ChangePoint Detection Using the Conditional Entropy of Ordinal Patterns
Abstract
:1. Introduction
2. Methods
2.1. Preliminaries
2.1.1. Ordinal Patterns
2.1.2. Stochastic Processes with Ordinal ChangePoints
2.1.3. Conditional Entropy of Ordinal Patterns
2.2. A Statistic for ChangePoint Detection Based on the Conditional Entropy of Ordinal Patterns
 $\pi \left(d\right),\pi (d+1),\dots ,\pi \left({t}^{*}\right)$ characterize the process before the change;
 $\pi ({t}^{*}+1),\pi ({t}^{*}+2),\dots ,\pi ({t}^{*}+d1)$ correspond to the transitional state;
 $\pi ({t}^{*}+d),\pi ({t}^{*}+d+1),\dots ,\pi \left(L\right)$ characterize the process after the change.
2.3. Algorithm for ChangePoint Detection via the CEofOP Statistic
 ${H}_{0}$:
 parts $\pi \left(d\right),\pi (d+1),\dots ,\pi ({\widehat{t}}^{*})$ and $\pi ({\widehat{t}}^{*}+d),\dots ,\pi \left(L\right)$ of the sequence ${\pi}^{d,L}$ come from the same distribution;
 ${H}_{A}$:
 parts $\pi \left(d\right),\pi (d+1),\dots ,\pi ({\widehat{t}}^{*})$ and $\pi ({\widehat{t}}^{*}+d),\dots ,\pi \left(L\right)$ of the sequence ${\pi}^{d,L}$ come from different distributions.
Algorithm 1 Detecting at most one changepoint 
Input: sequence $\pi ={\left(\pi \left(k\right)\right)}_{k={t}_{\mathrm{start}}}^{{t}_{\mathrm{end}}}$ of ordinal patterns of order d, nominal probability $\alpha $ of false alarm Output: estimate of a changepoint ${\widehat{t}}^{*}$ if changepoint is detected, otherwise return 0.

 Step 1:
 preliminary estimation of boundaries of the stationary segments with a threshold $h\left(2\alpha \right)$ computed for doubled nominal probability of false alarm (that is, with a higher risk of detecting false changepoints).
 Step 2:
 verification of the boundaries and exclusion of false changepoints: a changepoint is searched for a merging of every two adjacent intervals.
Algorithm 2 Detecting multiple changepoints 
Input: sequence $\pi ={\left(\pi \left(k\right)\right)}_{k=d}^{L}$ of ordinal patterns of order d, nominal probability $\alpha $ of false alarm. Output: estimates of the number ${\widehat{N}}_{\mathrm{st}}$ of stationary segments and of their boundaries ${\left({\widehat{t}}_{k}^{*}\right)}_{k=0}^{{\widehat{N}}_{\mathrm{st}}}$.

