kernelMatrix.RdkernelMatrix calculates the kernel matrix \(K_{ij} = k(x_i,x_j)\) or \(K_{ij} =
k(x_i,y_j)\).kernelPol computes the quadratic kernel expression \(H = z_i z_j
k(x_i,x_j)\), \(H = z_i k_j k(x_i,y_j)\).kernelMult calculates the kernel expansion \(f(x_i) =
\sum_{i=1}^m z_i k(x_i,x_j)\)kernelFast computes the kernel matrix, identical
to kernelMatrix, except that it also requires the squared
norm of the first argument as additional input, useful in iterative
kernel matrix calculations.
# S4 method for class 'kernel'
kernelMatrix(kernel, x, y = NULL)
# S4 method for class 'kernel'
kernelPol(kernel, x, y = NULL, z, k = NULL)
# S4 method for class 'kernel'
kernelMult(kernel, x, y = NULL, z, blocksize = 256)
# S4 method for class 'kernel'
kernelFast(kernel, x, y, a)the kernel function to be used to calculate the kernel
matrix.
This has to be a function of class kernel, i.e. which can be
generated either one of the build in
kernel generating functions (e.g., rbfdot etc.) or a user defined
function of class kernel taking two vector arguments and returning a scalar.
a data matrix to be used to calculate the kernel matrix, or a
list of vector when a stringkernel is used
second data matrix to calculate the kernel matrix, or a
list of vector when a stringkernel is used
a suitable vector or matrix
a suitable vector or matrix
the squared norm of x, e.g., rowSums(x^2)
the kernel expansion computations are done block wise
to avoid storing the kernel matrix into memory. blocksize
defines the size of the computational blocks.
Common functions used during kernel based computations.
The kernel parameter can be set to any function, of class
kernel, which computes the inner product in feature space between two
vector arguments. kernlab provides the most popular kernel functions
which can be initialized by using the following
functions:
rbfdot Radial Basis kernel function
polydot Polynomial kernel function
vanilladot Linear kernel function
tanhdot Hyperbolic tangent kernel function
laplacedot Laplacian kernel function
besseldot Bessel kernel function
anovadot ANOVA RBF kernel function
splinedot the Spline kernel
(see example.)
kernelFast is mainly used in situations where columns of the
kernel matrix are computed per invocation. In these cases,
evaluating the norm of each row-entry over and over again would
cause significant computational overhead.
kernelMatrix returns a symmetric diagonal semi-definite matrix.kernelPol returns a matrix.kernelMult usually returns a one-column matrix.
## use the spam data
data(spam)
dt <- as.matrix(spam[c(10:20,3000:3010),-58])
## initialize kernel function
rbf <- rbfdot(sigma = 0.05)
rbf
#> new("rbfkernel", .Data = function (x, y = NULL)
#> {
#> if (!is(x, "vector"))
#> stop("x must be a vector")
#> if (!is(y, "vector") && !is.null(y))
#> stop("y must a vector")
#> if (is(x, "vector") && is.null(y)) {
#> return(1)
#> }
#> if (is(x, "vector") && is(y, "vector")) {
#> if (!length(x) == length(y))
#> stop("number of dimension must be the same on both data points")
#> return(exp(sigma * (2 * crossprod(x, y) - crossprod(x) -
#> crossprod(y))))
#> }
#> }, kpar = list(sigma = 0.05))
#> <bytecode: 0x564204d155f8>
#> <environment: 0x56420664a5a0>
#> attr(,"kpar")
#> attr(,"kpar")$sigma
#> [1] 0.05
#>
#> attr(,"class")
#> [1] "rbfkernel"
#> attr(,"class")attr(,"package")
#> [1] "kernlab"
## calculate kernel matrix
kernelMatrix(rbf, dt)
#> An object of class "kernelMatrix"
