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AD-AlO8 523 DAVID W TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CE--TC F/6 12/1MODIFICATIONS To COMPUTER PROGRAM FOR PARAMETER ESTIMATION FOR --ETC(U)MAY 77 N K OCHI
UNCLASSIFIED DTNSRDC/S.772-01 L
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FORM DOCUMENT PROCESSING SHEETDTIC OCT 79 70A
DAVID W. TAYLOR NAVAL SHIP00 RESEARCH AND DEVELOPMENT CENTER9 I'C) Bethesda, Md. 20084
Ot
MCC MODIFICATIONS TO COMPUTERp PROGRAM FOR PARAMETER ESTIMATION
FOR THE GENERALIZED GAMMA DISTRIBUTION
by
MihlK. Ochl
LI.-U, C
12089
MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS
i DTNSRDC
COMMANDER
TECHNICAL DIRECTOR01
[ 7 ERINCHARG' OFFICE R-IN-CHARGECARDEROCK 05ANNAPOLIS 0
SYSTEMSDEVELOPMENTDEPARTMENT
SHIP PERFORMANCE AVIA ;ION AND
DEPARTMENT SURFACE EFFECTS
15 DEPARTMENT 16
STRUCTURES COMPUTATION.MATHEMATICS AND
DEPARTMENT ILOGISTICS DEPARTMENT
17_ 18
[SHIP ACOUSTICS 1 PROPULSION ANDDEPARTMENT AUXILIARY SYSTEMSD 19 DEPARTMENT 27
MATERIALS CENTRAL
DEPARTMENT INSTRUMENTATION28 DEPARTMENT 29
_-
UNCLASSIFIFDqECUMITY CLASSIFICATION OF TNIS PAGE (When Date Entered)
REPORT DOCUMENTATION PAGE READ INSTRUCTION9REPORT DOCUMENTATION PAGE3BEFORE COMPLETING FORM
I. REPORT NUMBER 12. GOVT ACUION NO: S. RECIPIENT'S CATALOG NUMBER
SPD-772-01OA/04. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
MODIFICATIONS TO COMPUTER PROGRAM FOR PARAMETERESTIMATION FOR THE GENERALIZED GAMMA DISTRIBUTION . ROMING ORG. REPORT NUNSER
7. AUTHOR(&) S. CONTRACT OR GRANT NUMEER(e)Michel K. Ochl
9. PERFORMING ORGANIZATION NAME AND ADRESS I0. pRGR SLEMN, PROJECT, TASK
Ship Performance Department Program Element 63534NPrgam Eleen 63534NT.ux
David W. Taylor Naval Ship R&D Center Work Unit No. 1-1506-012Bethesda, Maryland 20084 ,
If. CONTROLLING OFFICE NAME AND ADDRESS I2 REPORT DATE
Naval Sea Systems Command may 1977Surface Effect Ship Project Office, PMS 304 IS NUMMEROFPAGESBethesda, Maryland 20084 96
141- MONITORING AGENCY NAME & AODRESS(f different fom ContrOllinot Office) IS. SECURITY CLASS. (of tale report)
UnclassifiedISo. DEC ASSI FIC ATION/ DOWNGRADING
SN EDU LE
IS. DISTRIBUTION STATEMENT (of thle Report)
APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED
I?, DISTRIBUTION STATEMENT (of the ebotre¢t entered in Block 20, If diflerent froe Report)
I. SUPPLEMENTARY NOTES
It. KEY WORDS (Continue on reerse side If necoeeoy end Ifdentfy by block ntwbr)
Extreme values, Generalized gamma distribution, Computation,Maximum likelihood
20. ABSTRACT (Continue on reverse Wide If nocoeirw and identify by block namiber)An extensive study of an existing digital computer program for determining
parameters required to represent SES IOOB trials data by a generalized gammafunction Is carried out. The purpose of the study Is to resolve the difficul-ties encountered in earlier studies. In these earlier studies, approximately5 percent of the cases analyzed produced unrealistic values for the distri-bution parameters. In order to resolve this problem, data sensitivity studiesare carried out. Based on results of the study, criteria are developed fordetermining cases that should not be orocessed. Three otcr chnliaues besides IDD 1oAN, 1473 gooToo oi INovs SOLET isNCLSSIF
S/N 0102.LF.014660t UNCLASSIFIEDSECURITY CLAIFICATION OF THIS PAGE ( hen DOMe E1te.40
.U.NCLASS I FIEDSeCI'qTY CLASSiFIC ATION OF THIS PAIGE '%%on DO$& EnttOedl
the Stacy-Mlhram method are Investigated. The most successful alternativemethod Is the maximum likelihood method, which Is Incorporated Into the newdigital computer program. A program listing and an example of the generatedoutput are Included in this report.
UNCLASSIFIED O ISI[CUORITY CL.AIOP~ICATIOld OF-T;iS P&G(llrlttl Do.l Rnftm
TABLE OF CONTENTS
Page
ABSTRACT. .. ... ...... ...... ...........
ADMINISTRATIVE INFORMATION .. .. ...... ......... 1
INTRODUCTION. .. .... ..... ...... ........
MATHEMATICAL DEVELOPMENT. .. .... ..... ....... 4
OVERVIEW .. .... ...... ..... ....... 4
MAXIMUM LIKELIHOOD METHOD .. .. ...... ...... 5
ANALYSIS. .. .... ..... ...... .......... 8
SENSITIVITY OF RESULTS ON EXPERIMENTAL DATA .... 10
EVALUATION OF MAXIMUM LIKELIHOOD METHOD .. .. ..... 13
DIGITAL PROGRAM DESCRIPTION .. .... ...... .... 14
INPUT DATA .. .... ...... ..... ...... 16
PRINTED OUTPUT. .. ...... ...... ...... 22
PROGRAM DESCRIPTIONS .. .... ...... ...... 24
Main Program. .. ...... ..... ...... 24
Subroutine GETCML. .. .... ..... ...... 27
Subroutine MLHCML. .. .... ..... ...... 30
Subroutine DIST .. .. ..... ...... ... 32
Subroutine WEG. .. ...... ...... ... 33
Function PSIl .. .. ...... ...... ... 34
FunctionPS12 .. .. ..... ...... .... 35
Function PSI. .. ...... ...... .... 36
Subroutine MGAMMA. .. ... ...... ...... 37
Subroutine MDGAM .. .... ...... ...... 38
Page
Subroutine HISTO .. .. ..... .... ..... 38
Function G. .. .... .... .... ...... 39
Function F. .. .... .... .... ...... 40
CONCLUSIONS .. ........................... 41
ACKNOWLEDGEMENTS .. .. ..... ...... ........ 42
REFERENCES .. .. ..... ................ 43
APPENDIX A -PROGRAM LISTING. .. ... ...... .... A-1
APPENDIX B - EXAMPLE OF PRINTOUT .. .. ..... ...... B-1
LIST OF ILLUSTRATIONS
Page
FIGURE
I A COMPARISON OF THE GENERALIZED GAMMADISTRIBUTION WITH EXPERIMENTAL DATA .... ......... 9
2 A COMPARISON OF THE GENERALIZED GAMMADISTRIBUTION AS DETERMINED BY THE MAXIMUMLIKELIHOOD METHOD WITH EXPERIMENTAL DATA . . .... 15
3 CARD IMAGES FOR A TYPICAL COMPUTER RUN . ....... ... 21
TABLE
1 RESULTS OF SENSITIVITY STUDY ON FOURREPRESENTATIVE CASES ........ ............... 12
2 MAJOR FORTRAN SYMBOLS USED IN MAIN PROGRAM ... ...... 25
3 MAJOR FORTRAN SYMBOLS ..... ................ ... 28
Iii
ABSTRACT
An extensive study of an existing digital computer program
for determining parameters required to represent SES 100B trials data
by a generalized gamma function is carried out. The purpose of the
study is to resolve the difficulties encountered in earlier studies.
In these earlier studies, approximately 5 percent of the cases analyzed
produced unrealistic values for the distribution parameters. In order
to resolve this problem, datasensitivity studies are carried out.
Based on results of the study, criteria are developed for determining
cases that should not be processed. Three other techniques besides
the Stacy-Mihram method are investigated. The most successful alterna-
tive method is the maximum likelihood method, which is incorporated into
the new digital computer program. A program listing and an example
of the generated output are included in this report.
ADMINISTRATIVE INFORMATION
This work was conducted at the David W. Taylor Naval Ship
Research and Development Center and authorized by Naval Sea Systems
Command, Surface Effect Ships Project Office, Program Element Number
63534N. It is identified as Work Unit Number 1-1506-012.
INTRODUCTION
A computerized procedure for evaluating the parameters of the
generalized gamma distribution from a set of random data following the
method proposed by Stacy and Mihram (see Reference 1) has been previously
developed and extensively exercised (see Reference 2 and 3). During
these earlier studies it was discovered that what appeared to be unreal-
istic values for the distribution parameters were obtained in roughly 5
percent of the cases analyzed. The initial purpose of the study, whose
results are summarized in this report, was to explore in detail a repre-
sentative sample of cases resulting in unrealistic values of the param-
eters, and if possible, to develop an alternate method for processing
these cases.