3. Numerical Simulations and Results
 The ordinalpatternsbased method for detecting changepoints via the CMMD statistic [23,24]: A time series is split into windows of equal lengths $W\in \mathbb{N}$, empirical probabilities of ordinal patterns are estimated in every window. If there is a ordinal changepoint in the time series, then the empirical probabilities of ordinal patterns should be approximately constant before the changepoint and after the changepoint, but they change at the window with the changepoint. To detect this change, the CMMD statistic was introduced. (Note that the definition of the CMMD statistic in [23] contains a mistake, which is corrected in [24]. The results of numerical experiments reported in [23] also do not comply with the actual definition of the CMMD statistic (see Section 4.2.1.1 and 4.5.1.1 in [22] for details). In the original papers [23,24], authors do not estimate changepoints, but only the corresponding window numbers; for the algorithm of changepoint estimation by means of the CMMD statistic, we refer to Section 4.5.1 in [22].
 Two versions of the classical Brodsky–Darkhovsky method [11]: the Brodsky–Darkhovsky method can be used for detecting changes in various characteristics of a time series $x={\left(x\left(t\right)\right)}_{t=1}^{L}$, but the characteristic of interest should be selected in advance. In this paper, we consider detecting changes in mean, which is just the basic characteristic, and in correlation function $corr\left(x\right(t),x(t+1\left)\right),$ which reflects relations between the future and the past of a time series and seems to be a natural choice for detecting ordinal changepoints. Changes in mean are detected by the generalized version of the Kolmogorov–Smirnov statistic [11]:$${\mathrm{BD}}^{\mathrm{exp}}(t;x,\delta )={\left(\frac{t(Lt)}{{L}^{2}}\right)}^{\delta}\phantom{\rule{0.222222em}{0ex}}\frac{1}{t}\sum _{l=1}^{t}x\left(l\right)\frac{1}{Lt}\sum _{l=t+1}^{L}x\left(l\right),$$$${\mathrm{BD}}^{\mathrm{corr}}(t;x,\delta )={\mathrm{BD}}^{\mathrm{exp}}\left(t;{\left(x\left(t\right)x(t+1)\right)}_{t=1}^{L1},\delta \right).$$
3.1. Estimation of the Position of a Single ChangePoint
3.2. Estimating Position of a Single ChangePoint for Different Lengths of Time Series
3.3. Detecting Multiple ChangePoints
4. Conclusions and Open Points
 A method for computing a threshold h for the $\mathrm{CEofOP}$ statistic without shuffling the original time series is of interest, since this procedure is rather time consuming. One possible solution is to utilize Theorem A1 (Appendix A.1) and to precompute thresholds using the values of ${\mathsf{\Delta}}_{\gamma ,\theta}^{d}(P,Q)$. However, this approach requires further investigation.
 The binary segmentation procedure [43] is not the only possible method for detecting multiple changepoints. In [8,55], an alternative approach is suggested: the number of stationary segments ${\widehat{N}}_{\mathrm{st}}$ is estimated by optimizing a contrast function, then the positions of the changepoints are adjusted. Likewise, one can consider a method for multiple changepoint detection based on maximizing the following generalization of $\mathrm{CEofOP}$ statistic:$$\mathrm{CEofOP}\left(t\right)=(Ld{\widehat{N}}_{\mathrm{st}})\phantom{\rule{0.166667em}{0ex}}\mathrm{eCE}\left({\left(\pi \left(k\right)\right)}_{k=d}^{L}\right)\sum _{l=1}^{{\widehat{N}}_{\mathrm{st}}}\left({\widehat{t}}_{l}^{*}{\widehat{t}}_{l1}^{*}d\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{eCE}\left(\pi \left({\widehat{t}}_{l1}^{*}+d\right),\dots ,\pi \left({\widehat{t}}_{l}^{*}\right)\right),$$
 As we have seen in Section 3.2, CEofOP statistic requires rather large sample sizes to provide reliable changepoint detection. This is due to the necessity of the empirical conditional entropy estimation (see Section 2.3). In order to reduce the required sample size, one may consider more effective estimates of the conditional entropy—for instance, the Grassberger estimate (see [56] and also Section 3.4.1 in [22]). However, elaboration of this idea is beyond the scope of this paper.
 We did not use the full power of ordinal time series analysis, which often considers ordinal patterns taken from sequences of equidistant time points of some distance $\tau $. This generalization of the case $\tau =1$ with successive points allows for addressing different scales and so to extract more information on the distribution of a time series [57], also being useful for changepoint detection.
 In this paper, only onedimensional time series are considered, though there is no principal limitation for applying ordinalpatternsbased methods to multivariate data (see [28]). Discussion of using ordinalpatternsbased methods for detecting changepoint in multivariate data (for instance, in multichannel EEG) is therefore of interest.
 We have considered here only the “offline” detection of changes, which is used when the acquisition of a time series is completed. Meanwhile, in many applications, it is necessary to detect changepoints “online”, based on a small number of observations after the change [1]. Development of online versions of ordinalpatternsbased methods for changepoint detection may be an interesting direction of a future work.
Author Contributions
Funding
Conflicts of Interest
Appendix A. Theoretical Underpinnings of the CEofOP Statistic
Appendix A.1. Asymptotic Behavior of the CEofOP Statistic
${\mathit{\varphi}}_{\mathbf{1}}$  0.00  0.10  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  0.99  

