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An example plot, from model B, of the negative log-likelihood against number of changepoints for the purpose of identifying the number of changepoints.
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This article proposes a test to detect changes in general autocovariance structure in nonstationary time series. Our approach is founded on the locally stationary wavelet (LSW) process model for time series which has previously been used for classification and segmentation of time series. Using this framework we form a likelihood-based hypothesis t...
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... involves producing a plot of J(τ K , x) against K and visually identifying the point of maximum curvature. Figure 2(a) shows an example, from model B in Section 4, of the type of graphic produced by this method where the true number of changes is 2. For this example the values of l k are 150, 138, 29, 29, 20, . . . from which it is clear that l i ≫ l j for i = 2. ...
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Citations
... Compared to CUSUM-type methods, KCP can locate multiple change points simultaneously. Change point analysis in time series has been studied by [19] and [12] and the references therein. However, in the high-dimensional setting, where the dimension can be much larger than the sample size, the methods constructed under the low-dimensional setting either perform poorly or are not even well defined. ...
... We begin by reviewing the locally stationary wavelet (LSW) model (Nason et al., 2000) and its associated forecasting framework . The LSW approach for modelling nonstationary time series has been used in many fields, from climatology (Fryzlewicz, 2003) and ocean engineering (Killick et al., 2013) to biology (Hargreaves et al., 2019), medicine (Nason and Stevens, 2015;Embleton et al., 2022a) and finance (Fryzlewicz, 2005). ...
Accurate forecasting of the U.K. gross value added (GVA) is fundamental for measuring the growth of the U.K. economy. A common nonstationarity in GVA data, such as the ABML series, is its increase in variance over time due to inflation. Transformed or inflation-adjusted series can still be challenging for classical stationarity-assuming forecasters. We adopt a different approach that works directly with the GVA series by advancing recent forecasting methods for locally stationary time series. Our approach results in more accurate and reliable forecasts, and continues to work well even when the ABML series becomes highly variable during the COVID pandemic.
... Early studies were concerned with the stability of the mean (Page, 1954(Page, , 1955. The methodology has since been extended, and there exist procedures to test for, e.g., the stability of variances (Gao et al., 2019), regression coefficients (Horváth, 1995), or autocovariances (Berkes et al., 2009;Killick et al., 2013;Preuss et al., 2015). Our approach provides a unifying framework to study these problems for parameters θ u which may be written as a function of nonlinear moments. ...
... The case of a nonparametric mean function is investigated by Li and Zhao (2013), and multiple change-points are studied by Preuss et al. (2015). The non-stationary case is investigated by Killick et al. (2013), although without a rigorous analysis of the type I error. ...
Tests for structural breaks in time series should ideally be sensitive to breaks in the parameter of interest, while being robust to nuisance changes. Statistical analysis thus needs to allow for some form of nonstationarity under the null hypothesis of no change. In this paper, estimators for integrated parameters of locally stationary time series are constructed and a corresponding functional central limit theorem is established, enabling change-point inference for a broad class of parameters under mild assumptions. The proposed framework covers all parameters which may be expressed as nonlinear functions of moments, for example kurtosis, autocorrelation, and coefficients in a linear regression model. To perform feasible inference based on the derived limit distribution, a bootstrap variant is proposed and its consistency is established. The methodology is illustrated by means of a simulation study and by an application to high-frequency asset prices.
... Efficient decomposition algorithms (Mallat, 1989;Shensa, 1992) make multiresolution analysis (MRA) computationally feasible. Wavelet decomposition over time (Killick et al., 2013), while useful for detecting second-order structural change, does not support a sequential updating scheme. Instead, a two-dimensional decomposition over space is considered. ...
