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Probabilistic parameters of a hidden Markov model (example) x — states y — possible observations a — state transition probabilities b — output probabilities
Probabilistic parameters of a hidden Markov model (example)
x — states
y — possible observations
a — state transition probabilities
b — output probabilities

A hidden Markov model (HMM) is a statistical model in which the system being modeled is assumed to be a Markov process with unknown parameters, and the challenge is to determine the hidden parameters from the observable parameters. Statistical models are used in Applied statistics. Three notions are sufficient to describe all statistical models A Markov process, named after the Russian mathematician Andrey Markov, is a mathematical model for the random evolution of a memoryless system In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical The extracted model parameters can then be used to perform further analysis, for example for pattern recognition applications. Pattern recognition is a sub-topic of Machine learning. It is "the act of taking in raw data and taking an action based on the category of the data" An HMM can be considered as the simplest dynamic Bayesian network. A dynamic Bayesian network is a Bayesian network that represents sequences of variables

In a regular Markov model, the state is directly visible to the observer, and therefore the state transition probabilities are the only parameters. In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. In a hidden Markov model, the state is not directly visible, but variables influenced by the state are visible. Each state has a probability distribution over the possible output tokens. Therefore the sequence of tokens generated by an HMM gives some information about the sequence of states.

Hidden Markov models are especially known for their application in temporal pattern recognition such as speech, handwriting, gesture recognition, musical score following, partial discharges and bioinformatics. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Speech recognition (also known as automatic speech recognition or computer speech recognition) converts spoken words to machine-readable input (for example to keypresses Handwriting recognition is the ability of a computer to receive and interpret intelligible Handwritten input such as pendrives digital cameras and other devices Gesture Recognition is a topic in Computer science and Language technology with the goal of interpreting human Gestures via mathematical Algorithms Sheet music is a hand-written or printed form of Musical notation; like its analogs -- books pamphlets etc In Electrical engineering, a partial discharge (PD is a localised Dielectric breakdown of a small portion of a solid or liquid Electrical insulation system Bioinformatics is the application of information technology to the field of molecular biology

Contents

Architecture of a hidden Markov model

The diagram below shows the general architecture of an instantiated HMM. Each oval shape represents a random variable that can adopt a number of values. The random variable x(t) is the hidden state at time t (with the model from the above diagram, x(t) \in \{x_1, x_2, x_3\}). The random variable y(t) is the observation at time t (y(t) \in \{y_1, y_2, y_3, y_4\}). The arrows in the diagram (often called a trellis diagram) denote conditional dependencies. A trellis is a graph of which the nodes are ordered into vertical slices ( time) and each node at each time is connected to (at least one node at an earlier and (at

From the diagram, it is clear that the value of the hidden variable x(t) (at time t) only depends on the value of the hidden variable x(t − 1) (at time t − 1). This is called the Markov property. A Markov process, named after the Russian mathematician Andrey Markov, is a mathematical model for the random evolution of a memoryless system Similarly, the value of the observed variable y(t) only depends on the value of the hidden variable x(t) (both at time t).

Temporal evolution of a hidden Markov model

Probability of an observed sequence

The probability of observing a sequence Y=y(0), y(1),\dots,y(L-1) of length L is given by

P(Y)=\sum_{X}P(Y\mid X)P(X),

where the sum runs over all possible hidden node sequences X=x(0), x(1), \dots, x(L-1). Brute force calculation of P(Y) is intractable for most real-life problems, as the number of possible hidden node sequences is typically extremely high. The calculation can however be sped up enormously using the forward algorithm [1] or the equivalent backward algorithm. The Viterbi algorithm is a Dynamic programming Algorithm for finding the most likely sequence of hidden states &ndash called the Viterbi path

Using hidden Markov models

There are three canonical problems associated with HMM:

A concrete example

Assume you have a friend who lives far away and to whom you talk daily over the telephone about what he did that day. Your friend is only interested in three activities: walking in the park, shopping, and cleaning his apartment. The choice of what to do is determined exclusively by the weather on a given day. You have no definite information about the weather where your friend lives, but you know general trends. Based on what he tells you he did each day, you try to guess what the weather must have been like.

