Introducing Kolmogorov-Smirnov Tests under Uncertainty: An Application to Radioactive Data

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by  Muhammad Aslam

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−
Introducing Kolmogorov Smirnov Tests under Uncertainty: An
Application to Radioactive Data
Muhammad Aslam*
Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia
ABSTRACT: The Kolmogorov−Smirnov (K−S) tests based on
the assumptions of determined observations in the sample have
been popularly applied for the analysis of the data. The existing
s. K−S tests for one sample and two samples cannot be applied
e
cl whenthedatacontainsneutrosophicobservationsmeasuredfrom
arti the complex system or under uncertainty. In this paper, we
hed propose the generalization of the existing K−S tests under the
s
bli neutrosophic statistics. The proposed tests are known as
UTC).are pu ntheeutnroecsoespshaircyKmoelamsuorgeosroanv−dSpmroircneodvure(sNtKo−pSe)rfotermsts.thWeeprporpeosseendt
18:32 (ately sh tgeivstesn.Ainntehxeampappleera.ndadvantagesoftheproposedNK−Stestsare
7:m
21 at 1o legiti
April 19, 20ns on how t 1Tin.heaINlsltTafiRtiesOltdiDcsaUlfoCmrTeIttOhhoeNdsa/nteaclyhsnisiquoefsthhaevedbaetae,necsotmimmatoinolny,uasnedd teinhxdeteentneseriomuntirnooasfcocypl.ahTsischiceallongsetiuactt.riostTsiochspe,hwicnhesicutahtrtiocsostoincpssihdiisecraspsptthaliteeisdmtiwcesahseuinsrethaonef
9 on optio forecasting purposes. The data obtained from the system oalb.2s5e,r2v6atidoinscsuisnsetdhetdhaetaaadrveannteaugtersosoopfhimcentuhmodbsersb.aCsehdenoent
18or always follows some statistical distribution, which is unknown neutrosophic numbers. Previous work27,28 introduced several
117.64.elines f innoramdavlandcies.trUibsuutaiollny., iHtoisweavsseur,miendpthraacttitchee, idtatias fnoolltowalswtahyes bparesviciocuosncweoprtks2f9orprthoeponseeudtrtohseopnheiuctrsotastoispthicics.ARNecOenVtAly,taensto.ther
79.uid necessarythatthedatainhandfollowsthenormaldistribution. The existing K−S test under classical statistics and a fuzzy
via ngg Therefore, statisticians designed several tests to test some approach cannot be applied when the measure of indetermi-
oaded g/shari hinyvpeosttihgeatsieosn;asbeoeurteft1h.eAsdmisternibtiuontieodnbyofMtahsseey,d1a“tAattuemndpetsr nanacdythisenfeuezdzeyda.pBpyroeaxcphlo,rwinegdtihdenloitterfiantudreanoynwcolarsksiocanltshtaetiKst−icSs
Downls.acs.or dhaisvteribubteieonn dmoaedsenottodefipnedndtuepstonsteaittihsetircsthewehxopsleicitsafmorpmlinogf tneesutturonsdoeprhtihceKnoelumtroogsoorpohvi−cSstmatiirsntiocvs.(INnKth−isSp)atpeesrts,wfoerparosipnogslee
ub or the value of certain parameters in, the distribution of the sampleandtwosamples.ItisexpectedthattheproposedNK−
p
https:// pdoisptruiblautitoionn.-fSreuechtesttess.t”sThhaevKeoblmeeongocraollve−dSmnoirnn-opvar(aKm−etSr)icteosrt SuncteersttasinwidllataeffeccotmivpealyredantaolyztehetheexisitminpgrecKis−eS, vtaegsute,unadnedr
See idsisatnribalutetironnatitvoedneocni-dpearaabmoeuttricthteests,pwechiificchudsisetsritbhuetciounmuolfattihvee classical statistics.
data. The K−S test is found to be efficient for goodness of fit 2. RESULTS
purposes. Many authors worked on the K−S test; see, for
example, refs 1−8. For a radioactive source model, radioactive engineering is
The K−S test under classical statistics is applied when all interested in testing the assumption that the count rate per
secondfollowsaneutrosophicPoissondistribution.Thecount
observations in the data are determined, precise, and sure.
