Patent ID: 6463341
Filing Date: 2002-10-08
Classification: G06N

Abstract:
A method for choosing a set of orthogonal basis functions for a function approximation from empirical data described as {xt,yt}t=1P,comprising the steps of:(a) constructing a heterogeneous regressor set F={fi}i=1Nfrom a set of randomly selected basis functions;(b) defining &PSgr; as &PSgr;â‰¡[&phgr;1, &phgr;2, . . . , &phgr;N]=rearrangement (F) by at a first step k=1, denoting a first column of the &PSgr; matrix, &phgr;1â‰¡ft(1), selected from fi(1) where &LeftDoubleBracketingBar;ft(1)&RightDoubleBracketingBar;2=max&it;{&AutoLeftMatch;&LeftDoubleBracketingBar;fi(1)&RightDoubleBracketingBar;2&RightBracketingBar;i=1N}â€ƒand the first orthogonal basis is h1=&Sum;i=1N&it;&LeftAngleBracket;ft(1),fi(1)&RightAngleBracket;&LeftDoubleBracketingBar;ft(1)&RightDoubleBracketingBar;2&it;ft(1);(c) building an orthogonal basis matrix H by at a kth step, where k&gE;2, calculate fi(k) and hk as fi(k)=fi(k-1)&it;&LeftAngleBracket;fi(k-1),ft(k-1)&RightAngleBracket;&LeftDoubleBracketingBar;ft(k-1)&RightDoubleBracketingBar;2&it;ft(k-1),&NewLine;&it;hk=&Sum;i=1N&it;&LeftAngleBracket;ft(k),fi(k)&RightAngleBracket;&LeftDoubleBracketingBar;ft(k)&RightDoubleBracketingBar;2&it;ft(k),â€ƒsuch that hk can be simplified as hk=&Sum;m=1N&it;&LeftAngleBracket;(&varphi;k-&Sum;i=1k-1&it;&LeftAngleBracket;&varphi;k,&varphi;i&RightAngleBracket;&LeftDoubleBracketingBar;&varphi;i&RightDoubleBracketingBar;2&it;&varphi;i),&varphi;m&RightAngleBracket;&LeftDoubleBracketingBar;&varphi;k-&Sum;i=1k-1&it;&LeftAngleBracket;&varphi;k,&varphi;i&RightAngleBracket;&LeftDoubleBracketingBar;&varphi;i&RightDoubleBracketingBar;2&it;&varphi;i&RightDoubleBracketingBar;2&it;(&varphi;k-&Sum;i=1k-1&it;&LeftAngleBracket;&varphi;k,&varphi;i&RightAngleBracket;&LeftDoubleBracketingBar;&varphi;i&RightDoubleBracketingBar;2&it;&varphi;i),where&varphi;k=ft(k);(d) initializing by letting Hsubset=&phgr;, where &phgr; is an empty set and let k=1; (e) finding hi such that maxi&it;{(yT&it;hi)2Î»+(hi)T&it;hi};and,(f) including hi as an element of the Hsubset such that Hsubset=Hsubsetâˆªhi; (g) regularizing by modifying the generalized cross validation variable &lgr; by letting the index of selected ft(k) in the original F matrix be j, where (&LeftDoubleBracketingBar;ft(k)&RightDoubleBracketingBar;2=max&it;{&AutoLeftMatch;&LeftDoubleBracketingBar;fi(k)&RightDoubleBracketingBar;2&RightBracketingBar;)i=1N},such&it;â€ƒ&it;that&it;â€ƒ&it;&varphi;k=fj(1);and,â€ƒsuch that &phgr;k=fj(1); and, (h) stopping if &par;ft(k)&par;2&lE;&egr;, where &egr; is a preselected minimum value, otherwise letting k =k+1 and repeating beginning at step (e).