Location Of Change Points Research Articles

Bootstrap confidence intervals for multiple change points based on moving sum procedures

The problem of quantifying uncertainty about the locations of multiple change points by means of confidence intervals is addressed. The asymptotic distribution of the change point estimators obtained as the local maximisers of moving sum statistics is derived, where the limit distributions differ depending on whether the corresponding size of changes is local, i.e. tends to zero as the sample size increases, or fixed. A bootstrap procedure for confidence interval generation is proposed which adapts to the unknown magnitude of changes and guarantees asymptotic validity both for local and fixed changes. Simulation studies show good performance of the proposed bootstrap procedure, and some discussions about how it can be extended to serially dependent errors are provided.

Inference for change points in high-dimensional data via selfnormalization

This article considers change-point testing and estimation for a sequence of high-dimensional data. In the case of testing for a mean shift for high-dimensional independent data, we propose a new test which is based on U-statistic in Chen and Qin (Ann. Statist. 38 (2010) 808–835) and utilizes the self-normalization principle (Shao J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 343–366; Shao and Zhang J. Amer. Statist. Assoc. 105 (2010) 1228–1240). Our test targets dense alternatives in the high-dimensional setting and involves no tuning parameters. To extend to change-point testing for high-dimensional time series, we introduce a trimming parameter and formulate a self-normalized test statistic with trimming to accommodate the weak temporal dependence. On the theory front we derive the limiting distributions of self-normalized test statistics under both the null and alternatives for both independent and dependent high-dimensional data. At the core of our asymptotic theory, we obtain weak convergence of a sequential U-statistic based process for high-dimensional independent data, and weak convergence of sequential trimmed U-statistic based processes for high-dimensional linear processes, both of which are of independent interests. Additionally, we illustrate how our tests can be used in combination with wild binary segmentation to estimate the number and location of multiple change points. Numerical simulations demonstrate the competitiveness of our proposed testing and estimation procedures in comparison with several existing methods in the literature.

A method to inform team sport training activity duration with change point analysis.

Duration is a key component in the design of training activities in sport which aim to enhance athlete skills and physical qualities. Training duration is often a balance between reaching skill development and physiological targets set by practitioners. This study aimed to exemplify change point time-series analyses to inform training activity duration in Australian Football. Five features of player behaviour were included in the analyses: disposal frequency, efficiency, pressure, possession time and player movement velocity. Results of the analyses identified moments of change which may be used to inform minimum or maximum activity durations, depending on a practitioner’s objectives. In the first approach, a univariate analysis determined change points specific to each feature, allowing practitioners to evaluate activities according to a single metric. In contrast, a multivariate analysis considered interactions between features and identified a single change point, reflecting the moment of overall change during activities. Six iterations of a training activity were also evaluated resulting in common change point locations, between 196 and 252 seconds, which indicated alterations to player behaviour between this time period in the training activities conduction. Comparisons of feature segments before and after change points revealed the extent to which player behaviour changed and can guide such duration decisions. These methods can be used to evaluate athlete behaviour and inform training activity durations.

Cross-covariance isolate detect: A new change-point method for estimating dynamic functional connectivity.

Evidence of the non stationary behavior of functional connectivity (FC) networks has been observed in task based functional magnetic resonance imaging (fMRI) experiments and even prominently in resting state fMRI data. This has led to the development of several new statistical methods for estimating this time-varying connectivity, with the majority of the methods utilizing a sliding window approach. While computationally feasible, the sliding window approach has several limitations. In this paper, we circumvent the sliding window, by introducing a statistical method that finds change-points in FC networks where the number and location of change-points are unknown a priori. The new method, called cross-covariance isolate detect (CCID), detects multiple change-points in the second-order (cross-covariance or network) structure of multivariate, possibly high-dimensional time series. CCID allows for change-point detection in the presence of frequent changes of possibly small magnitudes, can assign change-points to one or multiple brain regions, and is computationally fast. In addition, CCID is particularly suited to task based data, where the subject alternates between task and rest, as it firstly attempts isolation of each of the change-points within subintervals, and secondly their detection therein. Furthermore, we also propose a new information criterion for CCID to identify the change-points. We apply CCID to several simulated data sets and to task based and resting state fMRI data and compare it to recent change-point methods. CCID may also be applicable to electroencephalography (EEG), magentoencephalography (MEG) and electrocorticography (ECoG) data. Similar to other biological networks, understanding the complex network organization and functional dynamics of the brain can lead to profound clinical implications. Finally, the R package ccid implementing the method from the paper is available from CRAN.

