R package 'bayesImageS': a case study in Bayesian computation using Rcpp and ...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism. I will also discuss the process of releasing this software as an open source R package on the CRAN repository.
bayesImageS: Bayesian computation for medical Image Segmentation using a hidd...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism.
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsMatt Moores
So-called “inverse” problems arise when the parameters of a physical system cannot be directly observed. The mapping between these latent parameters and the space of noisy observations is represented as a mathematical model, often involving a system of differential equations. We seek to infer the parameter values that best fit our observed data. However, it is also vital to obtain accurate quantification of the uncertainty involved with these parameters, particularly when the output of the model will be used for forecasting. Bayesian inference provides well-calibrated uncertainty estimates, represented by the posterior distribution over the parameters. In this talk, I will give a brief introduction to Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution and describe how they can be combined with numerical solvers for the forward model. We apply these methods to two examples of ODE models: growth curves in ecology, and thermogravimetric analysis (TGA) in chemistry. This is joint work with Matthew Berry, Mark Nelson, Brian Monaghan and Raymond Longbottom.
R package 'bayesImageS': a case study in Bayesian computation using Rcpp and ...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism. I will also discuss the process of releasing this software as an open source R package on the CRAN repository.
bayesImageS: Bayesian computation for medical Image Segmentation using a hidd...Matt Moores
There are many approaches to Bayesian computation with intractable likelihoods, including the exchange algorithm, approximate Bayesian computation (ABC), thermodynamic integration, and composite likelihood. These approaches vary in accuracy as well as scalability for datasets of significant size. The Potts model is an example where such methods are required, due to its intractable normalising constant. This model is a type of Markov random field, which is commonly used for image segmentation. The dimension of its parameter space increases linearly with the number of pixels in the image, making this a challenging application for scalable Bayesian computation. My talk will introduce various algorithms in the context of the Potts model and describe their implementation in C++, using OpenMP for parallelism.
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
Bayesian Inference and Uncertainty Quantification for Inverse ProblemsMatt Moores
So-called “inverse” problems arise when the parameters of a physical system cannot be directly observed. The mapping between these latent parameters and the space of noisy observations is represented as a mathematical model, often involving a system of differential equations. We seek to infer the parameter values that best fit our observed data. However, it is also vital to obtain accurate quantification of the uncertainty involved with these parameters, particularly when the output of the model will be used for forecasting. Bayesian inference provides well-calibrated uncertainty estimates, represented by the posterior distribution over the parameters. In this talk, I will give a brief introduction to Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution and describe how they can be combined with numerical solvers for the forward model. We apply these methods to two examples of ODE models: growth curves in ecology, and thermogravimetric analysis (TGA) in chemistry. This is joint work with Matthew Berry, Mark Nelson, Brian Monaghan and Raymond Longbottom.
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
Convex Optimization Modelling with CVXOPTandrewmart11
An introduction to convex optimization modelling using cvxopt in an IPython environment. The facility location problem is used as an example to demonstrate modelling in cvxopt.
Strum Liouville Problems in Eigenvalues and Eigenfunctionsijtsrd
This paper we discusses with Strum Liouville problem of eigenvalues and eigenfunctions, within the standard equation where p,q and r are given functions of the independent variable x is an interval The is a parameter and is the dependent variable. The method of separation of variable applied to second order liner partial differential equations, the equation is known because the Strum Liouville differential equation. Which appear in the overall theory of eigenvalues and eigenfunctions and eigenfunctions expansions is one of the deepest and richest parts of recent mathematics. These problems are associate with work of J.C.F strum and J.Liouville. B. Kavitha | Dr. C. Vimala "Strum - Liouville Problems in Eigenvalues and Eigenfunctions" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-5 , August 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31721.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/31721/strum--liouville-problems-in-eigenvalues-and-eigenfunctions/b-kavitha
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
A Novel Methodology for Designing Linear Phase IIR FiltersIDES Editor
This paper presents a novel technique for
designing an Infinite Impulse Response (IIR) Filter with
Linear Phase Response. The design of IIR filter is always a
challenging task due to the reason that a Linear Phase
Response is not realizable in this kind. The conventional
techniques involve large number of samples and higher
order filter for better approximation resulting in complex
hardware for implementing the same. In addition, an
extensive computational resource for obtaining the inverse
of huge matrices is required. However, we propose a
technique, which uses the frequency domain sampling along
with the linear programming concept to achieve a filter
design, which gives a best approximation for the linear
phase response. The proposed method can give the closest
response with less number of samples (only 10) and is
computationally simple. We have presented the filter design
along with its formulation and solving methodology.