The results of the present investigation can be summarized as
follows:
a. In the many cases studied, it appears that the predicted
extreme design values are most sensitive to experimental
measurements and not the numerical procedure for deter-
mining the parameters. This is particularly true when
one of the parameters is negative.
b. All four methods considered in this investigation
yield probability distribution functions which represent
very well the experimentally obtained histogram even
though some of the parameter values were judged unrealisitic
in previous studies. However, the Stacy-Mihram and maximum
likelihood methods are by far the most efficient and are
the preferred procedures based on probabilistic arguments.
c. In some cases the predicted extreme design values are
unrealistically large whereas the significant values are
2
realistic. Based on an analysis of over 100 cases, it
can be concluded that cases having a nondimensional third
logarithmic moment greater than -0.5 produce unrealistic
extreme design values. Results of a data sensitivity study
have revealed that in all these cases the data may not be
considered as a sample taken from a steady-state random
phenomenon, and hence it is recommended not to carry out
statistical analysis for these cases.
3
MATHEMATICAL FORMULATION
OVERVIEW
The generalized gamma density function under consideration in
this report has the form
where X, m, and c are estimation parameters and r(m) is the gamma function.
Many other distributions such as the exponential, gamma, chi-squared,
Weibul, hydrograph, chi, truncated normal, and Rayleigh are special cases
of the generalized gamma function. Due to the general nature of f(x; X,
m, c), it is the preferred distribution particularly when the exact form
of the distribution function is unknown.
There are several techniques that can be used to estimate the
three parameters. Four such techniques are considered in this effort.
These are the Stacy-Mihram, the maximum likelihood, the nonlinear
least squares, and grid search methods. The last two methods deter-
mine the parameters by searching for that set of parameters resulting
in a minimum error between the experimental and theoretical histograms.
Since statistical inference cannot be applied in a rigorous manner, the
last two methods are not recommended. The Stacy-Mihram method
determines the parameters by requiring that the first three theoretical
logarithmic moments agree with the logarithmic moments determined from
the experimental data. The maximum likelihood method determines param-
eters such that the joint probability density function is a maximum.
4
A detailed derivation of the Stacy-Mihram method is presented
in Reference 2 pages 6 - 8. A derivation of the nonlinear least squares
procedure is presented in Reference 4. The grid search procedure merely
involves a systematic search of the X, m, c space for parameters that
minimize the error between the predicted distribution and the measured
distribution. A derivation of the equations required in the maximum
likelihood parametric estimation method is presented in the next section.
MAXIMUM LIKELIHOOD METHOD
Briefly, the maximum likelihood estimation technique involves
finding the parameter (or parameters) which maximizes the joint proba-
bility density function of a random sample X, . . . , XN from a
distribution with f(x, e) as its probability density function and e as
an element of the parameter space. The joint probability density func-
tion, referred to as the likelihood function, is given by
L. T"C-..G) (2)
Assume one has N. samples at random variable xi. Then the
maximum likelihood function will have the form
7 T(3)
-...
5
The natural logarithm of the likelihood function is used in
maximizing since it is a maximum when the function itself is a maximum,
and it proves more convenient to use. The logarithm is given by,
L = i N 1 (4)
The maximum of lnL is obtained by searching for parameters e such that,
0.0 (5)
where N = the number of parameters.p
For the generalized gamma function given by Equation (1),
the likelihood function becomes
KI
L= _ 7 <"° p <,) ")) (6)
and InL becomes
SNL i. + -A v q x~C~(7)
In the above equation, c is assumed positive. If the above equation is
differentiated with respect to cm, X and m and set to zero, the following
equations can be derived (see Reference 5).
V) (8)
6
iU
I.IIn- C. Inc k, n . N. ,,o, , N + cm(,,)
(9)
- -- In f.\ -- k- n N 9
== 0
CNC
where the summation on i is from I to K. An expression for m can be
obtained from the above equations:
*- I" .X;
The parameter c can be obtained by solving the following equation
numerically.
, r ('.'"N k \r " Y. N (12)
N
7
Once a value of c is determined by solving Equation (12),
values for X and m can be obtained via Equations (8) and (11). The
numerical method used to solve Equation (12) for c is due to Wegstein
(see Reference 5).
ANALYSIS
During previous studies in which the Stacy-Mihram method was
used to process SES lOOB data, large design extreme values were
predicted in approximately 5 percent of the cases although the most
significant values agreed with the experimental measurements. This
difficulty seemed to be associated with large values of the parameter
m and with negative values of the parameter c. In order to resolve
this difficulty, several cases were examined in detail. The results
of this investigation follows.
The first case considered in this investigation is a set
of pitching motion data measured during the SES 100B full scale trials
program at 30 kts in a seastate of 3. This particular case, identified
as ARI 3E 30, is presented in Reference 3. A comparison of the
estimated probability density function with the experimental histogram is
presented in Figure 1. As is demonstrated by the results in the figure, the
agreement is excellent. All other cases examined show similar results.
Assuming that an objective of the estimation procedure is to develop
the best representation of the data that is possible, it is con-
cluded that large values of m(> 5) are not necessarily unreasonable.
8
Q04'I
ARI E 30
UI- ,i GENERALIZED GAMMA DISTRIBUTION
C4
RAYLEIGH DISTRIBUTION
a0
M L4.0 16.. 199 Z1.6 4.40
EXCURSIONS
FIGURE 1. A COMPARISON OF THE GENERALIZED GAMMADISTRIBUTION WITH EXPERIMENTAL DATA
9
Also, since the predicted representations always represent the data
very well for both large and small m, the numerical procedure used
to solve the transcendental equations for the estimation parameters
is judged sufficiently accurate.
Further investigations revealed that in some of the cases
the predicted design extreme values were unreasonably high as
compared with the most significant values. In order to determine
more precisely a reasonable value for the ratio of the design extreme
value to the most significant value (XD/Xs), an analysis of over 100
previously processed data cases were examined. The results indicated
that unrealistically large ratios were obtained for those cases for
which the normalized logarithmic moment (see Equation 2.4 in Reference
2) is greater than -0.5. Most significant values were realisitic as
compared with those measured in trials. Consequently, in the current
data base, cases for which T > -0.5 should not be processed to deter-
mine design extreme values using the generalized gamma procedure.
The experimental histogram of ARl 3E 30 has a substantial
tail which may contribute to the large design extreme value predic-
tion. The last four bins (out of 12) each have only one observation.
A data sensitivity study was conducted in order to determine how
sensitive the design values are to the tail portion of the measured
data. The results of this study are presented in the next section.
SENSITIVITY OF RESULTS ON EXPERIMENTAL DATA
A data sensitivity study on cases ARI 3E 30, S14 3H 10,
S14 3Q 10, and S82 3Q 10 was carried out in order to make an assess-
10
-- ... ...... . ... .... ... .i 'll l . .. .... . .. ... .. .. .i III II~ l ..... ... ... . .. ....... ... .. .
ment of how sensitive predicted results are to the experimental data.
Computations were carried out for several cases for which four obser-
vations from a rather distinct tail were dropped. The results are
presented in Table 1.
The results indicate that elimination of a small percentage
of the experimental data can result in large changes in the design
extreme value, particularly when T > -0.5 and c < 0.0. Based on
these results, it appears that a substantially large number of obser-
vations in the tail portion of the experimental data is desirable
in order to predict results that are insensitive to minor errors in
the observations.
It should be noted that the data sensitivity study as conducted
above is rather severe, i.e., four points were removed in a systematic
manner from the tail portion of the measured histogram. The proba-
bility that four points from the tail would be Incorrectly measured
is rather small. One'way of determining how long an experiment should
be conducted and consequently how many observations should be made
is to monitor various statistics such as moments during any particular
experiment. Once these statistical measures reach steady state, the
sample can be Judged adequate. Proceeding in this manner may have
been impractical during a single prototype run. However, combining
several runs under nearly identical experimental conditions may prove
possible.
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12
EVALUATION OF THE MAXIMUM LIKELIHOOD METHOD
The method previously employed to determine estimation param-
eters for the generalized gamma distribution was based on the Stacy-
Mlhram method (see Reference 1 and 2). One of the advantages of the
Stacy-Mlhram method is that negative values of the parameter c
are allowed. However, based on the current analysis of over 100 cases,
it appears that cagzs for which c is negative result in unrealistically
large values of the ratio of the design extreme value to the significant
value regardle-.s of thie processing technique (Stacy-Mihram method,
grid search, nonlinear least squares method). Consequently, development
of representations of the experimental via the generalized gamma
function should be restricted to cases for which the parameter c > 0.
More detailed analysis (see comments at the beginning of this section)
has revealed that processing should be restricted to those cases for
which the normalized logarithmic moment is less than -O.5*. If the
above restrictions are imposed, the preferred processing method is
the maximum likelihood method as developed in a previous section of
this report. One of the reasons the maximum likelihood method is
preferred is that it provides unbiased and efficient parameter esti-
mates in the asymptotic limit, i.e., as the number of samples becomes
*Examination of the data shows that it cannot be adequately representedby any of the standard distributions.
13
large, provided specified regularity conditions hold (see References
6 and 7 for details). Also, the method is applied to the actual experi-
mental measurements, i.e., grouping the data into histograms is not
necessary.
