1. Location Patterns -- Race

2. Segregation and Discrimination One of the major factors of the urban fabric is separation by race. Must distinguish between discrimination and segregation. Discrimination -- Different treatment Segregation -- Separate treatment What are the differences?

3. Measuring Segregation Suppose we have a city that has 4,000 people. Half are white, and half are minority. Suppose we have 4 neighborhoods. If the people were randomly distributed, each neighborhood would be half white and half minority.

4. Dissimilarity Index D = t 1 |p 1 - p*| +t 2 |p 2 - p*| + t 3 |p 3 - p*| + t 4 |p 4 - p*| D = 1000 |.2 -.5| + 1000 |.4 -.5| + 1000 |.6 -.5| + 1000 |.8 -.5| D = 300 + 100 + 100 + 300 = 800 Tells you how many people would have to be moved for there to be no segregation. Discuss 200 M 800 W 400 M 600 W 800 M 200 W 600 M 400 W

5. Dissimilarity Index What is MAXIMUM number? D = 1000 |.0 -.5| + 1000 |.0 -.5| + 1000 |1.0 -.5| + 1000 |1.0 -.5| D = 500 + 500 + 500 + 500 = 2000 Comparing our value of 800 to the maximum of 2000 gives an index of 800/2000 = 0.4 Multiply by 100  40. 200 M 800 W 400 M 600 W 800 M 200 W 600 M 400 W

6. Dissimilarity Index It can be shown that the maximum equals: 2T p*(1-p*) where T is the total population, p* is the minority %. So: MAX = 2 * 4000 * 0.5 * 0.5 = 2,000 D = {  t i |p i - p*|}/[2Tp*(1-p*)] What does D=0 mean? What does D=100 mean? 200 M 800 W 400 M 600 W 800 M 200 W 600 M 400 W

7. Other Indices D SEGREGATION INDEX IS DIFFICULT TO USE FOR SOME THINGS, BECAUSE OF THE ABSOLUTE VALUE OPERATORS. THERE ARE AT LEAST 2 OTHER TYPES: VARIANCE BASED - YOU LOOK AT VARIATIONS FROM THE GRAND MEAN S* = [  T i * (P i -P*) 2 ]/TP*(1-P*)

8. Variance Based Index VARIANCE BASED – Provides a quadratic loss from segregation. S* = [  T i * (P i -P*) 2 ]/TP*(1-P*) Can also decompose this in an analysis of variance format, for example, within municipalities m, since: (P i -P*) 2 = [(P i – P m ) + (P m – P*)] 2.  Variance within Variance among

9. Information-Based Both D and S constrain the number of groups to 2. Information theory allows more groups. Good for looking at more diverse populations. Basically, the measure suggests that if you know the grand mean, it gives you some information about a given neighborhood. The more segregation, the easier it is to make an inference.

10. H = 1 –  T i E i /TE Where: E i = p i log (1/p i ) + (1 – p i ) log (1/(1-p i ), and E = p* log (1/p*) + (1 – p*) log (1/(1-p*) With logs to the base 2, the index is constrained to between 0 and 1. Note: lim as p i  0 of p i log (1/p i ): Using L’Hopital’s Rule, take derivative: Information-Based

11. p log (1/p) = p (log 1 – log p) = -p log p. Let m = log p; n = -1/p. Then m = 1/p; n = -1/p 2. So m / n = -p. The limit, as p  0 is -p, or 0. Same with other term as it approaches 0 or 1.

12. Information-Based H = 1 –  T i E i /TE Where: E i = p i log (1/p i ) + (1 – p i ) log (1/(1-p i ), and E = p* log (1/p*) + (1 – p*) log (1/(1-p*) Intuition: If p i = p* everywhere, numerator = denominator, and H  0. As p i are 0, or 1, E i  0, 2 nd term gets smaller, and H increases. D, S, and H move together (generally) but they are different.