Some of the series we have looked at have had clear annual cycles, returning to high levels in the same month every year. In our analysis of such processes we have tried to model the mean as a periodic function . Sometimes we have fitted specific periodic functions to - writing .
Another process we studied, that of sunspot numbers, also seems to show a clearly periodic component, though now the frequency or period of the oscillation is not so obvious. In this section of the course we investigate the notion of decomposing a general stationary time series into simple periodic components. We will take these components to be cosines and sines. We will be focussing on problems in which the period is not prespecified, that is problems more like the sunspot data than the annual cycle examples.
For a statistician, the simplest description of what we will do is to say that we will examine the correlation between our series and sines and cosines of various periods. We will use these correlations in several ways:
A periodic function f on the real line has the property that f(t+d)=f(t) for some d and all t. The smallest possible choice of d is the period of f. The frequency of f in cycles per time unit, is 1/d. The most famous periodic functions are the trigonometric functions and its relatives. This function has period and frequency cycles per time unit. Often, for trigonometric functions it is convenient to refer to as the frequency; the units now are radians per time point.
The achievement of Fourier was to recognize that essentially any
function f with period 1 can be represented as a sum of functions
The tactic is to suppose that
Now multiply f(t) by say
and integrate from
0 to 1. Expanding the integral using the supposed expression of
f as a sum gives us
Mathematically the fact that we can derive a formula for the coefficients is far from proving that the resulting sum actually represents f; the key missing piece of the proof is that any function whose Fourier coefficients are all 0 is essentially the 0 function.
The integrals in the previous section can be thought of as analogous to
covariances and variances. For instance a Riemann sum for
Interpreting all the integrals above, then, as covariances we see that all the sines are uncorrelated with each other and with all the cosines and all the cosines are uncorrelated with each other.
Notice particularly that the sine
with frequency j and the cosine with frequency j are uncorrelated. This
has an important implication for looking for components at frequency j cycles
per time unit in a time series: if we want a certain frequency we have to consider
both the cosine and the sine at that frequency. An alternative summary of what we
need is to consider the trigonometric identity
Many of the identities in this subject are more easily derived using
complex variables. In particular, the identity
For instance we can write
For functions which are not periodic we can proceed by a further approximation
Suppose f is defined on the real line and fix a large value of T. Define
We now seek to apply these ideas with the function f being our stochastic process X. We have several difficulties:
The discrete nature of X leads us to the study of a discrete
approximation to the integral:
Suppose that X is a mean 0 stationary time series with autocovariance
function C. We define the discrete Fourier transform of X as
We begin by computing moments of .
is complex valued we
have to think about what these moments should be. One way to think about
this is to view
as a vector with two components, the real and
imaginary parts. This would give
a mean and a 2 by 2 variance
covariance matrix. Also of interest however will be the expected modulus
squared of ,
Since the Xs have mean 0 we see that
The right hand side of this expression is defined to be the spectral
density, or power spectrum, of X:
There are a number of ways to look at spectral densities and the discrete Fourier transform: