Temporal Extrapolation within a Static Clustering1

Predicting the behaviour of individuals is a core business of policy makers. This paper discusses a new way of predicting the “movement in time” of items through pre-defined classes by analysing their changing placement within a static, preconstructed 2-dimensional clustering. It employs the visualization realized in previous steps within item analysis, rather than performing complex calculations on each attribute of each item. For this purpose we adopt a range of well-known mathematical extrapolation methods that we adapt to fit our need for 2-dimensional extrapolation. It is common practice to visualize analysis results as a clustering within the 2-dimensional plane, approximating the correct, multi-dimensional distances. As long as the error made within the visualization is as small as possible there is no preferred method of obtaining such a clustering within our prediction system. However, the algorithm must allow for incremental addition of single elements from the sequence to be extrapolated, which is not supported by all of the known methods. All interpolation schemes are suitable starting points for the process of extrapolation. In most cases, it is sufficient to simply continue the fabricated interpolation function after the last existing data point. In the case of a spline interpolation, however, a choice can be made to continue the polynomial constructed for the last interval (which can lead to strange artifacts), or extrapolate with a straight line, constructed with the last known derivative of that polynomial (Figure 1 and 2). In our approach both x and y are coordinates and therefore inherently independent variables. They depend on the current visualization alone. Within our model, they do however depend on the time variable t. Because our methods aim to extrapolate x, y out of one other variable t, we need a form of 2-dimensional extrapolation.