H. Akaike. Markovian representation of stochastic processes. SIAM J. Control, 13(1):162--173, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....greater than or equal to that of y, i.e. p m. Now, it is well known that the same discrete time process y can be represented by minimal realizations of the type (1.1) which may either have a non singular or a singular D matrix. In fact, there may be realizations, like those used by Akaike in [1] and quite commonly encountered in the statistical literature, where D = 0. When the matrix D in the representation (1.1) is singular, the problem of estimating the state x based on the (past) observations of y is known as cheap (or singular) filtering . This problem is dual to the better known ....

H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM Journal on Contr. Optim., 13:162--173, 1975.

....of the finest scale variables, and their inclusion in the representation may serve purposes such as exposing the statistical structure of the phenomenon under study and or capturing more global quantities whose estimation is desired. For example, in analogy with stochastic realization theory [8, 9, 214] and the concept of state for dynamic systems, such variables may simply play the role of capturing the intrinsic memory in the signals that are observed or of primary interest. The models we describe also have close ties to Hidden Markov Models (HMM s) 272, 265, 222, 80, 261, 281, 302] in which ....

....related problems using an expectation maximization formalism. 6.2. 2 Internal Models and Approximate Stochastic Realization In this section we describe a more general and formal construction of linear MR models [161, 85, 122, 82] The approach makes use of concepts adapted from state space theory [8, 9, 214, 16]; however, the adaptation to trees uncovers some important di#erences with the temporal case. First, in contrast to the usual temporal state space framework and, for that matter, to the framework implicitly used in most graph theoretic studies we consider problems in which the random process or ....

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H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM J. on Control, 12(1):162--173, January 1975.

....subspace identification method of Van Overschee and De Moor [17, 18, 19] Q and R are chosen to be the inverted Cholesky factors of the EXPERIMENTAL EVIDENCE FOR FAILURE OF SUBSPACE IDENTIFICATION 5 block Toeplitz matrices of (2.10) and # 0 , # # 1 , # # 2, # # # respectively. As explained in [1] (also see [14] this choice ofw eights, tow#v# hw e shall refer as the canonical weights, is natural, since then the singular values # k are the canonical correlation coe#cients, i.e. the cosines of the angles betw een the spaces spanned by the future and past observations; see [14] All ....

....(z) 1 2 z 1 # z 1 # , 4.1) w#4. h is stable, having a pole of modulus less than one, but is not positive real for any # 0. Expanding V (z) as a Laurent series for z #1w e obtain V (z) 1 2 c 0 c 1 z 1 c 2 z 2 . w#w#N c 0 = 1 and c k = #(# 1) k 1 for k 0. Now# it isw ell know# [1] that c 0 ,c 1, # is a bona fide partial covariance sequence if and only if # k 1 k =0, 1, 2, 1, 4.2) w#4.2 # 0 ,# 1 ,# 2, are its Schur parameters. Obviously, since V (z) is not positive real, this is not the case for all #. It can be show# [8, 14] that these Schur parameters ....

H. Akaike. Markovian representation of stochastic processes bycanonical variables. SIAM J. Con ol, 13:162--173, 1975.

....may sometimes only obscure the basic issues. All this motivates us to study the geometric structure of stochastic models and to investigate the natural geometric formulations of some of the system theoretic properties mentioned above. This is basically the scope of the approach initiated in [1,44,47] and developed in [26 34,48 51] into a geometric theory of stochastic realization,leading to a extensive literature in the past fifteen years; see,e.g. 5,6,11,12,23,24] The introduction of coordinate free geometric descriptions is based on factoring out eq#, alent models with respect to a ....

H.Akaike,"Markovian representation of stochastic processes by canonical variables, " SIAM J.Control 13 (1975),162--173.

....which are particularly suited for dealing with multivariable systems are the so called subspace methods , which have been introduced recently and studied in the last ten years by a number of authors. The main ideas date back to the work of Hotelling [8] in the statistical community and Akaike [1] in system engineering. In this paper we consider a simplified version which, under reasonable simplifying assumptions, allows implementation with reasonable computational complexity also for very high dimensional data. In the second part of the paper we discuss possible extensions to capture ....

H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM J. of Control, 13:162--173, 1975.

..... 0 0 1 0 1 C C C C A for each t: This is one of several possible DLM forms of the TVAR model, and a natural extension of the DLM representation of standard AR models; the latter are obtained when t = and G t = G( are constant, and their state space forms originated with Akaike (1974). The time varying versions in equation (2) are special cases of the broader class of DLMs, also known as a state space models that date back at least to Kalman (1960) Our decomposition theory applies at that level of generality. The central decomposition result arises simply from standard ....

Akaike, H. (1974), \Markovian Representation of Stochastic Processes and its Application to the Analysis of Autoregressive Moving Average Processes," Ann. Inst. Statist. Math., 26, 363-387.