#> 10 11 12 13 14
#> 10 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 3.076372e-01
#> 12 0.000000e+00 0.000000e+00 1.000000e+00 5.576760e-184 0.000000e+00
#> 13 0.000000e+00 0.000000e+00 5.576760e-184 1.000000e+00 0.000000e+00
#> 14 0.000000e+00 3.076372e-01 0.000000e+00 0.000000e+00 1.000000e+00
#> 15 0.000000e+00 0.000000e+00 7.620012e-15 2.283705e-99 0.000000e+00
#> 16 0.000000e+00 0.000000e+00 4.354479e-135 6.728227e-05 0.000000e+00
#> 17 0.000000e+00 1.759111e-175 1.580090e-138 0.000000e+00 1.474439e-159
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 19 0.000000e+00 3.704851e-53 4.048455e-284 0.000000e+00 5.620206e-45
#> 20 0.000000e+00 0.000000e+00 2.375087e-140 1.144518e-142 0.000000e+00
#> 3000 0.000000e+00 0.000000e+00 0.000000e+00 9.823504e-126 0.000000e+00
#> 3001 0.000000e+00 4.157797e-25 0.000000e+00 0.000000e+00 1.629625e-19
#> 3002 0.000000e+00 1.584893e-07 0.000000e+00 0.000000e+00 1.153939e-04
#> 3003 0.000000e+00 2.485273e-06 0.000000e+00 0.000000e+00 3.459807e-09
#> 3004 0.000000e+00 0.000000e+00 3.802475e-19 5.318413e-295 0.000000e+00
#> 3005 0.000000e+00 6.080875e-113 2.419462e-191 0.000000e+00 2.324676e-100
#> 3006 0.000000e+00 2.141277e-36 0.000000e+00 0.000000e+00 5.957980e-30
#> 3007 0.000000e+00 1.347476e-05 0.000000e+00 0.000000e+00 7.850063e-08
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3010 1.836994e-45 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 15 16 17 18 19
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 0.000000e+00 1.759111e-175 0.000000e+00 3.704851e-53
#> 12 7.620012e-15 4.354479e-135 1.580090e-138 0.000000e+00 4.048455e-284
#> 13 2.283705e-99 6.728227e-05 0.000000e+00 0.000000e+00 0.000000e+00
#> 14 0.000000e+00 0.000000e+00 1.474439e-159 0.000000e+00 5.620206e-45
#> 15 1.000000e+00 1.700705e-64 3.198028e-211 0.000000e+00 0.000000e+00
#> 16 1.700705e-64 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 17 3.198028e-211 0.000000e+00 1.000000e+00 0.000000e+00 1.131156e-46
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 0.000000e+00
#> 19 0.000000e+00 0.000000e+00 1.131156e-46 0.000000e+00 1.000000e+00
#> 20 4.909368e-107 1.406121e-115 7.630353e-215 0.000000e+00 0.000000e+00
#> 3000 1.748673e-300 5.672366e-150 0.000000e+00 0.000000e+00 0.000000e+00
#> 3001 0.000000e+00 0.000000e+00 5.322112e-77 0.000000e+00 8.829844e-07
#> 3002 0.000000e+00 0.000000e+00 3.492160e-118 0.000000e+00 2.787511e-23
#> 3003 0.000000e+00 0.000000e+00 1.069177e-239 0.000000e+00 5.288180e-90
#> 3004 1.076909e-58 1.857644e-231 3.153752e-58 0.000000e+00 1.161063e-159
#> 3005 6.669696e-284 0.000000e+00 1.473126e-08 0.000000e+00 1.980755e-17
#> 3006 0.000000e+00 0.000000e+00 4.958692e-65 0.000000e+00 1.809605e-22
#> 3007 0.000000e+00 0.000000e+00 3.077777e-225 0.000000e+00 1.125459e-80
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 4.882814e-270 0.000000e+00
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 20 3000 3001 3002 3003
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 0.000000e+00 4.157797e-25 1.584893e-07 2.485273e-06
#> 12 2.375087e-140 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 13 1.144518e-142 9.823504e-126 0.000000e+00 0.000000e+00 0.000000e+00
#> 14 0.000000e+00 0.000000e+00 1.629625e-19 1.153939e-04 3.459807e-09
#> 15 4.909368e-107 1.748673e-300 0.000000e+00 0.000000e+00 0.000000e+00
#> 16 1.406121e-115 5.672366e-150 0.000000e+00 0.000000e+00 0.000000e+00
