xkML ] = E[ z ] = E[ x0 + w] = x0 + E[ w] = x0 ˆ Maximum A Posterior Estimate:  σ2  [ ] σ0 σ2 σ 02 2 Exˆ k= E 2 MAP x+ 2 z = 2 2  x+ 2 E[ z ] σ + σ 0 σ +σ0  σ +σ0 σ +σ0 2 2 2 σ2 σ 02 σ2 σ02 = 2 x+ 2 E[ x + w] = 2 x+ 2 ( x + E[ w]) σ +σ0 2 σ +σ0 2 σ +σ02 σ +σ0 2 σ2 σ0 2 = 2 x+ 2 x = x = E[ x] σ +σ02 σ +σ02
53. 53. Estimation Errors Non-Bayesian (Non-Random): { } 2 { } Var[ xk ( Z k )] = E[ xk ( Z k ) − E[ xk ( Z k )] ] = E[ xk ( Z k ) − x0 ] ˆ ˆ ˆ ˆ 2 Bayesian (Random): ˆ ˆ { } 2 [ˆ { } 2 MSE[ xk ( Z k )] = E[ xk ( Z k ) − x ] = E E[ xk ( Z k ) − x | Z k ] ] General Case: [x ( ) E ~k Z k 2 ] = ( ( )) var xk Z k ˆ x unbiased and x non − random ˆ ( ( )) ˆ k  MSE xk Z all cases
54. 54. Variances of ML and MAP Estimators Maximum Likelihood Estimate: ˆ ( ˆ ) var[ xkML ] = E[ xkML − x0 ] = E[( z − x0 ) ] = σ 2 2 2 Maximum A Posterior Estimate: [ ] [( var xkMAP = E xkMAP − x ˆ ˆ ) 2 ] σ 2σ 02 = 2 σ +σ0 2 [ ] < σ 2 = var xkML ˆ The MAPE error is less than the MLE error since the MAPE incorporates prior information.
55. 55. Cramer-Rao Lower Bound The Cramer-Rao Lower Bound states that a limit on the ability to estimate a parameter. [ ( )] [( ( ) )( ( ) ) MSE xk Z k = E xk Z k − x xk Z k − x ˆ ˆ ˆ T ]≥ J k −1 Not surprisingly, this lower limit is related to the likelihood function which we recall as the “evidence from the data”. This limit is called the Fisher Information Matrix. ∆ [   T ( ( J k = − E ∇ x ∇ x ln p Z k | x ) )]  x When equality holds, the estimator is called efficient. An example of this is the MLE estimate we have been working with.
56. 56. Filtering Fundamentals
57. 57. Filtering Fundamentals Linear Filtering – Linear Gaussian Assumptions, Kalman Filter, Kalman Properties – Direct Discrete-Time, Discretized Continuous-Time, Steady State Gains Non-Linear Filtering – Non-Linear Dynamics & Measurements, Extended Kalman Filter – Iterated Extended Kalman Filter Multiple-Model Filtering – Need for Multiple Models, Adaptive Filtering – Switching Multiple Model & Interacting Multiple Model Filter – Variable Structure IMM
58. 58. Linear Filtering
59. 59. Kalman-Bucy Problem A stochastic discrete-time linear dynamic system:     xk = Fk −1 xk −1 + Gk −1 uk −1 + Γk −1ν k −1  xk is the state vector of dimension “nx” at time “k”  Gk uk is the control input of dimension “nx” at time “k” Fk is the transition matrix of dimension “nx x nx” at time “k”  Γkν k is the plant noise of dimension “nx” at time “k” The measurement equation is expressed in the discrete form:    z k = H k xk + wk  zk is the measurement vector of dimension “nz” at time “k” Hk is the observation matrix of dimension “nz x nx” at time “k”  wk is the measurement noise of dimension “nz” at time “k”
60. 60. Kalman-Bucy Problem The Linear Gaussian Assumptions are:  E [ν k ] = 0 [  T ] E ν kν j = Qk δ jk  E [ wk ] = 0 [   ] E wk wT = Rk δ jk j The measurement and plant noises are uncorrelated:   E[ wkν k ] = 0 The conditional mean is: [ ] ∆ x j |k = E x j | Z k ˆ Z k = { zi , i ≤ k } ˆ xk | k Filtered State Estimate ˆ xk |k −1 Extrapolated State Estimate The estimation error is denoted by: ∆ ~ =x −x x j |k ˆ j |k j
