Dimension Reduction and Visualization of the Histogram Data

Transcription

1 The 4th Workshop n Symbolc Data Analyss (SDA 214): Tutoral Dmenson Reducton and Vsualzaton of the Hstogram Data Han-Mng Wu ( 吳漢銘 ) Department of Mathematcs Tamkang Unversty Tamsu 25137, Tawan hmwu@mal.tku.edu.tw DR for Hstogram Data June 13, / 27

2 Outlnes 1 The Hstogram Data DR for Hstogram Data June 13, / 27

3 Outlnes 1 The Hstogram Data 2 Prncpal Component Analyss for Hstogram Data () The Hstogram Arthmetc Approach () The Quantle Method () The Transformaton Approach (v) The Dstrbutonal Approach DR for Hstogram Data June 13, / 27

4 Outlnes 1 The Hstogram Data 2 Prncpal Component Analyss for Hstogram Data () The Hstogram Arthmetc Approach () The Quantle Method () The Transformaton Approach (v) The Dstrbutonal Approach 3 An Example DR for Hstogram Data June 13, / 27

5 The Hstogram data (1/2) I ( b) x ( b) x ( b+1) = (, ] p ( b) b... B DR for Hstogram Data June 13, / 27

6 The Hstogram data (1/2) I ( b) x ( b) x ( b+1) = (, ] p ( b) A hstogram: b... B h = {I, p} = {x I (b) = (x (b), x (b+1) ], p (b), b = 1,, B}. DR for Hstogram Data June 13, / 27

7 The Hstogram data (1/2) I ( b) x ( b) x ( b+1) = (, ] p ( b) A hstogram: b... B h = {I, p} = {x I (b) = (x (b), x (b+1) ], p (b), b = 1,, B}. 1 B b=1 p(b) = 1, B: the number of modaltes of h. DR for Hstogram Data June 13, / 27

8 The Hstogram data (1/2) I ( b) x ( b) x ( b+1) = (, ] p ( b) b... B A hstogram: h = {I, p} = {x I (b) = (x (b), x (b+1) ], p (b), b = 1,, B}. 1 B b=1 p(b) = 1, B: the number of modaltes of h. 2 I = b I (b) = (x (1), x (B+1) ]. DR for Hstogram Data June 13, / 27

9 The Hstogram data (2/2) Let O = {o 1, o 2,, o n } be gven objects. 1 The classcal data: x = (x 1 x p ) T, x R. DR for Hstogram Data June 13, / 27

10 The Hstogram data (2/2) Let O = {o 1, o 2,, o n } be gven objects. 1 The classcal data: x = (x 1 x p ) T, x R. 2 The hstogram data: H = (h ) n p, where h = {I, p } = {x I (b) = (x (b), x (b+1) ], p (b), b = 1,, B }. DR for Hstogram Data June 13, / 27

11 The Hstogram data (2/2) Let O = {o 1, o 2,, o n } be gven objects. 1 The classcal data: x = (x 1 x p ) T, x R. 2 The hstogram data: H = (h ) n p, where h = {I, p } = {x I (b) = (x (b), x (b+1) ], p (b), b = 1,, B }. Possble source: the result of an aggregaton, the descrpton of a populaton, or any other grouped collectve. DR for Hstogram Data June 13, / 27

12 PCA for Hstogram Data (1/2) The Hstogram Arthmetc Approach 1 Rodrguez, O., Dday, E., and Wnsberg, S., (2), Generalzaton of the prncpal component analyss to hstogram data, PKDD2, Lyon, 2. DR for Hstogram Data June 13, / 27

13 PCA for Hstogram Data (1/2) The Hstogram Arthmetc Approach 1 Rodrguez, O., Dday, E., and Wnsberg, S., (2), Generalzaton of the prncpal component analyss to hstogram data, PKDD2, Lyon, 2. The Quantle Method 1 Ichno, M., (28), Symbolc PCA for hstogram-valued data, n Proceedngs IASC. December 5-8, Yokohama, Japan. 2 Ichno, M., (211), The quantle method for symbolc prncpal component analyss, Statstcal Analyss and Data Mnng, 4(2): Verde, R. and Irpno, A., (213), Dmenson reducton technques for dstrbutonal symbolc data, SIS 213 Statstcal Conference, Advances n Latent Varables - Methods, Models and Applcatons, Unversty of Bresca, June, 19-21, 213. (Consder only one unvarate hstogram varable.) DR for Hstogram Data June 13, / 27

