Markov Chain in Python Tutorial

Overview

Has it ever crossed your mind how expert meteorologists make a precise prediction of the weather or how Google ranks different web pages? How they make the fascinating python applications in real world. These calculations are complex and involve several variables that are dynamic and can be solved using probability estimates.

Here lies the idea of Markov Chains; there are individual states (say, the weather conditions) where each state can randomly change into other states (rainy day can change into the sunny day), and these changes or transitions are probability-based. This article gives a brief introduction to the concept of Markov Chains and how Python Markov Chain can be utilized to code Markov Chain models in Python to solve real-world problems.

Content Overview

  • A brief introduction to the concepts of Markov Chain and Markov Property
  • Mathematical and graphical expression of Markov Chain
  • Python Markov Chain – coding Markov Chain examples in Python

Introduction to Markov Chain

To use Python Markov Chain for solving practical problems, it is essential to grasp the concept of Markov Chains. In 1906, Russian mathematician Andrei Markov gave the definition of a Markov Chain – a stochastic process consisting of random variables that transition from one particular state to the next, and these transitions are based on specific assumptions and probabilistic rules.

A fundamental mathematical property called the Markov Property is the basis of the transitions of the random variables. In other words, a Markov Chain is a series of variables X1, X2, X3,…that fulfill the Markov Property.

Principle of Markov Chain – Markov Property

A Markov Chain is based on the Markov Property. The theory of discrete-time Markov Property states that the probability of a random system changing from one particular state to the next transition state depends only on the present state and time and is independent of the preceding states.

The fact that the probable future state of a random process is independent of the sequence of states that existed before it makes the Markov Chain a memory-less process that depends only on the current state of the variable.

Read: Built in Data Structures in Python

The mathematical expression of the Markov Chain

In terms of a probability distribution, assume a system at time instance ‘n.’ Applying the principle of Markov property, the conditional distribution of the states at the following time instance, n+1, is independent of the states of the system at time instances 1, 2, …, n-1. 

Graphical representation of Markov Chain

Directed graphs are often used to represent a Markov Chain. In the directed graphs, the nodes indicate different likely states of the random variables while the edges denote the probability of the system moving from one state to another in the next time instance. To understand the representation, let us take the example of predicting the weather. Assume that the random variable is ‘weather,’ and it has three possible states viz. Weather = {sunny, rainy, snowy}. The Markov Chain for this scenario can be represented as:

markov chain python

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In the graphical representation shown above, say the current observed state of the random variable is sunny. The probability of the random variable taking the value sunny at the next time instance is 0.8. It can also take the value snowy with a probability of 0.01, or rainy with a probability of 0.19. An important thing to note here is that the probability values existing in a state will always sum up to 1.

Coding a Markov Chain in Python

To better understand Python Markov Chain, let us go through an instance where an example of Markov Chain is coded in Python. While solving problems in the real world, it is common practice to use a library that encodes Markov Chains efficiently. However, coding Markov Chain in Python is an excellent way to get started on Markov Chain analysis and simulation. Hence comes the utility of Python Markov Chain. Let us see how the example of weather prediction given in the previous section can be coded in Python. Begin by defining a simple class:

 

Having defined the MarkovChain class, let us try coding the weather prediction example as a representation of how Python Markov Chain works.

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Read: Operators in Python

Parameterising Markov Chains using Transition Matrix

In the previous section, the Python code parameterised the Markov Chain using a dictionary that contained the probability values of all the likely state transitions. An alternative way of representing the transition probabilities is using a transition matrix, which is a standard, compact, and tabular representation of a Markov Chain.

In situations where there are hundreds of states, the use of the Transition Matrix is more efficient than a dictionary implementation. The Markov Chain class is modified as follows for it to accept a transition matrix:

 

The dictionary implementation was looping over the states names. However, in case of a Transition Matrix, the probability values in the next_state method can be obtained by using NumPy indexing:

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Conclusion

Markov Chains are an essential mathematical tool that helps to simplify the prediction of the future state of complex stochastic processes; it solely depends on the current state of the process and views the future as independent of the past. Utilising the Markov Property, Python Markov Chain coding is an efficient way to solve practical problems that involve complex systems and dynamic variables.

Be it weather forecasting, credit rating, or typing word prediction on your mobile phone, Markov Chains have far-fetched applications in a wide variety of disciplines. Depending on the nature of the parameters and the application, there are different concepts of Markov Chains. Python Markov Chain is a logical and efficient way to implement Markov Chains by coding them in Python.

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