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1 MDS Analysis of the Visibility Data for the Combinations of Foreground and Background Colors by Maximum ikelihood Asymmetric MDS Using the Multiplicative Error Model Shingo Saburi A maximum likelihood method for asymmetric multidimensional scaling, which was proposed by Saburi and Chino (Comput. Stat. Data Anal., 52: , 2008), uses the additive error model in which the normally distributed error terms are added to the dissimilarities. In this study, we introduce in this method the multiplicative error model, where the log-normally distributed error terms are multiplied by the dissimilarities, and the corresponding representation of the dissimilarities. It was applied to the visibility data for the combinations of foreground and background colors, assuming the multiplicative as well as the additive error model. The optimal model was found with the multiplicative error model according to AIC. Key words: Asymmetric multidimensional scaling, Color scheme, og-normal distribution, Maximum likelihood method, Visibility 1. orld ide eb Consortium (2000) 125 i = 299Ri +587Gi +114Bi. (1) 1000 R i G i B i i RGB RGB UV Institute for Psychological and Physical Science, Aichi-Gakuin University Tel Fax saburi@dpc.agu.ac.jp UV U V (2008) Saburi & Chino (2008) MDS ASMMAXSCA Okada & Imaizumi (1987) UV (2008) ASMMAXS- CA Takane (1981) MDS
2 Takane (1981) AIC (Akaike, 1974) ASMMAXSCA Takane (1981) additive error model multiplicative error model ASMMAXSCA Okada & Imaizumi (1987) Ramsay (1977) Takane (1981) AIC ASMMAXSCA (2008) (2008) AIC (2008) 2. ASMMAXSCA Saburi & Chino (2008) Okada & Imaizumi (1987) OI g ij = d ij r i + r j. (2) d ij i j { A } 1/2 d ij = (x ia x ja) 2 a=1 (3) r i i x ia a i A g ij (g ij + g ji)/2 (g ij g ji)/2 i j Saburi & Chino (2008) g ij saturated representation model: SR OI Euclidean distance ED g ij = d ij g ij = g ji SR g ij Saburi & Chino (2008) { τ ij = g ij + e ij e ij N(0, σ 2 (4) ) the law of categorical judgment (Torgerson, 1958) = b 0 b 1 b M 1 b M =. M i j m p ijm = bm b m 1 f(τ ij) dτ ij (5) f g ij σ 2 (τ ij g ij)/σ z p ijm = aijm a ij(m 1) φ(z) dz. (6)
3 MDS 211 φ bm gij a ijm = σ bm 1 (7) gij a ij(m 1) = σ b m Saburi & Chino (2008) b 1 b 2 b M 1 (8) b m = αm + β (α >0). (9) = i,j M m=1 p ijm ijm (10) ijm i j m i j ln AIC p ijm = p jim H (cs) 0 : p ijm = p jim (1 i<j n; 1 m M 1), H (s/sr) 0 : g ij = g ji (1 i<j n), H (s/oi) 0 : r 1 = r 2 = = r n. n SR OI 3. ASMMAXSCA { τ ij = g ije ij ln e ij N(0, σ 2 ). p ijm p ijm = ln bm (11) ln b m 1 g( τ ij) d τ ij (12) τ ij =lnτ ij g ln g ij σ 2 b m ln b 0 ln b M ( τ ij ln g ij)/σ z (6) (7) ln bm ln gij a ijm = σ ln bm 1 ln (13) gij a ij(m 1) =. σ (2) OI g ij (2010) (2) g ij = d ij rj r i. (14) (, 2010, 2012) (2011) (14) g ij (2012) (14) (12) (14) OI ASMMAXSCA ln r i r i OI ln g ij ln g ij =lnd ij r i + r j. (15) OI d ij OI ln g ij =lnd ij ED b m
4 (8) Takane (1981) b m = βm α (α>0; β>0) (16) ln b m b m = β exp(m) α (α>0; β>0) (17) τ ij ln b m bm ) ln b m = α ln m + β (16) ) (18) αm + β (17) ). 4. SR SR b m ln g ij g ij g ij > 0 (9) (17) (16) b m = α ln m + β x ia OI ED g ij SR SR s(> 0) (7) b m α β σ r i OI s (13) b m β ln s OI r i OI g ij OI r i r i 5. ASUS EeePC 1015PEM-BK 10.1 inch TFT dpi 52 cm MS Gothic; 72 ; orld ide eb Consortium (2011) ASMMAXSCA = white red 5 SR SD = ± XZ CS ASMMAXSCA g ij (2008)
5 MDS XZ X Z aqua A black blue Be fuchsia F gray green Gn lime maroon M navy N olive Ol orange Or purple P red R silver S teal T white yellow i j (10) ijm i j m m ijm Saburi & Chino (2008) ED 2 3 AIC OI ED SR SR b m OI AIC 3 OI ED 4 H (s/oi) 0 H (s/sr) 0 5% (2008) UV UV 1 XZ standard RGB srgbebner (2007, pp.87 89) (1) U i = B i i U V i = R i i V U i V i d ij UV OI ED d ij ={wy( 2 i j) 2 +wu(u 2 i U j) 2 +wv(v 2 i V j) 2 } 1/2. (19) w y w u w v AIC 3) d ij OI ED { {( i j) 2 } 1/2 (A=1) d ij = {( i j) 2 + A 1 a=1 (xia xja)2 } 1/2 (A>1). (20) UV orld ide eb Consortium (2000) b m OI Gower & Dijksterhuis 2004, p b m OI AIC 2 r i Borg & Groenen 2005, pp r i 3
6 OI g ij = d ij r i + r j τ ij N(g ij,σ 2 ) ln g ij =lnd ij r i + r j τ ij N(ln g ij,σ 2 ) ED g ij = d ij τ ij N(g ij,σ 2 ) τ ij N(ln g ij,σ 2 ) UV OI UV ED UV OI UV ED SR SR (19) g ij = d ij r i + r j τ ij N(g ij,σ 2 ) ln g ij =lnd ij r i + r j τ ij N(ln g ij,σ 2 ) (19) g ij = d ij τ ij N(g ij,σ 2 ) τ ij N(ln g ij,σ 2 ) (20) g ij = d ij r i + r j τ ij N(g ij,σ 2 ) ln g ij =lnd ij r i + r j τ ij N(ln g ij,σ 2 ) (20) g ij = d ij τ ij N(g ij,σ 2 ) τ ij N(ln g ij,σ 2 ) g ij g ij = g ji g ij p ijm p ijm = p jim p ijm b m OI r i 4 2 Schönemann & Carroll (1970) r i b m 95% 7. 3 b m UV b m OI AIC (2008) OI AIC OI OI AIC OI b m OI UV (2008) orld ide eb Consortium (2000) 2 3
7 MDS AIC b m b m =αm+β b m ln b m =αm+ β ln b m =α ln m+ β OI ( 52) ( 49) ( 52) ( 49) ( 49) ( 66) ( 63) ( 66) ( 63) ( 63) ( 79) ( 76) ( 79) ( 76) ( 76) ( 91) ( 88) ( 91) ( 88) ( 88) (102) ( 99) (102) ( 99) ( 99) (112) (109) (112) (109) (109) (121) (118) (121) (118) (118) ED ( 36) ( 33) ( 36) ( 33) ( 33) ( 50) ( 47) ( 50) ( 47) ( 47) ( 63) ( 60) ( 63) ( 60) ( 60) ( 75) ( 72) ( 75) ( 72) ( 72) ( 86) ( 83) ( 86) ( 83) ( 83) ( 96) ( 93) ( 96) ( 93) ( 93) (105) (102) (105) (102) (102) UV OI ( 24) ( 21) ( 24) ( 21) ( 21) UV ED ( 8) ( 5) ( 8) ( 5) ( 5) UV ( 22) ( 19) ( 22) ( 19) ( 19) OI ( 38) ( 35) ( 38) ( 35) ( 35) ( 53) ( 50) 3.45 ( 53) ( 50) ( 50) ( 67) ( 64) 6.11 ( 67) ( 64) ( 64) ( 80) ( 77) ( 80) ( 77) ( 77) ( 92) ( 89) ( 92) ( 89) ( 89) (103) (100) (103) (100) (100) (113) (110) (113) (110) (110) UV ( 6) ( 3) ( 6) ( 3) ( 3) ED ( 22) ( 19) ( 22) ( 19) ( 19) ( 37) ( 34) 5.37 ( 37) ( 34) ( 34) ( 51) ( 48) 8.10 ( 51) ( 48) ( 48) ( 64) ( 61) ( 64) ( 61) ( 61) ( 76) ( 73) ( 76) ( 73) ( 73) ( 87) ( 84) ( 87) ( 84) ( 84) ( 97) ( 94) ( 97) ( 94) ( 94) b m b m = αm + βb m = α ln m + β (1360) SR (276) (273) (273) ( 680) SR (140) (137) (137) AIC 3630 black H S V HSV white silver S g ij i j
8 b m b m = αm + β χ 2 p χ 2 p H (s/oi) b m ln b m =αm+ β ln b m =α ln m+ β χ 2 p χ 2 p χ 2 p H (s/oi) b m b m = αm + β b m = α ln m + β χ 2 p χ 2 p χ 2 p H (s/sr) χ 2 p H (cs) Ol T R Be Gn P M N F Or 150 S A b m OI UV 1 (15) (15) r i 2 r i purple P red R r i silver S lime 5 C 2 C 3 C 4
9 MDS 217 A S F Or Be OlT R Gn P N M A F Be Ol T R P Gn M N S Or Dim. 1 Dim Dim. 2 A S P F N M Ol Gn T Be Dim. 1 Or R estimate of r ~ i A Be N Or R T F Gn M Ol P S r i r i 1 UV C 5 2 r i 4 r i UV 1 2 black maroon M white aqua A black 2 4 white aqua white aqua A Dim. 1 S Gn Be N P Ol T 1 UV M F R Or Dim. 2
10 A S F Or Ol T R Be P Gn N M A S F Or R Be T Ol P Gn M N Dim. 1 Dim Dim. 2 A S P F N M Ol Gn T Be Dim. 1 Or R estimate of r i A Be N Or R T F Gn M Ol P S r i 2 r i 1 UV C C C C C C b m 95%C 1,...,C 6 3
11 MDS 219 Indow, 1988 UV ln g ij r i Indow, 1988 Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control. 19, Borg, I., & Groenen, P.J.F. (2005). Modern multidimensional scaling: Theory and applications (2nd ed.). New ork: Springer. Ebner, M. (2007). Color Constancy. Chichester: iley. Gower, J.C., & Dijksterhuis, G.B. (2004). Procrustes problems. New ork: Oxford university press. Indow, T. (1988). Multidimensional studies of Munsell color solid. Psychological review, 95, (2005).., 29, Okada, A., & Imaizumi, T. (1987). Nonmetric multidimensional scaling of asymmetric proximities. Behaviormetrika, 21, (2010).. 38, (2011). MDS. 39, (2012). MDS. 40, Ramsay, J.O. (1977). Maximum likelihood estimation in multidimensional scaling. Psychometrika, 42, (2008).., 35, Saburi, S., & Chino, N. (2008). A maximum likelihood method for an asymmetric MDS model. Computational Statistics and Data Analysis, 52, Schönemann, P.H., & Carroll, R.M. (1970). Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 35, Takane,. (1981). Multidimensional successive categories scaling: A maximum likelihood method. Psychometrika, 46, Torgerson,.S. (1958). Theory and methods of scaling. New ork: iley. orld ide eb Consortium (2000). Techniques for accessibility evaluation and repair tools. ( ). orld ide eb Consortium (2011). Cascading style sheets level 2 revision 1 (CSS 2.1 ) specification - 4 Syntax and basic data types. ( w3.org/tr/2011/rec-css / syndata.html). (1983).., 7, (1986).., 40,
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