Dataplot Reference Manual, Volume 2, Chapter 5: Random Numbers

Dataplot Reference Manual
Volume 2: LET Subcommands and Library Functions

Chapter 5: Random Numbers

Introduction

The generation of random numbers is performed via subcommands under the LET command, as in

LET Y = NORMAL RUNDOM NUMBERS FOR I = 1 1 100

LET GAMMA = 2.5
LET Z = WEIBULL RUNDOM NUMBERS FOR I = 1 1 100

In the above, the FOR sequence has three numbers. The first defines the start row for the output variable, the second defines the row increment and the third defines the last row. The start row and row increment are usually set to 1. Also, in the FOR clause, you must use I as the dummy index (i.e., FOR I). If you use another name, the results may be unpredictable. If you enter

LET X = NORMA RANDOM NUMBERS FOR I = 3 2 21

you will get something like

That is, the output values start in row 3 and after that every other row is skipped. Any skipped rows will be set to 0 for a new variable and will keep their current value for a previously existing variable.

The output from the random number generation is always a variable (never a parameter or function). Random numbers can be generated from a variety of distributions. Some distributions represent a family of distributions. In this case, one or more parameters need to be specified (via the LET command) before generating the random numbers.

Setting the Seed

The SEED command is used to specify the seed for the random number generator. For example,

The purpose of the seed is to allow a specific set of random numbers to be replicated. That is, to obtain the same random numbers use the same seed value.

Available Uniform Random Number Generators

Random numbers for all distributions are ultimately based on generating uniform random numbers. Dataplot supports a number of uniform random number generators. The desired uniform random number generator is specified with the command

SET RANDOM NUMBER GENERATOR

where <generator> is one of the following

FIBONACCI - this is is a Fibonacci generator as defined by George Marsaglia in “Comments on the Perfect Random Number Generator” (unpublished notes from Washington State University). Dataplot uses an implementation of this algorithm from James Blue and David Kahaner. For this algorithm, the seed should have a minimum value of 305 and be an odd integer. Values less than 305 will use 305 and even integers will be equivalent to the nearest odd integer.

LINEAR CONGRUENTIAL - this implements a linear congruential generator. This generator is not recommended for routine use. It is included to allow for comparisons to other generators. It uses the RUNIF routine of Arthur Fullerton. This generator does not utilize a seed.

MULTIPLICATIVE CONGRUENTIAL - this implements the SUNIF multiplicative congruential generator given in ACM algorithm 599 (ACM-Trans. Math. Software, Vol.9, No. 2, June, 1983, pp. 255-257). This is considered a better generator than the linear congruential generator.

FIBONACCI CONGRUENTIAL - this is a combination of a Fibonacci and multiplicative congruential generator. The algorithm is from Kahaner and Marsagalia and given in Kahanar, Moler and Nash, "Numerical Methods and Software".

LUXURY - this implements the RANLUX generator of F. James. It is a modified version of the Marsaglia and Zaman rcarry method..

GENZ - this is a generator written by Alan Genz. It is based on Pierre L'Ecuyer (1996), "Combined Multiple Recursive Random Number Generator," Operations Research 44, pp. 816-822.

AS183 - this is the Applied Statistics Aalgorithm 183 generator.

GFSR - this is the generalized feedback shift register of Lewis and Payne as implemented by Monohan. It is documented in T. G. Lewis and W. H. Payne (1973), "Generalize Feedback Shift Register Pseudorandom Numbers", Journal of the ACM, Vol. 20, pp. 456-468.

FUSHIMI - this is the Fushimi-Tezuka generalized feedback shift register generator. It is documented in M. Fushimi and S. Tezuka (1983), "The J-Distribution of Generalized Feedback Shift Register Psuedorandom Numbers," Ccommunications of the ACM, Vol. 26, No. 7, pp. 516-523.

MERSENNE TWISTER - This generator is currently not available.

