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fastmath.kernel

Various kernel functions.

  • RBF (double -> double functions)
  • vector kernels (vector x vector -> double function; may be positive definite, conditional positive definite, positive semi-definite, mercer)
  • density estimation
  • some kernel operations

Categories

Other vars: kernel-density-list kernels-list rbf-list

approx

(approx k precision)(approx k)

Kernel wrapper. Round value returned by kernel using fastmath.core/approx function.

Examples

Usage

(let [ak (approx (kernel :gaussian))] (ak [1 2] [3 4]))
;;=> 0.02
(let [ak (approx (kernel :gaussian) 6)] (ak [1 2] [3 4]))
;;=> 0.018316

Plot of :gaussian kernel pproximated to one decimal place.

view source

cpd->pd

(cpd->pd k)

Convert conditionally positive definite kernel into positive definite.

Formula is based on this SO answer. x0 is equals 0.

Doesn’t work well.

Examples

Usage

(let [k (kernel :circular) pd (cpd->pd k)] (pd [0.0 -0.1] [0.2 0.4]))
;;=> 0.013011305

Plot of :periodic kernel converted with cpd->pd function.

view source

exp

(exp k)(exp k t)

Kernel wraper. exp of kernel k with optional scaling value t.

Examples

Usage

(let [k (exp (kernel :laplacian))] (k [1 2] [3 4]))
;;=> 1.060887420006057

Exp with scale=0.5

(let [k (exp (kernel :laplacian) 0.5)] (k [1 2] [3 4]))
;;=> 1.0299938931887205

Plot of exp of :dirichlet kernel with scaling=5.0

view source

fields

(fields k f)(fields k f1 f2)

Kernel wrapper. Apply vector field for each input before applying kernel function.

Examples

Usage

(let [k (kernel :laplacian)
      fld (fields/field :horseshoe)
      kf (fields k fld)]
  (kf (v/vec2 2 3) (v/vec2 1 2)))
;;=> 0.21437946979352915
view source

kernel

multimethod

Crated vector kernel.

Kernels can be Mercer, positive definite, conditional positive definite, positive semi-definite or other.

Optional parameters:

For :gaussian, :exponential, :laplacian, :rational-quadratic, :multiquadratic, :inverse-multiquadratic, :circular, :spherical, :wave, :power, :log, :cauchy, :generalized-t-student, :hyperbolic-secant, :thin-plate, :mattern-12, :mattern-32, :mattern-52 and ::hyperbolic-secant you can provide scaling parameter and distance (see fastmath.distance, default is euclidean).

Others:

  • :linear - alpha, scaling parameter
  • :polynomial - alpha (scaling), c (shift) and d (power)
  • :anova - sigma (scaling), k and d (power)
  • :hyperbolic-tangent - alpha (scaling), c (shift)
  • :periodic - sigma (scaling), periodicity and distance
  • :bessel - sigma (scaling), n and v (power factors) and distance
  • :generalized-histogram - alpha and beta (power factors)
  • :dirichlet - N
  • :pearson - sigma (scaling) and omega (power)

Additionally there are two special kernels build from funcitons:

  • :scalar-functions - provide one or two double->double functions
  • :variance-function - provide any variance function (smooth, vector->double type)

The rest of the kernels do not require parameters.

Examples

Usage

(let [k (kernel :gaussian)] (k [1 2 3 4] [-1 2 3 5]))
;;=> 0.08208499862389877
(let [k (kernel :gaussian 1.0 d/chebyshev)] (k [1 2 3 4] [-1 2 3 5]))
;;=> 0.1353352832366127
(let [k (kernel :thin-plate)] (k [1 2 3 4] [-1 2 3 5]))
;;=> 4.023594781085252
(let [k (kernel :mattern-52 0.5)] (k [1 2 3 4] [-1 2 3 5]))
;;=> 0.0020127302194701585

Plot of anova

Plot of bessel

Plot of cauchy

Plot of chi-square-cpd

Plot of chi-square-pd

Plot of circular

Plot of dirichlet

Plot of exponential

Plot of gaussian

Plot of generalized-histogram

Plot of generalized-t-student

Plot of hellinger

Plot of histogram

Plot of hyperbolic-secant

Plot of hyperbolic-tangent

Plot of inverse-multiquadratic

Plot of laplacian

Plot of linear

Plot of log

Plot of mattern-12

Plot of mattern-32

Plot of mattern-52

Plot of multiquadratic

Plot of pearson

Plot of periodic

Plot of polynomial

Plot of power

Plot of rational-quadratic

Plot of scalar-functions

Plot of spherical

Plot of spline

Plot of thin-plate

Plot of variance-function

Plot of wave

view source

kernel->rbf

(kernel->rbf k)(kernel->rbf k center)

Convert vector kernel to RBF kernel. center is fixed y vector (default contains EPSILON values).