${\mathit{\varphi}}_{\mathbf{2}}$  
0.00  0  0.02  0.07  0.15  0.26  0.40  0.56  0.74  0.95  1.18  1.44  
0.10  0.02  0  0.02  0.06  0.14  0.25  0.37  0.53  0.71  0.91  1.13  
0.20  0.07  0.02  0  0.02  0.06  0.13  0.23  0.36  0.51  0.68  0.88  
0.30  0.15  0.06  0.02  0  0.01  0.06  0.13  0.22  0.34  0.49  0.66  
0.40  0.26  0.14  0.06  0.01  0  0.01  0.06  0.12  0.22  0.33  0.48  
0.50  0.40  0.25  0.13  0.06  0.01  0  0.01  0.05  0.12  0.21  0.33  
0.60  0.56  0.37  0.23  0.13  0.06  0.01  0  0.01  0.05  0.12  0.21  
0.70  0.74  0.53  0.36  0.22  0.12  0.05  0.01  0  0.01  0.05  0.12  
0.80  0.95  0.71  0.51  0.34  0.22  0.12  0.05  0.01  0  0.01  0.05  
0.90  1.18  0.91  0.68  0.49  0.33  0.21  0.12  0.05  0.01  0  0.01  
0.99  1.44  1.13  0.88  0.66  0.48  0.33  0.21  0.12  0.05  0.01  0 
${\mathit{\varphi}}_{\mathbf{1}}$  0.00  0.10  0.20  0.30  0.40  0.50  0.60  0.70  0.80  0.90  0.99  

${\mathit{\varphi}}_{\mathbf{2}}$  
0.00  0  0.04  0.15  0.33  0.56  0.85  1.18  1.55  1.95  2.40  2.88  
0.10  0.04  0  0.04  0.14  0.31  0.53  0.80  1.12  1.48  1.89  2.34  
0.20  0.15  0.04  0  0.03  0.13  0.29  0.51  0.77  1.08  1.44  1.85  
0.30  0.33  0.14  0.03  0  0.03  0.13  0.28  0.49  0.75  1.06  1.43  
0.40  0.56  0.31  0.13  0.03  0  0.03  0.12  0.27  0.48  0.74  1.06  
0.50  0.85  0.53  0.29  0.13  0.03  0  0.03  0.12  0.27  0.48  0.74  
0.60  1.18  0.80  0.51  0.28  0.12  0.03  0  0.03  0.12  0.27  0.48  
0.70  1.55  1.12  0.77  0.49  0.27  0.12  0.03  0  0.03  0.12  0.28  
0.80  1.95  1.48  1.08  0.75  0.48  0.27  0.12  0.03  0  0.03  0.13  
0.90  2.40  1.89  1.44  1.06  0.74  0.48  0.27  0.12  0.03  0  0.03  
0.99  2.88  2.34  1.85  1.43  1.06  0.74  0.48  0.28  0.13  0.03  0 
Appendix A.2. CEofOP Statistic for a Sequence of Ordinal Patterns Forming a Markov Chain
Appendix A.3. ChangePoint Detection by the CEofOP Statistic and from Permutation Entropy Values
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Short Name  Complete Designation 

NL, $3.95\to 3.98$, $\sigma =0.2$  $\mathrm{NL}\left((3.95,3.98),(0.2,0.2),{t}^{*}\right)$ 
NL, $3.95\to 3.80$, $\sigma =0.3$  $\mathrm{NL}\left((3.95,3.80),(0.3,0.3),{t}^{*}\right)$ 
NL, $3.95\to 4.00$, $\sigma =0.2$  $\mathrm{NL}\left((3.95,4.00),(0.2,0.2),{t}^{*}\right)$ 
AR, $0.1\to 0.3$  $\mathrm{AR}\left((0.1,0.3),{t}^{*}\right)$ 
AR, $0.1\to 0.4$  $\mathrm{AR}\left((0.1,0.4),{t}^{*}\right)$ 
AR, $0.1\to 0.5$  $\mathrm{AR}\left((0.1,0.5),{t}^{*}\right)$ 
NL, $3.95\to 3.98$  NL, $3.95\to 3.80$  NL, $3.95\to 4.00$  

