Hidden Markov Model & It's Application in Python

Hidden Markov
Model & It's
Application.

Background
Three Elements of HMM
HMM Example
Application in Python
Presentation Outline

The Hidden Markov Model is dual Stochastic Process, where one
of the underlying process is Hidden.
The hidden process is a Markov chain moving from one state to
another but cannot be observed directly.
The other process is observable but its movement depends upon
hidden state.
Hidden Markov model is a branch of machine learning. it’s useful
in solving problem related to sequence.
Hidden Markov Model

HMM Example
Markov chain is discrete stochastic process, where probability of event occurring
only depend upon immediate previous event.
A Short History
Primarily it's use where there are sequence of event which take place. One of the
popular use case is in weather forecasting.
Popular Use Case
Let's analyse two states example of weather, here two states means it can be sunny
or rainy.
Two States Example

HMM Example : Weather Data
Sunny or Rainy Observations of 15 days.
Transition Probabilities
7/10 = 0.7
3/10 = 0.3
2/5 = 0.4
3/5 = 0.6
0.3
0.4
0.7 0.6

HMM Example : Weather Data
0.3
0.4
0.7 0.6
The Application of HMM would be based on real action, let's say there are three
actions person can do depend upon weather like "walk", "Shop", "Travel"
0.6 0.4
Initial Probability : Pi
Walk Shop Travel
0.1
0.4
0.1
0.3
0.6 0.5

HMM Example : Terminology & Calculations
X0 = Initial Probability Distribution :Probability that chain will start at some
state
X = Hidden State (Rain / Cloud)
Y = Observables (Walk, Shop, Travel)
a = Transition Probability (Moving from Rain to Sunny and vice versa).
b = Emmission Probability (Observation being generated from State)
Person goes for walk, first day it was rainy and second day it was sunny.
P((walk,walk),(rainy, sunny)) :
P((walk|rainy) * P(walk|sunny) * P(sunny|rainy) * P(rainy)
0.1 * 0.6 * 0.4 * 0.4 = 0.0096

HMM: Three Elements
In order to model HMM we need to run through three algorithms.
Forward-Backward Algorithm: It's evaluation phase where computation of
probability of Observation Sequence.
Baum-Welch Algorithm: It's learning phase for determination of parameter of
model.
Viterbi Algorithm: Decoding the most probable state sequence.

HMM: Three Elements : Forward-Backward Algorithm
The Forward-Backward algorithm computes the posterior (updated probability
of an event occurring after taking into consideration new information.)
marginals of all hidden state variables.
The Algorithm uses two passes, the first one goes forward in time and the
second pass goes backward.
The First pass computes set of probabilities which provides probability of
ending up in particular state.
The Second pass computes a set of backward probabilities which provide the
probability of observing remaining observation given any starting point.
Basically this algorithm is used to find the most likely state for any point in
time.

HMM: Three Elements : Baum-Welch Algorithm
This algorithm deals with unknown parameters of a hidden markov model.
It's case of E-M Algorithm (Expectation Maximization) which is a method to find
maximum estimation.
The "E" part of Expectation re-estimate the pi given current HMM parameter.
The "M" part of Maximization, re-estimating HMM parameter given current pi
state.
In Conclusion, the baum-welch algorithm attempt to find model that assigns the
training data the highest likelihood.

HMM: Three Elements : Viterbi Algorithm
The most useful algorithm when one wants to calculate the most likely path
through the state transition.
The observation made by this algorithm is that at any state at time T, there is
only one most likely path to that state.
Using this algorithm we can find the most likely sequence of hidden states
given the sequence of observations.

HMM Application in Python
So far we have observed how this entire machine learning algorithm works.
The Example which we have gone through (Weather Forecasting) is of discrete
in nature.
Stock returns are continuous in nature and random noise is already present
there.
In order to model random noise we need to use Gaussian model and it takes
two input mean and variance.
Observed set of data is of NIFTY (National Stock Exchange of India) Index.
Data set is for last 10 years (2011-2020) data of 15 minutes interval.

HMM Application in Python
Code here is very simple unoptimized code for illustration perspective.

HMM Final Prediction and Actual Values.

Acknowledgment:
Various paper and webpages helped to complete this project.
https://medium.com/analytics-vidhya/hidden-markov-model-a-
statespace-probabilistic-forecasting-approach-in-quantitative-finance-
df308e259856
https://towardsdatascience.com/probability-learning-vi-hidden-
markov-models-fab5c1f0a31d
Experimental Mathematics: hidden Markov models.pdf
https://rubikscode.net/2018/10/29/stock-price-prediction-using-
hidden-markov-model/

Hidden Markov Model & It's Application in Python

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