.... a shift structure generated by the matrices A h and C h (or similarity transforms of these matrices) Fact 3: The row space of the matrix E(Y future Y t past ) has a shift structure generated by the matrices A h and G (or similarity transforms of these matrices) Some relevant references are [1] [9] Fact 1 permits to estimate the order of the stochastic system. Fact 2 and 3 allow to estimate the matrices A h ; G and C h by constructing a block Hankel matrix with the output covariances (see the algorithm below) The fact that the matrix E(Y future Y t past ) is rank deficient implies ....

Akaike H. Markovian Representation of Stochastic Processes by Canonical Variables. SIAM Journal on Control, 13, 1, pp.162-173, January 1975.

....specified covariance structure. The problem of constructing such a model is the multiscale generalization of the problem of stochastic realization for time series, and the technique developed in [13] is based on the statistical concept of canonical correlations used in building time series models [1]. As discussed in [1, 13] the key to constructing a recursive model for a time series z(t) is the specification of the state x(t) at each time t. If z p (t) denotes the past of the process at time t and z f (t) the future then the components of x(t) represent a set of linear functionals of the ....

....structure. The problem of constructing such a model is the multiscale generalization of the problem of stochastic realization for time series, and the technique developed in [13] is based on the statistical concept of canonical correlations used in building time series models [1] As discussed in [1, 13], the key to constructing a recursive model for a time series z(t) is the specification of the state x(t) at each time t. If z p (t) denotes the past of the process at time t and z f (t) the future then the components of x(t) represent a set of linear functionals of the past, so that conditioned ....

H. Akaike. "Markovian Representation of Stochastic Processes by Canonical Variables." SIAM Journal of Control (13) #1, 1975.

....these methods can be explained as first estimating the state vector x(t) and then finding the state space matrices by a linear least squares procedure. The state vector in innovations type parameterizations like (2) is always found as linear combinations of k step ahead output predictors. See [1], 4] or [3] Appendix 4.A. In the subspace methods these k step ahead predictors are found by so called oblique projections, 7] While this constitutes algorithms that are very efficient, numerically, it gives the drawback that closed loop data cannot be handled in this way. The purpose of this ....

....these states. The literature on state space identification has very elegantly showed how the state can be estimated directly from the data by certain projections. We shall not use that language here but describe the process (equivalently) along the classical realization path, as developed by [2] [1] and [4] See Appendix 4.A in [3] for an account) The essence is as follows: Let fe(t)g be the innovations and let the impulse response matrix from (t) u(t) e(t) # (3) to y(t) be H k so that y(t) 1 X k=0 H k (t Gamma k) 4) H k is thus a p by p m matrix where p is the number ....

H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM Journal of Control and Optimization, 13:162--173, 1975.

.... general method for constructing the linear functionals L(s) to achieve exact or approximate conditional decorrelation required by the tree model is the so called canonical correlations realization (CCR) algorithm [56, 55] It represents a generalization of Akaike s canonical correlations algorithm [2] for the stochastic realization of time series, where one wants to define, at a point in time, a set of state variables, written as linear functionals of the past or of the future, that either exactly or approximately conditionally decorrelates the past and the future. Following the same idea, but ....

H. Akaike. "Markovian Representation of Stochastic Processes by Canonical Variables". SIAM Journal of Control, 13(1), January 1975.

....single stationary process y 1 . More generally one may want to construct state space models involving also exogenous (input) variables. The mathematical problem of constructing linear state space representations of a stationary process has been studied in some depth in the past three decades see [5, 17, 1, 2, 57, 13, 36, 37, 39]. On the other hand, system identification, i.e. the statistical problem of describing an observed time series by a linear dynamic model, in particular by a state space model of the type (1) has traditionally been regarded as a different problem. We shall argue in this paper that stochastic ....

....approach to stochastic modeling has been put forward in a series of papers by Lindquist, Picci, Ruckebusch et al. 35, 36, 37, 57, 58, 59] which aims at the representation of random processes in this more specific sense. This motivation is also present in the early papers by Akaike [1, 2]. A main point of the geometric approach is that a stochastic state space system is defined in terms of the conditional independence relation between past and future of the signals involved. This relation is intrinsically coordinatefree and in the present setting involves only linear subspaces of ....

H. Akaike (1975). Markovian representation of stochastic processes by canonical variables, SIAM J. Control, 13 , pp. 162--173.

.... Of enormous recent interest in the area of system identification methods designed for control theory applications has been the study of so called State Space Subspace Identification (4SID) 4] Despite this recent interest, the ideas involved actually go back many years, at least to Akaike [1] whose approach was targeted at stochastic estimation problems pertinent to this paper. For the purposes of explaining this, suppose one is presented with observations fy t g of a stationary stochastic process and is faced the task of estimating a state space representation of this process in ....

