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Application of the double kernel density approach to the multivariate analysis of attributeless event point datasets

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Abstract

Attributeless event point datasets (AEPDs) are datasets composed of discrete events or observations defined by their geographical location only and lacking any other additional attributes. Examples of such datasets include spotted criminal events, road accidents and residential locations of disease patients. A commonly used approach to the analysis of such datasets involves their aggregation into predefined areal units, such as neighborhoods or census tracts. However, this approach does not perform effectively when the events of interests are geographically localized and the number of areal units available for aggregation is small. An alternative approach to the analysis of AEPDs is based on double kernel density (DKD) smoothing, according to which events of interest are transformed into continuous density surfaces and then normalized by the density of the entire population from which the events of interest are drawn. In the present study, the applicability of the DKD approach to multivariate analysis is tested for estimation consistency, sensitivity to the number of input observations and potential bias attributed to the spatial dependency of neighboring observations. Our analysis indicates that the DKD approach provides reasonably stable and consistent estimates, if the following three preconditions are met: (a) the kernel estimation parameters are properly defined, (b) the number of reference points, used for transformation of continuous DKD surfaces into discrete observations, is sufficiently large, and (c) the spatial dependency of neighboring observations is taken into account using spatial analysis tools.

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Notes

  1. The industrial zone in question hosts several petrochemical facilities, and, therefore, aerial proximity to this zone was suspected to be a potential determinant of the observed cancer morbidity.

  2. We implicitly assume that individual exposure is determined by the place of residence. This assumption is required since working place and duration of living in a particular location were not available in our data set.

  3. As previously mentioned, the term “double kernel density” (DKD) refers to an analytical method that performs kernel estimation twice: first, for the events of interests, second, for the overall population from which the events of interest are drawn, and then by calculating the ratio between the two. The term DKD itself originates in an early theoretical paper by Devroye (1989).

  4. Several GIS software packages can perform pixel-on-pixel regressions, but such regressions are unable to account for spatial dependency of regression residuals. In our study, the GeoDa\(^{\mathrm{TM}}\) software (Anselin et al. 2006) was used for this task.

  5. Cross–validation test uses all the data and makes comparison by removing each data location once at a time and predicts the associated data values (Johnston et al. 2001).

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Author information

Authors and Affiliations

  1. Department of Natural Resources and Environmental Management, Graduate School of Management, University of Haifa, Mount Carmel, Haifa, 31905, Israel

    Marina Zusman & Boris A. Portnov

  2. Department of Economics, Natural Resource and Environmental Research Centre, University of Haifa, Mount Carmel, Haifa, 31905, Israel

    Dani Broitman

Authors
  1. Marina Zusman
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  2. Dani Broitman
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  3. Boris A. Portnov
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Corresponding author

Correspondence to Dani Broitman.

Appendices

Appendix 1: Descriptive Statistics

See Table 3.

Table 3 Descriptive statistics of the research variables
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Appendix 2: Normality tests

See Figs.10, 11.

Fig. 10
figure 10

Normality test statistics of unstandardized residuals for different kernel bandwidths (Kolmogorov-Smirnov test (a) and Shapiro-Wilk test (b)

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Fig. 11
figure 11

Normal Q-Q Plot of unstandardized residual for DKD of lung cancer patients (R\(^{2}= 0.79\); 1000-m bandwidth; 1000 reference points)

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Zusman, M., Broitman, D. & Portnov, B.A. Application of the double kernel density approach to the multivariate analysis of attributeless event point datasets. Lett Spat Resour Sci 9, 363–382 (2016). https://doi.org/10.1007/s12076-015-0166-y

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  • DOI: https://doi.org/10.1007/s12076-015-0166-y

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