25autocorrelation

In this section, we will make use of a few fundamental concepts from statistics. Knowing these concepts well is fundamental to make sense of stationarity.

25.1 mean and standard deviation

Let’s call our time series X, and its length N. Then:

\begin{aligned} \text{mean}& &\mu &= \frac{\displaystyle\sum_{i=1}^N X_i}{N} \\ \text{standard deviation}& &\sigma &= \sqrt{\frac{\displaystyle\sum_{i=1}^N (X_i-\mu)^2}{N}} \end{aligned}

The mean and standard deviation can be visualized thus:

25.2 expected value

The expected value (or expectation) of a variable X is given by E[X] = \sum_{i=1}^N X_i p_i.

p_i is the weight or probability that X_i occurs. For a time series, the probability p_i that a given point X_i is in the dataset is simply 1/N, therefore we can write the following measures in terms of expected values:

• mean, also called 1st moment: \mu = E[X].
• variance, also called 2nd moment: \sigma^2 = E[(X-E[X])^2] = E[(X-\mu)^2]. Of course, \sigma is called the standard deviation.

25.3 covariance

The covariance between two time series X and Y is given by

\begin{split} \text{cov}(X,Y) &= E[(X-E[X])(Y-E[Y])]\\ &= E[(X-\mu_X)(Y-\mu_Y)] \end{split}

Compare this to the definition of the variance, and it is obvious that the covariance \text{cov(X,X)} of a time series with itself is its variance.

25.4 correlation

We are almost there. I promise.

The fact that \text{cov(X,X)} = \sigma_X^2 begs us to define a new measure, the correlation:

\text{corr}(X,Y) = \frac{E[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X \sigma_Y}.

This is convenient, because now we can say that the correlation of a time series with itself is \text{corr}(X,X)=1.

This is also called the Pearson correlation coefficient, and the result has a value between 1 and -1.

Source: Wikimedia

25.5 autocorrelation

The autocorrelation of a time series X is the answer to the following question:

if we shift X by \tau units, how similar will this be to the original signal?

In other words:

how correlated are X(t) and X(t+\tau)?

The autocorrelation is expressed as \rho_{XX}(\tau) = \frac{E\left[ (X_t - \mu)(X_{t+\tau} - \mu) \right]}{\sigma^2}

In other disciplines, the autocorrelation is simply the autocovariance, i.e., it is not normalized by dividing by \sigma^2. In time series it is assumed that the autocorrelation is always normalized, therefore between -1 and 1.

The autocorrelation function \rho_{XX}(\tau) provides a useful measure of the degree of dependence among the values of a time series at different times.

A video is worth a billion words, so let’s see the autocorrelation in action: