刘仁沛教授-AppliedMultivariateQuantitativeMethods.ppt

Copyright by Jen-pei Liu, PhD and Wei-chu Chie, MD, PhD Applied Multivariate Quantitative Methods Multidimensional Scaling By Jen-pei Liu, PhD Division of Biometry, Department of Agronomy, National Taiwan University and Wei-Chie Chie, MD, PhD Department of Public Health National Taiwan University Multidimensional Scaling Introduction Procedures Examples Summary Introduction Objectives Given a distance matrix between objects To construct a diagram (map) showing the relationships between a number of objects Maps – one dimension, two dimensions, three dimensions or a high number of dimension (a simple geometrical representation is not possible) Introduction The mirror image of a map has the same distance matrix If n =3, the objects may not lie on a plane Multidimensional scaling is applied when the underlying relationship is unknown but the estimated distance matrix is available A tool for reduction of dimensions and visulaization An exploratory tool Introduction Distance matrix for 4 objects 1 2 3 4 1 0 2 6.0 0 3 6.0 9.5 0 4 2.5 7.5 3.5 0 Procedures Step 1: Start with a matrix of distance between n objects [n(n-1)/2] in p-dimensions, ?ij Step 2: Determine the number of dimension for multidimensional scaling, say t Step 3: A starting configuration is set up for the n objects in t dimensions Procedures Step 4: The Euclidean distances between the objects are computed for the assumed configuration. Let dij be object i and object j for this configuration Step 5: Fit a regression of dij on ?ij. Denote the fitted distances by d*ij which is called the disparities Procedures Step 6: Compute the STRESS to check the fit STRESS = (?(dij – dij*)2/ ?d2ij)1/2 Repeat 3-6 until the stress can not be further reduced Obtain the values of the n objects and to draw a map based on the new values Procedures The word stress is used because the statistic is a measure of the extent to which the spatial configuration of points has to be stressed to obtained the data distanc