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

As it was done for the TT-$\delta$ plots, examples from artificially produced point movement patterns are presented here. The original data can be seen in figures 5.3a-d.

The first pattern results from the animal walking forth and back along a line (figure  5.3a). The resulting TT-$\pi$ plot is presented in figure 5.14a. There are basically three characteristics that I would like to point out in this plot. First (A) a base diagonal line from the lower left to the upper right can also be found in this type of plot. Here it represents the difference in direction from a point to itself, which is always zero, i.e. they are exactly parallel. The second feature (B) is a blue/red pattern similar to a checker board with hardly any other colors in between. It shows that the animal almost exclusively uses two directions which are at a 180$^\circ$ angle. The third feature (C) are the thin green lines which indicate that there are short periods at which the animal uses another direction, in this case probably turn-points as the one indicated with a green arrow in figure 5.13. From the size of the blue squares it can be easily determined (visually or by calculations from the matrix) how long the animal keeps on going into the same or similar direction.

Figure 5.14: Example TT-$\pi$ plots from artificial data (from left to right): a) line, b) circle, c) 8-shaped, d) star movement patterns. Blue indicates parallel, green intermediate, red anti-parallel.

a

b

c

d

The second pattern illustrated in figure 5.14b comes from an animal running around in a circle. The resulting TT-$\pi$ plot is very different from the previous one. Here the width of the 'blue phases' is much smaller than in the previous example indicating a smaller tendency to go on walking into the same direction. The checker board pattern is replaced by a cluster of diagonal lines going repeatedly from parallel (blue) to a 90$^\circ$ angle (green) to anti-parallel (red) and then back to parallel (blue). It is important to know that this pattern is independent from the place where this circular path is located. When the animal is shifting to a direction, but maintains the circular path, the same TT-$\pi$ plot results.

The third example (figure 5.14c) includes a similar structure as the one above, the blue lines being parallel to the base diagonal line. Additionally we find blue parallel lines at an angle of 90$^\circ$ to the base diagonal line. They indicate a fact I shall illustrate with a simple situation. It is a movement which is, from the point of view of the direction, mirrored at a line. For example if the animal is making a half circle turning to the left and then performing a half-circle turning to the right, the resulting pattern in a TT-$\pi$ plot would be a blue x-shape. The two half circles can be transformed into each other by mirroring one of them at a line. At what angle from the horizontal the upper left/lower right diagonal line is appearing depends on the relative speed used in the two parts of the movement. The third structure that can be recognized are the more or less distorted red circles which are arranged in two alternating rows and columns. They indicate a regular movement which is divided into two parts as the one mentioned above, but in an anti-parallel direction. According to the size of the blue areas the tendency to keep on going into the same direction is in between the first and second example above.

The TT-$\pi$ plot for the last artificial movement presented here is shown in figure 5.14d. It is based on the star-shaped movement of the (computer-)mouse. First we can see at several places in the plot an alternation of parallel (blue) and anti-parallel (red) movements (e.g. lower left corner, second and last quarters of the observation period). There seems to be a tendency towards a pendular movement with an angle of approximately 180$^\circ$. The exact values would have to be extracted from the plot. The blue squares are not so regular as in the previous examples. From the statements above it can be deduced that the animal in concern has a higher variability in how long it keeps on going in the same direction.