Kernel Quantile Regression.

Description

The Kernel Quantile Regression algorithm kqr performs non-parametric Quantile Regression.

Usage

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## S4 method for signature 'formula'
kqr(x, data=NULL, ..., subset, na.action = na.omit, scaled = TRUE)

## S4 method for signature 'vector'
kqr(x,...)

## S4 method for signature '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 signature 'kernelMatrix'
kqr(x, y, tau = 0.5, C = 0.1, fit = TRUE, cross = 0)

## S4 method for signature 'list'
kqr(x, y, tau = 0.5, C = 0.1, kernel = "strigdot",
    kpar= list(length=4, C=0.5), fit = TRUE, cross = 0)

Arguments

x

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.

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment which kqr is called from.

y

a numeric vector or a column matrix containing the response.

scaled

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)

tau

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)

C

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)

kernel

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.

kpar

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").

reduced

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)

rank

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)

fit

indicates whether the fitted values should be computed and included in the model or not (default: 'TRUE')

cross

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

subset

An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)

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

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.

Value

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 :

alpha

The resulting model parameters which can be also accessed by coef.

kernelf

the kernel function used.

error

Training error (if fit == TRUE)

see kqr-class for more details.

Author(s)

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

References

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

See Also

predict.kqr, kqr-class, ipop, rvm, ksvm

Examples

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# 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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