Change Point Problems
- Detects points in data where statistical properties change (mean, variance, distribution).
- Common methods include CUSUM for mean shifts and the Generalized Likelihood Ratio Test (GLRT) for variance shifts.
- Useful for diagnosing causes of change and informing decisions (e.g., investments or quality control).
Definition
Section titled “Definition”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.
Explanation
Section titled “Explanation”Change point analysis focuses on locating times or indices where the behavior of a data sequence changes. For mean shifts in time series, one inspects the data for abrupt changes in average level; for variance shifts, one inspects for abrupt changes in variability. Statistical procedures assess whether observed deviations are consistent with a null hypothesis of no change versus an alternative with a change at some point.
Common methods described in the source:
- CUSUM (Cumulative Sum) method: calculates the cumulative sum of differences between observed and expected values under a null hypothesis; exceeding a predefined threshold signals a change point in the mean.
- Generalized Likelihood Ratio Test (GLRT): compares the likelihood of observed data under two hypotheses—constant variance versus a variance change at a given time; a substantially higher likelihood under the change hypothesis indicates a variance shift.
Examples
Section titled “Examples”Shift in mean (time series / stock market)
Section titled “Shift in mean (time series / stock market)”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. The CUSUM method can be used: calculate the cumulative sum of differences between observed and expected data under a null hypothesis, and if the cumulative sum exceeds a certain threshold, a change point is identified.
Shift in variance (manufacturing)
Section titled “Shift in variance (manufacturing)”Identifying shifts in the variance of a data set. In manufacturing data, there may be sudden changes in the variability of product defects due to changes in production processes or materials. The Generalized Likelihood Ratio Test (GLRT) can be used: compare the likelihood under a constant-variance hypothesis and a varying-variance hypothesis, and if the likelihood under the second hypothesis is significantly higher, a change point is identified.
Use cases
Section titled “Use cases”- Provide insight into underlying drivers of a time series (e.g., stock performance) to aid informed investment decisions.
- Identify underlying causes of changes in variability (e.g., manufacturing defects) to support corrective measures and improve product quality.
- Aid decision making by detecting shifts or discontinuities in data.
Notes or pitfalls
Section titled “Notes or pitfalls”- CUSUM signals a change point when the cumulative sum exceeds a certain threshold; selecting that threshold is part of the method setup.
- GLRT identifies a change point when the likelihood under the change hypothesis is significantly higher than under the no-change hypothesis.
Related terms
Section titled “Related terms”- CUSUM (Cumulative Sum)
- Generalized Likelihood Ratio Test (GLRT)
- Time series
- Mean
- Variance
- Distribution