A comparison of the maximum likelihood results with experi-
mental data is presented in Figure 2. As demonstrated by the figure,
the agreement between the generalized gamm~a representation and the
experimental data is excellent. The ratio of design extreme value to
most significant value is 3.65 which is less than the value obtained
by the Stacy-Mihram method is 5.58.
DIGITAL PROGRAM DESCRIPTION
The current digital program is based on a previous program
that implemented the Stacy and Mihram method. The following modifi-
cations have been made:
1. The maximum likelihood method has been incorporated
into the previous program. This is accomplished by
replacing subroutine GETCML with a new version. The
maximum likelihood method should be used only when the
numbers of observations is greater than 100, or the
number of bins is greater than 60. The Stacy-Mihram
method should be used when the number of bins is less
than 20 and the number of observations is greater than 100.
14
in
wi
KKC
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--
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E-4 E-' I
1-I-
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0 *in i L J
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15
2. As an added option, data can be read in the form of
densities or counts per bin.
3. Printer plots of Rayleigh, generalized gamma function,
and experimental data are generated.
4. Since the extreme values are a function of the number of
observations (time), time effect plots are generated along
with tabular results.
Three proprietary programs supplied by DTNSRDC are used in
the present system. These are PLOTPR, MGAMMA, and MDGAM. PLOTPR is
used to produce printer plots of data written on unit 3, MGAMMA computes
the gamma function, and MDGAM computes the incomplete gamma function.
Printer plot programs and programs for the gamma function are readily
available on most computer systems. Hence, replacing these with programs
available at different installations should present no major difficulties.
The incomplete gamma function can be obtained from the gamma function via
a simple integration procedure. However, due care must be exercised to
assure sufficient numerical accuracy.
A complete description of input data, printed output, and
subroutines follows. A listing of the program is presented in Appendix
A. An example of print out is presented in Appendix B.
INPUT DATA
A description of input data follows. Card images of a typical
run stream are illustrated in Figure 3.
16
Card Variable Cols. Format Description
1 IC(l) 1-2 12 option to convert inputdata from frequencies todensities.HNEW = HOLD/(N x DEL)
ON: IC(l) = 1
2 IC(2) 3-4 12 option to convert L valuesfrom time to sample/run timeL = (L /Run time) x NNEW OLD
ON: IC(2) = 1
3 IC(3) 5-6 12 option to change randomvariable values to midpointvaluesON: IC(3) = 1
I IC(4) 7-8 12 option to convert inputdata by factor CALON: IC(4) = 1
1 IC(5) 9-10 12 option to plot Histogram,Rayleigh and Gamma with"PLOTPR"ON: IC(5) = 1
1 IC(6) 11-12 12 option to plot above inMKS units (change byfactor CONVER)ON: IC(6) a I
1 IC(7) 13-14 12 option to plot L in hoursvs. XD, XG (Time effect
plots)ON: IC(7) = 1
1 IC(8) 15-16 12 option to plot above inMKS units
ON: IC(8) = 1
1 IC(9) 17-18 12 option to print gammafunctions for all L valuesOFF: IC(9) a 1ON: otherwise
17
Card Variable Cols. Format Description
1 IC(lO) 19-20 12 option to print above inMKS unitsON: IC(lO) = 1
1 IC(ll) 21-22 12 scale on Time Effect PlotsEach dot will be spaced by10 ** (IC(ll)).-9 < IC(ll) < 99Default: .05, 1-5 hrs.
1 IC(12) 23-24 12 IC(12)=O calculates designvalues and option for plots.IC(12)=l no design valuescalculated and no plotsprintedIC(12)=2 no design valuescalculated, but option forplots.
1 IC(15) 29-30 12 debug printout for Stacy-Mihrammethod
1 IC(16) 31-32 12 number of double pages onplotDefault: 1 double pageminimum is 1 double page(2 pages)
1 IC(20) 39-40 12 IC(20)=l: Use maximum likeli-hood method in estimatingparametersOtherwise: Use Stacy-Mihrammethod
1 ISTOPD 41-50 110 Number of increments inexcess of 200 to be usedin determining variousstatistical measures.Recommended value is zeroor one.
I IPLOT 51-55 15 IF IPLOT 0 0, Calcompplots will be generated
1 IRAY 56-60 15 IF IRAY # 0, Rayleighdistribution will begenerated
18
Card Variable Cols. Format Description
I CONVER 61-70 F10.0 Conversion factor to gofrom English to MKS units
1 CAL 71-80 F10.0 Conversion factor of inputdata to measurement desired
2 IOP(1) 1-2 12 IOP(1)=O: c = 2.0 asinitial guess for use inWegstein iterationIOP(1)=l: Initial c valueread in.
2 IOP(2) 3-4 12 IOP(2)=l: Write initialc value
2 IOP(3) 5-6 12 IOP(3)=1: Write c, m, Xestimates obtained bymaximum likelihood estimation
2 IOP(5) 9-10 12 IOP(5)1l: Debug printout
3 N 1-5 15 Total number of samples
3 K 6-10 15 Number of d#vsions
3 ALP 11-20 F10.0 Value of I - F(x) corres-ponding to the desiredextreme value
3 DEL 21-30 F10.0 Division size
3 CMNT(1-5) 31-80 5A10 Description of the setof data
4 NL 1-10 110 Number of L values to beconsidered. Up to fivecan be used (See Equation2.15 in Reference 2.)
4 ELL(1-5) 11-60 5F10.0 up to five values of L
5 X(l), H(1) 1-80 8F10.0 K pairs of ordinate andexperimental density valuesto be analyzed 4 pairs percard.
19
Multiple cases are processed by stacking the data sets with
the last card as a blank to serve as a flag indicating the last
case. However, the option cards (first two cards) are only read
once. A run consisting of two sets of data is illustrated in Figure 3.
20
XXXXXXX.C' T7n Oo T , 4.C14 A R GE 9 TTTT* NNNNNNNNNNCC K.AT I TACH.B I 1 Aq BBBBBBBBBbD. I tl TTTT-ATT &CHI9 R DC.ATTACH914SL.
Tir, T (LS SLNPC1 NAP.
.311 195 0.01 11.0 ;2 H 3
.0075 in. .0147 30., .0217 A;. nJ7454. .0101 FG. o01V,! 7A. in
"o " fl. .001A
1 2. .0005 114. OW 126. .000! t!P. .0150. .0 162. .1 174. .000'261 14 0.01 3.0 "? .r In
t.5 .01527 pick IP°1 I - •~ f)Frlq R 9 I n W %O4qfi2
13.5 .0559A 16.5 oA343i 13.0 .04071 2?.!' 001"A25.5 *f1272 >.5 .0076! 31. .i0'' 3., .0012737.5 .00127 40.5 0fl025 4
nLANK CARDI
FIGURE 3. CARD IMAGES FOR A TYPICAL COMPUTER RUN
21
PRINTED OUTPUT
The printed output is dependent upon the options that are
chosen. A general description of the printout follows. A sample run
is shown in Appendix B.
The first page consists of the user options chosen. The
options are only printed once for all the data sets. The initial guess
for c, the value of c at each iteration used in finding c and the
estimated parameters are next, if maximum likelihood method is chosen.
If Stacy-Mihram method is chosen the iterations for finding m are given.
The second page identifies the data set being processed. The
number of observations, number of bins, a, and bin size are next
printed followed by the bin widths, the histogram data. Next the log-
arithmic moments of the data follows; y, s, and T. The last items in
the header are the calculated generalized gamma parameters m, c, X as
computed by the maximum likelihood method or Stacy-Mihram method.
Eight columns follow: the values of the random variable, the
value of the random variable in MKS units if CONVER is given and Option
6 of the IC options is chosen, the corresponding Rayleigh predicted
frequencies, the corresponding measured densities, the corresponding
Rayleigh predicted densities and the corresponding generalized gamma
predicted densities.
The values of L, the density function, the cumulative distri-
butions, and corresponding random variables are printed for each value
22
of L. These occupy eight columns of output arranged as follows:
Column 1 - running index
Column 2 - value of the random variable.
Column 3 - f(x), (Equation 1 in Reference 1)
Column 4 - f(x), (Equation 11 in Reference 1)
Column 5 - f(x)L
Column 6 - G(x), (Equation 12 in Reference 1)
Column 7 - h(x), (Equation 14 in Reference 1)
Column 8 - H(x), (Equation 15 in Reference 1)
The following page gives the Rayleigh design values for all L values.
The fourth page gives the generalized gamma design values
for one L value. These include xM, xO , xS, XGP XGA, x0, and XDA. To
the right of these values, xO, xG, XGA, xDo and XDA as computed by
a table look-up procedure are printed. This page is repeated for
each L value.
After these pages there is a comparison of design values for
all L values. The next pages consist of a printer plot which show
the measured density, the density as predicted by the generalized
gamma function, and the density predicted by the Rayleigh density
function.
The following pages consist of a time effect printer plot
which give L in hours vs. x. and xG. The CalComp plots generated
replicate the printer plots.
23
L. - _
PROGRAM DESCRIPTIONS
Main Program
The main program serves as a driver for the entire digital
program. In addition, many of the required computations are performed
in the main program. A description of program flow follows.
All data required for the program is read from unit 5. A
description of this data is presented earlier in connection with the
input data.
After reading the data, the moments of the logarithms of the
random variables are computed along with the Rayleigh parameter and
maximum histogram value The normalized third logarithmic moment, T,
is tested against the value -0.5. If T >-0.5, the case is not processed.
The parameters m, c, and X are computed by a call to subprogram GETCML
via the maximum likelihood or Stacy-Mihram methods.
Once values for m, c, and A are determined, the various statis-
tical measures are computed for up to five values of L. The density
functions and cumulative distributions are printed for each L value
considered, followed by the extreme, most probable, and significant
values. The extreme, most probable, and significant values are obtained
by the Wegstein iterative method (see Subroutine WEG).
After the above computations are performed, CalComp and printer
plots are generated. These consist of comparisons of the measured den-
sity function with the generalized Gammna density function and with the
Rayleigh density function.
Descriptions of major FORTRAN symbols are presented in Table 2.
24
TABLE 2
MAJOR FORTRAN SYMBOLS USED IN MAIN PROGRAM
FORTRAN MathSymbol Symbol Description
N N total number of samplesK K number of divisionsALP a value of l-F(x) corresponding to
the desired extreme valueDEL AX bin sizeNL number of L valuesELL L array of L valuesISTOPD number of increments in excess of
200 to be used in determiningextreme values
X x array of ordinate valuesH f. experimental density values
S S value related to second logarithmicmoment
YBAR 7 first logarithmic momentT T value related to third logarithmic
momentR R Rayleigh parameterC c parameter cM m parameter m, realAMBDA X parameter lambdaXO x ordinate valueFOXO f(x;X,m,c) predicted density functionBFOXO F(x) cumulative distribution function
BFOXL F(x)L cumulative distribution raised tothe Lth power
GOFX g(x) density function corresponding toF(x)L
HLO h(x) asymptotic density functionHHO H(x) asymptotic cumulative distributionXSUBM* xv most probable valueXSUBO* x value at which F(x) = 2/3
0
XSUBS* x significant value
XSUBG* x most probable exteme valueg
*These symbols are used as labels in the printed output generated by
the program.
25
TABLE 2
MAJOR FORTRAN SYMBOLS USED IN MAIN PROGRAM (Cont.)
FORTRAN Math
Symbol Symbol Description
XSUBGA* x most probable asymptotic value
XSUBD* x design extreme value
XSUBDA* Xda extreme asymptotic value
* These symbols are used as labels in the printed output generated bythe program.
26
Subroutine GETCML
Purpose: To determine the parameters of a generalized gamma
distribution from statistical measures of input data by either
Stacy-Mihram method or maximum likelihood method.
Usage: CALL GETCML(AT,U,E,S,YBAR,GAMMA,OOC,T)
COMMON/PARAMS/,/PRNT/,/HIST/,/INPUT/,/OPT/,/IOUNIT/
Subroutines and Subprograms Called:
MLHCML Maximum likelihood estimation
WEG Wegstein iteration
PSI Digamma function
PSIl Trigamma function
DIST Evalutes measured and observed
frequencies and densities
MGAMMA Gamma function
Description of Parameters: See Table 3.
Remarks:
(1) If XD does not converge, a check value is sent
to calling routine.
27
f
TABLE 3
MAJOR FORTRAN SYMBOLS
Name Location Description
AMBDA /PARAMS/ X parameter of output distribution
C /PARAMS/ c parameter of output distribution
M /PARAMS/ m parameter of output distribution
X(I) /INPUT/ random variable values
ICKI /OPT/ Check value. If ICKl=l, c rootwas not found via maximumlikelihood method
IOP(1) /OPT/ options
YBAR argument logarithmic mean of input data
S argument second log moment of input data
T argument third log moment of data
CMNT(I) /INPUT/ header label for GETCML output
FOBS, FTH /PRNT/ computed values for observedand theoretical frequencies
DENO, DENT /PRNT/ computed values for observedand theoretical densities
OOC argument reciprocal of c
ICK argument check value. If XD does not
converge, ICK=l is sent backto calling routine.
U argument trigamma function of finalm value
E argument digamma function of finalm value
28
L ...
. . . . . . . . .o . . . . . . . . . . . . . .
TABLE 3
MAJOR FORTRAN SYMBOLS (Cont.)
Name Location Description
AT argument absolute value of the thirdlog moment of data
H(I) /HIST/ densities corresponding torandom variable values.
29
-I . . .. . ... . . . .i ililml -" • .. . . Hm , -- ...
Subroutine MLHCML
Purpose: To estimate parameters X, c, and m by the maximum
likelihood method for the generalized gamma function.
Usage: CALL MLHCML
COMMON/PARAMS/,/HIST/,/INPUT/,/IOUNIT/,/PRNT/,/OPT/
Subroutines and Subprograms Called:
WEG Wegstein iteration
PSI Digamma function
F Function of parameter c
G Function used by WEG to determine c
Description of Parameters:
Name Location Description
AMBDA /PARAMS/ X parameter of outputdistribution
C /PARAMS/ c parameter of outputdistribution
M /PARAMS/ m parameter of outputdistribution
X(I) /INPUT/ random variable values
0(0) /HIST/ densities corresponding torandom variable values
IOP(I) /OPT/ options
ICKI /OPT/ check value. If ICKl=I,c root was not found
Remarks:
(1) c is restricted to values greater than zero.
30
(2) If a root for F(c) is not found, a check value is
sent to the calling routine.
(3) An initial guess for c for use in WEG may be
read in. Otherwise a default value of 2.0
is used.
31
Subroutine DIST
Purpose: To calculate frequency and density distributions
for Rayleigh and generalized gamma distributions.
Usage: CALL DIST (all parameters and arguments are passed
in COMMON Blocks)
Subroutines and Subprograms called:
MOGAM evaluates incomplete Gamma function
Description of Variables:
Variable Location Description
X(I) /INPUT/ random variables values
H(I) /INPUT/ array of histogram ordinatesfor input data
K /HIST/ number of bins in inputhistogram
N /INPUT/ number of observationsin input histogram
FTH(I), /PRNT/ computed values for theFOBS(I) observed frequency distri-
bution and theoreticalfrequency distributions
DENO(I), /PRNT/ computed values for theDENTR(I) observed density distri-
bution and theoreticalfrequency distributions
Remarks: The input histogram need not be evenly spaced in
the abscissa x.
Method: Generalized gamma cumulative distribution is
evaluated by MDGAM. The density function is obtained by
numerically differentiating the obtained cumulative. The
Rayleigh density distribution is obtained in a like manner.
32
Subroutine WEG
Purpose: To determine root of x = f(x) by Wegstein iteration.
Usage: CALL WEG (I, X, J, N)
Description of parameters:
I - input, iteration count. Initialization action is
taken when I = 1.
X - input, estimate of root of x - f(x). On output,
refined estimate of root.
J - number of significant digits of accurary desired in x
for solution -- used only in initialization.
N - output - completion flag
0 convergence not obtained
1 convergence is within given accuracy
Remarks: The function whose root is to be found is computed
externally. The calling sequence is as follows.
(1) Initialize by issuing CALL WEG (1, XG, J, N) where
XG is the initial guess for the root and J is the
tolerance described above. N is irrelevant.
(2) Set up a loop to calculate XN=F(X) using output
X from WEG
(3) The simplest calling sequence would then be
X=XG Initialize X to guess valueDO 1 I=l,ITERCTCALL WEG(I,X,ITOL,N) First pass will inialize WEGir N.EQ.1) GO TO 2X - F(X)
I CONTINUEC MAXIMUM NUMBER OF ITERATIONS REACHED WITHOUT CONVERGENCE
STOP2 CONTINUEC ROOT FOUND WITHIN TOLERANCE
33
Method: The Wegsteln method refines the root of the equation
x = f(x) by calculating an improved estimate xi+1 which is the
intersection point of the line y = x and the secant line of
f(x) based on the previous two evaluations of f(x). This
method requires only one function evaluation per iteration.
The equation for the intersection point is given by
CX) it -
Completion code N is set 1 when
x x I
i.e., when the change in X between iterations is less than
one part in lOJ .
Function PSIl
Purpose: To evaluate the trigamma function, '(x)
Usage: PSII(x)
Description of Parameters:
PSIl(x) - value of the trigamma function at argument
X.(Output)
X - argument of the function (Input)
Remarks:
(1) Argument x must be greater than zero.
(2) The trigamma function is the second derivative of
the natural logarithm of the gamma function.
34
I I l . . .. . . . . .. . . . . . II lill I ll lll I . . .. .. . I . . .. . .II. .. . ... .. II . .. . . .. . .. . . . . .. .. . . . ..' ' " . . ... ... . .. "
I7
Method: For arguments greater than 13 an asymptotic expansion,
YaI-- - _LA rax 6 3 oy X 4Ly 3oXI
is used. The truncation error associated with the above expan-
sion is less than 1 x 10-10 for x > 13. For arguments between
0 and 13, the recursion relation
is used, where m is chosen as the smallest integer for which
x + m >.13. '(m + x) is evaluated by the previous formula.
The formulas are obtained from Reference 8.
Function PSI2
Purpose: To evaluate the tetragamma function, '"(x)
Usage: PSI2(x)
Description of Parameters:
PSI2(x) - value of tetragamma function at argument X
(Output).
X - argument of the function (Input).
Remarks:
(1) argument must be greater than zero.
(2) the tetragamma function is the third derivative of
the natural logarithm of the gamma function.
35
Method: For arguments greater than 13, the asymptotic
expansion,
is used. The truncation error associated with the above
expansion is less than 1 x 10-10 for x > 13.
For arguments between zero and 13, the recursion relation
YK
is used, where m is the smallest integer which satisfies
x + m > 13, and T"(x + m) is evaluated by the previous formula.
The above formulas are obtained from Reference 8.
Function PSI
Purpose: To evaluate the digamma function, '(x).
Usage: PSI(x)
Description of Parameters:
PSI(x) - value of digamma function at argument X (Output)
X - argument of function (Input)
Remarks:
(1) Argument X must be greater than zero.
(2) The digamma function is the derivative of the
natural logarithm of the gana function.
36
Method: For arguments greater than 13 an asymptotic expansion
is used:
The truncation error associated with the above expansion is
than 1 x 10-10 for x > 13. For arguments less than 13, the
recursion relation
is used, where m is the smallest integer which satisfies
x + m > 13, and T(x + m) is evaluated by the previous formula.
The above formulas are obtained from Reference 8.
Subroutine MGAMMA
Purpose: To evaluate Gamma function
Usage: CALL MGAMMA (X, GAMMA, IER)
Description of Parameters:
X - input, argument of Gamma function
GAMMA - output value of Gamma function at X
IER - error code
Remarks: This is a library routine supplied by DTNSRDC.
User must attach library (IMSL).
Error code meaning is unknown.
37
Subroutine MOGAM
Purpose: To evaluate incomplete gamma function
Usage: Call MDGAM(T, X, PR, IE)
Description of Parameters:
T input, limit of integral
X input, exponent in integral
PR output, value of incomplete gamma function
IE output, error code
Remarks:
(1) This is a library provided by DTNSRDC
(2) User must attach IMSL library
(3) Meaning of error code unknown
(4) Function evaluated is given by,
S f T "
P el
U
0
Subroutine HISTO
Purpose: To set up labels and plot the experimentally measured
density function.
Usage: Call HISTO
Subprograms and function called: SCALE, AXIS, SYMBOL, LINE
Remarks: This program takes the measured density, which is
passed in COMMON and uses these values to generate data for
a CalComp plot.
38
Function G
Purpose: To be used by WEG in evaluating the c parameter
by the maximum likelihood method.
Usage: G(c)
Subroutines and Subprograms Called: PSI
Description of Parameters:
G(c) - value of function (Output)
c - argument of the function (Input)
Remarks: c must be greater than zero.
Method: To implement the Wegstein technique the equation
f(x) = 0 must be put into the form x = g(x). The equation
used is given by
±L 0 1 4 S
v~04
+
V.A, Y. 0 Z
01 is the density
Xi is the value of random variable
39
Function F
Purpose: To evaluate the function used to find the c parameter
by the maximum likelihood method.
Usage: F(c)
Subroutines and Subprograms Called: PSI
Description of Parameters:
F(c) - value of the function used to find the
parameter c (Output)
c - argument of the function (Input)
Remarks: c must be greater than zero
Method: The equation used is given by
L \4 .. S.oi-
- 7~
01 is the density
Xi is the value of the random variable
40
4
CONCLUSIONS
An extensive study of an existing digital computer program
for determining parameters required to represent SES 100B trials data
by a generalized gammna function was carried out. Difficulties encountered
in earlier studies in which approximately 5 percent of the cases analyzed
produced unrealistic values for the distribution parameters have been
resolved. In order to resolve this problem, data sensitivity studies
were carried out and three other techniques besides the Stacy-Mihram
method were investigated. Based on results of the study, criteria are
developed for determining cases that should not be processed. A summnary
of major conclusions follows.
a. The Stacy-Mlhram, maximum likelihood, grid search, and
nonlinear least squares methods all yield estimated proba-
bility distribution functions which represent very well
the experimentally measured histograms.
b. Design extreme values are more sensitive to the experi-
mental measurements than to the numerical procedure used
in determining the estimation parameters.
C. In some cases the predicted extreme design values are
unrealistically large whereas the significant values are
realistic. Based on an analysis of over 100 cases, it
is concluded that cases having a nondimenslonal third
logarithmic moment greater than -0.5 produce unrealistic
extreme design values. Results of a data sensitivity
41
study have revealed that in all these cases the data
may not be considered as a sample taken from a steady-
state random phenomenon, and hence it is recommnended
that statistical analyses for these cases not be carried
out.
Kd. The Stacy-Mihram method should be used when the number
of bins (number of divisions used in the analyses) is
small (< 20) and when the number of samples is large (> 100).
e. The maximum likelihood method should be used when the
number of bins is large (> 60) or when the sample size is
large (> 100).
ACKNOWLEDGEMENTS
The author expresses appreciation to Operations Research, Inc.
for providing the analytical services necessary to carry out this
investigation.
42
REFERENCES
1. Stacy, E. W. and Mihram, G. A., "Parameter Estimationfor a Generalized Gamma Distribution," TechnometricsVol. 7, No. 3 (1965).
2. Ochi, M. K., "Computer Program for Estimation ofExtreme Values Based on the Generalized Gamma Distri-bution," DTNSRDC Report SPD-692-Ol (1976).
3. Ochi, M. K., "Extreme Values of Surface Effect Ship(SES) Responses in a Seaway, Part II Analysis of SES-100B Test Results," DTNSRDC Report SPD-690-02 (1976).
4. Whalen, J. E., "A Rapidly Convergent Nonlinear LeastSquares Procedure for Representing Hydrofoil Data,"Operations Research, Inc., Report 810, (1973),Contract No. N00167-73-C-0254.
5. Parr, B. van and Webster, J. T., "A Method for Discrim-inating Between Failure Density Functions Used InReliability Predictions," Technometrics, Vol. 7,No. 1 (1965).
6. Wegstein, J., "Algorithms 2," Communications of theAssociation for Computing Machinery, Vol. 3, Iss 2 (1960).
7. Johnson, N. and Katz, S., Distributions in Statistics,Continuous Univariate Distributions, John Wiley andSons (1970).
8. Zacks, S., Theory of Statistical Inference, John Wileyand Sons (171).
9. Abramowltz, M. and Stegun, L. A., Eds., Handbook ofMathematical Functions, National Bureau of Standards(1964).
43
APPENDIX A
COMPUTER PROGRAM LISTING
A-i
EAL vA~Z- . )- !' !) 9 Z fr -1 *F"' fDQ 11 ~ Cfl 2 ' A 0/ 4
!E"SIY V~lv0(7)*ELTf7)9 S()q "
'OCr~fqPLO:P.UNPLC.TeTP( ) 9ZTYtf,.r* TTV,,
~T -J' * p ~
C14 m~ -14 /0 14 C /1 laT IC 17
"'' I'4 T/ 12-9 1 T) TY'Tv.C%T 1 7- ~. P 1"Tv1;71/ .
40ITC(C.113) ILT1.t
2?1 GPCTIC- US'ED I* T-LS P1,.N SrT)
r1O 979 ,11CI I.!12
L?7q C3 TILEIF( C INvro.'a. . ) CO~IVT =1 .0r" ~ T (2 7 !.10 9215 9'F 1 .o
C C' TIN LEVNI I ) =.
C..* REAO IPUT 6NC PQpT *.
C
110 CR4q ATf2I5.2'1O.Q,5A1O)
C FA L 'JAL'ffS
OED(*12)'L fLL I~I=%'L
IM= 1. -ALP
c DEL. RE'. I'. c~r . AR')IYc IF STARTTI 011*T M.E. Gq! -SIG"' JILL P"rCFrE"C TH-E STARTT\,r VALLE "lYVE' AS OrL.
2~vW=DFLiIF(2IEL."'.1.01 ': TIjaST=-JEL
DSU u 0 (1v (I1) 5!
?I IN6 Q )=r( 11J SU Zsu'.CT
!c( 'CCI .E' . 0 c~Lj IT TFLr.E1.1) 'n Tn 1002'
q T-1 1.3
c IPTT3'J 1: roC;VEr:T !':PIJT I'" CE'lSITY IFTE r-rc
it -Q!) = '(l) I (=L'3ATrN)*'(l))
lt1'TC"'; 2: 'OOlcy L VALUFS !F %ECrS-,A~Y
'TI ME =ELL(1)~12 !=1,'L
C OT 2 12 !: I.DJUST T-PUT Y( T) TP3 MIrPnTNT Or !TFRVtL.S !r ''Cr'z5-Qv3 TrCIfr'.)NE.1) GC TP3
.0 1' T =I~ (f) =Y(fl 4 07/.C OPT1O0J 4: C'of.VFRT INPUT T3 "ESUREME'NT 'Es IREr4 IF( IC ( 4 Nc'.I ) 10 TI 9
DO 7 T I.K"M= wfl) 0 CAL
Off )=Off )*CAL7 XC!) = (l) * CAL5 CONTINLrC IF L VALLES ARE 0% O I" A! SAmOLFS DEQ TP". TOTEOVIL,C CIT M0 UTE TIT vE VALUES F2R PLMT
IF(ICC2).E*".l) "0 O P:?TI 'E:FLL (1)/ll':0 57 1 1.L
-5 r7LT(!) FLL(t) *P'E.