rate per second has the neutrosophic mean [5,7] counts per
However,inrealsituations,itmayhappenthatthedatacannot
second. To test the assumption, radioactive engineering
be represented by statistical terms or the data may be in an
collected a large number of count data. The neutrosophic
interval or imprecise data. For example, the ecology data, soil
Poisson distribution from ref 18 is given by
data, ocean data, and censored data may be fuzzy data rather
thanexactdata.Therefore,severalauthorsdevelopedtheK−S exp(−λ )λ xn
test for the analysis of fuzzy data; see, for example, refs 9−16. F0N(xn) = xN! N ; F0N(xnN)ϵ[F0L(xnL), F0N(xnU)]
The fuzzy logic is a special case of neutrosophic logic. The n
neutrosophiclogicisconsideredthemeasureofindeterminacy
in addition to the fuzzy logic; see ref 17. More applications of Received: November18,2019
the neutrosophic logic can be seen in refs 18−23. The Accepted: December18,2019
neutrosophicstatisticswasdevelopedbySmarandache24 using Published: December31,2019
©2019AmericanChemicalSociety 914 DOI:10.1021/acsomega.9b03940
ACSOmega2020,5,914−917
ACS Omega Article
Table 1. Necessary Computations for the NK−S Test
no. X Cu(X ) S (x ) F (x) D
iN iN nN nN 0N n N
1 [1,4] [1,4] [0.0128,0.0435] [0.0404,0.1730] [0.0276,0.1295]
2 [1,4] [2,8] [0.0256,0.0870] [0.0404,0.1730] [0.0148,0.0860]
3 [3,5] [5,13] [0.0641,0.1413] [0.2650,0.3007] [0.2009,0.1594]
4 [3,5] [8,18] [0.1026,0.1957] [0.2650,0.3007] [0.1625,0.1051]
5 [4,5] [12,23] [0.1538,0.2500] [0.4405,0.3007] [0.2866,0.0507]
6 [5,6] [17,29] [0.2179,0.3152] [0.6160,0.4497] [0.3980,0.1345]
7 [5,6] [22,35] [0.2821,0.3804] [0.6160,0.4497] [0.3339,0.0693]
8 [6,6] [28,41] [0.3590,0.4457] [0.7622,0.4497] [0.4032,0.0041]
9 [6,6] [34,47] [0.4359,0.5109] [0.7622,0.4497] [0.3263,0.0612]
10 [6,7] [40,54] [0.5128,0.5870] [0.7622,0.5987] [0.2494,0.0118]
11 [8,8] [48,62] [0.6154,0.6739] [0.9319,0.7291] [0.3165,0.0552]
12 [8,9] [56,71] [0.7179,0.7717] [0.9319,0.8305] [0.2141,0.0588]
13 [10,9] [66,80] [0.8462,0.8696] [0.9863,0.8305] [0.1402,0.0391]
14 [12,12] [78,92] [1.0000,1.0000] [0.9980,0.9730] [0.0020,0.0270]
and S (x ) = Cu(XiN) where Cu(X ) is the neutrosophic the sample belongs to the Poisson distribution. However, the
nN nN [78,92] iN indeterminate part shows that under uncertainty, the sample
commutative values, λNϵ[5,7] and nNϵ[78,92]. does not come from the Poisson distribution. From this
Theneutrosophiccountdataandneutrosophicstatisticsare comparison,weconcludethatthevaluesofthestatisticD can
showninTable1.Supposethelevelofsignificanceforthistest befrom0.1594to0.4032underuncertainty.Hence,thethNeory
is 0.01. The critical neutrosophic value from ref 32 is oftheproposedNK−Stestconcurswiththetheoriesofrefs25
D = [1.63/ 78, 1.63/ 92] = [0.1845,0.1699].Accord- and 26.
