By Andrew M. Fraser
This article offers an advent to hidden Markov types (HMMs) for the dynamical platforms neighborhood. it's a beneficial textual content for 3rd or fourth yr undergraduates learning engineering, arithmetic, or technological know-how that comes with paintings in likelihood, linear algebra and differential equations. The ebook offers algorithms for utilizing HMMs, and it explains the derivation of these algorithms. It provides Kalman filtering because the extension to a continuing nation area of a uncomplicated HMM set of rules. The e-book concludes with an program to biomedical signs. this article is detailed for offering crucial introductory fabric in addition to proposing adequate of the idea in the back of the elemental algorithms in order that the reader can use it as a consultant to constructing their very own variations.
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Extra info for Hidden Markov models and dynamical systems
The backward algorithm starts at the final time T with β set to one6 for each state, β(s, T ) = 1 for all s ∈ S, and solves for β at earlier times with the following recursion that goes backwards through time: β(˜s , t − 1) = β(s, t) s∈S PY (t)|S(t) (y(t)|s) · PS(t)|S(t−1) (s|˜s ) . 12) Note that γ (t) ≡ P y(t)|y1t−1 is calculated by the forward algorithm and that the terms in the numerator are model parameters. 13) P ytT |y1t−1 s∈S PYtT ,S(t)|S(t−1) ytT , s|˜s P ytT |y1t−1 . 11): T |y1t · P (y(t)|s) · P (s|˜s ) .
47 ✐ ✐ ✐ ✐ ✐ ✐ ✐ 48 main 2008/9/3 page 48 ✐ Chapter 3. 1. Histogram of Tang’s laser measurements. Even though neither y = 5 nor y = 93 occurs in y1600 , it is more plausible that y = 93 would occur in future measurements because of what happens in the neighborhood. Discarding the numerical significance of the bin labels would preclude such an observation. 1 Gaussian Observations Independent Scalar Observations A simple model for continuously distributed measurements is an HMM with an independent scalar Gaussian observation model associated with each state.
45)). 2 Singularities of the Likelihood Function and Regularization Running five iterations of the Baum–Welch algorithm on the observation sequence in Fig. 2(c) starting with the model in Fig. 3(a) produces the model in Fig. 3(b) in which the variance of the observations produced by the second state looks suspiciously small. In fact with additional iterations of the Baum–Welch algorithm that variance continues to shrink, and after the seventh iteration, the code stops with a floating point exception.
Hidden Markov models and dynamical systems by Andrew M. Fraser