#> 17 7.630353e-215 0.000000e+00 5.322112e-77 3.492160e-118 1.069177e-239
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 19 0.000000e+00 0.000000e+00 8.829844e-07 2.787511e-23 5.288180e-90
#> 20 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3000 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3001 0.000000e+00 0.000000e+00 1.000000e+00 1.352868e-06 4.823404e-51
#> 3002 0.000000e+00 0.000000e+00 1.352868e-06 1.000000e+00 2.576800e-23
#> 3003 0.000000e+00 0.000000e+00 4.823404e-51 2.576800e-23 1.000000e+00
#> 3004 2.356445e-154 0.000000e+00 7.945641e-224 3.305267e-299 0.000000e+00
#> 3005 9.713541e-301 0.000000e+00 3.387569e-37 1.378134e-67 8.914794e-166
#> 3006 0.000000e+00 0.000000e+00 2.663515e-15 2.320201e-19 1.902969e-65
#> 3007 0.000000e+00 0.000000e+00 3.061563e-44 1.242501e-19 1.643562e-02
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 5.593377e-93 0.000000e+00 0.000000e+00 0.000000e+00
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3004 3005 3006 3007 3008 3009
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 11 0.000000e+00 6.080875e-113 2.141277e-36 1.347476e-05 0 0.000000e+00
#> 12 3.802475e-19 2.419462e-191 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 13 5.318413e-295 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 14 0.000000e+00 2.324676e-100 5.957980e-30 7.850063e-08 0 0.000000e+00
#> 15 1.076909e-58 6.669696e-284 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 16 1.857644e-231 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 17 3.153752e-58 1.473126e-08 4.958692e-65 3.077777e-225 0 0.000000e+00
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 4.882814e-270
#> 19 1.161063e-159 1.980755e-17 1.809605e-22 1.125459e-80 0 0.000000e+00
#> 20 2.356445e-154 9.713541e-301 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 3000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 5.593377e-93
#> 3001 7.945641e-224 3.387569e-37 2.663515e-15 3.061563e-44 0 0.000000e+00
#> 3002 3.305267e-299 1.378134e-67 2.320201e-19 1.242501e-19 0 0.000000e+00
#> 3003 0.000000e+00 8.914794e-166 1.902969e-65 1.643562e-02 0 0.000000e+00
#> 3004 1.000000e+00 1.023004e-91 3.147192e-234 0.000000e+00 0 0.000000e+00
#> 3005 1.023004e-91 1.000000e+00 2.479034e-35 2.698841e-153 0 0.000000e+00
#> 3006 3.147192e-234 2.479034e-35 1.000000e+00 2.494577e-60 0 0.000000e+00
#> 3007 0.000000e+00 2.698841e-153 2.494577e-60 1.000000e+00 0 0.000000e+00
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1 0.000000e+00
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 1.000000e+00
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0 0.000000e+00
#> 3010
#> 10 1.836994e-45
#> 11 0.000000e+00
#> 12 0.000000e+00
#> 13 0.000000e+00
#> 14 0.000000e+00
#> 15 0.000000e+00
#> 16 0.000000e+00
#> 17 0.000000e+00
#> 18 0.000000e+00
#> 19 0.000000e+00
#> 20 0.000000e+00
#> 3000 0.000000e+00
#> 3001 0.000000e+00
#> 3002 0.000000e+00
#> 3003 0.000000e+00
#> 3004 0.000000e+00
#> 3005 0.000000e+00
#> 3006 0.000000e+00
#> 3007 0.000000e+00
#> 3008 0.000000e+00
#> 3009 0.000000e+00
#> 3010 1.000000e+00
yt <- as.matrix(as.integer(spam[c(10:20,3000:3010),58]))
yt[yt==2] <- -1
## calculate the quadratic kernel expression
kernelPol(rbf, dt, ,yt)
#> 10 11 12 13 14
#> 10 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 3.076372e-01
#> 12 0.000000e+00 0.000000e+00 1.000000e+00 5.576760e-184 0.000000e+00
#> 13 0.000000e+00 0.000000e+00 5.576760e-184 1.000000e+00 0.000000e+00
