Creation of TT-Plots

The construction of a TT-plot is outlined with the example of the TT-$\delta$ plot. The TT-plots are a new way to transform the three dimensional data consisting of two spatial and one temporal axis to a two dimensional representation by reducing the spatial component to an inter-event distance matrix and introducing a second time axis. This is performed by creating a matrix of spatial distances for every time-point to all other time-points. The plot can be constructed as follows: The x and y axis are both t (time), while the z axis at the point t₁,t₂ represents the geographical distance $\delta$ between the two locations at P_t1 and P_t2.
Let us have a look at figure 5.8 where four observations are plotted ( P_t1 - P_t4).

Figure 5.8: Example TT-$\delta$ plot construction. Left: Planar view of an animal's path with four observed locations P_t1 - P_t4. Right: 3-D view of a TT-$\delta$ plot construction. See text.

For x = t₁, y = t₁ the calculated z is 0 ( P_t1 = P_t1). For x = t₂, y = t₁, z would be the geographical distance between P_t1 and P_t2, for x = t₃, y = t₁, z would be the distance between P_t1 and P_t3 and so forth. Now the calculated distances are drawn as z values (arrows) at the coordinates t₁, t₁ and t₂, t₁ and t₃, t₁ (figure 5.8). This results in a xyz scatter plot that can be interpolated to a surface for easier pattern recognition. The z values lying above the indicated diagonal line from the lower left to the upper right in figure 5.8 can be viewed as mirrored from the lower triangle in TT-plots, because the distance P_t1 - P_t2 is equal to P_t2 - P_t1. This duplication of information is done for easier recognition of patterns, especially for the beginning (lower left) and the end (upper right) of the observation period. A second reason for this mirroring is to keep the plots consistent with the TT2-plots introduced in Section 5.10 which need the full quadrangle to represent all the information in the data. Constructing the TT-$\delta$ plot in this manner, the distance from a location at time t₁ to the location at time t_n can be 'seen'. In a general mathematical form, the z value is a function of two locations at two time-points of an object:

Z_TT = f(P_t1,P_t2) (5.1)

Z_TT = Calculated value at plot location t₁, t₂
P_tx = Object's location at time-point x

Figure 5.9 shows a three dimensional plot with the standard color scheme used throughout this thesis. Short distances are indicated in blue, medium distances in green and large distances are colored in red (table 5.1). This figure serves as a link picture between the illustration for the plot construction (figure 5.8) and the remaining illustrations of TT-, and TT2-plots, which will only contain the color shading without the 3D effect.

Figure 5.9: Example TT-$\delta$ plot (3-d perspective) from the artificial data 'walking on a line'. Short distances are indicated in blue, medium ones in green and large distances have a yellow to red color.

Figure 5.10 illustrates the basic features found in a TT-plot that will be used in the following sections: the base diagonal line (5.10:A), a line parallel to it (5.10:B), another line at a 90$^{\circ}$ angle to it (5.10:C), and the lower right triangle (5.10:T).

Figure 5.10: Basic features found in a TT-plot: A = base diagonal line, B = parallel line to a, C = line at a 90$^{\circ}$ angle to a, T = lower right triangle.