AD-A ON ONTAINING PHYSICAL MEANIHG FROM THE PARAMETERS IN / THE WEIBAL DISTN BU.A U AUNIVERSIT OF CENTNAL FLONIDA ONLANDO DEPT OF MATHEMATICS

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1 , AD-A ON ONTAINING PHYSICAL MEANIHG FROM THE PARAMETERS IN / THE WEIBAL DISTN BU.A U AUNIVERSIT OF CENTNAL FLONIDA ONLANDO DEPT OF MATHEMATICS AND. S JBEAN ET AL 4 841

2 I L25. I MICROCOPY RESOLUTION TEST CHART NATIONAL BUJREAU Of STANDAR1DS- i%-a

3 AFGL-TR ON OBTAINING PHYSICAL MEANING FROM THE PARAMETERS IN THE WEIBULL DISTRIBUTION S. J. Bean P. N. Somerville 00 University of Central Florida Department of Mathematics and Statistics Orlando, Florida Scientific Report No May 1983 Approved for public release; distribution unlimited SELECTE DTIC r MAR AIR FORCE GEOPHYSICS LABORATORY AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE HANSCOM AFB, MASSACHUSETTS A IC FILE COPY *( *. "-- _ *

4 This report has been reviewed by the ESD Public Affairs Office (PA) and is releasable to the National Technical Information Service (NTIS). This technical report has been reviewed and is approved for publication CIARLES F. Contract Monitor DONALD D. GRANTHAM Chief, Tropospheric Structure Branch FOR THE COMMANDER RO Rf A. MCCL/ T CHEY idirector, Meteo.rology Division V Qualiffed requestors may obtain additional copies from the Defense Technical Information Center. All others should apply to the National Technical Information Service. If your address has changed, or if you wish to be removed from the mailing list, or if the addressee is no longer employed by your organitstion, please notify AFGLIDAA, Hanscom AFB, MA This will assist us in maintaining a current mailing list. Do not return copies of this report unless contractual obligations or notices on a specific document requires that it be returned.

5 Unclassified SIECURITY CLASSIFICATION OF THS5 PAGE filssnda.. e e REPORT DOCUMENTATION PAGE 1IFF~F, CMNS'HU NGFlORM I. REPORT "UMBER jgovt ACC ESSION NO. 11. OECiPI1ENT*S CATALOG NUMBER AFGL-TR JA /; s7f 4. TOT LE (And Swbfft) y57 5PP on Obtaining Physical Meaning From thescetfcrprno4 Parameters in the Welbull DistributionScetfcRprNo4 4- PCRFORAON~G ORG. RIEPORT NUMBER. TYPE or REPORT & PERIOD COVERED 7. AUTHOR(.) 8. CONTRACT OR GRANT NUMBDER(s) S. J. Bean P. N. Somerville F K-OOO1 S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKC AREA 6 WORK UNIT NUMBERS University of Central Florida 62101F Department of Mathematics and Statistics AK Orlando, Florida It. CONTROLLING OFFICE NAMIE AND ADDRESS 12. REPORT DATE Air Force Geophysics Laboratory 13 May 1983 Hanscom AFB Massachusetts NUMBER OF PAGES Monitor/Charles F.-Burger/LYT MONITORING AGENCY NAME 6 AODRESSiI ifferent fromn Controlling Ollie*) IS. SECURITY CLASS. e r Iport) Unclassified IS&. DECO.ASSIFIC ATIONI DOWNGRADING SCHEDULE IS. DISTRIBUTION STATEMENT (of this Report) Approved for public release; Distribution unlimited I?. DIST RIOUTION STATEMENT (ofia Ch oet.ce entered in Block 20. it different help Report) IS. SUPPLEMENTARY NOTES IS. KEY WORDS (Coftlinu. on rosrofed aide It R*coe*917 anidentify by Wleek number) Weibull Distribution Standard deviation mean Approximate Prediction Intervals median4j t-,ht 10 ABSTRACT (ContIass. an reverse II if assesseser a"u Identify by Slekz / 7t The parametrw of the Weibull dist lut n do ~t lend themselves to p;hysical interpretation. This report illuitdtate how t0lie mean, median, mode, and other measures may be obtained from (4,0). ables are given which display the mean, median, mode, and standard deviati n as well as the cumulative probaility of the distribution for one standa deviation above and below the mean for a number of different combinations of and CavOITIO Or anv s is osolere Unclas if lied SECURITY U $,ft I^'" rsr~ TIS V.AI.Or not.. Fe neorow

