Subroutine WSHRT for generating matrix of variances and covariances of Wishart distributio

Hocking, R. R.

Smith, W. B.

English

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https://ntrs.nasa.gov/search.jsp?R=19710026194 2020-03-11T22:26:22+00:00Z
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Instituoe of bia is _c5
Tema s A&M UL002 s i iy
Tec inicu,_ "eport # b
Navional Aeronautics and ;pace Aumi"isvrwtion
Reseaxch Grant WaR 44-001-395
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by
W. B.	 .;itn and it. ii.
Institute of Sta^. ^^;;ics
Texas AScN. Univcrs:ty
Larguac;e
AOA Standard FORTRAN
Description u d Purpose
By L':i1.10 the Bartlett (1933) decomposition zubrou'dine W0:u11' L;C!ACi•:.Lk
-::he rra.rix of sample variances and covariaiices froir, a ranuorra 	 oi'
dy :,;,ributed as a p-variate normal, Np (µi 2 	The point uis;.r -A.buLion of
^,ar^ple variances a.ad covariances is called the Wioliurt aistribu-.-o^::.
By generating p(p-1)/2 independent vuriates a,, ; (1 < < i < p) al p „^	 ,^__;1 U
Identically as NI (0, 1) and p-variates aii (1 < i <p) independently s^, z^u„,^u
as XN_i+l, form the p x p matrix A = (ai j ) , where ai j = O l L < j) . The ..::..,.^
,,ransformation B = ( b i j ) = AA', w .LJacob L an
P
	
J 	 n aP-i+1
i=1
yields as the joint distribution of the unique b..l^ (B is sy:crietric)5
IBI(N-p-i)/2 e
xp (-2 tr(B))
p
N-i+1r i( 2 )
i=1
Equation (2) corresponds to the joint density function of a sample vCLiiwa.Vl:—
covariance matrix constructed from a random sample of size N from N, (0, i,.
2NP/2 4(p-l)/4
By c:ecoi7,po:; in,, E lnto the product of a lower	 D
tracisp^se (see r addeva (- 959) for a set o^• recursion . ori:ul^^s;
elci.:C::. 3 of tiie matrix (:iii) = D B D ^ wi" I nave a WiL; i:ai't Cli:;tr..t")liL.1 J:1
a	 t^a.r^,rcietors N and	 It should be noted that if w is ^u-iuV, n t:.cn
C' -^,i.erwise N = n - 1.
W:ijs:r a.n (1957) arrives
	
above decomposition by uL;L,,; ortilOL;C, _L.L
...^.^rices cieper:ding on certair. rardor.1 vectora, whF^reas, itshirsa a'r '15591/
g ives	 the sru;.e re.,ult by tr^Lnsforrrations rather ti;ar. by denJlz,les a:s
^acobians. To generate the standard univariate noriral deviate:; requirca,
t^'Iie Box and MLiller (1958) formulas are used, while X 2 va.ridtes are vi:i.l l'a^ l:u
by means of the Wilson and Hilferty (1931) approxirc:ation. Iie y iable uce of
t::e Wilson-Hilferty X2 deviates requires the degrees of freedom to exceeu
but the authors have had satisfactory Wishart variates from ,.-ch smaller vL,iuO.;.
.each formula, of course, depends upon the availability of a reliable uni-f;;r:..
( i ,e. rectangular (0, 1)) ra.ndam number generator. The authors have --lu
unsatisfactory results with the generator made available by the cc :puler
company and have resorted to the use of a subroutine (called RANDC) writen
by a colleague, Dr. C. E. Gates. The WSHRT output subroutine is highly
aependent upon the input; variate and any substitution of a local vnifori:.
pseudo random number generator should be preceded by careful checking of
t're reliability of that generator.
n Otaue:,,cr.;,
-fir n •i — ^^ •
	_-n /^^
	
- ^
	
^^	 ^A)
Zeal Array input:: correspo::d.; to :	 il:
N, T^
rteger Scalar in put 1. _ r.^ ,	 ii	 µ n6wa	 .if ur..L.^....
1'P Integer Scalar irp corrosdor,ds to in
Of t, a	 .. ormzil v;,xi;a.e:.
Sti Real Arruy output: corresponds to U. ",	 if.	 :;..eory.
:.ote: The input D is used instead of Z since each cycle woula require
recomputaLion of D.
Additional Comment
r:e above technique for generating the sample covaria.ce natrix
._;triUution) -requires only p(p_r-i)/2 twndard univariaLe normal deviaLt.
4 ,ereas, straightforward generation of such a random sample: would re u 	 .:^-
standard  nomal variates. For small sample sizes (r. < 10) and moderw,.,;
--zed p a straightforward calculation of the sample variance-covari .0.ce :.. " I I
ij -recommended. However, for the larger sample sizes the above algol
could be recommended as the Wilson-H ;.lferty approximation to k varia;; :.
would be valid.
Execution time of subroutine WSHRT is approximately .033 seconds c:_
- V" 360 67 computer using a WA FIV computer (developed by the
rdaterloo and modified for local use by the Data Processing Center, 	 =
Asti. University) .
iie fei elice s
\.i123/
	
