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Operational local join count statistics for cluster detection

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Abstract

This paper operationalizes the idea of a local indicator of spatial association for the situation where the variables of interest are binary. This yields a conditional version of a local join count statistic. The statistic is extended to a bivariate and multivariate context, with an explicit treatment of co-location. The approach provides an alternative to point pattern-based statistics for situations where all potential locations of an event are available (e.g., all parcels in a city). The statistics are implemented in the open-source GeoDa software and yield maps of local clusters of binary variables, as well as co-location clusters of two (or more) binary variables. Empirical illustrations investigate local clusters of house sales in Detroit in 2013 and 2014, and urban design characteristics of Chicago census blocks in 2017.

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Notes

  1. For a general discussion of spatial weights, see, for example, Bavaud (1998), Getis (2009), and Anselin and Rey (2014). Social interaction and social network extensions can be found in Dow et al. (1982), Akerlof (1997), Leenders (2002), Páez et al. (2008), and Papachristos and Bartomski (2018), among others.

  2. In some rare examples, data on the complete population are available, and a case-control design becomes equivalent to a lattice data setting. However, in a typical case-control setup, the controls are a sample, and thus, not all non-event locations are included.

  3. Rogerson (2006) also includes a local form of the statistic, which counts the number of cases among the neighbors for a given location. Except for the case-control setup, this is formally equivalent to the local join count statistic described below.

  4. This is formally the same as the Jacquez et al. (2005) local Q statistic for location i at time t with k-nearest neighbors, i.e., \(Q_{i,k,t} = c_i \sum _j n_{ijkt} c_j\), where \(c_{i,j} = 1\) for a case and \(= 0\) for a control, and \(n_{ijkt}\) are the nearest neighbor weights for k-nearest neighbors of location i at time t. It is also essentially the same as the local similarity relation in Farber et al. (2015), i.e., \(\Gamma _{d,i} = \sum _j I_{ij}\), where \(I_{ij} = 1\) when the values at i and j are “similar” for d nearest neighbors. In contrast to these measures, which are based on nearest neighbor relations, the local join count statistic is couched in a lattice data structure with spatial weights. Formally, the expressions are the same, but conceptually, they differ.

  5. Yet a different strand of local cluster statistics is based on the scan-statistic logic first outlined in Kulldorff (1997), and its many extensions. However, since this approach does not provide a link between a local and global statistic—a fundamental property of a LISA statistic as outlined in Anselin (1995)—it is not further considered here.

  6. Note that this is a conditional probability. It thus underestimates the actual uncertainty associated with the occurrence of a value of 1 and its particular configuration of neighbors. The unconditional probability would be the joint probability of observing \(x_i = 1\)andp neighbors \(x_j = 1\). This not what is considered here.

  7. In larger samples, the distinction between using \(N-1\) and \(P-1\) compared to N and P is likely negligible. Also, the distinction between sampling without replacement (the hypergeometric distribution) and sampling with replacement (the binomial distribution) is likely to be small for large data sets with few events.

  8. This is the logic behind the local z statistic for the case-control setting suggested in Rogerson (2006).

  9. In the limit, the neighbors would include all other observations.

  10. Note how a case-control setup can be couched in these terms, since a case and a control cannot occur at the same location. For example, \(x_i = 1\) for a case and \(z_j = 1\) for a control. The BJC statistic would then count the number of controls among the neighbors of i, or, with the roles reversed, the number of cases around a control a i.

  11. Since the conditional permutation is designed to draw tuples of existing pairs of x and z, the procedure respects the in-place association between x and z.

  12. Formally, we could also consider the situation where \(x_i = z_i = 1\) is surrounded by either \(z_j = 1\) or \(x_j = 1\), ignoring the value for the other variable. However, we see little practical application where there is a meaningful interpretation for this situation, and we do not consider it further.

  13. Repeat sales were removed from the data set (only the latest sale is recorded), so that there is no overlap between the two point patterns.

  14. Note that not all sales are standard transactions and many are the result of auctions, resulting in arbitrary sales prices, typically less than $1000. We ignore the actual sales value in our analysis, but keep all transactions in the data set.

  15. In the point pattern approach taken by Cromley et al. (2014) and Wang et al. (2017), this would be equivalent to a uniform adaptive kernel, in the sense that each neighbor gets equal weight and each observation has exactly 30 neighbors.

  16. Because of the resolution of the map, it is not possible to distinguish all individual points, since several pertain to close-by locations that tend to be plotted on top of each other.

  17. Recall that by construction, none of the points overlap between the two years.

  18. Again, due to the scale of the map, the figure only shows eight points. In three cases, two adjoining locations are found that cannot be individually distinguished in the map.

  19. The classification is derived from an extensive set of data, most notably the City of Chicago Business Licenses data for 2017. Most data are for 2017, a few are for 2016, and the sidewalk data are for 2012. The census block definition is from 2010. Details can be found in Talen and Jeong (2018, Table 1).

  20. Note that the highlighted blocks form the core of the cluster, but do not include the neighbors that also may show co-location. In this example, several blocks are neighbors as well, but this is not always the case. In other words, the highlighted blocks underestimate the spatial extent of the actual cluster.

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Acknowledgements

This research was funded in part by Award 1R01HS021752-01A1 from the Agency for Healthcare Research and Quality (AHRQ), “Advancing spatial evaluation methods to improve healthcare efficiency and quality.” Emily Talen and Hyesun Jeong provided the urban design classifications of the Chicago census block data. Comments by Julia Koschinsky and referees on an earlier version of the paper are greatly appreciated.

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  1. Center for Spatial Data Science, The University of Chicago, Chicago, IL, 60637, USA

    Luc Anselin & Xun Li

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  1. Luc Anselin
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Correspondence to Luc Anselin.

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Anselin, L., Li, X. Operational local join count statistics for cluster detection. J Geogr Syst 21, 189–210 (2019). https://doi.org/10.1007/s10109-019-00299-x

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