61. 61. Kalman-Bucy Problem  The estimate covariance is defined as: [ ] ∆ Pj|k = E ~ j|k ~ jT|k | Z k x x Pk |k Filtered Error Covariance Pk |k −1 Extrapolated Error Covariance  The predicted measurement is given by: [ ] [ ] [ ] [ ] ∆    z k |k −1 = E z k | Z k −1 = E H k xk + wk | Z k −1 = H k E xk | Z k −1 + E wk | Z k −1 = H k xk |k −1ˆ ˆ  The measurement residual or innovation is denoted by: ∆ ηk = zk − zk |k −1 = z k − H k xk |k −1 ˆ ˆ
62. 62. Kalman-Bucy Approach Recall that the MMSE is equivalent to the MAPE in the Gaussian case. Recall that the MAPE, with a Gaussian prior, is a linear combination of the measurement and the prior information. Recall that the prior information was, more specifically, the expectation of the random variable prior to receiving the measurement. If we consider the Kalman Filter to be a recursive process which applies a static Bayesian estimation (MMSE) algorithm at each step, we are compelled to consider the following linear combination.  ′ ˆ ˆ k |k = K k xk |k −1 + K k z k x Prior State Observation Information Information
63. 63. Kalman Filter - Unbiasedness We start with the proposed linear combination:  ′ ˆ xk |k = K k xk |k −1 + K k z k ˆ We wish to ensure that the estimate is unbiased, that is: E [ ~k |k ] = 0 x Given the proposed linear combination, we determine the error to be:  ~ = [K′ + K H − I ] x + K′ ~ + K w  xk | k k k k k k xk |k −1 k k Applying the unbiasedness constraint, we have:   E[ ~k |k ] = 0 = [ K k + K k H k − I ] E[ xk ] + K k E[ ~k |k −1 ] + K k E[ wk ] x ′ ′ x ′ Kk = I − Kk H k
64. 64. Kalman Filter – Kalman Gain So, we have the following simplified linear combination:  xk |k = xk |k −1 + K k η k ˆ ˆ We also desire the filtered error covariance, so that it can be minimized: [ Pk |k = E ~k |k ~k |k x x T ] Pk |k = ( I − K k H k ) Pk |k −1 ( I − K k H k ) + K k Rk K k T T If we minimize the trace of this expression with respect to the gain: K k = Pk |k −1 H k T [H Pk k |k −1 T H k + Rk ] −1
65. 65. Kalman Filter - Recipe Extrapolation:  xk |k −1 = Fk −1 ⋅ xk −1|k −1 + Gk −1uk −1 ˆ ˆ T Pk |k −1 = Fk −1 ⋅ Pk −1|k −1 ⋅ Fk −1 + Γk −1Qk −1ΓkT−1 Update:  xk |k = xk |k −1 + K kη k ˆ ˆ Pk |k = ( I − K k H k ) Pk |k −1 ( I − K k H k ) + K k Rk K k T T
66. 66. Kalman Filter – Innovations The innovations are zero-mean, uncorrelated (p. 213) and have covariance: [ ] S k = E η kη k = H k Pk |k −1 H k + Rk T T The normalized innovation squared or statistical distance is chi-square distributed: −1 d k2 = η k S k η k ~ χ nz T 2 So, we expect that the innovations should have a mean and variance of: { } E[ d i2 ] = nz { } var[ d i2 ] = 2nz The Kalman Gain can now be written as: K k = Pk |k −1 H k T [H P k T k |k −1 H k + Rk ] −1 T = Pk |k −1 H k S k −1 The state errors are correlated: [ E~ ~ x x T k |k k −1|k −1 ] = [ I − K H ]F k k P k −1 k −1|k −1
67. 67. Kalman Filter – Likelihood Function  We wish to compute the likelihood function given the dynamics model used: [ ] [ ]p z k | Z k −1 = p z k | xk |k −1 = N ( z k ; z k |k −1 , S k ) = N ( z k − z k |k −1 ;0, S k ) = N (η k ;0, S k ) ˆ ˆ ˆ  Which has the explicit form: [  1  T −1   exp− η k S k η k  ] [ Λ k = p z k | Z k −1 ] =  2  det[ 2π S k ]  Alternatively, we can write:  1  exp − d k2  [ Λ k = p z k | Z k −1 ] =  2  ⇒ ln Λ = − 1 d 2 − 1 ln( det[ 2π S ] ) [ det 2π S kr ] k 2 k 2 k
