Regression analyses are a popular statistical tool for illustrating trends. They are also used in various types of risk assessment such as the Jensen and Treynor ratios. The basic method of linear regression seeks to construct a straight line in a two-dimensional system of coordinates such that all data points within the system of coordinates lie as near as possible to this line.

The straight line constructed in this manner is described by the two variables alpha (intercept, intersection of the straight line with the y-axis) and beta (slope, gradient of the straight line). For every data point, the point n,y of the regression line can then be calculated by means of these two variables.

The formulas for the calculation of alpha and beta in the analysis of time series are:

In these equations, N is the number of data points in the time series, i.e. the number of days, for example, n is the number of the data point, i.e. 1 for the first and 1000 for the thousandth data point in the time series. That is to say, y-values exist only for the natural numbers (n) on the x-axis. The curve of the time series thus arises through the connection of all points n,y of the time series.

Hence, the calculation for time series of a constant length is relatively easy, since the denominators in the formulas for the calculation of alpha and beta all become constants.

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In the investigation of time series, the results of spectral analyses can reveal interesting details. Moreover, on the basis of the information thus extracted, it becomes possible to parameterize other transformation procedures in such a way that they act as ideal filters for suppressing irrelevant information.

The following examples are intended merely to give you an idea of what is possible by means of spectral analysis. A comprehensive discussion of this topic would easily fill an entire scientific volume. Hence, we will limit ourselves at this point to one field of application and two simple examples that can be grasped without in-depth knowledge of physics and mathematics: the investigation of cyclical behavior patterns and the extraction of parameters for information filters.

In order to explain our approach, we begin with an experimental time series. Trigonometric functions – such as sin(x), for example – are very suitable for generating experimental series with known periods. In the following illustration, we generated an experimental time series in this manner.

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