Global earth monitoring aims to identify and characterize land cover change like construction as it occurs. Remote sensing makes it possible to collect large amounts of data in near real-time over vast geographic areas and is becoming available in increasingly fine temporal and spatial resolution. Many methods have been developed for data from a single pixel, but monitoring pixel-wise spectral measurements over time neglects spatial relationships, which become more important as change manifests in a greater number of pixels in higher resolution imagery compared to moderate resolution. Building on our previous robust online Bayesian monitoring (roboBayes) algorithm, we propose monitoring multiresolution signals based on a wavelet decomposition to capture spatial change coherence on several scales to detect change sites. Monitoring only a subset of relevant signals reduces the computational burden. The decomposition relies on gapless data; we use 3 m Planet Fusion Monitoring data. Simulations demonstrate the superiority of the spatial signals in multiresolution roboBayes (MR roboBayes) for detecting subtle changes compared to pixel-wise roboBayes. We use MR roboBayes to detect construction changes in two regions with distinct land cover and seasonal characteristics: Jacksonville, FL (USA) and Dubai (UAE). It achieves site detection with less than two thirds of the monitoring processes required for pixel-wise roboBayes at the same resolution.
... Hence, there are periods in the summer months where the ice melts and Surface Temperature is recorded as 0 o C for prolonged periods. We detrend the sequences by applying a first difference to the AWS variables; an approach often used in the changepoint literature (Killick et al., 2013), and assume that the resulting sequences are each sampled independently from a Normal distribution with zero mean. Prior to differencing Surface Temperature, we perturb the values by adding independent noise, sampled from a N(0, 0.1 2 ) distribution, to avoid estimates of zero variance in the summer months caused by the upper bound of the measurements. ...
Different environmental variables are often monitored using different sampling rates; examples include half‐hourly weather station measurements, daily CO2$$ {\mathrm{CO}}_2 $$ data, and six‐day satellite data. Further when researchers want to combine the data into a single analysis this often requires data aggregation or down‐scaling. When one is seeking to identify changes within multivariate data, the aggregation and/or down‐scaling processes obscure the changes we seek. In this article, we propose a novel changepoint detection algorithm which can analyze multiple time series for co‐occurring changepoints with potentially different sampling rates, without requiring preprocessing to a standard sampling scale. We demonstrate the algorithm on synthetic data before providing an example identifying simultaneous changes in multiple variables at a location on the Greenland ice sheet using synthetic aperture radar and weather station data.
... For the detection of changes in µ we mention the nonparametric methods of Wolfe and Schechtman (1984); Horváth et al. (2008); Jarušková (2010); or Dehling et al. (2013), and in particular regarding change points on multiple time scales we refer to Spokoiny (2009); Fryzlewicz (2014); Matteson and James (2014). For the detection of changes in σ 2 we mention the articles of Hsu (1977); Inclán and Tiao (1994); Chen and Gupta (1997); Whitcher et al. (2000); Killick et al. (2013); Korkas and Fryzlewicz (2017) and a paper for changes in scale by Gerstenberger et al. (2020). Combining both, changes in µ and σ 2 , we refer to Pein et al. (2017) who aim at detecting changes in µ by allowing σ 2 to change simultaneously. ...
A method for change point detection is proposed. We consider a univariate sequence of independent random variables with piecewise constant expectation and variance, apart from which the distribution may vary periodically. We aim to detect change points in both expectation and variance. For that, we propose a statistical test for the null hypothesis of no change points and an algorithm for change point detection. Both are based on a bivariate moving sum approach that jointly evaluates the mean and the empirical variance. The joint consideration helps improve inference as compared to separate univariate approaches. We infer on the strength and the type of changes with confidence. Nonparametric methodology supports the analysis of diverse data. Additionally, a multi‐scale approach addresses complex patterns in change points and effects. We demonstrate the performance through theoretical results and simulation studies. A companion R‐package jcp (available on CRAN) is discussed.
... The problem of change-points detection has been studied by researchers for many decades. This problem has received much attention in various fields as statistics, engineering, economics, climatology, and bioscience (Braun and Müller 1998;Reeves et al. 2007;Killick et al. 2013). Specifically, given an ordered sequence of observations by time or location, a change-point is a location or time at which the structure of the sequence changes. ...
