Change Point Test Research Articles

Detection and localization of changes in a panel of densities

We propose a new methodology for identifying and localizing changes in the Fréchet mean of a multivariate time series of probability densities. The functional data objects we study are random densities fs,t indexed by discrete time t and a vector component s, which can be treated as a broadly understood spatial location. Our main objective is to identify the set of components s, where a change occurs with statistical certainty. A challenge of this analysis is that the densities fs,t are not directly observable and must be estimated from sparse and potentially imbalanced data. Such setups are motivated by the analysis of two data sets that we investigate in this work. First, a hitherto unpublished large data set of Brazilian Covid infections and a second, a financial data set derived from intraday prices of U.S. Exchange Traded Funds. Chief statistical advances are the development of change point tests and estimators of components of change for multivariate time series of densities. We prove the theoretical validity of our methodology and investigate its finite sample performance in a simulation study.

Drying and wetting trend in Xinjiang and related circulations background over the past 60 years

The study of climate variability and trend provides important evidence for water resource management strategies in Xinjiang. The scientific community has systematically studied the ‘warming and wetting’ process in the Xinjiang region. However, significant controversy persists regarding whether this trend has stagnated, and there remains a gap in understanding the decadal climate variability across different seasons within the subregions of Xinjiang. Here, we analyzed the spatiotemporal characteristics of seasonal-to-annual climate variations in Southern and Northern Xinjiang over the past 60 years by the Standardized Precipitation Evapotranspiration Index (SPEI-3) and identified the change points of mean and trend changes in the SPEI series and further explored the atmospheric circulation patterns before and after the transition from dry/wet conditions. Specifically, although the mean change point test indicated a wetter climate after 1986 at both the annual and seasonal scales, the trend analysis showed a shift from wet to dry from 1996, which has persisted until the present. There were significant interdecadal features in the spatial distribution of SPEI. Xinjiang exhibited a predominantly dry pattern from 1961 to 1980, which changed to a wetter pattern from 1981 to 2000, Xinjiang experienced its wettest decade from 1991 to 2000, and the whole region returned to a drier pattern from 2001 to 2020. From the annual and seasonal variation, the rate of change is greater in the south, with stronger change amplitude in summer and autumn and a higher frequency of anomalous SPEI. The results of the atmospheric circulation analysis demonstrated that the net water vapor flux in the Xinjiang region increased by about 8.30 g·kg−1·m·s−1 in the period 1987–2020 compared to 1961–1986 due to the combined interaction of high-, mid-, and low-latitude systems with the Central Asian low-pressure system. The results indicated a sharp increase in aridification in Xinjiang, providing valuable insights for disaster prevention, mitigation, and risk management related to regional drought hazards.

Understanding non‐stationarity patterns in basin‐scale hydroclimatic extremes

AbstractStationarity, a cornerstone in hydraulic design, is now under scrutiny due to anthropogenic activities and climate change. Numerous studies have sought to identify non‐stationarity (NS); however, a comprehensive assessment of time invariance in all statistical properties of a time series is less explored. This study presents a non‐overlapping block‐stratified random sampling (NBRS) framework leveraging the strengths of several nonparametric tests to assess NS. The NBRS approach exclusively detects NS and distinguishes between various forms of stationarity, including weak and strict. A variant of NBRS is proposed in this study to identify the underlying stochastic process(es) influencing NS in hydroclimatic extremes. Furthermore, a nonparametric clustering approach is used to unveil spatial clusters showcasing NS due to shifts in mean, variance, distribution of time series or a combination of these factors. A comparative assessment of the modified NBRS approach with traditional trend and change point methods is also performed. The proposed methodology is applied to assess the presence of NS in 28 hydroclimatic indices derived for the west‐central river basins of India, exhibiting diverse physio‐climatic settings, for the study period 1973–2021. The modified NBRS approach rigorously explores NS within extreme hydroclimatic indices, conclusively pinpointing its root causes and profound implications for hydrologic design. The applicability of the modified NBRS approach to gridded and point datasets is also demonstrated. The findings highlight the limitations of conventional trend and change point tests in capturing time‐invariant characteristics in heteroscedastic variables (such as streamflow and rainfall extremes) compared to the NBRS approach. The research reveals that NS in rainfall and streamflow extremes primarily results from distributional shifts, whilst temperature extremes are influenced by changes in mean and distribution properties. This research deepens our understanding of the evolving patterns in hydroclimatic extremes in a changing climate.

