A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

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The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

Using research designs patterned after randomized experiments, many recent economic studies examine outcome measures for treatment groups and comparison groups that are not randomly assigned. By using variation in explanatory variables generated by changes in state laws, government draft mechanisms, or other means, these studies obtain variation that is readily examined and is plausibly exogenous. This paper describes the advantages of these studies and suggests how they can be improved. It also provides aids in judging the validity of inferences they draw. Design complications such as multiple treatment and comparison groups and multiple pre- or post-intervention observations are advocated.

Comparing Predictive Accuracy

Statistics for Spatial Data.

Summary 1. While teaching statistics to ecologists, the lead authors of this paper have noticed common statistical problems. If a random sample of their work (including scientific papers) produced before doing these courses were selected, half would probably contain violations of the underlying assumptions of the statistical techniques employed. 2. Some violations have little impact on the results or ecological conclusions; yet others increase type I or type II errors, potentially resulting in wrong ecological conclusions. Most of these violations can be avoided by applying better data exploration. These problems are especially troublesome in applied ecology, where management and policy decisions are often at stake. 3. Here, we provide a protocol for data exploration; discuss current tools to detect outliers, heterogeneity of variance, collinearity, dependence of observations, problems with interactions, double zeros in multivariate analysis, zero inflation in generalized linear modelling, and the correct type of relationships between dependent and independent variables; and provide advice on how to address these problems when they arise. We also address misconceptions about normality, and provide advice on data transformations. 4. Data exploration avoids type I and type II errors, among other problems, thereby reducing the chance of making wrong ecological conclusions and poor recommendations. It is therefore essential for good quality management and policy based on statistical analyses.

https://besjournals.onlinelibrary.wiley.com/doi/epdf/10.1111/j.2041-210X.2009.00001.x

A protocol for data exploration to avoid common statistical problems

SUMMARY A bias correction to the Akaike information criterion, AIC, is derived for regression and autoregressive time series models. The correction is of particular use when the sample size is small, or when the number of fitted parameters is a moderate to large fraction of the sample size. The corrected method, called AICC, is asymptotically efficient if the true model is infinite dimensional. Furthermore, when the true model is of finite dimension, AICC is found to provide better model order choices than any other asymptotically efficient method. Applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed.

/pdf/regression-and-time-series-model-selection-in-small-samples-42cu3yxb6h.pdf

Regression and time series model selection in small samples

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed noise. For general ARMA$(p,q)$ equations these conditions are expressed in terms of the characteristic polynomials of the defining equations and moments of the driving noise sequence, while for $p=1$ an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

Summary

Estimation of scale and index parameters of positive stable laws is considered. Maximum likelihood estimation is known to be efficient, but very difficult to compute, while methods based on the sample characteristic function are computationally easy, but have uncertain efficiency properties.



In this paper an estimation method is presented which is reasonably easy to compute, and which has good efficiency properties, at least when the index α (0, 0.5). The method is based on an expression for the characteristic function of the logarithm of a positive stable random variable, and is derived by relating the stable estimation problem to that of location/scale estimation in extreme-value-distribution families, for which efficient methods are known.



The proposed method has efficiency which 1 as α,but on the other hand, efficiencies deteriorate after α >0.5, and in fact appear to 0 as α+ 1.

Estimation for the positive stable laws, i

In this chapter we shall examine the problem of finding an appropriate model for a given set of observations {x 1 ,..., x n } that are not necessarily generated by a stationary time series. If the data (a) exhibit no apparent deviations from stationarity and (b) have a rapidly decreasing autocovariance function, we attempt to fit an ARMA model to the mean-corrected data using the techniques developed in Chapter 5. Otherwise we look first for a transformation of the data that generates a new series with the properties (a) and (b). This can frequently be achieved by differencing, leading us to consider the class of ARIMA (autoregressive integrated moving average) models, defined in Section 6.1. We have in fact already encountered ARIMA processes. The model fitted in Example 5.1.1 to the Dow-Jones Utilities Index was obtained by fitting an AR model to the differenced data, thereby effectively fitting an ARIMA model to the original series. In Section 6.1 we shall give a more systematic account of such models.

Nonstationary and Seasonal Time Series Models

The analysis of time series data is an important aspect of data analysis across a wide range of disciplines, including statistics, mathematics, business, engineering, and the natural and social sciences. This package provides both an introduction to time series analysis and an easy-to-use version of a well-known time series computing package called "Interactive Time Series Modelling". The programs in the package are intended as a supplement to the text "Time Series: Theory and Methods, 2nd edition", also by Peter J. Brockwell and Richard A. Davis. Many researchers and professionals will appreciate this straightforward approach enabling them to run desk-top analyses of their time series data. Amongst the many facilities available are tools for: ARIMA modelling, smoothing, spectral estimation, multivariate autoregressive modelling, transfer-function modelling, forecasting, and long-memory modelling. This version is designed to run under Microsoft Windows 3.1 or later. It comes with two diskettes: one suitable for less powerful machines (IBM PC 286 or later with 540K available RAM and 1.1 MB of hard disk space) and one for more powerful machines (IBM PC 386 or later with 8MB of RAM and 2.6 MB of hard disk space available).

ITSM for Windows: A User's Guide to Time Series Modelling and Forecasting

We investigate the problem of minimum dispersion linear prediction of the components of a linear process driven by multivariate symmetric a-stable noise. The results generalize some of those obtained for univariate processes by Cline and Brockwell (1985) and Stuck (1978)

Peter J. Brockwell

Papers

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

Estimation for the positive stable laws, i

Nonstationary and Seasonal Time Series Models

ITSM for Windows: A User's Guide to Time Series Modelling and Forecasting

Linear prediction for a class of multivariate stable processes