Any time series can be expressed as a combination of cosine and sine waves with differing periods (how long it takes to complete a full cycle) and amplitudes (maximum/minimum value during the cycle). This fact can be utilized to examine the periodic (cyclical) behavior in a time series.

Properties of a Cosine Function

For discrete time (meaning time \(t\) = integer values), these definitions are useful for a cosine (or sine) wave:

Imagine fitting a single cosine wave to a time series observed in discrete time. Suppose that we write this cosine wave as

\(x_t = A \cos(2\pi \omega t + \phi)\)

• \(A\) is the amplitude. It determines the maximum absolute height of the curve.
• \(\omega\) is the frequency. It controls how rapidly the curve oscillates.
• \(\phi\) is the phase. It determines the starting point, in angle degrees, for the cosine wave.

To temporarily simplify things, suppose that \(\phi\)=0 and think about the quantity \(2\pi\omega t\). Recall that T = number of time periods for a full cycle and that \(\omega\) = 1/T. As we move through time from \(t\) = 0 to \(t\) = T, the value of \(2\pi\omega t\) = \(2\pi t\)/T ranges from 0 at \(t\) = 0 to \(2\pi\) at \(t\) = T. In angle degrees, this represents a full cycle of a cosine wave.

In Example 1.12 of the text, on pages 12-14, the authors give a plot of the function

\(x_t = 2 \cos\left(2\pi \frac{1}{50}t + 0.6\pi \right)\)

for t = 1, 2, ..., 500. In addition they add normally distributed errors with mean 0 and variance 1 to this function in a second plot, and add normally distributed errors with mean 0 and variance 25 in a third plot. The R code is given on page 13. Following are the first two plots, the basic cosine function and the function plus errors with variance 1. In the basic plot, the period = 50 and the frequency is 1/50. Thus it takes 50 time periods to cycle through the cosine function. Before errors are added, the maximum and minimum values are +2 and -2, respectively.

Following is what the plots look like when we change the period to 250 so that the frequency is 1/250 = 0.004. The function is

\(x_t = 2 \cos\left(2\pi \frac{1}{250}t + 0.6 \pi \right)\)

for t = 1, 500.

Notice above that longer period (250 for the second set of plots versus 50 in the first set of plots) leads to fewer cycles.

In the area of time series called spectral analysis, we view a time series as a sum of cosine waves with varying amplitudes and frequencies. One goal of an analysis is to identify the important frequencies (or periods) in the observed series. A starting tool for doing this is the periodogram. The periodogram graphs a measure of the relative importance of possible frequency values that might explain the oscillation pattern of the observed data.

Suppose that we have observed data at n distinct time points, and for convenience we assume that n is even. Our goal is to identify important frequencies in the data. To pursue the investigation, we consider the set of possible frequencies \(\omega_j\) = j/n for j = 1, 2,…, n/2. These are called the harmonic frequencies.

We will represent the time series as

\(x_t = \sum_{j=1}^{n/2}\left[\beta_1\left(\frac{j}{n}\right)\cos(2\pi \omega_j t) + \beta_2\left(\frac{j}{n}\right)\sin(2\pi \omega_j t)\right]. \)

This is a sum of sine and cosine functions at the harmonic frequencies. The form of the equation comes from the identity given above in the section entitled “A Useful Identity”).

Think of the \(\beta_1\)(j/n) and \(\beta_2\)(j/n) as regression parameters. Then there are a total of n parameters because we let j move from 1 to n/2. This means that we have n data points and n parameters, so the fit of this regression model will be exact.

The first step in the creation of the periodogram is the estimation of the \(\beta_1\)(j/n) and \(\beta_2\)(j/n) parameters. It’s actually not necessary to carry out this regression to estimate the \(\beta_1\)(j/n) and \(\beta_2\)(j/n) parameters. Instead a mathematical device called the Fast Fourier Transform (FFT) is used. We’ll skip the details of that – it’s advanced Calculus, something not required for this course.

After the parameters have been estimated, we define

\(P\left(\frac{j}{n}\right) = \widehat{\beta}_1^2\left(\frac{j}{n}\right)+\widehat{\beta}_2^2\left(\frac{j}{n}\right)\)

This is the value of the sum of squared “regression” coefficients at the frequency j/n.

This is the periodogram value at the frequency j/n, although the authors of our textbook (on page 169) say they will call this the scaled periodogram value. Thus, for them the scaled periodogram is a plot of P(j/n) versus j/n for j = 1, 2, …, n/2.

A relatively large value of P(j/n) indicates relatively more importance for the frequency j/n (or near j/n) in explaining the oscillation in the observed series. P(j/n) is proportional to the squared correlation between the observed series and a cosine wave with frequency j/n. The dominant frequencies might be used to fit cosine (or sine) waves to the data, or might be used simply to describe the important periodicities in the series.

The Fast Fourier Transform in R doesn’t quite give a direct estimate of the scaled periodogram. A small bit of scaling has to be done (and the FFT produces estimates at more frequencies than we need).  Code is given on page 170 of the text for the data from Example 4.1. 

Spectral analysis, described in Chapter 4 of our textbook, is the analysis of the dominant frequencies in a time series. In practice, spectral analysis imposes smoothing techniques on the periodogram. With certain assumptions, we can also create confidence intervals to estimate the peak frequency regions. Spectral analysis can also be used to examine the association between two different time series.