Singly and doubly ordered cumulative correspondence analysis.

Transcription

1 Singly and doubly odeed cumulative coespondence analysis. L. D Amba*, E. Beh** and I. Camminatiello* *Univesity of Naples Fedeico II (Italy) ** Univesity of Newcasle (Austalia) damba@unina.it

2 Outline A shot eview Singly odeed cumulative coespondence analysis: methodology and application in industial expeiment Doubly odeed cumulative coespondence analysis: some developments and application with Van Rijckevosel s data We also popose an unified appoach

3 Review (/3) In multidimensional data analysis fo studying the association between two categoical vaiables, Coespondence Analysis (CA) is one of the most popula tool. his method is based on chi-squaed Dawback It does not take in consideation the odeed natue of the categoies

4 Review (/3) hee ae some contibutions that deal with odinal categoical vaiables, including those of Pasa and Smith (993), Ritov and Gilula (993) and Schieve (983) hese pocedues involve constaining the output obtained fom applying singula value decomposition (SVD) so that the coodinates in the fist dimension have an odeed stuctue. An altenative appoach applies moment decomposition (MD - Beh, 997) o hybid decomposition (HD - Beh, 004) that involve using the othogonal polyonomials in ode to detect linea, quadatic, cubic components

5 Review (3/3) In some industial expeiments, sometimes the output consists of categoical data (contingency table ) with an odeing in the categoies. Fo analyzing such data, aguchi (966, 974) poposed the Accumulation Analysis method as an altenative to Peason's chisquaed test. His motivation fo ecommending this technique appeas to be its similaity to ANOVA fo quantitative vaiables. Moe ecently, Light and Magolin (97) poposed a method called CAANOVA by defining an appopiate measue of vaiation fo categoical data. Unlike these methods aguchi consides situations with odeed categoies and does ANOVA on the cumulative fequencies

6 Aim of ou pape In this pape we exploe the development of coespondence analysis which takes into account the pesence of odeed vaiables by consideing the cumulative sum of cell fequencies acoss the vaiables.

7 Singly odeed cumulative coespondence analysis Beh, D Amba, Simonetti (Came 007; Communication in Statistics 0) pefomed coespondence analysis when coss-classified vaiables have an odeed stuctue by consideing the aguchi s statistic. aguchi s statistic is an appopiate measue of nonsymmetic association fo two categoical vaiables of which one is on odinal scale. It takes into account the pesence of an odeed vaiable by consideing the cumulative sum of cell fequencies acoss the vaiable.

8 Notation (/3) the absolute two-way contingency table that coss-classifies n units accoding to I odeed ow categoies and J odeed column categoies the elative two-way contingency table N P n ij n n j the ow and column maginals., N n i A tiangula matix of s with the last J-th ow is emoved so that it is of dimension (J -) J. A tiangula matix of s of dimension J x J A tiangula matix of s of dimension I x I M J L M

9 Notation (/3) the vectos with the maginal fequencies of P and c the diagonal matices with the maginal fequencies of P D and D c the cumulative fequencies z n, z n n,, z n i i i i i ij i n ij the cumulative column popotions n n n n d, d,, dj n n n n J

10 aguchi s statistic (/). aguchi (966) poposed the following statistic w w J,, J I ij w j ni j i ni z d ae weights >0. wo choices ae possible d d w o w j J j j j j

11 aguchi s statistic (/). he popeties of, aguchi'(966, 974) "cumulative-sums' statistic obtained by assigning a weight to each column that is invesely popotional to its conditional expectation of the j-th tem (conditional on the given maginals) In this pape we use this weighting system. w j d d... j... J j j A simple statistic,, which assigns each column constant weights /J

12 he Peason chi-squaed statistic and aguchi s statistic Nai (987) demonstated that the link between the Peason chi-squaed statistic and aguchi s statistic is j J j j is the Peason chi-squaed statistic fo the contingency table obtained by aggegating column categoies to j, and aggegating the column categoies j+ to J. Fo this eason, it is also efeed to as cumulative chisquaed statistic.

13 aguchi s statistic in matix notation (/) he aguchi s statistic may be expessed in matix notation by tace n D WAN NA D W (J-,J-) is the diagonal matix of weights A (J-,J) is the matix involving the cumulative column popotions J J J J d d d d d d d d d d d d A

14 aguchi s statistic in matix notation (/) Consideing that he aguchi s statistic afte some algeba may be ewitten by M J d A c M d J J J J J J J J J tace n tace n n n n n tace n n n tace n D c P WM M c P D D D c P WM M c P D D d PM W d PM D D d W NM d NM D c tace n D c P D c P D (C.A.)

