Past, Present, and Future of Multidimensional Scaling Patrick J. F. Groenen *Econometric Institute, Erasmus University Rotterdam, The Netherlands, groenen@few.eur.nl, http://people.few.eur.nl/groenen/ Summary: 1 What is MDS? 2 Some Historical Milestones 3 Present 4 Future 5 Summary of highlights in MDS
1 What is MDS? • Table of travel times by train between 10 French cities: Bor- Mar- Strassb Tou- deaux Brest Lille Lyon seille Nice Parijs ourg louse Tours Bordeaux 0 Brest 9h58 0 Lille 6h39 7h11 0 Lyon 8h05 7h11 4h52 0 Marseille 5h47 8h49 6h12 1h35 0 Nice 8h30 13h36 8h20 4h33 2h26 0 Parijs 2h59 4h17 1h04 2h01 3h00 5h52 0 Strassbourg 8h08 10h16 6h54 4h36 7h04 11h15 4h01 0 Toulouse 2h02 13h52 9h42 4h25 3h26 6h29 5h14 10h56 0 Tours 2h36 5h38 4h17 4h21 5h13 9h04 1h13 6h03 6h06 0 Lille Strassbourg Brest Paris Strassbourg Brest Lille Paris Tours Tours Lyon Lyon Bordeaux Marseille Bordeaux Toulouse Nice Toulouse Nice Marseille MDS map of travel time by train. Geographic map of France. Past, Present, and Future of MDS – 2 –
dissimilarity matrix � O 1 O 2 O 3 � O n -1 O n O 1 0 O 2 δ 12 0 O 3 δ 13 δ 23 0 �� � � � � � O n -1 δ 1, n -1 δ 2, n -1 δ 3, n -1 � 0 O n δ 1 n δ 2 n δ 3 n δ 2 n � 0 � O coordinates matrix X 1 • O • n dim 1 dim 2 • � O 1 x 11 x 12 O n -1 O 2 x 21 x 22 O 3 x 31 x 32 O O 2 • 3 • � � � �� O n -1 x n -1,1 x n -1,2 O n x n 1 x n 1 Past, Present, and Future of MDS – 3 –
• First sentence in Borg and Groenen (2005): Multidimensional scaling (MDS) is a method that represents measurements of similarity (or dissimilarity) among pairs of objects as distances between points of a low- dimensional space. • Who uses MDS? – psychology, – medicine, – sociology, – chemistry, – archaeology, – network analysis – biology, – economists, etc. • Similarities and dissimilarities: – Large similarity approximated by small distance in MDS. – Large dissimilarity ( δ ij ) approximated by large distance in MDS. – General term: proximity. Past, Present, and Future of MDS – 4 –
2 Some Historical Milestones • 1635: van Langren: Provides a distance matrix and a map. Newcastle Durham Map of Durham county – Cartographer: Jacob van Langren – Date 1635 Past, Present, and Future of MDS – 5 –
• 1635: van Langren: Provides a distance matrix and a map. • 1958: Torgerson: Provides a solution for classical MDS based on eigendecomposition • 1966: Gower: Provides independently the same solution for classical MDS and gives connection to principal components analysis. ∆ (2) – D (2) ( X )) J || 2 Classical MDS: minimize Strain( X ) = 1/4|| J ( ∆ ∆ ∆ with J centering matrix by ∆ (2) J eigendecomposition of –½ J ∆ ∆ ∆ Past, Present, and Future of MDS – 6 –
• 1635: van Langren: Provides a distance matrix and a map. • 1958: Torgerson: Provides a solution for classical MDS based on eigendecomposition • 1966: Gower: Provides independently the same solution for classical MDS and gives connection to principal components analysis. • 1962: Shepard: Provides a heuristic for MDS. Past, Present, and Future of MDS – 7 –
• 1635: van Langren: Provides a distance matrix and a map. • 1958: Torgerson: Provides a solution for classical MDS based on eigendecomposition • 1966: Gower: Provides independently the same solution for classical MDS and gives connection to principal components analysis. • 1962: Shepard: Provides a heuristic for MDS. • 1964: Kruskal: Establishes least-squares MDS. Provides a minimization algorithm. Proposes ordinal MDS plus optimization ( ) � 2 ˆ − ( X ) d d ij ij i < j ˆ ) = Minimize Stress-I: σ I ( X , d � 2 d ( X ) ij i < j ˆ disparity satisfying a monotone with d ij relation with proximities. Past, Present, and Future of MDS – 8 –
– Classic example: Rothkopf (1957) Morse code confusion data + Is there some systematic way in which people confuse Morse codes? + 36 Morse code (26 for alphabet, 10 for numbers) + Subjects task: judge whether two Morse codes are the same or not. For example: + Is .- (N) the same as .-. (R)? Yes (1), or no (2) + Stimulus pair presented in two orders: pair NR and RN. + Each subject judges many combinations of Morse codes. + N = 598. + Morse code confusion table: proportion confused. + Data are similarities � A B C D 0 � .- A 92 4 6 13 3 � -... B 5 84 37 31 4 � -.-. C 4 38 87 17 12 � -.. D 8 62 17 88 6 � � � � � � � � � ----- 0 9 3 11 2 94 Past, Present, and Future of MDS – 9 –
