kernelMatrix: Kernel Matrix functions

Description Usage Arguments Details Value Author(s) See Also Examples

View source: R/kernels.R

Description

kernelMatrix 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) = ∑_{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.

Usage

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## S4 method for signature 'kernel'
kernelMatrix(kernel, x, y = NULL)

## S4 method for signature 'kernel'
kernelPol(kernel, x, y = NULL, z, k = NULL)

## S4 method for signature 'kernel'
kernelMult(kernel, x, y = NULL, z, blocksize = 256)

## S4 method for signature 'kernel'
kernelFast(kernel, x, y, a)

Arguments

kernel

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.

x

a data matrix to be used to calculate the kernel matrix, or a list of vector when a stringkernel is used

y

second data matrix to calculate the kernel matrix, or a list of vector when a stringkernel is used

z

a suitable vector or matrix

k

a suitable vector or matrix

a

the squared norm of x, e.g., rowSums(x^2)

blocksize

the kernel expansion computations are done block wise to avoid storing the kernel matrix into memory. blocksize defines the size of the computational blocks.

Details

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:

(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.

Value

kernelMatrix returns a symmetric diagonal semi-definite matrix.
kernelPol returns a matrix.
kernelMult usually returns a one-column matrix.

Author(s)

Alexandros Karatzoglou
alexandros.karatzoglou@ci.tuwien.ac.at

See Also

rbfdot, polydot, tanhdot, vanilladot

Examples

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## 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

## calculate kernel matrix
kernelMatrix(rbf, dt)

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)

## calculate the kernel expansion
kernelMult(rbf, dt, ,yt)

Example output

Gaussian Radial Basis kernel function. 
 Hyperparameter : sigma =  0.05 
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
                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
            [,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

kernlab documentation built on Nov. 12, 2019, 9:07 a.m.