Numerical results are used to substantiate the efficiency of
the proposed method.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2010. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The paper examines the problem of systems redesign within the context of passive electrical networks and through analogies provides also the means of addressing issues of re-design of mechanical networks. The problem addressed here are special cases of the more general network redesign problem. Redesigning autonomous passive electric networks involves changing the network natural dynamics by modification of the types of elements, possibly their values, interconnection topology and possibly addition, or elimination of parts of the network. We investigate the modelling of systems, whose structure is not fixed but evolves during the system lifecycle. As such, this is a problem that differs considerably from a standard control problem, since it involves changing the system itself without control and aims to achieve the desirable system properties, as these may be expressed by the natural frequencies by system re-engineering. In fact, this problem involves the selection of alternative values for dynamic elements and non-dynamic elements within a fixed interconnection topology and/or alteration of the network interconnection topology and possible evolution of the cardinality of physical elements (increase of elements, branches). The aim of the paper is to define an appropriate representation framework that allows the deployment of control theoretic tools for the re-engineering of properties of a given network. We use impedance and admittance modelling for passive electrical networks and develop a systems framework that is capable of addressing “life-cycle design issues” of networks where the problems of alteration of existing topology and values of the elements, as well as issues of growth, or death of parts of the network are addressed.
We use the Natural Impedance/ Admittance (NI-A) models and we establish a representation of the different types of transformations on such models. This representation provides the means for an appropriate formulation of natural frequencies assignment using the Determinantal Assignment Problem framework defined on appropriate structured transformations. The developed natural representation of transformations are expressed as additive structured transformations. For the simpler case of RL or RC networks it is shown that the single parameter variation problem (dynamic or non-dynamic) is equivalent to Root Locus problems.
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This presentation discuss a sufficient and necessary condition for quadratic stability of a class of switched systems including two modes. This result has been published in proceeding of the ECC Conference in Zürich, 2013.
Similar to State Equations Model Based On Modulo 2 Arithmetic And Its Applciation On Recursice Convolutional Coding (20)
State Equations Model Based On Modulo 2 Arithmetic And Its Applciation On Recursice Convolutional Coding
1. State equations model based on
modulo-2 arithmetic and its
application on recursive
convolutional coding
Ch. N. Tasiopoulos, A.A. Fotopoulos, K.P. Peppas,
P.H. Yannakopoulos
International Scientific Conference eRA-6
2. Introduction
According to the theory of control systems, a time-
invariant system can be represented by a block diagram
with feedback for error correction, such a system can
be modeled by state equations. These equations can be
derived from the transfer function of the given system,
expressed in zeta transformation.