14 PCA for Hstogram Data (2/2) The Transformaton Approach 1 Rodrguez, O., Dday, E., and Wnsberg, S., (2), Generalzaton of the prncpal component analyss to hstogram data, PKDD2, Lyon, 2. 2 Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: DR for Hstogram Data June 13, / 27

15 PCA for Hstogram Data (2/2) The Transformaton Approach 1 Rodrguez, O., Dday, E., and Wnsberg, S., (2), Generalzaton of the prncpal component analyss to hstogram data, PKDD2, Lyon, 2. 2 Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: The Dstrbutonal Approach 1 Wang, H., Chen, M., L, N., and Wang, L., (211), Prncpal component analyss of Modal nterval-valued data wth constant numercal characterstcs, Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 211, Dubln (Sesson CPS53), pp DR for Hstogram Data June 13, / 27

16 The quantle method (1/3) Defne the cumulatve dstrbuton functon F (x) of the hstogram: 1 Assume that n bns (sub-ntervals) have unform dstrbutons. DR for Hstogram Data June 13, / 27

17 The quantle method (1/3) Defne the cumulatve dstrbuton functon F (x) of the hstogram: 1 Assume that n bns (sub-ntervals) have unform dstrbutons. F (x) = for x < x 1, F (x) = p 1 x x 1 x 2 x 1 for x 1 x < x 2, F (x) = F (x 2 ) + p 2 x x 2 x 3 x 2 for x 2 x < x 3,. F (x) = F (x n ) + p n x x n x n+1 x n for x n x < x n+1, F (x) = 1 for x n+1 x. DR for Hstogram Data June 13, / 27

18 The quantle method (2/3) Obtan quantles for a hstogram-valued feature: 1 The object o s descrbed by an (m + 1)-tuple (q 1, q 2,, q m, q m+1 ). F (x mn ) =, q 1 = x mn F (q 2 ) = 1/m, F (q 3 ) = 2/m,, F (q m ) = (m 1)/m, and F (x max ) = 1, q (m+1) = x max DR for Hstogram Data June 13, / 27

19 The quantle method (2/3) Obtan quantles for a hstogram-valued feature: 1 The object o s descrbed by an (m + 1)-tuple (q 1, q 2,, q m, q m+1 ). F (x mn ) =, q 1 = x mn F (q 2 ) = 1/m, F (q 3 ) = 2/m,, F (q m ) = (m 1)/m, and F (x max ) = 1, q (m+1) = x max 2 Express each object o under a hstogram-valued varable X j by an (m + 1)-tuple: (q 1, q 2,, q (m), q (m+1) ). DR for Hstogram Data June 13, / 27

20 The quantle method (3/4) Output: 1 PCA on the correlaton matrx of X Q [n p] = (o Q 1 o Qn ) T, where o Q = q 11 q p1 q 12 q p2... q 1m q pm q 1(m+1) q p(m+1) T, n = n (m + 1) DR for Hstogram Data June 13, / 27

21 The quantle method (3/4) Output: 1 PCA on the correlaton matrx of X Q [n p] = (o Q 1 o Qn ) T, where Vsualzaton: o Q = q 11 q p1 q 12 q p2... q 1m q pm q 1(m+1) q p(m+1) T, n = n (m + 1) 1 In the factor plane, express each object as a seres of m connected arrow lnes. DR for Hstogram Data June 13, / 27

22 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. DR for Hstogram Data June 13, / 27

23 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. 2 D j = (α j, β j ): Doman of all possble values of bns of X j. DR for Hstogram Data June 13, / 27

24 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. 2 D j = (α j, β j ): Doman of all possble values of bns of X j. 3 δ j = nf b=1,,b {x (b+1) x (b) }: smallest length of the nterval I. DR for Hstogram Data June 13, / 27

25 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. 2 D j = (α j, β j ): Doman of all possble values of bns of X j. 3 δ j = nf b=1,,b {x (b+1) 4 Determne the lower lmt and the upper lmt 1 f I (b) x (b) }: smallest length of the nterval I. = (, a j ], then I (b) = (max{a j δ j, α j }, a j ]. DR for Hstogram Data June 13, / 27