Motor Behavior Changes Are Predictive of Acute Events in Skilled Nursing

Background: Common acute medical conditions among older adults with dementia in skilled nursing include falls, delirium, and pneumonia. This study utilized a sensor technology to examine how motor behaviors may predict these acute events. Methods: Radio frequency identification (RFID) technology continuously measured time and distance travelled, gait speed, and continuous walking with little/no breaks (paths) across 3 long-term facilities for up to 1 year (N=51). Change point analysis estimates the probability of whether a sudden change occurred and provides the location of the change point (in days prior to the event) in a time series model. Results: Gait speed had very low probability to detect a change point across all events (22 falls, 10 delirium and 8 pneumonia). Sensitivity estimates ranged from 63% (number of paths) to 90% (distance travelled) for a fall; 37.5% (number of paths) to 100% (rest of the motor behaviors) for pneumonia. Except for gait speed, all other motor behaviors had high probability (100%) to detect a delirium change point. There was intra-individual variability in the location of the change points (mean of 10 days). Linear regression models for time and distance travelled using baseline predictors of age, ethnicity, gait and balance explained 89% and 90% of the variance in change point locations. Conclusions: Prior to an acute event there is a significant change in motor behavior, suggesting these are an early signal that may be used to prevent a fall or provide for the earlier recognition and treatment of delirium and pneumonia.

Changepoint detection in seismic double-difference data: application of a trans-dimensional algorithm to data-space exploration

Abstract. Double-difference (DD) seismic data are widely used to define elasticity distribution in the Earth's interior and its variation in time. DD data are often pre-processed from earthquake recordings through expert opinion, whereby pairs of earthquakes are selected based on some user-defined criteria and DD data are computed from the selected pairs. We develop a novel methodology for preparing DD seismic data based on a trans-dimensional algorithm, without imposing pre-defined criteria on the selection of event pairs. We apply it to a seismic database recorded on the flank of Katla volcano (Iceland), where elasticity variations in time have been indicated. Our approach quantitatively defines the presence of changepoints that separate the seismic events in time windows. Within each time window, the DD data are consistent with the hypothesis of time-invariant elasticity in the subsurface, and DD data can be safely used in subsequent analysis. Due to the parsimonious behaviour of the trans-dimensional algorithm, only changepoints supported by the data are retrieved. Our results indicate the following: (a) retrieved changepoints are consistent with first-order variations in the data (i.e. most striking changes in the amplitude of DD data are correctly reproduced in the changepoint distribution in time); (b) changepoint locations in time correlate neither with changes in seismicity rate nor with changes in waveform similarity (measured through the cross-correlation coefficients); and (c) the changepoint distribution in time seems to be insensitive to variations in the seismic network geometry during the experiment. Our results demonstrate that trans-dimensional algorithms can be effectively applied to pre-processing of geophysical data before the application of standard routines (e.g. before using them to solve standard geophysical inverse problems).

Slow Momentum with Fast Reversion: A Trading Strategy Using Deep Learning and Changepoint Detection

Momentum strategies are an important part of alternative investments and are at the heart of the work of commodity trading advisors. These strategies have, however, been found to have difficulties adjusting to rapid changes in market conditions, such as during the 2020 market crash. In particular, immediately after momentum turning points, when a trend reverses from an uptrend (downtrend) to a downtrend (uptrend), time-series momentum strategies are prone to making bad bets. To improve the responsiveness to regime change, the authors introduce a novel approach, in which they insert an online changepoint detection (CPD) module into a deep momentum network pipeline, which uses a long short-term memory deep-learning architecture to simultaneously learn both trend estimation and position sizing. Furthermore, their model is able to optimize the way in which it balances (1) a slow momentum strategy that exploits persisting trends but does not overreact to localized price moves and (2) a fast mean-reversion strategy regime by quickly flipping its position and then swapping back again to exploit localized price moves. The CPD module outputs a changepoint location and severity score, allowing the model to learn to respond to varying degrees of disequilibrium, or smaller and more localized changepoints, in a data-driven manner. The authors back test their model over the period 1995–2020, and the addition of the CPD module leads to a 33% improvement in the Sharpe ratio. The module is especially beneficial in periods of significant nonstationarity; in particular, over the most recent years tested (2015–2020), the performance boost is approximately 66%. This is especially interesting because traditional momentum strategies underperformed in this period.

