Is Changepoint Detection valid if the process is not piecewise stationary?

Question

I have seen these two definitions:

Let us consider a multivariate non-stationary random process $y=\left\{y_1, \ldots, y_T\right\}$ that takes value in $\mathbb{R}^d(d \geq 1)$ and has $T$ samples. The signal $y$ is assumed to be piecewise stationary, meaning that some characteristics of the process change abruptly at some unknown instants $t_1^*<t_2^*<\cdots<t_{K^*}^*$. Change point detection consists in estimating the indexes $t_k^*$. Depending on the context, the number $K^*$ of changes may or may not be known, in which case it has to be estimated too.

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1 Change point detection A particular change point test seeks to identify the specific period of time that relates to a change in the probability distribution of a stochastic process or time series. In general, the problem concerns both detecting whether or not a change has occurred, or whether several changes might have occurred, and identifying the times of any such changes.

To establish a relatively general framework for the detection of a change point, we can assume that we have an ordered sequence of data, $y_t=\{y_1, \ldots, y_T\}$. A change point is then said to arise within this dataset when there exists a time, $\tau \in\{1, \ldots, T-1\}$, such that the statistical properties of $\{y_1, \ldots, y_\tau\}$ and $\{y_{\tau+1}, \ldots, y_T\}$ are different in some way. Extending this idea for a single change point to multiple changes, we can allow for $m$ change points, that are associated with positions, $\tau_{1: m}=\{\tau_1, \ldots, \tau_m\}$. Each change point position is then ordered so that $\tau_i<\tau_j$ if, and only if, $i<j$. Consequently the $m$ change points will split the data into $m+1$ segments, where the $i$ th segment may be summarised by a set of parameters. The parameters associated with the $i$ th segment will be denoted $\{\theta_i, \phi_i\}$, where $\phi_i$ is a possible set of nuisance parameters and $\theta_i$ is the set of parameters that may describe the change. In this case, we typically want to test how many segments are needed to provide the best representation of the data generating process.

They also discuss tests for finding changes in mean and/or variance.

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So the 1st definition specifically states the signal is assumed to be piecewise stationary. The 2nd one does not, but it does say changepoint detection algorithms have been developed to look for changes in the mean,variance, etc. But a weakly stationary process has a constant mean and variance, so it seems like that definition also implicitly assumes it will split it into a piecewise stationary process.

I have also tried some changepoint detection on a time series and each segment was still nonstationary (under the ADF test).

Is Changepoint Detection valid if the process is not piecewise stationary?

Also if you had another form of nonstationarity(such as unit roots or deterministic trends), and then a structural break/changepoint, how would you deal with that?

Answer 1

Generally for changepoint detection the null case is there is no change in the quantity of interest (e.g. mean or variance) and the alternative case is the quantity of interest is piecewise constant. If you run a changepoint detection method which assumes a piecewise constant mean on something that is not piecewise constant you will likely detect many spurious changepoints.

There are some methods which relax this assumption and assume piecewise linear (Bardwell et al. 2019) or piecewise smooth (Wu and Zhou 2023) for the mean function. This is sometimes referred to as jump detection.