ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang

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1 Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang Department of Mathematical Science Tinghua Univerity Beijing , CHINA ABSTRACT Mathematical model are often decribed by multivariate function, which are uually approximated by a um of lower dimenional function. A major problem i the approximation error introduced and the factor that affect it. Thi paper invetigate the error of approximating a multivariate function by a um of lower dimenional function in the etting of high dimenional model repreentation. Two kind of approximation are tudied, namely, the approximation baed on the ANOVA (analyi of variance) decompoition and the approximation baed on the anchored decompoition. We prove new theorem for the expected error of approximation baed on anchored decompoition when the anchor i choen randomly and etablih the relationhip of the expected approximation error with the global enitivity indice of Sobol. The expected approximation error give indication on how good or how bad could be the approximation baed on anchored decompoition and when the approximation i good or bad. The influence of the anchor on the goodne of approximation i tudied. Method for chooing good anchor are preented. 1 INTRODUCTION Mathematical model are often decribed by multivariate function f (x), where x = (x 1,...,x ). We tudy repreentation of a multivariate function f (x) writing a a finite um of lower-dimenional function f (x) = f /0 + f { j} (x j ) + + f {1,...,} (x 1,...,x ), 1 i< j f {i, j} (x i,x j ) + where f /0 i a contant, f { j} (x j ) repreent the individual contribution of the variable x j wherea f {i, j} (x i,x j ) repreent the the cooperative effect of x i and x j and o on. Such decompoition are called high dimenional model repreentation, ee Rabitz et al (1999) and Sobol (2003). It i argued in Rabitz et al (1999) that quite often in high dimenional model in practice low order interaction of input variable have the main impact on the output (uually up to the third order). In many application (for intance, in mathematical finance), a multivariate function often ha mall effective dimenion, ee Caflich et al and Wang and Fang (2003). Roughly, thi mean that the function depend mainly on a mall number of variable or i nearly a um of function that depend on a mall number of variable. A common practice i to ue f (x) = f /0 + L m=1 i 1 < <i m f {i1,...,i m }(x i1,...,x im ) (1) a an approximation to f (x) (where L < ). Two decompoition, namely, the ANOVA decompoition and the anchored decompoition, are popular in practice. ANOVA decompoition ha ome good propertie and i widely ued in practice. However, the ANOVA term and their variance are difficult to compute due to the high dimenional integral involved (which may require Monte Carlo or quai-monte Carlo method). The anchored decompoition might be le attractive in theoretical apect, but it i much eaier to obtain. Due to the complexity in computing the ANOVA term, it i uggeted to ue the low order term in anchored decompoition to approximate the model function f (x), ee Rabitz et al (1999). A major problem here i the approximation error and the factor that affect it. The anchored decompoition contain an arbitrary reference point (called anchor). The choice of a good anchor i almot neglected in literature. It i pointed out in Sobol (2003) that in ome cae the choice of the anchor i important in order to obtain a good approximation and a carele choice may lead to unacceptable approximation. Thi paper invetigate the error of approximating a multivariate function f (x) by a um of lower dimenional function a in (1). We compare the approximation baed on ANOVA decompoition and on anchored decompoi /08/$ IEEE 453

