A Brief Study about Nonparametric Adherence Tests

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1 A Brief Study about Noparametric Adherece Tests Viicius R. Domigues, Lua C. S. M. Ozelim Abstract The statistical study has become idispesable for various fields of kowledge. Not ay differet, i Geotechics the study of probabilistic ad statistical methods has gaied power cosiderig its use i characterizig the ucertaities iheret i soil properties. Oe of the situatios where egieers are costatly faced is the defiitio of a probability distributio that represets sigificatly the sampled data. To be able to discard bad distributios, goodess-of-fit tests are ecessary. I this paper, three o-parametric goodess-of-fit tests are applied to a data set computatioally geerated to test the goodess-of-fit of them to a series of kow distributios. It is show that the use of ormal distributio does ot always provide satisfactory results regardig physical ad behavioral represetatio of the modeled parameters. Keywords Smirov, Aderso-Darlig, Cramer- Vo-Mises, Noparametric adherece tests. I. INTRODUCTION S i most braches of sciece, the use of probabilistic Aad statistical methods has become extremely importat i the developmet of moder Geotechics. I particular, the cocept of reliability of a veture has attracted cosiderable attetio i recet years, boostig therefore the study of statistics for egieers. It is kow that oe of the key poits of the correct modelig of geotechical data is the choice of a probability distributio represetig the behavior of the data aalysis. I geeral, the ormal distributio has bee the default choice of the vast majority of egieers. This ca be explaied by the wide applicability of this distributio; however, there are cases where the use of such radom variable does ot preserve the meaig ad the physical behavior of the variables uder cosideratio. The arises the questio o which distributio best fits a give sample of the target populatio. To provide aswers to this questio, adherece tests are used. II. ADHERENCE TESTS Adherece tests are statistical tests used to measure how well a give probability distributio is able to model the data set beig aalyzed. Adherece tests are also commoly called goodess of fit tests [1]. I geeral this type of testig is based o a hypothesis test, i which the ull hypothesis, H0, is that data cosidered follow a give distributio test, while the alterative hypothesis, H1, cosiders that the data do ot follow that distributio. There are two large groups of adherece tests with respect to prior kowledge of the Viicius R. Domigues ad Lua C. S. M. Ozelim are with the Departmet of Civil ad Evirometal Egieerig, Uiversity of Brasilia, Brasilia , Brazil ( viicius.rdomigues@gmail.com, luaoz@gmail.com). parameters ad distributio of data, amely parametric ad o-parametric adherece tests. A. Parametric Adherece Tests Parametric adherece tests are those i which the distributio of the studied populatio is kow or selected i some way ad ot ito questio, so that the hypotheses to be tested oly ivolve populatio parameters. For example, i the case of a ANOVA, the rigid assumptios for the variables uder compariso (ormal variables, for example) icur i this type of testig ot providig aswers o how well other distributios fit to the data. B. Noparametric Adherece Tests Noparametric adherece tests, o the other had, are valid tests for a broad rage of distributios, so that their applicatio provides a evaluatio of the hypothesis of a give radom variable beig distributed i a distributio differet from Normal. Still, this type of adherece test provides the meas to as-certai the distributio that best fits the data aalysis. I this article, it will be explored the use of three kow adherece tests i the evaluatio of the probability distributio that best fits a set of data geerated computatioally. The three tests to be explored are: Smirov test; Aderso- Darlig test ad test Cramer-Vo-Mises. III. CONSIDERED ADHERENCE TESTS The three adherece tests cosidered belog to the class of tests that uses the empirical distributio fuctio (EDF) i the calculatio of their statistics. Thus, it