Abstract: . In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first-order and second-order autoregressive as well as first-order moving average processes.

Abstract: Abstract In the most widely applied models for stationary time series the spectral density function is bounded at the frequency A = 0 and the autocorrelation function decays exponentially. This is true in stable autoregressive moving average models, and it is characteristic of estimates obtained by nonparametric spectral methods, for example.

Abstract: [1] One of three criteria to demonstrate self-organized criticality (SOC) for a critical phenomenon is that time arrival of events displays a frequency dependence which is inversely proportional to frequency (f) to some power That is, for SOC, the power spectrum in the frequency (f) domain is supposed to fall off as 1/fβ, where β is typically a number between 1 and 2 Avalanche phenomena have been used as prototypes for illustrating SOC, and therefore it is of interest as to whether snow avalanches follow the criterion In this paper, time series analyses of mass arrivals from 20 years of records constituting ∼10,000 avalanches are presented for Bear Pass and Kootenay Pass, British Columbia The results suggest that the autocorrelation functions and partial autocorrelation functions of the series fall off in an exponential manner so that the implied power spectra in the frequency domain, given by the Fourier transforms of the autocorrelation functions, decay with frequency in a manner which is not strictly consistent with SOC In common with SOC, the power spectra are suggested to have most content in the low-frequency events and the spectra do not constitute white noise However, given the limitations on the data sampling and recording, it cannot be definitively stated that the power spectra fall off with 1/f® as required for SOC

Abstract: The paper discusses the effects of spatial, temporal, and joint spatio-temporal autocorrelation. The purpose of the study of spatio-temporal autocorrelation is to optimize the efficiency of dynamic regression estimates, in a spatio-temporal univariate regression model. After calculating and examining the effects of global indicators of spatial and temporal autocorrelation, a new spatio-temporal global autocorrelation index is proposed. The index can be view as a preliminary proposal for a dynamic version of Moran's I.

Abstract: Time series are demeaned when sample autocorrelation functions are computed. By the same logic it would seem appealing to remove seasonal means from seasonal time series before computing sample autocorrelation functions. Yet, standard practice is only to remove the overall mean and ignore the possibility of seasonal mean shifts in the data. Whether or not time series are seasonally demeaned has very important consequences on the asymptotic behavior of autocorrelation functions. The effect on the asymptotic distribution of seasonal mean shifts and their removal is investigated and the practical consequences of these theoretical developments are discussed. We also examine the small sample behavior of autocorrelation function estimates through Monte Carlo simulations.

Abstract: In this study we seek to resolve one of the most important problems in time series, the identification of the model, using the Box-Jenkins method. Our goal is to obtain an expert system based on paradigms of artificial intelligence, such as fuzzy logic and genetic algorithms, so that the model can be identified automatically, without the necessity for a human expert to intervene. A set of rules based on fuzzy logic is constructed, using as the main source of information the evolution and behaviour of the coefficients of autocorrelation and partial autocorrelation obtained from the time series. Each rule of the expert system is assigned a weight that determines the importance of this rule in the phase of model identification. A priori, the relevance of the rules is unknown, and so the rule system constructed is optimised by means of genetic algorithms.

Abstract: In this article we deal with the Arov–Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy. This description is based on a one-to-one correspondence between the set of all solutions of the Covariance Extension Problem and the set of all contractive analytic functions H from the open unit disk with values on the space of q × q matrices. With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H\equiv0. Also, from the information that the Arov–Grossman functional model provides we obtain a version of the Levinson algorithm. The partial autocorrelation coefficient matrices are computed directly from Levinson’s recursions.

Abstract: Analysis of increments dependence is an actual problem in testing of data series. The classic way is estimating the autocorrelation function of time series increments. This estimates are rather small and mutually independent in the case of independent increments, while it can be large in the case of dependent increments. But sequential values of the autocorrelation function are small and dependent in many cases. To prove dependence of increments, we repeat autocorrelation calculations: we calculate estimates for autocorrelation of autocorrelation function. Values of this twice-autocorrelation function are appeared to be rather large. That is, probabilities to have such values under the independence hypothesis are very small. We calculate it using a theorem on large deviations. We apply these results to text analysis: the better a text the lesser this probability.

Abstract: Mathematical bios and heartbeat series show an inverse relation between frequency and power; the time series of differences between successive terms of cardiac and mathematical chaos shows a direct relation between frequency and power. Other statistical analyses differentiate these biotic series from stochastically generated 1/f noise. The time series of complex biological and economic processes as well as mathematical bios show asymmetry, positive autocorrelation, and extended partial autocorrelation. Random, chaotic and stochastic models show symmetric statistical distributions, and no partial autocorrelation. The percentage of continuous proportions is high in cardiac, economic, and mathematical biotic series, and scarce in pink noise and chaos. These findings differentiate creative biotic processes from chaotic and stochastic series. We propose that the widespread 1/f power spectrum found in natural processes represents the integration of the fundamental relation between frequency and energy stated in Planck's law. Natural creativity emerges from determined interactions rather than from the accumulation of accidental random changes.

Abstract: INTRODUCTION DIAGNOSTIC CHECKS FOR UNIVARIATE LINEAR MODELS Introduction The Asymptotic Distribution of the Residual Autocorrelation Distribution Modifications of the Portmanteau Statistic Extension to Multiplicative Seasonal ARMA Models Relation with the Lagrange Multiplier Test A Test Based on the Residual Partial Autocorrelation test A Test Based on the Residual Correlation Matrix test Extension to Periodic Autoregressions THE MULTIVARIATE LINEAR CASE The Vector ARMA model Granger Causality Tests Transfer Function Noise (TFN) Modeling ROBUST MODELING AND ROBUST DIAGNOSTIC CHECKING A Robust Portmanteau Test A Robust Residual Cross-Correlation Test A Robust Estimation Method for Vector Time Series The Trimmed Portmanteau Statistic NONLINEAR MODELS Introduction Tests for General Nonlinear Structure Tests for Linear vs. Specific Nonlinear Models Goodness-of-Fit Tests for Nonlinear Time Series Choosing Two Different Families of Nonlinear Models CONDITIONAL HETEROSCEDASTICITY MODELS The Autoregressive Conditional Heteroscedastic Model Checks for the Presence of ARCH Diagnostic Checking for ARCH Models Diagnostics for Multivariate ARCH models Testing for Causality in the Variance FRACTIONALLY DIFFERENCED PROCESS Introduction Methods of Estimation A Model Diagnostic Statistic Diagnostics for Fractional Differencing MISCELLANEOUS MODELS AND TOPICS ARMA Models with Non-Gaussian Errors Other Non-Gaussian time Series The Autoregressive Conditional Duration Model A Power Transformation to Induce Normality Epilogue