6 CON TIN LE
YlvA X =0
*9* 0 -
C
20 1 1=19K
IFcT!.'!T.YVAV)YvAY=Tl,
P=RTII)2
YT3 ='rI -V 12
! CCNTI*Y!
C *.** CV"UTE YPAI% S. T .*
Y94 0 S I
T4=v AO.YBA2
T= T2*rS3- 3.*YEAO.T3.VPIQ*f3..T4-T ) )/ .'1 ,
1237 'CRAT (1"O .A5'T IS Vj"EA TEP TH4AN -. F 'e - Tn NrYT 'ITS .~
4T=AeS (T)
c c~
!FUTCK.EG.1) GO TO zoIF~ifI.r0.tl 7,0 TO ig
Q-114 r" mTf1.V1,22l4PYLEVIr4 r ETIGM VALtJE ~
'JPYTEf6.P013) FLLCJ)*XRD*YQD4SQnl?2 CO0.JT !01 r:
16:13 .4)IF(ICf!21.F'.l) r0 Tr I C4
C*** C04PUTE Cv9LAMoOA OO"CTS FnO LATER U~r ,.
111=TI-1.
C 10000 Li 0 -41OO TwE' L VALUES
c INflITIAL GUESSES FCQ X AN~D v *
C:Y
TYS C.
Tyr)Azi ,a.
cL=ELL FIL)
IF (E2L .1T. 303) GO Tl 0 011
'.I T' A^1
I I C (Y' E C 'JP I T T0; 71'X1'4) )4
710 FZRAAT (iP/* FYTRAPO3LATE VALUE! */I)rSTOP = 200*!ST'Pl)
c * SET UP w1ST.nGRA% PADAr D!s
r =0
TAX X= -)" *Y .LEN(:0
W LL=: 3D 120.
R ES=(FI A~0 TSC I ff2 11(p0
YTO = YC . 11000 L000.
C "* C0vcUTE SIOALL cfX), PQOOAETLITY 'rkSry 9P! ..
C
T45=( A Y9 A -Y 7**CYSzFvt(-T40)
Oyo= -*4TC
IIIP.E7.5JO0) GC TO I ~C2C TOI~
ST? =XO *)( TINX =2. *Y TI *r YOC YT2)
r( Rv') .3. A )YA='
11002 C04 T !!4Lr
C *..COvPUTE CA'rT A L F (Y. 0' CofoI TTY 'IS T~lI LIT C11--'
CALL T * 9 *
S 1 FCQ'AAT.:-H FORCR IN "A~f rOIVTRT~JTC2 CL'PXTI)N u~r~
IF (C .LT. 0.)'I= .P
Tv (TTT.LT.-!03.) 't' T'" PtG3
-^G! rFOYL =^A 14 ONT!J)LF
C *...',COUTFC SV LL C-MX, 1;"ALL P(Y), CA0ITL wfX)
C
a4 r L*f(VFOYfL/l3F0Wn)fl).p)0
Ic (EX.LT.-31P.) rf- T- .4010
-4L3 EL.C'Iyr*Ev
10 Tn 111
14LOO = L .-'H0 m-E v I
IF (TF *GT. 1) flO TO 12001X WNS CI ) = X * CCN V'FOXOMSCI) =FOYT I CO X'VrR')FGf 1) Rr%2
r§Ffl) =G"FY / CO*JVE0,M.'MStl) = HL9/CNVE'HMO0"SCI) = liMO
YCOM I
Icy
c I*** [IT IALI ? T09! .**
TX=A FS ferOVL.414 A)7!r ( x-TYC) 000,99019,001
!~ (.CE.~X)Gf3 Tr, 9012
9 02 CCI NL E
IF f'X-TY9) 9903q9V>,t043 0 3 7 a T X
'4 04 POITTIF (TX^A *rE. 6AL) to7 an
300S TYA X
90C9~ C04T NL-
TV =: IO.L
4101 TA '!X
E12 y 3 A v .1104 S'
11003 CCNITIA!LrXS=YC(0).llr ) 0 LL
71 FC~A( f9FX9LE.Q4 991H'2 W. *~TO /1
211 O';IT-- 6711c J,(,STJ)I X'Dt).F f . F L
11C0 C0OIT14LE
17~G TO 17NL
94ITEC(123Eq2)0 0TO17Dc1C tc~
C iI F 6p j ) Jp S J o O n*-( ) t( ) r % ( ) !c v t~
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C 0TMc
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XSJFVS = X0'J Ay C~l VE
CC t** SOLVE --CR X USJP~ WISTF!I ITE04TTVE vl:Twnr *..
TT=ALD#F~L
CALL dEC. '1 U,*,O)C ** ST,2T !l'E~rEVl LO~ *
23 T45 =(AoQtrA*U)*.C17ALL ~fa..,TIF (C *Ll. 0.) - O I.-DO
C 'f=J.TT#5UL!CP*)CALL WE~f?,U90."C)
C 4OITE fg;92'5) U*P,T'TT
IF ('!E *LT. c0) ro TO llt
30CI CON T !JMEt'('C.F.~'C*C T^~ 33
C32 V 7j 2r) =
CC EVILUATE G&vUA DIST019UT~rk FuLIPM!
C
C CALL *JCr~A'( TTo.,pF.TE)
712 rOR'AT (* 00*OQ Gv
C *'.SOlLVE rlP 4 I!ST'S WFrlSTFIPI TT-?C&II!vf:T'CC
CA =C*A080A
"L'4 =EL-I.130 = O ^U =UO
**STAOT ''FOATTOJ LOt)5
43 IU Aon2AUC 8I'..C
TF(AUC *LT- 20.) SO TA 9ACO
1; T C 6:r~,fot C 0RiT If14L
C&LL 11"05A!AALCw9PR91E
6 1 Cn'JT'JUrCtLL 'Jr!2,1),O,0C
C WQ~IT (GTG3) U.PR,F.C.EYPTY'CvAUC*ALq,T7
.)3
COMA (* DP!-47
TF U! r'E. 1. 10 Tri 6003
PD TI 435
'! * IIF ("E .LT. 1!91 SO Tm 1-1100
4000 CO' NL -
c~ C'%JD ITFOATT-0-4 97" If
'a 7 r IN T I LC
C.. SOLVE !'Q USING 'JESTEI': 1TcPAT1'E mETwV'*..
C
VK~g( /3IJO0 x (K
='iL 'Jr lt~l
C .* ST40T JTTP5T'TO9 LOOt';.07 AU YAU
AIJC IU**CCALL MtCGAl'ALC",PRTE,Y' (C .LT. 0.) PP
It .TTu-PRCALL WEG(2*li.O,'!C)
C iPIT!:(69703) Uoc'IF (U.2E.0) SO TO SOO&
C " 0J')= . E-13
6004 CO34TP4(!7' - ME 4'1
IF (NE.Ll.50) q0 TO 6O1!'4RITE (SP2100)
20aO CORMATrAH 'ff COP.VERO-'Ettr fITAPJE FO P Xr!J9C)U 0
oTr- 'n0
S 1 9 CDVTTNLE
C .. ENO ITrPATIlt F! Y
lJlS = * C -F
3 v 1
S *TYSUFI *,rl2.4,3Xq*TsUPOM4WS)= ,1~A
C *'*.* CC14UTE SICNIFICANT VAL"E
IF: Cj .E9. 1.) 0='EXfl
CALL MAMAETGA~f1E*)
,j F:)0mlT(4w -'QCR 114 GAm41 FWJCTT"I CEOQIJATTJ TQ
A.UC a,.
IF IC *LT. * '
UvS = L * CO'VER
3 l~ roqAAT(?X7HX4SU~Er: ,Fl2 .4q3W,2HYSU!"(mwr) *1.9y
1 9.TYSLF:= qF12.4,3XqI3"TXStF'Gl KS)= 9 F]'.4)C
SO~LVE FC SUTTCVALUV F Y USING
EU = L/Ga "AtF (C *LT. 1.) rLOG=-EL~r
IJSE JXG AA.,
CALL 'F1;(jU,vO)
C ** STigT ITERATION Ln-Y ''.310 IF fU *LT. 300) GO TO QIE01
UT =3 .1,0 TI Pr
A9'31 UJT = U*"*Exr(-LU)-91 'J=Cv~'C*FLCl,*UT
CALL WEG(29U,09NC)
JO:UD+C( 1):2 Uo
J~I . c- 36011 CONTINUE
'5312 C(40'J"L
11C )c A( W! CNER~irD T ITAN FO FK l~A
'44TE69OX U9^0CCO"POE0
':n TI E113I )UUS YG "A
-1 Ca~J'PJLE
C1C 1*4* SOLV *~ *11T TI CC'J "A -Q I''j jPli
C M *CNE
"3T!~C3% U."S)OA*,(
631 cR"A l' q*XU9 A
,PTF"(IC *Ls. ) PPfOYU~~E~~n
615 ':RAf~,YU~. *F24jY*S.0"v) gU?&
CA=LLm(2UO,")
"JE =E4I
TcN.L.0 *G ST Ta EAT' LO*
IFI C LT0..P .r!I' =V [JO4
'A* F(J.' r! TO 6041
C *' ENO ITCQATG' TO 6044
IJIE (909
-V U * : -:EEO EXCA * C^NV~v
6-45 itTr( 9~44) LU.UmqEVCAEXDAm6046 Oq'ATf2X,.YlUCA= *9Fl2.4.2x,*XSU trPmw) *.F11.4.1Y,
$ TYStUefYA: *9j..X*~rU"( ) -,F!2.4/~//)
6047 FOR'A M~(1119T3,q"TIM- c!T?0yg~, 4 T! HP I 9
..7 VPLfl,1U2,?