68. 68. Kalman Filter – Measurement Validation Suppose our Kalman filter has the following output at a given time step: 5 0  1 0  10 Pk +1|k = H k +1 =  ˆ k +1|k =   x  0 16   0 1  15 Suppose that we now receive 3 measurements of unknown origin:  4 0 1  7  2 16  3 19  R i = z k +1 =  , z k +1 =  , z k +1 =   0 9 k +1   20 5 25 Evaluate the consistency of these measurements for this Kalman filter model. This procedure is called gating and is the basis for data association. ( ) 1  ( )  ( ) d k2+1 z k +1 = 2 d k2+1 z k2+1 = 8 d k2+1 zk3+1 = 13 χ 2 ( 95% ) = 6 2 χ 2 ( 99% ) = 9.2 2
69. 69. Kalman Filter – Initialization The true initial state is a random variable distributed as: x0 = N ( x0|0 , P0|0 )  ˆ It is just as important that the initial covariance and estimate realistically reflect the actual accuracy. Thus, the initial estimate should satisfy: ~ T P −1 ~ ≤ χ 2 ( 95% ) x0|0 0|0 x0|0 nx If the initial covariance is too small, then the Kalman gain will initially be small and the filter will take a longer time to converge. Ideally, the initial state estimate should be within one standard deviation (indicated by the initial covariance) of the true value. This will lead to optimal convergence time.
70. 70. Kalman Filter – Initialization In general, a batch weighted least-squares curve fit can be used (Chapter 3): [ −1 x0|0 = H init Rinit H init ˆ T ] −1 T −1 H init Rinit zinit [ T −1 P0|0 = H init Rinit H init ] −1   [  T zinit = z0 , , z nx −1 ] [ ( H init = H 0 ,  , H nx −1 Fnx − 2 ) ] n x −1 T  R0  0    Rinit =      0  Rn −1   x  This initialization will always be statistically consistent so long as the measurement errors are properly characterized.
71. 71. Kalman Filter – Summary The Kalman Gain: – Proportional to the Predicted Error – Inversely Proportional to the Innovation Error The Covariance Matrix: – Independent of measurements – Indicates the error in the state estimate assuming that all of the assumptions/models are correct The Kalman Estimator: – Optimal MMSE state estimator (Gaussian) – Best Linear MMSE state estimator (Non-Gaussian) – The state and covariance completely summarize the past
72. 72. Kalman Filter – Summary
73. 73. Kalman Filter – Summary
74. 74. Kalman Filter: Direct Discrete Time Example Consider the simplest example of the nearly constant velocity (CV) dynamics model: T 2 2  ξ k   1 T   xk =   xk −1 +  ν k −1 xk =    [ ] E ν k2 = q 0 1   T  ξ k  Discrete WhiteNoise Acceleration  z k = [1 0] xk + wk  [ ] E wk = r 2 The recursive estimation process is given by the Kalman equations derived above. How do we select “q”? q ≅ amax 2
75. 75. Kalman Filter: Other Direct Discrete Time Models For nearly constant acceleration (CA) models, the Discrete Weiner Process Acceleration (DWPA) model is commonly used: 1 T T 2 2 T 2 2 ξ k    x k = 0 1    T  xk −1 +  T ν k −1    xk = ξ k  [ ] E ν k2 = q 0 0 1   1  ξk         T 4 4 T 3 3 T 2 2   Qk = qΓk ΓkT = q T 3 3 T 2 T  q ≅ ∆amax 2 T 2 2 T 1    Notice the simple relationship between the “q-value” and the physical parameter that is one derivative higher than that which is estimated.
