R: Additive Quantile Regression Smoothing

rqss {quantreg} | R Documentation

## Additive Quantile Regression Smoothing

### Description

Fitting function for additive quantile regression models with possible univariate and/or bivariate nonparametric terms estimated by total variation regularization. See `summary.rqss` and `plot.rqss` for further details on inference and confidence bands.

### Usage

    rqss(formula, tau = 0.5, data = parent.frame(), weights, subset, na.action,
    	method = "sfn", lambda = NULL, contrasts = NULL, ztol = 1e-5, control, ...)

### Arguments

`formula` | a formula object, with the response on the left of a ‘~’ operator, and terms, separated by ‘+’ operators, on the right. The terms may include `qss` terms that represent additive nonparametric components. These terms can be univariate or bivariate. See `qss` for details on how to specify these terms.
`tau` | the quantile to be estimated, this must be a number between 0 and 1,
`data` | a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument.
`weights` | vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous.
`subset` | an optional vector specifying a subset of observations to be used in the fitting. This can be a vector of indices of observations to be included, or a logical vector.
`na.action` | a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with `na.fail`) is to create an error if any missing values are found. A possible alternative is `na.omit`, which deletes observations that contain one or more missing values.
`method` | the algorithmic method used to compute the fit. There are currently two options. Both are implementations of the Frisch–Newton interior point method described in detail in Portnoy and Koenker(1997). Both are implemented using sparse Cholesky decomposition as described in Koenker and Ng (2003). Option `"sfnc"` is used if the user specifies inequality constraints. Option `"sfn"` is used if there are no inequality constraints. Linear inequality constraints on the fitted coefficients are specified by a matrix `R` and a vector `r`, specified inside the `qss` terms, representing the constraints in the form Rb >= r. The option `method = "lasso"` allows one to penalize the coefficients of the covariates that have been entered linearly as in `rq.fit.lasso`; when this is specified then there should be an additional `lambda` argument specified that determines the amount of shrinkage.
`lambda` | can be either a scalar, in which case all the slope coefficients are assigned this value, or alternatively, the user can specify a vector of length equal to the number of linear covariates plus one (for the intercept) and these values will be used as coordinate dependent shrinkage factors.
`contrasts` | a list giving contrasts for some or all of the factors default = `NULL` appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any full-rank matrix with as many rows as there are levels in the factor), or else a function to compute such a matrix given the number of levels.
`ztol` | A zero tolerance parameter used to determine the number of zero residuals in the fitted object which in turn determines the effective dimensionality of the fit.
`control` | control argument for the fitting routines (see `sfn.control`
`...` | Other arguments passed to fitting routines

### Details

Total variation regularization for univariate and bivariate nonparametric quantile smoothing is described in Koenker, Ng and Portnoy (1994) and Koenker and Mizera(2003) respectively. The additive model extension of this approach depends crucially on the sparse linear algebra implementation for R described in Koenker and Ng (2003). There are extractor methods `logLik` and `AIC` that is relevant to lambda selection. A more detailed description of some recent developments of these methods is available from within the package with `vignette("rq")`. Since this function uses sparse versions of the interior point algorithm it may also prove to be useful for fitting linear models without `qss` terms when the design has a sparse structure, as for example when there is a complicated factor structure.

If the MatrixModels and Matrix packages are both loadable then the linear-in-parameters portion of the design matrix is made in sparse matrix form; this is helpful in large applications with many factor variables for which dense formation of the design matrix would take too much space.

Although modeling with `rqss` typically imposes smoothing penalties on the total variation of the first derivative, or gradient, of the fitted functions, for univariate smoothing, it is also possible to penalize total variation of the function itself using the option `Dorder = 0` inside `qss` terms. In such cases, estimated functions are piecewise constant rather than piecewise linear. See the documentation for `qss` for further details.

### Value

The function returns a fitted object representing the estimated model specified in the formula. See `rqss.object` for further details on this object, and references to methods to look at it.

### Note

If you intend to embed calls to `rqss` inside another function, then it is advisable to pass a data frame explicitly as the `data` argument of the `rqss` call, rather than relying on the magic of R scoping rules.

### Author(s)

Roger Koenker

### References

[1] Koenker, R. and S. Portnoy (1997) The Gaussian Hare and the Laplacean Tortoise: Computability of Squared-error vs Absolute Error Estimators, (with discussion). Statistical Science 12, 279–300.

[2] Koenker, R., P. Ng and S. Portnoy, (1994) Quantile Smoothing Splines; Biometrika 81, 673–680.

[3] Koenker, R. and I. Mizera, (2003) Penalized Triograms: Total Variation Regularization for Bivariate Smoothing; JRSS(B) 66, 145–163.

[4] Koenker, R. and P. Ng (2003) SparseM: A Sparse Linear Algebra Package for R, J. Stat. Software.

### See Also

`qss`

### Examples

    n <- 200
    x <- sort(rchisq(n,4))
    z <- x + rnorm(n)
    y <- log(x)+ .1*(log(x))^2 + log(x)*rnorm(n)/4 + z
    plot(x, y-z)
    f.N  <- rqss(y ~ qss(x, constraint= "N") + z)
    f.I  <- rqss(y ~ qss(x, constraint= "I") + z)
    f.CI <- rqss(y ~ qss(x, constraint= "CI") + z)

    lines(x[-1], f.N $coef[1] + f.N $coef[-(1:2)])
    lines(x[-1], f.I $coef[1] + f.I $coef[-(1:2)], col="blue")
    lines(x[-1], f.CI$coef[1] + f.CI$coef[-(1:2)], col="red")

    ## A bivariate example
    if(requireNamespace("tripack")){
    if(requireNamespace("akima")){
    data(CobarOre)
    fCO <- rqss(z ~ qss(cbind(x,y), lambda= .08), data=CobarOre)
    plot(fCO)
    }}

[Package quantreg version 5.93 Index]