2.3 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)=(1β|z|) for |z|<1,
Quadratic or Epanechnikov, K(z)=(3/4)(1βz^2) for |z|<1,
Quartic, K(z)=(15/16)(1βz^2)^2 for |z|<1|z|<1, and
Gaussian. K(z) = (2Ο)^{1/2}exp(-z^2/2)
For more information, please read: https://geodacenter.github.io/workbook/4c_distance_functions/lab4c.html#kernel-weights
CONTENTS
kernel_weights()
kernel_knn_weights() -- fixed and adaptive bandwidth
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. kernel_weights()
Synopsis
Short version
bytea kernel_weights(integer gid, geometry the_geom,
float dist_band, character varying kernel)Full version
Arguments
Input Arguments
Type
Description
gid
integer
the feature id of geometry: e.g. gid, fid, ogcfid, cartodb_id
the_geom
geometry
the geometry (only points and polygons are supported)
dist_band
float
the distance band/threshold that makes sure each observation has at least one neighbor
kernel
character varying
a varchar value of kernel method, which has to be one of {'triangular', 'uniform', 'epanechnikov', 'quartic', 'gaussian'}
use_kernel_diagonals
boolean
if apply kernel on the diagonal of weights matrix. Default: FALSE.
power
float
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
bytea
the weights structure for each observation in binary format, which is defined in table 2.1.
Examples
2. kernel_knn_weights()
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.
Synopsis
Short version
Fixed bandwidth
Adaptive bandwidth
Full version
Arguments
Input Arguments
Type
Description
gid
integer
the feature id of geometry: e.g. gid, fid, ogcfid, cartodb_id
the_geom
geometry
the geometry (only points and polygons are supported)
k
integer
the k nearest neighbors
kernel
character varying
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
float
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
bytea
the weights structure for each observation in binary format, which is defined in table 2.1.
Examples
Last updated
Was this helpful?