Location Of Change Points Research Articles

A Bayesian multiple changepoint model for marked poisson processes with applications to deep earthquakes

A multiple changepoint model for marked Poisson process is formulated as a continuous time hidden Markov model, which is an extension of Chib’s multiple changepoint models (J Econ 86:221–241, 1998). The inference on the locations of changepoints and other model parameters is based on a two-block Gibbs sampling scheme. We suggest a continuous time version of forward-filtering backward-sampling algorithm for simulating the full trajectories of the latent Markov chain without utilizing the uniformization method. To retrieve the optimal posterior path of the latent Markov chain, i.e. the maximum a posteriori estimation of changepoint locations, a continuous-time version of Viterbi algorithm (CT-Viterbi) is proposed. The set of changepoint locations is obtainable either from the CT-Viterbi algorithm or the posterior samples of the latent Markov chain. The number of changepoints is determined according to a modified BIC criterion tailored particularly for the multiple changepoint problems of a marked Poisson process. We then perform a simulation study to demonstrate the methods. The methods are applied to investigate the temporal variabilities of seismicity rates and the magnitude-frequency distributions of medium size deep earthquakes in New Zealand.

Simultaneous Credible Regions for Multiple Changepoint Locations

Within a Bayesian retrospective framework, we present a way of examining the distribution of changepoints through a novel set estimator. For a given level, α, we aim at smallest sets that cover all changepoints with a probability of at least 1 − α. These so-called smallest simultaneous credible regions, computed for certain values of α, provide parsimonious representations of the possible changepoint locations. In addition, combining them for a range of different α’s enables very informative yet condensed visualizations. Therewith we allow for the evaluation of model choices and the analysis of changepoint data to an unprecedented degree. This approach exhibits superior sensitivity, specificity, and interpretability in comparison with highest density regions, marginal inclusion probabilities, and confidence intervals inferred by stepR. While their direct construction is usually intractable, asymptotically correct solutions can be derived from posterior samples. This leads to a novel NP-complete problem. Through reformulations into an Integer Linear Program we show empirically that a fast greedy heuristic computes virtually exact solutions. Supplementary material for this article is available online.

Detecting Changes in Slope With an L0 Penalty

ABSTRACTWhile there are many approaches to detecting changes in mean for a univariate time series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope is much more challenging: simple binary segmentation procedures do not work for this problem, while existing dynamic programming methods that work for the change in mean problem cannot be used for detecting changes in slope. We present a novel dynamic programming approach, CPOP, for finding the “best” continuous piecewise linear fit to data under a criterion that measures fit to data using the residual sum of squares, but penalizes complexity based on an L0 penalty on changes in slope. We prove that detecting changes in this manner can lead to consistent estimation of the number of changepoints, and show empirically that using an L0 penalty is more reliable at estimating changepoint locations than using an L1 penalty. Empirically CPOP has good computational properties, and can analyze a time series with 10,000 observations and 100 changes in a few minutes. Our method is used to analyze data on the motion of bacteria, and provides better and more parsimonious fits than two competing approaches. Supplementary material for this article is available online.

Application of the bootstrap method for change points analysis in generalized linear models

In this paper, we focus on the construction methods of the prediction model, estimation methods of the change point locations, and the confidence intervals for the generalized linear model with piecewise different coefficients. As a standard approach for multiple change point analysis, the application of the hierarchical splitting algorithm is widely used. However, the hierarchical splitting algorithm has a high risk in that the standard error of the change point estimators become large and, therefore, the prediction accuracy of the estimated model decreases. To deal with this problem, we consider the application of a bootstrap method based on the hierarchical splitting algorithm. Through simulation studies, we compare the algorithms in terms of the prediction accuracy of the estimated model, bias and variance of the change point estimators, and the accuracy of the confidence intervals of the change points. From the result, we confirmed the utility of the bootstrap-based methods for change point analysis, especially the increased prediction accuracy of the obtained model, decreased standard error of the change point estimators, and construction of better confidence intervals depending on the situation. We also present the results of a simple example to demonstrate the utility of the method.

