kAJ NE - DTICThis report shows the expected results with estimates, referred to as gates, of varying...
77
kAJ NE REPORT NO. QAR-Q-O11- COMPUTER SIMULATION -OF MAXIMUM' LIKELIHOOD OUTPUT FOR STANDARD- LANGLIE TEST DONALD C. RAPPAPORT DECEMBER 1977 ARTILLERY SYSTEMS DIVISION PRODUCT ASSURANCE DIRECTORATE US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND DOVER, NEW JERSEY DISTRIBUTION SITXNT A Approved for public wa1ea~ Distriibution Unlimited
Transcript of kAJ NE - DTICThis report shows the expected results with estimates, referred to as gates, of varying...
kAJ NE
REPORT NO. QAR-Q-O11-
COMPUTER SIMULATION -OF MAXIMUM'
LIKELIHOOD OUTPUT FOR STANDARD-
LANGLIE TEST
DONALD C. RAPPAPORT
DECEMBER 1977
ARTILLERY SYSTEMS DIVISION
PRODUCT ASSURANCE DIRECTORATE
US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDDOVER, NEW JERSEY
DISTRIBUTION SITXNT AApproved for public wa1ea~
Distriibution Unlimited
0 _0 E ~NOREPORT NO. QAR-Q-011
Id
-C M UTER JU M LA ION OF -MA IMUzfIKELIHOODUTPUT FOR S NDARD l::
i= LA =LIE T EST '
DONALD C. RAPPAPORT D D Q
........ -'- F -
Artillery/Tank Systems DivisionProduct Assurance Directorate
US Army Armament Research and Development Command /Dover, New Jersey 07801
AppwoV0 4 - "
'd
TABLE OF CONTENTS
PAGE
I. Discussion ........................................... 1
II. Sample Statistics: Maximum Likelihood Method ........ 2
III. Sample Statistics: Five Percent Random MalfunctionRate ............................. 3
IV. Sample Statistics: Ten Percent Random MalfunctionRate ............................. 4
V. Sample Statistics: Least Squares Fit of CumulativeNormal Probability Distribution.. 5
VI. Sample Statistics: Least Squares Linear Fit forAverage followed by Least SquaresFit of Cumulative Normal Pro-bability Deviation for StandardDeviation ........................ 6
VII. Comparison of Three Estimating Methods .............. 7
VIII. Applications of Derived Statistics ................. 8
Appendix A: Procedures for Determining TheAverage (V50) ........................... Al - A2
IX ~ 'te Sectionl,:j". section 0
'it13
te.. I'i
Discussion
The purpose ef this report is to give an accounting of theefficiency and reliability of the sample statistics derivedfrom the Langlie Sensitivity Test using the maximum likelihoodmethod in obtaining an average penetration velocity (V50) anda standard deviation of the penetration velocity.
The assumption is made that the probability of obtaininga penetration is distributed as the cumulative normal proba-bility distribution. The maximum likelihood method in theorywill give the most likely estimates of the average and standarddeviation of this normal curve.,
The test procedure begins wich estimates of those levelsat which failure and success have highly likely probabilitiesof occurrence. These estimates have little credibility. Ifthey were good estimates, the test would not have to be conducted.This report shows the expected results with estimates, referredto as gates, of varying size around the true average.
The Langlie Test Procedure is defined in detail in Appendix A.A computer program was prepared simulating test results usingthis procedure. The true population was defined as being acumulative normal distribution with an average of zero and astandard deviation of one. Each of the sample distributionsshown were derived by using different starting estimates interms of the standard devi.ition.
Hopefully, these tables will provide insight in the useand application of this procedure.
II. Sample Statistics: Maximum Likelihood Method
Most of this section is devoted to computer printouts ofdistributions of sample averages and sample standard deviations.An attempt was made in Figure 2-1 to summarize some of theaccumulated data. It is the author's opinion that the summaryhides much of the information contained in the printoutsbecause the distributions do not conform to standard proba-bility distribution curves, therefore although these printoutsare bulky, they are included.
The author has been advised by Ms. KodatA of AberdeenProving Ground, that the standard deviation (for the standardtest) should be divided by .745 in order to unbias the results.
From Figure 2-1, it is noted that as the initial gatesget closer the variation or standard deviation of the standarddeviation increases per unit average standard deviation (underStandard Deviation, see Standard Deviation/Average). Thestatistics were obtained from the grouped data on the printouts.It is assumed that part of the reason the Average of theestimated averages all tend to be negative is that the estimatesthat are exactly zero are in the group labeled -.10 to .A0 andthat some of the tables were computed using the same randomnumber set.
