Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals

Introduction

Time Series is a set of observations, x (t) each one being recorded at time t where time could be equally spaced discrete t = 1, 2, 3…, or unequally spaced discrete or continuous t > 0.

Time-series analysis comes under the purview of statistics. Applications of real world time series are plentiful in the fields as economics, business, engineering, natural sciences (commonly in the meteorology, geophysics, astrophysics, ecology and biology), social and behavioral sciences, etc. Phenomena like human breath rate, human electrocardiogram, earthquake, stock prices etc. are some other examples of real world time series.

Jean Baptiste Joseph Fourier accompanied Napoleon to Egypt, and was Governor of Lower Egypt in the years just before his formulation of harmonic analysis, in 1807 which eventually appeared as the book “Théorie Analytique de la Chaleur” in the year 1822 (Fourier, 1822) which dealt with infinite periodic functions. Only periodic time series have Fourier series representations. This is supposed to be a milestone in the history of development of computational sciences and this can be considered as the inception of time series analysis.

In 1880, Thiele (Lauritzen, 1981, 1999) formulated and analyzed a model for the time series consisting of a sum of a regression component, a Brownian motion and a white noise.

In the field of Economics, in particular, the first application of the method of correlation to a timeseries, was published in 1901, viz. Hooker's Study of the Marriage Rate and Trade (Annals of the American Academy of Political and Social Science). Statistical analysis of time series was incepted by Yule in the year 1927 (Yule, 1927).

In 1942, Weiner produced a monograph, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Tsay, 2000). Doing the calculations was a challenge then. Several of the geophysical students at MIT recognized that Weiner's method could be a powerful tool for identify the structural interfaces. They started doing time series analysis.

Champernowne went to Oxford as Director of the Oxford Institute of Statistics, becoming Professor of Statistics in the year 1948. In the same year he read an important paper to the Royal Statistical Society on the time series analysis of autoregressive processes. This paper represented the first serious attempt at the application to time-series analysis of the techniques of Thomas Bayes, an 18^th century Presbyterian minister at Tunbridge Wells (Tsay, 2000).

Generally speaking, eight important technical advances generated in the second half of last century in the field of time series. The first advancement is the Kalman filtering (Kalman and Bucy, 1961). The second one is the publication of Time Series Analysis: Forecasting and Control by Box and Jenkins in 1970 (Box and Jenkins, 1970) which one was an important milestone for time series analysis. The third one is the use of state-space parameterization (Harrison and Stevens, 1976). The fourth one is the use of Markov Chain Monte Carlo (MCMC) methods (Geltand and Smith, 1990), especially Gibbs sampling and the concept of data augmentation (Tanner and Wong, 1987). The fifth one is the pattern identification methods and the extended autocorrelation function that is capable of handling both stationary and unit root non-stationary time series (Gray et. al, 1978; Tsay et. al, 1984). The sixth one comes in this regard in the form of simple exponential smoothing (Makridakis et.al, 1982). The seventh one came in the form of fractal analysis of time series. Higuchi (Higuchi, 1988; 1990) developed a new method for calculating the fractal dimension of a given time series. The eighth one came in the form of forecasting of time series (Ferraz-Mello, 1981; Scargle, 1980; Bai & Cliver, 1990; Roy et. al, 1999). For the forecasting purpose time variation (periodicity) analysis is a very significant approach.