qsammon {qkerntool}

qKernel Sammon Mapping

Description

The qkernel Sammon Mapping is an implementation for Sammon mapping, one of the earliest dimension reduction techniques that aims to find low-dimensional embedding that preserves pairwise distance structure in high-dimensional data space. qsammon is a nonlinear form of Sammon Mapping.

Usage

## S4 method for signature 'matrix'
qsammon(x, kernel = "rbfbase", qpar = list(sigma = 0.5, q = 0.9),
          dims = 2, Initialisation = 'random', MaxHalves = 20,
          MaxIter = 500, TolFun = 1e-7, na.action = na.omit, ...)

## S4 method for signature 'cndkernmatrix'
qsammon(cndkernel, x, k, dims = 2, Initialisation = 'random',
          MaxHalves = 20,MaxIter = 500, TolFun = 1e-7, ...)

## S4 method for signature 'qkernmatrix'
qsammon(qkernel, x, k, dims = 2, Initialisation = 'random',
          MaxHalves = 20, MaxIter = 500, TolFun = 1e-7, ...)

Arguments

x

the data matrix indexed by row or a kernel matrix of cndkernmatrix or qkernmatrix.

kernel

the kernel function used in training and predicting. This parameter can be set to any function, of class kernel, which computes a kernel function value between two vector arguments. qkerntool provides the most popular kernel functions which can be used by setting the kernel parameter to the following strings:

• rbfbase Radial Basis qkernel function "Gaussian"

• nonlbase Non Linear qkernel function

• laplbase Laplbase qkernel function

• ratibase Rational Quadratic qkernel function

• multbase Multiquadric qkernel function

• invbase Inverse Multiquadric qkernel function

• wavbase Wave qkernel function

• powbase d qkernel function

• logbase Log qkernel function

• caubase Cauchy qkernel function

• chibase Chi-Square qkernel function

• studbase Generalized T-Student qkernel function

• nonlcnd Non Linear cndkernel function

• polycnd Polynomial cndkernel function

• rbfcnd Radial Basis cndkernel function "Gaussian"

• laplcnd Laplacian cndkernel function

• anocnd ANOVA cndkernel function

• raticnd Rational Quadratic cndkernel function

• multcnd Multiquadric cndkernel function

• invcnd Inverse Multiquadric cndkernel function

• wavcnd Wave cndkernel function

• powcnd d cndkernel function

• logcnd Log cndkernel function

• caucnd Cauchy cndkernel function

• chicnd Chi-Square cndkernel function

• studcnd Generalized T-Student cndkernel function

The kernel parameter can also be set to a user defined function of class kernel by passing the function name as an argument.

qpar

the list of hyper-parameters (kernel parameters). This is a list which contains the parameters to be used with the kernel function. Valid parameters for existing kernels are :

• sigma, q for the Radial Basis qkernel function "rbfbase" , the Laplacian qkernel function "laplbase" and the Cauchy qkernel function "caubase".

• alpha, q for the Non Linear qkernel function "nonlbase".

• c, q for the Rational Quadratic qkernel function "ratibase" , the Multiquadric qkernel function "multbase" and the Inverse Multiquadric qkernel function "invbase".

• theta, q for the Wave qkernel function "wavbase".

• d, q for the d qkernel function "powbase" , the Log qkernel function "logbase" and the Generalized T-Student qkernel function "studbase".

• alpha for the Non Linear cndkernel function "nonlcnd".

• d, alpha, c for the Polynomial cndkernel function "polycnd".

• gamma for the Radial Basis cndkernel function "rbfcnd" and the Laplacian cndkernel function "laplcnd" and the Cauchy cndkernel function "caucnd".

• d, sigma for the ANOVA cndkernel function "anocnd".

• c for the Rational Quadratic cndkernel function "raticnd" , the Multiquadric cndkernel function "multcnd" and the Inverse Multiquadric cndkernel function "invcnd".

• theta for the Wave cndkernel function "wavcnd".

• d for the d cndkernel function "powcnd" , the Log cndkernel function "logcnd" and the Generalized T-Student cndkernel function "studcnd".

Hyper-parameters for user defined kernels can be passed through the qpar parameter as well.

qkernel

the kernel function to be used to calculate the qkernel matrix.

cndkernel

the cndkernel function to be used to calculate the CND kernel matrix.

k

the dimension of the original data.

dims

Number of features to return. (default: 2)

Initialisation

"random" or "pca"; the former performs fast random projection and the latter performs standard PCA (default : "random")

MaxHalves

maximum number of step halvings. (default : 20)

MaxIter

the maximum number of iterations allowed. (default : 500)

TolFun

relative tolerance on objective function. (default : 1e-7)

na.action

A function to specify the action to be taken if NAs are found. The default action is na.omit, which leads to rejection of cases with missing values on any required variable. An alternative is na.fail, which causes an error if NA cases are found. (NOTE: If given, this argument must be named.)

...

additional parameters

Details

Using kernel functions one can efficiently compute principal components in high-dimensional feature spaces, related to input space by some non-linear map.
The data can be passed to the qsammon function in a matrix, in addition qsammon also supports input in the form of a kernel matrix of class qkernmatrix or class cndkernmatrix.

Value

dimRed

The matrix whose rows are embedded observations.

kcall

The function call contained

cndkernf

The kernel function used

all the slots of the object can be accessed by accessor functions.

Author(s)

Yusen Zhang
[e-mail]

References

Sammon, J.W. (1969) A Nonlinear Mapping for Data Structure Analysis. IEEE Transactions on Computers, C-18 5:401-409.

See Also

qkernmatrix, cndkernmatrix

Examples

data(iris)
train <- as.matrix(iris[,1:4])
labeltrain<- as.integer(iris[,5])
## S4 method for signature 'matrix'
kpc2 <- qsammon(train, kernel = "rbfbase", qpar = list(sigma = 2, q = 0.9), dims = 2,
                Initialisation = 'pca', TolFun = 1e-5)
plot(dimRed(kpc2), col = as.integer(labeltrain))
cndkernf(kpc2)