The SPP Procedure

Nonparametric Intensity Estimation

The KERNEL option in the PROCESS statement enables you to perform nonparametric intensity estimation. You can use five different kernel types: Epanechnikov, Gaussian, uniform, triangular, and quartic (Silverman 1986), whose kernel functions are as follows, where t equals StartRoot left-parenthesis s Subscript x Baseline minus x right-parenthesis squared plus left-parenthesis s Subscript y Baseline minus y right-parenthesis squared EndRoot slash h, s Subscript x, s Subscript y are the grid point coordinates, x and y are the point coordinates, and h is the bandwidth parameter:

  • Epanechnikov

    upper K left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column three-fourths left-parenthesis 1 minus StartFraction t squared Over 5 EndFraction right-parenthesis StartFraction 1 Over StartRoot 5 EndRoot EndFraction 2nd Column StartAbsoluteValue t EndAbsoluteValue less-than StartRoot 5 EndRoot 2nd Row 1st Column 0 2nd Column otherwise EndLayout

  • Gaussian

    upper K left-parenthesis t right-parenthesis equals StartFraction e Superscript minus StartFraction t squared Over 2 EndFraction Baseline Over StartRoot 2 pi EndRoot EndFraction

  • uniform

    upper K left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column one-half 2nd Column StartAbsoluteValue t EndAbsoluteValue less-than 1 2nd Row 1st Column 0 2nd Column otherwise EndLayout

  • triangular

    upper K left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 minus StartAbsoluteValue t EndAbsoluteValue 2nd Column StartAbsoluteValue t EndAbsoluteValue less-than 1 2nd Row 1st Column 0 2nd Column otherwise EndLayout

  • quartic

    upper K left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction 15 Over 16 EndFraction left-parenthesis 1 minus t squared right-parenthesis squared 2nd Column StartAbsoluteValue t EndAbsoluteValue less-than 1 2nd Row 1st Column 0 2nd Column otherwise EndLayout

Given the preceding kernel definitions, the nonparametric intensity estimate can be computed as

lamda left-parenthesis s right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts h Superscript negative 2 Baseline times upper K left-parenthesis StartFraction s minus s Subscript i Baseline Over h EndFraction right-parenthesis

where h is the fixed bandwidth. In practice, nonparametric intensity estimation also involves an edge correction. By default, PROC SPP divides the nonparametric estimate lamda left-parenthesis s right-parenthesis by an edge correction factor

rho left-parenthesis s right-parenthesis equals integral Underscript upper A Endscripts h Superscript negative 2 Baseline times upper K left-parenthesis StartFraction s minus s Subscript i Baseline Over h EndFraction right-parenthesis

where A is the study area. The choice of the bandwidth parameter that nonparametric intensity estimation requires is more important than the choice of the kernel type itself (Silverman 1986). The bandwidth can be spatially fixed or spatial varying. If the bandwidth is spatially varying, it is called adaptive kernel estimation. For adaptive kernel estimation, the SPP procedure uses the technique suggested in Silverman (1986, p. 101) and Diggle, Rowlingson, and Su (2005, p. 426), which is computed in two steps:

  1. Use an initial bandwidth h to compute pilot estimates of the first-order intensity as

    lamda 0 left-parenthesis s right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts h Superscript negative 2 Baseline times upper K left-parenthesis StartFraction s minus s Subscript i Baseline Over h EndFraction right-parenthesis

    where K(.) is a kernel.

  2. Compute bandwidth factors as

    h Subscript i Baseline equals h times left-parenthesis StartFraction lamda 0 left-parenthesis s right-parenthesis Over ModifyingAbove g With caret EndFraction right-parenthesis Superscript negative 0.5

    where ModifyingAbove g With caret is the geometric mean of the pilot estimates lamda 0 left-parenthesis s right-parenthesis.

Based on the computed bandwidth estimates, h Subscript i, the nonparametric intensity estimates are computed as

lamda left-parenthesis s right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts h Subscript i Superscript negative 2 Baseline times upper K left-parenthesis StartFraction s minus s Subscript i Baseline Over h Subscript i Baseline EndFraction right-parenthesis

In PROC SPP, adaptive kernel estimation does not incorporate edge correction.

Last updated: December 09, 2022