Change point problems involve identifying shifts or discontinuities in a given data set. These shifts can occur in various forms, such as a sudden change in the mean or variance of the data, or a change in the underlying distribution of the data.

One example of a change point problem is identifying shifts in the mean of a time series data set. For instance, in stock market data, there may be sudden changes in the average daily price of a stock due to market events or shifts in investor sentiment. Identifying these shifts can provide valuable insight into the underlying drivers of the stock’s performance, and can aid in making informed investment decisions.

To solve this problem, statistical techniques such as the CUSUM (Cumulative Sum) method can be used. This method involves calculating the cumulative sum of the differences between the observed data and the expected data, based on a given null hypothesis. If the cumulative sum exceeds a certain threshold, this indicates a shift in the mean of the data and a change point is identified.

Another example of a change point problem is identifying shifts in the variance of a data set. In manufacturing data, for instance, there may be sudden changes in the variability of product defects due to changes in production processes or materials. Identifying these shifts can help identify underlying causes of the changes in variance, and can aid in implementing corrective measures to improve product quality.

To solve this problem, statistical techniques such as the Generalized Likelihood Ratio Test (GLRT) can be used. This method involves comparing the likelihood of the observed data under two different hypotheses: one where the variance remains constant, and one where the variance changes at a given point in time. If the likelihood under the second hypothesis is significantly higher, this indicates a shift in the variance of the data and a change point is identified.

In both of these examples, change point analysis can provide valuable insights into shifts or discontinuities in a given data set. This can aid in decision making, and can help identify underlying causes of changes in the data.