Scalable multiple changepoint detection for functional data sequences

AbstractWe propose the multiple changepoint isolation (MCI) method for detecting multiple changes in the mean and covariance of a functional process. We first introduce a pair of projections to represent the variability “between” and “within” the functional observations. We then present an augmented fused lasso procedure to split the projections into multiple regions robustly. These regions act to isolate each changepoint away from the others so that the powerful univariate CUSUM statistic can be applied region‐wise to identify the changepoints. Simulations show that our method accurately detects the number and locations of changepoints under many different scenarios. These include light and heavy tailed data, data with symmetric and skewed distributions, sparsely and densely sampled changepoints, and mean and covariance changes. We show that our method outperforms a recent multiple functional changepoint detector and several univariate changepoint detectors applied to our proposed projections. We also show that MCI is more robust than existing approaches and scales linearly with sample size. Finally, we demonstrate our method on a large time series of water vapor mixing ratio profiles from atmospheric emitted radiance interferometer measurements.

Comparative Analysis of Change-Point Techniques for Nonlinear Photovoltaic Performance Degradation Rate Estimations

A linear performance drop is generally assumed during the photovoltaic (PV) lifetime. However, operational data demonstrate that the PV module degradation rate ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> ) is often nonlinear, which, if neglected, may increase the financial uncertainty. Although nonlinear behavior has been the subject of numerous publications, it was only recently that statistical models able to detect change-points and extract multiple <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> values from PV performance time-series were introduced. A comparative analysis of six open-source libraries, which can detect change-points and calculate nonlinear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> , is presented in this article. Since the real <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> and change-point locations are unknown in field data, 960 synthetic datasets from six locations and two PV module technologies have been generated using different aggregation and normalization decisions and nonlinear degradation rate patterns. The results demonstrated that coarser temporal aggregation (i.e., monthly vs. weekly), temperature correction, and both PV module technologies and climates with lower seasonality can benefit the change-point detection and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> extraction. This also raises a concern that statistical models typically deployed for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> analysis may be highly climatic- and technology-dependent. The comparative analysis of the six approaches demonstrated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">median</i> mean absolute errors (MAE) ranging from 0.06 to 0.26%/year, given a maximum absolute <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rd</i> of 2.9%/year. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">median</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MAE</i> in change-point position detection varied from 3.5 months to 6 years.

A change-point approach for the identification of financial extreme regimes

Inference over tails is usually performed by fitting an appropriate limiting distribution over observations that exceed a fixed threshold. However, the choice of such threshold is critical and can affect the inferential results. Extreme value mixture models have been defined to estimate the threshold using the full dataset and to give accurate tail estimates. Such models assume that the tail behavior is constant for all observations. However, the extreme behavior of financial returns often changes considerably in time and such changes occur by sudden shocks of the market. Here the extreme value mixture model class is extended to formally take into account distributional extreme change-points, by allowing for the presence of regime-dependent parameters modelling the tail of the distribution. This extension formally uses the full dataset to both estimate the thresholds and the extreme change-point locations, giving uncertainty measures for both quantities. Estimation of functions of interest in extreme value analyses is performed via MCMC algorithms. Our approach is evaluated through a series of simulations, applied to real data sets and assessed against competing approaches. Evidence demonstrates that the inclusion of different extreme regimes outperforms both static and dynamic competing approaches in financial applications.

A multiple changepoint approach to hydrological regions delineation

Hydrologic regionalization consists of regrouping stations and catchments in pools based on a similarity measure. Regionalization is commonly used to extract a robust signal that can be used to describe the hydrology of the region or extrapolated to a location without measured information. Obviously, the similarity measure used affects the type of hydrological behavior one would expect from stations within a region. Most regionalization methods assume a stable and/or linear relationship between parameters of interests while it is well known that the physical processes driving the behavior of hydrometeorological variables are inherently non-linear and non-stationary. In this paper, we propose a similarity measure that is based on the location of changepoints in hydrological time series. The proposed method has the unique advantage over other hydrological region delineation methods to detect regions where hydrological member stations are non-linearly correlated, and where the strength of the relation varies with time. It therefore has the potential to uncover similarities that would not have been detected by existing regionalization techniques. The proposed method is applied to the Tensift watershed located in Morocco, North Africa. The coherence of the detected regions is checked using wavelet coherence.