ESCM = ELTfI) / 60.0
$!77.F1 0. 4,Tq?%*FIl'4)
cC ^D~TTJN 9: OL^T HISTOGGA" WITHI *L'MT DIe~f:
q7! !;fI t(c).LT.1) ' ' TI3 "7
C C'2PUTVE HT4STO'RA' VALUES
ICN
,Y) 93 1 1.1cC4T =ICOJT + 1
IO 92I I1I
~IF( 'P.L.lS O8
I(1 2lL.~)S O9
91 M ) .
4ANSCA. 2 1I~SP
ITP(2) 2 6IE' *3A
TT )2 6w RAYL
TTP(9) 2 E0-r!tHT T2 f ) = 6HL^TSTTW(I) =2 WAUITX(2) = 6-09~ OAN
!TY()) = 6wOC" Va I
ITX~f) = 2P
T TY fI) I PI S!rTY (2 )=SH
TTY ( I) = 2 C UTTY f4) = 2 N "
TTY ( ) = 411
c 'Cf5)2 6R1TE VALUES -FR OLOTTT!'
TX 1 ) = 6u 'wDo9 1 = 2 V
7w 1 ) 7(l / C:%IVER
OrNMS ( I z F'111) /CCZ';VEQ
'!I Tn112~I
C ONT T'J ~ ! T"sWCT:~T ~ G~C
1 9G1 1 0 1
7 CC" TO N L
CALL PLdO~TlqI14IA
c IPTONT: TIM EFFFCTE PLOTS ~ LO~AUTI u~ro
I F1I~''.0I " 39
'JPL2 I2T
C OPTION 1 1: ?CALE FmR r"OEr-r5.0E'T VARTIFL7 T'I PLOT( 'EA'!LT. I I
"AXSCA =
ITI .1= jFFECT
IT*(!) =6fMOLOTS
TTX '?) 6-fl HOUP
1TYC1) ri1 !',1 11TY(") 6H VALUE
:VA P (,) 4vf'lVARe3) SHXSUR
TVAP(4) =fC 10TION A : TT"c' EFFE:CT PLOTS T%' '4VS !JITTIICC
lFf1CtF).NE.1) rnT^~ "0
!VAR f2) -,WYfl..F(s
ITY CA)
S . 1T' (E 7C (jI9 X(jL'xfJ
Sl Tn E5
65 CC4INLECALL OLOTPO(709IVAR)
!F(TDLCT.E^.0) Gn T' IqC ENO. l' O 000 L^IF***C
C PLOT* RC.SULTS
'ILL 0Lr'Tr"LLp.fl.-3)
CALL HISTC
XN( lP.0)=LNELX
ON( T P+ 2) =0 9: L
CALL LllIE(XNJqFllolP,1q,p' )!r(!QAY.Eq.0) Gfv TO 10014
IC014 C3NTINILIFGO TO 59
11013 TF(IPLCT.E0.C) STOPCALL PLOT(CXLL90.,999-)STOP
SU9ROUT I 'E !ETCML( AT UlrqSqYQARVGAm!nO -C4 t .Z1
"11!E S1CCI: YEW.1 CC ,F "Sf 1
C* IC *** S)L'IE C' " USTIJ' V7GSTrVP TTFQATTVFr vETL.4-
C
r .. ST12T rTT TION LOO
CALL WFJ(2q*Co C
IC(UC.Lr.,s~i cc To 22o,^
6% rOqu8T(2tfM * Cl'!VE1rENCE 'RTAj,"! F*V D l
2COO IF(v.GE.P.) O T' 2' 01
?1.1 C04 T IN LEIF(PIC.-'I.9 -0 T' 3
C ".10. ITERATION pF'D
C .**COvPUTE CC
rONT!TJtr
'J2 P I IC SIRTfU2/S)
Ic=1. 1CC
C.. CCuPUTF, LAwC!A *.
4 AP31 =EXot-Y8AR+U1*or)5 CONJTINUE
'F(IC(2:).E0.11 CALL 'LMM~IF$ ICKI.Eg.1 I ETUR'U
IA~(* )CN
1 O "~ AT Q i-I . A10
CMA t .- AL1,,iIF(r.l6 CC IVER *COL
c COM PUTr W((I( ) VIC.' ^EL("'(')
1' XMi(StIi) = (JT) * ~rC F1'JD MKS VaLfl$. OF~ YE
wRIT (,001 NvvALO.
297 F014AT(l1X,*rEL ARRAY*)
F 3R'AT(1l- ,14tA2v!2v1W4)97.4,2y))
296 0R ikTWX4.EL ARRAY(MV1 ).)!F(C^4VE'D.PE.1) WRIT(o?9?) ('"rAC(I i,'(f')r1
E'"ALUATF 1,A"mA rUIC TY 31
103 cO&"T(464 FQFCO IN C6kt"A FU*!CiTZ'I CPAUTATTJV. E:Plp .1.
CILL 011T
!OC:1.dCCALL lrC,-4m 8 "GA"MA, TEI
Pm< = * (C0NVER-1~)
IFlIC12').E2.1) 0ET~l.HwAYImU" LI
I !12.4#2v.1 L&4DA('e~l ('719.4*93140
= .'1?.4,Q4(mvlZ =t/,74 9 IA AqAR,7WM(TS4I'4YORDO81-P AYLE H
110 FOR'lAT(T'.;'HVALE Oc.Tlq9Al'VALUr OFTloPrA1R~TjIA8 ,T'65-HGA 'A.TS 2,eU4rA ,!JPEC ,OP ,AP eTll&* 5'4'ZA"vk~
$"H'Q EDICTED ,T82.7HDE-"SITT0T ,*HPQEr) TCTE' ,Tl ~qPC"IT: 1
1210 FOq"&T T!,FP.4.T199,Q.8.T37.r'12.4A7ACO.rlr.4.%.C1~r2o.
' T9 'I2 4 94 l' 4, ll . 1 . )-ETUTo
SIU3R'UTT'iF HISTC14 10 '~ I l. "0) .Y'lAV XL :.YLF.i.C"J'e
D1wE: S 10%~ Xf2f0 ),Y C2n) 9R VAf9)
3jD (N2)=YAAX
C Cl'qPLTE SCILE CO v-AY!SC CR0 IS THE APPAY OF W4ElrT4S FnQ THEI FTNS
*C YLEN. IS T E LENGTH 1'% TICHES (-F To4 v-ylX 7EF;5,tfl p1
C a DATA STATE"C'IT 11! THE 'AATP F'r3ip To -- a C CH-C N? IS TH4E N)Umn'E. OF PC!IJTS 11: (on~f ip; *i SAYS FVEDY '0PE? I^,C TO 9E PLOTTEDC FIRSTv TH4E FT 9ST Y 'ILUE ANNOTATED ('5; THE Y-AYIS IS RETTJ2' EOc III T~ 1 2+1 L"CATIZON OF ')pr
C OELY THE SCALE VALUE - AvOUNT (NF ATA PF INCu- I11;~~EC DY 'SCALr IN THF '!2+' Lc!CATfI'. ^F CPC
CALL SCALE(OPO ,YLENNZ. *1)FI.RSTY-oQUE)N2+1O)ELY=O9DfNJ'42)TITLX=IP4EXCv!KS IONSTTTLY=7HI)F'SITY')RD ('12 )=I.
