Estimation using an IFS

The confidence function is in fact a conditional probability function dependent on actual and past (but recent) conditions of the signal (the observable time series coming from the system being estimated). For practical purposes the signal we wish to estimate is sampled and a certain number of these samples is kept in memory. When certain conditions appear they are used as an index in a multidimensional table that selects a particular probability function (the confidence function given actual conditions). The dynamical system with highest probability is the one selected to perform the estimation (see Figures 5 and 6).

Figure 6: Confidence Function

We define the accuracy $\gamma$ , of a prediction scheme as the root mean square of the difference between the predicted time-series and the observed one. Figure 7 shows how $\gamma$ varies with time when we use this scheme to predict a quasi-triangular sequence.

Figure 7: $\gamma$ as a function of time.

It can be seen in Figure 7 that the estimator ``learns" as times go by. The almost constant line at $\gamma$ = 4 corresponds to the error in prediction we would have, had we chosen to predict, at every time, that the signal would not change. It is included in the graph because it represents the cheapest form of prediction and it can be used for comparison. The predictions themselves can be improved by an appropriate selection of the probabilistic table's dimension. Nevertheless, there is an optimal dimension beyond which no improvement occurs [33].