is ecessary to first defie this cocept. A. Empirical Fuctio Cosider the ordered data set {x1, x, x3,..., xn}, whose empirical cumulative distributio fuctio, F(X), oe wats to calculate. Mathematically, the EDF may be give by: N 1 1sig x xi F ( x) N i 1 (1) where x idicates the ceilig mathematical fuctio that provides the greatest iteger less tha x ad sig (x) is the sig fuctio, i which the result is +1 if x is positive ad -1 otherwise. I other words, the EDF a variable x is rages from 0 to 1 ad is icreased by 1 / N whe X passes each value i the ordered set of data. The EDF beig de-fied, the adherece tests of iterest ca be studied. B. Smirov Test The Smirov belogs to the highest class of EDF based statistics, give the fact that it works with the 685

2 biggest differece betwee the empirical distributio F(x) ad the hypothetical F(x). The hypothetical cumulative distributio comes from the test distributio, over which the ull hypothesis of the hypothesis test resides. Thus, i the case of the Smirov test, the ull hypothesis is that the data are distributed accordig F (x) ad the alterative hypothesis is that the data do ot follow such distributio Mathematically, the KS statistics of the Smirov test may be defied as: KS sup F ( x) F( x) () x I order to be able to accept or reject the ull hypothesis, the value of KS ad its distributio should be assessed. Note that, ituitively, if the distributio patter approximates the empirical distributio, the value of KS should be small. O the other had, whe KS is big, it is a idicative that the test does ot characterize the distributio ad the variable of iterest. Put i aother way, H0 ca be rejected if the statistic value KS is greater tha or equal to a give limit value, KSmax, the last of which deped o the cofidece level adopted, α. Aother importat cocept that arises from this evaluatio process is called the p-value, which gives the probability of obtaiig a statistic at least as extreme as the oe calculated, assumig that the ull hypothesis is true. Thus, oe ca accept H0 if the p-value associated is greater tha the sigificace level. I the preset paper, the software Mathematica will be used to evaluate the statistics ad p-values of the three tests cosidered. C. Cramer-Vo-Mises Test Ulike the Smirov test, Cramer-Vo Mises test is a quadratic test of the empirical distributio fuctio. This desigatio stems from the fact that this test works with the squared differeces betwee the empirical ad the hypothetical distributios []. This test has bee applied to study a wide variety of problems i sciece. I [3], a Cramer vo Mises type test based o local time of switchig diffusio process has bee studied. O the other had, i [4] a compariso betwee the Cramer-vo Mises test ad adaptive tests has bee performed. I the preset paper, o the other had, we restrict our attetio to the traditioal test. Mathematically, the CM statistics of the Cramèr-Vo-Mises test may be defied as: CM N F ( ) ( ) ( ) x F x df x (3) By kowig the data ad the target distributio, the use of (3) becomes ready. As i the case of the Smirov test, the Cramer-Vo-Mises test provides results which cofirm whether or ot H0, accordig to the statistical distributio of the CM or accordig to the p-value. D. Aderso-Darlig Test As the Cramer-Vo-Mises test, Aderso-Darlig test is a quadratic test of the empirical distributio fuctio. Furthermore, ulike what happes i (3), a weight is give to each observatio iside the itegral [5]. This test has bee modified ad applied to several distributios, such as powerlaw types [6] ad extreme-value distributios [7]. O the other had, as the case of the CM test, we restrict our aalysis to the classical Aderso-Darlig test. Mathematically, the AD statistics of the Aderso-Darlig test may be defied as: AD N F ( ) ( ) 1 ( ) x F x F x F x df x (4) Iterestigly, the mai differece betwee Ader-so- Darlig ad Cramèr-Vo-Mises tests is that the first gives greater weight to the data comig from the tails of the distributios. Similarly, to the other tests already metioed, the hypothesis H0 should be rejected by compariso with extreme values. Beig defied the adherece tests to be used; the ext step is to characterize a process of geeratig radom data through the software Mathematica ad subsequet applicatio of the adherece tests. IV. RANDOMIZES SAMPLES GENERATION AND ADHERENCE TESTS APPLICATION I the preset paper, three radomized samples of 10 4 elemets will be cosidered. I order to evaluate the applicability of the adherece tests, each oe of the radomized samples will display a tedecy that ca be foud i the practice of geotechical egieerig. The cosidered tedecies are: variables distributed accordig to a ormal distributio; variables with asymmetric distributio ad symmetric variable with log tail. A. Normal Sample (D1) Accordig to a ormal distributio, the Radom Variate fuctio of the software Mathematica was used for the geeratio of the distributed sample. The histogram of the respective geerated data ca be foud i Fig. 1 The histogram is show i Fig. 1 was geerated from a ormal distributio with a mea of ad stadard deviatio of 3. Mathematically, oe ca defie the probability desity fuctio of a ormal variable with mea μ ad stadard deviatio σ by meas of the followig equatio: ( x ) e f( x) (5) 686

3 Fig. 1 Radom sample from a ormal distributio with 10 4 elemets B. Asymmetric Sample (D) I order to geerate the sample that follows a skewed distributio the RadomVariate fuctio of the software Mathematica was also used. It was cosidered that the data follow a Levy distributio for the geeratio process. The histogram of the respective geerated data ca be foud i Fig.. I this case the Levy distributio with locatio parameter 4 ad dispersio parameter was used as source. Fig. Radom sample from a Levy distributio with 104 elemets Mathematically, oe ca defie the probability desity fuctio of a Levy variable with locatio parameter μ ad dispersio parameter σ from (6): e f ( x) ( x ) x C. Sample with Log Tail (D3) Fially, for the geeratio of the sample distributed accordig to a distributio with log tail the RadomVariate fuctio of the software Mathematica was used agai. The histogram of the data geerated ca be foud i Fig. 3. To geerate the data preseted i Fig. 3, a studet s t distributio with locatio parameter 0, scale parameter 3 ad 4 degrees of freedom was used. 3/, x (6) Fig. 3 Radom sample of a Studet's t distributio with 104 elemets Mathematically, it is possible to defie the probability desity fuctio of a t-studet variable with locatio parameter μ, scale parameter σ ad ν degrees of freedom, through (7): ( x ) f ( x) 1 Beta, where Beta [x, y] is the special fuctio defied by the followig itegral: As the samples ad its source distributios are set, it is possible to proceed to the dataa aalysis ad applicatio of the adherece tests. D. Basic Data Aalysis Based o the geerated data it is possible to calculate the average, stadardd deviatio ad skewess for each oe of the cosidered sets. Table I cotais those iformatio for each dataa group. Data D1 D D3 Beta x, y Average TABLE I BASIC DATA ANALYSIS Stadard deviatio As expected, the D1 ad D3 data are practically symmetric, while the D data is extremely asymmetrical. Still, accordig to the distributios used to geerate the data, averages ad stadard deviatios are compatible. After the basic data aalysis, the ext step is the determiatio of which distributios are goig to be ested accordig to the goodess-of-fit. 1 0 t (1 t) x1 y1 dt Skewess (7) (8) 687

4 E. s to Be Tested As previously argued, to use the adherece tests the test distributios must be determied. For simplicity, the classes of test distributios will be the same used for the geeratio of data that is the Normal, Levy ad Studet s t distributios. Each of the data will be tested agaist three distributios whose parameters were previously determied through the maximum likelihood estimatio provided i Mathematica software. The parameters for each distributio to be tested are show i Table II. TABLE II PARAMETERS AND DISTRIBUTIONS TO BE TESTED Data Normal Levy T-studet (μ, σ) (μ, σ) (μ, σ, ν) D1 (, 3) (-9.9, 10.8) (, 3, 41) D (6037, ) (4, ) (10 11, 10 11, 7368) D3 (1, 4.) (-9, 9) (1, 3, 4) F. Adherece Tests By applyig the adherece tests discussed i