#> 14 0.000000e+00 3.076372e-01 0.000000e+00 0.000000e+00 1.000000e+00
#> 15 0.000000e+00 0.000000e+00 7.620012e-15 2.283705e-99 0.000000e+00
#> 16 0.000000e+00 0.000000e+00 4.354479e-135 6.728227e-05 0.000000e+00
#> 17 0.000000e+00 1.759111e-175 1.580090e-138 0.000000e+00 1.474439e-159
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 19 0.000000e+00 3.704851e-53 4.048455e-284 0.000000e+00 5.620206e-45
#> 20 0.000000e+00 0.000000e+00 2.375087e-140 1.144518e-142 0.000000e+00
#> 3000 0.000000e+00 0.000000e+00 0.000000e+00 -9.823504e-126 0.000000e+00
#> 3001 0.000000e+00 -4.157797e-25 0.000000e+00 0.000000e+00 -1.629625e-19
#> 3002 0.000000e+00 -1.584893e-07 0.000000e+00 0.000000e+00 -1.153939e-04
#> 3003 0.000000e+00 -2.485273e-06 0.000000e+00 0.000000e+00 -3.459807e-09
#> 3004 0.000000e+00 0.000000e+00 -3.802475e-19 -5.318413e-295 0.000000e+00
#> 3005 0.000000e+00 -6.080875e-113 -2.419462e-191 0.000000e+00 -2.324676e-100
#> 3006 0.000000e+00 -2.141277e-36 0.000000e+00 0.000000e+00 -5.957980e-30
#> 3007 0.000000e+00 -1.347476e-05 0.000000e+00 0.000000e+00 -7.850063e-08
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3010 -1.836994e-45 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 15 16 17 18 19
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 0.000000e+00 1.759111e-175 0.000000e+00 3.704851e-53
#> 12 7.620012e-15 4.354479e-135 1.580090e-138 0.000000e+00 4.048455e-284
#> 13 2.283705e-99 6.728227e-05 0.000000e+00 0.000000e+00 0.000000e+00
#> 14 0.000000e+00 0.000000e+00 1.474439e-159 0.000000e+00 5.620206e-45
#> 15 1.000000e+00 1.700705e-64 3.198028e-211 0.000000e+00 0.000000e+00
#> 16 1.700705e-64 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 17 3.198028e-211 0.000000e+00 1.000000e+00 0.000000e+00 1.131156e-46
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 0.000000e+00
#> 19 0.000000e+00 0.000000e+00 1.131156e-46 0.000000e+00 1.000000e+00
#> 20 4.909368e-107 1.406121e-115 7.630353e-215 0.000000e+00 0.000000e+00
#> 3000 -1.748673e-300 -5.672366e-150 0.000000e+00 0.000000e+00 0.000000e+00
#> 3001 0.000000e+00 0.000000e+00 -5.322112e-77 0.000000e+00 -8.829844e-07
#> 3002 0.000000e+00 0.000000e+00 -3.492160e-118 0.000000e+00 -2.787511e-23
#> 3003 0.000000e+00 0.000000e+00 -1.069177e-239 0.000000e+00 -5.288180e-90
#> 3004 -1.076909e-58 -1.857644e-231 -3.153752e-58 0.000000e+00 -1.161063e-159
#> 3005 -6.669696e-284 0.000000e+00 -1.473126e-08 0.000000e+00 -1.980755e-17
#> 3006 0.000000e+00 0.000000e+00 -4.958692e-65 0.000000e+00 -1.809605e-22
#> 3007 0.000000e+00 0.000000e+00 -3.077777e-225 0.000000e+00 -1.125459e-80
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 -4.882814e-270 0.000000e+00
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 20 3000 3001 3002 3003
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 11 0.000000e+00 0.000000e+00 -4.157797e-25 -1.584893e-07 -2.485273e-06
#> 12 2.375087e-140 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 13 1.144518e-142 -9.823504e-126 0.000000e+00 0.000000e+00 0.000000e+00
#> 14 0.000000e+00 0.000000e+00 -1.629625e-19 -1.153939e-04 -3.459807e-09
#> 15 4.909368e-107 -1.748673e-300 0.000000e+00 0.000000e+00 0.000000e+00
#> 16 1.406121e-115 -5.672366e-150 0.000000e+00 0.000000e+00 0.000000e+00
#> 17 7.630353e-215 0.000000e+00 -5.322112e-77 -3.492160e-118 -1.069177e-239
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 19 0.000000e+00 0.000000e+00 -8.829844e-07 -2.787511e-23 -5.288180e-90
