A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

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Ensemble empirical mode decomposition: a noise-assisted data analysis method

Statistics for Spatial Data.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has been generalized by constructing a “blocks of blocks” resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distribution of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-time series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural...

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The stationary bootstrap

In this article, the construction of confidence regions by approximating the sampling distribution of some statistic is studied. The true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data. In the i.i.d. context, the method has been studied by Wu in regular situations where the statistic is asymptotically normal. The goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions. Essentially, all that is required is that the statistic, suitably normalized, possesses a limit distribution under the true model. Unlike the bootstrap, the convergence to the limit distribution need not be uniform in any sense. The method is readily adapted to parameters of stationary time series or, more generally, homogeneous random fields. For example, an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field.

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Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions

We review the different block bootstrap methods for time series, and present them in a unified framework. We then revisit a recent result of Lahiri [Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods, Ann. Statist. 27:386–404] comparing the different methods and give a corrected bound on their asymptotic relative efficiency; we also introduce a new notion of finite-sample “attainable” relative efficiency. Finally, based on the notion of spectral estimation via the flat-top lag-windows of Politis and Romano [Politis, D. N., Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16:67–103], we propose practically useful estimators of the optimal block size for the aforementioned block bootstrap methods. Our estimators are characterized by the fastest possible rate of convergence which is adaptive on the strength of the correlation of the time series as measured by the correlogram.

/pdf/automatic-block-length-selection-for-the-dependent-bootstrap-399w4bew8g.pdf

Automatic Block-Length Selection for the Dependent Bootstrap

Sparked by Efron's seminal paper, the decade of the 1980s was a period of active research on bootstrap methods for independent data--mainly i.i.d. or regression set-ups. By contrast, in the 1990s much research was directed towards resampling dependent data, for example, time series and random fields. Consequently, the availability of valid nonparametric inference procedures based on resampling and/or subsampling has freed practitioners from the necessity of resorting to simplifying assumptions such as normality or linearity that may be misleading.

The Impact of Bootstrap Methods on Time Series Analysis

A correction on the optimal block size algorithms of Politis and White (2004) is given following a correction of Lahiri's (Lahiri 1999) theoretical results by Nordman (2008).

http://www.math.ucsd.edu/~politis/PAPER/SBblockCORRECTION.pdf

Dimitris N. Politis

Papers

The stationary bootstrap

Large Sample Confidence Regions Based on Subsamples under Minimal Assumptions

Automatic Block-Length Selection for the Dependent Bootstrap

The Impact of Bootstrap Methods on Time Series Analysis

Correction to “Automatic Block-Length Selection for the Dependent Bootstrap” by D. Politis and H. White