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class Kernel( object )	Source

A kernel is an even real-valued integrable function.

It is satisfying the following two requirements

1. The integral over [-Inf,+Inf] exists ( < Inf ).
2. It is an even function: K( x ) = K( -x ).

A kernel is called bound when it is 0 everywhere except when |x| < range < inf.

All kernels are scaled such that its value at x=0 is 1.0.

Several kernel functions, K( x ) are defined in this package

Name	Definition	Integral	FWHM	range	comment
Biweight	( 1-x2 )2	16/15	1.08	1.0	aka Tukey
CosSquare	cos2(0.5*π*x)	1.0	1.00	1.0
Cosine	cos( 0.5*π*x )	4/π	1.33	1.0
Gauss	exp( -0.5*x2 )	√(2*π)	1.22	inf
Huber	min( 1, 1/|x| )	inf	4.00	inf	improper
					aka Median
Lorentz	1 / ( 1 + x2 )	π	2.00	inf
Parabola	1 - x2	4/3	1.41	1.0
Sinc	sin(π*x)/(π*x)	1.0	1.21	inf
Triangle	1 - |x|	1.0	1.00	1.0
Tricube	( 1 - |x|3 )3	81/70	1.18	1.0
Triweight	( 1 - x2 )3	32/35	0.91	1.0
Uniform	1.0	2.0	2.00	1.0	aka Clip
Tophat 0	1.0	1.0	1.00	0.5	like Uniform
Tophat 1	1 - |x|	1.0	1.00	1.0	aka Triangle
Tophat 2	2nd order polynome	1.0	1.26	1.5
Tophat 3	3rd order polynome	1.0	1.44	2.0
Tophat 4	4th order polynome	1.0	1.60	2.5
Tophat 5	5th order polynome	1.0	1.73	3.0
Tophat 6	6th order polynome	1.0	1.86	3.5

For all bound Kernels the definition in the table is true for |x| < range; elsewhere it is 0.

Huber is not a proper Kernel as the integral is inf. However it is important in robust fitting (RobustShell) to get a madian-like solution for the outliers.

Attributes

• integral : float
       the integral over the valid range
• fwhm : float
       the full width at half maximum
• range : float
       the region [-range..+range] where the kernel is non-zero.

Author Do Kester

Category mathematics/Fitting

Kernel( integral=1.0, fwhm=1.0, range=1.0 )

Constructor

Parameters

• integral : float
       over [-inf, +inf]
• fwhm : float
       full width at half maximum
• range : float
       the region [-range,+range] where the kernel is nonzero

result( x )

Return the result for input values.

Parameters

• x : array-like
       input values

resultsq( xsq )

Return the result for squared input values.

Parameters

• x : array-like
       the squares of the input values

partial( x )

Return the partial derivative wrt the input values.

Parameters

• x : array-like
       the input values

isBound( )

Return true when the kernel is bound, i.e. all non-zero values are between -1 and +1

name( )

Return the name of the kernel.