Efficient change-points detection for genomic sequences via cumulative segmented regression.

Knowing the number and the exact locations of multiple change points in genomic sequences serves several biological needs. The cumulative-segmented algorithm (cumSeg) has been recently proposed as a computationally efficient approach for multiple change-points detection, which is based on a simple transformation of data and provides results quite robust to model mis-specifications. However, the errors are also accumulated in the transformed model so that heteroscedasticity and serial correlation will show up, and thus the variations of the estimated change points will be quite different, while the locations of the change points should be of the same importance in the original genomic sequences. In this study, we develop two new change-points detection procedures in the framework of cumulative segmented regression. Simulations reveal that the proposed methods not only improve the efficiency of each change point estimator substantially but also provide the estimators with similar variations for all the change points. By applying these proposed algorithms to Coriel and SNP genotyping data, we illustrate their performance on detecting copy number variations. The proposed algorithms are implemented in R program and the codes are provided in the online supplementary material. Supplementary data are available at Bioinformatics online.

Two-stage data segmentation permitting multiscale change points, heavy tails and dependence

The segmentation of a time series into piecewise stationary segments is an important problem both in time series analysis and signal processing. In the presence of multiscale change points with both large jumps over short intervals and small jumps over long intervals, multiscale methods achieve good adaptivity but require a model selection step for removing false positives and duplicate estimators. We propose a localised application of the Schwarz criterion, which is applicable with any multiscale candidate generating procedure fulfilling mild assumptions, and establish its theoretical consistency in estimating the number and locations of multiple change points under general assumptions permitting heavy tails and dependence. In particular, combined with a MOSUM-based candidate generating procedure, it attains minimax rate optimality in both detection lower bound and localisation for i.i.d. sub-Gaussian errors. Overall competitiveness of the proposed methodology compared to existing methods is shown through its theoretical and numerical performance.

Structural change detection in ordinal time series.

Change-point detection in health care data has recently obtained considerable attention due to the increased availability of complex data in real-time. In many applications, the observed data is an ordinal time series. Two kinds of test statistics are proposed to detect the structural change of cumulative logistic regression model, which is often used in applications for the analysis of ordinal time series. One is the standardized efficient score vector, the other one is the quadratic form of the efficient score vector with a weight function. Under the null hypothesis, we derive the asymptotic distribution of the two test statistics, and prove the consistency under the alternative hypothesis. We also study the consistency of the change-point estimator, and a binary segmentation procedure is suggested for estimating the locations of possible multiple change-points. Simulation results show that the former statistic performs better when the change-point occurs at the centre of the data, but the latter is preferable when the change-point occurs at the beginning or end of the data. Furthermore, the former statistic could find the reason for rejecting the null hypothesis. Finally, we apply the two test statistics to a group of sleep data, the results show that there exists a structural change in the data.

Greedy Segmentation for a Functional Data Sequence

We present a new approach known as greedy segmentation (GS) to identify multiple changepoints for a functional data sequence. The proposed multiple changepoint detection criterion links detectability with the projection onto a suitably chosen subspace and the changepoint locations. The changepoint estimator identifies the true changepoints for any predetermined number of changepoint candidates, either over-reporting or under-reporting. This theoretical finding supports the proposed GS estimator, which can be efficiently obtained in a greedy manner. The GS estimator’s consistency holds without being restricted to the conventional at most one changepoint condition, and it is robust to the relative positions of the changepoints. Based on the GS estimator, the test statistic’s asymptotic distribution leads to the novel GS algorithm, which identifies the number and locations of changepoints. Using intensive simulation studies, we compare the finite sample performance of the GS approach with other competing methods. We also apply our method to temporal changepoint detection in weather datasets.

