R: Kernel Quantile Regression.

kqr {kernlab} | R Documentation

## Kernel Quantile Regression.

### Description

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

### Usage

    ## 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 `NA`s 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

    # 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]

[Package kernlab version 0.9-25 Index]