Early works on the Change-Point analysis mainly appears from the papers relating to quality control. CUSUM detection method uses the cumulative sum of the deviations of the observations from the model to determine the change points. The method is later generalized by David V. Hinkley (1970) that the problem is equivalent to detecting whether the differences in the maximum likelihood with or without a specific change point is significant. Later, with the need of analyzing a more complex series, approaches for Optimal multiple change point detection problems were developed to deal with the computational challenges. For example, the binary segmentation method introduced by Scott and Knott (1974) and an improved Wild Binary Segmentation method which combines the idea of binary segmentation and CUSUM statistics using randomly extracted subsamples from the time series.
We introduce how the NOT procedure works in general. The most important idea adopted in this approach is a combination of ‘global’ and ‘local’ analysis of the time series data. At the ‘global’ stage, we first draw M sub-intervals along the total time span, This could be simply achieved by extracting p uniformly from {0, . . . , T − 1} and extracting q uniformly from {1, . . . , T }, then M valid sets of p and q satisfying p − q ≥ 2d (since we typically require at least d data to decide a d-dimension parametric vector Θ) were drawn and recorded. Next, we calculate the generalized likelihood ratio statistic for all the points (i) within one sub-intervals (p, q]: