A combined test for randomness of spatial distribution of composite microstructures

Transcription

1 ISSN Revsta Matéra, v., n. 4, pp , A combned test for randomness of spatal dstrbton of composte mcrostrctres ABSTRACT João Domngos Scalon Departamento de Cêncas Exatas, Unversdade Federal de Lavras CP: 3037, Lavras, MG. CEP: e-mal: scalon@fla.br A new methodology s presented for characterzng the spatal dstrbton of second-phase partcles n planar sectons of mlt-phase materals. It s based on the sse of statstcally smmarzng the reslts of ndependent tests aganst the hypothess of randomness of the partcles. The methodology was appled n mltple planar sectons of an almnm alloy renforced wth slcon carbde partcles and leaded to a rejecton of the hypothess of randomness even when the tests from sngle planar sectons were ambgos. Keywords: Partclate renforced composte; spatal dstrbton; randomness; combned test. INTRODUÇÃO The mproved materal propertes obtaned wth partclate metal matrx composte wll depend on the spatal dstrbton of the partcles n the matrx materal [, ]. The basc methodology for characterzng the spatal dstrbton of second-phase partcles n sngle planar sectons of composte materals s now wellestablshed [3]. In ths context, drng the last few years, statstcal methods have been appearng, to a certan extent, n materal lteratre to provde the analyss of spatal dstrbton of partcles n composte materals. Methods sch as qadrat conts [4], nearest neghbor dstances [5], Drchlet tessellaton [6] and spatal pattern descrptors [7] have been appled to test departre aganst the hypothess of randomness of the spatal dstrbton of the renforcng partcles. A sngle sample rarely provdes a defntve answer to research qestons and, therefore there s an nterest among materal researchers to develop methods based on mltple samples for provdng a more rgoros qanttatve analyss of the spatal dstrbton of second-phase partcles n two-dmensonal mltyphase systems. When mltple samples of composte materal are avalable, n two or more expermental grops, parametrc and non-parametrc methods can be sed to the analyss of data n the form of replcated spatal pont patterns [8]. We are consderng here the staton n whch we have one expermental grop that contans several planar sectons of a composte materal. The man am of ths paper s to present a method to combne P-vales from several ndependent tests aganst the hypothess of spatal randomness of the partcles. The basc dea of the method s to test whether collectvely they can reject the hypothess of spatal randomness. TESTING AGAINST SPATIAL RANDOMNESS FOR INDIVIDUAL SAMPLES For practcal prposes, each partcle s treated as a pont defned by ts coordnates and, therefore the spatal data can be assmed a map of all partcle locatons n an essental planar regon. Usng some fnctonal spatal pattern descrptors sch as F, G and K-fnctons s a natral way to proceed wth the statstcal analyss of spatal dstrbton of partcle centers n ndvdal planar sectons of composte materals [7]. In ths paper, we se the J-fncton, ntrodced by LIESHOUT et al. [9], snce t performs very well n detectng departre from randomness towards both reglarty and clsterng alternatves. For a statonary pont process, the J-fncton s gven by the eqaton: Ator Responsável: João Domngos Scalon Data de envo: 3/08/06 Data de acete: 04//06

2 SCALON, J.D.; Revsta Matéra, v., n. 4, pp , 007. Gx ( ) J( x) = F( x) () for all dstances x 0, sch that F(x) <, where F(x) s the dstrbton fncton of the dstance from an arbtrary fxed pont to the nearest partcle center of the planar secton and G(x) s the dstrbton fncton of the nearest dstance between two partcle centers. A stable edge-corrected estmator for F(x) s provded by the eqaton: m I ( x, r) x ˆ ( ) = = m I x( r) = F x () where m s the nmber of sample ponts n the planar secton, pont to the nearest of the n partcle centers n the analyzed pattern, x I ( ) denotes the dstance from the th chosen x r s an ndcator fncton that takes the vale when s less than or eqal to x, r s the dstance from each partcle center the nearest pont on x the bondary of the planar secton and less than or eqal to x and r (, ) x I x r s an ndcator fncton that takes the vale when s s greater than or eqal to x. An estmator for G(x) s provded by an eqaton analogos to eqaton (), sbstttng the dstance x by the nearest dstance between two partcle centers [ 0]. The smplest estmator for J(x) s obtaned by plggng nto eqaton () the estmates of F(x) and F ˆ ( x ) = G ˆ ( x ) J ˆ( x ) = J ˆ( x) G(x). It s easly seen that nder the randomness hypothess,, so. Vales of smaller than ndcate clsterng whle vales larger than ndcate reglarty. In order to defne a statstcal test for detectng departre aganst randomness, t s sal to choose a measre of dscrepancy that examnes the degree of agreement between the observed and the expected emprcal dstrbton fnctons nder the nll hypothess of randomness. A sensble measre to evalate these dfferences over a range of dstances ( x ) s gven by the eqaton: x x0 ˆ { J( x) } dx (3) 0 = The statstc does not have known samplng dstrbton. DIGGLE [ 0] sggests the se of the followng Monte Carlo based method to perform the test aganst randomness. Let J ˆ ( x ) be the J-fncton of an observed pont pattern wth n events and Jˆ ( x),..., Jˆ ( x) the J-fnctons from s smlatons of random patterns wth n events. Calclate the statstc for the observed and smlated patterns. Then, the vale for the observed pattern s compared wth vales for the smlated patterns. If ranks among the,..., s largest of, t ndcates departre from randomness. Sppose J = J ( j) for some j s ( s+ j) then reject the hypotheses of CSR f P = α, where P s the one-taled P-vale. For example, s based on 99 smlatons (s = 00), rejecton at the 5% level occrs f J (96) J J (00 ). s,..., s {,..., } 3 COMBINED TEST AGAINST SPATIAL RANDOMNESS Snce the ndvdal P-vales of each sample are avalable, we can carry ot a combned test for a statstcal generalzaton to be made wth respect to the combned evdence of a random dstrbton of 598

3 SCALON, J.D.; Revsta Matéra, v., n. 4, pp , 007. partcles from all samples. For ths, we consder each Monte Carlo test as an ndvdal and ndependent stdy to test departre from randomness. Under ths spposton, we advocate to se a technqe for poolng reslts across dfferent statstcal tests aganst the hypothess of randomness. Ths technqe can provde one sngle P-vale that allows s to decde the natre of the spatal dstrbton of the partcles wthn the metal matrx. When data come n the form of one-taled P-vales, FISHER [] sggests that they can be combned by formng a statstc that s ther prodct. If we have k ndependent stdes that gve P as the tal probabltes, the statstc smmarzng the reslt s the prodct P= PP... Pk. If the nll hypothess s tre, then P have nform dstrbton on the nterval from 0 to. FISHER [] noted that f P s dstrbted accordng to the nform dstrbton on the nterval from 0 to, then conseqently log e P j s dstrbted lke a ch-sqare dstrbton wth degrees of freedom. If all of the nll hypotheses of randomness the k tests are tre, then the statstc: χ o k = log Pj (4) j= e wll have a ch-sqare dstrbton wth k degrees of freedom and so sgnfcance s tested by fndng the probablty of a larger vale of the statstc χ. o 4 EXPERIMENTAL DATA We have appled the combned test to eghteen metallographc samples of an almnm slcon carbde composte materal prodced by the Department of Engneerng of Materals, Unversty of Sheffeld, England, where slcon carbde second-phase partcles had a volme fracton eqal to %. The metallographc sample areas were 9 88 μm. The eghteen metallographc samples were placed on a compter controlled optcal mcroscope stage analyzer (Polyvar) whch allowed flly atomatc adjstment, focsng, postonng and scannng of the samples. The overall magnfcaton sed was 600 tmes, yeldng a pxel sze of μm. Ths, the Polyvar prodced eghteen dgtal mages wth area frame eqal to sqare pxels. The two-dmensonal dgtal mages were analyzed by sng mage-processng technqes to extract the coordnates (centre) of each partcle wthn the mages. The actal mages were 5 x 767 pxels, ot of whch we sed only the partcles located n the left top sqare of 5 x 5 pxels. The 5 x 5 mages were transformed nto patterns wth nt sqare area to facltate the spatal analyss. 5 RESULTS AND DISCUSSION The man am of the present work has been to provde a statstcal analyss of the spatal dstrbton of second-phase partcle centers n two-dmensonal dstrbted mlt-phase materals. We have sed eghteen samples of an almnm slcon carbde composte materal to answer the man scentfc qeston: whether or not the composte materal presents partcles that are randomly dstrbted. We advocate that ths qeston can be adeqately answered by sng a combned test from ndependent tests aganst the hypothess of randomness. The analyss start by performng a Monte Carlo test aganst the hypothess of randomness of the partcle centers n each planar secton of the composte materal. To carry ot these tests, we sed 99 smlatons (s = 00) from a statonary Posson process of ntensty n (actal nmber of partcles). As DIGLLE [0] ponts ot s = 00 s sally sffcent snce for greater s the power of the test ncreases only margnally wth s. We se m sample ponts n a reglar grd v v to estmate the vales of F ˆ ( x) n eqaton (), where v n. The ntegral n eqaton (3) was approxmately calclated by a Remann sm at 50 ntervals between 0 and Table presents the reslts (one-taled P-vales) of the tests aganst randomness for the ndvdal samples of the composte materal. 599