6 On Obtaining Physical Meaning From the Parameters. in the Weibull Distribution by,. S. J. Bean and P. N. Somerville :,, University of Central Florida 1.0 Introduction The Weibull distribution has been used in a number of applications. In climatological studies Somerville and Bean (1981, 1979) have found the Weibull distribution to fit visibility and windspeed well. The cumulative distribution function of the two parameter Weibull used in these studies is given by F(x) 1-e. (1) Other equivalent forms may be found in Johnson and Kotz (1970). The parameters (*,B) do not lend themselves directly to interpretation 2 such as mean () and variance (a ) in the normal distribution. However, with a few calculations (e,6) can be used to obtain descriptive quantities for location and variability. -ax Measures of Location and Variability The median, mode, percentiles, and interquartlle range are directly available with only a few calculations. These quantities are found using the following formulas: Median [In(2)/a] /a (2) Mode - ((B -1)/r1Ji / $, B ' 1 (3) x p (-1n(1-p)/] l " (4) where xp is the 1 00.pth percentile. This expression for xp is found by setting Ii

7 -2- p-i - exp(-axp 8 ) (5) and solving for xp. The median is found by letting p -.5. The mode may be found by taking the derivative of the frequency distribution and setting it equal zero and solving for x. The interquartile range may be used as a measure of variability. The interquartile range is the difference between the third and first quartiles or equivalently the 75th and 25th percentiles. That Is Interquartile Range = x - x (6) The percentiles may be used for obtaining prediction intervals. example, 90% prediction Intervals for a future observation of the variable x are given by (Xo 5, x 9 5 ) or (Xo I, x 91 ). There are numerous other similar 90% prediction intervals. Each interval has the property that with approximate probability.9, the future observation will be contained within it. The moments of the Weibull distribution are not as easily obtainable as the median, mode, and percentiles. The moments are defined in terms of gamma functions. The gamma function is defined as r(p) -" jx p '1 e- x dx, p 1 0 (7) The mean (expected value) and variance of a Weibull variable x are given by For EEx] - r(1 + 1/)/a / (8) and Var[x] = r(1 + 2/B) - (r(i + 1/o)) 2 3/ 2 /. (9) The mean, median, mode, and standard deviation for various combinations of (a,o) are given in Tables 2.1, 2.2, 2.3 and 2.4 respectively. For the normal distribution, the cumulative probability distribution of one standard deviation, a, below the mean, P, is always F(P-o) - P(X s P-) Also the cumulative probability distribution of one standard deviation above the mean Is always F(.+o) - P (X s V+o) Hence the probability of having any normal random variable within one standard deviation of its mean

8 -3- is always F(va+a) - F(u-0) Unfortunately this is not true of the Weibull distribution. Since the mean and standard deviation can be obtained with a computer for the Weibull and since its cumulative distribution F(x) " C X is easily calculated, the comparable measures for the Weibull (i.e., F(i-o), F(p+o) and F(,+o) - F(u-o)) can be easily computed. Table 2.5 gives these values for a number of different combinations of a and An Example Visibility has been modeled for Fulda AAF, West Germany by Somerville and Bean (1981). At Fulda, for March h local time.094, 0 a This yields the following mode ((.247)/(.094)(1.247))1/1.247 * 1.82 miles median = x 50 - (-ln(.5)f.094) 1/ = 4.96 miles mean * r(1 + 1/1.247)/(.094) 1 / miles At the same location for June h.0027, 0 = In this case we have the following estimates Mode miles Median miles Mean - r(1.4)/(.0027) 1 / miles

9 -4- TABLE 2.1 Mean for Weibull Distribution for Various Values of and S x xI TABLE 2.2 Median for Weibull Distribution for Various Values of a and 0 B x

10 I5 TABLE 2.3 Mode for Weibull Distribution for Various Values of and o a (NOTE: There is no mode when ) TABLE 2.4 Standard Deviation for Welbull Distribution for Various Values of and B B x

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14 9-4.0 References Somerville, P. N. and S. J. Bean, "Some Models for Visibility for German Stations," AFGL-TR , 15 April AD A Somerville, P. N. and S. J. Bean, "Stochastic Modeling of Climatic Probabilities," AFGL-TR , November AD A080i559. Johnson, N. and S. Kotz, Continuous Univariate Distributions, Houghton Mifflin Company, Boston, Massachusetts, 1970.