Time Series Analysis

Journal Articles

1. Mean shift testing in correlated data

   M. W. Robbins, C. M. Gallagher, R. B. Lund, and A. Aue

   Journal of Time Series Analysis, 2011

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   Several tests for detecting mean shifts at an unknown time in stationary time series have been proposed, including cumulative sum (CUSUM), Gaussian likelihood ratio (LR), maximum of F(Fmax) and extreme value statistics. This article reviews these tests, connects them with theoretical results, and compares their finite sample performance via simulation. We propose an adjusted CUSUM statistic which is closely related to the LR test and which links all tests. We find that tests based on CUSUMing estimated one-step-ahead prediction residuals from a fitted autoregressive moving average perform well in general and that the LR and Fmax tests (which induce substantial computational complexities) offer only a slight increase in power over the adjusted CUSUM test. We also conclude that CUSUM procedures work slightly better when the changepoint time is located near the centre of the data, but the adjusted CUSUM methods are preferable when the changepoint lies closer to the beginning or end of the data record. Finally, an application is presented to demonstrate the importance of the choice of method.

2. Changepoints in the North Atlantic tropical cyclone record

   M. W. Robbins, R. B. Lund, C. M. Gallagher, and Q. Lu

   Journal of the American Statistical Association, 2011

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   This article examines the North Atlantic tropical cyclone record for statistical discontinuities (changepoints). This is a controversial area and indeed, our end conclusions are opposite of those made in Dr. Kelvin Droegemeier’s July 28, 2009 Senate testimonial. The methods developed here should help rigorize the debate. Elaborating, we develop a level-α test for a changepoint in a categorical data sequence sampled from a multinomial distribution. The proposed test statistic is the maximum of correlated Pearson chi-square statistics. This test statistic is linked to cumulative sum statistics and its null hypothesis asymptotic distribution is derived in terms of the supremum of squared Brownian bridges. The methods are used to identify changes in the tropical cyclone record in the North Atlantic Basin over the period 1851–2008. We find changepoints in both the storm frequencies and their strengths (wind speeds). The changepoint in wind speed is not found with standard cumulative sum mean shift changepoint methods, hence providing a dataset where categorical probabilities shift but means do not. While some of the identified shifts can be attributed to changes in data collection techniques, the hotly debated changepoint in cyclone frequency circa 1995 also appears to be significant.

3. Changepoint detection in daily precipitation data

   C. M. Gallagher, R. B. Lund, and M. W. Robbins

   Environmetrics, 2012

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   This paper introduces a method to identify an undocumented changepoint time in a daily precipitation series. A two-state Markov chain is used to induce dependence in the precipitation amounts; our dynamics allow for seasonality in the daily observations, a structure inherent to many nonequatorial region series. No current precipitation changepoint techniques exist that consider day-to-day dependencies, the zero support set aspect (the fact that most measurements are zero), and the periodic dynamics of the problem. The test statistic is constructed by applying cumulative sum methods to a strategically devised set of one-step-ahead prediction residuals. The methods are robust to distributional assumptions, requiring only seasonal mean and transition probability estimators. Simulations are presented that demonstrate the efficacy of the methods; application to two daily precipitation series is made.

4. Changepoint detection in climatic time series with long-term trends

   C. Gallagher, R. B. Lund, and M. W. Robbins

   Journal of Climate, 2013

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   Climate time series often have artificial shifts induced by instrumentation changes, station relocations, observer changes, etc. Climate time series also often exhibit long-term trends. Much of the recent literature has focused on identifying the structural breakpoint time(s) of climate time series—the so-called changepoint problem. Unfortunately, application of rudimentary mean-shift changepoint tests to scenarios with trends often leads to the erroneous conclusion that a mean shift occurred near the series’ center. This paper examines this problem in detail, constructing some simple homogeneity tests for series with trends. The asymptotic distribution of the proposed statistic is derived; en route, an attempt is made to unify the asymptotic properties of the changepoint methods used in today’s climate literature. The tests presented here are linked to the ubiquitous t test. Application is made to two temperature records: 1) the continental United States record and 2) a local record from Jacksonville, Illinois.

5. A general regression changepoint test for time series data

   M. W. Robbins, C. M. Gallagher, and R. B. Lund

   Journal of the American Statistical Association, 2016

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   This article develops a test for a single changepoint in a general setting that allows for correlated time series regression errors, a seasonal cycle, time-varying regression factors, and covariate information. Within, a changepoint statistic is constructed from likelihood ratio principles and its asymptotic distribution is derived. The asymptotic distribution of the changepoint statistic is shown to be invariant of the seasonal cycle and the covariates should the latter obey some simple limit laws; however, the limit distribution depends on any time-varying factors. A new test based on ARMA residuals is developed and is shown to have favorable properties with finite samples. Driving our work is a changepoint analysis of the Mauna Loa record of monthly carbon dioxide concentrations. This series has a pronounced seasonal cycle, a nonlinear trend, heavily correlated regression errors, and covariate information in the form of climate oscillations. In the end, we find a prominent changepoint in the early 1990s, often attributed to the eruption of Mount Pinatubo, which cannot be explained by covariates. Supplementary materials for this article are available online.

