# Change-Point Detection in Autoregressive Processes via the Cross-Entropy Method
## 4 Numerical Experiments
### Example 2: Simulated AR(1) Processes with a Single Change-Point
#### 4.2.2 Example 2.2  
In scenario 2, we generated three sets of 100 AR(1) sequences of length 201 with a homogeneous mean \(\delta=0\) and one abrupt change-point at position \(\tau_{1}=100\) caused by the variation in autocorrelation coefficient. Three settings are displayed in Table 3; in the first set, a weak positive correlation is followed by a moderate correlation; the change occurs from weak to strong, and from moderate to vigorous, in the second and third sets respectively. In general, the performance of strucchange is relatively better than AR1 seg and CE.AR. AR1 seg is not explicitly designed for this scenario since the autoregressive parameter changes between segments. It is interesting to see that CE.AR method performs very well only when \(\rho\) varies dramatically. When \(\rho\) does not change that much, the algorithm tends to underestimate the number of change-points, preferring a model with a smaller number of change-points. Figure 4 displays that the estimation performance of each method under scenario 2. The strucchange shows better performance than the other two. Compared with scenario 1, when the autoregressive parameter \(\rho\) varies weakly or moderately, it is more difficult \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Mean Shift** & \multicolumn{3}{c}{\(\delta_{2}=1\)} & \multicolumn{3}{c}{\(\delta_{2}=2\)} & \multicolumn{3}{c}{\(\delta_{2}=3\)} \\ \hline  & \multicolumn{3}{c}{\(\hat{N}\)} & \multicolumn{3}{c}{\(\hat{N}\)} & \multicolumn{3}{c}{\(\hat{N}\)} \\
**Algorithm** & 0 & \(1^{*}\) & \(\geq 2\) & 0 & \(1^{*}\) & \(\geq 2\) & 0 & \(1^{*}\) & \(\geq 2\) \\ \hline AR1 seg & 58\% & 29\% & 13\% & 20\% & 65\% & 15\% & 10\% & 68\% & 22\% \\ strucchange & 60\% & 40\% & 0\% & 2\% & 97\% & 1\% & 0\% & 97\% & 3\% \\ CE.AR & 97\% & 3\% & 0\% & 41\% & 58\% & 1\% & 2\% & 97\% & 1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The estimated number of change-points under scenario 1. Figure 3: Frequency of the estimated locations filtered by the correct number of change-points; the true number was 100. **(a)** The results from mean shifts \(\delta_{2}=1\). **(b)** The results from mean shifts \(\delta_{2}=2\). **(c)**The results from \(\delta_{2}=3\). estimate the correct number of and the locations of change-points. Lastly, the histogram shows that the estimated location of strucchange has a broader spread than CE.AR method. In addition, we run more tests when \(\rho\) changes from negative to positive; three settings are displayed in Table 4. The detection rates of strucchange and CE.AR methods increased in comparison with the previous settings. In Example 2, we have considered two scenarios of single change-point detection. The first scenario assumes that there are no coefficient changes while the mean changes between different segments. In the second scenario, the coefficient parameter \(\rho\) changes, but the mean is assumed to be 0. Individually, the coefficients are simulated to change weakly, moderately, and strongly; we have considered six cases, which are graphically illustrated in Figure 5 by plotting one time series out of the 100 simulated AR(1) sequences. The frequency charts are used to measure the performances of the \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Coefficient Shift** & \(\rho_{1}=0.5\), \(\rho_{2}=0.9\) & \(\rho_{1}=-0.5\), \(\rho_{2}=0.5\) & \(\rho_{1}=-0.5\), \(\rho_{2}=0.1\) \\ \hline \multirow{3}{*}{**Algorithm**} & \multicolumn{3}{c}{\(\bar{N}\)} & \multicolumn{3}{c}{\(\bar{N}\)} & \multicolumn{3}{c}{\(\bar{N}\)} \\  & 0 & \(1^{*}\) & \(\geq 2\) & 0 & \(1^{*}\) & \(\geq 2\) & 0 & \(1^{*}\) & \(\geq 2\) \\ \hline ARIseg & 2\% & 0\% & 98\% & 47\% & 9\% & 44\% & 84\% & 5\% & 11\% \\ strucchange & 0\% & 97\% & 3\% & 0\% & 98\% & 2\% & 25\% & 74\% & 1\% \\ CE.AR & 0\% & 99\% & 1\% & 3\% & 95\% & 2\% & 87\% & 13\% & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The estimated number of change-points under scenario 2.