26 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. 2 D j = (α j, β j ): Doman of all possble values of bns of X j. 3 δ j = nf b=1,,b {x (b+1) 4 Determne the lower lmt and the upper lmt x (b) }: smallest length of the nterval I. 1 f I (b) = (, a j ], then I (b) = (max{a j δ j, α j }, a j ]. 2 f I (b) = (b j, ), then I (b) = (b j, max{β j, b j + δ j }]. DR for Hstogram Data June 13, / 27

27 The Transformaton Approach (1/2) Makosso-Kallyth, S. and Dday, E., (212), Adaptaton of nterval PCA to symbolc hstogram varables, Adv Data Anal Classf, 6: Preprocessng: 1 Assume bns of the X j varable s the same for all ndvduals. 2 D j = (α j, β j ): Doman of all possble values of bns of X j. 3 δ j = nf b=1,,b {x (b+1) 4 Determne the lower lmt and the upper lmt x (b) }: smallest length of the nterval I. 1 f I (b) = (, a j ], then I (b) = (max{a j δ j, α j }, a j ]. 2 f I (b) = (b j, ), then I (b) = (b j, max{β j, b j + δ j }]. 5 f bns of varables don t have same unt, replace (a, b ] by an adjusted nterval (a/(b a ), b /(b a )]. DR for Hstogram Data June 13, / 27

28 The Transformaton Approach (2/2) Quantfcaton: 1 The transformed varable: s = (s (1), s (2),, s (B j ) ) 1 Codng 1: s (b) = (x (b) + x (b+1) )/2: center of the adjusted ntervals. DR for Hstogram Data June 13, / 27

29 The Transformaton Approach (2/2) Quantfcaton: 1 The transformed varable: s = (s (1), s (2),, s (B j ) ) 1 Codng 1: s (b) = (x (b) + x (b+1) )/2: center of the adjusted ntervals. 2 Codng 2: s (b) : ranks, the scores of the modaltes of the bns. DR for Hstogram Data June 13, / 27

30 The Transformaton Approach (2/2) Quantfcaton: 1 The transformed varable: s = (s (1), s (2),, s (B j ) ) Output: 1 Codng 1: s (b) = (x (b) + x (b+1) )/2: center of the adjusted ntervals. 2 Codng 2: s (b) : ranks, the scores of the modaltes of the bns. 1 PCA on G = (g ), g = B j b=1 s(b) p (b), where p (b) may be normalzed by a angular transformaton arcsn p (b). DR for Hstogram Data June 13, / 27

31 The Transformaton Approach (2/2) Quantfcaton: 1 The transformed varable: s = (s (1), s (2),, s (B j ) ) Output: 1 Codng 1: s (b) = (x (b) + x (b+1) )/2: center of the adjusted ntervals. 2 Codng 2: s (b) : ranks, the scores of the modaltes of the bns. 1 PCA on G = (g ), g = B j b=1 s(b) p (b), where p (b) may be normalzed by a angular transformaton arcsn p (b). Vsualzaton: 1 Use hypercube for representaton of concepts. 2 Use nterval lengths for representng the concepts. DR for Hstogram Data June 13, / 27

32 The Dstrbutonal Approach (1/1) Reference: 1 Bllard, L. and dday, E., (23), From the statstcs of data to the statstcs of knowledge: symbolc data analyss, Journal of the Amercan Statstcal Assocaton, 98(462), Wang, H., Chen, M., L, N., and Wang, L., (211), Prncpal component analyss of Modal nterval-valued data wth constant numercal characterstcs, Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 211, Dubln (Sesson CPS53), pp DR for Hstogram Data June 13, / 27

33 The Dstrbutonal Approach (1/1) Reference: 1 Bllard, L. and dday, E., (23), From the statstcs of data to the statstcs of knowledge: symbolc data analyss, Journal of the Amercan Statstcal Assocaton, 98(462), Wang, H., Chen, M., L, N., and Wang, L., (211), Prncpal component analyss of Modal nterval-valued data wth constant numercal characterstcs, Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 211, Dubln (Sesson CPS53), pp Assumpton: 1 Wthn each sub-nterval I (b), the random varable x s unformly dstrbuted across the sub-nterval. DR for Hstogram Data June 13, / 27

34 The Dstrbutonal Approach (2/1) The emprcal densty functon (Bllard and Dday, 23): 1 The emprcal densty functon for jth hstogram-valued varable X j : f j (u) = 1 n B n =1 b=1 x (b+1) p (b) x (b), x (b) u x (b+1), j = 1,, p. DR for Hstogram Data June 13, / 27

