Extensions for Two Objects: TT2-Plots

In the previous sections about the TT-plots intra-dataset features were considered as the point of interest. Now I would like to extend this concept to the analysis of inter-dataset characteristics, i.e. characteristics originating from two datasets. The movements of an animal in its environment is also influenced by the movements of its neighbors. This can be accomplished by simultaneously observing the animals, but it is very difficult to find time lag effects with the traditional methods as for example Markov chain analysis in behavior research. Spatial effects are hardly ever considered. The extension of TT-plots to enable the analysis of two animals' movements can provide insights into the spatial components of the behavior of two animals.

TT2-plots can be constructed in a similar way as TT-plots. The difference is that the measurement of interest does not refer to one single animal, but from one animal to the other. In a TT2-$\delta$ plot, which describes spatial distances, the distance from one animal to the other is measured for all time lags (figure 5.17).

Figure 5.17: Creation of a TT2-$\delta$ plot.

This results in a distance matrix which can be visualized in the same way as the TT-plots, where the x and y axis are used for time, and the z axis represents the distance between the two animals. Also the same types of TT2-plots can be constructed as the ones mentioned in the TT-plot section (e.g. distance, parallelity etc.).

There are two main differences between TT- and TT2-plots which need to be outlined. First the TT2-plots do not contain a symmetry axis from the lower left to the upper right of the graph anymore. The distance from animal A at time point t₁ to the animal B at time point t₂ is not equal to the distance A(t₂)-B(t₁). Hence the full plot area is needed to represent the whole information. Second there is no base diagonal line visible because of the same reason.

The interpretation of TT2-plots can be easily deduced from the interpretation of TT-plots. To avoid repeating the same statements made earlier I will concentrate in the remainder of this section on the illustration of a single artificial dataset illustrated in figure 5.18a-c.

Figure 5.18: Example TT2-$\delta$ and TT2-$\pi$ plots from artificial data: a: original data. b: TT2-$\delta$ plot. Short distances are indicated in blue, medium ones in green and large distances have a yellow to red color. c: TT2-$\pi$ plot. Blue indicates parallel, green intermediate, red anti-parallel.

a
b
c

There are two (artificial) animals walking around simultaneously. The first one is the same one used in the previous examples walking on a straight line. The second one is walking around in a meandering movement (figure 5.18a). The corresponding TT2-$\delta$ plot is shown in figure 5.18b. Having a look at the (formerly called) base diagonal line we can see that the animals have six occasions where they come relatively close to each other, indicated by the blue phases lying on the diagonal line. The first animal (black) often passes again locations it crossed before, and which are additionally lying on the other animal's path. The second animal on the contrary does seldomly pass a location twice where it encountered the other animal's track. This can be seen when building vertical and horizontal transects through the TT2-$\delta$ plot. Possibly two phases can be distinguished by the blue '<-shapes' and '>-shapes'. In the TT2-$\pi$ plot (figure 5.18c) a distinct pattern can be seen at the beginning of the observation period. The two animals walk parallel to each other, then anti-parallel, and afterwards the first animal keeps on performing the same parallel-anti-parallel walking pattern while the second changes its direction mainly to a 90$^\circ$ direction from the first animal. In the second half of the observation period it changes its direction to an angle of approximately 45$^\circ$ from the first animals direction until in the last phase it approaches again similar directions (parallel and anti-parallel) as the first animal.