Applied Multivariate Quantitative Methods Multidimensional Scaling By Jenpei
Applied Multivariate Quantitative Methods Multidimensional Scaling By Jen-pei Liu, Ph. D Division of Biometry, Department of Agronomy, National Taiwan University and Wei-Chie, MD, Ph. D Department of Public Health National Taiwan University 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 1
Multidimensional Scaling n n Introduction Procedures Examples Summary 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 2
Introduction n Objectives n n n 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) 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 3
Introduction n n 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 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 4
Introduction n Distance matrix for 4 objects n n 1 2 3 4 10/7/2020 1 0 6. 0 2. 5 2 3 4 0 9. 5 7. 5 0 3. 5 0 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 5
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Procedures n n n Step 1: Start with a matrix of distance between n objects [n(n-1)/2] in pdimensions, 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 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 8
Procedures n n 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 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 9
Procedures n n n Step 6: Compute the STRESS to check the fit STRESS = ( (dij – dij*)2/ d 2 ij)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 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 10
Procedures n n 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 distances ij. Small value of stress is desirable Increase the number of dimension when stress is already less than 0. 05 Reduce the number of dimension to the extent if stress exceeds 0. 1 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 11
Examples n Road distances between 13 New Zealand Towns n n n Road distances are proportional to geographical distances Recover the true map by 2 -dimensional analysis Stress value is 0. 043 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 12
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Examples New Values after multidimensional scaling (2) Town 1 2 Alexandra 0. 11 -0. 07 Invercargill 0. 26 0. 01 Balclutha 0. 19 0. 08 Milford 0. 36 -0. 13 Blenheim -0. 38 0. 16 Nelson -0. 45 0. 08 Christchurch -0. 15 0. 11 Queenstown 0. 13 -0. 12 Dunedin 0. 13 0. 10 Te Anau 0. 28 -0. 08 Franz Josef -0. 18 -0. 20 Timaru -0. 03 -0. 13 Greymouth -0. 27 -0. 06 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 14
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Examples n n Very successful in recovering the real map Town are in correct relationship except Milford which can be reached only by road through Te Anau 2 -dimension map: Milford is close to Te Anau Geographical map: Milford is close to Queenstown 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 16
Examples n n Voting behavior of 15 New Jersey congressmen Distances are based on the number of voting disagreement on 19 bills concerned with environmental matters 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 17
Examples n n Metric multidimensional scaling: doubling a distance value is equivalent to configuration distance between 2 objects is also doubled. Assume a linear relationship between dij and ij. Nonmetric multidimensional scaling: only assume monotonic relationship between dij on ij. 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 18
Examples Stress n Dimmension 1 2 3 10/7/2020 Metric Nonmetric 0. 237 0. 130 0. 081 0. 113 0. 066 0. 044 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 19
Examples Values after multidimensional Scaling Congressmen 1 2 3 Hunt (R) 0. 33 0. 00 0. 09 Sand (R) 0. 26 0. 18 Howard (D) -0. 21 0. 05 0. 11 Thompson (D) -0. 12 0. 22 -0. 03 Frelinghuysen (R) 0. 20 -0. 06 -0. 24 Forsythe (R) 0. 13 -0. 06 Windnall (R ) 0. 33 0. 00 -0. 11 Roe (D) -0. 21 -0. 05 0. 09 Helstoski (D) -0. 22 0. 02 -0. 01 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 20
Examples Values after multidimensional Scaling Congressmen 1 2 Rodino (D) -0. 22 -0. 07 Minish (D) -0. 16 -0. 03 Rinaldo (R) -0. 18 0. 01 Maraziti (R) 0. 19 -0. 20 Daniels (D) -0. 02 -0. 09 Pattern (D) -0. 16 0. 05 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 3 0. 00 -0. 02 -0. 01 0. 10 0. 03 -0. 12 21
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Examples n Voting pattern of 15 NJ congressmen n n Dimension 1: party difference Dimension 2: number of abstentions from voting Dimension 3: no simple or obvious interprestation Configuration distance vs. disparities n n 10/7/2020 Not a straight linear relationship Data distances do not increase with the configruation distances Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 24
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Examples n Microarray data n 32 patients with prostate cancers n n n 23 patients from primary sites with no known metastatasis 9 patients from metastatic disease site Expression data from 12626 genes 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 26
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Summary n n Objectives Procedures n n n Stress Effect of scale Real Data n n n Distance between cities Voting pattern Gene expression levels 10/7/2020 Copyright by Jen-pei Liu, Ph. D and Wei-chu Chie, MD, Ph. D 30
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