Transfer probability matrix)

What is transfer probability matrix?

Transfer probability matrix: each element in the matrix is non-negative, and the sum of elements in each row is equal to 1. Each element is represented by probability. Under certain conditions, it is transferred to each other. Therefore, it is called the transfer probability matrix. For example, when used for market decision-making, the elements in the matrix are the retention, acquisition, or loss probability of the market or customer.P(K) Indicates the K-step transfer probability matrix.

 

Features of transition probability matrix

The transfer probability matrix has the following features:

①, 0 ≤PIJ ≤ 1

②, That is, the sum of the transfer probabilities of each row in the matrix is equal to 1.

 

Transfer Probability and transfer probability matrix [1]

Assume that a university has 10 thousand students, each of whom uses one toothpaste every month, and only uses one of the "China" and "black sister" toothpaste. According to this month (December) survey, 3000 people use black sister toothpaste and 7000 use Chinese toothpaste. According to the survey, 3000 of the 60% people who use black sister toothpaste will continue to use black sister toothpaste next month, and 40% will switch to Chinese toothpaste. Among the 7000 people who use Chinese toothpaste, 70% of people will continue to use Chinese toothpaste next month, and 30% will use black sister toothpaste. Accordingly, the statistical table shown in table-1 can be obtained.

Table-1 transfer probability between two toothpaste types

Intended Use	Heimei toothpaste	Chinese toothpaste
Current Use
Heimei toothpaste	60%	40%
Chinese toothpaste	30%	70%

The four probabilities in the above table are called the transition probability of the state, and these four transfer probabilities constitute a matrix.

　　

It is called the transfer probability matrix. It can be seen that one feature of the transfer probability matrix is that the sum of elements in each row is 1. In this example, the economic significance is that the sum of the percentage of people who use toothpaste of various brands in the future is 1.

2. Use the transfer probability matrix to predict market share changes

With the transfer probability matrix, we can predict the number of people using black sister toothpaste and Chinese toothpaste in the next month (March). The calculation process is as follows:

　　

That is, the number of people who use black sister toothpaste in March will be 3900, and the number of people who use Chinese toothpaste will be 6100.

Assuming that the transfer probability matrix remains unchanged, we can continue to predict the situation by January as follows:

　　

　　

　　

This is called the two-step transfer matrix, that is, the transfer from January to January. The two-step transfer probability matrix is the square of the one-step transfer probability matrix. Generally, K-step transfer probability matrix

This is exactly the k power of the step-by-step transfer probability matrix. It can be proved that in the K-step transfer probability matrix, all the elements of each row are also 1.