Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....given all the previous observations, t = p(o t jo 1 ; o t 1 ) can be obtained by t = N (e(t) 0; t) 64) 17 and the joint likelihood of an observation sequence is simply p(O) Q T t . The parallel backward recursion, also known as Kalman or Rauch Tung Streibel smoother [36, 37], is derived in Appendix E.2. By de ning x(t) Efx t jOg and R(t) Efx t x t jOg the estimates of the required statistics can now be initialised as x(T ) x (T ) T ) and R(T ) T ) x(T ) x (T ) The rest of the estimates can be obtained by using the ....

H.E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371-372, 1963.

.... the expected complete data sucient statistics u mentioned in Theorem 1(a) We now turn to the VE step: computing Q(x 1:T ) Since SSMs are singly connected belief networks Corollary 1 tells us that we can make use of belief propagation, which in the case of SSMs is known as the Kalman smoother [29]. We therefore run the Kalman smoother with every appearance of the natural parameters of the model replaced with the following corresponding expectations under the Qdistribution: h i c i i, h i c i c i i, hAi, hA Ai. We omit the details here. Results from this model are presented in [7] ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371-372, 1963.

....all the previous observations, t = p(o t jo 1 ; o t 1 ) can be obtained by t = N (e(t) 0; e) t) 63) 17 and the joint likelihood of an observation sequence is simply p(O) Q T t=1 t . The parallel backward recursion, also known as Kalman or Rauch Tung Streibel smoother [36, 37], is derived in Appendix E.2. By de ning x(t) Efx t jOg and R(t) Efx t x 0 t jOg the estimates of the required statistics can now be initialised as x(T ) x (T ) T ) T ) T ) T ) and R(T ) T ) x(T ) x 0 (T ) The rest of the estimates can be obtained by ....

H.E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371-372, 1963.

....sources needs to be modied to take into account the dependency between successive source signals, however. If the probability distribution p i (d i (t) were Gaussian, the posterior distributions of the source signals could be solved analytically, the resulting method being called Kalman smoothing [10, 14]. As the distributions are non Gaussian, we will have to resort to an iterative approach. When computing the distribution of the source s i (t) the values x(t) s i (t Gamma 1) and s i (t 1) from the previous iteration are used for computing the new estimate of the posterior distribution of ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371:372, 1963.

.... the expected complete data sucient statistics u mentioned in Theorem 1(a) We now turn to the VE step: computing Q(x 1:T ) Since SSMs are singly connected belief networks Corollary 1 tells us that we can make use of belief propagation, which in the case of SSMs is known as the Kalman smoother [29]. We therefore run the Kalman smoother with every appearance of the natural parameters of the model replaced with the following corresponding expectations under the Qdistribution: h i c i i, h i c i c i i, hAi, hA Ai. We omit the details here. Results from this model are presented in ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371-372, 1963.

....inputs and outputs up to time T , where T t. The Kalman filter recursions are used in the forward direction to compute the probability of X t given fY g t 1 and fUg t 1 . A similar set of backward recursions from T to t complete the computation by accounting for the observations after time t (Rauch, 1963). We will refer to the combined forward and backward recursions for smoothing as the Kalman smoothing recursions (also known as the RTS or Rauch TungStreibel smoother ) Finally, the goal of prediction is to compute the probability of future states and observations given observations upto time t. ....

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

....it is easy to show that these are the first and the second order statistics: mean and covariance among hidden states x t ;x t,1 ;s t ;s t,1 . If there were no switching dynamics, the inference would be straightforward we could infer X T from Y T using LDS inference (RTS smoothing [25]) However, the presence of switching dynamics embedded in matrix # makes exact inference more complicated. To see that, assume that the initial distribution of x 0 at t =0is Gaussian, at t =1the pdf of the physical system state x 1 becomes a mixture of S Gaussian pdfs since we need to marginalize ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Trans. Automatic Control, AC-8(4):371--372, October 1963.

....of two parts: a forward recursion which uses the observations from y 1 to y t , known as the Kalman lter [13] and a backward recursion which uses the observations from y T to y t 1 . The combined forward and backward recursions are known as the Kalman or Rauch Tung Streibel (RTS) smoother [14]. There are three key insights to understanding the Kalman lter. The rst insight is that the Kalman lter is simply a method for implementing Bayes rule. Consider the very general setting where we have a prior p(x) on some state variable and an observation model p(yjx) for the noisy outputs ....