C SCALES SET UZI BY AXIS TO HE USED -Y LINE TO PLOT
C >mINTS - Y-AYIS LENGTH IS CCSTANIT AHOC IS oEFINjEj 1 1C DATA STATPO~tIT IN THE M"L! PPe,-AM Tj N E 11 T"C'~cSC THJE IXIS ANNfTAT17N 'JILL QE DONE WITHOUT TH~E USE IFC AXIS SO THAT THE NU~~cERS LI~4r UP WITw TPE :11'S IIA,*
'(OJNT I
IF ('RINLC.EFl.%.C) GO TO 5
(('I = INLrO
5 1 =O 102.!,
'(OUNT o!T110 COVT1NL'r
CALL SCALE(XXLE'IvN2.4I)FIqSTX= X('12+110ELX = (J2+2)
C OUT flF'1 67 nRIGIIN
CALL PL.3T(0..93)JO 20 I=l.P?2STEP X(I)*LNIT
c DRA. AXIS PORTION F~fO" CUDRENT LOCATI'ng TO ENE) OF NEXT BI%CALL PL'T(STF040.,2)
c DRA'A TICK( 4ARKCALL PLOT(STEP9-.1,2)
C APINCTATE AVIS WITH NUIAERALSCALL 'UM'ERISTEP-.21 .-. 'I..1,X(I),0.tqO')
C RFTLRP' PEN To~ AXIS L!NF' WITHOUT ORAWI'.G LT'IFC CONtTtIUE DRAW1ING AXIS! AND AN%,nTATP1IG IT FOR THE 097*4AI'InEf: OF TME VALUES
20 CONTINUE
C TTTC TH AXISCALL SY"'FOL(4.299-.52,. 14,TTTLX,3.l,10)CALL AA1S(0.,0.,TITLY,.7,YLENo? .,F!'PSTY,'J)ELY)QA'V(1)=1TH*RAYLEYG4
RAYC!) =IO!4-CC1'PIJTEDPAY(4) = lasiALUEPAYf5) = CG4X~A GEzXLr'-? .g
CALL SkOLv0A(EYOAGE..14,OA(1),0.0.11,)YPk3E=XP AGE*1.4CALL SY COL PA'EYPAG!,*.14,RA(2)f1l .lt)YPAGE=ALENJ-? *
YPASE=YLEM-1.CALL SY EOL(YPAG6!,Y
0AGE,.14,OAY(1),0 .0910)
YP ̂ E=X AGE+ 1 . ACALL SY 01 L(PArgYPAGE..149AY(4v),0 .0,1")XPA GE=XPAGF+. .CALL SYAP0IL(YPAr .Y AGE,.14, AY(0-),C .0 91n)
YPA4 YO A GE-.5
CALL SYtAC0LCXPAC-EYPAGE,.14,CVJT).0.5')
X (2) =6 I'
AT X A(J-1 * TON I)Y (J =2XTV =J1) (I)
A (J = Y T
CO'J'INLFX(J+ I) =P TRSTY
v(*I U F IR STYY(J+2):0'LyCALL LV!JEtX, YJ ,I,0,0)PET IR-1
c UI~~ FUPICT10J PSI EVtLUATES TWE OILFAF'M CU'lCT1.V; DST~w) F^R
C Aq~~,JU*EKT M. PSIQ') IS THE DERIVATIVE YPT M4 OF TH4E L14 -IFC THE CAvvA FUNCTIONJ.
')ATA CIG ,Ci126/.1666666E7,.O79!65OA/
Tj 0.
IF(.J.GT.13.) GC' TO 2
14 . -W'4 .1
I TI TI I ./tr~ + I)
2 O'cw 1./V
DSI ALC,('W) * .5*'QCW~f-I. 4 PCW(-Cil * RC?.4*..1CIl - C!!".4RCWj2))) -TI
'4ETURJ.EN
C FL'NCTT'T PST1 EVALI)ATFS Twr TQt"-AmM- FtUh'CTION CST (m)c C* ARGU"E'"T M. PSlT'4M) tS T4r IO[QTVATrvs WPT .C3.12.I67."3 TW$72~
C 0108y!'A FUNCTICM.
TI 0
1Ff ..~T.1!)'~C TO 2
4 '
I TI Ti * a.WI)**:?c V. tT .13
? 0 cW 1.,,Y4C42 RC'4..2
"Sri 2 C ''CJ2*f)ioT'
! FuP1
FU4~CTrtflH PS12(M)
c FL'JCTION PS12 EVALUATES THE TETPAGAMIAA FUNCTION CSr W
c FCR 4RGU"E*IT M4. PSI2(m) IS THE ?NrO rERIVATIVE WRY m mr
C T14E O!5l'P'A FUNCTICNi.
REAL MLATA C16,C!6/.166666'4,? ..83 333333/
cTI 0.
IF(4.GT.13.1 'O TO 2C w.LE.13
DO 1 1=1.4.
I TI =TI - 2.1(0 * 1)**3
C o.IT.132 qCd~1./W
PS12 = PCY2*f-1. *RC J*( -1. *PC'*(-.1, +PCW2*(ClI *C'J.(Sj+:?CW2.f.! -CS6* PCW2)')) -Ti
PrTUOP4
SUaBnUTI"!41E 4 tJ~
C WVlSTEPi ITERATIVE flLUTICP METHO~D FOR X =FIK
c INY1TIALrZ4TiopiTF(T.NE.1) 311 Tf' 2
XTOv
RET U N
FIRST ITERATIONI2 tF(.(.NE.1) IC TO 4
KI =
PETURIJ
SUC 'Ik! TTEO&Tr?"IS4 XSNPI=
IF( A8SrKXP-V2AlP1)~ ~7 (XTFuc*XP) :r
S) T 1 7
OFT U R 4
__D_
SUPR('u71%E AXIS
EFJTP Y L
7N~TR Y PLOENTR YY F80T
EN1T PY PLi'SRET UP
~UNCT!CJ G(Cl
'rOSUIG.YL.OSUM=0O
YL3StJ*= iVLOS'J - AL1'rfX())*(Y(I).'C).O(I)
I? CONTINLEARG r(LnSw./WSU) -(LSUMSU0fl)
$ S!V1./CISP)))AOGC)
OET UO
* PFUNCTIC'! F(C)
CO0 !w")N 0 1tSPUT~IL, / ~ y 0 L HA O5 1r O I( L0 Tr qIYA.LEC J t '5
VfSU"=O.X(LOSUM=(:CIL4 SU"=O .
:)o 10 T~lg'vXISUM= YCSUMq (()*C&'T
F' LOf).A=LOG(XOSU"'()-A ALGEQG-L0GU(YP(1.(C)R
AR'.=C.OL SUM'jSUMO-C~FIlSPO
R F U q N
SU9ROUTIN : 'LHC*'LcC "Axl4)JV L!'(ELY'4C00 ESTT"A!O':
CO-m'N /OPT/T^P (20) ,IC9K1
1 C't IP (I)P45.1 ) 0 TO 10C PAYLEIGH4 PAPA-ETEOS
1) REAU(Tq5*I21) C
30 !'1 ().'1 RITCrI!1;102) C
~5 5 r-,RMA!Tf' 9,ITERAT1O4j'S FIR F1NCPll~G C Pn^T)'4ITr'! ,551 P
"0 1 1=1 .91CALL WFI(ICA.IKE)
CC=AP(C)
I ^''JT I NU!C "AY'U" %IUMr-ED ^F TTER.4.'!O'S QEiCHE0 V!THr'UT C'NJV!'1GE'Cc'
4-71 Tr( 1069A441
444 C(OmATf1'0916HC POOT R'OT FOUND)
22 C3'lT!%t2rC C RIOT FOLNC WITHI4N TOLERVICEC FTKC 'I
X LOSLVA 0.SS 1) 0
SUWOt =P
'LOSUXLOSUP * AO(()*vT*0O?
'sUml= CcI).SU"C'3 CONTI'JLE
C FlUND 'U--SOLVr FOR LAM8OA
YFI VPf3).E').1) ;prT!T0A9100)APf)hpA9"qc10') FIR"ATf1U0, 'PAPAMETED
0; ECTT-ATE) 9- V4VYT12- LT'(ELY'9'-ln
SIlV' ES'T "&TTC%//1iOETURN
SUoBoUf1i .E OISTREAL '4
COP404/0LVDRA"C/N .AMBOA,C.P
POTT~m = 0.0. 'lz1.0
D2m1 . 0.)O ID I =1 PK
C OPT 10*1 1P: TF 3 9 1.NS ITY =IN')./0 IN)~I
!F( ICEI1S).E1.0) CIV=D(r)!Ff ICC 1P).E(1.l7 02=0)(I)
CALL mCi.S4(AR'q,"oTO0 . JEP)
CC (IRSERVEO FOEQUtEjCY
CC 0QSER',fD DENSITY
C THEORETICAL 0AYLFf'3H(EXPO FNTTAL) FREOUEr.CY
C RAYLEISM(EX01"ENTIAL) 0ENTTYDE4~TQ(I) =(qUjfIN - 0)f!
CC THEORETI CAL GAw~f FREQUtENCY
IFC1ER.497.n) ':0 TO If!FT'4(1) = TIO - PCTT"4M).
C GA'AMA CR.EO!CTEC OE.SITY
V')TT.314 TO2P1i C0'JTIJE
QETUDPJ
EN
APPENDIX B
SAMPLE OUTPUT
B-1
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(1) DTNSRDC REPORTS. A FORMAL SERIES PUBLISHING INFORMATION OFPERMANENT TECHNICAL VALUE, DESIGNATED BY A SERIAL REPORT NUMBER.
(2) DEPARTMENTAL REPORTS. A SEMIFORMAL SERIES, RECORDING INFORMA-TION OF A PRELIMINARY OR TEMPORARY NATURE, OR OF LIMITED INTEREST ORSIGNIFICANCE, CARRYING A DEPARTMENTAL ALPHANUMERIC IDENTIFICATION.
(3) TECHNICAL MEMORANDA, AN INFORMAL SERIES. USUALLY INTERNALWORKING PAPERS OR DIRECT REPORTS TO SPONSORS, NUMBERED AS TM SERIESREPORTS; NOT FOR GENERAL DISTRIBUTION.
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