T-$\sigma$ Plot

The last category of T-plots I will bring in is represented by the T-$\sigma$ plot. It is used to describe the change in spread of the point object. As a contrast to the T-r plot, which describes one spatial aspect, and the T-$\nu$ plot, which describes a derivative of the spatial information using 2 observations, the T-$\sigma$ works with a measurement derived from several observations (in the case here with the standard deviation).

The approach used here is using the temporal data frames introduced in chapter 4. The plot uses time for the x-axis and as one example of a spread index the standard deviation on the y-axis. For calculating the standard deviation at time-point x, all observations within the time frame $x \pm \frac{1}{2}timeframe$ are considered. This is done repeatedly for all time-points. Such a plot is shown in figure 5.7, indicating the change in spread during the observation period. One may assume that a critical and also essential point is how to determine the width of the time frame. This is not the case here, because as with all exploratory data analysis (EDA) techniques (Tukey, 1977) it is important to be able to change the perspectives and views quickly to search for patterns within the data instead of imposing a rigid pseudo correct corset. As the scientific saying goes, one man's data is another man's noise. There is no 'correct' value, instead you need the ability to change and compare different values.

The approach used here - an equal period of time for the selection of objects considered for the calculation of the statistics - could also be replaced by a fixed number of observations resulting in varying time frame widths. This would result in more homogeneous results with scarce data and more inhomogeneous values in dense data.

Figure 5.7: T-$\sigma$ plot for the artificial movement pattern walking along a line.