INTRODUCE SOME CONCEPTS OF THE FRAMES IN HILBERT AND BANACH SPACES IN RANDOM VARIABLES

In this work, we establish various functional-analytical features of these decompositions and demonstrate their applicability to wavelet and gabor systems. First, we demonstrate the stability of atomic decompositions and frames under minor perturbations. This is motivated by analogous classical perturbation outcomes for bases, such as the Kato perturbation theorem and the Paley-Wiener basis stability requirements. The methodological contributions are concentrated on creating confidence bands, change-point tests, and two-sample tests because these procedures seem appropriate for the suggested situation. The reason for its selection and analysis in the thesis is its connection to operators. Frame sequences are created, and an investigation is conducted into a class of operators connected to a specific Bessel sequence, which turns it into a frame for every operator in the class.

Change Point Test for Length-Biased Lognormal Distribution under Random Right Censoring

The length-biased lognormal distribution is a length-biased version of lognormal distribution, which is developed to model the length-biased lifetime data from, for example, biological investigation, medical research, and engineering fields. Owing to the existence of censoring phenomena in lifetime data, we study the change-point-testing problem of length-biased lognormal distribution under random censoring in this paper. A procedure based on the modified information criterion is developed to detect changes in parameters of this distribution. Under the sufficient condition of the Fisher information matrix being positive definite, it is proven that the null asymptotic distribution of the test statistic follows a chi-square distribution. In order to evaluate the uncertainty of change point location estimation, a way of calculating the coverage probabilities and average lengths of confidence sets of change point location based on the profile likelihood and deviation function is proposed. The simulations are conducted, under the scenarios of uniform censoring and exponential censoring, to investigate the validity of the proposed method. And the results indicate that the proposed approach performs better in terms of test power, coverage probabilities, and average lengths of confidence sets compared to the method based on the likelihood ratio test. Subsequently, the proposed approach is applied to the analysis of survival data from heart transplant patients, and the results show that there are differences in the median survival time post-heart transplantation among patients of different ages.

Robust change-point detection for functional time series based on U-statistics and dependent wild bootstrap

The aim of this paper is to develop a change-point test for functional time series that uses the full functional information and is less sensitive to outliers compared to the classical CUSUM test. For this aim, the Wilcoxon two-sample test is generalized to functional data. To obtain the asymptotic distribution of the test statistic, we prove a limit theorem for a process of U-statistics with values in a Hilbert space under weak dependence. Critical values can be obtained by a newly developed version of the dependent wild bootstrap for non-degenerate 2-sample U-statistics.

The maximally selected likelihood ratio test in random coefficient models

Summary In a recent contribution, we developed a family of cumulative sum-based change-point tests in the context of a random coefficient autoregressive model of order 1. In the current paper, we complement the results in that contribution by studying the (maximally selected) likelihood ratio statistic. We show that this has power versus breaks occurring even as close as periods from the beginning/end of sample; moreover, the use of quasi-maximum likelihood-based estimates yields better power properties, with the added bonus of being nuisance-free. Our test statistic has the same distribution—of the Darling–Erdős type—irrespective of whether the data are stationary or not, and can therefore be applied with no prior knowledge of this. Our simulations show that our test has very good power and, when applying a suitable correction to the asymptotic critical values, the correct size. We illustrate the usefulness and generality of our approach through applications to economic and epidemiological time series.

Analysis of Meteorological Element Variation Characteristics in the Heilongjiang (Amur) River Basin

Located in the Heilongjiang (Amur) River in north-east Asia, spanning four countries, plays a crucial role as an international border river, and its meteorological changes significantly impact the variation in water resources in the basin. This study utilizes daily average temperature and precipitation data from 282 meteorological stations in the Heilongjiang (Amur) River Basin and its surrounding areas for the period 1980–2022. The analysis employs spatial interpolation, change point testing, and model construction prediction methods. The results indicate a significant increasing trend in both overall temperature and precipitation changes within the Heilongjiang (Amur) River Basin. At the spatial scale, the annual warming rate increases gradually from the southeastern coastal region to the northwestern plateau region, while the rate of precipitation increase decreases from the southern area towards its surroundings. Temporally, the warming amplitude during the growing season decreases gradually from east to west, and the trend in precipitation changes during the growing season aligns with the overall annual precipitation trend. During the non-growing season, the warming trend shows a decrease in the plains and an increase in the plateau, while precipitation increase concentrates in the central and southern plains, and precipitation decrease predominantly occurs in the northwestern plateau region. Temperature and precipitation change points occurred in the years 2001 and 2012, respectively. In precipitation prediction, the Long Short-Term Memory (LSTM) model exhibits higher accuracy, with R (Pearson correlation coefficient) and NSE (Nash-Sutcliffe efficiency coefficient) values approaching 1 and lower NRSME values. This study provides a research foundation for the rational development and utilization of water resources in the Heilongjiang (Amur) River Basin and offers valuable insights for research on climate change characteristics in large transboundary river systems.

Hepatitis B time series in Xinjiang, China (2006-2021): change point detection based on the Mann-Kendall-Sneyers test.

Hepatitis B is a major global challenge, but there is a lack of epidemiological research on hepatitis B incidence from a change point perspective. This study aimed to fill this gap by identifying significant change points and trends in hepatitis time series in Xinjiang, China. The datasets were obtained from the Xinjiang Information System for Disease Control and Prevention. The Mann-Kendall-Sneyers (MKS) test was used to detect change points and trend changes on the hepatitis B time series of 14 regions in Xinjiang, and the effectiveness of this method was validated by comparing it with the binary segmentation (BS) and segment regression (SR) methods. Based on the results of change point analysis, the prevention and control policies and measures of hepatitis in Xinjiang were discussed. The results showed that 8 regions (57.1%) with at least one change fell within the 95% confidence interval (CI) in all 14 regions by the MKS test, where five regions (Turpan (TP), Hami (HM), Bayingolin (BG), Kyzylsu Kirgiz (KK), Altai (AT)) were identified at one change point, two change points existed for two regions (Aksu (AK), Hotan (HT)) and three change points was detected in 1 region (Bortala (BT)). Most of the change points occurred at both ends of the sequence. More change points indicated an upward trend in the front half of the sequence, while in the latter half, many change points indicated a downward trend prominently. Finally, in comparing the results of the three change point tests, the MKS test showed a 61.5% agreement (8/13) with the BS and SR.

Temporal changes in the frequency of flood types and their impact on flood statistics

Standard flood frequency analysis assumes stationarity of flood conditions, i.e., no change of the distribution over time. However, long-term variability in climate and anthropogenic impacts question this assumption. Consequently, more and more non-stationary models are considered in flood frequency analyses. Yet, most of them only consider a change-point or trend in the magnitude of flood peaks while ignoring changes in the underlying flood geneses. Recent climate reports suggest such a change in frequency of certain flood-generating factors, e.g., the increase of frequency of heavy-rainfall events. In this study, flood types are applied to detect changes in the meteorological drivers of flood regimes. By application of a robust change-point test for the variance based on Gini’s Mean Difference, significant changes in the frequency of occurrence of certain flood types are detected. A clear tendency to more frequent heavy-rainfall floods and less snowmelt-induced floods is observed for many catchments in Central Europe. A special focus is laid on the shifts in winter floods, which occur less often and are replaced by rainfall-driven floods. The impacts of such changes on flood statistics are demonstrated by several approaches. Though the magnitude of flood peaks does not (necessarily) change, the changing frequency of floods leads to changing flood quantiles. Quantile estimations from traditional statistical analyses of annual series are compared to results of type-based flood statistics. It is shown how standard models are more affected by these changes because they are not able to compensate for changes in the frequency of individual flood types.