You believe that the weather operates as a discrete Markov chain. In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. There are two states, "Rainy" and "Sunny", but you cannot observe them directly, that is, they are hidden from you. On each day, there is a certain chance that your friend will perform one of the following activities, depending on the weather: "walk", "shop", or "clean". Since your friend tells you about his activities, those are the observations. The entire system is that of a hidden Markov model (HMM).

You know the general weather trends in the area, and what your friend likes to do on average. In other words, the parameters of the HMM are known. You can write them down in the Python programming language:

states = ('Rainy', 'Sunny') observations = ('walk', 'shop', 'clean') start_probability = {'Rainy': 0. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability 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},   }

In this piece of code, start_probability represents your belief about which state the HMM is in when your friend first calls you (all you know is that it tends to be rainy on average). The particular probability distribution used here is not the equilibrium one, which is (given the transition probabilities) actually approximately {'Rainy': 0. 571, 'Sunny': 0. 429}. The transition_probability represents the change of the weather in the underlying Markov chain. In this example, there is only a 30% chance that tomorrow will be sunny if today is rainy. The emission_probability represents how likely your friend is to perform a certain activity on each day. If it is rainy, there is a 50% chance that he is cleaning his apartment; if it is sunny, there is a 60% chance that he is outside for a walk.

This example is further elaborated in the Viterbi algorithm page. The Viterbi algorithm is a Dynamic programming Algorithm for finding the most likely sequence of hidden states &ndash called the Viterbi path

Applications of hidden Markov models

History

Hidden Markov Models were first described in a series of statistical papers by Leonard E. Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for Speech recognition (also known as automatic speech recognition or computer speech recognition) converts spoken words to machine-readable input (for example to keypresses Machine translation, sometimes referred to by the abbreviation In Electrical engineering, a partial discharge (PD is a localised Dielectric breakdown of a small portion of a solid or liquid Electrical insulation system Baum and other authors in the second half of the 1960s. The 1960s decade refers to the years from the beginning of 1960 to the end of 1969 One of the first applications of HMMs was speech recognition, starting in the mid-1970s. Speech recognition (also known as automatic speech recognition or computer speech recognition) converts spoken words to machine-readable input (for example to keypresses This article is about the Decade 1970-1979 For the Year 1970 see 1970. [2]

In the second half of the 1980s, HMMs began to be applied to the analysis of biological sequences, in particular DNA. The 1980s was the decade spanning from January 1 1980 to December 31 1989. Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known Since then, they have become ubiquitous in the field of bioinformatics. Bioinformatics is the application of information technology to the field of molecular biology [3]

See also

Notes

  1. ^ Rabiner, p. Bayesian inference is Statistical inference in which evidence or observations are used to update or to newly infer the Probability that a hypothesis may be true Estimation theory is a branch of Statistics and Signal processing that deals with estimating the values of parameters based on measured/empirical data The Hierarchical hidden Markov model (HHMM is a Statistical model derived from the Hidden Markov model (HMM The layered Hidden Markov model (LHMM is a Statistical model derived from the hidden Markov model (HMM A hidden semi-Markov model (HSMM is a statistical model with the same structure as a Hidden Markov model except that the unobservable process is semi-Markov rather Variable-order Markov (VOM models are an important class of models that extend the well known Markov chain models Sequential dynamical systems (SDSs are a class of Discrete Dynamical systems which generalize many aspects of systems such as Cellular automata, and provide A conditional random field (CRF is a type of discriminative probabilistic model most often used for the labeling or Parsing of sequential data such as 262
  2. ^ Rabiner, p. 258
  3. ^ Durbin et al.

References

The path-counting algorithm, an alternative to the Baum-Welch algorithm:

External links

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