# Kernel Loop Start

This node allows the user to select one or more Kernel Estimators and one or more Kernel Symmetries (if the 'Is Multi-dimensional?' option is selected). All combinations are looped through, with the values in each in the 'Kernel Estimator' and 'Kernel Symmetry' flow variables.

## Kernel Estimators

A variety of kernel estimators are available, as shown in the table:

NameFunction
UNIFORMK(u) = 0.5 (|u| ≤ 1), 0 (|u) > 1); aka 'Uniform' or 'Boxcar'
TRIANGLEK(u) = 1-|u| (|u| ≤ 1), 0 (|u) > 1)
EPANECHNIKOVK(u) = 3•(1-u²)/4 (|u| ≤ 1), 0 (|u) > 1)
QUARTICK(u) = 15•(1-u²)²/16 (|u| ≤ 1), 0 (|u) > 1)
TRIWEIGHTK(u) = 35•(1-u²)³/32 (|u| ≤ 1), 0 (|u) > 1)
TRICUBEK(u) = 70•(1-|u|³)³/81 (|u| ≤ 1), 0 (|u) > 1)
GAUSSIANK(u) = e^(-u²/2) / √(2π)
COSINUSK(u) = (π/4)•cos(πu/2) (|u| ≤ 1), 0 (|u) > 1)
LOGISTICK(u) = 1/(e^u + 2 + e^-u)
SIGMOIDK(u) = 2/(π•(e^u + e^-u))
SILVERMANK(u) = 0.5•e^(-|u|/√2)•sin((|u|/√2) + (π/4))

In the 2D case, u is a vector. The 'Kernel Symmetry' option controls how the 1-dimensional 'Kernel Estimator' is applied, as shown in the table

NameFunction
RADIAL_MULTIPLICATIVEThe kernel estimator is applied multiplicatively across dimensions, e.g. K(u) = K(u(x)) • K(u(y)), where u(x) is the x-dimension component of u, and u(y) the y-dimension component
SPHERICALThe kernel estimator is applied spherically symmetrically - i.e. any point of the same distance from the kernel estimator center has the same value. This is equivalent to K(u) = K(√uᵀu)

## Options

Kernel Estimators
The Kernel Estimator(s) to loop through
Kernel Symmetries
The kernel Symmetries looped through if 'Is Multi-dimensional?' is selected

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## Workflows

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