1. Delineating Metropolitan Housing Submarkets with Fuzzy Clustering Methods Julie Sungsoon Hwang Department of Geography, University of Washington Jean-Claude Thill Department of Geography, State University of New York at Buffalo November 10, 2005 North American Meetings of Regional Science Association International

2. Outlines Research objectives Methodology: specification Methodology: illustration Evaluating the performance of fuzzy clustering Conclusions

3. Research objectives Demonstrate the use of fuzzy c-means (FCM) algorithm for delineating housing submarkets –Comparison to K-means Discuss empirical characteristics of FCM applied to given applications, in particular choice of parameters –Cluster validity index

4. Challenges Are the boundaries of clusters crisp? Cluster A Cluster C X1X1 X2X2 Housing market in metropolitan area q Cluster B Cluster A Cluster B Cluster C X1X1 X2X2 Housing market in metropolitan area p

5. Methodology: specification

6. Our task is to group census tracts to homogeneous housing submarkets within a metropolitan area Using fuzzy c-means algorithm In order to examine whether fuzzy set-based clustering can do the better job Implemented in 85 metropolitan areas Most of data set are public (e.g. 2000 Census) The whole procedure is automated in GIS

7. Methodology: flow chart National Regional Local … Census Tract Layer #x1x1 x2x2 x3x3 …xmxm 1 2 3 … n #y1y1 y2y2 …ykyk 1 2 3 … n Cluster Analysis #U1U1 U2U2 …UcUc 110…0 201…0 …01…0 n00…1 #U1U1 U2U2 …UcUc 10.850.05…0.10 20.120.80..0.05 …0.020.74…0.12 n0.400.03…0.50 K-means Fuzzy C- means Candidate variables Significant variables Stepwise regression(k ≤ m) Metro Hard Cluster Layer (c ≤ n) Fuzzy Cluster Layer … 12c12c k: # selected variables c: # submarkets For each metropolitan area U j : membership to cluster j

8. Explanatory variables for house price Var_NameVariable DefinitionDataYearSpatial Unit Socioeconomic/demographic Characteristics of Residents pcincomeper capita incomeCensus2000Census Tract college% college degreeCensus2000Census Tract managep% management workersCensus2000Census Tract prodp% production workersCensus2000Census Tract famcpchl% family with childrenCensus2000Census Tract nfmalone% nonfamily living aloneCensus2000Census Tract black_p% blackCensus2000Census Tract nhwht_p% non-hispanic whiteCensus2000Census Tract nativebr% native bornCensus2000Census Tract Structural Characteristics of Housing Units medroommedian number of roomCensus2000Census Tract hudetp% detached housing unitCensus2000Census Tract yrhubltmedian year structure builtCensus2000Census Tract Locational Characteristics (Amenities) of Neighborhoods ptratiopupil to teacher ratioNCES*2002School District schexpschool expenditure per studentNCES2002School District vrlcrimeviolent crime rateFBI**2003Designated Place prpcrimeproperty crime rateFBI2003Designated Place jobacmjob accessibility (Hansen 1959)CTPP***2000Census Tract *National Center for Education Statistics; **FBI annual report “Crime in the U.S. 2003”; *** CTPP: Census Transportation Planning Package Dependent variables: median home value of owner-occupied housing units

9. Study set: 85 metropolitan areas

10. Clustering method that minimizes the following objective function: Updates cluster means v i and membership degree u ik until the algorithm converges Vectors of data point, 1 ≤ k ≤ n Center of cluster i, 1 ≤ i ≤ c Membership degree of data point k with cluster i; [0,1] Fuzziness amount associated with assigning data point k to cluster i, 1≤ m ≤ ∞ Source: Bezdek 1981 x1x1 x2x2 What is fuzzy c-means (FCM)? (III-3a) (III-3b)

11. FCM: missing elements Optimal number of clusters c* Optimal fuzziness amount m* m c FCM

12. Extended fuzzy c-means algorithm Step 1: Initialize the parameters related to fuzzy partitioning: c = 2 (2 ≤ c  cmax), m = 1 (1 ≤ m  mmax), where c is an integer, m is a real number; Fix minc where minc is incremental value of m ( 0 < minc ≤ 0.1); Fix cut-off threshold  L; Choose validity index v Step 2: Given c and m, initialize U(0) so that it becomes the fuzzy matrix. Then at step l, l = 0, 1, 2, ….; Step 3: Calculate the c fuzzy cluster centers {vi(l)} with (III-3a) and U(l) Step 4: Update U(l+1) using (III-3b) and {vi(l)} Step 5: Compare U(l) to U(l+1) in a convenient matrix norm; if || U(l+1) – U(l) || ≤  L to go step 6; otherwise return to Step 3. Step 6: Compute the validity index for given c and m Step 7: If c < cmax, then increase c  c + 1 and go to step 3; otherwise go to step 8 Step 8: If m < mmax, then increase m  m + minc and go to step 3; otherwise go to step 9 Step 9: Obtain the optimal validity index from, optimal number of clusters c*, and optimal amount of fuzziness exponent m*; The optimal fuzzy partition U is obtained given c* and m*

13. Cluster validity indices Partition coefficientPartition entropy Xie-Beni index SVi index where w is set to 2 in this study

14. Selected validity indices are calibrated over the study set Xie-Beni index is recommended as a validity index Average m* is 1.38 Determining c* and m*

15. Histogram of m* for FCM

16. Methodology: illustration

17. Median home value of Buffalo, NY

18. Dimensionality of Buffalo housing market PredictorCoefficientStandard Errort-statisticsp-value Constant-1455768164417-8.850.000 Per capita income2.36670.27918.480.000 % college degree88221113467.780.000 % family: couple with children65735187753.500.001 % detached housing unit-312605527-5.660.000 Housing age (year)692.8880.268.630.000 % non-hispanic white1118639142.860.005 % native born status130039311114.180.000 Job accessibility-0.052660.02227-2.360.019 Hedonic regression equation of median home value in Buffalo, NY Adjusted R sq = 84.3%

19. Optimal number of housing submarkets c*, Optimal fuzziness amount m*, Buffalo, NY c m1.01.11.2 1.3 1.41.51.61.71.81.9 20.47350.45700.43808.098310.411512.547814.433416.063417.464518.6721 3 0.41360.38890.3460 0.3385 10.786412.913714.793916.421717.829019.0553 40.78020.71160.60800.52411.31546.88377.48078.04418.56329.0391 50.55600.56220.59400.61210.46830.34040.64890.68500.72060.7555 60.62230.75781.01870.81730.69071.33931.40741.48191.55951.6382 70.88360.69030.68810.60160.61480.95152.43972.63062.83173.0383 80.59810.58880.57030.52320.39920.73810.89101.23881.29261.3538 90.96450.61600.48360.48660.84491.40201.41981.83171.86391.9161 100.70530.60040.66190.58730.58681.34651.50811.68751.82151.8591 c*3333855555 Values in the cell represent Xie-Beni index given c and m

20. c* = 3; m* = 1.3 Membership to Cluster 1Membership to Cluster 2 Membership to Cluster 3Defuzzified Clusters Buffalo housing submarkets

21. Evaluating the performance of fuzzy clustering

22. Compare the sum of squared error derived from KM (m=1) and FCM (m=m*) given c* Fuzzy clustering outperforms crisp clustering Compare FCM with K-means (KM)

23. Conclusions Fuzzy set theory provides a mechanism for uncertainty handling involved in classification task Fuzzy c-means algorithm is of practical use in delineating housing submarkets Fuzzy set theory needs further attention in social science fields More works on the choice of parameters are needed