The Calculation of Multidimensional Hermite Polynomials and Gram-Charlier Coefficients*

mathematics of computation, volume 24, number 111, july, 1970

By S. Berkowitz and F. J. Garner

Abstract. The paper documents derivations of: (a) a recurrence relation for calculating values of multidimensional Hermite polynomials, (b) a recurrence relation for calculating an approximation to the Gram-Charlier coefficients of the probability density distribution associated with a random process, based on (a), (c) an efficient algorithm to utilize the formulae in (a) and (b).

I. Introduction. The paper documents derivations of: (a) a recurrence relation for calculating values of multidimensional Hermite polynomials; (b) a recurrence relation for calculating an approximation to the Gram-Charlier coefficients of the probability density distribution associated with a random process, based on (a); (c) an efficient algorithm to utilize the formulae in (a) and (b). The relations for producing Hermite polynomials are generalizations of formulae derived in Appell and Kampé de Fériet [1]. The recurrence relation mentioned in (a) above appears without derivation in Vilenkin [5], which further references a dissertation by Sirazhdinov [4]. Since Hermite polynomials form complete, biorthogonal systems with respect to the Gaussian probability density, one basic use of the polynomials is to expand a near-Gaussian probability density distribution in terms of the polynomials in a so- called Gram-Charlier series. As we shall demonstrate below, the coefficients of the series can be calculated directly from the time series generated by a random process. Finally, we present and prove an algorithm which computes a Hermite polynomial or Gram-Charlier coefficient of vector order m by means of the above recurrence relations. The algorithm requires the smallest number of suborder polynomials and/or coefficients possible. The derivation of the recurrence relations is the responsibility of S. Berkowitz, and the algorithm represents the work of both authors. The algorithm has been im- plemented in FORTRAN subroutines by Mr. Garner.

II. Multidimensional Hermite Polynomials. A. Definitions. 1. m = (m,, • ■• , m„) is an order vector of degrees m¿. The polynomial or argument x and of index m is said to be of degree m, in the argument xt (/ = 1, •••,«) and of total degree m = XX i m<-

Received August 22, 1969. AMS Subject Classifications. Primary 3340; Secondary 6020, 6525. Key Words and Phrases. Multidimensional Hermite polynomials, Gram-Charlier series, recurrence relations, Gaussian probability density, normal distributions. * This paper was written in the Applied Mathematics Laboratory of the Naval Ship Research and Development Center for Project No. 1-830-918-02. Copyright © 1971, American Mathematical Society 537

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2. x = (x,, • • • , xn) is a vector of arguments x,. 3. R = Q~l is a positive definite matrix. One calls R a covariance matrix for the particular application of expanding joint probability density distributions in a Gram- Charlier series. R =- [r,,]; Q m \qit\. 4. The complete biorthogonal systems of Hermite polynomials {//m(x)| and |Gm(x)¡ are defined by the generating functions:

(J) exp [sYQx - WO*-) = ¿ cm(a)ffm(x),

OS (2) exp [aTx - ia**a] = E cm(a)Gm(x), »-0 where (O cm(a>= n:-i «CMl, (ii) the a, are arbitrary. The sets {77m(x)| and |Gm(x)S are biorthogonal with respect to (det Ö)1/2(2x) "'2 •exp [—§xrgx] over the entire real «-space. In the case of unidimensional polynomials (« = 1), both \Hmix)\ and {C7m(x)¡ reduce to a single orthogonal, complete set {Hjix)}, which is generated by the following relation (cf. Erdélyi [2]):

» m (3) exp [ax - \a2] = Y3,£ —ml Hm(x). The polynomials {Hm(x)\ are orthogonal with respect to exp [—§x2] over the entire real line. B. Recurrence Relations for Hmix) and Gm(x). Differentiating the generating functions (1) with respect to ak produces

¿i n cc — ¡exp [arÖx - a7£?a]¡ = X

Hmix) - ( X) generating function (2) produces:

Gm(x) = xtGm-tki±) — Jl rk¡mjGm-tt-t¡(x) — imk — l>tiGm_2et(x), (6) "' , . k = 1, • ■• , n. For any particular //m(x) or Gm(x), one has a choice of « recurrence relations. The value of any term in (5) or (6) with a negative index is zero and the jth relation must not be used if m, = 0 as one would falsely conclude Hm(x) = 0. If one chooses the pXh recurrence relation to compute Hmix) or Gm(x), then mv is called the pivot order, and p is called the pivot.

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III. Calculation of Gram-Charlier Series Coefficients. Consider the «-dimen- sional vector time series x

(7) x.

