R: Estimating Conditional Probability Density Using Smoothing...

sscden {gss} | R Documentation

## Estimating Conditional Probability Density Using Smoothing Splines

### Description

Estimate conditional probability densities using smoothing spline ANOVA models. The symbolic model specification via `formula` follows the same rules as in `lm`.

### Usage

    sscden(formula, response, type=NULL, data=list(), weights, subset,
           na.action=na.omit, alpha=1.4, id.basis=NULL, nbasis=NULL,
           seed=NULL, ydomain=as.list(NULL), yquad=NULL, prec=1e-7,
           maxiter=30, skip.iter=FALSE)

    sscden1(formula, response, type=NULL, data=list(), weights, subset,
            na.action=na.omit, alpha=1.4, id.basis=NULL, nbasis=NULL,
            seed=NULL, rho=list("xy"), ydomain=as.list(NULL), yquad=NULL,
            prec=1e-7, maxiter=30, skip.iter=FALSE)

### Arguments

`formula` | Symbolic description of the model to be fit.
`response` | Formula listing response variables.
`type` | List specifying the type of spline for each variable. See `mkterm` for details.
`data` | Optional data frame containing the variables in the model.
`weights` | Optional vector of counts for duplicated data.
`subset` | Optional vector specifying a subset of observations to be used in the fitting process.
`na.action` | Function which indicates what should happen when the data contain NAs.
`alpha` | Parameter defining cross-validation scores for smoothing parameter selection.
`id.basis` | Index of observations to be used as "knots."
`nbasis` | Number of "knots" to be used. Ignored when `id.basis` is specified.
`seed` | Seed to be used for the random generation of "knots." Ignored when `id.basis` is specified.
`ydomain` | Data frame specifying marginal support of conditional density.
`yquad` | Quadrature for calculating integral on Y domain. Mandatory if response variables other than factors or numerical vectors are involved.
`prec` | Precision requirement for internal iterations.
`maxiter` | Maximum number of iterations allowed for internal iterations.
`skip.iter` | Flag indicating whether to use initial values of theta and skip theta iteration. See `ssanova` for notes on skipping theta iteration.
`rho` | rho function needed for sscden1.

### Details

The model is specified via `formula` and `response`, where `response` lists the response variables. For example, `sscden(~y*x,~y)` prescribe a model of the form

log f(y|x) = g_{y}(y) + g_{xy}(x,y) + C(x)

with the terms denoted by `"y"`, `"y:x"`; the term(s) not involving response(s) are removed and the constant `C(x)` is determined by the fact that a conditional density integrates to one on the `y` axis. `sscden1` does keep terms not involving response(s) during estimation, although those terms cancel out when one evaluates the estimated conditional density.

The model terms are sums of unpenalized and penalized terms. Attached to every penalized term there is a smoothing parameter, and the model complexity is largely determined by the number of smoothing parameters.

A subset of the observations are selected as "knots." Unless specified via `id.basis` or `nbasis`, the number of "knots" q is determined by max(30,10n^{2/9}), which is appropriate for the default cubic splines for numerical vectors.

### Value

`sscden` returns a list object of class `"sscden"`. `sscden1` returns a list object of class `c("sscden1","sscden")`.

`dsscden` and `cdsscden` can be used to evaluate the estimated conditional density f(y|x) and f(y1|x,y2); `psscden`, `qsscden`, `cpsscden`, and `cqsscden` can be used to evaluate conditional cdf and quantiles. The methods `project.sscden` or `project.sscden1` can be used to calculate the Kullback-Leibler or square-error projections for model selection.

### Note

Default quadrature on the Y domain will be constructed for numerical vectors on a hyper cube, then outer product with factor levels will be taken if factors are involved. The sides of the hyper cube are specified by `ydomain`; for `ydomain$y` missing, the default is `c(min(y),max(y))+c(-1,1)*(max(y)-mimn(y))*.05`.

On a 1-D interval, the quadrature is the 200-point Gauss-Legendre formula returned from `gauss.quad`. For multiple numerical vectors, delayed Smolyak cubatures from `smolyak.quad` are used on cubes with the marginals properly transformed; see Gu and Wang (2003) for the marginal transformations.

The results may vary from run to run. For consistency, specify `id.basis` or set `seed`.

For reasonable execution time in high dimensions, set `skip.iter=TRUE`.

### References

Gu, C. (1995), Smoothing spline density estimation: Conditional distribution. Statistica Sinica, 5, 709–726. Springer-Verlag.

Gu, C. (2014), Smoothing Spline ANOVA Models: R Package gss. Journal of Statistical Software, 58(5), 1-25. URL http://www.jstatsoft.org/v58/i05/.

### Examples

    data(penny); set.seed(5732)
    fit <- sscden1(~year*mil,~mil,data=penny,
                  ydomain=data.frame(mil=c(49,61)))
    yy <- 1944+(0:92)/2
    quan <- qsscden(fit,c(.05,.25,.5,.75,.95),
                    data.frame(year=yy))
    plot(penny$year+.1*rnorm(90),penny$mil,ylim=c(49,61))
    for (i in 1:5) lines(yy,quan[i,])
    ## Clean up
    ## Not run: rm(penny,yy,quan)

[Package gss version 2.2-2 Index]