2 International Scientific Conference eRA-6
3. Concepts of Digital Theory
A discrete time controller can be described from the
transfer function:
1
U ( z) b( z ) b0 b1 z b2 z 2 bn z n
Gc ( z ) 1
E( z) a( z ) a0 a1 z a2 z 2 an z n
3 International Scientific Conference eRA-6
4. Concepts of Digital Theory
Direct hardware
realization model
Fig. 10.2.1, pg. 284 “Control Systems”, P.N. Paraskeuopoulos
4 International Scientific Conference eRA-6
5. Concepts of Digital Control Theory
State Equations
x(k+1)= Ax(k)+be(k) where:
u(k) = cTx(k)+de(k) 0 1 0 ... 0 0 bn anb0
0 0 1 ... 0 0 bn 1 an 1b0
A= , b= , c=
0 0 0 1 0 b2 a2 b0
an an 1 an 2 a1 1 b1 a1b0
and the Constant d=b0
5 International Scientific Conference eRA-6
6. Concepts of Information
& Control Theory
Group definition
A group (G,*) is a pair consisting of a set G and an operation
* on that set , that is a function from the Cartesian product
GxG to G , with the result of operating on a and b denoted
by a*b , which satisfies
1. Associativity : a*(b*c)= (a*b)*c for all a, b, c G
2. Existence of identity: There exists e G such that
e*a=a and a*e=a for all a G
1
3.Existence of inverses: For each a G there exists a G
such that a * a 1 =e and a 1 * a =e
6 International Scientific Conference eRA-6
7. Concepts of Information
& Control Theory
Cyclic Groups definition
For each positive integer p, there is a group called the cyclic
group of order p, with set of elements
Zp {0,1,,( p 1)}
and operation defined by i j i j
If i j p , where (+ )denotes the usual operation
of addition of integers, and i j i j p
If i j p ,where (-) denotes the usual operation of subtraction of integers.
The operation in the cyclic group is addition modulo p. We shall use the sign +
instead of to denote this operation in what follows and refer to “the cyclic
Group ( Z p , ) ” , or simply the cyclic group Z
p
7 International Scientific Conference eRA-6
8. Concepts of Information
& Control Theory
Ring definition
A ring is a triple ( R, , ) consisting of a set R, and two operations + and , referred
to as addition and multiplication, respectively, which satisfy the following conditions:
1.Associativity of +: a (b c) (a b) c, for all a,b,c R
2.Commutativity of +: a b b a for all a,b R
3.Existence of additive identity: there exists 0 R such that 0 a a and a 0 a for all a R
a ( a) 0
4.Existence of additive inverses: for each a R there exists a R such that and ( a) a 0
5.Associativity of : a (b c) (a b) c, for all a, b, c R
6. Distributivity of over +: a (b c) (a b)(a c), for all a, b, c R
8 International Scientific Conference eRA-6
9. Concepts of Information
& Control Theory
Cyclic Groups definition
For every positive integer p, there is a ring (Z p , , ) , called the cyclic ring of order p,
with set of elements
Zp {0,1,,( p 1)}
and operations + denoting addition modulo p, and denoting multiplication modulo p
9 International Scientific Conference eRA-6
10. Modulo-2 Arithmetic
From the cyclic groups definition we have: Zp {0,1,,( p 1)}
The equation becomes for p=2: Z2 {0,1}
Where Z 2 is a cyclic group in modulo-2 arithmetic.
The operations stands as referred previously.
10 International Scientific Conference eRA-6
11. State equations & modulo-2
Assume the following transfer function:
1
U ( z) b( z ) b0 b1 z b2 z 2 bn z n
Gc ( z ) 1
E( z) a( z ) a0 a1 z a2 z 2 an z n
with b0b1b2bn GF (2) and a0 a1a2an GF (2)
The differential equation will be:
1 2
U ( z)(a0 a1z a2 z ... an z n ) e( z)(b0 b1z 1
b2 z 2
... bn z n )
11 International Scientific Conference eRA-6
12. State equations & modulo-2
1 2 n 1 2 n
u( z)a0 u( z)a1z u( z)a2 z ... u(z)an z e(z )b0 e(z )b1z e(z )b2 z ... e(z )bn z
Applying inverse Z transform with k 0,1, 2,...m GF (10)
we have:
u(k )a0 u(k 1)a1 u(k 2)a2 ... u(k n)an e(k )b0 e(k 1)b1 e(k 2)b2 ... e(k n)bn
12 International Scientific Conference eRA-6
13. State equations & modulo-2
The above differential
equation has the
following
direct hardware
realization:
Fig. 10.2.1, pg. 284 “Control Systems”, P.N. Paraskeuopoulos
13 International Scientific Conference eRA-6
14. State equations & modulo-2
Based on the above figure we derive the following
recursive signal equations:
xn ( k 1) e( n ) a1 xn ( k ) a2 xn 1 (k ) ... an x1 (k )
xn ( k ) xn 1 (k 1)
xn 1 (k ) xn 2 (k 1)
x2 ( k ) x1 ( k 1)
x1 ( k )
14 International Scientific Conference eRA-6
15. State equations & modulo-2
0 1 0 ... 0 0 bn anb0
x( k 1) Ax ( k ) be( k ) 0 0 1 ... 0 0 bn 1 an 1b0
A= , b= , c=
u (k ) cT x( k ) de( k ) 0 0 0 1 0 b2 a2 b0
an an 1 an 2 a1 1 b1 a1b0
where: , b, c, d GF (2)
15 International Scientific Conference eRA-6
18. Description of Recursive
Convolutional Encoder
o The convolutionally encoding data, requires the use of n memory
registers, each holding 1 input bit, unless otherwise specified. All
memory registers start with a value of zero (0).
