# 10.1.3 Principles of Forecasting Techniques with Extrapolation of Time Series and the Definition of Variables

### Intended learning outcomes: Present an example of a time series. Explain possible and common demand models. Produce an overview on statistical methods to determine mean and dispersion. Identify definitions of variables, each calculated at the end of a statistical period.

Particularly for forecasting based on historical data, statistical techniques are used that are based on a series of observations along the time axis (here see [BoJe15], [IBM73], or [WhMa97]). The following values are fundamental to the determination of stochastic requirements:

```A time series is the result of measurement of particular quantifiable variables at set observation intervals equal in length.

The statistical period or observation interval is a time unit, namely, the period of time between two measurements of the time series (e.g., 1 week, 1 month, 1 quarter).

The forecast interval is the time unit for which a forecast is prepared ([APIC16]). This time unit best corresponds to the statistical period.

The forecast horizon is the period of time into the future for which a forecast is prepared ([APIC16]). It is generally a whole number multiple of the statistical period.```

As an example, Figure 10.1.3.1 shows the frequency distribution[note 1001] of the observed variable “customer order receipts” during the most recent statistical period as a histogram. [note 1002]

Fig. 10.1.3.1      Example of a time series.

```A demand model attempts to represent demand by drawing the curve that shows the least scattering of the measured values.

Curve fitting is the process performed to obtain that curve, by means of a straight line, polynomial, or another curve.```

We assume that the scattering (dispersion) of values is random and, most often, distributed normally. This presupposes that while demand values do indeed have a fluctuating pattern, it is possible to make fairly good appro­xi­­mations. Figure 10.1.3.2 presents some common cases of demand models.

Fig. 10.1.3.2      Possible and common demand models.

Matching a particular demand model to a particular time series leads to the choice of a forecasting technique. The forecasting technique is thus based on a concept or a model of the course of demand. This concept forms the basis for the perception of regularity or a regular demandand the model is

• Either an econometric model, mostly defined by a set of equations, formulating the interrelation of collected data and variables of the model of the course of the demand as a mathematical regularity,
• Or an intuitive model as an expression of the perception of an intuitive regularity.

It is quite possible that for a single time series several models will overlap.

```(Statistical) decomposition or time series analysis is a breakdown of time series data into various components by analysis; for example, into:
- (Long-term) trend component
- Seasonal component
- Nonseasonal, but (medium-term) cyclical component
- Marketing component (advertising, price changes, etc.)
- Random component (nonquantifiable phenomena), e.g., due to noise, that is random variation or a random difference between the observed data and the “real” event. ```

Mathematical statistics offers various methods for determining the mean, deviation, expected value, and dispersion (scattering)[note 1003] of measured values for a time series. Its ability to reproduce the demand for a demand model accurately depends upon the situation. Figure 10.1.3.3 shows a morphology of possible statistical features and the statistical methods that they characterize.

Fig. 10.1.3.3      Statistical methods to determine mean and dispersion.

1. Calculation of dispersion. Two basic methods are used:
• Extrapolation, or estimation by calculation of deviations of individual values in the previous statistical periods from the mean, postulated by the demand model.
• Direct, that is, retrospective determination of the forecast error as the difference between actual demand and projected demand according to the demand model.
2. Measure of dispersion. There are two standards here:
• Mean square deviation σ: (sigma, i.e., standard deviation)
3. Weighting of values. Most commonly encountered are:
• Equal weighting of all measured value
• Exponential weighting of measured values in the direction of the past

In most cases, we only measure satisfied demand for all models. This equates consumption with demand. The basic problem with this measure­ment is that real demand is not taken into account. The customer order receipts mentioned in Figure 10.1.3.1 may have been higher, for example, if a better demand model had resulted in better availability. Strictly speaking, the customer orders that could not be filled should have been measured as well. The problem with this, however, is that the customer orders may be filled at a later time period. At that time, there may be other orders that will then be unfilled, etc. Determining the exact amount of demand in the past by employing a “what would have happened if” method rapidly proves itself redundant; later demand on the time axis is most likely dependent on satisfied demand in the preceding periods on the time axis.

The following sections use the variables defined in Figure 10.1.3.4. The nomenclature was chosen in such a way that the index always shows the point at the end of the statistical period in which a value is calculated. The period to which the value refers is shown in parentheses.

Fig. 10.1.3.4      Definitions of variables, each calculated at the end of a statistical period.