2 tion. For the latter one, we introduce the expected error of approximation a a quality meaure when the anchor i choen randomly. We prove new theorem about the expected error of the approximation and etablih the relationhip of the expected approximation error with the global enitivity indice. The expected error give indication on how good or bad could be the approximation, and when and why the approximation i good or bad. Thi paper i organized a follow. In Section 2, we introduce high dimenional model repreentation, namely, the ANOVA decompoition and the anchored decompoition. In Section 3, we tudy the approximation error, with focu on the expected error of anchored approximation. In Section 4, we preent method to chooe good anchor. In the lat ection we preent concluion. 2 HIGH DIMENSIONAL MODEL REPRESENTATION 2.1 The ANOVA decompoition For any ubet u {1,...,}, let u denote the complementary et and u denote the cardinality of u. Conider a quare-integrable function f (x) defined on [0,1]. We ay that the repreentation f (x) = u {1,...,} i an ANOVA decompoition of f (x) if 1 0 u (x) (2) u (x)dx j = 0 for j u. (3) The condition (3) uniquely define the term in (2). Indeed, integrating both ide of (2) over [0,1] and uing (3), we obtain that f /0 anova (x) = f (x)dx. [0,1] Integrating both ide of (2) with repect to all variable except x j and uing (3), we obtain that [0,1] 1 f (x)dx { j} = /0 (x) + { j} (x), Similarly, integrating both ide of (2) with repect to variable x u and uing (3), we obtain that [0,1) u f (x)dx u = v (x), (4) thu we obtain the recurive formula for the ANOVA term fu anova (x) = f (x)dx u [0,1) u v u fv anova (x), where denote trict incluion, x u denote the vector compried of the component of x with j in u. The term fu anova (x) depend only on the variable x u. A an invere linear relationhip of (4), we have an explicit formula for the ANOVA term u (x) = ( 1) [0,1) u v f (x)dx v, u {1,...,}. v It follow from (3) that the ANOVA decompoition i orthogonal [0,1] u (x) fv anova (x)dx = 0 for u v. Baed on thi orthogonal property, we have the decompoition for the total variance = σu 2 ( f ), (5) /0 u {1,...,} where and σu 2 ( f ) are the variance of f (x) and fu anova (x), repectively, with x being a random vector uniformly ditributed on [0,1]. Define D u = σv 2 ( f ), u {1,...,}. An invere linear relationhip of thi equality i σu 2 ( f ) = ( 1) D v, u {1,...,}. (6) To meaure the importance of variable, we defined the global enitivity indice a thu f{ anova j} (x) = f (x)dx [0,1] 1 { j} f /0 anova (x). S u ( f ) = σ 2 u ( f )/, u {1,...,}. Global enitivity indice can be ued to identify the key variable and to fix the uneential variable, ee Sobol (2003). We may alo define R l ( f ) = 1 σu 2 ( f ), l = 1,...,, (7) u =l 454

3 which meaure the importance of all order-l term taken together. In particular, the um of all firt order global enitivity indice R 1 ( f ) i referred a the degree of additivity, which meaure the additive tructure inherent in the function. If R 1 ( f ) = 1 (or R 1 ( f ) 1), then the function i perfectly additive (or nearly additive). If R 1 ( f ) 1, then the function i dominated by higher order term. A good property of the ANOVA decompoition i it optimality (ee Section 3). A main limitation of ANOVA decompoition i the complexity in computing the ANOVA term, ince high-dimenional integral are involved (which may require Monte Carlo or quai-monte Carlo method). 2.2 The anchored decompoition An alternative decompoition the anchored decompoition i available. Let a = (a 1,...,a ) [0,1] be an arbitrary point. Thi point i called the anchor or the reference point. Iteratively define the component term f a u (x) a and generally f a /0 = f (a), f a { j} (x j) = f (x j,a { j} ) f (a), fu a (x) = f (x u,a u ) fv a (x) for /0 u {1,...,}, v u where (x u,a u ) denote an -dimenional vector whoe jth component i x j if j u and i a j if j u. For example, (x j,a { j} ) = (a 1,...,a j 1,x j,a j+1,...,a ). Since the lat term f{1,...,} a (x) i determined by the difference between f (x) and all lower order term, we have Since f (x) = u {1,...,} f (x u,a u ) = fv a (x), f a u (x). (8) imilarly a for ANOVA term, we have an explicit formula (invere linear relationhip) for the anchored term fu a (x) = ( 1) u v f (x v,a v ), u {1,...,}. Another explicit expreion for the anchored term for function with continuou mixed firt derivative i a follow. Let u = { j 1,..., j m }, where m = u > 0, then (ee Sobol 2003) h fu a j1 h jm m f (a + t) (x) = dt j1 dt jm (9) 0 0 x u where t = (0,...,0,t j1,0,...,0,t jm,0,...,0) and h i = x i a i. The decompoition of the kind (8) i conidered in Rabitz et al (1999) under the name cut-hdmr. Since it i related to the anchored function pace (ee Dick et al. 2004), we called it anchored decompoition. Note that for u > 0, f a u (x) vanih when any of it own variable x j with j u take the value a j, i.e., fu a (x) = 0, j u. x j =a j There exit a imple relationhip between the term of ANOVA and anchored decompoition fu anova (x) = f a [0,1] u (x)da. In contrat to ANOVA term, the term in the anchored decompoition (8) are extremely imple to compute (at leat for low-order term), ince they do not neceitate the computation of high-dimenional integral. It i natural to ue the truncated anchored decompoition a an approximation of f (x). It remain open whether uch an approximation i reaonably good and when it i good. 3 THE APPROXIMATION ERRORS For an ANOVA or anchored decompoition of a function f (x) contain 2 term of component. We are intereted in approximating f (x) by the um of the term up to order L f L (x) := f /0 + L t=1 u =t f u (x u ), L. (10) Thi i called the ANOVA or anchored order-l approximation, and i denoted by fl anova (x) or fl a (x) (with anchor a), repectively. If L = 1, the approximation i called the (ANOVA or anchored) additive approximation. A major problem i the approximation error introduced and the factor that affect the approximation error. The goodne of approximation i meaured by e( f, f L ) = 1 [0,1] [ f (x) f L(x)] 2 dx. 3.1 The ANOVA approximation error For the ANOVA order-zero approximation f0 anova = I( f ), we have e( f, f0 anova ) = 1. Thi erve a a benchmark of the 455

4 goodne of approximation. A good approximation hould have an approximation error that i much maller than 1. For the ANOVA additive approximation f1 anova (x), baed on the orthogonal property, we have e( f, f1 anova ) = 1 σu 2 ( f ) = 1 R 1 ( f ). (11) u >1 Similar reult hold for the ANOVA order-l approximation e( f, L ) = 1 R 1 ( f ) R L ( f ). The ANOVA order-l approximation i optimal. More preciely, let h(x) be any other function of x expreible a a uperpoition of lower-dimenional function, each of which depend on at mot L component of x, then e( f,h) e( f, L ). (12) In fact, ince f (x) fl anova (x) only contain ANOVA term with order larger than L, while fl anova (x) h(x) are a um of function with at mot L component, then from (3) we have anova f (x) f [ L (x)][ fl anova (x) h(x)]dx = 0. Thu [0,1] [0,1] [ f (x) h(x)]2 dx = implying the inequality (12). [0,1] + anova f (x) f [ L (x)] 2 dx [0,1] 3.2 The anchored approximation error anova f [ L (x) h(x)] 2 dx, Although the ANOVA decompoition ha the optimal property, it i difficulty to compute the ANOVA term. We thu turn to anchored order-l approximation f a L (x) = f (a) + L t=1 u =t f a u (x u ), L. The goodne of approximation depend on the anchor and on the order L. It i obviou from (12) that for any anchor a and any order L we have e( f, f a L ) e( f, L ). We are intereted in the quetion: How good