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When performing incremental learning steps of linear rule consequent parameters in TSK fuzzy models, it may happen that the recursive weighted least squares guides the linear parameters to a wrong solution. This would be the case if the newly loaded data points or recently recorded measurements stay (almost) constant for a certain period. To demonstrate this in a simple practical example, see Figure 5.4, where the rst 200 data samples (light dots) are indeed well distributed over the whole input space, but the next 1300 samples (represented by the big dark dot) are concentrated around the speci c point (1200,2). Obviously, the car motor was steered with a constant rotation speed; the slight noise variance can be explained by sensor inaccuracies during recording. This example represents the situation whenever a process is in steady state, that is, staying at one speci c operating condition for a long time. When doing an initial learning phase with the rst 200 points (no matter whether in incremental or batch mode) and performing an adaptation of the fuzzy model with the later 1300 points (in steady state), an undesired unlearning effect of already-learned relationships outside this small constant region occurs. This can be recognized in the left image in Figure 5.4, where to the left and right of the small constant region the shape and behavior of the adapted model (dotted line) tend to be completely different from the shape and behavior of the original one (solid line), even though no new measurements were recorded for that area. The reason for these effects is that the parameters of all linear consequent functions are adapted for each incoming data sample, no matter which ring degree the rules have. In fact, rules with a very low ring degree (i.e., rules that are far away from the constant region) are adjusted very slightly for each point, but this sums up with a high amount of data recorded during steady state. Finally, this means that the consequent parameters are
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pressure and hypertension incidence in overweight people with high-normal blood pressure: The Trials of Hypertension, Phase II. Archives of Internal Medicine, 157, 657 667. Tunstall-Pedoe, H. (2000). Estimation of contribution of changes in coronary care to improving survival, event rates, and coronary heart disease mortality across the WHO MONICA Project populations. Lancet, 355, 688 700. Verrier, R. L., & Mittleman, M. A. (1996). Life-threatening cardiovascular consequences of anger in patients with coronary heart disease. Cardiology Clinics, 14, 289 307. Vogt, T., Pope, C., Mullooly, J., & Hollis, J. (1994). Mental health status as a predictor of morbidity and mortality: A 15-year follow-up of members of a health maintenance organization. American Journal of Public Health, 84, 227 231. Ward, R. (1990). Familial aggregation and genetic epidemiology of blood pressure. In J. H. Laragh & B. M. Brenner (Eds.), Hypertension pathophysiology, diagnosis and management (pp. 81 100). New York: Raven Press. Wassertheil-Smoller, S., Applegate, W. B., Berge, K., Chang, C. J., Davis, B. R., Grimm, R., et al. (1996). Change in depression as a precursor of cardiovascular events. Archives of Internal Medicine, 156, 553 561. Weder, A. B., & Julius, S. (1985). Behavior, blood pressure variability and hypertension. Psychosomatic Medicine, 47, 406 414. Weinberger, M. H. (1996). Salt sensitivity of blood pressure in human. Hypertension, 27, 487 490. Weiner, H., & Sapira, J. D. (1987). Hypertension: A challenge to behavioral research. In S. Julius & D. R. Bassett (Eds.), Handbook of hypertension: Behavioral factors in hypertension (Vol. 9, pp. 59 74). Amsterdam: Elsevier. Wells, J. A. (1985). Chronic life situations and life change events. In A. M. Ostfeld & E. D. Eaker (Eds.), Measuring psychosocial variables in epidemiologic studies of cardiovascular disease (NIH Publication No. 85 2270). Washington, DC: U.S. Department of Health and Human Services. Wenneker, M. B., & Epstein, A. M. (1989). Racial inequalities in the use of procedures for patients with ischemic heart disease in Massachusetts. Journal of the American Medical Association, 261, 253 257.
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