76. 76. Kalman Filter: Discretized Continuous-Time Models These models are derived from continuous time representations using the matrix superposition integral. Ignoring the control input:   x ( t ) = Ax (t ) + Dv (t )  ~ E [ v (t )v (τ )] = q δ ( t − τ ) ~ ~ ~   xk = Fk −1 xk −1 + vk −1 where F =e AT and T ~ vk = ∫ e A(T −τ ) Dv (τ ) dτ 0 Thus, the process noise covariance is found by: [Qk = E v v = ∫T k k ] T 0 ∫ 0 T Fk ( T − τ 1 ) D E [ v (τ 1 )v (τ 2 )] DT FkT ( T − τ 2 ) dτ 1dτ 2 ~ ~ = ∫ Fk ( T − τ 1 ) D q DT FkT ( T − τ 1 ) dτ 1 ~ T 0
77. 77. Kalman Filter: Discretized Continuous-Time Models Continuous White Noise Acceleration (CWNA) for CV Model: ~ ~ T 3 3 T 2 2  ~ ξ(t ) = v (t )  Qk = q  2  q ≅ amaxT 2 T 2 T  Continuous Weiner Process Acceleration (CWPA) for CA Model: T 5 20 T 4 8 T 3 6  ~ ξ(t ) = v (t )  ~ Qk = q  T 4 8 T 3 3 T 2 2  ~ q ≅ ∆amax T 2  T3 6 T2 2 T    Singer [IEEE-AES, 1970] developed the Exponentially Correlated Acceleration (ECA) for CA model (p. 187 & pp.321-324):   ~ ξ(t ) = −αξ(t ) + v (t )
78. 78. Kalman Filter: Time Consistent Extrapolation So, what is the difference between the Direct Discrete-Time and Discretized Continuous-Time models for CV or CA models. Which one should be used? [ FPF T + Q DWNA ] T =2 [ {[ ≠ F FPF T + Q DWNA ] }F T =1 T + Q DWNA ] T =1 [ FPF T + Q CWNA ] = [ F {[ FPF T + Q CWNA ] }F T + Q CWNA ] T =2 T =1 T =1 Thus, for the Continuous-Time model, 2 extrapolations of 1 second yields the same result as 1 extrapolation of 2 seconds. In general, the Continuous-Time models have this time consistent property. This is because the process noise covariance is derived using the transition matrix, while the Direct Discrete-Time is arbitrary.
79. 79. Kalman Filter: Steady State Gains If we iterate the Kalman equations for the covariance indefinitely, the updated covariance (and thus the Kalman gain) will reach steady state. This is only true for Kalman models that have constant coefficients. In this case, the steady-state solution is found using the Algebraic Matrix Riccati Equation (pp. 211 & 350): [ Pss = F Pss − Pss H ( HPss H + R ) HPss F T + Q T T −1 ] The steady state Kalman gain becomes: K ss = Pss H T [ HP H ss T +R ] −1
80. 80. Kalman Filter: Steady State Biases If a Kalman filter has reached steady-state, then it is possible to predict the filter’s bias resulting from un-modeled dynamics. Consider the CV model with an un-modeled constant acceleration (p. 13):    ξ k  T 2 2  α  xk = F xk −1 + Γν k −1 + Gλ xk =    G =   K ss =   ξk   T  β T  Un-Modeled Acceleration The steady-state error is found to be:  1−α 2   β T  ~ = ( I − K H ) F ~ + ( I − K H ) Gλ ⇒ ~ =  xss xss xss λ ss ss  2α − β  T  β   
81. 81. Kalman Filter – Summary #2 The Kalman Gain: – Reaches steady-state for constant coefficient models – Can determine steady-state errors for un-modeled dynamics The Covariance Matrix: – Is only consistent for when model matches true – Has no knowledge of the residuals The Kalman Estimator: – We need modifications for more general models – What about non-linear dynamics?
