00439 57683fba1bad90493f04d56b521874a

444

Russell

map movies can aid in uncovering a number of problems that are encountered in trying to fit autoregressive time senes models to dynamical Systems.

To begin answering these questions, consider an autoregressive process of order 1, an AR(1) process, where we dellne

X,=+*X„+s,

Now if we ignore the noise term, then we find that the expected slope of the points plotted on the univariate phase map movie for an AR(1) is <p. This is because the slope of the linę through any two plotted points at times (t + 1, t + 2), and (t, t + I) is given by

_m &_m-x_m)

m =-

W_M-x,)

which simplifies to <p.

This slope is realized only if there is essentially no noise in the process. If the noise term needs to be included then there is no ready simplification to a single number.

However if we consider an autoregressive process of order 2 defined as

X, =4>S^x,-i +łi*X,-_t + ^e, there is no readily interpretable reduction of the slopes of the lines in the univariate phase map movie. For instance, ignoring the noise term as before the slopes of the two joined lines through the plotted points at times (t + 2, t + 3), (t + 1, t + 2), and (t, t + 1) are given by

{X_M-X_M)

and m.

These slopes simplify to: