TT-$\delta$ Plots Generated from Artificial Point Data

To become used to reading TT-$\delta$ plots, the following section illustrates several plots resulting from the artificially produced point patterns (figure 5.3). This will help us sharpen our ability to recognize patterns in temporal point data.

The first pattern represents an animal walking forth and back along a line (figure 5.3a). The resulting TT-$\delta$ plot is shown in figure 5.12a. The color scheme used in the following figures indicates the distance according to table 5.1.

Table 5.1: Color scheme for TT-$\delta$ plots

Color	Distance (space)
blue	short
green	medium
red	large

Now we can start the first steps reading a TT-$\delta$ plot. Let us take a look at a thin horizontal transect through such a pattern provided in figure 5.11.

Figure 5.11: Transect for the example TT-$\delta$ plot from figure 5.12a.

Starting at the left side of the line the animal starts moving. As time passes by when we shift the view in direction to the right, the animal moves away from the starting point. This results in a color change to green and later to red, because the distance represented by the colors on this horizontal transect is measured in relation to the starting point. Now the animal turns around and moves back in direction to the starting point - the color changes again back to green - and finally reaches it, as it is indicated by the recurrence of the blue color. As the original movement path of the animal is 'walking stereotypically to and fro on a line' (as often seen in animals in captivity), the pattern is restarted from the beginning and continues until the end of the transect.

With this knowledge we can now start the interpretation of a whole TT-$\delta$ plot.

The first thing (A) we can note in figure 5.12a is the blue diagonal line from the lower left to the upper right corner. On this line, all the distances calculated are from location at t_x (horizontal axis) to the location at t_x (vertical axis), i.e. to the location itself and is therefore zero, as indicated by the blue color. This line will hereafter be referenced as the base diagonal line. Remember that in TT-plots the lower right triangle delimited by this line is mirrored to the upper left triangle.

As a second pattern (B) we can recognize the blue lines parallel to the before mentioned diagonal. They indicate that the animal walked the same path in the same direction as it did some time ago.

The third features (C) we can see are the blue diagonal lines from the top left to the bottom right at an angle of 90$^{\circ}$ to the diagonals described above. They show that the animal walked the same path as before, but in the opposite direction.

As the last feature (D) I would like to mention the equally spaced red spots. They indicate that the animal is far away at regular intervals, not only for certain points, but for all points visited.

Figure 5.12: Example TT-$\delta$ plots from artificial data (from left to right): a) line, b) circle, c) 8-shaped, d) star movement patterns. Short distances are indicated in blue, medium ones in green and large distances have a yellow to red color.

a

b

c

d

The second TT-$\delta$ plot (figure 5.12b) is an artificial movement from a (computer-)mouse running around in a circle (as shown in figure 5.3b). Looking at a horizontal transect through the figure we have a similar pattern as in the horizontal transect from figure 5.12a, starting close (blue), then going away (to green to red) and then returning to the original point. The patterns (A) and (B) above can also be found in this figure. But as an obvious difference there are no diagonal lines from the upper left to the lower right, indicating that the animal never walks the same way in the opposite direction.

The third example is a bit more complex. The movement describes an 8-shape. We can see the parallel blue lines from the lower left to the upper right as in the previous examples, showing that the animal walks the same pathway in the same direction at a later time. In addition to these lines we can see blue spots at regular intervals in between the parallel lines. They result from the intersection point in the middle of the 8-shape.

The last artificial dataset used here is the movement path shown in figure 5.3d which is describing a star shape. Here we meet again several patterns we know from the previous examples, sometimes only as fragments. There are two new characteristics I would like to point out. First, there is a relatively regular blue dot pattern, sometimes overlapping with other structures. The second pattern are the X-shapes in the lower right part of the plot. Such a shape indicates that the animal not only used the same path as some time ago, it also walked back on the same path as it did the first time.

This leads to another characteristic that can be seen within these plots. When an animal is using the same path twice, one can see the speed it used the second time relative to the speed it had the first time. More concretely, when a T-$\delta$ plot shows a blue line in a 45$^\circ$ direction, the animal used the same path in the same speed as the first time. A distortion towards a larger angle shows a faster speed, one towards a smaller angle a slower speed than the first time. The same is true when the angle is around 135$^\circ$, but the animal is then using the opposite direction on this path.

These four examples shall serve as a short introduction on how to read TT-$\delta$ plots. In the next section I will try to compile a first catalogue of features and patterns that can be discovered in this type of plots.