Trends in hydrological extremes in the Senegal and Niger Rivers

In recent years, West Africa has witnessed an increasing number of damaging floods that raise the question of a possible intensification of the hydrological hazards in the region. In this study, the evolution of extreme floods is analyzed over the period 1950–2015 for seven tributaries in the Sudano-Guinean part of the Senegal River basin and four data sets in the Sahelian part of the Niger River basin. Non-stationary Generalized Extreme Value (NS-GEV) distributions including twelve models with time-dependent parameters plus a stationary GEV are applied to annual maxima of daily discharge (AMAX) series. An original methodology is proposed for comparing GEV models and selecting the best for use. The stationary GEV is rejected for all stations, demonstrating the significant non-stationarity of extreme discharge values in West Africa over the past six decades. The model of best fit most commonly selected is a double-linear model for the central tendency parameter (μ), with the dispersion parameter (σ) modeled as either stationary, linear, or a double-linear. Change points in double-linear models are relatively consistent for the Senegal basin, with stations switching from a decreasing streamflow trend to an increasing streamflow trend in the early 1980s. In the Niger basin the trend in μ is generally positive since the 1970s with an increase in slope after the change point, but the change point location is less consistent. The recent increasing trends in extreme discharges are reflected in an especially marked increase in return level magnitudes since the 1980s in the studied Sahelian rivers. The rate of the increase indicated by the study results raises urgent considerations for stakeholders and engineers who are in charge of river basin management and hydraulic works sizing.

Simultaneous multiple change-point and factor analysis for high-dimensional time series

We propose the first comprehensive treatment of high-dimensional time series factor models with multiple change-points in their second-order structure. We operate under the most flexible definition of piecewise stationarity, and estimate the number and locations of change-points consistently as well as identifying whether they originate in the common or idiosyncratic components. Through the use of wavelets, we transform the problem of change-point detection in the second-order structure of a high-dimensional time series, into the (relatively easier) problem of change-point detection in the means of high-dimensional panel data. Also, our methodology circumvents the difficult issue of the accurate estimation of the true number of factors in the presence of multiple change-points by adopting a screening procedure. We further show that consistent factor analysis is achieved over each segment defined by the change-points estimated by the proposed methodology. In extensive simulation studies, we observe that factor analysis prior to change-point detection improves the detectability of change-points, and identify and describe an interesting ‘spillover’ effect in which substantial breaks in the idiosyncratic components get, naturally enough, identified as change-points in the common components, which prompts us to regard the corresponding change-points as also acting as a form of ‘factors’. Our methodology is implemented in the R package factorcpt, available from CRAN.

Retrospective analysis for phase I statistical process control and process capability study using revised sample entropy

This study explored a new nonparametric analytical method for identifying heterogeneous segments in time-series data for data-abundant processes. A sample entropy (SampEn) algorithm often used in signal processing and information theory can also be used in a time series or a signal stream, but the original SampEn is only capable of quantifying process variation changes. The proposed algorithm, the adjusted sample entropy (AdSEn), is capable of identifying process mean shifts, variance changes, or mixture of both. A simulation study showed that the proposed method is capable of identifying heterogeneous segments in a time series. Once segments of change points are identified, any existing change-point algorithms can be used to precisely identify exact locations of potential change points. The proposed method is especially applicable for long time series with many change points. Properties of the proposed AdSEn are provided to demonstrate the algorithm’s multi-scale capability. A table of critical values is also provided to help users accurately interpret entropy results.

A probabilistic approach to remaining useful life prediction of rolling element bearings

In this article, we present a probabilistic approach for fault detection and prognosis of rolling element bearings based on a two-phase degradation model. One of the main issues in dealing with bearing degradation is that the degradation mechanism is unobservable and can only be inferred through appropriate surrogate measures obtained from indirect sensory measurements. Furthermore, the stochastic nature of the degradation path renders fault detection and estimating the end-of-life characteristics from such data extremely challenging. When such components are a part of a larger system, the exact degradation path depends on both the operating and loading conditions, which means that the most effective condition monitoring approach should estimate the degradation model parameters under operational conditions, and not solely from isolated component testing or historical information. Motivated by these challenges, a two-phase degradation model using surrogate measures of degradation from vibration measurements is proposed and a Bayesian approach is used to estimate the model parameters. The underlying methodology involves using priors from historical data, while the posterior calculations are undertaken using surrogate measures obtained from a monitored unit combined with the aforesaid priors. The problem of fault detection is posed as a change point location problem. This allows the prior knowledge obtained from the past failures to be integrated for maintenance planning of a currently working unit in a systematic way. The correlation between the degradation rate and the time of occurrence of the change point, an often overlooked aspect in prognosis, is also considered in here. A numerical example and a case study are presented to illustrate the overall methodology and the results obtained using this approach.