35 The Dstrbutonal Approach (2/1) The emprcal densty functon (Bllard and Dday, 23): 1 The emprcal densty functon for jth hstogram-valued varable X j : f j (u) = 1 n B n =1 b=1 x (b+1) p (b) x (b), x (b) u x (b+1), j = 1,, p. The symbolc sample mean and varance: 1 The symbolc sample mean for jth hstogram-valued varable X j : µ j = E(X j ) = = 1 n = 1 2n B n =1 b=1 =1 b=1 (b+1) x x (b) ξ f j (ξ) dξ ξ x (b+1) p (b) x (b) dξ B n p (b) (x (b+1) + x (b) ), j = 1,, p. DR for Hstogram Data June 13, / 27

36 The Dstrbutonal Approach (3/1) 2 The symbolc sample varance for jth hstogram-valued varable X j : Var(X j ) = E(Xj 2 ) µ 2 j B n = 1 3n =1 1 4n 2 b [ p (b) =1 b=1 (x (b+1) B n p (b) (x (b+1) + x (b) ) 2 + x (b+1) x (b) + (x (b) ) 2] ) 2, j = 1,, p. DR for Hstogram Data June 13, / 27

37 The Dstrbutonal Approach (4/1) The emprcal jont densty functon (Bllard and Dday, 23): 1 Assume that (X j, X j ) values are unformly dstrbuted across the rectangles (x (b j ), x (b j +1) ] (x (b j ), x (b j +1) ]. 2 Assume h and h are ndependent. DR for Hstogram Data June 13, / 27

38 The Dstrbutonal Approach (4/1) The emprcal jont densty functon (Bllard and Dday, 23): 1 Assume that (X j, X j ) values are unformly dstrbuted across the rectangles (x (b j ), x (b j +1) ] (x (b j ), x (b j +1) ]. 2 Assume h and h are ndependent. 3 The emprcal jont densty functon for jth and j th hstogram-valued varables (X j, X j ): f jj (u, v) = 1 n B n =1 B b j =1 b j =1 p (b j ) p (b j ) Z (b j,b j ), where Z (b j,b j ) s the area of the rectangle Z (b j,b j ) = (x (b j ), x (b j +1) ] (x (b j ), x (b j +1) ]. (u, v) Z (b j,b j ), DR for Hstogram Data June 13, / 27

39 The Dstrbutonal Approach (5/1) The symbolc sample covarance: 1 The symbolc sample covarance functon s Cov(X j, X j ) = 1 n B B 4n =1 b j =1 b j =1 1 n B 4n 2 =1 b j =1 B n =1 b j =1 p (b j ) p (b j ) (x (b j ) + x (b j +1) ) (x (b j ) + x (b j +1) p (b j ) (x (b j ) + x (b j +1) ) p (b j ) (x (b j ) + x (b j +1) ) ), j, j = 1,, p, j j. DR for Hstogram Data June 13, / 27

40 The Dstrbutonal Approach (6/1) Lnear combnaton of p ntervals Gven p nterval-valued varables X 1,, X p, all wth n observatons and real numbers α j, a lnear combnaton of p nterval-valued varables s defned as Y = p α j X j = ([y 1, y 1 ], [y 2, y 2 ],, [y n, y n ]) T, j=1 where y = p α j (τx + (1 τ)x ), and y = j=1 p α j ((1 τ)x + τx ), j=1 wth τ = I(α j > ). DR for Hstogram Data June 13, / 27

41 The Dstrbutonal Approach (7/1) Lnear combnaton of p hstograms Gven p hstogram varables X 1,, X p, and real numbers α j, defne a hstogram varable Y as a lnear combnaton of X 1,, X p Y = p α j X j = (y 1,, y n ) T, j=1 where y = {I, p } = {y I (b) where B = max j {B }. = (y (b), y (b+1) ], p (b), b = 1,, B }, DR for Hstogram Data June 13, / 27

42 The Dstrbutonal Approach (8/1) 1 For the th hstogram h, wth the number of modaltes B, each sub-nterval has ts densty functon p (b j ), b j = 1,, B. DR for Hstogram Data June 13, / 27

