Cracking the Code: Transition Matrices & Markov Chains

Cracking the Code: Transition Matrices & Markov Chains

Table of Contents:

1. Introduction to Transition Matrices and Markov Chains
2. Definition of a Markov Chain
3. Key Concepts 3.1 Transition Diagram 3.2 Transition Matrix 3.3 Initial State Matrix 3.4 Finding Future States 3.5 Long-Term Steady State
4. Calculating State Changes
5. Finding Future States
6. Determining the Long-Term Steady State
7. Algebraic Technique for Finding the Steady State
8. Practice Questions

Introduction to Transition Matrices and Markov Chains

Transition matrices and Markov chains are crucial components of the AI HL course. In Topic 4: Statistics and Probability, specifically under the subtopic of probability, these concepts play a significant role. This article aims to provide a comprehensive understanding of transition matrices and Markov chains, their definitions, key concepts, and techniques for calculating future states and identifying the long-term steady state.

Definition of a Markov Chain

A Markov chain is a system that undergoes transitions from one state to the next Based on fixed probability rules. These rules are defined by a transition matrix, denoted as T. Markov chains can encompass various scenarios, such as population movement between cities, daily weather changes, or market share dynamics. To illustrate the concept, let's consider the changes in market share between two local cafes on a monthly basis.

Key Concepts

Transition diagrams, transition matrices, initial state matrices, finding future states, and determining the long-term steady state are five key concepts essential to understanding transition matrices and Markov chains.

3.1 Transition Diagram

A transition diagram visually represents the transitions between states in a Markov chain. Each state is represented as a node, and the transitions between states are depicted by arrows. In our example, the transition diagram showcases the monthly changes in market share between two local cafes.

3.2 Transition Matrix

A transition matrix, denoted as T, quantifies the probabilities of transitioning between states in a Markov chain. The columns of the matrix represent the Current states, while the rows represent the next states. The elements of the matrix denote the probabilities of transitioning from the current state to the next state. In our example, the transition matrix describes the changes in market share between the two cafes.

3.3 Initial State Matrix

The initial state matrix, denoted as S0, represents the starting distribution of market share between the two cafes. Assuming equal market share at the beginning, the initial state matrix ensures each cafe starts with 50% of the market.

3.4 Finding Future States

To determine the future state of the Markov chain after a specific period, we use a formula. The formula, given in the formula booklet, involves raising the transition matrix to the power of the desired period and multiplying it by the initial state matrix. For instance, calculating the market share after three months involves finding the value of S3 using the transition matrix raised to the power of 3 and the initial state matrix.

3.5 Long-Term Steady State

In a Markov chain, the long-term steady state represents the point where the market shares reach an equilibrium and no longer change significantly. Two methods can be employed to find the long-term steady state. One method involves calculating the future state for a large value of n, such as 100 or 500, to observe if the market shares stabilize. The Second method is to solve an algebraic equation, where the multiplication of the transition matrix by the long-term steady state matrix equals the long-term steady state matrix itself.

Calculating State Changes

Let's dive into the calculations involved in determining state changes in the market share of the two cafes. By using the transition matrix and the initial state matrix, we can accurately predict the changes in market share over time.

Finding Future States

Using the formula provided in the formula booklet, we can calculate the future state of the Markov chain after a specific period. By applying the transition matrix to the initial state matrix, we can determine the market share of each cafe.

Determining the Long-Term Steady State

Identifying the long-term steady state is critical to understanding the stability and equilibrium of the market share between the two cafes. The long-term steady state can be found either by calculating the future state for a significantly large value of n or by solving an algebraic equation involving the transition matrix and the long-term steady state matrix.

Algebraic Technique for Finding the Steady State

An alternate method for finding the long-term steady state involves solving an algebraic equation. By setting the multiplication of the transition matrix by the long-term steady state matrix equal to the long-term steady state matrix itself, we can determine the market share distribution that reaches a stable equilibrium.

Practice Questions

To enhance your understanding and proficiency in transition matrices and Markov chains, it is recommended to practice various questions related to these concepts. The practice questions will allow you to Apply the knowledge gained and reinforce your comprehension.