4 SCALON, J.D.; Revsta Matéra, v., n. 4, pp , 007. Table : One-taled P-vales of the hypothess tests aganst randomness of the partcle centers for the eghteen samples of the almnm alloy renforced wth slcon carbde partcles SAMPLE P-vale SAMPLE P-vale The one-taled P-vales provded n Table show that t s not easy to decde whether the composte materal presents evdence that the second-phase partcles are randomly dstrbted. Observe that f one had chosen, for example, sample for hs analyss, he had rejected the hypothess of randomness. Otherwse, f he had chosen sample 3, he had reached an opposte conclson. Ths, we sggest combnng the one-taled P-vales wth the prpose of obtanng a smmary overall P-vale for testng the same hypothess of randomness for the whole grop of samples. Applyng eqaton (4) to the reslts presented n Table, we obtaned = 6.3. Becase there are eghteen ndependent tests, one for each sample, there are 36 degrees of freedom and = 6.3 s assocated wth P = Ths, the combned evdence from these eghteen samples ndcates a strong rejecton of the nll hypothess of a random dstrbton of partcles n the composte materal. Observe that the method leads to a rejecton of the hypothess of randomness even when the tests from sngle samples were ambgos. The combned test works well when the alternatve dstrbton has a densty that s, approxmately, a reversed J shape becase then t s lkely that P fall near 0 and prodce a small prodct, and we are lkely to reject the nll hypothess. The strength of ths approach s that t has good power aganst the randomness for reverse J shaped alternatves []. The more seros dsadvantage of ths combned test s that t treats large and small P-vales asymmetrcally. It s asymmetrcally senstve to small P-vales compared to large P-vales [3]. There are several other statstcal methods avalable for combnng P-vales of ndependent stdes. They range from varos contng procedres to a varety of smmaton approaches nvolvng ether sgnfcance levels or weghted statstcal tests sch as t and z tests [, 3]. Despte the avalable methods, the combned test presented here remans one of the best known and appled becase t s the smplest and most asymptotcally effcent of them [3]. One observaton that s mportant to add to any dscsson abot the method for combnng P-vales s that t can be appled to ask whether the accmlatve nformaton among tests on smlar nll hypotheses can reject that shared nll hypothess. Ths, ths procedre may be appled not only to spatal analyss of partcle centers bt also to any statstcal analyss n materals research where there s an nterest to test the sgnfcance of aggregate ndependent hypothess tests. χ o χ o 6 CONCLUSION The statstcal analyss of the spatal dstrbton of partcles can be mproved by takng and analyzng mltple samples of composte materals. The goal of ths paper has been to advocate the se of a combned test for provdng an overall assessment aganst the hypothess of spatal randomness of the dstrbton of partcles n composte materals. We have demonstrated that the combned test s effcent to test spatal randomness of partcles even when the tests from sngle planar sectons were ambgos. 7 ACKNOWLEDGMENTS The athor thanks Professor Helen Atknson (Department of Engneerng, Unversty of Lecester, UK) for provdng the data of the composte materal. 600

5 SCALON, J.D.; Revsta Matéra, v., n. 4, pp , BIBLIOGRAPHY [] LEWANDOWSKI, J.J., LIU, C., HUNT, W.H.Jr., Effects of matrx mcrostrctre and partcles dstrbton on fractre of almnm metal matrx compostes, Materals Scence Engneerng A, v. 07, pp. 4-55, 989. [] PYRZ, R., Qanttatve descrpton of the mcrostrctre of compostes. Part I: Morphology of ndrectonal composte systems, Composte Scence and Technology, v. 50, pp , 994. [3] OSHER, J., MUCKLICH, F., Statstcal analyss of mcrostrctres n materals scence, Chchester, John Wley & Sons, 000. [4] CANS, M., HOUGARDY, H., Qanttatve descrpton of the dstrbton of non-metallc nclsons n steels on a bass of statstcal calclatons, Acta Stereologca, v. 5, pp. 8-85, 986. [5] SPITZIG, W.A., KELLY, J.F., RICHMOND, O., Qanttatve characterzaton of second-phase poplatons, Metallography, v. 8, pp. 35-6, 985. [6] PARSE, J.B., WERT, J.A., A geometrcal descrpton of partcle dstrbton n materals. Modellng and Smlaton n Materals Scence and Engneerng, v., pp , 993. [7] SCALON, J.D., FIELLER, N.R.J., STILLMAN, E.C., et al., Spatal pattern analyss of second-phase partcles n composte materals. Materals Scence and Engneerng A, v. 356, pp , 003. [8] DIGGLE, P.J., MATEU, J., CLOUGH, H.E., A comparson between parametrc and non-parametrc approaches to the analyss of replcated Spatal Pont Patterns, Advances n Appled Probablty, v. 3, pp , 000. [9] VAN LIESHOUT, M.N.M., BADDELEY, A.J., A nonparametrc measre of spatal nteracton n pont patterns, Statstca Neerlandca, v. 3, pp , 996. [0] DIGGLE, P.J., Statstcal analyss of Spatal Pont Patterns, London, Arnold, 003. [] FISHER, R.A., Combnng ndependent tests of sgnfcance, Amercan Statstcan, v., pp. 3-33, 948. [] HEDGES, L.V., OLKIN, I., Statstcal Methods for Meta Analyss, New York, Academc Press, 985. [3] WOLF, F.M., Meta-Analyss: Qanttatve Methods for Research Syntheses, London, Sage,