00431 œ2bd2b6c7f7803f642438372603e664

436

Russell

between dynami cal Systems and time senes analysis becomes obvious when the iterates of a dynamical system are tracked over time. In the language of these Systems, these iterates are said to tracÄ™ out an orbit. Frequently the orbits themselves become an object of study.

In this paper we will briefly review some of the morÄ™ traditional graphical aids for viewing time series data. Then we will examine some background concepts that have been developed in the study of fractals and deterministic chaos. Next, we will develop the concept and usage of the univariate phase map movie. For this we will examine some synthetic time-series data followed by some traditional time-series data sets. Finally we will apply the univariate phase map movie in examining some new data sets, using the univariate phase map movie both as an exploratory tool and as a model development aid.

Traditional Graphical Display of Time Series Data

In this paper, we will primarily be dealing with autocorrelated time series. A time series that is autocorrelated has observations that are correlated with prior observations. This is easiest to understand in it's simplest case: the model for an autoregressive time series of order 1, denoted by AR(1), is given by

^x, = <**,-. + *,

In generaÅ‚, X_t is the observed value at time t, <J> is the parameter we are trying to fit, and ^ is the noise term at time t, often assumed to be normally distributed with 0 mean and constant variance. Time series of higher order, such as the AR(2) and AR(3) models discussed in this paper, have morÄ™ of the "older" observations included in the model. For instance the AR(2) and AR(3) models are respectively given by

x, =    + e_t and x, = ^x,_, + (j>₂x,_₂ + ^x,_, + e,

One of the traditional plots that is examined in times series analysis is a plot of the data against itself lagged by one or morÄ™ time periods, i.e.: the coordinate pairs plotted are (X_t, X_t+j). This plot is typically presented in the form of a scatter plot of all of the data against itself as seen in FigurÄ™ 1 below.

It is well known that this type of scatterplot can readily show that neighboring observations are autocorrelated (Box and Jenkin, 1976). Another