0.01,14
ing to eq 4, the statistic in the indeterminacy interval can be
written as 0.1594 + 0.4032I; I ϵ[0,0.6046]. The neutrosophic 4. CONCLUDING REMARKS
N
statistic from Table 1 is DNϵ[0.4032,0.1594]. Note here that In this paper, we presented the modifications of the
the lower value of the indeterminacy interval denotes the Kolmogorov−Smirnov (K−S) test under the neutrosophic
determined part. By comparing the values of DN with D0.01,14, statistics. We proposed the neutrosophic Kolmogorov−
we note that the determinate part follows the Poisson Smirnov (NK−S) tests, which are the generalization of the
distribution, but the indeterminate part of the data does not K−S tests. The proposed NK−S test under the neutrosophic
follow the Poisson distribution. statistical interval method is more adequate, informative, and
effective to be applied when the data have neutrosophic
3. DISCUSSION numbers. The proposed test provides the results in the
indeterminacyinterval,whichisdesirableunderuncertaintyor
In this section, we compare the performance of the proposed
NK−S test over the K−S test under classical statistics. when the data is measured from the complex system. We
presented an example and found that the proposed test is
According to refs 25 and 26, a method that provides the
betterthantheexistingK−Stest.Werecommendapplyingthe
results in the indeterminacy interval when the data have the
proposedNK−Stestsfortheanalysisofthedatainbiomedical
neutrosophic numbers is said to be more adequate and
effective than the method that provides the results in the sciences, big data analysis, engineering, and statistics. More
determined form. To compare the proposed NK−S test with properties using the simulation data and/or the development
ofsoftwarefortheanalysisoftheproposedNK−Stestscanbe
theexistingNKtest,wewillusethesamedatathataregivenin
considered for future research.
Table 1. Note here that the data given in Table 1 reduces to
thedeterminedpartunderclassicalstatisticsifnoobservations
ofuncertaintyarerecorded.Forexample,forsample1,thefirst 5. COMPUTATIONAL METHODS
value, which is 1, represents the indeterminate part of the AssumethatX =a +b I beaneutrosophicnumber(NN)
N N N N
indeterminacy interval. The second value of this sample where a is the determinate part and b I ; I ϵ[I , I ] is the
N N N N L U
represents the determinate part of the interval. From Table 1, indeterminatepartoftheNN.LetX =X+I X;X ϵ[X ,X ]
N N N L U
we note that the proposed test provides the results in the be a random variable based on the NN where X and X are
L U
indeterminacy interval rather than the determined values. lower and upper values of the indeterminacy interval. Note
Usingeq4,thevaluesofthestatisticintheindeterminacyform here that the NN and X ϵ[X , X ] reduce to a number and a
N L U
can be written as 0.1594 + 0.4032I; I ϵ[0,0.6046]. Note here variable under classical statistics if I = 0 or X = X ,
N L L U
that the proposed test provides a good measure of respectively.TheneutrosophicvariableX ϵ[X ,X ]presented
N L U
indeterminacy. At a level of significance 0.01, the probability the NNs in a sample selected from the population having
thatthenullhypothesiswillbeacceptedis0.99,theprobability imprecise, uncertain, and indeterminate values or parameters.
ofrejectingthenullhypothesiswhenitistrue is0.01,andthe More details about neutrosophic statistics can be seen in ref
probability of indeterminacy is 0.60. For example, in the 17. The main aim is to propose the K−S tests under the
statistic D ϵ[0.4032,0.1594], the value D = 0.1594 presents neutrosophicstatisticstodeterminethespecificdistributionof
N L
thedeterminedpartundertheclassicalstatistics,andthevalue the data in the presence of neutrosophy.
D = 0.4032 shows the indeterminate part under the 5.1. Neutrosophic Kolmogorov−Smirnov Tests. The
U
uncertainty. By comparing both tests, we note that D < Kolmogorov−Smirnov (K−S) test was originally derived by
L
0.1845, which shows that the existing NK test indicates that Kolmogorov30 and Smirnov31 and has been used in non-
915 DOI:10.1021/acsomega.9b03940
ACSOmega2020,5,914−917
ACS Omega Article
parametric testing of the hypothesis. In classical statistics, the observations/parameters of two populations should be
K−Stesthasbeencommonlyusedtotestwhetherthesample determined and precise. In this section, we present the NK−