Available Distributions

Random number generation is available for the following distributions. Random numbers are generated for the standard form of the distribution (i.e., location parameter = 0, scale parameter = 1). For non-standard forms of the distribution, you can do something like the following

with LOC and SCALE denoting the location and scale parameters, respectively. Note that this applies to all distributions, not just the normal distribution (sometimes a parameter will be called a scale parameter when it is in fact a shape parameter).

Distribution Name on the LET command Name of Required Shape Parameter(s)

Normal NORMAL None

Uniform UNIFORM None

Logistic LOGISTIC None

Double Exponential DOUBLE EXPONENTIAL (or LAPLACE) None

Cauchy CAUCHY None

Semi-Circular SEMICIRCULAR None

Triangular TRIANGULAR None

Half-Normal HALF NORMAL (or HALFNORMAL) None

Exponential EXPONENTIAL None

Gumbel EXTREME VALUE TYPE I (or GUMBEL) None

Half-Cauchy HALF CAUCHY None

Cosine COSINE None

Anglit ANGLIT None

Arcsine ARCSIN None

Hyperbolic Secant HYPERBOLIC SECANT None

Half-Logistic HALF LOGISTIC None

Slash SLASH None

Rayleigh RAYLEIGH None

Maxwell MAXWELL None

Landau LANDAU None

Lognormal LOGNORMAL SIGMA

Discrete Uniform DISCRETE UNIFORM N

Leads in Coin Tossing (or Discrete Arcsine) LEADS IN COIN TOSSING N

t T NU

Chi-Square CHI SQUARE (or CHISQUARE) NU

Folded t FOLDED T NU

Tukey-Lambda TUKEY LAMBDA LAMBDA

Poisson POISSON LAMBDA

Skew Normal SKEW NORMAL LAMBDA

Log-Skew Normal LOG SKEW NORMAL LAMBDA and SD

Skewed Double Exponential SKEWED DOUBLE EXPONENTIAL LAMBDA

Borel-Tanner BOREL TANNER LAMBDA

Lagrange-Poisson LAGRANGE POISSON LAMBDA and THETA

F F NU1 and NU2

Beta BETA ALPHA and BETA

Kumaraswamy KUMARASWAMY ALPHA and BETA

Power Law POWER LAW ALPHA and BETA

Alpha ALPHA ALPHA and BETA

Power Exponential POWER EXPONENTIAL ALPHA and BETA

Inverted Beta INVERTED BETA ALPHA and BETA

Log-Beta LOG BETA ALPHA, BETA, C and D

Beta-Negative Binomial BETA NEGATIVE BINOMIAL ALPHA, BETA and K

Generalized Topp and Leone GENERALIZED TOPP AND LEONE ALPHA and BETA

Reflected Generalized Topp and Leone REFLECTED GENERALIZED TOPP AND LEONE ALPHA and BETA

Hermite HERMITE ALPHA and BETA

Beta-Geometric BETA GEOMETRIC ALPHA and BETA

Katz KATZ EXPONENTIAL ALPHA and BETA

Brittle Fracture BRITTLE FRACTURE ALPHA and BETA

Exponential Power EXPONENTIAL POWER BETA

Alpha ALPHA ALPHA

McLeish MCLEISH ALPHA

Generalized McLeish GENERALIZED MCLEISH ALPHA and A

Zeta ZETA ALPHA

Zipf ZIPF ALPHA and N

Gamma GAMMA GAMMA

Double Gamma DOUBLE GAMMA GAMMA

Inverted Gamma INVERTED GAMMA GAMMA

Log Gamma LOG GAMMA GAMMA

Weibull WEIBULL GAMMA

Double Weibull DOUBLE WEIBULL GAMMA

Extreme Value Type 2 EXTREME VALUE TYPE II (or EXTREME VALUE TYPE 2 or FRECHET) GAMMA