The state of cumulative sum sequential changepoint testing 70 years after Page

Abstract Quality control charts aim at raising an alarm as soon as sequentially obtained observations of an underlying random process no longer seem to be within stochastic fluctuations prescribed by an in-control scenario. Such random processes can often be modelled using the concept of stationarity, or even independence as in most classical works. An important out-of-control scenario is the changepoint alternative, for which the distribution of the process changes at an unknown point in time. In his seminal 1954 Biometrika paper, E. S. Page introduced the famous cumulative sum control charts for changepoint monitoring. Innovatively, decision rules based on cumulative sum procedures took the full history of the process into account, whereas previous procedures were based only on a fixed and typically small number of the most recent observations. The extreme case of using only the most recent observation, often referred to as the Shewhart chart, is more akin to serial outlier than changepoint detection. Page’s cumulative sum approach, introduced seven decades ago, is ubiquitous in modern changepoint analysis, and his original paper has led to a multitude of follow-up papers in different research communities. This review is focused on a particular subfield of this research, namely nonparametric sequential, or online, changepoint tests that are constructed to maintain a desired Type-1 error as opposed to the more traditional approach seeking to minimize the average run length of the procedures. Such tests have originated at the intersection of econometrics and statistics. We trace the development of these tests and highlight their properties, mostly using a simple location model for clarity of exposition, but we also review more complex situations such as regression and time series models.

A test for change points under the roughness of stochastic volatility: the case of the VIX index

ABSTRACT In light of the recent empirical studies with high-frequency data, the logarithm of the stock market volatility data behaves as a fractional Brownian motion (fBm) with the Hurst exponent smaller than 0.5. It thus leads to extensive research in the so-called rough volatility (RV). This paper introduces a novel non-parametric test for its change points detection problem, which combines the proposed autoregressive rough volatility (ARRV) model and an increment ratio (IR)-based filtering function. The empirical results of the VIX index show that our method can detect more accurate change points without dependence on initial conditions/parameters, work efficiently for trends and forecast robustly better.

Analysis of driving forces on ecohydrological regime and environmental flow changes in Hongze Lake, China

Environmental flow refers to the fundamental hydrological process that sustains the ecosystem of rivers and lakes. Safeguarding environmental flow is crucial for optimal water resource utilization and lake ecosystem preservation. The study delivers a comprehensive exploration of ecohydrology, focusing on the common analysis of runoff and water level (RWL) to evaluate ecohydrological conditions in Hongze Lake (HZL) over five decades (1970–2020). Techniques such as the change point test and wavelet analysis were employed to discern trends and periodicity in hydrological elements. A series of ecologically pertinent indicators was applied to measure the changes in RWL, assessing their impacts on the lake's ecosystem. The Budyko hypothesis and cumulative slope rate of change (CSRC) were integrated for a quantitative analysis that shed light on the ecological response to RWL variations. Differential effects of human activities and climate change on runoff were discerned through subsurface parameter analysis across different decades. The study's results indicate that human activities, particularly gate dam operations and agricultural irrigation, have precipitated a significant reduction in environmental flow components (EFCs). The gate dam manipulation during the dry season has notably influenced the lake's hydrological regime, with a substantial alteration of RWL by 37% and 41%, respectively. This hydrological modification aligns with observed ecological trends such as species population reduction, habitat fragmentation, and organism miniaturization within HZL. Analysis of subsurface parameters also reflected an increasing influence of climatic factors on runoff over time. Human activities, evapotranspiration, and precipitation were found to contribute 75.5%, 10.3%, and 14.2% to runoff alterations, and 70.1%, 11.4%, and 18.5% to water level (WL) changes, respectively.

