Fuzzy adaptive rules in the forecasting of short memory time series

Fuzzy rule extraction is performed on an artificial time series with memory generated with a given covariance matrix using the inverse whitening transformation. The covariance matrix is defined with a definite range of memory using the short memory form of exponential decay. Vector quantization is performed on this real-valued time series to convert it into a digitized sequence of finite number of classes. The sequence is then divided into two subsets: training and testing sets, and the problem of forecasting the time series given the past data corresponds to the construction of a set of prediction rules that will make a classification on the class of the data today given the past sequence. We then construct an adaptive classifier using simple genetic algorithm with fixed selection ratio and construct a set of hierarchical rules for the classification of patterns. Since fuzziness exists for data close to the boundary between two classes, we modify our classifier by introducing in the triangular membership function associated with each class of data. The fuzzy region between neighboring classes is the overlapped region of these triangular functions and is parameterized by the degree of fuzziness, f. After training, the best rule from the genetic algorithm is measured for a given degree of fuzziness. Two distinct phases in the degree of fuzziness, separated by a critical value at f=0.18 for a short memory time series with decay constant of 5 days are found and understood as the result of two distinct best rules in two different phases. Application of this fuzzy adaptive classifier to real financial time series is discussed.