模糊最小二乘法模型

LEAST SQUARES MODELS FITTING TO FUZZY DATA

本文引入模糊数间的一种距离,把类似的最小二乘法用于对模糊观测数据的分析,提出两类拟合模糊数据的模糊线性回归模型,证明确定模型的条件,给出计算模型参数的简明方法。

Tow fuzzy linear regression models are proposed for data analysis of triangular fuzzy numbers using analogues of simply linear least squares with a new metric on J (R). (Let J (R) denote the set of triangular fuzzy numbers and P(R)be that subspace of J(R) all of whose elements have nonnegative support) .The methods are rigorously justified by theorems 1 to 5.Simple algebraic criteria are given in terms of the data for when it is or is not appro priate to fit a model to a given data set. Detinition 1:If X.Y ∈J( R) ,λ∈ (0,1) .the metric on J(R) is deffined by dλ(X. Y). whereTHEOREM 1:Let C be a closed cone in P(R), For any X in P(R) there is a unique triangular fuzzy number Vo in C such thatdλ(X.Vo)≤dλ(X,V). for all V in C.THEOREM 2: If B> 0 . let m= {λ∈ [ 0 , 1]: B + ( 1 -λ)D + ( 1 -λ)2 (G+ F ) > 0 } for λ∈m, letand use (4)to get aλ+,Then the model (λ-F1)fitting problem has a unique solution aλ+and bλ+.THEOREM 3:If B<0.let N= { [0,1]:B+ (1-λ) D+(1-λ)2 (G+E)<0} , forλ∈N, letbλ+= [B(1-λ)D+(1-λ)2(G+F)] /Tλand use (4)to get a λ..Then the model (λ-F1) fitting problem has a unique solution aλ-and bλ-.THEOREM 4:If B'>0,λ∈ [0,1]m' = {λ:B'+( 1 -λ)D/+( 1 -λ)2(G'+E' )>0} ∩{A:B'+ ( 1 -λ)D' + ( 1 -λ ) 2 ( G' -E' )> 0 }, for λ∈m', letbλ-=CB'+(1 -λ)D'+ ( 1 -λ)2(G'+E')]/Tλ and if bλ+ use ( 9 ) , ( 10 ) to solve c, r, to get Eλ + =( c, r, r ) T. Then the model ( λ-F2 ) fitting problem has unique solution E λ+,, bλ+ THEOREL 5 : If B'<0, λ∈[0, 1]N' = {λ: B' + ( 1 -λ ) D' + ( 1 -λ )2 ( G' +E' ) <0 } ∩ {λ : B' + ( 1 -λ ) D'+ ( 1 -λ)2(G'-E' )<0}, for λ∈N, letbλ-=[B' + ( 1-λ)D'+ ( 1 -λ)2(G'-E')]/Tλand if - bλ- ,use(9), ( 12 ) to solve c, r, to get Er-=(c, r, r ) T.Then the model ( A- F2 ) fitting problem has unique solution Eλ-,bλ-.