Colin M. Gallagher

Changepoint Detection in Periodic and Autocorrelated Time Series

Abstract Undocumented changepoints (inhomogeneities) are ubiquitous features of climatic time series. Level shifts in time series caused by changepoints confound many inference problems and are very important data features. Tests for undocumented changepoints from models that have independent and identically distributed errors are by now well understood. However, most climate series exhibit serial autocorrelation. Monthly, daily, or hourly series may also have periodic mean structures. This article develops a test for undocumented changepoints for periodic and autocorrelated time series. Classical changepoint tests based on sums of squared errors are modified to take into account series autocorrelations and periodicities. The methods are applied in the analyses of two climate series.

Discrimination of Chinook Salmon, Coho Salmon, and Steelhead Redds and Evaluation of the Use of Redd Data for Estimating Escapement in Several Unregulated Streams in Northern California

Abstract We developed and evaluated a stratified index redd area method to estimate Chinook salmon Oncorhynchus tshawytscha, coho salmon O. kisutch, and steelhead O. mykiss escapement in several coastal streams in northern California based on the assumption that redd size is related to the number of redds a female builds. Sources of error in redd counts were identified, including the use of logistic regression to classify redd species (necessary due to temporal overlap in the spawning of these species in coastal northern California). Redd area escapement estimates were compared with estimates from more conventional methods and releases above a counting structure. Observer efficiency in redd detection ranged from 0.64 (SE = 0.10) to 0.75 (SE = 0.14) and was significantly associated with streamflow and water visibility (analysis of variance (ANOVA): F = 41.8; P < 0.001). Logistic regression reduced uncertainty in redd identification. Redd area and date observed were significant in predicting coho salmon and...

Mean Shift Testing in Correlated Data

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(F) 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 F 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.

Changepoints in the North Atlantic Tropical Cyclone Record

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.

New Weighted Portmanteau Statistics for Time Series Goodness of Fit Testing

We exploit ideas from high-dimensional data analysis to derive new portmanteau tests that are based on the trace of the square of the mth order autocorrelation matrix. The resulting statistics are weighted sums of the squares of the sample autocorrelation coefficients that, unlike many other tests appearing in the literature, are numerically stable even when the number of lags considered is relatively close to the sample size. The statistics behave asymptotically as a linear combination of chi-squared random variables and their asymptotic distribution can be approximated by a gamma distribution. The proposed tests are modified to check for nonlinearity and to check the adequacy of a fitted nonlinear model. Simulation evidence indicates that the proposed goodness of fit tests tend to have higher power than other tests appearing in the literature, particularly in detecting long-memory nonlinear models. The efficacy of the proposed methods is demonstrated by investigating nonlinear effects in Apple, Inc., and Nikkei-300 daily returns during the 2006–2007 calendar years. The supplementary materials for this article are available online.

A new test for sphericity of the covariance matrix for high dimensional data

In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N=n+1. Under the assumptions that (A) 0~ for i=1,...,16 and (B) p/n->c ~. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature.

Departures from Gibrat's Law, Discontinuities and City Size Distributions

Cities are complex, self-organising, evolving systems and the emergent patterns they manifest provide insight into the dynamic processes in urban systems. This article analyses city size distributions, by decade, from the south-eastern region of the US for the years 1860—1990. It determines if the distributions are clustered into size classes and documents changes in the pattern of size classes over time. A statistical hypothesis test was also performed to detect dependence between city size and growth using discrete probability calculations under the assumption of Gibrats law. The city size distributions for the south-eastern region of the US were discontinuous, with cities clustering into distinct size classes. The analysis also identified departures from Gibrats law, indicating variable growth rates at different scales.

A method for fitting stable autoregressive models using the autocovariation function

We use the sample covariations to estimate the parameters in a univariate symmetric stable autoregressive process. Unlike the sample correlation, the sample covariation can be used to estimate the tail decay parameter of the process. The fitted model will be consistent with the dependence as measured by the covariation. The limit distribution of the sample covariation can be used to derive confidence intervals for the autoregressive parameter in a first order process. Simulations show that confidence intervals coming from the covariation have better coverage probabilities than those coming from the sample correlations. The method is demonstrated on a time series of sea surface temperatures.

Changepoint Detection in Climate Time Series with Long-Term Trends

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 oftenleadstotheerroneousconclusionthatameanshiftoccurredneartheseries’center.Thispaperexamines this problem in detail, constructing some simple homogeneity tests for series with trends. The asymptotic distributionoftheproposedstatisticisderived;enroute,anattemptismadetounifytheasymptoticproperties 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.

VARIANCE ESTIMATION IN NONPARAMETRIC MULTIPLE REGRESSION

ABSTRACT We consider the variance estimation in a general nonparametric regression model with multiple covariates. We extend difference methods to the multivariate setting by introducing an algorithm that orders the design points in higher dimensions. We also consider an adaptive difference estimator which requires much less strict assumptions on the covariate design and can significantly reduce mean squared error for small sample sizes.