43 The Dstrbutonal Approach (8/1) 1 For the th hstogram h, wth the number of modaltes B, each sub-nterval has ts densty functon p (b j ), b j = 1,, B. 2 Hence, the th observaton, y, contans B p j=1 B hypercubes n p dmensonal space. DR for Hstogram Data June 13, / 27

44 The Dstrbutonal Approach (8/1) 1 For the th hstogram h, wth the number of modaltes B, each sub-nterval has ts densty functon p (b j ), b j = 1,, B. 2 Hence, the th observaton, y, contans B p j=1 B hypercubes n p dmensonal space. 3 Let p u denote the densty of each hypercube of the th observaton, that s the product of the densty of correspondng subntervals (I (b 1) 1,, I (bp) p ), u = 1,, B, p u p u(i (b 1) 1,, I (bp) p ) = p j=1 p (b j ), b j {1,, B }, j = 1,, p. DR for Hstogram Data June 13, / 27

45 The Dstrbutonal Approach (9/1) 4 Calculatng lnear combnaton to all the hypercubes, (I (b 1) 1,, I (bp) p ), we obtan I u, u = 1,, B : I u I u(i (b 1) 1,, I (bp) p ) = p j=1 α j I (b j ), b j {1,, B }, j = 1,, p. DR for Hstogram Data June 13, / 27

46 The Dstrbutonal Approach (9/1) 4 Calculatng lnear combnaton to all the hypercubes, (I (b 1) 1,, I (bp) p ), we obtan I u, u = 1,, B : I u I u(i (b 1) 1,, I (bp) p ) = p j=1 α j I (b j ), b j {1,, B }, j = 1,, p. 5 Let I = (y (1) mn u {I u }, y (B +1) max u {I u }]. The subnterval I (b) = (y (b), y (b+1) ], b = 1,, B, can be obtaned by dvdng I nto B parts: y (b) = y (1) + y (B +1) y (1) (b 1), b = 1,, B. B DR for Hstogram Data June 13, / 27

47 The Dstrbutonal Approach (1/1) 6 Projectng I u to I (b), we obtan p (b) = B u=1 p I u I (b) u I u, b = 1,, B, = 1,, n. DR for Hstogram Data June 13, / 27

48 The Dstrbutonal Approach (1/1) 6 Projectng I u to I (b), we obtan p (b) = B u=1 p I u I (b) u I u, b = 1,, B, = 1,, n. Vsualzaton: 1 none. (Wang, Chen, L, and Wang, 211) DR for Hstogram Data June 13, / 27

49 Vsualzaton: A hstogram of hstograms A hstogram of all of the observed hstograms (Bllard and Dday, 23): Let {(I (g), p (g) ), g = 1,, r} represent the relatve frequency hstogram for the combned set of observed hstograms. DR for Hstogram Data June 13, / 27

50 Vsualzaton: A hstogram of hstograms A hstogram of all of the observed hstograms (Bllard and Dday, 23): Let {(I (g), p (g) ), g = 1,, r} represent the relatve frequency hstogram for the combned set of observed hstograms. 1 Let I = (mn,b {x (b) }, max,b {x (b+1) }]: the nterval that spans all of the observed values of X. DR for Hstogram Data June 13, / 27

51 Vsualzaton: A hstogram of hstograms A hstogram of all of the observed hstograms (Bllard and Dday, 23): Let {(I (g), p (g) ), g = 1,, r} represent the relatve frequency hstogram for the combned set of observed hstograms. 1 Let I = (mn,b {x (b) }, max,b {x (b+1) }]: the nterval that spans all of the observed values of X. 2 Partton I nto r subntervals, I (g) = (ξ (g), ξ (g+1) ], g = 1,, r. DR for Hstogram Data June 13, / 27

52 Vsualzaton: A hstogram of hstograms A hstogram of all of the observed hstograms (Bllard and Dday, 23): Let {(I (g), p (g) ), g = 1,, r} represent the relatve frequency hstogram for the combned set of observed hstograms. 1 Let I = (mn,b {x (b) }, max,b {x (b+1) }]: the nterval that spans all of the observed values of X. 2 Partton I nto r subntervals, I (g) = (ξ (g), ξ (g+1) ], g = 1,, r. 3 The relatve observed frequency for the nterval I (g) s p (g) = 1 n n B =1 b=1 I (b) I (g) I (b) p (b), g = 1,, r, where I (b) (x (b), x (b+1) ]. DR for Hstogram Data June 13, / 27