Generalized Extreme Value GENERALIZED EXTREME VALUE GAMMA

Pareto PARETO GAMMA

Inverse Gaussian INVERSE GAUSSIAN GAMMA

Reverse Inverse Gaussian REVERSE INVERSE GAUSSIAN GAMMA

Generalized Inverse Gaussian GENERALIZED INVERSE GAUSSIAN CHI, LAMBDA and THETA

Fatigue Life FATIGUE LIFE GAMMA

Wald WALD GAMMA

Generalized Pareto GENERALIZED PARETO GAMMA

Geometric Extreme Exponential GEOMETRIC EXTREME EXPONENTIAL GAMMA

Generalized Logistic GENERALIZED LOGISTIC GAMMA

Generalized Half Logistic GENERALIZED HALF LOGISTIC GAMMA

Pearson Type 3 PEARSON TYPE 3 GAMMA

Power POWER C

Reflected Power REFLECTED POWER C

Asymmetric Double Exponential ASYMMETRIC DOUBLE EXPONENTIAL (or ASYMMETRIC LAPLACE) K

Generalized Asymmetric Double Exponential GENERALIZED ASYMMETRIC DOUBLE EXPONENTIAL (or GENERALIZED ASYMMETRIC LAPLACE) K

Classical Matching MATCHING K

Folded Normal FOLDED NORMAL MU and SD

Normal Mixture NORMAL MIXTURE MU1, SD1, MU2 and SD2

Non-Central Chi-Square NON CENTRAL CHI SQUARE NU and LAMBDA

Non-Central t NON CENTRAL CT NU and LAMBDA

Skew t SKEW CT NU and LAMBDA

Log-Skew t LOG SKEW CT NU, LAMBDA and SD

Doubly Non-Central t DOUBLY NON CENTRAL T NU, LAMBDA1 and LAMBDA2

Non-Central F NON CENTRAL F NU1, NU2 and LAMBDA

Non-Central Beta NON CENTRAL BETA NU1, NU2 and LAMBDA

Doubly Non-Central F DOUBLY NON CENTRAL F NU1, NU2, LANBDA1 and LAMBDA2

Generalized Tukey-Lambda GENERALIZED TUKEY LAMBDA LAMBDA3 and LAMBDA4

Power POWER P

Geometric GEOMETRIC P

Yule YULE P

Binomial BINOMIAL P and N

Negative Binomial NEGATIVE BINOMIAL P and K

Hypergeometric HYPERGEOMETRIC L, K, N and M

Log-Logistic LOG LOGISTIC DELTA

Log Double Exponential LOG DOUBLE EXPONENTIAL ALPHA

Error ERROR ALPHA

Zipf ZIPF ALPHA

Logistic-Exponential LOGISTIC EXPONENTIAL BETA

Bradford BRADFORD BETA

Topp and Leone TOP AND LEONE BETA

Muth MUTH BETA

Reciprocal RECIPROCAL B

Gompertz GOMPERTZ C and B

Logarithmic Series LOGARITHMIC SERIES THETA

Generalized Logarithmic Series GENERALIZED LOGARITHMIC SERIES THETA and BETA

Geeta GEETA THETA and BETA

Generalized Negative Binomial GENERALIZED NEGATIVE BINOMIAL THETA, BETA and M

Truncated Generalized Negative Binomial TRUNCATED GENERALIZED NEGATIVE BINOMIAL THETA, BETA, M and N

