Change Point Detection Time Series

 

Change Point Detection Methods

Kernel Change Point Detection:

Kernel change point detection method detects changes in the distribution of the data, not just changes in the mean or variance.

Kernel Method is utilized to map the data into a high-dimensional feature space, where changes are more easily detectable. This approach uses the Maximum Mean Discrepancy (MMD) to measure the difference between the distributions of segments of the time series.

Steps:

1-      Data and Kernel Function: Consider a univariate time series {x1,x2,…,xn} We start by choosing a kernel function k(x,y) to measure similarity between points.

2-      Construction of Kernel Matrix: kernel matrix K is constructed, where each element K_ij=k(xi,xj)

For the linear kernel, this is: Kij=xi⋅xj​ (X^TX)

 

3-      Maximum Mean Discrepancy (MMD):

 

MMD measures how different two groups of data are by comparing the average of all pairwise similarities within each group and between the groups or compares two distributions to see if they are different.

MMD is used to measure the difference between the distributions before and after a candidate change point t.

For each candidate change point t

In the above equation:

-          The first term measures the similarity within the first segment.

-          The second term measures the similarity within the second segment.

-          The third term measures the similarity between the two segments.

 

4-      To detect the change point, we compute the MMD values are computed for all possible change points t and choose the one that maximizes the MMD value:

Excel Example :https://docs.google.com/spreadsheets/d/1IdC-ss1VjaL2QVQdABNwuIPfRphDtlZi/edit?usp=sharing&ouid=115594792889982302405&rtpof=true&sd=true