Examples

Usage

(let [k (kernel :mattern-52) r (kernel->rbf k)] (r 0.234))
;;=> 0.9567485662415672
view source

kernel-density

multimethod

Create kernel density estimator.

Parameters:

  • kernel name, see kernel-density-list.
  • sequence of data values
  • optional: bandwidth (by default, bandwidth is estimated using nrd method)

Examples

Usage

(let [k (kernel-density :epanechnikov (repeatedly 1000 rand))] (k 0.5))
;;=> 1.0341156066465114
(let [k (kernel-density :gaussian (repeatedly 1000 rand) 2)] (k 0.5))
;;=> 0.1974963692421274

Plot of cauchy

Plot of cosine

Plot of default

Plot of epanechnikov

Plot of gaussian

Plot of laplace

Plot of logistic

Plot of quartic

Plot of sigmoid

Plot of silverman

Plot of smile

Plot of triangular

Plot of tricube

Plot of triweight

Plot of uniform

Plot of wigner

view source

kernel-density-ci

(kernel-density-ci method data)(kernel-density-ci method data bandwidth)(kernel-density-ci method data bandwidth alpha)

Create function which returns confidence intervals for given kde method.

Check 6.1.5 http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xlghtmlnode33.html

Parameters:

  • method - kernel name
  • data - sequence of data values
  • bandwidth
  • alpha - confidence level parameter

Returns three values: density, lower confidence, upper confidence

Examples

Usage

(let [k (kernel-density-ci :epanechnikov (repeatedly 1000 rand))]
  (k 0.5))
;;=> [0.9608554698785219 0.7933879002342569 1.128323039522787]
(let [k (kernel-density-ci :gaussian (repeatedly 1000 rand) 2)] (k 0.5))
;;=> [0.19741724778198436 0.18707480388528747 0.20775969167868125]

Kernel density with confidence intervals

view source

kernel-density-list

Examples

List of density kernels

kernel-density-list
;;=> (:cauchy :cosine
;;=>          :default :epanechnikov
;;=>          :gaussian :laplace
;;=>          :logistic :quartic
;;=>          :sigmoid :silverman
;;=>          :smile :triangular
;;=>          :tricube :triweight
;;=>          :uniform :wigner)
view source

kernels-list

Examples

List of vector kernels

kernels-list
;;=> (:anova
;;=>  :bessel
;;=>  :cauchy :chi-square-cpd
;;=>  :chi-square-pd :circular
;;=>  :dirichlet :exponential
;;=>  :gaussian :generalized-histogram
;;=>  :generalized-t-student :hellinger
;;=>  :histogram :hyperbolic-secant
;;=>  :hyperbolic-tangent :inverse-multiquadratic
;;=>  :laplacian :linear
;;=>  :log :mattern-12
;;=>  :mattern-32 :mattern-52
;;=>  :multiquadratic :pearson
;;=>  :periodic :polynomial
;;=>  :power :rational-quadratic
;;=>  :scalar-functions :spherical
;;=>  :spline :thin-plate
;;=>  :variance-function :wave)
view source

mult

(mult k1)(mult k1 k2)(mult k1 k2 k3)(mult k1 k2 k3 & r)

Kernel wrapper. Multiply two or more kernels.

Examples

Usage

(let [k1 (kernel :laplacian)
      k2 (kernel :periodic)
      res (mult k1 k2)]
  (res [1 2 3 4] [-1 2 3 5]))
;;=> 0.042910348739392076

Product of two kernels

view source

rbf

multimethod

RBF kernel creator. RBF is double->double function.

Parameters:

All kernels accept scale parameter (as last parameter).

Following kernels also accept beta: :multiquadratic, :inverse-multiquadratic, :truncated-power, :radial-powers and :thin-plate.

