Forecasting Theory

A time series is a sequence of observations of a periodic random variable. Examples are the monthly demand for a product, the annual freshman enrollment in a department of the university and the daily flows in a river. Time series are important for operations research because they are often the drivers of decision models. An inventory model requires estimates of future demands, a course scheduling and staffing model for a university department requires estimates of future student inflow, and a model for providing warnings to the population in a river basin requires estimates of river flows for the immediate future.
Time series analysis provides tools for selecting a model that describes the time series and using the model to forecast future events. Modeling the time series is a statistical problem because observed data is used in computational procedures to estimate the coefficients of a supposed model. Models assume that observations vary randomly about an underlying mean value that is a function of time.
On these pages we restrict attention to using historical time series data to estimate a time dependent model. The methods are appropriate for automatic, short term forecasting of frequently used information where the underlying causes of time variation are not changing markedly in time. In practice, the forecasts derived by these methods are subsequently modified by human analysts who incorporate information not available from the historical data.
Our primary purpose in this section is to present the equations for the four forecasting methods used in the Forecasting add-in: moving average, exponential smoothing, regression and double exponential smoothing. These are called smoothing methods. Methods not considered include qualitative forecasting, multiple regression, and autoregressive methods (ARIMA). Those interested in more extensive coverage should visit the Forecasting Principles site or read one of the several excellent books on the topic. We used the book Forecasting, by Makridakis, Wheelwright and McGee, John Wiley & Sons, 1983.
To use the Excel Examples workbook, you must have the Forecasting add-in installed. Choose the Relink command to establish the links to the add-in.

	Time Series	This page describes the models used for simple forecasting and the notation used for the analysis.
	Moving Average	This simplest forecasting method is the moving average forecast. The method simply averages of the last m observations. It is useful for time series with a slowly changing mean.
	Exponential Smoothing	This method considers the entire past in its forecast, but weighs recent experience more heavily than less recent. The computations are simple because only the estimate of the previous period and the current data determine the new estimate. The method is useful for time series with a slowly changing mean.
	Regression	The moving average method does not respond well to a time series that increases or decreases with time. Here we include a linear trend term in the model. The regression method approximates the model by constructing a linear equation that provides the least squares fit to the last m observations.
	Double Exponential Smoothing	This method estimates both the constant term and the linear coefficient for a linear forecasting equation that models trends. Exponential smoothing is estimates to both the constant term and the linear coefficient.
	Seasonality	We model seasonality with a multiplicative seasonal index. The data is adjusted by dividing by the index.and the adjusted data is used to obtain forecasts using one of the methods above.