應用模糊最小平方法分析二階迴歸方程式之研究

Analysis and Application of the Fuzzy Least Squares : A Quadratic Regression Study

蔡振彥(Zhen-Yan Tsai)

指導教授 ： 陳雲岫

元智大學/工程學院/工業工程與管理學系/碩士(1995年)

摘要

本文乃針對二階模糊線性迴歸方程式之模式，提出三種模糊方法，分
別解出迴歸式 中參數的估計值，以改善現存模糊線性迴歸模式之估計
效果。一般在探討模糊線性迴歸 問題時，皆假設模糊數值為對稱型，
不敷實際之需求，因而降低了應用性，所以本文探 討之範圍則推廣至
模糊數值之隸屬函數(Membership Function)可為對稱型或非對稱型
的三角函數。在求解的過程中，本文分別利用OERI(Overall Existence
Ranking Index) 及Diamond距離準則之修正，Diamond*，作為模糊數值(
Fuzzy Number)距離的二種衡量準 則。OERI距離衡量準則為準之解法是
採非線性規劃模式，簡稱為One-Step OERI。 若利用Diamond*之
距離準則，則求解方法便採用模糊最小平方法。然One-Step OERI於
輸出變數之中心值的估計不若模糊最小平方法來得理想，因此本研究則提
出一Two-Step OERI方法來改善估計效果。Two-Step OERI之觀念是先以
最小平方法估計參數之中心值， 再採以非線性規劃模式求取參數之分佈
寬度值。最後以模擬數據進行三種方法的效果評 估，評估準則採用估
計值與觀測值差距之絕對值平均為之參考。由模擬實驗的結果可推 知
，當樣本數少時，採用Two-Step OERI之估計效果較好，而當樣本數漸增
時，則模糊 最小平方法之表現漸佳。

關鍵字

模糊最小平方法 ； 模糊數值

並列摘要

In this research, three methods are used to estimate the
fuzzy parameters in quadratic fuzzy linear regression models.
The Overall Existence Ranking Index(OERI) and the
modification of Diamond''s distance measurement, Diamond*, are
used. The assumptions of membership functions of fuzzy numbers
are extended to be symmetric or asymmetric triangular
functions which can broaden the applications of the fuzzy
regressions. Non-Linear Programming(NLP) is adopted to
solve the parameter values based on the OERI distance
measurement and the method is referred as One-Step OERI,
while the Diamond* is the distance rule for the fuzzy
least squares method. Owing to the performance of One-Step
OERI is not good at estimating the central values of the output
variables, we propose a two-stage method, Two-Step OERI, to
enhance the estimation effects. The concept of two-step
OERI is to estimate the central values of the parameters by
least squares and then solve the estimation values of the
spread widths by NLP. Simulation is used to evaluate the
performance among three methods, One-step OERI, Two-step OERI
and fuzzy least squares. The criterion used to judge the
performance is the average of absolute difference
between estimated and observed values. The merits and
shortcomings of the three methods are discussed . Simulation
results show that the Two-Step OERI works well for the
small sample sizes and estimation effects of fuzzy least
squares gets better when the sample size gets larger.

並列關鍵字

One-Step OERI ； Two-Step OERI ； Diamond距離準則 ； Fuzzy Least Squares ； Fuzzy Number ； One-Step OERI ； Two-Step OERI ； Diamond Distance Measurement