82. 82. Non-Linear Filtering
83. 83. Nonlinear Estimation Problems Previously, all dynamics and measurement models were linear. Now, we consider a broader scope of estimation problems:     ~ x = f ( x , t ) + Du (t ) + v (t )    z (t ) = h( x , t ) + w(t ) Nonlinear Dynamics: – Ballistic Dynamics (TBM exo-atmospheric, Satellites, etc…) – Drag/Thust Dynamics (TBM re-entry, TBM Boost, etc…) Nonlinear Measurements: – Spherical Measurements – Angular Measurements – Doppler Measurements
84. 84. EKF - Nonlinear Dynamics  The state propagation can be done using numerical integration or a Taylor Series Expansion (linearization):  However, the linearization is necessary in order to propagate the covariance:    xk ≅ Fk −1 xk −1 + Gk −1uk −1 + Γk −1vk −1Fk −1 = e [  f ( x ) ( t k −t k −1 ) ]  x = xk −1|k −1 ˆ ∂f ≅ I + (t k − t k −1 )  ∂x  + (t k − t k −1 ) 2 ∂ 2 f 2  ∂x 2  + x = xk −1|k −1 ˆ x = xk −1|k −1 ˆ Jacobian Matrix Hessian Matrix  The state and covariance propagation are precisely as before:   ˆk |k −1 = Fk −1 ⋅ xk −1|k −1 + Gk −1uk −1 x ˆ   T Pk |k −1 = Fk −1 ⋅ Pk −1|k −1 ⋅ Fk −1 + Γk −1Qk −1ΓkT−1
85. 85. EKF - Nonlinear Measurements  We compute the linearization of the observation function:       h( xk ) = h( xk |k −1 ) + H k ( xk − xk |k −1 ) + H k ( xk − xk |k −1 ) 2 +  ˆ ˆ ˆ  ∂h   ∂ 2hJacobian Matrix Hk =  Hk = 2 Hessian Matrix ∂x x = xk|k −1  ˆ ∂x x = x  ˆ k |k −1  The residual is thus: ∆  ηk = zk − z k |k −1 = z k − h( xk |k −1 ) = H k ~k |k −1 + wk ˆ ˆ x  The covariance update and Kalman gain are precisely as before (381- 386):  ˆk |k = xk |k −1 + K kη k x ˆ Pk |k = ( I − K k H k ) Pk |k −1 ( I − K k H k ) + K k Rk K k T T −1 K k = Pk |k −1 H k S k T
86. 86. Polar Measurements Previously, we dealt with unrealistic observation models that assumed that measurements were Cartesian. Polar measurements are more typical. In this case, the observation function is nonlinear:    r   x 2 + y 2  x k = [ xk  yk ]  z k = h( xk ) + wk h( xk ) =   =  −1  xk  yk  T b  tan ( x y )   The Kalman Gain and Covariance Update only require the Jacobian of this observation function:  x y  0 0 ∂h  x 2 + y 2 x2 + y2 H =  =  ∂x  y −x   x2 + y2 0 0  x2 + y2  This Jacobian is evaluated at the extrapolated estimate.