o The encoder has n modulo-2 adders and n generator polynomials
o Using the generator polynomials and the existing values in the
remaining registers, the encoder outputs n bits.
o I(n) consists the Input data and C1,C2,Cn the output data
18 International Scientific Conference eRA-6
19. Recursive Convolutional Encoder
Based on the figure we derive
the following recursive signal
equations:
X1 n = X1 n
X1 n-1 = X 2 n
X 2 n-1 = X 3 n = X1 n-2
X 3 n-1 =X 4 n =X1 n-3
X n-1 n-1 =X n n
Xn n-1
19 International Scientific Conference eRA-6
20. Algebraic Form of State Equations of
Recursive Convolutional Encoder in modulo-2
From the recursive signal equations
we have the algebraic form of state
equations: m
xm (n 1) xi (n) I (n) where m is the number of registers
i 1
m
cj xi (n) dI (n) where 1 j n
i 1
n, m, i, j GF (10)
with:
X , I , c, d GF (2)
20 International Scientific Conference eRA-6
21. Vector Form of State Equations of Recursive
Convolutional Encoder in modulo-2
From the recursive signal equations
we have the vector form of state
equations:
0 1 0 ... 0 0 bn anb0
X (n 1) AX (n) bI (n) 0 0 1 ... 0 0 bn 1 an 1b0
A= , b= , c=
Cj cT X (n) dI (n) 0 0 0 1 0 b2 a2 b0
an an 1 an 2 a1 1 b1 a1b0
n, j, e GF (10)
with:
A, b, c, d , I , a, b GF (2)
21 International Scientific Conference eRA-6
23. Numerical Example
By applying the recursive signal formula for this particular model
we have:
X1(n)=X1(n)
X1(n-1)=X2(n)
X2(n-1)=X3(n)
X3(n-1)=X1(n)-I(n)-X3(n)-X2(n)
23 International Scientific Conference eRA-6
24. Numerical Example
Using these equations we can derive the state equations:
x1( n 1)
0 1 0 x1( n ) 0
x2( n 2) = 0 0 1 x2( n ) + 0 I(n)
x3( n 3)
1 1 1 x3( n ) 1
24 International Scientific Conference eRA-6
25. Numerical Example
Now we will study the controllability & observability of the given
system:
The determinant of s is given by the
s B AB A2 B following formula:
0 0 0 1 0 1 0 1
1 3 3
AB 1 s 0 1 1 det( s) 0 0 ( 1) det( s) 1
1 1 1 1
1 1 1 0
1 det(s) 1 0 1 1 0
2
AB 1
0 The system is controllable
25 International Scientific Conference eRA-6
26. Numerical Example
For finding if the system is observable we have:
C1 1 0 1 C2 1 0 0
R1 C1 A det( R) 1 0 1 det( R) 0 R2 C2 A det( R ) 0 1 0 1 0
C1 A2 1 1 1 C2 A 2 0 0 1
The system is observable only for the C2 output
26 International Scientific Conference eRA-6
27. Thank you for your attention
Evariste Galois
1811-1832
27 International Scientific Conference eRA-6