Manganaris [Man97] and Pednault [Ped89] discuss techniques for converting a sequence of continuous values into a piecewise polynomial model - in other words, you provide a continuous valued function, much like a continuous channel in our representation, and it provides you with a set of polynomials, that together, approximate the channel. Each polynomial is specified in terms of the start time, the end time and the coefficients of the polynomial. The number of polynomials applied is not defined a priori, rather it is decided by the application of an MDL-style heuristic, as is the degree of the polynomial used over a particular range. For simplicity, we will consider only first-degree polynomials, i.e. straight lines.
Figure 5.3: An example of a piecewise line approximation
to some real data
For example, consider figure 5.3. The input channel, indicated by an unbroken line, can also be converted to a sequence of ``straight-line'' approximations, as indicated by the dashed lines. In this particular case, the original sequence of 40 data points, is reduced to four straight lines.
Each of these straight lines can be considered as an instance of a general event primitive which is a straight line.
Each straight line can be defined in terms of the following parameters:
Note that this is not the only way to describe the line. The line could equally be described by: start value, end value, start time, end time. Also note that time has been explicitly encoded as part of the description of the event. Once we have created a description in terms of these parameters, we can make a table of these parameters. For example, the lines shown in figure 5.3 could be described as in table 5.1.
Table 5.1: Line approximation parameters
The finding function used above is based on a similar heuristic to that used by Manganaris [Man97]. Giving a closed-form definition of this finding function is difficult.