could be the anchored order-l approximation? How doe it goodne depend on the anchor? How do we chooe an anchor to obtain a good approximation? A uniform error bound for anchored order-l approximation i derived in Sobol (2003). Aume that the mixed derivative of the form u f / x u are piecewie continuou for /0 u {1,...,}. Let A u := up x u f / x u and A := u >L A u, then from (9) we have Therefore, f (x) fl a (x) fu a (x) A u = A. u >L u >L e( f, f a L ) A 2 /. A ufficient condition for a good anchored order-l approximation with an arbitrary anchor i that A 2. However, for function which do not atify the condition above the choice of anchor can be ignificant. To tudy the influence of the anchor on the goodne of anchored approximation, let the anchor a be random and uniformly ditributed over [0,1]. We introduce the expected error of the anchored order-l approximation IE[e( f, f a L )]. Thi quantity meaure the goodne of approximation on the average when the anchor i choen randomly. If the expected error IE[e( f, fl a )] i mall, then we are hopefully to achieve mall approximation error with a random choice of the anchor. Indeed, baed on Markov inequality, for b > 0 we have Pr(e( f, f a L ) < bie[e( f, f a L )]) 1 1 b. (13) For example, if IE[e( f, fl a )] = 1/100, then ( Pr e( f, fl a ) < 1 ) On the other hand, if the expected error i large, then there exit ome choice of anchor uch that the error of the correponding approximation i at leat a large a the expected error. Moreover, the probability of large error of anchored approximation depend on the variance D := Var(e( f, fl a)). Baed on Chebyhev inequality, we have ( Pr e( f, fl a ) IE[e( f, fl a )] < ε ) D 1 1 ε 2. It follow that ( Pr e( f, fl a ) > IE[e( f, fl a )] ε ) D 1 1 ε 2. (14) 456

5 Thu if the expected error IE[e( f, fl a )] i large and the variance D i mall, then there i a large probability that the error of the anchored approximation i large. For example, if IE[e( f, fl a)] = 1 + c with c > 0 and 3 D < c, then ( Pr(e( f, fl a ) > 1) Pr e( f, fl a ) > IE[e( f, fl a )] ε ) D The expected error of anchored order-zero approximation We firt conider the implet cae, namely, the anchored order-zero approximation Since f0 a (x) = f (a). e( f, f0 a ) = 1 f (x) f (a) 2 dx [0,1] = [I( f ) f (a)]2, (15) the expected error of the anchored order-zero approximation i IE[e( f, f0 a )] = [I( f ) f (a)]2 da = 2, [0,1] which i exactly twice a the error of the ANOVA order-zero approximation. Thu there are ome choice of anchor uch that the correponding approximation error are at leat a large a 2. Thi indicate that a random choice of anchor may reult in poor anchored order-zero approximation. We point out that if f i continuou over [0,1], then there exit a point a uch that f (a ) = I( f ). Thu the anchor a achieve the minimal error of the anchored order-zero approximation, which i the error of the ANOVA order-zero approximation. On the other hand, from (15) the maximal error of the anchored order-zero approximation i achieved when the anchor i choen a to max f (a) I( f ). a [0,1] Whether there i a large probability of large approximation error depend on the variance of e( f, f0 a ), which i given by D =: Var([e( f, f a 0 )]) = b 4 σ 4 ( f ) 1, where b 4 = [0,1] [ f (x) I( f )]4 dx. Thi variance can be mall or large. Baed on (14), if D < 1/3, then Pr(e( f, f a 1 ) > 1) The expected error of the anchored additive approximation Now we conider the anchored additive approximation (i.e., the approximation (10) with L = 1). Let f1 a (x) denote the anchored additive approximation with the anchor a. From the definition of anchored decompoition, the firt order term are f a { j} (x j) = f (x j,a { j} ) f (a), and the anchored additive approximation i f1 a (x) = f (a) + f{ a j} (x j) = ( 1) f (a) + f (x j,a { j} ). (16) The next theorem how how the expected error of the anchored additive approximation depend on the global enitivity indice of variou order. Theorem 1 Aume that the anchor a i random and i uniformly ditributed over [0,1]. Then the expected error of the anchored additive approximation i where IE[e( f, f a 1 )] = b 2R 2 ( f ) + + b R ( f ), (17) b l = l 2 l + 2, l = 2,...,, and R l ( f ) i the um of global enitivity indice of order l defined in (7). Remark On the average the anchored additive approximation eliminate the variance contribution due to the firt order term. The formula (17) hould be compared with the error of ANOVA additive approximation (11). Proof We introduce everal notation. For any ubet u {1,...,}, let D u := σv 2 ( f ), (18) where σv 2 denote the variance of the ANOVA term f v (anova) (x). Moreover, let { j} and {i, j} denote complementary et of { j} and {i, j}, repectively. 457

6 From (10) and (16), the expected error of the anchored additive approximation i e( f, f1 a ) = 1 [0,1] [ f (x) + ( 1) f (a) f (x j,a { j} )] 2 dx. (19) Expanding the quare inide the integral on the right hand ide of (19) and then integrating both ide with repect to the anchor a over [0,1], we have (after ome calculation) IE[e( f, f a 1 )] = 1 [(2 + 2) 2 σ 2 { j} ( f ) 2( 1) D { j} + 2 D {i, j} ], i< j where we have ued [0,1] 2 f (x) f (a)dxda = [I( f )]2 and the following formula (ee Sobol 2001) D u = f (x) f (x u,a u )dxda u [I( f )] 2. [0,1] 2 u In the derivation of (20), we have ued thi formula for u = { j},u = { j} and u = {i, j}, repectively; moreover, the number of [I( f )] 2 i exactly a required. Since the total variance and the quantitie D { j} and D {i, j} are all the um of ome variance term (ee (5) and (18)), we may write (20) a IE[e( f, f1 a )] = 1 [b 1 σu 2 ( f ) u =1 +b 2 σu 2 ( f ) + + b σu 2 ( f )]. u =2 u = The fact that the variance term σu 2 ( f ) with the ame order u = l have the ame coefficient i baed on the ymmetry. It remain to determine the coefficient b l. To determine b 1, we jut count the number of σ{1} 2 contained in each um on the right hand ide of (20). We have b 1 = ( 2 + 2) 2 2( 1) 2 + ( 1)( 2) = 0. The coefficient of σ{1} 2 ( f ) and σ { 2 j} ( f ) i exactly zero. Similarly, by counting the number of σ{1,2} 2 ( f ) contained in each um on the right hand ide of (20), we have b 2 = ( 2 + 2) 2( 1)( 2) + ( 2)( 3) = 4. In general, we have for l = 2,..., b l = ( 2 + 2) 2( 1)( l) + ( l)( l 1) = l 2 l + 2. Thu we have IE[e( f, f a 1 )] = b 2R 2 ( f ) + + b R ( f ). The theorem i proven. Corollary 2 Aume that the anchor a i random and i uniformly ditributed over [0,1]. Then we have 4e( f, 1 ) IE[e( f, f a 1 )] (2 + 2)e( f, 1 ), where e( f, 1 ) i the error of the ANOVA additive approximation given in (11). If the variance of ANOVA term higher than L are zero (L < ), then we have 4e( f, 1 ) IE[e( f, f a 1 )] (L2 L + 2)e( f, 1 ). Theorem 1 and Corollary 2 how what determine the goodne of the anchored approximation, and anwer how good and how bad could be the anchored additive approximation and when the approximation i good or bad. On the average the anchored additive approximation eliminate the variance due to the firt order term. Thi property i imilar with that of the ANOVA additive approximation. Thu if R 1 ( f ) dominate and if R 2 ( f ),...,R ( f ) are all very mall (epecially when the higher order one are very mall), then the expected error of the anchored additive approximation can be mall. Baed on (13), it i likely to get good anchored additive approximation. Note that the ituation of very mall higher order indice i not rare in practice. For example in financial application the firt order term dominate the function and the higher order one are negligible (Wang and Sloan 2005). In ome cae, the dominancy of