87. 87. Ballistic Dynamics As a common example of nonlinear dynamics, consider the ballistic propagation equations specified in ECEF coordinates: [x   y x   z ] T = y  z [x  2 ω y + ω 2 x + Gx  y − 2ω x + ω 2 y + Gy   z Gz  ] T ~ + Dv (t ) The gravitational acceleration components (to second order) are:  − µ x  3  R 2   z   2  3 1 + J 2  e  1 − 5      R  2 R     R     Gx        − µ y  3  Re  2  z 2   G = G y  =  3 1 + J 2   1 − 5     R  2 R     R     G z      − µ z  3  R 2   z    2  3 1 + J 2  e   3 − 5     R  2 R      R     
88. 88. Thrust and Re-entry Dynamics As a common example of nonlinear dynamics, consider the ballistic propagation equations specified in ECEF coordinates: [x   y x   z  a y  z  T β = ]  x  y  z   ~  Ballistic x ax  +a  Ballistic y ay +a z Gz + a  aβ − β 2  + D v (t )  s s s  The new states are the relative axial acceleration “a” and the relative mass depletion rate “”:  1   T + C D AC ρ ( v ⋅ v ) m(t )    2 a (t ) = athrust (t ) − adrag (t ) = β (t ) = m(t ) m(t ) The process noise matrix (if extended to second order) becomes a function of the speed. Thus, a more rapidly accelerating target while have more process noise injected into the filter.
89. 89. Pseudo- Measurements In the case of the TBM dynamics, the ECEF coordinates are the most tractable coordinates. However, typically the measurements are in spherical coordinates. Furthermore, the Jacobian for the conversion from ECEF to RBE is extremely complicated. Instead, we can convert the measurements into ECEF as follows:    zk′ = I 3 x 3 xk + w′ k Rk = J meas Rk J meas T However, since this is a linearization, we must be careful to make sure that this approximation holds.
90. 90. Pseudo- Measurements The linearization is valid so long as (pp. 397-402): r σ b2 < 0.4 σr
91. 91. Iterated EKF The IEKF iteratively computes the state “n” times during a single update. This recursion is based on re-linearization about the estimate: ∂h ∂h H ki =  where H k0 =  i = 0,1,..., n ∂x ˆi xk|k ∂x ˆ xk|k −1 The state is updated iteratively with a re-linearized residual and gain:  xk+k1 = xk |k −1 + K kiη ki ˆi | ˆ   ηki = z k − h( xk |k ) − H ki ˆi ˆi xk|k [x ˆ k |k −1 − xk | k ˆi ] −1  H i  H i Hi  + R  T T K k = Pk |k −1  k ˆ i   k i Pk |k −1  k ˆ i  k  xk|k   ˆi xk|k  xk|k   Finally, the covariance is computed based upon the values of the final iteration:  + K n R ( K n )T T Pk |k =  I − K kn H kn  P  I − K n H n n  k |k −1  k k  k k k  xk|k  ˆ  xk|k  ˆn
92. 92. Multiple-Model Filtering
93. 93. Why Multiple Models? When the target dynamics differ from the modeled dynamics, the state estimates are subject to: – Biases (Lags) in the state estimate – Inconsistent covariance output – In a tracking environment, this increases the probability of mis- association and track loss In most tracking applications, the true target dynamics have an unknown time dependence. To accommodate changing target dynamics, one can develop multiple target dynamics models and perform hypothesis testing. This approach is called hybrid state estimation.
94. 94. Why Multiple Models? 6-State Velocity Error (Meters/Sec) (x=red, y=blue, z=green, True Error=black) 100 Assuming a 80 True state estimate error ConstantVelocity target 60dynamics, the estimation Confidence interval 40 Given by Kalmanerrors become Covariance inconsistent 20 during an acceleration 0 -20 -40 -60 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
95. 95. Why Multiple Models? 9-State Velocity Error (Meters/Sec) (x=red, y=blue, z=green, True Error=black) 200 A Constant True state estimate error Acceleration 150 Confidence intervalmodel remains 100 Given by Kalman consistent, Covariance however the 50 steady-state estimation 0error is larger -50 -100 -150 -200 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
96. 96. Adaptive Process Noise Since the normalized innovations squared indicate the consistency of the dynamics model, it can be monitored to detect deviations (pp. 424- 426). At each update, the perform the following threshold test:  T −1  d k2 = η k S k η k ≥ ε max { } P d k2 ≥ ε max = α Then, the process noise value is adjusted such that the statistical distance is equal to this threshold value:  T −1  η k S k η k = ε max q The disadvantage is that false alarms result in sudden increases in error. We can use a sliding window average of these residuals, however this can delay the detection of a maneuver (pp. 424-426).
