Hidden Markov Model: Fundamentals and Applications

By Fouad Sabry

What Is Hidden Markov Model

A hidden Markov model, often known as an HMM, is a type of statistical Markov model. In an HMM, the system being represented is considered to be a Markov process, which we will refer to as it, with states that cannot be observed (thus the name "hidden"). In order to fulfill one of the requirements for the definition of HMM, there must be a measurable process whose results are "influenced" by those of another process in a certain way. Since it is not possible to directly see, the objective here is to learn about via observing.  HMM contains the additional criterion that the result of an event that occurs at a certain time must be "influenced" solely by the outcome of an event that occurs at that time, and that the outcomes of an event that occurs at and at must be conditionally independent of at provided that it occurs at a particular time.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Hidden Markov model

Chapter 2: Markov chain

Chapter 3: Viterbi algorithm

Chapter 4: Expectation-maximization algorithm

Chapter 5: Baum-Welch algorithm

Chapter 6: Metropolis-Hastings algorithm

Chapter 7: Bayesian network

Chapter 8: Gibbs sampling

Chapter 9: Mixture model

Chapter 10: Forward algorithm

(II) Answering the public top questions about hidden markov model.

(III) Real world examples for the usage of hidden markov model in many fields.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of hidden markov model.

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• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jul 1, 2023

Book preview

Chapter 4: Hidden Markov model

A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it X — with unobservable (hidden) states.

A component of this definition, HMM requires that there be an observable process Y whose outcomes are influenced by the outcomes of X in a known way.

Since X cannot be observed directly, the goal is to learn about X by observing {\displaystyle Y.} HMM has an additional requirement that the outcome of Y at time {\displaystyle t=t_{0}} must be influenced exclusively by the outcome of X at {\displaystyle t=t_{0}} and that the outcomes of X and Y at {\displaystyle t

In addition to their use in speech and language processing, hidden Markov models have been used in the fields of thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory, and pattern recognition, Let X_{n} and Y_{n} be discrete-time stochastic processes and n\geq 1 .

The pair {\displaystyle (X_{n},Y_{n})} is a hidden Markov model if

X_{n} is a Markov process whose behavior is not directly observable (hidden);

{\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )},}

for every {\displaystyle n\geq 1,} {\displaystyle x_{1},\ldots ,x_{n},} and every Borel set A .

Let X_{t} and Y_{t} be continuous-time stochastic processes.

The pair (X_{t},Y_{t}) is a hidden Markov model if

X_{t} is a Markov process whose behavior is not directly observable (hidden);

{\displaystyle \operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid \{X_{t}\in B_{t}\}_{t\leq t_{0}})=\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid X_{t_{0}}\in B_{t_{0}})}

, for every {\displaystyle t_{0},} every Borel set A, and every family of Borel sets {\displaystyle \{B_{t}\}_{t\leq t_{0}}.}

The states of the process X_{n} (resp.

{\displaystyle X_{t})} are called hidden states, and {\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\mid X_{n}=x_{n}{\bigr )}} (resp.

{\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{t}\in A\mid X_{t}\in B_{t}{\bigr )})} is called emission probability or output probability.

A hidden Markov process, in its discrete version, may be thought of as an extension of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). Take this scenario, in which a genie is hidden away in an impenetrable chamber. There are urns labeled X1, X2, X3,... throughout the room, and within each urn is a collection of balls with the labels y1, y2, y3,... The genie selects an urn from the room and pulls a ball at random. The ball is then placed on a conveyor line, so the viewer can see where each ball came from but not the urn it came from. A random number plus the urn selected for the (n 1)th ball are all the genie needs to choose which urn to use for the nth ball. A Markov process describes a situation in which the next urn selected does not rely on the selections that came before it. Its top half is seen in Figure 1.

This configuration is known as a hidden Markov process since only the sequence of labeled balls can be viewed, and not the Markov process itself. Balls y1, y2, y3, and y4 may be drawn at each state, as indicated in the bottom portion of the picture in Figure 1. It is impossible for a spectator to tell from which urn (i.e., at which state) the genie has pulled the third ball on the conveyor belt, even if the observer knows the composition of the urns and has just watched a series of three balls, such as y1, y2, and y3. Other details, such as the probability that the third ball originated from each urn, are, nonetheless, discernible to the spectator.

Take Alice and Bob, for example, two friends who live far away but yet manage to communicate every day about their days over the phone. Walking in the park, going to the mall, and tidying up his flat are the only things that fascinate Bob. The weather forecast for that day is the only factor in deciding what to do. Alice doesn't know the weather for sure, but she can predict it based on historical patterns. Alice attempts to infer the weather from Bob's daily descriptions in order to prepare for their outings.

According to Alice, the weather is governed by a series of discrete Markov processes. She is unable to immediately see the existence of the two conditions, Rainy and Sunny, since they are obscured from her view. Depending on the day's forecast, Bob may go for a stroll, go shopping, or clean the house. Bob's actions are observed because he reports them to Alice. Everything operates like a covert Markov model (HMM).

