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3. Stationarity."

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Table of contents

Key takeawaysCollaboration platforms are essential to the new way of workingEmployees prefer engati over emailEmployees play a growing part in software purchasing decisionsThe future of work is collaborativeMethodologyA Hidden Markov model is a statistical model that can be utilized for the purpose of describing the evolution of observable events that depend on internal factors, which are not directly observable. It is essentially an augmentation of the Markov chain that includes observations.

A Hidden Markov Model is made up of two stochastic processes. These are: an invisible process of hidden states and a visible process of observable symbols.

The observed event is known as a symbol, while the invisible factor underlying the observation is known as a state.

Similar to the state transition of the Markov chain, Hidden Markov models also include observations of the state.

These can be partial observations where different states can be mapped to the same observation and noisy in that the same state can be stochastically mapped to different observations at different times.

The hidden states make up a Markov chain, and the probability distribution of the observed symbol is dependent on the underlying state.

A Hidden Markov Model (HMM) is also referred to as a doubly-embedded stochastic process.

Modeling observations in these two layers, with one layer being visible and the other layer being invisible can be very beneficial because a large number of real-world problems and applications involve the classification of raw observations into a number of categories, or class labels, which are more meaningful to us.

Markov chains are mathematical properties systems that jump from one state to another. They were named after Andrey Markov.

A Markov chain contains a state space. This is essentially a list of all the possible states. The transitions are determined by a probability distribution and satisfy the Markov property. The Markov property renders Markov processes memoryless. Because of that, they do not have the capability to produce context-dependent content because they cannot take into account the full chain of prior states.

The Markov chain also gives you the probability of hopping or transitioning from one state to any other state.

Real modelers, however, do not necessarily draw out Markov chains. Rather, they employ a transition matrix to tally the transition probabilities. Every state that is listed in the state space is included once as a row and again as a column.

Every cell in the column shows you the possibility of transitioning from the state represented in the row to the state represented in the column.

One of the main applications of Markov chains is the inclusion of real-world phenomena in computer simulations.

A Markov chain is essentially no different than the hidden part of a Hidden Markov Model. The biggest difference between a Markov chain and a Hidden Markov Model is that in a Hidden Markov Model, there is a matrix that is used to link observations to the states, while in a Markov chain, no observation is considered.

The three most important assumptions of Hidden Markov models are:

The current state of the unobserved node *H*t is solely dependent on the previous state of the unobserved variable.

The current state of the observed node *Ot *is completely dependent on the current state of the

The transition probabilities are independent of time.

The Hidden Markov Model is very widely used in the domain of engineering. It is also used in digital communication and has been utilized for decades in the process of speech recognition.

In speech recognition, we seek to predict the uttered word from a recorded speech signal. To achieve this, the speech recognizer attempts to identify the sequence of phonemes (states) that gave rise to the actual uttered sound (observations). Since the actual pronunciation can vary to an extremely large extent, the original phonemes (and ultimately, the uttered word) cannot be directly observed, and need to be predicted.

Hidden Markov Models are also used for the purpose of modeling biological sequences like proteins and DNA sequences.

Profile Hidden Markov Models or Profile-HMMs are Hidden Markov Models which have a particular architecture that is very useful for modeling sequence profiles. Profile Hidden Markov Models have a strictly linear left-to-right structure that does not comprise of any cycles.

They use three types of hidden layers repetitively: match states *Mk*, insert states *Ik*, and delete states *Dk* for the purpose of describing position-specific symbol frequencies, symbol insertions, and symbol deletions, respectively.

Since they are extremely convenient and effective at representing sequence profiles, Profile Hidden Markov Models are extensively used for the purpose of modeling and analyzing biological sequences.

When Profile Hidden Markov Models were first proposed and introduced, people quickly started using them to model the characteristics of a range of protein families, like globins, immunoglobulins, and kinases. They have proved themselves to be rather useful when it comes to many tasks like protein classification, motif detection, and identifying multiple sequence alignments.

However, even though Profile Hidden Markov Models have been extensively used for the purpose of representing sequence profiles, their application is not, in any way whatsoever, limited to modeling amino acid or nucleotide sequences.

Di Francesco et al. made use of Profile-HMMs to model sequences of protein secondary structure symbols: helix (H), strand (E), and coil (C).

Feature-based profile-HMMs were proposed in order to improve the performance of remote protein homology detection. Rather than emitting amino acids, the emissions of these Hidden Markov Models are based on `features' that capture the biochemical properties of the protein family of interest.

These features are extracted through the performance of a spectral analysis of a number of selected `amino acid indices and by making use of principal component analysis (PCA) to reduce the redundancy in the resulting signal

The Jumping Profile Hidden Markov Model (jpHMM) is a probabilistic generalization of the jumping-alignment approach, which is a strategy used to compare a sequence with a multiple alignment, where the sequence is not aligned to the alignment as a whole, but it is able to `jump' between the sequences that constitute the alignment. Thus, different parts of the sequence can be aligned to different sequences in the given alignment.

A jpHMM makes use of multiple match states for every column to represent different sequence subtypes.

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