Kernel Hebbian Algorithm is a nonlinear iterative algorithm for principal component analysis.
# S4 method for formula
kha(x, data = NULL, na.action, ...)# S4 method for matrix
kha(x, kernel = "rbfdot", kpar = list(sigma = 0.1), features = 5,
eta = 0.005, th = 1e-4, maxiter = 10000, verbose = FALSE,
na.action = na.omit, ...)
An S4 object containing the principal component vectors along with the corresponding normalization values.
a matrix containing the principal component vectors (column wise)
The normalization values
The original data matrix
all the slots of the object can be accessed by accessor functions.
The data matrix indexed by row or a formula describing the model. Note, that an intercept is always included, whether given in the formula or not.
an optional data frame containing the variables in the model (when using a formula).
the kernel function used in training and predicting.
This parameter can be set to any function, of class kernel, which
computes the inner product in feature space between two
vector arguments (see kernels).
kernlab provides the most popular kernel functions
which can be used by setting the kernel parameter to the following
strings:
rbfdot Radial Basis kernel function "Gaussian"
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 Spline kernel
The kernel parameter can also be set to a user defined function of class kernel by passing the function name as an argument.
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 inverse kernel width for the Radial Basis
kernel function "rbfdot" and the Laplacian kernel "laplacedot".
degree, scale, offset for the Polynomial kernel "polydot"
scale, offset for the Hyperbolic tangent kernel
function "tanhdot"
sigma, order, degree for the Bessel kernel "besseldot".
sigma, degree for the ANOVA kernel "anovadot".
Hyper-parameters for user defined kernels can be passed through the kpar parameter as well.
Number of features (principal components) to return. (default: 5)
The hebbian learning rate (default : 0.005)
the smallest value of the convergence step (default : 0.0001)
the maximum number of iterations.
print convergence every 100 iterations. (default : FALSE)
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
Alexandros Karatzoglou
alexandros.karatzoglou@ci.tuwien.ac.at
The original form of KPCA can only be used on small data sets since it requires the estimation of the eigenvectors of a full kernel matrix. The Kernel Hebbian Algorithm iteratively estimates the Kernel Principal Components with only linear order memory complexity. (see ref. for more details)
Kwang In Kim, M.O. Franz and B. Schölkopf
Kernel Hebbian Algorithm for Iterative Kernel Principal Component Analysis
Max-Planck-Institut für biologische Kybernetik, Tübingen (109)
https://is.mpg.de/fileadmin/user_upload/files/publications/pdf2302.pdf
kpca, kfa, kcca, pca
# another example using the iris
data(iris)
test <- sample(1:150,70)
kpc <- kha(~.,data=iris[-test,-5],kernel="rbfdot",
kpar=list(sigma=0.2),features=2, eta=0.001, maxiter=65)
#print the principal component vectors
pcv(kpc)
#plot the data projection on the components
plot(predict(kpc,iris[,-5]),col=as.integer(iris[,5]),
xlab="1st Principal Component",ylab="2nd Principal Component")
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