The Fuzzy Inference System with Least Square Optimization for Time Series Forecasting

Abstract

The rule base on the fuzzy inference system (FIS) has a major role since the output generated by the system is highly dependent on it. The rule base is usually obtained from an expert but in this study proposed the rule base generated based on input-output data pairs with generating rule bases using lookup table scheme, then consequent part of each rule optimized with ordinary least square(OLS), so finally formed rule base from model FIS Takagi-Sugeno orde zero. The exchange rate dataset of EURO to USD is used for the development and validation of the system. In this study, 12 FISs were developed from a combination of linguistic values of n = 3,5,7, 9 with the number of lag (k) assumed to have an effect on output for k = 2,3,5. In training data, values R<sup>2</sup> ranged between 0.989 and 0.993, MAPE values ranged between 0.381% and 0.473% where the FIS with the combination of n = 9 and k = 5 has the best performance. In the testing data, values R<sup>2</sup> ranged between 0.203 and 0.7858, MAPE values ranged between 0.5136% and 0.9457% where FIS n = 3 and k = 2 perform best.

Generating of Fuzzy Rule Bases with Gaussian Parameters Optimized via Fuzzy C-Mean and Ordinary Least Square

“…At then ncl = 3 and ncl = 5 visually can still be recognized the Gaussian curves of each cluster. The result is supported by the FIS implementation carried out by Handoyo and Marji [9] that the best-performing FIS is on ncl = 3 or 5.…”

“…The bell-shaped Gaussian curve is characterized by two parameters, namely the center of the curve and spread (standard of deviation). In some FIS implementations such as in Handoyo and Marji [9], Handoyo et al [10], Arief et al [15] and Wang [1], the two parameters are determined in a simple way. The spread parameter is a constant which amount is calculated that divides range by (n-1), where n is the number of fuzzy sets, while the curve centers are obtained through the minimum value plus k * the spread value, for k = 0,1,2,3, ...., (n-1), in this case, the minimum value automatically becomes the center of the lowest curve, and the maximum value automatically becomes the center of the highest one.…”

“…The fuzzy rule base generated by the lookup table scheme such as in Handoyo and Marji [9] tends to produce a large number of rules. Of course, the more basic rules on the basis of fuzzy rules will have the effect of slowing down the FIS to produce output because the fuzzification process and its inference will take a long time.…”

“…The main difference with the generation of FRB with FCM is that this method (FCM and OLS Optimization) employs 42 parameter spread values while at FCM only one spread value is employed for 42 Gaussian functions. Rule number 7 if using FCM is as follows: More extreme is the use of the Gaussian membership functions found in Handoyo and Marji [9] where each linguistic values only has a deterministic parameters of the Gaussian membership function. In the case, the rule number seven above can be stated as follows:…”

“…The main motivation of the Fuzzy Inference System (FIS) is to map an input space in the form of fuzzy numbers obtained from the fuzzification process to produce output in the form of crips number yielded by inference and defuzification process [1][2][3]. The FIS has been widely applied to solve problems in various fields including engineering [4][5][6], health [7], [8] and finance [9][10][11]. The inference process in the component inference engine uses Fuzzy Rule Bases (FRB) to produce an output.…”