As stated previously, these tables were generated toprovide some insight in the analysis of sample estimates ofpenetration data to the user of the Langlie Test Procedure.Some applications are discussed in Section VIII of this report.
A. Author of the modified computer program used to calculatethe maximum likelihood average and standard deviation.
2
0
Selected Summary of Computer Printouts
Initial Estimates (In standard deviations)-5 -2 -2 -1 0 -0.5
to to to to to to -,
+5 +3 +2 +2 +2 +0.5
Averages
Average -.01 -.01 -.03 -.03 -.05 -.04Standard Deviation .40 .39 .39 .38 .61 .42Number of valuesused from printout 8A9 827 816 849 467 757
Number of extremevalues not used 0 0 0 1 21 4
Standard Deviations
Average .77 .69 .71 .61 .77 .39Standard Deviation .42 .38 .51 .49 .85 .47
Number of valuesused from printout 8A9 825 884 844 463 753
Number of extremevalues not used 0 2 2 6 25 8
StandardDeviation/Average .55 .70 .72 .80 1.10 1.21
Total SimulationsAttempted 1000 1000 1000 1000 1000 1000
Number of simulationsnot used due to
No overlap 135 95 65 53 68 52
Iterative processfailed 36 78 49 97 444 187
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III. Sample Statistics: Five Percent Random Malfunction Rate
In this section data is presented with the same assumptionsas in Section II except that it is assumed that there exists arandom five percent malfunction rate which can occur at anyvelocity level.
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IV. Sample Statistics: Ten Percent Random Malfunction Rate
In this section data is presented with same assumptions asin Section II except that it is assumed that there exists arandom ten percent malfunction rate which can occur at anyvelocity level.
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V. Sample Statistics: Least Squares Fit of Cumulative NormalProbability Distribution
An alternative method of etimating the average and standarddeviation was used with the hope that it would produce betterestimates. Although theoretically a least squares fit should notgive better results, it was felt that within the narrow constraintsof our sample space a better estimate could be made.
Two sample distributions were run. The printouts are shownin the figures following. The results were sufficiently poorin comparison to the maximum likelihood method that furtherstudy was abandoned.
5
atI. ix!$
4mMM 4434 44, ,,,
>'0 L * 01.............
00 0 00 0 0' L 0 o '0 0 0 0 M4 00 f4 W 000 00-0 0 r'4nc- 010 0 00 j)
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VI. Sample Statistics: Least Squares Linear Fit for Averagefollowed by Least Squares Fit of Cumulative NormalProbability Ri for Standard Deviation
Hopefully, within the narrow constraints of the StandardLanglie Test Procedure another method may be of value inestimating the average and standard deviation rather than themaximum likelihood method. The method attempted here used aleast squares linear fit of the data to determine the average.This average was then used in the determination of a standarddeviation which would minimize the sum of squared deviationsfor a cumulative normal distribution curve. This method hassome merit in comparison with the maximum likelihood method.The sample distributions are shown on the following page.
6j
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I2 [
VII. Comparison of Three Estimating Methods
On the following pages gross comparisons of the threeestimating methods attempted are made. In general, the standardmethod of maximum likelihood is best, although there are times,as shown in Figure 7-1, when the maximum likelihood computerprogram does not give answers while the method described inSection VI does.
Comparison of Methods Used to Estimate theAverage and Standard Deviation
Initial Guesses (In Standard Deviations)
0 to +2Maximum Least SquaresLikelihood Linear Average
and Least SquaresNormal Std, Dev. a
Averages
Average -.05 .09Standard Deviation .61 .51Number of Valuesused from printout 467 865Number of extremevalues not used 21 33
Standard Deviations
Average .77 .57
Standard Deviation .85 .57
Number of Valuesused from printout 463 858
Number of extremevalues not used 25 40
Standard Deviation/Average 1.10 1.00
Total Simulations Attempted 1000 1000
Number of simulationsnot used due to
No overlap 68 81Iterative process failed 444 21
Fig. 7-1
Comparison of Methods Used to Estimate theAverage and Standard Deviation
Initial Guesses (In Standard Deviations)
-1 to 1-2Maximum Least SquaresLikelihood Linear Average
and Least SquaresNormal Std. Dev.
