Kernel Weights

Kernel Weights applies kernel function to determine the distance decay in the derived continuous weights kernel. The kernel weights are defined as a function K(z) of the ratio between the distance dij from i to j, and the bandwidth hi, with z=dij/hi. The kernel functions include {triangular , uniform, quadratic , epanechnikov, quartic, gaussian}.

• Uniform, K(z)=1/2 for |z|<1,

• Triangular, K(z)=(1−|z|) for |z|<1,

• Quadratic or Epanechnikov, K(z)=(3/4)(1−z^2) for |z|<1,

• Quartic, K(z)=(15/16)(1−z^2)^2 for |z|<1|z|<1, and

• Gaussian. K(z) = (2π)^{1/2}exp(-z^2/2)

For more information, please read: https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html#kernel-weights

There are two types of fixed bandwidths for kernel weights. One is the max-min distance used earlier (the largest of the nearest-neighbor distances): kernel_weights(). The other is the maximum distance for a given specification of k-nearest neighbors: kernel_knn_weights().

1. getKernelWeights()

function getKernelWeights(
    String mapUid,
    Number distBand, 
    String kernel,
    Number power, 
    Boolean isInverse,
    Boolean isArc,
    Boolean isMile,
    Boolean useKernelDiagonals)

Arguments

Input Arguments	Type	Description
mapUid	String	the unique map id
distBand	Number	the distance band/threshold that makes sure each observation has at least one neighbor
kernel	String	a varchar value of kernel method, which has to be one of {'triangular', 'uniform', 'epanechnikov', 'quartic', 'gaussian'}
useKernelDiagonals	Boolean	if apply kernel on the diagonal of weights matrix. Default: FALSE.
power	Number	the power/exponent corresponds to the number of times the base (dist_band) is used as a factor. Default: 1.
isInverse	Boolean	if apply inverse on distance value. Default: False.
isArc	Boolean	if compute arc distance between two observations. Default: FALSE.
isMile	Boolean	if convert distance unit from mile to kilometer(KM). Default: TRUE.

Return

Value	Description
WeightsResult	the weights structure for each observation in binary format.

2. getKernelKnnWeights()

With knn set to a given value, the maximum distance between the selected k-nearest neighbors' pairs is used as a "fixed" bandwidth. However, a drawback of fixed bandwidth kernel weights is that the number of non-zero weights can vary considerably, especially when the density of the point locations is not uniform throughout space. The argument adaptive_bandwidth is provided to allow adaptive bandwidth in knn kernel weights: instead of a fixed distance bandwidth, the distance to the k-th nearest neighbor is used in the kernel function for each observation.

API

function getKernelKnnWeights(
    String map_uid, 
    integer k, 
    String kernel,
    Boolean adaptive_bandwidth,
    Number power, 
    Boolean is_inverse,
    Boolean is_arc,
    Boolean is_mile,
    Boolean use_kernel_diagonals)

Arguments

Input Arguments	Type	Description
map_uid	String	the unique map id
k	Number	the k nearest neighbors
kernel	String	a varchar value of kernel method, which has to be one of {'triangular', 'uniform', 'epanechnikov', 'quartic', 'gaussian'}
adaptive_bandwidth	Boolean	if use adaptive bandwidth (distance to k-th nearest neighbor for each observation), or use max knn distance of all observations. Default: FALSE.
use_kernel_diagonals	Boolean	if apply kernel on the diagonal of weights matrix. Default: FALSE.
power	Number	the power/exponent corresponds to the number of times the base (dist_band) is used as a factor. Default: 1.
is_inverse	Boolean	if apply inverse on distance value. Default: False.
is_arc	Boolean	if compute arc distance between two observations. Default: FALSE.
is_mile	Boolean	if convert distance unit from mile to kilometer(KM). Default: TRUE.

Return

Value	Description
WeightResult	the weights structure for each observation.

Try it yourself in the playground (jsgeoda + deck.gl):