A time series plot is similar to a scatterplot with 'time' as the variable on the horizontal axis. Successive values are usually joined by lines to emphasise systematic movements in the series.
A few examples show time series with trend, seasonal pattern and other oscillations.
An example shows four different time series on the same graph.
Moving averages are a simple way to smooth out irregularities in a time series.
When an even number of values is used for the moving averages, the smoothed values are at times half-way between those of the raw data. The method can be modified to give smoothed values at the times of the original values.
Using running medians instead of means avoids the effect of outliers, but the result often looks 'stepped'. The use of running medians, followed by running means, is a good compromise.
Exponential smoothing replaces each value with a weighted average of it and previous values. Unlike moving averages, it can be used right up to the end of the series.
An alternative smoothing method that provides smoothed values to both ends of a time series is to obtain the smoothed value from a least squares line fitted to adjacent values.
A similar method can be used to draw a 'smooth' curve to represent a nonlinear relationship on a scatterplot.
Many time series steadily increase or decrease over time and a linear model may describe the trend. This kind of model can be fitted by least squares.
If the trend is nonlinear, a quadratic model might describe the trend better and can also be fitted by least squares.
Linear and quadratic models can be used to forecast future values.
If a quadratic model does not adequately describe a nonlinear trend, it is possible to add higher powers of time to the model. These polynomial models may give a smooth picture of past trend but should not be used to forecast into the future.
Polynomial models are only useful to describe a smooth long-term trend in a time series. A time series plot of least squares residuals (actual values minus trend) highlights shorter-term movements in the series.