Examples

Usage

(let [k (rbf :gaussian)] (k 1))
;;=> 0.36787944117144233
(let [k (rbf :gaussian 0.5)] (k 1))
;;=> 0.01831563888873418

Thin-plate RBF with beta=2 and scale=1

(let [k (rbf :thin-plate 2.0 1.0)] (k 0.5))
;;=> -0.04332169878499659

Plot of above RBF

Plot of gaussian

Plot of gaussians-laguerre-11

Plot of gaussians-laguerre-12

Plot of gaussians-laguerre-21

Plot of gaussians-laguerre-22

Plot of inverse-multiquadratic

Plot of linear

Plot of mattern-c0

Plot of mattern-c2

Plot of mattern-c4

Plot of multiquadratic

Plot of poisson-2

Plot of poisson-3

Plot of poisson-4

Plot of radial-powers

Plot of thin-plate

Plot of truncated-power

Plot of wendland-10

Plot of wendland-20

Plot of wendland-21

Plot of wendland-30

Plot of wendland-31

Plot of wendland-32

Plot of wendland-41

Plot of wendland-42

Plot of wendland-52

Plot of wendland-53

Plot of whittaker-02

Plot of whittaker-03

Plot of whittaker-12

Plot of whittaker-13

Plot of wu-00

Plot of wu-10

Plot of wu-11

Plot of wu-20

Plot of wu-21

Plot of wu-22

Plot of wu-30

Plot of wu-31

Plot of wu-32

Plot of wu-33

view source

rbf->kernel

(rbf->kernel rbf-kernel)(rbf->kernel rbf-kernel distance)

Treat RBF kernel as vector kernel using distance function (default euclidean.

Examples

Usage

(let [r (rbf :mattern-c4) k (rbf->kernel r)] (k [1 2 3 4] [-1 2 3 5]))
;;=> 1.5719823264954609
view source

rbf-list

Examples

List of RBF kernels

rbf-list
;;=> (:gaussian
;;=>  :gaussians-laguerre-11 :gaussians-laguerre-12
;;=>  :gaussians-laguerre-21 :gaussians-laguerre-22
;;=>  :inverse-multiquadratic :linear
;;=>  :mattern-c0 :mattern-c2
;;=>  :mattern-c4 :multiquadratic
;;=>  :poisson-2 :poisson-3
;;=>  :poisson-4 :radial-powers
;;=>  :thin-plate :truncated-power
;;=>  :wendland-10 :wendland-20
;;=>  :wendland-21 :wendland-30
;;=>  :wendland-31 :wendland-32
;;=>  :wendland-41 :wendland-42
;;=>  :wendland-52 :wendland-53
;;=>  :whittaker-02 :whittaker-03
;;=>  :whittaker-12 :whittaker-13
;;=>  :wu-00 :wu-10
;;=>  :wu-11 :wu-20
;;=>  :wu-21 :wu-22
;;=>  :wu-30 :wu-31
;;=>  :wu-32 :wu-33)
view source

scale

(scale k scale)

Kernel wrapper. Scale kernel result.

Examples

Usage

(let [k (kernel :laplacian)
      sk (scale k 2.0)]
  {:kernel (k [1 2 3 4] [-1 2 3 5]),
   :scaled-kernel (sk [1 2 3 4] [-1 2 3 5])})
;;=> {:kernel 0.10687792566038574, :scaled-kernel 0.2137558513207715}
view source

smile-mercer

(smile-mercer k)

Create Smile Mercer Kernel object

Used to pass to Smile constructors/functions.

Examples

Usage

(smile-mercer (kernel :mattern-52))
;;=> fastmath.kernel$smile_mercer$reify__14530@6a4b9305
view source

smile-rbf

(smile-rbf rbf-fn)

Create RBF Smile object.

Used to pass to Smile constructors/functions.

Examples

Usage

(smile-rbf (rbf :mattern-c2))
;;=> fastmath.kernel$smile_rbf$reify__14268@2d0b946d
view source

wadd

(wadd kernels)(wadd weights kernels)

Kernel wrapper. Add kernels (weighted).

Examples

Usage

(let [k1 (kernel :laplacian)
      k2 (kernel :periodic)
      res (wadd [k1 k2])]
  (res [1 2 3 4] [-1 2 3 5]))
;;=> 0.5083672741320578

Weighted sum

(let [k1 (kernel :laplacian)
      k2 (kernel :periodic)
      res (wadd [0.2 0.8] [k1 k2])]
  (res [1 2 3 4] [-1 2 3 5]))
;;=> 0.3425670639094149

Weighted sum of two kernels

view source