Denote the Gaussian probability density by the formula:

p0{x) = (det 0),/2(27r)-/2 exp [-±(x - u)T0(x - »)],

where (a) y = Pi , ■■ • , m»is usually taken as the vector of ensemble averages of the variâtes xx , • ■• , x„ , respectively. (b) Q is a positive definite n by n matrix, usually taken as the inverse covariance matrix of x. One can expand pix) in a Gram-Charlier series as follows:

co pix) = p,fx) XI AmH,Jy),

where (a) y = yu • • • , yn is a normalization of x such that y¡ = (x< — p,)

(8) / p„ix)Gmix)Hp(x) dx = bm H 8-iPi. J s ¿-1 where: (a) S indicates the entire real «-space, (b) bm= n:.i wr\ (c) 5mp is the Kronecker delta. By means of the biorthogonality property (8) and the Gaussian weighting function Paix), one can formulate the Gram-Charlier coefficients as follows:

Am = bm / Gm(y)p(x) dx, J s or, simply,

(9) Am = bmE[Gmiy)].

From (9) and the recurrence relation (6), one deduces :

(10) Am = bm^E[ykGm.eiiy)] - ¿ r„m<£[Gm_.è_.,(y)] + rlt£[Gm 2et(y)]J.

For a sample set of M vectors of the type indicated by (7), the first term on the right-hand side of (10) may be written as

1 M (ID £Lv*Gm_ei(y)]« — 2Z yi°Gm^t(Y°). M ,«,,

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The expectations pk and

i M (12) p»«t;£*"\

(13) ^«¿Ztó" "M2.

Thus, by substituting (11) and (9) into (10) one has a recurrence relation for A^ (which, in turn, employs the recurrence relation (6) for Gm(x)) :

(14) Am = jrz [ôm £ ^i)Gm_ei(y<") - ¿ rtimY Am.et.etJ , k - 1. ■ ri.

Similarly, for the expansion

eo /?(x) = p0(x) X BmGm(y),

one can derive the recurrence relation:

j I- JT * n "I

m« M L i-x i-x i-i J k = 1, • ■• , «.

As in the recurrence relations for Hermite polynomials, any term in (14), or (15) with a negative index is to be regarded as zero. In fact, since the indexing in (14) and (15) corresponds (with a bit of manipulation) to the indexing in (5) and (6), the problems of sequencing the recursive generation of Gram-Charlier coefficients are the same as those for sequencing Hermite polynomials, with the exception that only the vectors m — ek — eh f = I, ■■ ■ , n, need be generated for the Gram-Charlier coefficients. Therefore, in the remainder of the paper, we restrict ourselves to a discussion of Hermite polynomials and make only parenthetical restrictions where necessary for Gram-Charlier coefficients.

IV. An Algorithm for Generating Values of Hermite Polynomials (or Gram- Charlier Coefficients). In recursively generating the value of Hmix) or Gm(x) from (5) or (6) one could conceivably generate all i7,(x) or Gv(x) such that v, g mt(i= 1, • • ■ , n). However, there is a test that permits one to determine, purely on the basis of the order vector v, whether or not a polynomial value need be calculated from the recurrence relation. The test is embodied in an algorithm which we present below and prove in the next section. The algorithm requires a prior decision on the generating sequence as follows: The order vector m is permuted according to an arbitrary but fixed pivot sequence vector

(16) 6 = o-(l), ••• ,

where the set {o-(y')j is a permutation of 1, • • • , «. The classes Vv, defined by

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Vv = {v = v¡, • ■■ ,vn | 0 g vrii) = mcU), 1 ú i Ú P — 1;

0 < »,(,) Ú m,M;u,u) =0,p+lè/^«), p = 1, •• • , n,

are each generated according to an ordering Sv which is defined by

Sv = •Sv.v' E ^p | v = (o,(1), • • • , D,(P), 0, • • • , 0),

v' = (oi(I), - • * . »í(p>» 0, • • • ,0), and

v < v' iff X ("»<» — fíen)**-1 < 0, where b > max imt)f.

The class Vv stipulates that the a(j))th recurrence relation is to be used for calculating //"„(x) if v G Vv. The sequence Sv is merely an ordering of the real numbers {(^i(i)f

n p—1 X m«u) + im,{p) - v,M) ^ 2 (m,,,) - D,(i)), Í-H+I Í-1 compute //T(x) (or GT(x)) from the a(j>)th recurrence relation and store. (For the computation of a Gram-Charlier coefficient, we require also that X)"-i (w> — Di) be even.) (b) Otherwise, proceed directly to 6. 6. (a) If Vv is not exhausted, update s by one and return to 4 to increment Vv according to Sp. (b) If VPis exhausted, and p < n, update/> by one and return to 3 to initialize 5„. (c) If p = «, exit. The number of polynomial values required to compute the value of a given i/m(x) or Gm(x) depends heavily upon the manner in which one chooses the order sequence. The order sequence (16) which minimizes the number of required polynomial values is

(17) júk iff «i„(0 ^ mff(t). We will justify this assertion after presenting the proof for the algorithm.