the firt order term may not be encountered, but we may increae thi dominancy by a proper ue of dimenion reduction technique (ee Sobol and Kucherenko 2005 and Wang 2006). However, comparing with the error of the ANOVA additive approximation, the expected error of anchored additive approximation i at leat 4 time large a the error of the ANOVA additive approximation. Moreover, the expected error of anchored additive approximation ha a tronger dependence on higher order global enitivity indice than on lower order one. Thu the expected error of anchored additive approximation can eaily become very large. Even when the error of the ANOVA additive approximation i mall, the expected error of anchored additive approximation can be large. For example, if = 100,R 1 ( f ) = 0.999,R 2 ( f ) = R 3 ( f ) = = R 1 ( f ) = 0,R ( f ) = 0.001, then the error of the ANOVA additive approximation i 0.001, which i mall. But the expected error of anchored additive approximation 458

7 i IE[e( f, f a 1 )] = b R ( f ) = 9.902, which i unacceptable. In thi cae, there are ome choice of anchor uch that the correponding anchored additive approximation are poor. The probability of large error of anchored additive approximation depend on the variance of e( f, f1 a ) (ee the argument at the end of Section 3.2). For an anchored order-l approximation with L > 1, there i no imple expreion for the expected approximation error in general. However, for the anchored highet order approximation and for function with eparate variable, it i till poible to derive a imple expreion for the expected error of the anchored approximation. 3.5 The highet order approximation for multiplicative function Conider function of the multiplicative form f (x) = g j (x j ), (20) where all g j (x j ) are quare integrable. Denote 1 1 µ j := g j (x)dx, λ j 2 := (g j (x) µ j ) 2 dx. 0 0 For a function f (x) of the form (20), both the ANOVA decompoition and the anchored decompoition can be found eaily. Indeed, for all ubet u {1,...,}, the ANOVA term are u (x) = [g j (x j ) µ j ] µ j, j u j u and the term in anchored decompoition are fu a (x) = [g j (x j ) g j (a j )] g j (a j ), j u where a = (a 1,...,a ) [0,1] i the anchor. Clearly, if the anchor a i choen uch that j u g j (a j ) = µ j, j = 1,...,, (21) then the anchored decompoition coincide with the ANOVA decompoition. Thi indicate that for function of the form (20), if each g j (x j ) i continuou, then it i alway poible to make the anchored decompoition being the ame a the ANOVA decompoition by properly chooing the anchor. The variance of the ANOVA term fu anova (x) i and the total variance i σu 2 ( f ) = λ j 2 µ 2 j, j u j u = (λ 2 j + µ 2 j ) µ 2 j. Now we conider the highet order approximation (i.e., the approximation (10) with L = 1). The error of the ANOVA highet order approximation i e( f, 1 ) = R ( f ) = 1 λ 2 j, where R l ( f ) i the um of the global enitivity indice of order l. The quantity e( f, f 1 anova ) i alo the minimal error of the anchored highet order approximation, when the anchor a i choen to atify (21). For an arbitrary anchor a [0,1], the error of the anchored highet order approximation i e( f, f 1 a ) = 1 f (x) f a [0,1] 2 1 (x) 2 dx, where f 1 a (x) i the anchored order-( 1) approximation. Since f (x) f a 1 (x) = f a {1,...,} (x) = it follow that e( f, f 1 a ) = 1 [0,1] = 1 [g j (x j ) g j (a j )], [g j (x j ) g j (a j )] 2 dx [λ 2 j + (g j (a j ) µ j ) 2 ]. (22) Let the anchor a be random and uniformly ditributed over [0,1]. Then from (22) the expected error of the anchored highet order approximation i IE [ e( f, f a 1 )] = 1 2λ 2 j = 2 R ( f ) = 2 e( f, 1 ). Once more, on the average the anchored highet order approximation eliminate the variance contribution due to ANOVA term up to order 1. Due to the factor 2, the expected error of the anchored highet order approximation can be much larger than the error of the ANOVA highet order approximation. The variance of e( f, f a 1 ) determine 459

