The Kernel Quantile Regression algorithm kqr performs
non-parametric Quantile Regression.
# S4 method for formula
kqr(x, data=NULL, ..., subset, na.action = na.omit, scaled = TRUE)# S4 method for vector
kqr(x,...)
# S4 method for matrix
kqr(x, y, scaled = TRUE, tau = 0.5, C = 0.1, kernel = "rbfdot",
kpar = "automatic", reduced = FALSE, rank = dim(x)[1]/6,
fit = TRUE, cross = 0, na.action = na.omit)
# S4 method for kernelMatrix
kqr(x, y, tau = 0.5, C = 0.1, fit = TRUE, cross = 0)
# S4 method for list
kqr(x, y, tau = 0.5, C = 0.1, kernel = "strigdot",
kpar= list(length=4, C=0.5), fit = TRUE, cross = 0)
e data or a symbolic description of the model to be fit.
When not using a formula x can be a matrix or vector containing
the training data or a kernel matrix of class kernelMatrix
of the training data or a list of character vectors (for use
with the string kernel). Note, that the intercept is always
excluded, whether given in the formula or not.
an optional data frame containing the variables in the model.
By default the variables are taken from the environment which
kqr is called from.
a numeric vector or a column matrix containing the response.
A logical vector indicating the variables to be
scaled. If scaled is of length 1, the value is recycled as
many times as needed and all non-binary variables are scaled.
Per default, data are scaled internally (both x and y
variables) to zero mean and unit variance. The center and scale
values are returned and used for later predictions. (default: TRUE)
the quantile to be estimated, this is generally a number strictly between 0 and 1. For 0.5 the median is calculated. (default: 0.5)
the cost regularization parameter. This parameter controls the smoothness of the fitted function, essentially higher values for C lead to less smooth functions.(default: 1)
the kernel function used in training and predicting.
This parameter can be set to any function, of class kernel, which computes a dot product between two
vector arguments. 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
stringdot String 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".
lenght, lambda, normalized for the "stringdot" kernel
where length is the length of the strings considered, lambda the
decay factor and normalized a logical parameter determining if the
kernel evaluations should be normalized.
Hyper-parameters for user defined kernels can be passed
through the kpar parameter as well. In the case of a Radial
Basis kernel function (Gaussian) kpar can also be set to the
string "automatic" which uses the heuristics in 'sigest' to
calculate a good 'sigma' value for the Gaussian RBF or
Laplace kernel, from the data. (default = "automatic").
use an incomplete cholesky decomposition to calculate a
decomposed form \(Z\) of the kernel Matrix \(K\) (where \(K = ZZ'\)) and
perform the calculations with \(Z\). This might be useful when
using kqr with large datasets since normally an n times n
kernel matrix would be computed. Setting reduced to TRUE
makes use of csi to compute a decomposed form instead and
thus only a \(n \times m\) matrix where \(m < n\) and \(n\) the sample size is
stored in memory (default: FALSE)
the rank m of the decomposed matrix calculated when using an
incomplete cholesky decomposition. This parameter is only
taken into account when reduced is TRUE(default :
dim(x)[1]/6)
indicates whether the fitted values should be computed and included in the model or not (default: 'TRUE')
if a integer value k>0 is specified, a k-fold cross validation on the training data is performed to assess the quality of the model: the Pinball loss and the for quantile regression
An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)
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.
An S4 object of class kqr containing the fitted model along with
information.Accessor functions can be used to access the slots of the
object which include :
The resulting model parameters which can be also accessed
by coef.
the kernel function used.
Training error (if fit == TRUE)
In quantile regression a function is fitted to the data so that
it satisfies the property that a portion \(tau\) of the data
\(y|n\) is below the estimate. While the error bars of many
regression problems can be viewed as such estimates quantile
regression estimates this quantity directly. Kernel quantile regression
is similar to nu-Support Vector Regression in that it minimizes a
regularized loss function in RKHS. The difference between nu-SVR and
kernel quantile regression is in the type of loss function used which
in the case of quantile regression is the pinball loss (see reference
for details.). Minimizing the regularized loss boils down to a
quadratic problem which is solved using an interior point QP solver
ipop implemented in kernlab.
Ichiro Takeuchi, Quoc V. Le, Timothy D. Sears, Alexander J. Smola Nonparametric Quantile Estimation Journal of Machine Learning Research 7,2006,1231-1264 http://www.jmlr.org/papers/volume7/takeuchi06a/takeuchi06a.pdf
predict.kqr, kqr-class, ipop, rvm, ksvm
# NOT RUN {
# create data
x <- sort(runif(300))
y <- sin(pi*x) + rnorm(300,0,sd=exp(sin(2*pi*x)))
# first calculate the median
qrm <- kqr(x, y, tau = 0.5, C=0.15)
# predict and plot
plot(x, y)
ytest <- predict(qrm, x)
lines(x, ytest, col="blue")
# calculate 0.9 quantile
qrm <- kqr(x, y, tau = 0.9, kernel = "rbfdot",
kpar= list(sigma=10), C=0.15)
ytest <- predict(qrm, x)
lines(x, ytest, col="red")
# calculate 0.1 quantile
qrm <- kqr(x, y, tau = 0.1,C=0.15)
ytest <- predict(qrm, x)
lines(x, ytest, col="green")
# print first 10 model coefficients
coef(qrm)[1:10]
# }
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