Estimating change-point latent factor models for high-dimensional time series

We consider estimating a factor model for high-dimensional time series that contains structural breaks in the factor loading space at unknown time points. We first study the case when there is one change point in factor loadings, and propose a consistent estimator for the structural break location, whose convergence rate is shown to depend on an interplay between the dimension of the observed time series and the strength of the underlying factor structure. Our results reveal that the asymptotic behavior of the proposed estimator can be asymmetric in the sense that a larger estimation error can occur toward the regime with weaker factor strength. Based on the proposed estimator for the structural break location, we also consider the problem of estimating the factor loading spaces before and after the structural break. We show that the proposed estimators for change-point location and loading spaces are consistent when the numbers of factors are correctly estimated or overestimated. The algorithm for multiple change-point detection is also developed in the paper. Compared with existing results on change-point factor analyses of high-dimensional time series, a distinguished feature of the current paper is that the noise process is not necessarily assumed to be idiosyncratic and as a result we allow the noise process with potentially strong cross-sectional dependence. Another advantage for the proposed method is that it is specifically designed for the changes in the factor loading space and the stationarity assumption is not imposed on either the factor or noise process, while most existing methods for change-point detection of high-dimensional time series with/without a factor structure require the data to be stationary or ’close’ to a stationary process between two change points, which is rather restrictive. Numerical experiments including a Monte Carlo simulation and a real data application are presented to illustrate the proposed estimators perform well.

BOOTSTRAP INFERENCE FOR MULTIPLE CHANGE-POINTS IN TIME SERIES

In this paper, we propose two bootstrap procedures, namely parametric and block bootstrap, to approximate the finite sample distribution of change-point estimators for piecewise stationary time series. The bootstrap procedures are then used to develop a generalized likelihood ratio scan method (GLRSM) for multiple change-point inference in piecewise stationary time series, which estimates the number and locations of change-points and provides a confidence interval for each change-point. The computational complexity of using GLRSM for multiple change-point detection is as low as $O(n(\log n)^{3})$ for a series of length n. Extensive simulation studies are provided to demonstrate the effectiveness of the proposed methodology under different scenarios. Applications to financial time series are also illustrated.

Detecting Abrupt Changes in the Presence of Local Fluctuations and Autocorrelated Noise

While there are a plethora of algorithms for detecting changes in mean in univariate time-series, almost all struggle in real applications where there is autocorrelated noise or where the mean fluctuates locally between the abrupt changes that one wishes to detect. In these cases, default implementations, which are often based on assumptions of a constant mean between changes and independent noise, can lead to substantial over-estimation of the number of changes. We propose a principled approach to detect such abrupt changes that models local fluctuations as a random walk process and autocorrelated noise via an AR(1) process. We then estimate the number and location of changepoints by minimizing a penalized cost based on this model. We develop a novel and efficient dynamic programming algorithm, DeCAFS, that can solve this minimization problem; despite the additional challenge of dependence across segments, due to the autocorrelated noise, which makes existing algorithms inapplicable. Theory and empirical results show that our approach has greater power at detecting abrupt changes than existing approaches. We apply our method to measuring gene expression levels in bacteria. Supplementary materials for this article are available online.

The CUSUM statistics of change-point models based on dependent sequences

In this paper, we investigate the mean change-point models based on associated sequences. Under some weak conditions, we obtain a limit distribution of CUSUM statistic which can be used to judge the mean change-mount is satisfied or dissatisfied . We also study the consistency of sample covariances and change-point location statistics. Based on Normality and Lognormality data, some simulations such as empirical sizes, empirical powers and convergence are presented to test our results. As an important application, we use CUSUM statistics to do the mean change-point analysis for a financial series.

Multithreshold change plane model: Estimation theory and applications in subgroup identification.

We propose a multithreshold change plane regression model which naturally partitions the observed subjects into subgroups with different covariate effects. The underlying grouping variable is a linear function of observed covariates and thus multiple thresholds produce change planes in the covariate space. We contribute a novel two-stage estimation approach to determine the number of subgroups, the location of thresholds, and all other regression parameters. In the first stage we adopt a group selection principle to consistently identify the number of subgroups, while in the second stage change point locations and model parameter estimates are refined by a penalized induced smoothing technique. Our procedure allows sparse solutions for relatively moderate- or high-dimensional covariates. We further establish the asymptotic properties of our proposed estimators under appropriate technical conditions. We evaluate the performance of the proposed methods by simulation studies and provide illustrations using two medical data examples. Our proposal for subgroup identification may lead to an immediate application in personalized medicine.