Statistic  $\mathit{\sigma}=0.2$  $\mathit{\sigma}=0.3$  $\mathit{\sigma}=0.2$  
$\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  $\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  $\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  
CMMD  0.34  698  1653  0.50  −51  306  0.68  −13  206 
$\mathrm{CEofOP},d=2$  0.46  147  1108  0.62  −3  267  0.81  33  147 
$\mathrm{CEofOP},d=3$  0.61  53  397  0.65  1  256  0.88  20  99 
$\mathrm{CEofOP},d=4$  0.47  −2  982  0.46  −41  1162  0.83  2  130 
${\mathrm{BD}}^{\mathrm{exp}}$  0.62  78  351  0.78  −6  145  0.89  43  96 
${\mathrm{BD}}^{\mathrm{corr}}$  0.44  85  656  0.71  13  202  0.77  43  189 
Statistic  AR, $0.1\to 0.3$  AR, $0.1\to 0.4$  AR, $0.1\to 0.5$  

$\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  $\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  $\mathbf{sE}$  $\mathbf{B}$  $\mathbf{RMSE}$  
CMMD  0.32  616  1626  0.54  −14  368  0.68  −48  184 
$\mathrm{CEofOP},d=2$  0.42  74  1096  0.67  6  244  0.82  3  129 
$\mathrm{CEofOP},d=3$  0.39  126  1838  0.68  0  234  0.86  0  110 
$\mathrm{CEofOP},d=4$  0.08  1028  6623  0.46  −176  1678  0.74  −27  214 
${\mathrm{BD}}^{\mathrm{exp}}$  0.00  >${10}^{3}$  >${10}^{4}$  0.00  >${10}^{4}$  >${10}^{4}$  0.00  >${10}^{4}$  >${10}^{4}$ 
${\mathrm{BD}}^{\mathrm{corr}}$  0.79  31  151  0.92  21  73  0.97  21  50 
Statistic  Number of False ChangePoints  Fraction ${\mathbf{sE}}_{\mathit{k}}$ of Satisfactory Estimates  

1st Change  2nd Change  3rd Change  Average  
cMMD  1.17  0.465  0.642  0.747  0.618 
$\mathrm{CEofOP}$  0.62  0.753  0.882  0.930  0.855 
${\mathrm{BD}}^{\mathrm{corr}}$  1.34  0.296  0.737  0.751  0.595 
Statistic  Number of False ChangePoints  Fraction ${\mathbf{sE}}_{\mathit{k}}$ of Satisfactory Estimates  

1st Change  2nd Change  3rd Change  Average  
CMMD  1.17  0.340  0.640  0.334  0.438 
$\mathrm{CEofOP}$  1.12  0.368  0.834  0.517  0.573 
${\mathrm{BD}}^{\mathrm{corr}}$  0.53  0.783  0.970  0.931  0.895 
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Unakafov, A.M.; Keller, K. ChangePoint Detection Using the Conditional Entropy of Ordinal Patterns. Entropy 2018, 20, 709. https://doi.org/10.3390/e20090709
Unakafov AM, Keller K. ChangePoint Detection Using the Conditional Entropy of Ordinal Patterns. Entropy. 2018; 20(9):709. https://doi.org/10.3390/e20090709
Chicago/Turabian StyleUnakafov, Anton M., and Karsten Keller. 2018. "ChangePoint Detection Using the Conditional Entropy of Ordinal Patterns" Entropy 20, no. 9: 709. https://doi.org/10.3390/e20090709