8 whether we have large or mall probability of approximation error which i cloe to the expected value. From (22) the variance of e( f, f a 1 ) i D := Var [ e( f, f 1 a [ )] 1 = σ 4 ( f ) [3λ j 4 + b j ] 4λ j 4 where b j = 1 0 [g j(x) µ j ] 4 dx. Baed on the Chebyhev inequality, we have for any ε > 0 e( Pr( f, f a 1 ) IE[e( f, f 1 a )] < ε ) D 1 1 ε 2. To ee how bad the anchored highet order approximation could be, conider an extreme cae. From (22), if the anchor a = (a 1,...,a ) i choen uch that a j olve the optimization problem max (g j(a j ) µ j ) 2 := c 2 j, j = 1,...,, (23) a j [0,1] then the error of the anchored highet order approximation achieve the maximal value. The maximal error i max e( f, f a a [0,1] 1 ) = 1 ] [λ 2 j + c 2 j] = GR ( f ), where G := [1 + c2 j λ j 2 ]. From (23), it follow that c 2 j λ j 2, thu G 2. Summarizing the reult above we have the following. Theorem 3 Aume that the function f (x) ha the multiplicative form (20). Let the anchor a be random and uniformly ditributed over [0,1], then the expected error of the anchored highet order approximation i IE [ e( f, f a 1 )] = 2 R ( f ) = 2 e( f, 1 ). The minimal error of the anchored highet order approximation i min e( f, f a a [0,1] 1 ) = 1 λ j 2 = R ( f ) = e( f, f 1 anova ). which i achieved when the anchor i choen to atify (21). The maximal error of the anchored highet order approximation i max e( f, f a a [0,1] 1 ) = 1 [λ 2 j + c 2 j] = Ge( f, 1 )., which i achieved when the anchor i choen to olve (23), where G = [1 + c2 j λ 2 j ]. From Theorem 3, for any anchor a [0,1], we have e( f, f 1 anova ) e( f, f 1 a anova ) Ge( f, f 1 ). Both the equalitie hold with ome particular choice of the anchor. The error of anchored approximation e( f, f 1 a ) varie from R ( f ) to GR ( f ), depending on the anchor (with the expected error to be 2 R ( f )). Whether the expected error i mall (ay, le than 1) depend on whether the error of the ANOVA highet order approximation i very mall (ay le than 1/2 ). Clearly, if Ge( f, f 1 anova ) i mall, then any choice of anchor lead to mall error of anchored highet order approximation; if e( f, f 1 anova ) i large, then any choice of anchor lead to large error of anchored highet order approximation. In other cae, the expected error and the variance determine the probability of large or mall approximation error. 4 FINDING A GOOD ANCHOR For an arbitrary anchor a [0,1], the anchored decompoition (8) i alway an exact repreentation of f (x), but the quality of the order-l approximation depend on the anchor and the approximation error can be quite different. In ome cae the choice of the anchor become important. A bad choice of the anchor may lead to a large approximation error. We hope to chooe a uitable anchor uch that the error of the anchored order-l approximation i a mall a poible. Thi lead to a general principle: finding an anchor a uch that the error of the anchored order-l approximation i minimized, i.e., olving the optimization problem min e( f, f a a [0,1] L ). Thi general principle include the method of Sobol (2003) a a pecial cae. Method A. For L = 0, the anchored order-zero approximation i f0 a = f (a). In thi cae, we need to find the minimizer for min a [0,1] [0,1] [ f (x) f (a)]2 dx. (24) Since [0,1] [ f (x) f (a)]2 dx = = [0,1] [ f (x) I( f )]2 dx + [I( f ) f (a)] 2, 460