Alice is familiar with the regular weather patterns in the region and Bob's typical pastimes. That is to say, we are familiar with the HMM's parameters. In Python, they look like this::

states = (Rainy, Sunny)

observations = (walk, shop, clean)

start_probability = {Rainy: 0.6, Sunny: 0.4}

transition_probability = {

Rainy: {Rainy: 0.7, Sunny: 0.3}, Sunny: {Rainy: 0.4, Sunny: 0.6}, }

emission_probability = {

Rainy: {walk: 0.1, shop: 0.4, clean: 0.5}, Sunny: {walk: 0.6, shop: 0.3, clean: 0.1},}

Here, start probability is Alice's estimation of the HMM's state at the time Bob dials her number (all she knows is that it tends to be rainy on average). Given the transition probabilities, an equilibrium probability distribution would look something like Rainy: 0.57, Sunny: 0.43, however this is not the case here. If the underlying Markov chain experiences a weather change, this would be reflected in the transition probability variable. If it rains today, there is only a 30% probability that the sun will shine tomorrow in this scenario. Bob's daily likelihood of engaging in a certain behavior is reflected in the emission probability. If it's raining, there's a 50% chance he'll be tidying up at home, but if the sun is shining, there's a 60% chance he'll be taking a stroll.

Graphical representation of the given HMM

The page devoted to the Viterbi method provides further details on a comparable case.

The following schematic depicts a typical HMM implementation.

There are many possible outcomes since each oval represents a random variable.

The following diagram is a model, where the random variable x(t) represents the hidden state at time t, x(t) ∈ { x1, x2, x3 }).

The random variable y(t) is the observation at time t (with y(t) ∈ { y1, y2, y3, y4 }).

Conditional dependencies are shown by the arrows in the figure.

The conditional probability distribution of the hidden variable x(t) at time t, given the values of the hidden variable x at all times, is shown to rely solely on the value of the hidden variable x(t 1); the values at time t 2 and previously have no bearing on the conditional probability distribution. The Markov property describes this phenomenon. In a similar vein, y(t) can only be determined through x(t), a hidden variable (both at time t).

The typical hidden Markov model is taken into account here, the concealed variables have a discrete state space, whereas the observables themselves may be either categorical (coming from a predetermined list of categories) or continuous (typically from a Gaussian distribution).

A hidden Markov model's parameters may be classified as either, probability of both transition and emission (also known as output probabilities).

The transition probabilities control the way the hidden state at time t is chosen given the hidden state at time t-1 .

It is presumed that there are N potential states concealed from view, categorical distribution modelled.

(For other options, see the Extensions section.) This implies that for each of the N states a hidden variable may be in at time t, there are N!, there is a transition probability from this state to each of the N possible states of the hidden variable at time t+1 , for a total of N^{2} transition probabilities.

Please be aware that the total of all transition probabilities from any condition must equal 1.

Thus, the N\times N matrix of transition probabilities is a Markov matrix.

Since the probabilities of each transition can be calculated if we know the probabilities of the others,, there are a total of N(N-1) transition parameters.

In addition, to account for the N conceivable states, Given the state of the hidden variable at any given moment, there is an associated set of emission probabilities that governs the distribution of the observed variable at that time.

The number of items in this collection is proportional to the kind of observed variable.

For example, if there are M distinct values for the observable variable, controlled by a system of discrete classes, there will be M-1 separate parameters, for a total of N(M-1) emission parameters over all hidden states.

To the contrary, When the observed variable is a vector in M dimensions and follows some arbitrary multivariate Gaussian distribution, there will be M parameters controlling the means and {\frac {M(M+1)}{2}} parameters controlling the covariance matrix, for a total of

N\left(M+{\frac {M(M+1)}{2}}\right)={\frac {NM(M+3)}{2}}=O(NM^{2})

emission parameters.

(If this happens), Unless M is quite tiny,, Covariances between components of the observation vector should be more usefully constrained, e.g.

by supposing that nothing is reliant on anything else, or less limiting, rely on nothing other than their immediate neighbors)

Temporal evolution of a hidden Markov model

The following are some of the inference issues related to hidden Markov models:.

The goal is to find the most efficient approach to calculate the probability of a certain output sequence given the model's parameters. Calculating this entails adding up every potential state sequence:

probabilities of seeing certain patterns

Y=y(0),y(1),\dots ,y(L-1)\,

for any length L:

P(Y)=\sum _{X}P(Y\mid X)P(X),\,

where the total evaluates to true for every and all sequences of hidden nodes

X=x(0),x(1),\dots ,x(L-1).\,

The forward method, which is based on the principles of dynamic programming, may be effectively used to solve this issue as well.

Several tasks in this category inquire as to the likelihood that one or more latent variables are present, given the model's parameters and a sequence of observations y(1),\dots ,y(t).

The objective is to calculate, in light of the model's parameters and a string of data, Statistical dispersion across latent states, with the final latent variable, i.e.

to compute P(x(t)\ |\ y(1),\dots ,y(t)) .

This activity is often used when a process's sequence of latent variables is seen