Averages
Average -.03 .06Standard Deviation .38 .39Number of Valuesused from printout 849 933
Number of extremevalues not used 1 6
Standard Deviations
Average .61 .62Standard Deviation .49 .53Number of Valuesused from printout 844 930Number of extremevalues not used 6 9
Standard Deviation/Average .80 .85
Total Simulations Attempted 1000 1000Number of simulationsnot used due to
No overlap 53 60Iterative process failed 97 1
Fig. 7-2
Comparison of Methods Used to Estimate theAverage and Standard Deviation
Initial Guesses (In Standard Deviations)
-2 to +2
Maximum Least SquaAes UnconstrainedLikelihood Linear AverAge Least Squares
and Least Square Normal Distri-Normal Std. Dev. bution
Averages
Average -.03 .01 .21Standard Deviation .39 .40 .65
Number of valuesused from printout 886 935 46
Number of extremevalues not used 0 0 0
Standard Deviations
Average .71 .72 .81Standard Deviation .51 .55 .98Number of valuesused from printout 884 933 45Number of extremevalues not used 2 2 1
Standard Deviation/Average .72 .76 1.21
Total Simulations Attempted 1000 1000 50
Number of simulationsnot used due to
No overlap 65 65 2
Iterative process failed 49 0 2
Fig. 7-3
Comparison of Methods Used to Estimate theAverage and Standard Deviation H':
Initial Guesses (In Standard Deviations)
-5 to +5Maximum Least Squares UnconstrainedLikelihood Linear Average Least Squares
and Least Squares Normal Distri-Normal Std. Dev. bution
Averages
Average -.01 .02 .07Standard Deviation .40 .42 77Number of valuesused from printout 8& 088 41Number of extreme i Ivalues not used 00 0
Standard Deviations
Average .77 .84 .49 IStandard Deviation .42 .51 .64
Number of valuesused from printout 8M 888 41
Number of extreme Avalues not used 0 0 0
Standard Deviation/Average .55 .61 1.31
Total Simulations Attempted 1000 1000 50Number of simulationsnot used due toNo overlap 135 112 5Iterative process failed 36 0 4
Fig. 7-4
I
VIII. Applications of Derived Statistics
Shown below are two examples using the derived statistics.
1. Given that an ammunition item has a true average pene-tration velocity of 800 and a standard deviation of 60, what isthe best estimate of the lowest sample average that can be obtainedwith 90% confidence. It is assumed the upper and lower gates are680 and 920.
Convert the upper and lower limit to standard deviations bytaking the differences from the mean and dividing by the standarddeviation. This gives values of -2.0 and 2.0. Look up the graph Ion Fig. 2-16. The 90% confidence level will be indicated by thatpoint on the graph which has 10 percent or less chance of occurrence.This value in standard deviations is roughly -.5. Multiplying this iinumber by the standard deivation and summing with the average gives
the solution as 770.
2. Given the same problem as above but assuming a 10% randommalfunction rate. Look up graph on Fig. 4-6. The value in standarddeviations is roughly .45, giving a solution of 773. A highervelocity which will yield a lower effective range.
A
8
APPENDIX A
PROCEDURES FOR DETERMINING THE AVERAGE (V5 0 )
1. Establish a lower and upper velocity such that:
a. The probability of obtaining a complete penetration atthe higher velocity is highly likely.
b. The probability of obtaining a complete penetration atth( lower velocity is highly unlikely.
2. Fire the first round at a velocity midway between thesetwo limits.
3. If the first round results in aNaT penetration, fire thesecond round halfway between the first round velocity and theupper limit velocity; otherwise, halfway between the first andlower limit velocity.
4. If the first two rounds result in a reversal (one partial,one complete), fire the third round midway between the velocityof the first two rounds. If the first two rounds result in twopartials, fire the third round at a velocity midway between thesecond round velocity and the Upper limit velocity. If thefirst two rounds result in two completes, fire the third roundmidway between the second round velocity and the A Mlimitvelocity.
5. Succeeding rounds are fired using the following rules:
a. If the preceding pair of rounds resulted in a reversal(one partial, one complete), fire at a velocity midway betweenthem.
b. If the last two rounds did not produce a reversal,look at the last four rounds. If the number of completes andpartials are equal, fire the next round midway between thevelocity of the first and last round of the group. If the lastfour did not produce equal numbers of partials and completes,look at the last six, eight, etc., until the number of partialsand completes is equal. Always fire at a velocity midwaybetween the first and last round of the group you examined.
c. In the event that conditions in b above cannot besatisfied, if the last round resulted in a complete, fire thenext round at a velocity midway between the last round and the40*r velocity limit; otherwise (last round is a partial) midwaTbetween the velocity of the last round and the Ipper limit.