on	 tl,cory Gi JVtA.ViJLiVul re, ^;1'1::;;;^^.'...	 - ---
.^!:ti,	 L.	 Elul .^.uller, 1%.	 iloLe o!. v:.oVe;-i'T' li.iv:^ Ul
I^^.Ir:i.it.► l.l^}rii.l,lLJ, t:. .•	 •.u.,. .• .J..iZl.^ w	 ., ^, U.,.l^-li...^.
V.	 (1959' • Co::a	 : :^:ethou v.	 ::C1.i..0  
(1959) . 3artle ^t decor. pos * t - or and Wizi 6rL U^ 6rlbu , ...
., ,.,,a ,,^ st . , 30 ) 239
R. A. (19;7). Rwilor.1 ort110 .,1,0nal Lrwasforira ion :i. -
?c^,
«o^:^, i''. :^. ac, i. ' ,ferry. M. M. (1j31) . he d -'A.  ;;Lrib^Aviu:"A
''roc. .rct^. Acaci, ac:., L'.:^.t1., 17, 684- "88.
►vSIMT (D, 'N JP, SA)
I:/.i'LICIT
II!'uiTWIS 	 VARIA F^ GENZRATOR
D11VIEWSIG n(IVT ) NTP) I C(i? -,V) ):' i,1i),SA(Ni? ) NP) )A(Ni',.r".')
CIiSW IS A FUANCTION STATEti:LNT OF TIE, WILSON-iilLFERTY (1931)
CI'l-SQUARE Ai'PiIOXIYJUION
CiiSQ'r'(E,F)=E^(1.-2.% 19•-^E)+i'^DSQRl(2.^19•^E)' )^'^ 3
:tir OP%Y X41D R.ROI^P
 ARZ FUNCTION C'1 ►' ,N11:i^'1'S Ol' TIM '30X-b:uI,L1;R (197G i
6TAN; )ARD NOFiVIAL TRANSFOItN.UiOlVS
iixO,RF(EjF)=(DSUiT(-2.*DLOG(E) ))-DCOS(2.*P -L*F)
R,X a (F,P)=(DSQRT(-2. DLOG( ;))) DSii1i(2.' xl'I I')
PI=3.11+152927
^,OAD A(I,J),I<J,WITH INDEPENDENT N(O,1) VARI,42ES
DO 1 I=l,NP
DO 1 J=1,NP
IF(NP*(NP-1)/2) 1+,5,2
DO 4 I=2,NP
_ T
1l^ l-
DO 4 J=1,II
CALL RANDC (ZY,Ui )
CALL RANDC (i{Y,U2 )
RAl\MC GENERATES A UNIFORM VARIATE (DUE TO DR. C . E .GATES )
u
	
...	 .^^\ l:V TV
	
^ (J	 r
ul 62
Y
	
LOAD	 W1Tli TIE SQ^Uki' ROOT 0:' l-XDEl'L Du^,'2
^,iiI -S^^,^.'u^% (IV— I+1 D. r'. } VAR -ATLS
i^ELUllC (iti, ul )
Yi=RNO:U-' (,ji, U2 }
l2 -1:10A11	 U2^
A(I,I)=DSC^RT(CliSW(Dl',Yl} )
I .^' ( I—: T^ . } ; , 8,14
Did =,O — Ii i
;i(I, i) =DSO (CHSQj-' (DF, Y2 } }
G 
Ir (i — i ? } 6) 8)14
8 C O T-,1- -UE
DO 10 V,--I) p
.DO 10 K=1,:/
3(N)K)=0.
'O 10 J=1,K
y0 :s(M,K)=B(M,K)+A(M)J)*A(K,J)
,j0 11 M=2,NP
o:l=M-1
DO 11 K=1,M1
^y 3(K,M)=3(M,K)
Do- 12 L=I)KP
DO 12 u=1, NP
C(L,M)=0.
DO 12 K=1,L
12 C(L,:4)=C(L)M)+D(L,K)*B(K,N.)
DO 13 I:-13 NP
DG 13 J=l)NP
DO 13 K=1,J
SA(I,J)=SA(I)J)+C(I,K)*D(J)K)/.LUi
SA IS ARRAY 'WHOSE ELEIVIENTS HAVE A 'KISHART 'N,SIGMA) DISTRIBUTION
END


Wishart variate generators

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