#> 20 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3000 0.000000e+00 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3001 0.000000e+00 0.000000e+00 1.000000e+00 1.352868e-06 4.823404e-51
#> 3002 0.000000e+00 0.000000e+00 1.352868e-06 1.000000e+00 2.576800e-23
#> 3003 0.000000e+00 0.000000e+00 4.823404e-51 2.576800e-23 1.000000e+00
#> 3004 -2.356445e-154 0.000000e+00 7.945641e-224 3.305267e-299 0.000000e+00
#> 3005 -9.713541e-301 0.000000e+00 3.387569e-37 1.378134e-67 8.914794e-166
#> 3006 0.000000e+00 0.000000e+00 2.663515e-15 2.320201e-19 1.902969e-65
#> 3007 0.000000e+00 0.000000e+00 3.061563e-44 1.242501e-19 1.643562e-02
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3009 0.000000e+00 5.593377e-93 0.000000e+00 0.000000e+00 0.000000e+00
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> 3004 3005 3006 3007 3008
#> 10 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 11 0.000000e+00 -6.080875e-113 -2.141277e-36 -1.347476e-05 0
#> 12 -3.802475e-19 -2.419462e-191 0.000000e+00 0.000000e+00 0
#> 13 -5.318413e-295 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 14 0.000000e+00 -2.324676e-100 -5.957980e-30 -7.850063e-08 0
#> 15 -1.076909e-58 -6.669696e-284 0.000000e+00 0.000000e+00 0
#> 16 -1.857644e-231 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 17 -3.153752e-58 -1.473126e-08 -4.958692e-65 -3.077777e-225 0
#> 18 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 19 -1.161063e-159 -1.980755e-17 -1.809605e-22 -1.125459e-80 0
#> 20 -2.356445e-154 -9.713541e-301 0.000000e+00 0.000000e+00 0
#> 3000 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 3001 7.945641e-224 3.387569e-37 2.663515e-15 3.061563e-44 0
#> 3002 3.305267e-299 1.378134e-67 2.320201e-19 1.242501e-19 0
#> 3003 0.000000e+00 8.914794e-166 1.902969e-65 1.643562e-02 0
#> 3004 1.000000e+00 1.023004e-91 3.147192e-234 0.000000e+00 0
#> 3005 1.023004e-91 1.000000e+00 2.479034e-35 2.698841e-153 0
#> 3006 3.147192e-234 2.479034e-35 1.000000e+00 2.494577e-60 0
#> 3007 0.000000e+00 2.698841e-153 2.494577e-60 1.000000e+00 0
#> 3008 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1
#> 3009 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 3010 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0
#> 3009 3010
#> 10 0.000000e+00 -1.836994e-45
#> 11 0.000000e+00 0.000000e+00
#> 12 0.000000e+00 0.000000e+00
#> 13 0.000000e+00 0.000000e+00
#> 14 0.000000e+00 0.000000e+00
#> 15 0.000000e+00 0.000000e+00
#> 16 0.000000e+00 0.000000e+00
#> 17 0.000000e+00 0.000000e+00
#> 18 -4.882814e-270 0.000000e+00
#> 19 0.000000e+00 0.000000e+00
#> 20 0.000000e+00 0.000000e+00
#> 3000 5.593377e-93 0.000000e+00
#> 3001 0.000000e+00 0.000000e+00
#> 3002 0.000000e+00 0.000000e+00
#> 3003 0.000000e+00 0.000000e+00
#> 3004 0.000000e+00 0.000000e+00
#> 3005 0.000000e+00 0.000000e+00
#> 3006 0.000000e+00 0.000000e+00
#> 3007 0.000000e+00 0.000000e+00
#> 3008 0.000000e+00 0.000000e+00
#> 3009 1.000000e+00 0.000000e+00
#> 3010 0.000000e+00 1.000000e+00
## calculate the kernel expansion
kernelMult(rbf, dt, ,yt)
#> [,1]
#> [1,] -1.0000000
#> [2,] -1.3076210
#> [3,] -1.0000000
#> [4,] -1.0000673
#> [5,] -1.3075217
#> [6,] -1.0000000
#> [7,] -1.0000673
#> [8,] -1.0000000
#> [9,] -1.0000000
#> [10,] -0.9999991
#> [11,] -1.0000000
#> [12,] 1.0000000
#> [13,] 1.0000005
#> [14,] 0.9998858
#> [15,] 1.0164331
#> [16,] 1.0000000
#> [17,] 1.0000000
#> [18,] 1.0000000
#> [19,] 1.0164221
#> [20,] 1.0000000
#> [21,] 1.0000000
#> [22,] 1.0000000