Multidimensional Scaling in R: SMACOF

Transcription

1 Multidimensional Scaling in R: SMACOF Patrick Mair Institute for Statistics and Mathematics WU Vienna University of Economics and Business Jan de Leeuw Department of Statistics University of California, Los Angeles (UCLA)

2 Workshop Content Throughout the workshop we use the R package smacof available on CRAN. Materials at Symmetric SMACOF Spherical SMACOF 3-Way SMACOF Constrained SMACOF Rectangular SMACOF De Leeuw, J. & Mair, P. (2009). Multidimensional Scaling using Majorization: SMACOF in R. Journal of Statistical Software, 31(3), p URL:

3 Multidimensional Scaling (MDS) Family of data-analytic methods which represent distances between objects in a low-dimensional space. Input structure: Dissimilarity matrix. Computation: Optimize corresponding target function. Output: Congurations in low-dimensional space. Visualization: Conguration plot.

4 MDS Models Classical (metric) scaling (Torgerson, 1952): data on interval level; strain loss, stress loss. Nonmetric MDS (Shepard, 1962): data on ordinal level; monotonicity constraint on dissimilarities. Individual dierences: K dissimilarity matrices. Unfolding: Preference data; n 1 raters, n 2 objects. Other related models: Procrustes, CA, Gi.

5 R Implementation: SMACOF package The R Project for Statistical Computing R is an open source software environment for statistical computing and graphics packages available The smacof package is available on CRAN: PsychoR:

6 Symmetric SMACOF Distance matrix of dimension n n with elements δ ij. Problem to solve: Locate points (congurations) in a p- dimensional space such that the distances d ij (X) between the points approximate δ ij. Conguration distances: d ij (X) = p s=1 (x is x js ) 2 Minimize stress (Majorization; de Leeuw, 1977): σ(x) = i<j w ij (δ ij d ij (X)) 2 min! Example 1: French Map

7 Nonmetric symmetric SMACOF Dissimilarities δ ij on ordinal scale. Monotonic transformation f: δ ij < δ i j f(δ ij) < f(δ i j ). Achieved by monotone regression t in each iteration (PAVA). Nonmetric stress: σ(x, D) = i<j w ij ( ˆd ij d ij (X)) 2 Disparities ˆd ij = f(δ ij ) Example 2: Political statements

8 3-Way SMACOF SMACOF for individual dierences: k = 1,..., K separate symmetric distance matrices. Stress: σ(x ) = K k=1 i<j w ij,k (δ ij,k d ij (X k )) 2. INDSCAL (unrestricted; Carrol & Chang, 1970). Restrictions on the conguration weight matrix X k = ZC k (diagonal, identity). IDIOSCAL (Carrol & Wish, 1974). Example 3: Wine tasting

9 Wine Tasting: Descriptives Price Alcohol Mean Rating Jurtschitsch Chardonnay Ziniel Chardonnay Markowitsch Chardonnay Ritinitis Noble Retsina Retsina Krems Chardonnay Castel Nova Chardonnay

10 Spherical SMACOF Restrictions on the congurations (weakly constrained MDS, Cox & Cox, 1991). x i Λx i + 2x i β + γ = 0 R 2 : circle, ellipse, hyperbola, parabola. R 3 : sphere, ellipsoid, hyperboloid, paraboloid, cylinder. Optimization: Primal and dual methods available. Example 4: Trading volume

11 Constrained SMACOF Restrictions by means of external constraints (conrmatory MDS, de Leeuw & Heiser, 1980). X = ZC where Z is a known predictor matrix. Decomposition: X = [ X 1 ZC 1 C 2 ]. X 1 is the unrestricted part and of dimension n q 1. ZC 1 is the linearly restricted part of dimension n q 2. C 2 is a diagonal matrix of order n.

12 Constrained SMACOF Triplet notation (q 1, q 2, q 3 ): q 1 is the number of unrestricted dimensions. q 2 the number of linearly restricted dimensions. q 3 is either zero or one (absence/presence). Important special case: Bentler-Weeks uniqueness model (q, 0, 1). Additional options: Simplex and circumplex tting. Example 5: Rectangles

13 Rectangular SMACOF (Unfolding) Rectangular n 1 n 2 preference matrix. Stress becomes σ(x) = n 1 n 2 i=1 j=1 Judge n 1 p conguration matrix. Object n 2 p conguration matrix. Example 6: Company rating w ij (δ ij d ij (X 1, X 2 )) 2 min!

14 References De Leeuw, J. & Mair, P. (2009). Multidimensional Scaling using Majorization: SMACOF in R. Journal of Statistical Software, 31(3), p URL: Borg, I., & Groenen, P. J. F. (2005). Modern Multidimensional Scaling. New York: Springer. Cox, T. F., & Cox, M. A. A. (2001). Multidimensional Scaling (2nd edition). Boca Raton, FL: Chapman & Hall/CRC.

15 Links and Contact PsychoR project: Website: Additional PsychoR topics: correspondence analysis (anacor), Gi optimal scaling (homals), isotone optimization (isotone), aspect framework (aspect). Patrick Mair Institute for Statistics and Mathematics WU Vienna University of Economics Augasse Vienna Website: