3.2 Gaussian (Normal) Random Numbers and Vectors

Transcription

1 3.2 Gaussian (Normal) Random Numbers and Vetors A. Purpose Generate pseudorandom numbers or vetors from the Gaussian (normal) distribution. B. Usage B.1 Generating Gaussian (normal) pseudorandom numbers B.1.a REAL B.1.b Program Prototype, Single Preision SRANG, X X = SRANG() Argument Definitions SRANG [out] The funtion returns a pseudorandom number from the Gaussian (normal) distribution with mean zero and unit standard deviation. B.2 Generating Gaussian (normal) pseudorandom vetors Given an N-vetor, µ, and a symmetri positive-definite N N matrix, A, the objetive is to ompute pseudorandom N-vetors, x, from the N-dimensional Gaussian (normal) distribution having mean vetor, µ, and ovariane matrix, A. On the first all to SRANGV with a new ovariane matrix, A, the user must set HAVEC =.false. to indiate that the Cholesky fator of A has not yet been omputed. SRANGV will replae A in storage by its Cholesky fator, C, and set HAVEC =.true. to indiate that the array A() now ontains C rather than A. On eah all to SRANGV an N-vetor, x, is returned as a pseudorandom sample from the speified distribution. B.2.a INTEGER REAL LOGICAL Program Prototype, Single Preision NDIM, N, IERR A(NDIM N, N), U( N), X( N) HAVEC Assign values to A(,), NDIM, N, U(), and HAVEC. CALL SRANGV(A, NDIM, N, U, X, HAVEC, IERR) Computed values will be returned in X() and IERR, and the ontents of A(,) and HAVEC may be hanged Calif. Inst. of Tehnology, 2015 Math à la Carte, In. B.2.b Argument Definitions A(,) [inout] When HAVEC is.false., A(,) ontains a user-supplied N N symmetri positive-definite ovariane matrix, A. Only the diagonal and subdiagonal elements of A need be given. This subroutine will not aess the super-diagonal loations in the array, A(,). This subroutine will replae A by its lower-triangular Cholesky fator, C, and set HAVEC =.true. When HAVEC is.true., A(,) is assumed to ontain the Cholesky fator, C. NDIM [in] First dimension of the array A(,). Require NDIM N. N [in] Order of the ovariane matrix A and dimension of the vetors U and X. Require N 1. U() [in] Contains the N-dimensional mean vetor, µ. X() [out] On return will ontain the N-dimensional generated random vetor. HAVEC [inout] See desription above for A(,). IERR [out] IERR will only be referened when the subroutine is entered with HAVEC =.false. If the Cholesky fatorization is suessful, the subroutine will set HAVEC =.true. and IERR = 0. Otherwise, it will leave HAVEC =.false. and set IERR to the index of the row of A in whih failure was noted. In this latter ase the results returned in A(,) and X() will not be useful. B.3 Modifiations for Double Preision For double-preision usage, hange the REAL statements to DOUBLE PRECISION and hange the initial S of the funtion and subroutine names to D. Note partiularly that if the funtion name, DRANG, is used it must be typed DOUBLE PRECISION either expliitly or via an IMPLICIT statement. C. Examples and Remarks The program DRSRANG demonstrates the use of SRANG to ompute Gaussian random numbers and uses SSTAT1 and SSTAT2 to ompute and print statistis and a histogram based on a sample of numbers delivered by SRANG. To ompute Gaussian random numbers with mean XMEAN and standard deviation STDDEV, one an use the statement X = XMEAN + STDDEV SRANG() July 11, 2015 Gaussian (Normal) Random Numbers and Vetors 3.2 1

2 The program DRSRANGV demonstrates the use of SRANGV to ompute pseudorandom vetors from a 3- dimensional Gaussian distribution with mean vetor and ovariane matrix speified as µ = and A = To feth or set the seed used in the underlying pseudorandom integer sequene use the subroutines desribed in Chapter 3.1. D. Funtional Desription Method The algorithm for generation of Gaussian random numbers is based on [1] [3]. This method draws pairs of uniform random numbers x from [0, 1] and y from [ 1, 1] until a pair is obtained that satisfies x 2 + y 2 1. The probability of satisfying this onstraint is π/ It then draws another uniform random number u from [0, 1] and omputes s = x 2 + y 2 t = 2 log u /s g 1 = t(x 2 y 2 ) g 2 = 2xyt The numbers g 1 and g 2 are independent samples from the Gaussian distribution with mean zero and unit standard deviation. The number g 1 is returned when it is omputed and g 2 is saved and returned the next time a Gaussian random number is requested. The saved value will be disarded if the underlying uniform sequene is reinitialized by a all to RAN1 or RANPUT of Chapter 3.1. This method is a mathematially exat transformation from the uniform distribution to the Gaussian distribution so the statistial quality of the delivered numbers depends entirely on the quality of the uniform pseudorandom numbers used. The uniform numbers are obtained by alling SRANUA or DRANUA, using the array in ommon blok /RANCMS/ or /RANCMD/ as a buffer as desribed in Chapter 3.1. For the generation of Gaussian random vetors we are given a mean vetor µ and a symmetri positive-definite ovariane matrix A. The Cholesky method is used to fator the given ovariane matrix A as A = CC t where C is a lower triangular matrix. Then for eah vetor to be delivered the method first onstruts a vetor g whose omponents are independent samples from the Gaussian distribution with mean zero and unit standard deviation. It then omputes x = µ + Cg. whih is a sample from the N-dimensional Gaussian distribution with mean µ and ovariane matrix A. Values returned as double-preision random numbers will have random bits throughout the word, however the quality of randomness should not be expeted to be as good in a low-order segment of the word as in a highorder part. Referenes 1. James R. Bell, Algorithm 334 Normal random deviates, Comm. ACM 11, 7 (July 1968) G. Box and M. Muller, Note on the generation of normal deviates, Ann. Math. Stat. 28 (1958) John von Neumann, Various tehniques used in onnetion with random digits in the National Bureau of Standards, Applied Mathematis Series 12, National Bureau of Standards (1959) 36 pages. E. Error Proedures and Restritions While omputing the Cholesky fatorization of A, subroutine SRANGV or DRANGV may determine that the matrix is not positive-definite. In that ase it will return with IERR set to the index of the row of the matrix at whih the problem was deteted. In this non-positivedefinite ase the ontents of the array A(,) will have been altered, HAVEC will still have the value.false., and X() will not ontain useful results. When the Cholesky fatorization is suessful, IERR will be set to zero. The onditions that N 1, and NDIM N are required but not heked. F. Supporting Information The soure language is ANSI Fortran 77. Entry Required Files DRANG DRANG, ERFIN, ERMSG, RANPK1, RANPK2 DRANGV DRANG, DRANGV, ERFIN, ERMSG, RANPK1, RANPK2 SRANG ERFIN, ERMSG, RANPK1, RANPK2, SRANG SRANGV ERFIN, ERMSG, RANPK1, RANPK2, SRANG, SRANGV Based on subprograms written for JPL by Carl Pitts, Heliodyne Corp., April, Adapted to Fortran 77 for the JPL MATH77 library by C. L. Lawson and S. Y. Chiu, JPL, April November 1991: Lawson reorganized and renamed ommon bloks. Improved oordination between Gaussian and uniform sequenes on reinitializations Gaussian (Normal) Random Numbers and Vetors July 11, 2015

3 DRSRANG program DRSRANG >> DRSRANG Krogh Minor hange f o r making. f90 v e r s i o n. >> DRSRANG Krogh Added e x t e r n a l statement. >> DRSRANG Krogh Changes t o use M77CON >> DRSRANG Lawson I n i t i a l Code. S r e p l a e s? : DR?RANG,?RANG,?STAT1,?STAT2 Demonstration d r i v e r f o r SRANG. Generates random numbers from t h e Gaussian d i s t r i b u t i o n. integer NCELLS parameter (NCELLS = 12+2) external SRANG real SRANG, STATS( 5 ), Y1, Y2, YTAB( 1 ), ZERO integer I, IHIST (NCELLS), N parameter (ZERO = 0. 0 E0) data N / 10000/ data Y1, Y2 / 3.0E0, +3.0E0/ STATS( 1 ) = ZERO do 20 I =1,N GET RANDOM NUMBER YTAB( 1 ) = SRANG( ) Aumulate s t a t i s t i s and histogram. a l l SSTAT1(YTAB( 1 ), 1, STATS, IHIST, NCELLS, Y1, Y2) 20 ontinue Print t h e s t a t i s t i s and histogram. write (, (13 x, a //) ) Gaussian random numbers from SRANG a l l SSTAT2(STATS, IHIST, NCELLS, Y1, Y2) stop end July 11, 2015 Gaussian (Normal) Random Numbers and Vetors 3.2 3

4 ODSRANG Gaussian random numbers from SRANG BREAK PT COUNT PLOT OF COUNT Count Minimum Maximum Mean Std. Deviation E Gaussian (Normal) Random Numbers and Vetors July 11, 2015

5 DRSRANGV program DRSRANGV >> DRSRANGV Krogh Minor hange f o r making. f90 v e r s i o n. >> DRSRANGV Krogh Changes t o use M77CON >> DRSRANGV Lawson I n i t i a l Code. S r e p l a e s? : DR?RANGV,?RANGV Demonstration d r i v e r f o r t h e Gaussian random v e t o r generator, SRANGV. C. L. Lawson & S. Y. Chiu, JPL, Apr integer I, J, N, IERR, NMAX, NSAMPL parameter (NMAX = 3) real A(NMAX,NMAX), U(NMAX), X(NMAX) l o g i a l HAVEC data (U( J ), J=1,NMAX)/ 1. 0 E0, 2. 0 E0, 3. 0 E0 / data (A( 1, J ), J=1,NMAX)/ E0, E0, E0 / data (A( 2, J ), J=1,NMAX)/ E0, E0, 0.03E0 / data (A( 3, J ), J=1,NMAX)/ E0, 0.03E0, E0 / N = NMAX NSAMPL = 20 HAVEC =. f a l s e. print (1 x, a/1x ), Pseudorandom v e t o r s omputed by SRANGV do 10 I = 1,NSAMPL a l l SRANGV(A, NMAX, N, U, X, HAVEC, IERR) i f (IERR. ne. 0) then print (1 x, i3, i 6 ),N, IERR else print (1 x, i3, 5 x, 3 f ), I,X endif 10 ontinue stop end July 11, 2015 Gaussian (Normal) Random Numbers and Vetors 3.2 5

6 ODSRANGV Pseudorandom v e t o r s omputed by SRANGV Gaussian (Normal) Random Numbers and Vetors July 11, 2015