97. 97. State Estimation – Multiple Models We can further assume that the true dynamics is one of “n” models: r r r r xk = Fk −1 xk −1 + vk −1 r = 1,2,  , n   r z kr = H kr xkr + wk −1 r = 1,2,  , n Using Kalman filter outputs, each model likelihood function can be computed: [  1  T −1   exp− η kr S kr η kr   2 ]  ; r = 1,2, , n Λrk = [ det 2π S kr ] At each filter update, the posterior model probabilities “ ki ” are computed recursively using Baye’s Theorem. The proper output can be selected using these probabilities (pp. 453-457).
98. 98. State Estimation – SMM Hypothesis 1 x1 ˆkMeasurement ˆ2 xk Λik µ k −1 i Hypothesis 2 µ ki = n Hypothesis ∑Λ j =1 j k µ j k −1 Selection Hypothesis “n” ˆ xk ˆn xk Most Probable State Estimate Each Kalman filter is updated independently and has no knowledge about the performance of any other filter. This approach assumes that the target dynamics are time-independent
99. 99. State Estimation – IMM Measurement Conditional Probability Update/ Hypothesis 1 x1 ˆk 1ˆx State Estimate Interaction k −1 ˆ2 xk Probabilityˆ2xk −1 Hypothesis 2 Updates Estimate Mixing ˆn xkˆx n k −1 ˆ xk Hypothesis “n” IMM Estimate Each Kalman filter interacts with others just prior to an update This interaction allows for the possibility of a transition This approach assumes that the target dynamics will change according to a Markov process. pp. 453-457.
100. 100. State Estimation – IMM x1k −1|k −1 , P1k −1|k −1 ˆ x 2 k −1|k −1 , P 2 k −1|k −1 ˆ  µ k −1  1 Interaction  2   µ k −1  x 0,1k −1|k −1 , P 0,1k −1|k −1 ˆ x 0, 2 k −1|k −1 , P 0, 2 k −1|k −1 ˆ Kalman Prob. Kalman zk Λ1k Λ2k zk Filter Updates Filter x1 k | k , P1 k | k ˆ x 2 k |k , P 2 k | k ˆ {µ 1 k −1 , µ k2−1 } Estimate Mixing ˆ xk |k , Pk |k
101. 101. State Estimation – Applied IMM 6-State Velocity Error (Meters/Sec) (x=red, y=blue, z=green, True Error=black) 100IMM adapts to 80 True state estimate error changes in Confidence interval target 60 Given by Kalmandynamics and 40 Covariance provides a consistent 20 covariance 0 during these transitions -20 -40 -60 -80 -100 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800
102. 102. State Estimation – IMM Markov Matrix The particular choice of the Markov Matrix is somewhat of an art. Just like any filter tuning process, one can choose a Markov Matrix simply based upon observed performance. Alternatively, this transition matrix has a physical relationship to the Mean Sojourn Time of a given dynamics state. 1 1 Tscan E[ Ni ] = pii = 1 − = 1− 1 − pii E[ N i ] E [τ i ]
103. 103. State Estimation – VSIMM Air Targets: Adaptive Grid Coordinated Turning Model TBM Targets: Constant Axial Thrust, Ballistic, Singer ECA Air IMM xkAir ˆ Measurement TBM SPRT Hypothesis Selection ˆ xk ˆ x k TBM IMM The SPRT is performed as follows:  xkAir ; > T2 ˆµ kAir  1− β β ⇒ mixed ; > T1 and < T2 T2 = and T1 =µk TBM  α 1−α  ˆ TBM ; < T1 xk