Time Series

A time series is a set of data collected over time: the ordering of the time points matters. For many data sets (for example, heights of a set of people) the order of the data points does not matter and one order is as good as another. For time series ordering is crucial as it imposes structure on the data. Later we will look at predicting future observations which absolutely relies on the ordering.

Figure left: Financial Time Series FTSE100

The picture (left) shows values of the FTSE100 share index plotted over time. Actually, we have plotted what are called log returns of the series, rather than the values themselves. Returns are commonly used in financial time series: they describe the proportional return that you might get on a given day compared to the previous day.

The FTSE series oscillates, but it oscillates around a given level. It does not drift too far above or below that level. In fact, if one calculates the mean of all the observations one obtains the number 0.000935, a number very close to zero. Indeed, for this kind of data the true mean of the underlying process is often modelled to be exactly zero. The blue dashed line illustrates this.

Underlying Process

IThe statement `process thought to underly the series' is a model statement. For the FTSE100 data, nobody knows, what process, if any, actually underlys and generates the FTSE data. However, we can formulate a model that we think is appropriate for this data and such a model is a commonly held assumption about the data. For example, we might assume that the model mean is exactly zero. One could, if one wished, choose another model, e.g. one where the mean is precisely 0.000935. That model might fit this data set better, but analysis of many other similar series, or even future observations of this series, might show means that were slightly positive and some slightly negative so that, overall, a mean zero model is typical. An aim of statistics, including time series analysis, is to choose simple models that fit the data well.

Stationary Time Series

Loosely speaking, a stationary time series is one whose statistical properties are constant over time. What do we mean by statistical properties? We mean things like mean value (or average level) of the series, the variance (variation of the time series around the mean level) and the autocorrelation (explained later). One might consider the FTSE 100 time series (above) to be appropriately modeled by a stationary model, but almost certainly the measles series which we exhibit next, is not.

Measles...