9 The firt term on the right-hand ide i independent of the anchor, thu the optimization problem (24) i equivalent to min f (a) I( f ). a [0,1] That i to ay, the anchor a hould be choen uch that f (a ) I( f ). Thi provide a jutification for the uggetion of Sobol (2003). The earching proce can be proceeded a follow: (i) Generate a low dicrepancy point et P n := {x j [0,1], j = 1,...,n} and calculate a rough etimate for I( f ), i.e., Î( f ) := 1 n n f (x j). (ii) Select a point a from P n a the anchor, uch that f (a ) Î( f ) = min 1 j n f (x j ) Î( f ). Method B. For an order L > 0 (ay L = 1 or 2), we find the anchor a uch that the error of the anchored order-l approximation i minimized, i.e., to find the minimizer min e( f, f a a [0,1] L ). Thi can be achieved imilarly a in Method A: (i) Generate a low dicrepancy point et P n := {x j [0,1], j = 1,...,n} and calculate the error of the anchored order-l approximation e( f, fl a ). The involved integral in e( f, fl a ) can be calculated numerically by quai-monte Carlo. (ii) Select a point a from the et P n, uch that the error of the anchored order-l approximation i minimized, i.e., e( f, f a L ) = min a P n e( f, f a L ). The comparion of the error of anchored approximation with good or bad choice of anchor for ecurity pricing problem are preented in Wang (2007). It turn out that with a good choice of anchor, the anchored additive approximation i quite atifactory. Such a good anchored additive approximation can be ued to contruct good importance denity or control variate to increae the efficiency of quai-monte Carlo method for ecurity pricing problem (ee Wang 2007). 5 CONCLUDING REMARKS The approximation of a function by a um of lower dimenional function i an important problem in the theory and application of high dimenional model repreentation. We tudied and compared the error of approximating a function by the um of the low order ANOVA or anchored term in the etting of ANOVA decompoition and anchored decompoition. We tudied the expected error of the anchored approximation when the anchor i choen randomly. We proved new theorem about the expected error of the anchored additive approximation and the anchored highet order approximation. In particular, we how that on the average the anchored additive approximation eliminate the variance contribution due to the firt order term. Thee theorem indicate the uefulne of an anchored approximation, but it hould be ued with care. We preented procedure to find a good anchor. ACKNOWLEDGMENTS The upport of the National Science Foundation of China i gratefully acknowledged. REFERENCES Caflich, R. E., W. Morokoff, A. Owen Valuation of mortgage backed ecuritie uing Brownian bridge to reduce effective dimenion. J. Comput. Finance, 1, Dick J., I. H. Sloan, X. Wang and H. Woźniakowki Liberating the weight. J. Complexity, 20, Rabitz, H., O. F. Ali, J. Shorter, K. Shim Efficient input-output model repreentation. Comput. Phy. Commun., 117, Sobol, I. M Global enitivity indice for nonlinear mathematical model and their Monte Carlo etimate. Math. Comput. Simulation, 55, Sobol, I. M Theorem and example on high dimenional model repreentation. Reliability Engineering and Sytem Safety. 79, Sobol, I. M., S. S. Kucherenko On global enitivity analyi of quai-monte Carlo algorithm. Monte Carlo Method Appl., 11, Wang, X On the effect of dimenion reduction technique on ome high-dimenional problem in finance. Oper. Re., 54, Wang, X High dimenional model repreentation in quai-monte carlo method for ecurity pricing. Working paper. Wang, X., K.-T. Fang The effective dimenion and quai-monte Carlo integration. J. Complexity, 19, Wang, X., I. H. Sloan Why are high-dimenional finance problem often of low effective dimenion?. SIAM J. Sci. Comput., 27, AUTHOR BIOGRAPHY XIAOQUN WANG i a Profeor of Department of Mathematical Science, Tinghua Univerity, Beijing, 461

10 China. Hi reearch interet include computational finance and financial engineering, Monte Carlo and quai-monte Carlo method, information-baed complexity. Hi addre i <xwang@math.tinghua.edu.cn>. Wang 462