*Intended learning outcomes: Explain mean, standard deviation, and forecast error in linear regression.*

Regression analysis, orlinear regression, is often described as trend analysis. It is based on the assumption that demand values appear as a particular function of time, such as a linear function.

This means that a number of points represented on the *x*–*y* plane can be approximated by a line. Figure 10.3.0.1 shows demand as a function of time period. Given a *y*-axis value of a and a slope of b, we can determine the mean line (regression line) sought between the two pairs of values. Figure 10.3.1.1 provides the formulas for determining this, along with the values a and b. To perform the calculation, we need to know the values for at least n periods preceding time t. See also the definitions of indexes and variables in Figure 10.1.3.4. The derivation of the formulas is taken from [Gahs71], p. 67 ff.

**Fig. 10.3.1.1** Mean, standard deviation, and forecast error in linear regression.

Because of uncertainty in the determination of a and b, the forecast error is larger than the standard deviation, as shown in Figure 10.3.1.1. The term 1/n in the formula for forecast error represents the uncertainty in determining a, while the other term represents slope b. The influence of the slope b increases with increased forecast distance k. In this situation, therefore, we determine the forecast error by extrapolation of the deviations of individual values from the past value of the regression curve. Figure 10.3.1.2 shows a sample calculation of linear regression with n = 14.

**Fig. 10.3.1.2** Linear regression: sample calculation with n = 14.

## Course section 10.3: Subsections and their intended learning outcomes

##### 10.3 Historically Oriented Techniques with Trend-Shaped Behavior

Intended learning outcomes: Explain the regression analysis forecast and the second-order exponential smoothing forecast. Describe the Trigg and Leach adaptive smoothing technique. Produce an overview on seasonality.

##### 10.3.1 Regression Analysis Forecast

Intended learning outcomes: Explain mean, standard deviation, and forecast error in linear regression.

##### 10.3.2 Second-Order Exponential Smoothing Forecast

Intended learning outcomes: Disclose the determination of trend lines in second-order exponential smoothing. Explain the formulas for calculation of the trend line and forecast error in second-order exponential smoothing. Present an example of determination of forecast value using second-order exponential smoothing.

##### 10.3.3 Trigg and Leach Adaptive Smoothing Technique

Intended learning outcomes: Identify forecast errors and their exponential weighting (mean deviation). Explain the tracking signal following Trigg and Leach. Describe the determination of the smoothing constant in first-order exponential smoothing.

##### 10.3.4 Seasonality Forecast

Intended learning outcomes: Identify the seasonal index Sf. Explain forecasting that considers seasonality. Differentiate between “Additive seasonality” and “Multiplicative seasonality” formulation.