6. A fully flexible changepoint test for regression models with stationary errors

   M. W. Robbins

   Statistica Sinica, 2020

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   Temporal discontinuities in time series represent one of the classic problems of time series. Such discontinuities are often analyzed by detecting changes at specific times in the parameters governing a regression model fit to the series. The regression framework examined here contains three classes of predictors: functional form, seasonal, and stochastic. Regression errors are allowed to observe a general stationary structure. Methods are proposed that provide the analyst with full flexibility in selecting which set of regression parameters are allowed to change under the alternative hypothesis. Here, we also examine several mathematical complications that arise in the development of such procedures. A simulation study illustrates the efficacy of the proposed methodology, where a test statistic based on the residuals from an ARMA model is shown to perform most favorably. The methods are applied to a carbon dioxide time series measured at Mauna Loa Observatory, where a shift in the seasonal variations is detected (in addition to a known shift in trend), and to a series of monthly temperatures at Barrow, Alaska, where only a shift in trend is found.

7. An improved measure for lack of fit in time series models

   T. J. Fisher and M. W. Robbins

   Statistica Sinica, 2018

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   The correlation structure of time series is of fundamental importance in diagnostic procedures. The squared autocorrelation function of the residuals of a fitted model is generally used as a measure of the goodness-of-fit; multivariate analogues are available for vector time series. As an alternative, we propose a logarithmic transformation of the determinant of a constructed Toeplitz matrix containing the typical measure of correlation. We show that the proposed measure is asymptotically more powerful than the typical measure of correlation (when used with or without the Ljung-Box correction) in the detection of a variety of residual dependence structures. The proposed method is shown to have utility when applied in conjunction with a host of methods used to diagnose the fit of strong and weak autoregressive moving average models and generalized autoregressive conditional heteroskedastic models. A simulation study demonstrates the effectiveness of the proposed method and illustrates its improvement over the existent procedures.

8. A cheap trick to improve the power of a conservative hypothesis test

   T. J. Fisher and M. W. Robbins

   The American Statistician, 2019

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   Critical values and p-values of statistical hypothesis tests are often derived using asymptotic approximations of sampling distributions. However, this sometimes results in tests that are conservative (i.e., understate the frequency of an incorrectly rejected null hypothesis by employing too stringent of a threshold for rejection). Although computationally rigorous options (e.g., the bootstrap) are available for such situations, we illustrate that simple transformations can be used to improve both the size and power of such tests. Using a logarithmic transformation, we show that the transformed statistic is asymptotically equivalent to its untransformed analogue under the null hypothesis and is divergent from the untransformed version under the alternative (yielding a potentially substantial increase in power). The transformation is applied to several easily-accessible statistical hypothesis tests, a few of which are taught in introductory statistics courses. With theoretical arguments and simulations, we illustrate that the log transformation is preferable to other forms of correction (such as statistics that use a multiplier). Finally, we illustrate application of the method to a well-known dataset. Supplementary materials for this article are available online.

9. Cross-correlation matrices for tests of independence and causality between two multivariate time series

   M. W. Robbins and T. J. Fisher

   Journal of Business and Economic Statistics, 2015

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   An often-studied problem in time series analysis is that of testing for the independence of two (potentially multivariate) time series. Toeplitz matrices have demonstrated utility for the related setting of time series goodness-of-fit testing—ergo, herein, we extend those concepts by defining a nontrivial block Toeplitz matrix for use in the setting of independence testing. We propose test statistics based on the trace of the square of the matrix and determinant of the matrix; these statistics are connected to one another as well as known statistics previously proposed in the literature. Furthermore, the log of the determinant is argued to relate to a likelihood ratio test and is proven to be more powerful than other tests that are asymptotically equivalent under the null hypothesis. Additionally, matrix-based tests are presented for the purpose of inferring the location or direction of the causality existing between the two series. A simulation study is provided to explore the efficacy of the proposed methodology—the methods are shown to offer improvement over existing techniques, which include the famous Granger causality test. Finally, data examples involving U.S. inflation, trade volume, and exchange rates are given. Supplementary materials for this article are available online.

Book Chapters

1. A statistical analysis of North Atlantic tropical cyclone changes

   T. J. Fisher, R. B. Lund, and M. W. Robbins

   In Quantitative Approaches to Evaluating Climate Change Impacts, 2020

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