Intersection as key locations for bearded capuchin monkeys (Sapajus libidinosus) traveling within a route network.

There is evidence that wild animals are able to recall key locations and associate them with navigational routes. Studies in primate navigation suggest most species navigate through the route network system, using intersections among routes as locations of decision-making. Recent approaches presume that points of directional change may be key locations where animals decide where to go next. Over four consecutive years, we observed how a wild group of bearded capuchin monkeys used a route network system and Change Point locations (CPs) in the Brazilian ecotone of Cerrado-Caatinga. We built 200 daily routes of one wild bearded capuchin group. We used ArcGIS, the Change Point Test, Spatial Analysis in Macroecology (SAM), and statistical models to test the hypothesis that wild bearded capuchins use CPs located along routes in a different fashion than they use the CPs located at intersections of routes. A logistic regression model was used to determine the landscape variables affecting capuchins' directional changes at intersections or along routes. CPs at intersections were important points of travel path changes, whereas CPs along routes represented a zig-zag movement along the routes following the landscape features. CPs at intersections were associated with steeper terrains and shorter distances from important resources, along with better visibility of the home range. Our results support the hypothesis that intersections among routes in the route network system are located at points where monkeys have the best visibility available to make decisions on where to visit next.

Change-point analysis of geophysical time-series: application to landslide displacement rate (Séchilienne rock avalanche, France)

The rank-sum multiple change-point method is a robust statistical procedure designed to search for the optimal number and the location of change points in an arbitrary continue or discrete sequence of values. As such, this procedure can be used to analyse time-series data. Twelve years of robust data sets for the Sechilienne (French Alps) rockslide show a continuous increase in average displacement rate from 50 to 280 mm per month, in the 2004–2014 period, followed by a strong decrease back to 50 mm per month in the 2014–2015 period. When possible kinematic phases are tentatively suggested in previous studies, its solely rely on the basis of empirical threshold values. In this paper, we analyse how the use of a statistical algorithm for change-point detection helps to better understand time phases in landslide kinematics. First, we test the efficiency of the statistical algorithm on geophysical benchmark data, these data sets (stream flows and Northern Hemisphere temperatures) being already analysed by independent statistical tools. Second, we apply the method to 12-yr daily time-series of the Sechilienne landslide, for rainfall and displacement data, from 2003 December to 2015 December, in order to quantitatively extract changes in landslide kinematics. We find two strong significant discontinuities in the weekly cumulated rainfall values: an average rainfall rate increase is resolved in 2012 April and a decrease in 2014 August. Four robust changes are highlighted in the displacement time-series (2008 May, 2009 November–December–2010 January, 2012 September and 2014 March), the 2010 one being preceded by a significant but weak rainfall rate increase (in 2009 November). Accordingly, we are able to quantitatively define five kinematic stages for the Sechilienne rock avalanche during this period. The synchronization between the rainfall and displacement rate, only resolved at the end of 2009 and beginning of 2010, corresponds to a remarkable change (fourfold increase in mean displacement rate) in the landslide kinematic. This suggests that an increase of the rainfall is able to drive an increase of the landslide displacement rate, but that most of the kinematics of the landslide is not directly attributable to rainfall amount. The detailed exploration of the characteristics of the five kinematic stages suggests that the weekly averaged displacement rates are more tied to the frequency or rainy days than to the rainfall rate values. These results suggest the pattern of Sechilienne rock avalanche is consistent with the previous findings that landslide kinematics is dependent upon not only rainfall but also soil moisture conditions (as known as being more strongly related to precipitation frequency than to precipitation amount). Finally, our analysis of the displacement rate time-series pinpoints a susceptibility change of slope response to rainfall, as being slower before the end of 2009 than after, respectively. The kinematic history as depicted by statistical tools opens new routes to understand the apparent complexity of Sechilienne landslide kinematic.

A MOSUM procedure for the estimation of multiple random change points

In this work, we investigate statistical properties of change point estimators based on moving sum statistics. We extend results for testing in a classical situation with multiple deterministic change points by allowing for random exogenous change points that arise in Hidden Markov or regime switching models among others. To this end, we consider a multiple mean change model with possible time series errors and prove that the number and location of change points are estimated consistently by this procedure. Additionally, we derive rates of convergence for the estimation of the location of the change points and show that these rates are strict by deriving the limit distribution of properly scaled estimators. Because the small sample behavior depends crucially on how the asymptotic (long-run) variance of the error sequence is estimated, we propose to use moving sum type estimators for the (long-run) variance and derive their asymptotic properties. While they do not estimate the variance consistently at every point in time, they can still be used to consistently estimate the number and location of the changes. In fact, this inconsistency can even lead to more precise estimators for the change points. Finally, some simulations illustrate the behavior of the estimators in small samples showing that its performance is very good compared to existing methods.

Nonparametric multiple change-point estimation for analyzing large Hi-C data matrices

We propose a novel nonparametric approach to estimate the location of block boundaries (change-points) of non-overlapping blocks in a random symmetric matrix which consists of random variables whose distribution changes from block to block. Our change-point location estimators are based on nonparametric homogeneity tests for matrices. We first provide some theoretical results for these tests. Then, we prove the consistency of our change-point location estimators. Some numerical experiments are also provided in order to support our claims. Finally, our approach is applied to Hi-C data which are used in molecular biology to study the influence of chromosomal conformation on cell function.

Inference under biased sampling and right censoring for a change point in the hazard function

Length-biased survival data commonly arise in cross-sectional surveys and prevalent cohort studies on disease duration. Ignoring biased sampling leads to bias in estimating the hazard-of-failure and the survival-time in the population. We address estimating the location of a possible change-point of an otherwise smooth hazard function when the collected data form a biased sample from the target population and the data are subject to informative censoring. We provide two estimation methodologies, for the location and size of the change-point, adapted to two scenarios of the truncation distribution: known and unknown. While the estimators in the first case show gain in efficiency as compared to those in the second case, the latter is more robust to the form of the truncation distribution. In both cases, the change-point estimators can achieve the rate $\mathcal{O}_{p}(1/n)$. We study the asymptotic properties of the estimates and devise interval-estimators for the location and size of the change, paving the way towards making statistical inference about whether or not a change-point exists. Several simulated examples are discussed to assess the finite sample behavior of the estimators. The proposed methods are then applied to analyze a set of survival data collected on elderly Canadian citizen (aged 65$+$) suffering from dementia.

How Strong is the Relationship Among Gold and USD Exchange Rates? Analytics Based on Structural Change Models

This study examines the dynamic relationship among gold and USD exchange rates. Since one single time series model can suffer from structural (or parameter) changes in underlying models, we consider those models with structural breaks. We first employ the cumulative sum of squared residual test to determine the number and locations of change points in the volatility of time series and then divide the whole period by the change points to investigate the relationship between gold and USD exchange rates in each sub-period, based on the time-varying correlations obtained from dynamic conditional correlation models. We show that a negative correlation exists in almost all periods and that the correlation coefficients have higher absolute values during the global financial crisis period than in other periods. Furthermore, the correlation becomes much greater along with downside moves of USD versus upside moves, indicating that a depreciating trend of USD typically has more influence on gold than an appreciating trend. This phenomenon is in line with the leverage effect in financial markets. After comparing the two methods of with/without structural changes, our findings from an empirical study provide evidence that ignoring structural changes can lead to a false conclusion and confirm that our method offers a functional tool to analyze gold prices and USD exchange rates.

