In an earlier article we described Time series forecasting in 4 simple terms for business users. In that article we observed that time series data can sometimes show a trending behavior. This trend may be downward or upward. In addition to a trend, there may also be cyclic or seasonal variations. What happens when data shows both trend and seasonality? Then the question is how to obtain predictions or forecasts using this data?

Time series data can contain trends which may be either linear or exponential or mixed. Smoothing on this data is required to predict the values for forecasting. Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways:

• Curve fitting often involves the use of an explicit function form for the result, whereas the immediate results from smoothing are the "smoothed" values with no later use made of a functional form if there is one
• The aim of smoothing is to give a general idea of relatively gradual changes in value with little attention paid to the close matching of data values, while curve fitting concentrates on achieving as close a fit as possible.
• Smoothing methods often have an associated tuning parameter which is used to control the extent of smoothing. Curve fitting will adjust any number of parameter of the function to obtain the 'best' fit.

Forecast profiles from exponential smoothing adapted from E. Gardner, Journal of Forecasting, Vol. 4 (1985)

There are multiple methods for time series forecasting based on trend as well as seasonality. These methods could be classified as described in this table below.

Additive Model – During the development of additive models there is an implicit assumption that the different components affect the time series additively

Data = Seasonal effect + Trend + Cyclical + Residual

For example, for monthly data, an additive model assumes that the difference between the January and July values is approximately the same each year. In other words, the amplitude of the seasonal effect is the same each year.

The model similarly assumes that the residuals are roughly the same size throughout the series -- they are a random component that adds on to the other components in the same way at all parts of the series.

Multiplicative models - In many time series involving quantities (e.g. money, wheat production, etc.), the absolute differences in the values are of less interest and importance than the percentage changes.

For example, in seasonal data, it might be more useful to model that the July value is the same proportion higher than the January value in each year, rather than assuming that their difference is constant. Assuming that the seasonal and other effects act proportionally on the series is equivalent to a multiplicative model,

Data = Seasonal effect * Trend * Cyclical * Residual

Example - For example, suppose we wanted to forecast from month to month the number of households that purchase a particular consumer electronics device (e.g., TV). Every year, the number of households that purchase a TV will increase, however, this trend will be damped (i.e., the upward trend will slowly disappear) over time as the market becomes saturated. In addition, there will be a seasonal component, reflecting the seasonal changes in consumer demand for TVs from month to month (demand will likely be smaller in the summer and greater during the December holidays). This seasonal component may be additive, for example, a relatively stable number of additional households may purchase TVs during the December holiday season.

Understanding the nature of your data is important before applying any of the forecasting techniques. See how time series forecasting is employed to make annual budget forecasts in this case study.