Hidden Markov ModelsA Comparative Analysis of implementing algorithms

ByUmair, Mazhar, Shaheer

To Sir Nadeem and Fellow students

@MAJU

SCOPE• We will try to cover following briefly:• Markov Model and its elements• Hidden Markov Model, and why it is hidden.• HMM Examples• Forward to resolve problem in any HMM.

Introduction• Markov model is named after Russian mathematician Andrew Markov

(1856-1922). • A Markov model is stochastic, because it can model the system which

is represented by randomly changing variables. (Weather, Word types in any human language sentence, Voice Features, In short any type of signals)

Example # 1. Coin Toss Process• In any coin toss process, output is either Head or Tale.

Example # 2: Coin Toss by Gambler in a Room• Gambler has some coins. • He only informs about the output (Head / Tale).• Never tell you which coin he used.

The first problem: what is the most likelihood of an observed sequence (H-T-H)?

Example # 3: Ice cream and Weather• Now we will consider an example of a person who eats ice cream

daily. And based upon his eating ice cream habit, we want to predict the weather either it was hot or cold on that day.• Ice cream eating related to weather.

• Chances of Weather today based upon yesterday.

  Cold Hot

Ice creams

1 0.7 0.1

2 0.2 0.2

3 0.1 0.7

  Cold Hot

Cold 0.8 0.2

Hot 0.2 0.8

Day # Ice cream count

1 2

2 3

3 3

Applying Forward Algorithm on example 3• We denote partial probability as (α), and:• α (day x) = P(observation at day-x| hidden state at day-x ) * P(all

paths to hidden state on day-x))• We will sum all the partial probabilities for state path to solve the

problem.

Complete Implementation of forward algorithm for 3 days.

  Cold Hot

Ice creams

1 0.7 0.1

2 0.2 0.2

3 0.1 0.7

  Cold Hot

Cold 0.8 0.1

Hot 0.1 0.8α (day x) = P(observation at day-x| hidden state at day-x ) * P(all paths to hidden state on day-x))