under study belongs to a specific distribution or not. In other S test for comparing two neutrosophic populations. Let X ,
1N
words,theK−Stestisappliedtodecidewhethertheobserved X , ..., X and Y , Y , ..., Y be two neutrosophic
distribution significantly differs from the specified population in2dNependennt1Nsamples1oNfsize2sNn ϵ[nn2N,n ]andn ϵ[n ,n ]
distribution.32 The existing K−S test is applied under the from a specified population,1Nresp1eLctiv1Uely. Let2NS (1Lx 1)Uϵ-
assumption that all observations/parameters in the observed n1N n1N
[S (x ), S (x )] and S (y )ϵ[S (y ), S (y )] be
sampleandinthepopulationaredeterminedandprecise.The nL n1L nU n1U n2N n2N nL n2L nU n2U
data that came from complex systems such as the ocean, the neutrosophic sample cumulative distribution functions. Then,
human brain data, and power grid or under uncertainty may the neutrosophic maximum difference statistic based on
not have all determined observations. In these situations, the S (x )ϵ[S (x ), S (x )] and S (y )ϵ[S (y ),
K−Stestunderclassicalstatisticscannotbeappliedfortesting Sn1N(y n1)N] is gniLvenn1bLy nU n1U n2N n2N nL n2L
whether the data belong to a specific distribution. We modify nU n2U
the existing K−S test under classical statistics using the D = |S (y ) − S (x )|; D ϵ[D , D ]; S (x )
neutrosophicstatistics.TheproposedneutrosophicKolmogor- N n2N n2N n1N n1N N L U n1N n1N
ov−Smirnov (NK−S) test is the generalization of the existing ϵ[S (x ), S (x )], S (y )
K−S test proposed by Kolmogorov30 and Smirnov.31 The nL n1L nU n1U n2N n2N
proposed NK−S test will be applicable under the following ϵ[SnL(yn ), SnU(yn )] (3)
2L 2U
assumptions:
The proposed test for two populations in the form of
1. The data consists of uncertain, imprecise, and
indeterminacy can be written as
indeterminate values.
2. The two neutrosophic samples should be mutually DN = CN + ENIN; INϵ[IL, IU]; DNϵ[DL, DU] (4)
independent.
Note here that C and E I are the determined and
TheK−Stestcanbeappliedindependentofthecumulative indeterminate parts Nof the testN. TNhe proposed test reduces to
distributionfunction.Woodruffet al.33 useditfortheWeibull the tests in refs 30 and 31 if no indeterminacy is found in the
distribution. Papadopolous and Qiao34 and Frey35 presented data. Note also here that the proposed NK−S test reduces to
the K−S test for the Poisson distribution. the tests in refs 30 and 31 when S (x ) = S (x ) and
Suppose that X , X , ..., X be a neutrosophic random nL n1L nU n1U
1N 2N nN S (y )=S (y ).Theneutrosophicnullhypothesisthattwo
sample from a neutrosophic population having a neutrosophic nL n2L nU n2U
cumulative frequency distribution function, say F (x ). By samplescamefromthesameneutrosophicspecifiedpopulation
following ref 1, the null hypothesis that the ne0uNtrosnophic is accepted if DNϵ[DL, DU] > DαN where DαNϵ[DαL, DαU] is a
sample came from the specified neutrosophic distribution is n■eutrosophic critical value.
rejected if the neutrosophic cumulative frequency distribution
functionisnotclosetothespecifiedneutrosophicdistribution AUTHOR INFORMATION
function. Suppose now that F (x ); F (x )ϵ[F (x ), Corresponding Author
F (x )] and S (x ); S (x )ϵ0N[S n(x )0,NS n(Nx )]0LbentLhe *E-mail: [email protected].
0N nU nN nN nN nN nL nL nU nU
neutrosophic population cumulative distribution function and ORCID
the observed neutrosophic sample distribution function, Muhammad Aslam: 0000-0003-0644-1950
respectively. Then, the neutrosophic maximum difference
Notes
statistic based on F0N(xn); F0N(xnN)ϵ[F0L(xnL), F0N(xnU)] and T■he author declares no competing financial interest.
S (x )ϵ[S (x ), S (x )] is given by
nN nN nL nL nU nU
ACKNOWLEDGMENTS
D = |F (x) − S (x)|; D ϵ[D , D ]; F (x )
N 0N n nN n N L U 0N nN
The author is deeply thankful to the editor and the reviewers
ϵ[F (x ), F (x )]; S (x )ϵ[S (x ), S (x )]
0L nL 0N nU nN nN nL nL nU nU for their valuable suggestions to improve the quality of this
(1) manuscript. This work was funded by the Deanship of
The proposed test in the indeterminacy interval can be ScientificResearch(DSR),KingAbdulazizUniversity,Jeddah,
written as under grant no. 130-132-D1441. The author, therefore,
gratefully acknowledges the DSR technical and financial
DN = AN + BNIN; INϵ[IL, IU]; DNϵ[DL, DU] (2) s■upport.
Note here that A and B I are the determined and
N N N
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