– Classic example: Rothkopf (1957) Morse code confusion data ----- ----. .---- ---.. ..--- .--- --.- --- --... -.-- --.. .--. - --. . ...-- -.-. -.... -..- .-- -... .-.. -- -.- -. ....- ..-. .-. -.. ...- .... .- .... ..- .. ... Past, Present, and Future of MDS – 10 –
• 1635: van Langren: Provides a distance matrix and a map. • 1958: Torgerson: Provides a solution for classical MDS based on eigendecomposition • 1966: Gower: Provides independently the same solution for classical MDS and gives connection to principal components analysis. • 1962: Shepard: Provides a heuristic for MDS. • 1964: Kruskal: Establishes least-squares MDS. Provides a minimization algorithm. Proposes ordinal MDS plus optimization • 1964: Guttman: Facet theory and regional interpretation in MDS. – In facet theory, extra information (external variables) is available on the objects according to the facet design by which the objects are generated: Past, Present, and Future of MDS – 11 –
• 1964: Guttman: Facet theory and regional interpretation in MDS. + Every object i belongs to a category on one or more facets. + See, e.g., Guttman (1959), Borg & Shye (1995), Borg & Groenen (1997, 1998) Dissimilarity matrix � : Facet design � Facet O1 O2 O3 � O n -1 O n 1 2 3 O1 O1 0 1 1 3 O2 O2 δ 12 0 1 2 3 O3 O3 δ 13 δ 23 0 2 1 3 �� �� � � � � � � � � � O n -1 δ 1, n -1 δ 2, n -1 δ 3, n -1 � O n -1 0 3 1 1 O n O n δ 1 n δ 2 n δ 3 n δ 2 n � 0 3 2 1 – The extra facet information is used to partition the objects in the MDS space in regions. – Facets are used for regional hypotheses about the empirical structure of the data. c c c c c a a b a a a c c b b c c b a a c a c a a a a a b b b b c b c b a b a c a b c b c b b c b axial modular polar Past, Present, and Future of MDS – 12 –
• 1964: Guttman: Facet theory and regional interpretation in MDS. – For the Morse code data, we have additional information available: 1. Length of the signal (.05 to .95 seconds). 2. Signal type (ratio of long versus short beeps). Morse Morse Letter code Length Signal type Letter code Length Signal type A .- 25 1=2 S ... 25 1 B -... 45 1>2 T - 15 2 C -.-. 55 1=2 U ..- 35 1>2 D -.. 35 1>2 V ...- 45 1>2 E . 05 1 W .-- 45 1<2 F ..-. 45 1>2 X -..- 55 1=2 G --. 45 1<2 Y -.-- 65 1<2 H .... 35 1 Z --.. 55 1=2 I .. 15 1 1 .---- 85 1<2 J .--- 65 1<2 2 ..--- 75 1<2 K -.- 45 1<2 3 ...-- 65 1>2 L .-.. 45 1>2 4 ....- 55 1>2 M -- 35 2 5 ..... 45 1 N -. 25 1=2 6 -.... 55 1>2 O --- 55 2 7 --... 65 1>2 P .--. 55 1=2 8 ---.. 75 1<2 Q --.- 65 1<2 9 ----. 85 1<2 R .-. 35 1>2 0 ----- 95 1 S ... 25 1 Past, Present, and Future of MDS – 13 –
• 1964: Guttman: Facet theory and regional interpretation in MDS. – Borg and Groenen (2005): Regional restrictions through Proxscal, by specifying: + two dimensions + two external variables, + each variable is transformed ordinally using the primary approach ties. 1 1>2 1=2 2>1 2 1 1>2 1=2 2>1 2 22211 95 22222 95 12222 11122 11222 22111 22222 22221 12222 2122 85 2212 21111 22211 22221 1222 11222 2211 11112 11122 85 75 22111 2212 2112 1221 75 2211 11111 1222 222 2121 21111 2122 11112 2111 65 1221 1112 2112 65 1211 1121 221 222 11111 2111 2121 221 55 1112 1211 212 1121 55 1111 122 212 45 211 1111 122 211 45 112 121 35 112 121 22 111 22 111 2 35 15 25 21 12 1 12 05 11 21 25 2 05 15 11 1 Unconstrained Regionally constrained Past, Present, and Future of MDS – 14 –
• 1635: van Langren: Provides a distance matrix and a map. • 1958: Torgerson: Provides a solution for classical MDS based on eigendecomposition • 1966: Gower: Provides independently the same solution for classical MDS and gives connection to principal components analysis. • 1962: Shepard: Provides a heuristic for MDS. • 1964: Kruskal: Establishes least-squares MDS. Provides a minimization algorithm. Proposes ordinal MDS plus optimization • 1964: Guttman: Facet theory and regional interpretation in MDS. • 1969: Horan Dimension weighting models in 3-way MDS 1970: Carroll and Chang: Introduction (INDSCAL, IDIOSCAL) Past, Present, and Future of MDS – 15 –
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