A shape metric for evolving time series models

Most applications of Genetic Programming to time series modeling use a fitness measure for comparing potential solutions that treat each point in the time series independently. This non-temporal approach can lead to some potential solutions being given a relatively high fitness measure even though they do not correspond to the training data when the overall shape of the series is taken into account. This paper develops two fitness measures which emphasize the concept of shape when measuring the similarity between a training and evolved time series. One approach extends the root mean square error to higher dimensional derivatives of the series. The second approach uses a simplified derivative concept that describes shape in terms of positive, negative and zero slope.Unpublished[Chatfield, 1989] Chatfield, C. (1989). The Analysis of Time Series: An Introduction. Chapman and Hall, 4 edition. [Eldredge, 1989] Eldredge, N. (1989). Macroevolutionary Dynamics: Species, Niches and Adaptive Peaks. McGraw-Hill. [Jones, 1995] Jones, T. (1995). Evolutionary Algorithms, Fitness Landscapes and Search. PhD thesis, Computer Science Dept., University of New Mexico, Albuquerque, New Mexico. [Keogh and Pazzani, 1999] Keogh, E. and Pazzani, M. (1999). Relevance feedback retrieval of time series data. In The Twenty-Second Annual International ACM-SIGIR Conference on Research and Development in Information Retrieval. [Lee, 1999] Lee, G. (1999). Genetic recursive regression for modeling and foreasting real-world chaotic time series. In Spector, L., Langdon, W., and Angeline, U. O. P., editors, Advances in Genetic Programming 3, chapter 17, pages 401–423. USA:MIT Press. [Makridakis et al., 1983] Makridakis, S., Whellwright, S., and McGee, V. (1983). Forecasting: Methods and Applications. John Wiley & Sons, Inc., 2 edition. [Manderick et al., 1991] Manderick, B., de Weger, M., and Spiessens, P. (1991). The genetic algorithm and the structure of the fitness landscape. In Proceedings of the 4th International Conference on Genetic Algorithms, pages 143–149. Morgan Kaufmann Publishers. [Tansel et al., 1993] Tansel, A., Clifford, J., Gadia, S., Jajodia, S., Sesgev, A., and Snodgrass, R. (1993). Temporal databases: theory, design and implementation. Redwood City, Calif.:Benjamin/Cummings Pub. Co. [Whigham and Crapper, 1999] Whigham, P. and Crapper, P. (1999). Time series modelling using genetic programming: An application to rainfall-runoff models. In Spector, L., Langdon, W., O’Reilly, U., and Angeline, P., editors, Advances in Genetic Programming 3, chapter 5, pages 89–104. USA:MIT Press. [Zhang et al., 1997] Zhang, B.-T., Ohm, P., and Mu ̈hlenbein, H. (1997). Evolutionary induction of sparse neural trees. Evolutionary Computation, 5(2):213–236