H. E. Rauch, \Solutions to the linear smoothing problem," IEEE Transactions on Automatic Control, vol. 8, pp. 371-372, 1963.

....[Saul and Jordan, 1 We use the notation O T 1 to denote the sequence of random variables O t from time 1 to time T . 3 1996] The authors show that the E step can be approximated by decoupling into forward backward recursions on a HMM [Baum et al. 1970] and Kalman smoothing recursion [Rauch, 1963] on each state space model, which are the relevant versions of the E step for hidden Markov models and linear dynamical systems 2 . Once the posterior probabilities have been approximated, it is easy to derive re estimations of the parameters Theta. The parameters of the HMM are reestimated ....

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

....it is easy to show that these statistics are h#x t s t #i, h#x t s t ##x t s t # 0 i,and h#x t s t ##x t,1 s t,1 # 0 i 2 . If there were no switching dynamics, the inference would be straightforward we could infer X T from Y T using LDS inference (RTS smoothing [21]) However, the presence of switching dynamics embedded in matrix # makes exact inference more complicated. To see that, assume that the initial distribution of x 0 at t =0is Gaussian, at t = 1 the pdf of the physical system state x 1 becomes a mixture of S Gaussian pdfs since we need to ....

...., i # t = # t;i # t 1 : 12) Switching model s sufficient statistics are now simply hs t i = e i # t and# s t s 0 t,1 # = e i # t e 0 i # t,1 . Giventhe best switching state sequence the sufficient LDS statistics can be easily obtained using the Rauch Tung Streiber smoothing [21]. For example, hx t ;s t #i#i = # x tjT ,1;i # t i = i # t 0 otherwise for i =0; S, 1. The Viterbi inference algorithm for complex DBNs can now be summarized as: Initialize LDS state estimates x 0j,1;i and # 0j,1;i ; Initialize cost J 0;i . for t =1:T , 1 for i =1:S for j ....

H. E. Rauch, "Solutions to the linear smoothing problem," IEEE Trans. Automatic Control, vol. AC-8, pp. 371--372, October 1963.

.... algorithms allow to compute P (q t jx t 1 ; y t 1 ) in a forward recursion (thus solving the filtering problem) Similarly to Markov switching Neural Computing Surveys 2, 129 162, 1999, http: www.icsi.berkeley.edu jagota NCS 153 models, a backward recursion (the Rauch equations [128]) allows to compute the posterior probabilities P (q t jx T 1 ; y T 1 ) for T t (thus solving the smoothing problem) In the context of real time control and other applications where learning must be on line, numerical maximization of the likelihood can be performed recursively with a ....

H. Rauch, "Solutions to the linear smoothing problem," IEEE Transactions on Automatic Control, vol. 8, pp. 371--372, 1963.

....and outputs up to time T , where T t. The Kalman filter recursions are used in the forward direction to compute the probability of X t given fY g t 1 and fUg t 1 . A similar set of backward recursions from T to t complete the computation by accounting for the observations after time t (Rauch, 1963). We will refer to the combined forward and backward recursions for smoothing as the Kalman smoothing recursions (also known as the RTS or Rauch Tung Streibel smoother ) Finally, the goal of prediction is to compute the probability of future states and observations given observations upto time t. ....

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

.... values of a recursive method used to compute (6) give the desired distributions (8) or (9) Filtering and smoothing have been extensively studied for continuous state models in the signal processing community, starting with the seminal works of Kalman (Kalman, 1960; Kalman and Bucy, 1961) and Rauch (Rauch, 1963; Rauch et al. 1965) although this literature is often not well known in the machine learning community. For the discrete state models much of the literature stems from the work of Baum and colleagues (Baum and Petrie, 1966; Baum and Eagon, 1967; Baum et al. 1970; Baum, 1972) on hidden Markov ....

....and control communities for decades. The emphasis has traditionally been on the inference problems: the famous Discrete Kalman Filter (Kalman, 1960; Kalman and Bucy, 1961) gives an efficient recursive solution to the optimal filtering and likelihood computation problems, while the RTS recursions (Rauch, 1963; Rauch et al. 1965) solve the optimal smoothing 10 Recall that if C is p Theta k with p k and is rank k then left multiplication by C T (CC T ) Gamma1 (which appears not to be well defined because (CC T ) is not invertible) is exactly equivalent to left multiplication by (C T C) ....