....observations fy t g of a stationary stochastic process and is faced the task of estimating a state space representation of this process in innovations form: x t 1 = Ax t Ke t ; 6) y t = Cx t e t (7) where fe t g is a stationary white noise process. Via the idea of predictor space , Akaike [1] made clear for the first time that such a representation always exists, and in so doing suggested a way that it may be estimated from observations of fy t g. This estimation method is now known with some simple modifications (involving user chosen weighting matrices) as 4SID estimation. To ....

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H. Akaike. Markovian representation of stochastic processes. SIAM J. Control, 13(1):162--173, 1975.

.... sequence of the linear system characterised by the triplet (P; P Gammae; T Gamma) The stochastic identification problem comprises more than the order estimation: the matrices P , C and G are estimated and used in the computation of the state space description of the linear system (see [1, 4, 11, 18, 25, 26]) The complete identification of the MMPP using this approach is difficult because of the model restrictions of the MMPP (see [23] Yi and De Moor [28] only try to find the order of the MMPP, using the relevant parts of the stochastic realization algorithms. The autocorrelation function R (see ....

....per fl M i is equal to t i : Theta t 1 t 2 t 3 t 4 = Theta 6 2 5 7 and the total number of pieces is equal to the model order of the CMPP, NC . The resulting first order parameters of the CMPP model are: fl C = 5 5 5 5 5 5 17 17 53 53 53 53 53 112 112 112 112 112 112 112 ] T C = [ 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 ] T . 6.3 A more practical version for the translation If the translation of the first order MMPP parameters into the first order CMPP parameters is carried out as explained in Section 6.1, the resulting order of the CMPP can become very large. A more practical solution is to impose a certain ....

H. Akaike. Markovian Representation of Stochastic Processes by Canonical Variables. SIAM Journal on Control, vol. 13, 1975, pp. 162-173.

....y k (including the determination of the system order n) and the determination of the noise covariance matrices. The state space model should be equivalent up to within second order statistics of the output. The main contributions of this paper are the following : Since the pioneering papers by Akaike (1975), canonical correlations (which were first introduced by Jordan (1875) in linear algebra and then by Hotelling (1936) in the statistical community) have been used as a mathematical tool in the stochastic realization problem. In this paper we show how the approach by Akaike (1975) and others (e.g. ....

....papers by Akaike (1975) canonical correlations (which were first introduced by Jordan (1875) in linear algebra and then by Hotelling (1936) in the statistical community) have been used as a mathematical tool in the stochastic realization problem. In this paper we show how the approach by Akaike (1975) and others (e.g. Arun et al. 1990, Larimore, 1990) boils down to applying canonical correlation analysis to two matrices that are double infinite (i.e. have an infinite number of rows and columns) A careful analysis reveals the nature of this double infinity and we manage to reduce the ....

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Akaike H. (1975). Markovian Representation of Stochastic processes by Canonical Variables.

....to compute the Kalman filter for (1) Note that all of the above holds without changes for multivariable systems, i.e. when the output and input signals are vectors. The problem is where to get the state vector sequence x from. For that we turn to basic realization theory, as developed by [3] [1] and [7] See Appendix 4.A in [5] for an account) The basic results are as follows (see Lemmas 4A.1 and 4A.2 in [5] and their proofs) Let a system be given by the impulse response representation y(t) 1 X j=0 (h u (j)u(t Gamma j) h e (j)e(t Gamma j) 3) where u is the input and e the ....

H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM Journal of Control and Optimization, 13:162--173, 1975.

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H. Akaike. Markovian representation of stochastic processes. SIAM J. Control, 13(1):162--173, 1975.

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H. Akaike, "Markovian Representation of Stochastic Processes by Canonical Variables, " SIAM J. Control, 1975, vol. 13, no. 1, pp. 162-173.

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Akaike H. (1975). Markovian representation of stochastic processes by canonical variables. Siam J. Control, Vol. 13, no.1, pp.162-173.

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Akaike H. Markovian representation of stochastic processes by canonical variables. SIAM Journal of Control, 13, 1, p.162-172, 1975.

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Akaike H. Markovian Representation of Stochastic processes by Canonical Variables. SIAM Journal on Control, 13, 1, pp. 162173, January 1975.

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H. AKAIKE, Markovian representation of stochastic processes by canonical variables, SIAM Journal of Control, 13 (1975), pp. 162--173.

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Akaike,H. (1974): "Markovian Representation of Stochastic Processes and Its Application to the Analysis of Autoregressive Moving Average Processes", An. Inst. Stat. Math., 26, 363-387.

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Akaike, H., (1974), "Markovian Representation of Stochastic Processes and its Applications to the Analysis of Autoregressive Moving Average Processes", Annals of the Institute of Statistical Mathematics, 26, 363-- 387.

No context found.

H. Akaike. Markovian representation of stochastic processes by canonical variables. SIAM Journal of Control, 13(1):162--173, January 1975.

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AKAIKE H.,1975. Markovian representations of stochastic processes by canonical variables. SIAM J. Control, Vol.13 (pp 162-173).

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