15 Appoach poposed Beh, D Amba, Simonetti (Came 007 and Communication in Statistics 0) caied out CA when coss-classified vaiables have an odeed stuctue by consideing the aguchi s statistic. In tems of the aguchi's statistic, Beh et al. (00) pefom SVD on W M c P D D W M c P D X J J Matix X is centeed

16 Special cases and Popeties of Cumulative Coespondence analysis Fo I > and in the case of EQUIPROBABLE categoies the eigenvectos ae given by CHEBYCHEV POLYNOMIALS Fo I >, and in the equipobable case, the fist component (location o linea ) is popotional the Kuskal-Wallis statistic fo contingency tables Similaly the second component ( dispesion o quadatic ) is the genealizzation of the gouped data vesion of Mood's (954) statistic. In geneal case this is no tue In the case of xj table we have two components: the fist component ( linea ) of aguchi statistics is equivalent to Wilcoxon statistics he second component (Quadatic ) is equivalent to Mood s test (954) (See Nai 987) See Beh- D Amba- Simonetti in Communication in Statistics 0 Coodinates Distances Popeties of decomposition of aguchi s Statistic and Non Symmetical Coespondence Analysis (NSCA)

17 Relationship between the coodinates in the cumulative analysis and in the classical C.A (/) Fo cumulative analysis we may wite the ow coodinates by O D P c M W V Fo classical CA the ow coodinates ae defined by ~ ~ O D P c V V, V ~ theefoe ae the matices containing the ight singula vectos fo cumulative analysis and classical CA, espectively. O ~ MJ ~ O V W his shows that you may be able to go fom classical CA coodinates to cumulative coodinates easily. V J

18 Relationship between the coodinates in the cumulative analysis and in the classical analysis (/) Using the same agument we can obtain the classical coodinates fom the cumulative coodinates fom the elationship ~ O V MJW O ~ V

19 Example: Phadke s data (/) o illustate the cumulative coespondence analysis using the taguchi s statistic, D Amba, Köksoy, Simonetti (00) use Phadke s data (989). he contol factos (6) and thei levels (3) of polysilicon deposition pocess Levels 3 A. Deposition tempeatue ( o C) B. Deposition pessue (mtto) P 0 00 P 0 P C. Nitogen flow (sccm) N 0 N 0 50 N 0 75 D. Silane flow (sccm) S 0 00 S 0 50 S 0 E. Setting time (min) t 0 t 0 +8 t 0 +6 F. Cleaning method None CM CM 3

20 Example: Phadke s data (/) Categoies of poduct s quality Categoies Desciption Cumulative categoies I : 03 defects No suface defect (I) = I (03 defects) II : 430 defects Vey few defects (II) = I+II (030 defects) III : 3300 defects Some defects (III) = I+II+III (0300 defects) IV : defects Many defects (IV) = I+II+III+IV (0000 defects) V : 00 and moe defects oo many defects (V) = I+II+III+IV+V (0 defects)

21 Example: Phadke s data (3/) Facto effects fo the categoized suface defect data Numbe of obsevations by categoies Pobabilities fo the cumulative caegoies Facto Levels (I) (II) (III) (IV) (V) (I) (II) (III) (IV) (V) A A A B B B C C C D D D E E E F F F

22 Example: Phadke s data (4/) he aguchi s statistic =38,5669 he Peason chi-squaed statistic fo the fou contingency tables obtained by aggegating column categoies to j, and aggegating the column categoies j+ to J. I II+III+IV+V I+II III+IV+V I+II+III IV+V I+II+III+IV V 83,09 79,65 95, ,43

23 he patition of aguchi s statistic fom contingency table in Peason chi-squaed statistics Aggegated Column Categoies Facto (I) (II+III+IV+V) (I+II) (III+IV+V) (I+II+III) (IV+V) (I+II+III+IV) (V) A A A B B B C C C D D D E E E F F F otal , 9 60 O 38, 6,, 4,

24 Example: Phadke s data (5/) Souce AA A 4,3 B 0,8 he table shows the ANOVA esults. following aguchi ( see Nai ) A and B ae the two most impotant factos affecting poduct s quality C,3 D,8 E, F,9

25 Example: Phadke s data (6/) Figue shows the gaphical epesentation of the esults. able shows the distances fom the oigin to the column points in Figue above. I (II+III+IV+V) I+II (III+IV+V) I+II+III (IV+V) I+II+III+IV (V) 30,68 4,995,658 0,03 We note that the point I vs (II+III+IV+V) is the most impotant because it epesents a lage contibution (30,68), which is measued by the distances fom the oigin.

26 Example: Phadke s data (7/) Row codinates of cumulative odinal coespondence analysis Colomn codinates of cumulative odinal coespondence analysis Colomn codinates of cumulative odinal coespondence analysis Supplementay point of factos: A, B, C, D, E, F Categoies I fom coespondence analysis Categoies II fom coespondence analysis Categoies III fom coespondence analysis Categoies IV fom coespondence analysis Categoies V fom coespondence analysis Figue shows the ow and column categoies of Singly odeed cumulative coespondence analysis, the supplementay points of the factos (A,B,C,D,E,F) and the column categoies of classical analysis A,B ae impotant factos

27 Example: Phadke s data (8/) Level A B C D E F 5,97,6 3,353 3,35,963 5,749 5,536,88 6,59,75,836 0, ,33,53 0,700 5,54 5,77 4,035 Level A B C D E F wo tables show the distances between the ow points and I vs (II+III+IV+V) column point on the fist and second factoial axes. 5,345 4,43 5,777 5,38 4,854 7,438,9 4,908 4,088,476 4,39 4,8 3 5,59 3,745,93 3,538 3,803,056

28 Example: Phadke s data (9/) I vs (II+III+IV+V) Axis Axis A A B B 3 able shows the fist and second axes solutions based on the minimum distance epots. So we choose this optimal combination C 3 C 3 D D E E 3 F F 3

29 Example: Phadke s data (0/) Compaative esults fo the optimal facto settings Pobabilities fo the cumulative categoies Method Solution (I) (II) (III) (IV) (V) MEL A B C 3 D E F SCORE A B C 3 D E F WSNR A B C D E F AA A B C D 3 E F MSD A B C 3 D E F SARING A B C D 3 E F PROPOSED: Axis A B C 3 D E F Axis A B 3 C 3 D E 3 F Plane solution A B C 3 D E F MEL=Asiaba and Ghomi (006), SCORE=Nai(986), WSNR=Wu and Yeh (006), AA=Phadke s accumulation Analysis (989), MSD=Jeng and Guo (996), SARING= Stating, PROPOSED=D Amba, Köksoy and Simonetti

30 Example: Phadke s data (/) he last table shows the compaative esults fo the solution methods to optimize facto settings accoding to thei pedicted pobabilities fo the cumulative categoies. o calculate the optimal pobabilities fo the cumulative categoies aguchi uses the omega tansfom, also known as the logit tansfom. he omega tansfom fo pobability p is defined by p w(p) 0log0 p he optimum settings ecommended by the fist factoial axes solution of cumulative coespondence analysis is A, B, C3, D, E, F. By the invese omega tansfom, the pedicted pobability fo categoy (I) is he second axis solution (i.e., A, B3, C3, D, E3, F3) does not seem so poweful and the pobabilities fo the cumulative categoies ae not high enough. he plane solution (i.e., A, B, C3, D, E, F3) especially povides a vey low pobability in categoy (I). As a esult, we suggest to pick the fist axes solution as the optimal solution fo the Phadke s polysilicon deposition pocess.

31 Example: Phadke s data (/) We obseve that the fist axis solution is equivalent to the MEL (i.e., minimization of expected loss) solution poposed by Asiaba and Ghomi (006). he solution seems a nice candidate among the othes since the pobabilities fo the cumulative categoies ae highe. Asiaba and Ghomi (006) suggested a technique, which is called MEL that minimizes the expected loss fo the analysis of odeed categoical data. Afte an expeiment and data collection authos define a pobability distibution function of data in categoies. In the final step of MEL algoithm, expected loss in each level of factos is calculated and the decision is made by the fact that the optimum level of a facto is the one whee the expected loss is lowe than the expected loss at othe levels of that facto.

32 Doubly odeed cumulative coespondence analysis. Now, we exploe a genealization of aguchi s statistic which takes into account the pesence of both odeed vaiables by consideing the cumulative sum of cell fequencies acoss the vaiables.