Al
d. Proceed as in a and b above.
e. Firing will terminate when 12 rounds have been fired.
f. In the event that the firing does not produce a zone ofmixed results, the average will be computed by averaging thehighest complete and lowest partial.
g. In the event that the firing produces a zone of mixedresults, computation of the average will be performed by usingthe cumulative normal and the principle of maximum likelihood.
A2
SUPPLEMENTARY
INFORMATION
I REPORT NO. QAR-O-OIi
COMPUTER SIMULATION OF MAXIMUM
'LIKELIHOOD OUTPUT FOR STANDARD
, LANGLiE TEST
K. . DONALD C. RAPPAPORT
DECEMBER 1977
ARTILLERY SYST.IS DIVISIONPRODU, T StJANCE DIRECTORATE \/
US AR,,,Y ARlM,1AMENT rz :'HAND DEVELOPMENT C 0'A'DOVE-S, N W JERSEY
PROCEDURES FOR DETERMINING THE AVERAGE (V5 0 )
1. Establish a lower and upper velocity such that:
a. The probability of obtaining a complete penetration atthe higher velocity is highly likely.
b. The probability of obtaining a complete-penetration atthe lower velocity is highly unlikely.
2. Fire the first round at a velocity midway between thesetwo limits.
3. If the first round results in a# penetration, fire the
second round halfway between the first iound velocity and theupper limit velocity; otherwise, halfway between the first andlower limit velocity.
4. If the first two rounds result in a reversal (one partial,
one complete), fire the third round midway between the velocityof the first two rounds. If the first two rounds result in twopartials, fire the third round at a velocity midway between thesecond round velocity and the Upper limit velocity. If thefirst two rounds result in two completes, fire the third roundmidway between the second round velocity and the kVlimitvelocity.
5. Succeeding rounds are fired using the'following rules:
a. If the preceding pair of rounds resulted in a reversal(one partial, one complete), fire at a velocity midway betweenthem.
b. If the last two rounds did not produce a reversal,look at the last four rounds. If the number of completes andpartials are equal, fire the next round midway between thevelocity of the first and last round of the group. If the lastfour did not produce equal numbers of partials and completes,look at the last six, eight, etc.,*until the number of partials
and completes is equal. Always fire at a velocity midwaybetween the first and last round of the group you examined.
c. In the event that conditions in b above cannot besatisfied, if the last round resulted in a complete, fire thenext round at a velocity midway between the last round and the40or velocity limit; otherwise (last round is a partial) midwaybetween the velocity of the last round and the jPper limit.
AlAII
.4/
fir, - ,. "
172 /7 f, _
Elp01%
( .TAT,, '; o
DEM4BER 1977
Selected Surmary of Computer Printouts
Initial Estimates (In standard deviations)-5 -2 -2 -! 0 -0.5to to to to to to+5 +3 - 2 +2 +2 +0.5
Averages
Average -.01 -.01 -.03 -.03 -.05 -.04Standard Deviation .40 .39 .39 .38 .61 .42Number of values .
used from printout 827 849 467 757Number of extremevalues not used 0 0 0 1 21 4
Standard Deviations
es Average .77 .69 .71 .61 .77 .39Standard Deviation .42 .38 .51 .49 .85 .47
Number of values .
used from printout 825 884 844, 463 753
Number of extremevalues not used 0 2 2 6 25 8
StandardDeviation/Average .55 .70 .72 .80 1.10 1.21
Total SimulationsAttempted 1000 1000 1000 £000 1000 1000
Number of simulationsnot used due to
No overlap 135 95 65 53 68 52Iterative processfailed 36 78 49 97 444 187
Fig. ,2-1
K
Comparison of 'ethods Used co Estimate the
Average and Standard Deviation
Initial Guesses (in Standard Deviatiors)
-5 to +5-aximun Least SGuares Uncons trains-dLikelihood Linear Average Least Scr"--,s
and Least Squares Normal Distri-Normal Std, Dev. bucion
Averages
Average -. 0! .02 .07
Standard Deviation .40 .42 .11Number of values Ssused from printout 9 888 41
Number of extremevalues not used 0 0
Standard Deviations
Average .77 .84 .49
Standard Deviation .42 .51 .64Number of values 219_used from printout 888 41
Nuber of extremevalues not used 0 0 0
- Standard Deviation/Average .55 .61 1.31
Total Simulations Attempted 1000 1000 50
Number of simulationsnot used due to
No overlap 135 112 5Iterative process failed 36 0 4
Fig. 7-4
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