Exponentiated Weibull Exponentiated Weibull GAMMA and THETA

Two-Sided Power TWO SIDED POWER THETA and N

G G G

G and H GH G and H

Gompertz-Makeham GOMPERTZ MAKEHAM XI, LAMBDA and THETA

Bi-Weibull BIWEIBULL SCALE1, GAMMA1, LOC2,SCALE2 and GAMMA2

Trapezoid TRAPEZOID A, B, C and D

Generalized Trapezoid GENERALIZED TRAPEZOID A, B, C, D, NU1, NU3 and ALPHA

Waring WARING A and C

Truncated Pareto TRUNCATED PARETO GAMMA and A

Mielke Beta-Kappa MIELKE BETA KAPPA K and THETA

Kappa KAPPA K and H

Burr Type 2 BURR TYPE 2 R

Burr Type 3 BURR TYPE 3 R and K

Burr Type 4 BURR TYPE 4 R and C

Burr Type 5 BURR TYPE 5 R and K

Burr Type 6 BURR TYPE 6 R and K

Burr Type 7 BURR TYPE 7 R

Burr Type 8 BURR TYPE 8 R

Burr Type 9 BURR TYPE 9 R and K

Burr Type 10 BURR TYPE 10 R

Burr Type 11 BURR TYPE 11 R

Burr Type 12 BURR TYPE 12 C and K

Doubly Pareto Uniform DOUBLY PARETO UNIFORM M, N, ALPHA and BETA

Asymmetric Log-Laplace ASYMMETRIC LOG LAPLACE ALPHA and BETA

Uneven Two-Sided Power UNEVEN TWO SIDED POWER A, B, D, NU1, NU3 and ALPHA

Slope SLOPE ALPHA

Two-Sided Slope TWO SIDED SLOPE ALPHA and THETA

Ogive OGIVE N

Two-Sided Ogive TWO SIDED OGIVE N and THETA

Wakeby WAKEBY BETA, GAMMA and DELTA

Quasi Binomial Type 1 QUASI BINOMIAL TYPE I P and PHI

Consul CONSUL THETA and M

Discrete Weibull DISCRETE WEIBULL Q and BETA

Lost Games LOST GAMES P and R

Generalized Lost Games GENERALIZED LOST GAMES P, J and A

Bessel I BESSEL I S1SQ, S2SQ and NU

Several distributions generate a matrix, as oppossed to a vector, of random numbers.

Multivariate normal distribution:

Note that M will be an NxP matrix. N is the number of rows generated for each component and their are P components to the multivariate normal. SIGMA is the pxp variance-covariance matrix of the multivariate normal. SIGMA will be checked to ensure that it is a positive definite matrix. MU is a vector specifying the means of the p components.

Multivariate t distribution:

The variables are the same as for the multivariate normal random numbers with the exception that there is an additional vector, NU, that specifies the degrees of freedom for the p components.

Uniform distribution:

The first syntax generates independent uniform random numbers with LOWL and UPPL denoting vectors that contain the lower and upper limits for the uniform distributions, respectively. The scalar NP denotes the number of rows to generate.

The second syntax generates correlated uniform random numbers. The matrix SIGMA is the variance-covariance matrix of a multivariate uniform distribution and N denotes the number of rows to generate.

Multinomial distribution:

The P variable defines the probabilities for each of the outcomes, N defines the number of trials, and NEVENTS defines the number of multinomial experiments to simulate. The returned M will be a matrix with NEVENTS rows and the number of columns equal to the number of rows in P.

Dirichlet distribution:

The ALPHA variable contains the shape parameters of the Dirichlet distribution and N denotes the number of rows to generate.

Wishart distribution:

Note that W will be a PxP matrix. N is a scalar that specifies the sample size. SIGMA is the pxp variance-covariance matrix of the multivariate normal. SIGMA will be checked to ensure that it is a positive definite matrix. MU is a vector specifying the means of the p components.

Program Example 1

multiplot 2 2
multiplot corner coordinates 5 5 95 95
multiplot scale factor 2
title case asis
let y = normal random numbers for i = 1 1 100
title Histogram of 1000 Normal Random Numbers
histogram y
title Normal Probability Plot of 1000 Normal Random Numbers
normal probability plot y
let ynew = 50 + 20*y
title Location: 50, Scale: 20
histogram ynew
let z = laplace random numbers for i = 1 1 100
title Normal Probability Plot of Laplace Random Numbers
normal probability plot z
end of multiplot

Program Example 2

multiplot 2 2
multiplot corner coordinates 5 5 95 95
multiplot scale factor 2
title case asis
let y1 = uniform random numbers for i = 1 1 100
let y2 = exponential random numbers for i = 1 1 100
let gamma = 5.2
let y3 = weibull random numbers for i = 1 1 100
let gamma = 2.7
let y4 = frechet random numbers for i = 1 1 100
.
title Uniform Random Numbers
histogram y1
title Exponential Random Numbers
histogram y2
title Weibull Random Numbers (Gamma = 5.2)
histogram y3
title Frechet Random Numbers (Gamma = 2.7)
histogram y4
end of multiplot