this article o the geerated data ad usig as test distributios those preseted i Table II, Tables III-V ca be geerated. TABLE III GOODNESS-OF-FIT TESTS RESULTS (P-VALUES) FOR THE D1 DATA Smirov Cramèr-Vo-Mises Aderso-Darlig Normal Levy T-studet It is worth otig that i cases where the D1 data set is cosidered, adherece tests clearly show that oly the Normal ad Studet s t distributios ca represet the sample. It is further oted that the fact that both distributios metioed fit well to the data follows from the cosideratio that the Normal distributio is the limitig case of t distributio whe it has a large umber of degrees of freedom. Thus, it ca be oted that adherece tests used are accurate o the characterizatio of the distributio process from which the D1 data set belogs. TABLE IV GOODNESS-OF-FIT TESTS RESULTS (P-VALUES) FOR THE D DATA Smirov Cramèr-Vo-Mises Aderso-Darlig Normal Levy T-studet By lookig at Table IV, it is easy to perceive that all tests suggest the Levy distributio as the best distributio of the data. I fact, the p-value is 0 for the other distributios, implyig the rejectio of H0 for cases of Normal ad Studet s t distributios. Therefore, all the tests are effective i characterizig the D set of data. TABLE V GOODNESS-OF-FIT TESTS RESULTS (P-VALUES) FOR THE D DATA Smirov Cramèr-Vo-Mises Aderso-Darlig Normal Levy T-studet It is easy to otice from the Table V that the adherece tests idicate that the D3 data are distributed accordig to a Studet s t distributio. This fact shows, oce agai, that the cosidered goodess-of-fit tests are effective as regards the determiatio of the probability distributio of a data set. V. CONCLUSIONS The role of Statistics i exact scieces has bee more cosidered with passig of time. Especially i the geotechical egieerig field, the iheret variability of soil parameters fids a great ally i this area of the kowledge. Oe of the most commo situatios whe a egieer is aalyzig a set of data is the proper choice of probability distributio that represets it. This choice take must be as reliable as it ca possibly be, so that the physical behavior of the modeled variable is ot lost. The adherece tests come as powerful allies to help determie which distributio is better adjusted to the database. Therefore, it is imperative that there is complete kowledge of them for a successful statistic modelig process. Oe of the aims of this paper was a discussio about some the features of three of the most kow o-parametric adherece testes, such as: Smirov test, Cramèr- Vo-Mises test ad Aderso-Darlig test. From computatioal experimets, it was show how these tests ca be used to determie the probability distributio that adapts better to the data i aalysis. This way, it is believed that a cotributio to the diffusio of traditioal statistics tools i the field of geotechical egieerig has bee doe. ACKNOWLEDGMENTS The authors would like to thak FAP-DF for the fiacial support, provided as grats (umber /015) to preset the paper at the coferece. REFERENCES [1] Torma, V. B. L.; Coster, R.; Riboldi, J. 01. Normalidade de variáveis: métodos de verifcação e comparação de algus testes ão paramétricos por simulação, Rev. HCPA, Porto Alegre, v.3,., p [] Aderso, T. W. ad Darlig, D. A A Test of Goodess-of-Fit. Joural of the America Statistical Associatio, 49: [3] Gassem, A O Cramér vo Mises type test based o local time of switchig diffusio process. Joural of Statistical Plaig ad Iferece, Vol 141(4), P [4] Iglot, T. ad Ledwia, T O cosistet miimax distiguishability ad itermediate efficiecy of Cramér vo Mises test. Joural of Statistical Plaig ad Iferece, Vol. 14 (), P [5] D'Agostio, R. B. ad Stephes, M. A Goodess-of-Fit Techiques. New York: Marcel Dekker. ISBN , [6] Coroel-Brizio, H. F. ad Herádez-Motoya, A. R The Aderso Darlig test of fit for the power-law distributio from left- 688

5 cesored samples. Physica A: Statistical Mechaics ad its Applicatios, Vol. 389 (17), P [7] Heo, J-H., Shi, H., Nam, W., Om, J., Jeog, C Approximatio of modified Aderso Darlig test statistics for extreme value distributios with ukow shape parameter. Joural of Hydrology, Vol. 499 (30), P