Background

Time series come in all shapes and sizes. They arise in many walks of life - almost anything you can think of can be associated with a time series. Notable examples are: (i) economic time series, e.g. stock prices, economic indices such as inflation or unemployment, sales figures; (ii) medical time series, e.g. various vital statistics such as ECG, EEG, MRI scans, spread of disease; (iii) environmental series such as temperature, water level, pollution level; (iv) network series: transportation networks, internet traffic, social networks, epidemics; (v) energy series: demand, supply... the list is endless.

Time series have unique characteristics that are not shared with `ordinary' data. First, time series data have an order, you can't jumble the observations up and obtain the same series. Second, one can forecast the future with time series methods and obtain some idea of the uncertainty of the forecast. Generally speaking, we are not good at forecasting the future! In some domains (e.g. energy) we are very good. However, typically in these areas small improvements can yield huge dividends.

Much classical research investigates an area called stationary time series (see box, right). However, these models are often not appropriate for many real time series where the environment or underlying processes change their form over time. My main research into time series is in locally stationary time series.

Stationary Time Series

Stationary time series are those whose underlying statistical properties do not change with time. So, although the values of the series can change, quantities such as the mean and variance do not change.

For example, at rest, your heart rate will go up and down a bit but essentially the mean level, and the variation around the mean will remain constant. Of course, if you change the environment (you start running) then the underlying mean heart rate will change.

More on stationary time series.

Tests for Stationarity

How can you tell if your series is stationary, or exhibits second-order nonstationary characteristics? This section tells you how you can do this.

More...

Costationarity

If you have two locally stationary time series can you combine them in some way to get a (second-order) stationary series? For example, wind speeds.

The answer is yes, sometimes. However, in general, the way in which you combine the series has to vary with time also.

Two series that can combine to make a stationary series are termed costationary.

You can find a paper on this here.

Free software to find them is here.

Local Autocovariance

Autocovariance tells you about the internal association in a time series: what is the linear covariance between time t and t+k? For a stationary series this is constant over all time. For nonstationary series it can change over time. This section shows a method for estimating and displaying this.

More...

Is a stationary time series just white noise?

Some new wavelet-based tests for white noise for stationary time series and methods for computing their theoretical power. Describes R functionality for the tests and plotting theoretical power.

More ...