A Non-Stationarity Analysis of Annual Maximum Floods: A Case Study of Campaspe River Basin, Australia

A design flood is an essential input for water infrastructure design and flood protection. A flood frequency analysis has been traditionally performed under stationarity assumption indicating that the statistical properties of historical flooding will not change over time. Climate change and variability challenges the stationarity assumption, and a flood frequency analysis without consideration of non-stationarity can result in under- or overestimation of the design floods. In this study, non-stationarity of annual maximum floods (AMFs) was investigated through a methodology consisting of trend and change point tests, and non-stationary Generalized Extreme Value (NSGEV) models, and the methodology was applied to Campaspe River Basin as a case study. Statistically significant decreasing trends in AMFs were detected for almost all stations at the 0.01 significance level in Campaspe River Basin. NSGEV models outperformed the stationary counterparts (SGEV) for some stations based on statistical methods (i.e., Akaike information criterion (AIC) and Bayesian information criterion (BIC)) and graphical approaches (i.e., probability and quantile plots). For example, at Station 406235, AIC and BIC values were found to be 334 and 339, respectively, for the SGEV model, whereas AIC and BIC values were calculated as 330 and 334, respectively, for the NSGEV 15 model with time-varying location and scale parameters. Deriving a design flood from conventional stationary models will result in uneconomical water infrastructure design and poor water resource planning and management in the study basin.

Flexible nonlinear inference and change-point testing of high-dimensional spectral density matrices

This paper studies a flexible approach to analyze high-dimensional nonlinear time series of unconstrained dimension based on linear statistics calculated from spectral average statistics of bilinear forms and nonlinear transformations of lag-window (i.e. band-regularized) spectral density matrix estimators. That class of statistics includes, among others, smoothed periodograms, nonlinear statistics such as coherency, long-run-variance estimators and contrast statistics related to factorial effects as special cases. Especially, we introduce the class of nonlinear spectral averages of the spectral density matrix. Having in mind big data settings, we study a sampling design which includes a sparse sampling scheme. Gaussian approximations with optimal rate are derived for nonlinear time series of growing dimension for these frequency domain statistics and the underlying lag-window (cross-) spectral estimator under non-stationarity. For change-testing (self-standardized) CUSUM statistics are examined. Further, a specific wild bootstrap procedure is proposed to estimate critical values. Simulation studies and an application to SP500 financial returns are provided in a supplement to this paper.

Assessing the Vegetation Dynamics and Its Influencing Factors in Central Asia from 2001 to 2020

As vegetation plays a critical role in terrestrial ecosystems, understanding its status and variation is vital for preserving the stability of an ecosystem. Central Asia serves as a representative example of an arid and semi-arid region characterized by sparse vegetation and poor soils, making its vegetation particularly fragile and sensitive. To investigate the vegetation condition in the region, this study examined the spatial and temporal characteristics of vegetation variation from 2001 to 2020, utilizing the normalized difference vegetation index (NDVI) as an indicator. Meanwhile, trend analysis, Mann–Kendall abrupt change point test, geodetector, and correlation analysis were used to quantitatively analyze the natural and anthropogenic drivers of these variations over the past two decades. The results suggest that vegetation coverage in Central Asia was relatively low, with an annual average NDVI of 0.16 over the past 20 years. Moreover, the spatial distribution of NDVI in Central Asia exhibited significant spatial heterogeneity, with vegetation coverage declining from north to south and from east to west. Furthermore, the NDVI exhibited a slightly increasing trend during the period of 2001 to 2020 with an increased rate of 0.00025/yr. However, we detected an abrupt change point in vegetation dynamics in Central Asia around 2010, which indicated a significant shift in vegetation variation in the region. Land-use type has a great influence on the spatial heterogeneity of NDVI in Central Asia, which can explain 46% of the vegetation distribution in this region. Moisture factors such as precipitation and soil water content followed with 35% and 32% contributions, respectively. Regarding the temporal variation of NDVI, it is mainly driven by the fluctuation in precipitation, with the degree of influence of precipitation on NDVI varying for different regions in various geographical conditions. This study offers a more comprehensive insight into the spatial and temporal dynamics of NDVI in Central Asia and indicates that precipitation plays a significant role in driving the spatial heterogeneity and temporal variation of NDVI. These findings are essential for predicting vegetation changes in arid regions under future environmental conditions and formulating effective strategies to prevent and alleviate vegetation degradation.

