1、Designation: D5124 96 (Reapproved 2013)Standard Practice forTesting and Use of a Random Number Generator in Lumberand Wood Products Simulation1This standard is issued under the fixed designation D5124; the number immediately following the designation indicates the year oforiginal adoption or, in the
2、 case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice gives a minimum testing procedure ofcomputer generation routines for the
3、standard uniform distri-bution. Random observations from the standard uniformdistribution, RU, range from zero to one with every valuebetween zero and one having an equal chance of occurrence.1.2 The tests described in this practice only support thebasic use of random number generators, not their us
4、e incomplex or extremely precise simulations.1.3 Simulation details for the normal, lognormal,2-parameter Weibull and 3-parameter Weibull probability dis-tributions are presented.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is therespon
5、sibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use. See specificwarning statement in 5.5.3.2. Referenced Documents2.1 ASTM Standards:2E456 Terminology Relating to Quality and Statistics3.
6、 Terminology3.1 Definitions:3.1.1 periodthe number of RUdeviates the computergenerates before the sequence is repeated.3.1.2 seed valuea number required to start the computergeneration of random numbers. Depending upon the computersystem, the seed value is internally provided or it must be userspeci
7、fied. Consult the documentation for the specific randomnumber generator used.3.1.3 serial correlationthe statistical correlation betweenordered observations. See 5.2.2.3.1.4 standard normal deviate, RNa computer generatedrandom observation from the normal probability distributionhaving a mean equal
8、to zero and standard deviation equal toone.3.1.5 standard uniform deviate, RUa random observationfrom the standard uniform distribution.3.1.6 standard uniform distributionthe probability distri-bution defined on the interval 0 to 1, with every value between0 and 1 having an equal chance of occurrenc
9、e.3.1.7 triala computer experiment, and in this standard thegeneration and statistical test of one set of random numbers.4. Significance and Use4.1 Computer simulation is known to be a very powerfulanalytical tool for both practitioners and researchers in the areaof wood products and their applicati
10、ons in structural engineer-ing. Complex structural systems can be analyzed by computerwith the computer generating the system components, giventhe probability distribution of each component. Frequently thecomponents are single boards for which a compatible set ofstrength and stiffness properties are
11、 needed. However, theentire structural simulation process is dependent upon theadequacy of the standard uniform number generator required togenerate random observations from prescribed probabilitydistribution functions.4.2 The technological capabilities and wide availability ofmicrocomputers has enc
12、ouraged their increased use for simu-lation studies. Tests of random number generators in com-monly available microcomputers have disclosed serious defi-ciencies (1).3Adequacy may be a function of intended end-use.This practice is concerned with generation of sets of randomnumbers, as may be require
13、d for simulations of large popula-tions of material properties for simulation of complex struc-tures. For more demanding applications, the use of packagedand pretested random number generators is encouraged.1This practice is under the jurisdiction of ASTM Committee D07 on Wood andis the direct respo
14、nsibility of Subcommittee D07.05 on Wood Assemblies.Current edition approved April 1, 2013. Published April 2013. Originallyapproved in 1991. Last previous edition approved in 2007 as D5124 96 (2007).DOI: 10.1520/D5124-96R13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcon
15、tact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3The boldface numbers in parentheses refer to the list of references at the end ofthis standard.Copyright ASTM International, 100 Barr H
16、arbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States15. Uniformity of Generated Numbers5.1 Test of the MeanThe mean of the standard uniformdistribution is12 . Generate 100 sets of 1000 random uniformnumbers and conduct the following statistical test on each set.Z 5X 2 0.500.0091
17、29(1)where:Z = test statistic,X= RU/1000,the standard deviation is assumed to be 112, andthe summation over 1000 values is implied.If the absolute value of Z exceeds 1.28 for more than 10 %and less than 30 % of the trials, the random number generatorpasses. If the random number generator fails the t
18、est using 100sets, then the number of sets can be increased or the randomnumber generator can be rejected.NOTE 1The assumption of standard deviation being equal to 112may be examined with a Chi-Square test wheres 5 (RU22 1000 X2!999(2)where:X= estimated means = estimated standard deviation of the 10
19、00 RUvalues, andthe summation over 1000 values is implied.A significant difference between s and 112, suggests anon-random generator.5.2 Test for Patterns in PairsThe purpose of this visualtest is to evaluate the tendency of pairs of deviates to formpatterns when plotted. Generate 2000 pairs of stan
20、dard uniformdeviates. Plot each pair of deviates on an x-y Cartesiancoordinate system. Inspect the resulting plot for signs ofpatterns, such as “strips.” Fig. 1 is one example of “stripes”generated by a BASIC function on a personal computer. Inmore than two dimensions, all generated random numbers f
21、allmainly on parallel hyperplanes, a fact discovered by Marsaglia(3).5.2.1 The following shuffling technique is an effectiveremedy for the general problem of “stripes” and randomnumbers falling on planes. Fill a 100-element array withstandard uniform deviates. Select a deviate from the arrayusing th
22、e integer portion of the product of a random deviateand 100. Replace the selected deviate with a new uniformdeviate. Repeat the process until the desired number ofdeviates has been generated. The plot of Fig. 2 resulted fromusing the shuffling technique on the random number generatorwhich produced F
23、ig. 1.5.2.2 Unless the RUgenerator is extensively tested bystringent tests (4, 5, 6) a shuffling procedure comparable to thatdescribed in 5.2.5 should be used.5.3 Visual Test for Uniform Distribution Conformance:5.3.1 The purpose of the visual test for distribution confor-mance is to detect some odd
24、 behavior of the random numbergenerator beyond what might be detected by the method in 5.4.It is impossible to predict the various shapes of the histogramswhich might indicate a problem with the generator. However,a few examples given here may alert the user of the generalform of a problem.5.3.2 His
25、togram PreparationFig. 3 is a histogram of 1000generated standard uniform numbers. The theoretical densityfunction is a horizontal dashed line crossing the ordinate at 1.0.The interval width is 0.1. The values of the ordinates for eachinterval were calculated as follows:fi5NiWI3T(3)where:fI= adjuste
26、d relative frequency,NI= number observed in interval i,WI= interval width, andT = total number generated.FIG. 1 Plotted Pairs of Random Numbers Showing “Stripes”NOTE 1The plot resulted from using the shuffling technique on thegenerator which produced Fig. 1.FIG. 2 Plotted Pairs of Random Numbers wit
27、h no DetectablePatternsD5124 96 (Reapproved 2013)2Since the interval width, WI, in this case equalled 0.1 and1000, values were generated as follows:fi5Ni0.1 31000(4)fi5Ni100NOTE 2If different sample sizes are used, bias may exist in makingvisual interpretations from histograms. One way to lessen thi
28、s bias is toapply the Sturgess Rule (7) to determine the number of cells for thehistograms.Nc5 113.3log10Ng! (5)where:Nc= number of histogram cells, andNg= number of generated numbers.5.3.3 Histogram EvaluationThe histogram of Fig. 3 has avery typical appearance for a sample as large as 1000. If one
29、would increase the sample size, less variation in fiis expected.On the contrary, by decreasing the sample size to perhaps 50,tremendous variation in fican be expected.Aproblem would beevidenced, if for a sample size of 1000, one of the followingoccurs: (1)iffiequalled zero or near zero for one class
30、 interval,(2) if one class interval had an fivalue 50 % greater than anyother interval, or (3) if there is any noticeable trend in the fivalue such as an increase in fifrom left to right, a decrease, orwhatever. The fivalues should vary about 1.0 in a randomfashion. The data must span the entire ran
31、ge from 0 to 1.5.4 Formal Test of Distribution Conformance:5.4.1 The Kolmogorov-Smirnov (KS) goodness-of-fit testgiven in Ref (5) should be used to test the conformance of therandom numbers to the standard uniform distribution. The KStest should be conducted on 100 sets of generated randomnumber dat
32、a each containing 1000 observations.5.4.2 Kolmogorov-Smirnov TestGenerate the RUnumbersand store in an array. Rank the data from smallest to largest.Calculate the following:Dn15 maxFiN2 XiG i 5 1, N! (6)Dn25 maxFXi2i 2 1NGi 5 1, NDn5 maxDn1, Dn2#where:N = sample size, (1000),XI=ithvalue of the ranke
33、d array, andDn= Kolmogorov-Smirnov (K-S) test statistic.For the test in 5.4, N equals 1000. X1is the smallest value ofthe ranked array, X2is the second smallest and so on. Dnascalculated is the largest vertical distance between the sampledensity function and the hypothesized distribution, in this ca
34、sethe standard uniform distribution. If Dnis greater than1.07/=N!for more than 10 % and less than 30 % of the trials, the ran-dom number generator passes. If the generator fails the testsusing 100 sets, then the number of sets can be increased orthe generator can be rejected.5.5 Correlations Among G
35、enerated Numbers:5.5.1 The computer generated values of RUmust appear tobe random and independent. The word “appear” is used sincethe numbers are actually being generated by a mathematicalalgorithm and all such algorithms have a cycle. Provided thenumbers have the appropriate distribution function (
36、as tested in5.3 and 5.4) and the numbers are not serially correlated, thenthe generated numbers are most useful for simulation purposes.Since the generated numbers are not truly random they areoften called “pseudo random.”5.5.2 PeriodSome personal computer brands have a uni-form number generator wit
37、h an extremely short period depend-ing upon the seed. Some machines repeat the same sequence ofnumbers after approximately 200 numbers. Depending uponthe simulation application, the user must determine if theperiod of the machine is adequate. Reference (1) is useful forevaluating the period of vario
38、us random number generators.5.5.3 Test for Lag-1 Serial CorrelationLag-1 serial corre-lation is a measure of association between the Xiobservationand the following Xi+1. Lag-2 serial correlation is a measure ofassociation between Xiand Xi+2or all pairs of observationsseparated by one observation. In
39、 theory, it is possible to haveany lag-k serial correlation. For random number generators, itis necessary for all lag-k to be zero for k less than the period.For k equal the period, lag-k serial correlation equals 1.0. Thefollowing statistical test from Ref. (2) is for lag-1 serialcorrelation and it
40、 is recommended as a minimum test forstatistical independence.NOTE 3Warning: Random number generators that pass the tests inthis standard can display very bad behavior in more than two dimensions.There are existing random number generators that can pass the tests in thisstandard but whose values fal
41、l on a small number of hyperplanes.5.5.3.1 Let Xibe an array of generated RUvalues. X1beingthe first generated, X2the second and so on. Generate 1000values of Xi. Calculate:FIG. 3 Histogram of Random Numbers with Theoretical DensityFunction SuperimposedD5124 96 (Reapproved 2013)3r1! 5(XiXi112 (Xi!2/
42、1000(Xi22 (Xi!2/1000(7)where:r(1) = lag one serial correlation, and denotes an impliedsummation from 1 to 1000.Xi+1must be replaced by X1when i equals 1000; take X1001= X1. If the calculated r(1) falls outside of the following limitsfor 0.042 r(1) 0.040 more than 10 % and less than 30 %of the trials
43、, the random number generator passes. If therandom number generator fails the test using 100 sets, then thenumber of sets can be increased or the random numbergenerator can be rejected. (Assuming there is no lag-1 serialcorrelation, 20 % of the calculated r(1) values would beexpected to fall outside
44、 of the specified range as the numbertrials increased indefinitely.)NOTE 4Serial correlations greater than lag-1 may affect modelingprocedures. It is the responsibility of the investigator to assess, in anappropriate manner, the significance of these correlations.5.6 Selection of RUGeneratorProvided
45、 the RUgeneratorpasses the tests and provisions in 5.1-5.4, it can be considereduseful for purposes of computer simulation.The tests in 5.1-5.4are considered as minimum for qualification; an individual usermay want to increase the number of trials.5.7 Rejection of RUGeneratorIn the tests of 5.1-5.4,
46、 thereis a chance of falsely rejecting a good generator. For thisreason, one may choose to repeat all tests (using different seedvalues) if a given generator failed on the first series of tests.5.8 Generation of RUIn BASIC programs the generatormay produce different results depending on whether thep
47、rogram is compiled or interpreted. On some systems consid-erable differences have been observed, between the modes,because of differences in how the generator is seeded. In anycase, the results from both methods of program executionshould be checked when using BASIC. For a comprehensivediscussion on
48、 the various methods of generating RU, Chapter 6of Ref (5) is recommended.6. Simulation from Selected Distributions6.1 Random values from a prescribed distribution will benoted by y which is often referred to as a random deviate. Thissection assumes that the parameters of the various probabilitydist
49、ributions have been estimated by the various methodsavailable and are now known quantities.6.2 Simulation from the Normal DistributionIn general,when simulating lumber and wood product properties from thenormal distribution, truncation is required since the normaldistribution is defined from minus infinity to plus infinity. Withsimulation it is possible to generate extremely small ornegative values. Therefore it is the responsibility of the user todiscard all values below a user specified minimum. Thedefinition of the minimum is a diffic
