Detection of Change Points in Spatiotemporal Data in the Presence of Outliers and Heavy-Tailed Observations

Abstract

This work improves the estimation algorithm of a general spatiotemporal autoregressive model proposed by Wu et al. (Br J Environ Clim Chang 7(4):223–235, 2017). We substitute their least squares technique in the EM-type algorithm by M-estimation and also present an M-estimation based change-point detection procedure. In addition, data examples are provided.

Appendix

A specific example is given to show how to calculate $y_{T_{1}(k-1)+t}^{(1)}$, $y_{T_{1}(k-1)+t}^{(2)}$, $\tilde {y}_{T_{1}(k-1)+t}^{(3)}$ and $\tilde {\epsilon }_{T_{1}(k-1)+t}$ in Sect. 2.1.

Consider t ₁ = 1, $\mathcal {S}_{1}=\{1,2,\ldots ,90,307,308,\ldots ,366\}$ for Winter, $\mathcal {S}_{2}=\{91,92,\ldots ,152\}$ for Spring, $\mathcal {S}_{3}=\{153,\ldots ,244\}$ for Summer and $\mathcal {S}_{4}=\{245,\ldots ,306\}$ for Fall. In this case, T = 366 and T ₁ = 181. For winter,

$$\displaystyle \begin{aligned} \begin{array}{rcl} y_{181(k-1)+t-1}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t}-\bar{y}_{366(k-1)+75-t}\\ y_{181(k-1)+t-1}^{(2)} &\displaystyle \equiv &\displaystyle 2\cos{}(2t\pi/s_{1}),\\ y_{181(k-1)+t-1}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t-1}-\bar{y}_{366(k-1)+75-t-1}\\ \tilde{\epsilon}_{181(k-1)+t-1} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+t}-\epsilon_{366(k-1)+75-t},\quad t=2,\ldots,37.\\ y_{181(k-1)+37+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t+75}-\bar{y}_{366(k-1)+366-t}\\ y_{181(k-1)+37+t}^{(2)} &\displaystyle \equiv &\displaystyle -2\cos{}(2t\pi/s_{1}),\\ y_{181(k-1)+37+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t+75-1}-\bar{y}_{366(k-1)+366-t-1}\\ \tilde{\epsilon}_{181(k-1)+37+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+t+75}-\epsilon_{366(k-1)+366-t},\quad t=0,1,2,\ldots,15.\\ y_{181(k-1)+37+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t+291}-\bar{y}_{366(k-1)+366-t}\\ y_{181(k-1)+37+t}^{(2)} &\displaystyle \equiv &\displaystyle -2\cos{}(2t\pi/s_{1}),\\ y_{181(k-1)+37+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+t+291-1}-\bar{y}_{366(k-1)+366-t-1}\\ \tilde{\epsilon}_{181(k-1)+37+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+t+291}-\epsilon_{366(k-1)+366-t},\quad t=16,17,\ldots,37. \end{array} \end{aligned} $$

For Spring,

$$\displaystyle \begin{aligned} \begin{array}{rcl} y_{181(k-1)+74+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+90+t}-\bar{y}_{366(k-1)+121-t}\\ y_{181(k-1)+74+t}^{(2)} &\displaystyle \equiv &\displaystyle 2\cos{}(2t\pi/s_{2}),\\ y_{181(k-1)+74+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+90+t-1}-\bar{y}_{366(k-1)+121-t-1}\\ \tilde{\epsilon}_{181(k-1)+74+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+90+t}-\epsilon_{366(k-1)+121-t},\quad t=1,2,\ldots,15.\\ y_{181(k-1)+90+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+121+t}-\bar{y}_{366(k-1)+152-t}\\ y_{181(k-1)+90+t}^{(2)} &\displaystyle \equiv &\displaystyle -2\cos{}(2t\pi/s_{2}),\\ y_{181(k-1)+90+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+121+t-1}-\bar{y}_{366(k-1)+152-t-1}\\ \tilde{\epsilon}_{181(k-1)+90+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+121+t}-\epsilon_{366(k-1)+152-t},\quad t=0,1,2,\ldots,15. \end{array} \end{aligned} $$

For Summer,

$$\displaystyle \begin{aligned} \begin{array}{rcl} y_{181(k-1)+105+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+152+t}-\bar{y}_{366(k-1)+198-t}\\ y_{181(k-1)+105+t}^{(2)} &\displaystyle \equiv &\displaystyle 2\cos{}(2t\pi/s_{3}),\\ y_{181(k-1)+105+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+152+t-1}-\bar{y}_{366(k-1)+198-t-1}\\ \tilde{\epsilon}_{181(k-1)+105+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+152+t}-\epsilon_{366(k-1)+198-t},\quad t=1,2,\ldots,22.\\ y_{181(k-1)+128+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+198+t}-\bar{y}_{366(k-1)+244-t}\\ y_{181(k-1)+128+t}^{(2)} &\displaystyle \equiv &\displaystyle -2\cos{}(2t\pi/s_{3}),\\ y_{181(k-1)+128+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+198+t-1}-\bar{y}_{366(k-1)+244-t-1}\\ \tilde{\epsilon}_{181(k-1)+128+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+198+t}-\epsilon_{366(k-1)+244-t},\quad t=0,1,2,\ldots,22. \end{array} \end{aligned} $$

For Fall,

$$\displaystyle \begin{aligned} \begin{array}{rcl} y_{181(k-1)+150+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+244+t}-\bar{y}_{366(k-1)+275-t}\\ y_{181(k-1)+150+t}^{(2)} &\displaystyle \equiv &\displaystyle 2\cos{}(2t\pi/s_{4}),\\ y_{181(k-1)+150+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+244+t-1}-\bar{y}_{366(k-1)+275-t-1}\\ \tilde{\epsilon}_{181(k-1)+150+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+244+t}-\epsilon_{366(k-1)+275-t},\quad t=1,2,\ldots,15.\\ y_{181(k-1)+166+t}^{(1)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+275+t}-\bar{y}_{366(k-1)+306-t}\\ y_{181(k-1)+166+t}^{(2)} &\displaystyle \equiv &\displaystyle -2\cos{}(2t\pi/s_{4}),\\ y_{181(k-1)+166+t}^{(3)} &\displaystyle \equiv &\displaystyle \bar{y}_{366(k-1)+275+t-1}-\bar{y}_{366(k-1)+306-t-1}\\ \tilde{\epsilon}_{181(k-1)+166+t} &\displaystyle \equiv &\displaystyle \epsilon_{366(k-1)+275+t}-\epsilon_{366(k-1)+306-t},\quad t=0,1,2,\ldots,15. \end{array} \end{aligned} $$