Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.4. Univariate Time Series Models
A common assumption in many time series techniques is that the
data are stationary.
A stationary process has the property that the mean, variance and autocorrelation structure do not change over time. Stationarity can be defined in precise mathematical terms, but for our purpose we mean a flat looking series, without trend, constant variance over time, a constant autocorrelation structure over time and no periodic fluctuations (seasonality).
For practical purposes, stationarity can usually be determined from a run sequence plot.
|Transformations to Achieve Stationarity||
If the time series is not stationary, we can often transform it
to stationarity with one of the following techniques.
|Example||The following plots are from a data set of monthly CO\(_2\) concentrations.|
|Run Sequence Plot||
The initial run sequence plot of the data indicates a rising trend. A visual inspection of this plot indicates that a simple linear fit should be sufficient to remove this upward trend.
This plot also shows periodical behavior. This is discussed in the next section.
|Linear Trend Removed||
This plot contains the residuals from a linear fit to the original data. After removing the linear trend, the run sequence plot indicates that the data have a constant location and variance, although the pattern of the residuals shows that the data depart from the model in a systematic way.