A Typology of Spatial Patterns The Kulldorff's D plots are very informative with respect to clustering/avoidance patterns for a particular retail type in a particular city. To compare and contrast the patterns of 108 plots across 54 retail types in two cities without information overload, we next classify the results into a limited number of categories. Two complications arise when doing this. First, the absolute value of the Kulldorff's D statistic that is likely to be outside a Monte Carlo confidence interval for a given distance varies systematically with the proportion of cases to controls. The more cases—the number of outlets of a store type—the smaller the value of D that is likely to be viewed as just barely significant clustering or avoidance. We develop a measure (Mi) that addresses this problem by normalizing the value of the Kulldorff's D statistic by the upper or lower 90% confidence limits. Specifically, for D taken at distance i, the measure Mi is calculated as (4) DiU is the 95th percentile confidence level for the Kulldorff's D statistic, and DiL is the fifth percentile confidence level at distance i. Thus, Mi measures the proportion of the confidence interval attained by Di. A second problem is that the Kulldorff's D statistic is a continuous measure over distance. Creating a typology is facilitated by converting it to a more discrete measure. We address this issue by calculating Mi at 200 meter intervals for each retail category in each city. This results in 50 values for each category, which, while retaining interesting detail, is too many to easily summarize with a parsimonious categorization scheme. Inspection of how rapidly the plots change with distance and how much they differ from each other suggest that the main discriminating features of the spatial distributions could be captured with a data point every two kilometers. This resolution reduces the number of values characterizing each store type to five, which is both manageable and retains the overall location structure of the store types. Thus, the Mi values were averaged over each 2 kilometer quintile of the 10 kilometer range. These five values are then used to classify retail store types.
1. Empirical Research Study Data Type Agglomeration Measure Demand Control City Store Types and Results Rogers (1965) Density Distribution fit; scalar None Stockholm From most to clustered: antiques, clothing, furniture, grocery, tobacco liquor Rogers (1969) Density Distribution fit; scalar None Ljubjana, San Francisco From most to least clustered: clothing, all stores, food stores, grocery stores Rogers and Martin (1971) Density Distribution fit; scalar Population Ljubjana Food stores, ambiguous results Lee and Koutsopoulos (1976) Density Distribution fit; scalar Regress results on population density (not point), demographics; wear or no results Denver Convenience stores of same chain cluster, different chains avoid each other Lee (1979) Point Nearest neighbor distribution fit; scalar None Hong Kong, Phoenix, Atlanta Western and Chinese grocery stores cluster; convenience stores random overall, chains each other Lee and Schmidt (1980) Point Nearest neighbor distribution fit; scalar Regress station density on households, demographics, trips into area densities (not point); weak or no effects Hong Kong, Denver Gasoline stations cluster Fischer and Harrington (1996) Point Nearest Neighbor None Baltimore From most to least clustered: shoes, antiques, computers, automobiles, clinics, gasoline stations, video stores, supermarkets, theaters Netz and Taylor (2002) Regression coefficient; relative within type; scalar Population demographics Los Angeles Gasoline stations avoid each other more as competitive intensity increases (not an absolute measure) Jensen, Boisson, and Larralde (2005) Point Average store counts in contiguous sites None Lyon From most to least clustered: motorbikes, banks, groceries, hairdressers, laundries, drugstores, savings banks Picone, Ridely, and Zandbergen (2009) Point Nearest neighbor scalar and vector; relative within store type Population demographics on scalar; assume constant effects on compared stores for vector Birmingham, Chicago, Minneapolis- St. Paul, Oakland, Tampa On site alcohol retailers cluster more than off site alcohol retailers
Measure Definition Source K-Function A function specifying the expected (theoretical) or empirical (average) number of points (retail locations) within a radius d around any single (retail location) as a function of the radius d from the central point. For a random Poisson process, for example, K increases with the square of d. Ripley (1981) Kulldorff's D A measure of clustering or avoidance that compares the difference in the local density of a set of “cases” (retail outlets of a focal type) to the local density of a set of “controls” (all other retailers). Kulldorff (1998) Mi A normalization of Kulldorff's D statistic that better enables comparisons across retail categories by the 90% Monte Carlo confidence intervals to scale Kulldorff's D values. This paper Table 4. Pattern Membership Criteria Pattern Membership Criteria Hyper agglomeration Mi > 1 for all quintiles agglomeration Mi > 1 for the first and/or second quintile, but not all five, and no quintiles have Mi < -1 Auto mall Mi > 1 for the first and/or second quintile, and Mi < -1 for at least one of the third, fourth, or fifth quintiles Avoidance Mi < -1 for at one quintile, and Mi is never greater than one. No pattern Default, the criteria for membership is not met for any pattern. 5. Agglomeration Patterns Exhibited by Different Retail Types Retail with Same Nonrandom Spatial Pattern in Vancouver and Calgary Hyper Agglomeration Local Agglomeration Auto Mall Pattern Avoidance Furniture Carpets New Gas Stations Antiques Children’s Clothing Boat Dealers Supermarkets Coffee Shops Clothing Health and Diet Art Galleries Pizza Parlors Smoking Supplies Liquor, Beer, Wine