53 Vsualzaton: a jont hstogram for Z 1 and Z 2 Let {R (g 1,g 2 ), p (g 1,g 2 ) } be the jont hstogram for two hstogram-valued varables, Z 1 and Z 2 (Bllard and Dday, 23): DR for Hstogram Data June 13, / 27

54 Vsualzaton: a jont hstogram for Z 1 and Z 2 Let {R (g 1,g 2 ), p (g 1,g 2 ) } be the jont hstogram for two hstogram-valued varables, Z 1 and Z 2 (Bllard and Dday, 23): 1 R (g 1,g 2 ) (ξ (g 1) 1, ξ (g 1+1) 1 ] (ξ (g 2) 2, ξ (g 2+1) 2 ], g 1 = 1,, r 1, g 2 = 1,, r 2. DR for Hstogram Data June 13, / 27

55 Vsualzaton: a jont hstogram for Z 1 and Z 2 Let {R (g 1,g 2 ), p (g 1,g 2 ) } be the jont hstogram for two hstogram-valued varables, Z 1 and Z 2 (Bllard and Dday, 23): 1 R (g 1,g 2 ) (ξ (g 1) 1, ξ (g 1+1) 1 ] (ξ (g 2) 2, ξ (g 2+1) 2 ], g 1 = 1,, r 1, g 2 = 1,, r 2. 2 p (g 1,g 2 ) = 1 n n B 1 B 2 =1 b 1 =1 b 2 =1 Π (b 1,b 2 ) R (g 1,g 2 ) Π (b 1,b 2 ) where Π (b 1,b 2 ) (x (b 1) 1, x (b 1+1) 1 ] (x (b 2) 2, x (b 2+1) 2 ]. p (b 1) 1 p (b 2) 2, DR for Hstogram Data June 13, / 27

56 Example: rs data (15 4, y=(5, 5, 5)) PCA of rs data PCA Setosa Verscolor Vrgnca PCA1 DR for Hstogram Data June 13, / 27

57 Example: rs data (15 4, y=(5, 5, 5)) PCA of rs data PCA Setosa Verscolor Vrgnca PCA1 Repeat (Wthn each class, 3 observatons were randomly sampled to generate hstograms for four varables) 5 tmes. DR for Hstogram Data June 13, / 27

58 Hstogram-PCA..3 DR for Hstogram Data June 13, / 27

59 Jont hstogram of hstogram-pca Jont Hstogram of Hstograms: ( DR1, DR2) [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) DR for Hstogram Data June 13, / 27

60 Jont hstogram for each speces Jont Hstogram of Hstograms: ( DR1, DR2) [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) DR for Hstogram Data June 13, / 27

61 Jont hstogram for each speces Jont Hstogram of Hstograms: ( DR1, DR2) [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) Jont Hstogram of Hstograms: ( DR1, DR2), cluster=1 [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8).8.4 [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7).2.1 [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) DR for Hstogram Data June 13, / 27

62 Jont hstogram for each speces Jont Hstogram of Hstograms: ( DR1, DR2) [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) Jont Hstogram of Hstograms: ( DR1, DR2), cluster=1 [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8).8.4 [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7).2.1 [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) Jont Hstogram of Hstograms: ( DR1, DR2), cluster=2 [1.9,2.3) [2.3,2.8).1.1 [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) DR for Hstogram Data June 13, / 27

63 Jont hstogram for each speces Jont Hstogram of Hstograms: ( DR1, DR2) [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) Jont Hstogram of Hstograms: ( DR1, DR2), cluster=1 [1.9,2.3) [2.3,2.8) [2.8,3.3) [3.3,3.8).8.4 [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7).2.1 [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) Jont Hstogram of Hstograms: ( DR1, DR2), cluster=2 Jont Hstogram of Hstograms: ( DR1, DR2), cluster=3 [1.9,2.3) [1.9,2.3) [2.3,2.8).1.1 [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) [2.3,2.8).1.1 [2.8,3.3) [3.3,3.8) [3.8,4.2) [4.2,4.7) [4.7,5.2) [5.2,5.7) [5.7,6.2) [1.9,2.7) [3.6,4.4) [5.3,6.1) [6.9,7.8) [8.6,9.5) DR for Hstogram Data June 13, /