33 Appoach of Cuadas (/) Cuadas (00) poposed the following appoach based on double cumulative fequencies D c M W USV L P J U is the matix containing the left singula vectos S is the diagonal matix containing the singula values V is the matix containing the ight singula vectos. W J is the J x J diagonal matix of weights /J L is lowe tiangula matix M is uppe tiangula matix

34 Appoach of Cuadas (/) Disadvantages: his appoach does not decompose any known index. his appoach has not the popety to be the sum of the Peason chi-squaed statistic fo the contingency table obtained by patitioning and pooling the oiginal data.( see aguchi)

35 Doubly Cumulative Coespondence Analysis (/4) Stating fom the poposal of Beh et al. (007-0), we pesent a moe geneal appoach based on double cumulative fequencies which ovecomes these poblems and pesents some inteesting popieties. Notation R the (I-)xI matix obtained by altenating the ows of an (I-)xI lowe tiangula matix of ones without the ow of all ones and the ows of an (I-)xI uppe tiangula matix of ones without the ow of all ones. C the Jx(J-) matix obtained by altenating the columns of an Jx(J-) uppe tiangula matix of ones without the column of all ones and the columns of an Jx(J-) lowe tiangula matix of ones without the column of all ones. D R and D C the diagonal matices with the maginal fequencies of doubly cumulative table.

36 Doubly Cumulative Coespondence Analysis (/4) he CA can be appoached by using cumulative fequencies fo ows and columns D c CD USV R P R C he ow and column coodinates ae espectively G DR RP c CDC V, Gc DC C P c R DR U

37 Doubly Cumulative Coespondence Analysis (3/4) he inetia Q n R C R R ni J I J tace D RP c CD C P c D can be consideed a genealization of aguchi s statistic because takes into account the pesence of both odinal vaiables. k s k It is easy to veify that tace of Q is identical to doubly cumulative chi-squaed statistic defined by Hiotsu (986) ( used fo compaing teatments and change point analysis) his appoach peseves same popety of aguchi s statistics I J i j ij ij is the Peason chi-squaed statistic fo the x contingency table obtained by patitioning and pooling the oiginal table

38 Doubly Cumulative Coespondence Analysis (4/4) he CA on the on doubly cumulative table, which we call Doubly Cumulative Coespondence Analysis, pesents the following popeties he appoach maximizes the fi-squaed statistic of each by table and, apat the constant, (I-)x(J-), of doubly cumulative table. All the weighted ow and the column coodinates ae cented he weighted ow and the column coodinates ae cented fo the by tables his appoach allows of epesenting the vaiations of ow and column categoies athe than the categoies on the space geneated by cumulative fequencies. Successively, it is possible to poject on the same space the ow and column categoies as supplementay points.

39 An unified appoach In ode to epesent the ows and columns of N we can conside the following SVD depending on fou matices F, B, D, E and the vecto a FB P ac D E USV Oveall appoach. F D, a, B I, D I, E D c Coespondence Analysis. F D, a, B I, D M, E WJ Cuadas appoach 3. F D, a,. B I, D M J, E W aguchi decomposition (Beh, D Amba, Simonetti, 007-0) 4. F D, a, B L, D M, E WJ Doubly Cumulative Coespondence Analysis (Cuadas appoach) 5. F D R, a, B R, D C, E D C Doubly Cumulative Coespondence Analysis (ou appoach Hiotsu decomposition) 6. F D, a, B I, D I, E DC Non Symmetical Coespondence Analysis

40 Example: Van Rijckevosel s data (/8) A data matix that is both RR (ow egession dependence ) and CR (ow egession dependence) following Schieve 983, Waen-Heise 009 he appeciations of five ed Bodeaux wines by 00 judges using a fou categoy system: fom excellent to boing (Van Rijckevosel, 987, p. 60) C C C3 C4 excellent good medioce boing R gand cu classè R cu Bougeois R3 Bodeaux d'oigine R4 vin de maque R5 vin de table he ows and columns of able have been pemuted using the scoes of the fist CA dimension. Since able is both RR and CR, thee exists a stong odinal association between the categoies of two vaiables and the five wines can be pefectly odeed fom excellent to boing. hen, we use such table fo illustating the doubly cumulative coespondence analysis

41 Example: Van Rijckevosel s data (/8) Calculating the doubly cumulative table R= N= C= RxNxC=

42 Example: Van Rijckevosel s data (3/8) he doubly cumulative chi-squaed statistic defined by Hiotsu It is easy to veify R R-R5 R-R R3-R5 R-R3 R4-R5 R-R4 R5 Peason chi-squaed statistic fo the by tables 609, ij C C-C4 C-C C3-C4 C-C3 C4 7, , , , , , , , , , , , i j ij max ij min ij

43 Example: Van Rijckevosel s data (4/8) Eigenvalues and pecentages of inetia of doubly cumulative coespondence analysis F F F3 otal inetia Eigenvalue 0,3 0,004 0,00 0,7 Cumulative % 97,749 99,764 00,000 It is easy to veify that, apat the constant, n I J he total inetia is identical to doubly cumulative chisquaed statistic defined by Hiotsu n I J 00 0,7 s k k 609,089

44 F (,0 %) Pincipal Axis ( 5.05 %) Example: Van Rijckevosel s data (5/8) Plot of Doubly odeed cumulative C.A. Max Dist fom oigin is c-c c3-c Max Chi-squaed 448,67. We note diffeent vaiations c, c-c c c3 Fom - -3 Diff. vaiations R5 R4-R5 R3-R5 fom C4 C3-C4 C-C4 Plot of coespondence analysis Coespondence Plot (Pofile Coodinates) Symmetic plot (axes F and F: 99,76 %) 0,5 C R R5 C4 CR C4R5 R C 0 C-C R-R R-R3 C-C3 R-R4 R-R5 R4-R5 C-C4 R3-R5 C3-C4 R3 C3 R4-0,5 - -0,5 0 0,5,5 F (97,75 %) Columns Rows Pincipal Axis ( 80.7 %) OAL D ASSOC %

45 Example: Van Rijckevosel s data (6/8) Lets look fist at the position of C and R. Since they ae situated nea each othe in this plot, this suggests that this ow categoy and column categoy ae associated with each othe. So if we wee to look at thei position in the classical plot they would be located nea each othe. Looking at the position of C and C-C: hese two points ae situated faily close to one anothe indicating that thee is a small diffeence between C and C. Since C-C is slightly close to the oigin than C this suggests that C is also slightly close to the oigin (in the classical CA plot) than C. Simila comments can be made by consideing the elatively shot distance between R and R-R. Such a distance implies that, in the classical CA plot R and R ae located nea each othe. If we conside the elative distance between (C, C-C) and (R, R-R) we can see that these two distances ae simila. Since we have discussed that C is associated with R, these simila distances imply that C and R ae also similaly positioned in the classical CA plot. he elatively simila distance between R, R-R and R-R3 suggests that the elative distance between R, R and R3 in the classical CA plot ae the same. Lets look at the ight hand side of ou cumulative plot. C4 and R5 ae situated close to each othe implying that in the classical CA plot they will also be situated close to one anothe. he distance between R5 and R4-R5 tells me that R4 is quite diffeent to R5. Since R4-R5 is situated close to the oigin than R5 then R4 will be situated close to the oigin that R5. he elative equal distance between R-R5 (close to the oigin), R3-R5 and R4-R5 (futhe fom the oigin) tells me that R, R3 and R4 ae oughly the same distance apat fom each othe in the classical CA plot. What is inteesting is that the distance between the pais (R, R-R5), (R-R, R3-R5), (R-R3, R4-R5) and (R-R4, R5) ae about the same indicating that the cumulative natue of ou analysis is peseving the elative diffeence (o similaity) of R, R, R3, R4 and R5 that the classical CA plot would eflect. All of these conclusions egading the intepetation of the cumulative coespondence plot is eflected in the classical CA plot.

46 Refeences Beh, E. J. (004), Simple coespondence analysis: A bibliogaphic eview, Intenational Statistical Review, 7, Beh, E. J., D'Amba, L., Simonetti B. (0), Cumulative coespondence analysis fo odeed categoial data using aguchi's Statistic, Communication in Statisticcs Cuadas, C. M. (00), Coespondence analysis and diagonal expansions in tems of distibution functions, J. of Statistical Planning and Infeence 03, pp D Amba L., Köksoy O., Simonetti B (009) Cumulative coespondence analysis of odeed categoical data fom industial expeiments, Jounal of applied statistics, 36, Hiotsu C. (986), Cumulative Chi-squaed Statistic as a ool fo esting Goodness of Fit, Biometika, 73, pp Nai, V. N. (987), Chi-squaed type tests fo odeed altenatives in contingency tables, Jounal of the Ameican Statistical Association, 8, aguchi, G. (974), A new statistical analysis fo clinical data, the accumulating analysis, in contast with the chi-squae test, Saishin Igaku, 9, Waens M. J., Heise W. J (009), Diagnostics fo egession dependence in tables e-odeed by the dominant coespondence analysis solution, Computational Statistics and Data Analysis, 53,