V. Proof of the Algorithm. A. Definitions. The reverse sequence of indices in the following definitions facili- tates the description of index increments at the top of the partial ordering of order vectors required by the recurrence relation.

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1. Order Vectors. (a) Target Order Vector h = h», • • • , hu = m,U), ■■ ■ , mHn). (b) Test Order Vector w = wn, ■• • , wu = u,(1), • ■• , v.lni. (c) Test Difference d, = «,- — w,, j = 1, • • • , n. (d) Masking Vector e', s e„_i+1 = (5„,-, • • • , 5,,-), J = 1, • ■• , n. 2. Piuor Classes. (a) C„ = K„_p+1,p = 1, • • • , n. (Note that the Cv form a partition of the set of suborder vectors to be considered in computing h.) (b) T, = {w|wG Cpand ¿J-i - 1, • • • , «}. (c) /Î = Í w|w is needed in a recurrence relation to compute Hmix) or Gm(x) |. (d) The pivot associated with CP or Tv is cfn — p + 1); that is, if w E TVC\ R, then the o(n — p + l)th recurrence relation is used to compute /Tv(x) or GT(x). 3. Increment Classes. (a) According to the recurrence relation (5) or (6), there are three types of increments: Type I—The pivot order wp is incremented by one. Type II-—The pivot order wp is incremented by two. (This type is equivalent to two Type I increments, and will not be used below.) Type III—The pivot order wp is incremented by one, and some other order w, if > P) is also incremented by one. (b) A vector a = an, • ■• , a, is in LJ,t, w) if and only if a is the result of zero or more increments from w with pivot ain — p + 1) and increment Type(s) / = I, II, or III. B. The Proof. The following two notes labelled (Nl) and (N2) will be employed in the proof. When a pivot remains constant throughout incrementing from w to a, (Nl) the recurrence relations demand that av — wp ^ ^iHp (a,- — w¡), where the pivot is ain — p + 1). (N2) If any chain of increments can be constructed from w to h, then w E R. We seek to prove that jR = \Jk Tk. However, since the Ck form a partition of the order vectors under consideration, R = \Jk (i? C\ Ck), it suffices to prove that R(~\Ck= Tk, k = 1, ••• ,n, by induction on k. 1. For A:= 1. (a) Assume w G i? H Q. The pivot is o(ri), since w£C,. Since, by (NI), d, ^ X)"-2 rf» lt is true that w G 7\ and thus T, C -R<~> C,. (b) Assume w E 2\. Thus, í/i ^ X/*-a ^;- (i) If the right-hand side of the inequality is zero, then h E ¿a(L w) and, by (N2), w E R- Since, r, C C„ w E ^ H C,. (ii) If the right-hand side of the inequality is positive, then 3 w, = h — (¿i - Zï-» ¿í)eí E ¿,(111,w). But h E ¿i(I, w,) and, by (N2), w E *- As in (i) then, r, C R r\ C,. .-. T, = Rn C,. 2. Suppose 7\ = Ä H Ct, 1 ^ fc ^ w - 1 < «. (Hl)

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We show now that Tm = RP\ Cm. (a) Assume wG^ C„. There are three cases: (i) w = h; (ii) 3 a - (a., ■• • , ap+1, wp + 1, 0, • • • , 0) E ¿„(I or III, w), for p < m. By definition, a E C„. Moreover, if a is also in R, then, by (HI), a E TV If one defines d'¡ = «j: — a,, j = 1, • • • , n. then one has

£ 4 = ¿ «/, = £ ¿i + 1 í-1 í-1 í-1

n II ^ E¿!+ E (a,-w,) (by (HI) and (NI)) J-P+l î-P+1

è £ «/,.