Computationally efficient and data-adaptive changepoint inference in high dimension

Abstract High-dimensional changepoint inference that adapts to various change patterns has received much attention recently. We propose a simple, fast yet effective approach for adaptive changepoint testing. The key observation is that two statistics based on aggregating cumulative sum statistics over all dimensions and possible changepoints by taking their maximum and summation, respectively, are asymptotically independent under some mild conditions. Hence, we are able to form a new test by combining the p-values of the maximum- and summation-type statistics according to their asymptotic null distributions. To this end, we develop new tools and techniques to establish the asymptotic distribution of the maximum-type statistic under a more relaxed condition on componentwise correlations among all variables than those in existing literature. The proposed method is simple to use. It is adaptive to different levels of the sparsity of change signals, and is comparable to or even outperforms existing approaches as revealed by our numerical studies.

Change analysis of All India and regional rainfall data series at annual and monsoon scales

AbstractRainfall characteristics are changing due to several reasons and change/trend detection is required. Literature survey reveals many relevant studies whose outcomes are divergent, possibly because different data series and different methodologies have been applied. This paper presents a critical appraisal of past studies and methodologies for trend analysis. Results of trend analysis of Indian rainfall data are presented. Data for all of India and for five homogenous regions (North-West, Central North-East, North-East, West Central, and Peninsular India) for 1871–2016 were used. The Pettitt change point test, regression, Mann-Kendall (MK), and Wavelet Decomposition were used to study different aspects of changes. Results of the change point test showed that most rainfall series had change points around 1957–65, possibly due to large-scale land use, cultivation, irrigation, and industrial changes in this period. Generally, rainfall for most homogenous regions and sub-divisions show falling trends; some are statistically significant. Series was also decomposed by the wavelet method. Approximate and detailed components of some decomposed series showed a significant declining trend. This work has focused on the magnitude of rainfalls; trends in rainfall intensities are also important. It is necessary to establish denser observation networks to collect short-term data and analyze.

Quality assessment of hydrometeorological observational data and their influence on hydrological model results in Alpine catchments

ABSTRACT Hydrometeorological observational data can be affected by trends, change points and inhomogeneities. It is often difficult to identify the cause of these statistical properties of the time series and therefore they represent a source of uncertainty that can propagate to the hydrological model results. We examine two case studies of high-elevation Alpine catchments for which 59 hydrometeorological time series are analysed to show how often trends, change points and inhomogeneities in the time series occur. We consider daily and monthly aggregated data, as well as a set of nine homogeneity tests, six trend tests and three change-point tests. The results show that inhomogeneities in meteorological time series have a stronger and more direct influence on the cryosphere (snow and glacier processes) than on runoff generation. The non-linear behaviour of hydrological processes greatly complicates the judgement about the consequences of including or excluding suspicious time series in hydrological model set-ups.

District-Level Seasonal Rainfall Characteristics over Andhra Pradesh and its Global Teleconnections in Changing Climate

Investigating the trends and changes in rainfall over vulnerable regions is of huge importance in this global warming era. The present study intensively investigates the rainfall over the Indian state, Andhra Pradesh (AP), and its 13 districts using a high-resolution (0.25°×0.25°) gridded rainfall analysis dataset from India Meteorological Department (IMD) for the study period of 118 years (1901-2018). For this, normality, homogeneity, persistence, and change-point tests are performed and changes in the district-level rainfall in the present global warming period (1991-2018) as compared to the pre-global warming period (1901-1990) is also analyzed.
 The results suggest that the long-term average annual rainfall over AP is 882 mm and most of the rainfall is contributed by the monsoon (55.7%) and the post-monsoon rainfall (32.8%). The coefficient of variation is low (high) during monsoon (winter). The coastal region receives more rainfall than the inland districts. The post-monsoon rainfall over AP is more consistent than in other seasons, and the persistence is only during the southwest monsoon season. The southwest monsoon and post-monsoon rainfall have increased (by about 10%) over most of the districts in the recent period. The Nino3.4 region SST (South Oscillation Index; SOI) has a significant negative (positive) relationship with southwest summer monsoon rainfall in most of the districts. The relationship of Nino 3.4 SST and DMI is strikingly similar for post-monsoon and has significantly weakened in recent decades. This study is useful for proper planning and mitigation measures for the agricultural and water resources sector at the district level over AP in this global warming era.