This study considers the problem of multiple change-points detection. For this problem, we develop an objective Bayesian multiple change-points detection procedure in a normal model with heterogeneous variances. Our Bayesian procedure is based on a combination of binary segmentation and the idea of the screening and ranking algorithm (Niu and Zhang in Ann Appl Stat 6:1306–1326, 2012). Using the screening and ranking algorithm, we can overcome the drawbacks of binary segmentation, as it cannot detect a small segment of structural change in the middle of a large segment or segments of structural changes with small jump magnitude. We propose a detection procedure based on a Bayesian model selection procedure to address this problem in which no subjective input is considered. We construct intrinsic priors for which the Bayes factors and model selection probabilities are well defined. We find that for large sample sizes, our method based on Bayes factors with intrinsic priors is consistent. Moreover, we compare the behavior of the proposed multiple change-points detection procedure with existing methods through a simulation study and two real data examples.
... In practice, however, the assumptions imposed by modelling an observed time series as stationary can be restrictive and unrealistic. Nonstationary time series, i.e. series with time-varying second order structure, are observed in various fields including finance (Stȃricȃ and Granger 2005;Fryzlewicz et al. 2006), life sciences (Cranstoun et al. 2002;Hargreaves et al. 2019) and oceanography (Killick et al. 2013). Hence, the ability to impute missing values in multivariate, nonstationary time series has potential widespread benefits. ...
Many multivariate time series observed in practice are second order nonstationary, i.e. their covariance properties vary over time. In addition, missing observations in such data are encountered in many applications of interest, due to recording failures or sensor dropout, hindering successful analysis. This article introduces a novel method for data imputation in multivariate nonstationary time series, based on the so-called locally stationary wavelet modelling paradigm. Our methodology is shown to perform well across a range of simulation scenarios, with a variety of missingness structures, as well as being competitive in the stationary time series setting. We also demonstrate our technique on data arising in a health monitoring application.
... If a change is detected, then the location of the changepoint,p, is estimated as the value of p that maximises λ. Note that a similar test statistic (with zero mean and changepoint matrix Σ) was applied to the problem of detecting changes in a piecewise constant autocovariance function, in Killick, Eckley, and Jonathan (2013). Under the simplifying assumption of IID Gaussian errors with known variance, this reduces to the standard likelihood test for a change in mean, equivalent to the cumulative sum (CUSUM) test statistic. ...
There has been much attention in recent years to the problem of detecting mean changes in a piecewise constant time series. Often, methods assume that the noise can be taken to be independent, identically distributed (IID), which in practice may not be a reasonable assumption. There is comparatively little work studying the problem of mean changepoint detection in time series with non‐trivial autocovariance structure. In this article, we propose a likelihood‐based method using wavelets to detect changes in mean in time series that exhibit time‐varying autocovariance. Our proposed technique is shown to work well for time series with a variety of error structures via a simulation study, and we demonstrate its effectiveness on two data examples arising in economics.
... The properties of the classical CUSUMs statistic in the variance break detection in general setting were examined by several authors. We can cite for example [6] and [9] for ARCH models, [11] for standardized residuals from an estimated GARCH model, [15] for a class of nonstationary and nonparametric time series models, [4] for ongoing monotring process and [10] for detecting changes in general autocovariance structure in nonstationary time series using locally stationary wavelet process. Our approach is inspired by the application of the discrete wavelet transform in [3] for testing homogeneity of variance in a time series with long memory structure. ...
... Under the assumption that X t , t = 1, · · · , N is a sequence of independent and identically distributed (i.i.d) normal random variables. This assumption was later relaxed to allow for various form of dependence and heterogeneity in X t , see [7], [10] and [13]. The limiting distribution of this statistic depend on the long-run variance σ 2 = ∞ k=−∞ γ k where γ k is the k-th order autocovariance of the series X 2 t which should be estimated. ...
... where K j (β) is a constant and J j=1 K j (β)L j = B J (β). Using Minkowski inequality |Z J,t | β ≤ 2 β−1 (|C J,t | β +|µ C J | β ) it follows that sup t E|Z J,t | β < ∞ and then (10) Let Y j,t = W 2 j,t − E(W 2 j,t ), then it follows from proof of theorem 1 in [17], page 195 that α Yj (k) ≤ α Wj (k) for any 1 ≤ j ≤ J. Therefore (11) follows and ...