Thus, w £ rm if any a of the above type is in R; (iii) 3 b = (6„, • • • , bm+l,hm, 0, • • • , 0) E ¿„(I and/or II and/or Jill, w). By definition, b E Cm. Consider any c = (c„, • • • , cp+1, 1,0, • • ■ , 0) E £„(I or III, b) for p < m.lfcE R, then, as in (ii) above, b E r„. Let d'¡ = h, — b,, j = 1, • • • , «. The following inequalities show that w E Tm (if c E J?):

m m —1 » E¿,^I<+ Z ibi - w,) (by(NI)) i m ) / -1 ; « m + I

n n è E 4 + £ (*i - »/) (since b E T„) m +1 î=«îm+1

= £ d,. j =m+l

But increments of b for pivot

m — l n (18) £¿; è £ d,. i -1 J = m + 1

Case 0«e. If the right-hand side of (18) is zero, then either w = h (in which case v?<=Rr\ Cx), or 3p = max {j \ h¡ > 0,1 ^ j < m\, and 3c = w + e'v E Lp(L w). But c E Tp, since £?_, K■= 1- Hence, by (Hl), cGÄAC,, and, by (N2), w E R- Since r„ C Cm, w E Ä H C„. Case Two. On the other hand, if the right-hand side of the inequality (18) is positive, w 5¿ h and 3 k, m + 1 = k ^ n, such that 0. Thus, for p chosen as in Case One—p exists, since otherwise w = h—there is a vector g = g„ • ■• , gt =

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w + e'k + e£ E ¿»(III, w). If one defines d'¡ = g¡ — w¡, 1 g j g «, then

Í-P+1 Thus, as in Case One, w E R C\ Cm. Therefore, for dm = 0, Tm C R r> Cm. (ii) Suppose that I.C^HCJoiO g ¿, < r. (H2) We show now that if w E Tm and if dm = t, so that

to —1 n (19) £ di: + r è £ d„ i-l i-m+1 then w E ^ H Cm. Case One. If the inequality in (19) is strict, then 3 y = yn, ••• , J>i = w + e¿ E L„(I, w). If d\ = ft, - y» I ^ f = n. then

£ Cm. Case Two. If, in (19), the inequality is in fact an equality, then 3 k, m + 1 ^ k g n, such that dk > 0, and 3z = z„, ■■ ■ , z, = w + e¿ + e't E Lm (III, w). If d'¡ m h, — Zj, 1 Ú j ú n, then

£ d\ = £ d, + a - i) - £ rf, - i - £ ¿jI ÍÜT+i Thus, z E Tm, and, exactly as in Case One of (ii) w E RF\ Cm. Therefore, LCÄH C„, and, indeed, Tm = RC\ Cm. In applying the algorithm to Gram-Charlier coefficients, it was necessary that £"-i imi ~~ vi) De even. Let u be a type / decrement of w if, and only if, w is a type t increment of u. Then, in the recurrence relation (14) or (15), decrements of types II or III were needed to designate required order vectors for the Am or Bm. Hence for any required order vector, £"_, d, must be even. Q.E.D. Let us now justify the assertion that the order sequence (16) which minimizes the number of order vectors in R is

j ^ k iff m,U) è m.w (cf. (17)). According to the algorithm, the vectors w E R C> Cp are those specified by

ft (20) Wp — £ w,- g kv, p = 1, •■ • , n, i-p+i where

V n K= £ «, - £ h„ h, à w, à 0, j = I, ■■■, n. i-l j'-p+l Consider a Euclidean (n — p + l)-space Ep with orthogonal axes wp, • • • , w„. If one replaces the inequality in (20) by an equality, one has a hyperplane Rp of (positive) unity slope with Wp-intersect at wP = kp. The points in R C\ Cp form a closed subset

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of Ep, bounded by the hyperplanes Rp, w, = 0, and w4 = «, (/ = p, • ■■ , n), where for any i, vv,-^ 0 if j j± i. The number of points in R C\ Cp is minimum for least Wp-intersectkv, and thus for the minimum £*_i A,-,a fact that justifies the assertion (17).

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1. P. Appell & J. Kampé de Fériet, Fonctions Hyper géométriques et Hypersphériques: Polynômes d'Hermite, Gauthier-Villars, Paris, 1926. 2. A. Erdélyi, et al., Higher Transcendental Functions, McGraw-Hill, New York, 1953. MR 15, 419. 3. J. Kampé de Fériet, The Gram-Charlier Approximation of the Normal Imw and the Statistical Description of a Homogeneous Turbulent Flow near Statistical Equilibrium, Ap- plied Mathematics Laboratory, NSRDC Report No. 2013, March, 1966. 4. S. Kh. Sirazhdinov, Contribution to the Theory of Hermite Multidimensional Poly- nomials, Candidate Dissertation, Tashkent State University, USSR, 1949. (Russian) 5. S. Ya. Vilenkin, "Determination of the maximum time (critical path time) distribution," Avtomat. i Telemeh., v. 1965, no. 7, pp. 1247-1252 = Automat. Remote Control, v. 1965,no. 7, pp. 1333-1337.

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