Statistical process monitoring via image data using wavelets

AbstractImage data plays an important role in manufacturing and service industries because each image can provide a huge set of data points in just few seconds with relatively low cost. Enhancement of machine vision systems during the time has led to higher quality images, and the use of statistical methods can help to analyze the data extracted from such images efficiently. It is not efficient from time and cost point of views to use every single pixel in an image to monitor a process or product performance effectively. In recent years, some methods are proposed to deal with image data. These methods are mainly applied for separation of nonconforming items from conforming ones, and they are rarely applied to monitor process capability or performance. In this paper, a nonparametric regression method using wavelet basis function is developed to extract features from gray scale image data. The extracted features are monitored over time to detect process out‐of‐control conditions using a generalized likelihood ratio control chart. The proposed approach can also be applied to find change point and fault location simultaneously. Several numerical examples are used to evaluate performance of the proposed method. Results indicate suitable performance of the proposed method in detecting out‐of‐control conditions and providing precise diagnostic information. Results also illustrate suitable performance of our proposed method in comparison with a competitive approach.

Detecting Change-Point via Saddlepoint Approximations

AbstractIt’s well-known that change-point problem is an important part of model statistical analysis. Most of the existing methods are not robust to criteria of the evaluation of change-point problem. In this article, we consider “mean-shift” problem in change-point studies. A quantile test of single quantile is proposed based on saddlepoint approximation method. In order to utilize the information at different quantile of the sequence, we further construct a “composite quantile test” to calculate the probability of every location of the sequence to be a change-point. The location of change-point can be pinpointed rather than estimated within a interval. The proposed tests make no assumptions about the functional forms of the sequence distribution and work sensitively on both large and small size samples, the case of change-point in the tails, and multiple change-points situation. The good performances of the tests are confirmed by simulations and real data analysis. The saddlepoint approximation based distribution of the test statistic that is developed in the paper is of independent interest and appealing. This finding may be of independent interest to the readers in this research area.

Unit Root Tests for Long Memory Series in the Presence of Structural Breaks in Variance

This paper extends the unit root tests to long memory observations in the existence of variance breaks. Given for the case of non-constant variance, the asymptotic properties of commonly used unit root tests are derived under the null hypothesis. It is shown that the non-constant variance can both inflate and deflate the rejection frequency, thus the statistic tests are not robust. The simulation results also indicate the extent of size distortion is heavily sensitive to the location and magnitude of change points, long memory index and sample size.

A note on estimating the bent line quantile regression model

This paper considers a new estimating method for the bent line quantile regression model. By a simple linearization technique, the proposed method can simultaneously obtain the estimates of the regression coefficients and the change-point location. Moreover, it can be readily implemented by current software. Simulation studies demonstrate that the proposed method has good finite sample performance. Two empirical applications are also presented to illustrate the method.

Computationally Efficient Changepoint Detection for a Range of Penalties

ABSTRACTIn the multiple changepoint setting, various search methods have been proposed, which involve optimizing either a constrained or penalized cost function over possible numbers and locations of changepoints using dynamic programming. Recent work in the penalized optimization setting has focused on developing an exact pruning-based approach that, under certain conditions, is linear in the number of data points. Such an approach naturally requires the specification of a penalty to avoid under/over-fitting. Work has been undertaken to identify the appropriate penalty choice for data-generating processes with known distributional form, but in many applications the model assumed for the data is not correct and these penalty choices are not always appropriate. To this end, we present a method that enables us to find the solution path for all choices of penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitable penalty choice. The computational complexity of this approach can be linear in the number of data points and linear in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values. Supplementary materials for this article are available online.

Inference for a mean-reverting stochastic process with multiple change points

The use of an Ornstein-Uhlenbeck (OU) process is ubiquitous in business, economics and finance to capture various price processes and evolution of economic indicators exhibiting mean-reverting properties. The time at which structural transition representing drastic changes in the economic dynamics occur are of particular interest to policy makers, investors and financial product providers. This paper addresses the change-point problem under a generalised OU model and investigates the associated statistical inference. We propose two estimation methods to locate multiple change points and show the asymptotic properties of the estimators. An informational approach is employed in detecting the change points, and the consistency of our methods is also theoretically demonstrated. Estimation is considered under the setting where both the number and location of change points are unknown. Three computing algorithms are further developed for implementation. The practical applicability of our methods is illustrated using simulated and observed financial market data.