[Article contains additional citation context not shown here]

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

.... values of a recursive method used to compute (6) give the desired distributions (8) or (9) Filtering and smoothing have been extensively studied for continuous state models in the signal processing community, starting with the seminal works of Kalman (Kalman, 1960; Kalman and Bucy, 1961) and Rauch (Rauch, 1963; Rauch et al. 1965) although this literature is often not well known in the machine learning community. For the discrete state models much of the literature stems from the work of Baum and colleagues (Baum and Petrie, 1966; Baum and Eagon, 1967; Baum et al. 1970; Baum, 1972) on hidden Markov ....

....and control communities for decades. The emphasis has traditionally been on inference problems: the famous Discrete Kalman Filter (Kalman, 1960; Kalman and Bucy, 1961) gives an efficient recursive solution to the optimal filtering and likelihood computation problems, while the RTS recursions (Rauch, 1963; Rauch et al. 1965) solve the optimal smoothing problem. Learning of unknown model parameters was studied by Shumway and Stoffer (1982) C known) and by Ghahramani and Hinton (1996a) and Digalakis et al. 1993) all parameters unknown) Figure 1 illustrates this model, and the appendix gives ....

[Article contains additional citation context not shown here]

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

....sequence of inputs and outputs up to time T , where T t. The Kalman lter is used in the forward direction to compute the probability of X t given fY g t 1 and fUg t 1 . A similar set of backward recursions from T to t complete the computation by accounting for the observations after time t (Rauch, 1963). We will refer to the combined forward and backward recursions for smoothing as the Kalman smoothing recursions (also known as the RTS or Rauch Tung Streibel smoother) Finally, the goal of prediction is to compute the probability of future states and observations given observations upto time t. ....

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371-372.

....be expressed as in equation 10. The Kalman filter [113] is in fact such a model, and the associated algorithms allow to compute P (q t jx t 1 ; y t 1 ) in a forward recursion (thus solving the filtering problem) Similarly to Markov switching models, a backward recursion (the Rauch equations [114]) allows to compute the posterior probabilities P (q t jx T 1 ; y T 1 ) for T t (thus solving the smoothing problem) In the context of real time control and other applications where learning must be on line, numerical maximization of the likelihood can be performed recursively with a ....

H. Rauch, "Solutions to the linear smoothing problem," IEEE Transactions on Automatic Control, vol. 8, pp. 371--372, 1963.

....our use of EKS, to estimate just the hidden state as part of the E step of EM. For linear dynamical systems with Gaussian state evolution and observation noises, this conditional density is Gaussian and the recursive algorithm for computing its mean and covariance is known as Kalman smoothing [4, 8]. Kalman smoothing is directly analogous to the forward backward algorithm for computing the conditional hidden state distribution in a hidden Markov model, and is also a special case of the belief propagation algorithm. 5 For nonlinear systems this conditional density is in general ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

.... be seen as a continuous state analogue of the hidden Markov model (HMM; see Rabiner and Juang, 1986, for a review) The forward part of the forward backward algorithm from HMMs is computed by the well known Kalman filter in LDSs; similarly, the backward part is computed by using Rauch s recursion (Rauch, 1963). Together, these two recursions can be used to solve the problem of inferring the probabilities probabilities of the states given the observation sequence (known in engineering as the smoothing problem) These posterior probabilities form the basis of the E step of the EM algorithm. Finally, ....

Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

....model given the model parameters and a sequence of observations fY 1 ; Y t ; YT g. It consists of two parts: a forward recursion which uses the observations from Y 1 to Y t , known as the Kalman filter [29] and a backward recursion which uses the observations from Y T to Y t 1 [43]. 8 We have already seen that in order to compute the marginal probability of a variable in a Bayesian network one must take into account both the evidence above and below the variable. In fact, the Kalman smoother is simply a special case of the belief propagation algorithm we have already ....

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

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Rauch, H. E. (1963). Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372.

No context found.

H. E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

No context found.

H.E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

No context found.

H.E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

No context found.

H.E. Rauch. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8:371--372, 1963.

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