File: rlm.R

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# file MASS/R/rlm.R
# copyright (C) 1994-2016 W. N. Venables and B. D. Ripley
#
#  This program is free software; you can redistribute it and/or modify
#  it under the terms of the GNU General Public License as published by
#  the Free Software Foundation; either version 2 or 3 of the License
#  (at your option).
#
#  This program is distributed in the hope that it will be useful,
#  but WITHOUT ANY WARRANTY; without even the implied warranty of
#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
#
#  A copy of the GNU General Public License is available at
#  http://www.r-project.org/Licenses/
#
rlm <- function(x, ...) UseMethod("rlm")

rlm.formula <-
    function(formula, data, weights, ..., subset, na.action,
             method = c("M", "MM", "model.frame"),
             wt.method = c("inv.var", "case"),
             model = TRUE, x.ret = TRUE, y.ret = FALSE, contrasts = NULL)
{
    mf <- match.call(expand.dots = FALSE)
    mf$method <- mf$wt.method <- mf$model <- mf$x.ret <- mf$y.ret <- mf$contrasts <- mf$... <- NULL
    mf[[1L]] <- quote(stats::model.frame)
    mf <- eval.parent(mf)
    method <- match.arg(method)
    wt.method <- match.arg(wt.method)
    if(method == "model.frame") return(mf)
    mt <- attr(mf, "terms")
    y <- model.response(mf)
    offset <- model.offset(mf)
    if(!is.null(offset)) y <- y - offset
    x <- model.matrix(mt, mf, contrasts)
    xvars <- as.character(attr(mt, "variables"))[-1L]
    if ((yvar <- attr(mt, "response")) > 0L)
        xvars <- xvars[-yvar]
    xlev <- if (length(xvars) > 0L) {
        xlev <- lapply(mf[xvars], levels)
        xlev[!sapply(xlev, is.null)]
    }
    weights <- model.weights(mf)
    if(!length(weights)) weights <- rep(1, nrow(x))
    fit <- rlm.default(x, y, weights, method = method,
                       wt.method = wt.method, ...)
    fit$terms <- mt
    ## fix up call to refer to the generic, but leave arg name as `formula'
    cl <- match.call()
    cl[[1L]] <- as.name("rlm")
    fit$call <- cl
    fit$contrasts <- attr(x, "contrasts")
    fit$xlevels <- .getXlevels(mt, mf)
    fit$na.action <- attr(mf, "na.action")
    if(model) fit$model <- mf
    if(!x.ret) fit$x <- NULL
    if(y.ret) fit$y <- y
    fit
}

rlm.default <-
  function(x, y, weights, ..., w = rep(1, nrow(x)),
           init = "ls", psi = psi.huber,
           scale.est = c("MAD", "Huber", "proposal 2"), k2 = 1.345,
           method = c("M", "MM"), wt.method = c("inv.var", "case"),
           maxit = 20, acc = 1e-4, test.vec = "resid", lqs.control=NULL)
{
    irls.delta <- function(old, new)
        sqrt(sum((old - new)^2)/max(1e-20, sum(old^2)))
    irls.rrxwr <- function(x, w, r)
    {
        w <- sqrt(w)
        max(abs((matrix(r * w, 1L, length(r)) %*% x)/
                sqrt(matrix(w, 1L, length(r)) %*% (x^2))))/sqrt(sum(w * r^2))
    }
    wmad <- function(x, w)
    {
        o <- sort.list(abs(x)); x <- abs(x)[o]; w <- w[o]
        p <- cumsum(w)/sum(w)
        n <- sum(p < 0.5)
        if (p[n + 1L] > 0.5) x[n + 1L]/0.6745 else (x[n + 1L] + x[n + 2L])/(2*0.6745)
    }

    method <- match.arg(method)
    wt.method <- match.arg(wt.method)
    nmx <- deparse(substitute(x))
    if(is.null(dim(x))) {
        x <- as.matrix(x)
        colnames(x) <- nmx
    } else x <- as.matrix(x)
    if(is.null(colnames(x)))
        colnames(x) <- paste("X", seq(ncol(x)), sep="")
    if(qr(x)$rank < ncol(x))
        stop("'x' is singular: singular fits are not implemented in 'rlm'")

    if(!(any(test.vec == c("resid", "coef", "w", "NULL"))
         || is.null(test.vec))) stop("invalid 'test.vec'")
    ## deal with weights
    xx <- x
    yy <- y
    if(!missing(weights)) {
        if(length(weights) != nrow(x))
            stop("length of 'weights' must equal number of observations")
        if(any(weights < 0)) stop("negative 'weights' value")
        if(wt.method == "inv.var") {
            fac <- sqrt(weights)
            y <- y*fac; x <- x* fac
            wt <- NULL
        } else {
            w <- w * weights
            wt <- weights
        }
    } else wt <- NULL

    if(method == "M") {
        scale.est <- match.arg(scale.est)
        if(!is.function(psi)) psi <- get(psi, mode="function")
        ## match any ... args to those of psi.
        arguments <- list(...)
        if(length(arguments)) {
            pm <- pmatch(names(arguments), names(formals(psi)), nomatch = 0L)
            if(any(pm == 0L)) warning("some of ... do not match")
            pm <- names(arguments)[pm> 0L]
            formals(psi)[pm] <- unlist(arguments[pm])
        }
        if(is.character(init)) {
            temp <- if(init == "ls") lm.wfit(x, y, w, method="qr")
            else if(init == "lts") {
                if(is.null(lqs.control)) lqs.control <- list(nsamp=200L)
                do.call("lqs", c(list(x, y, intercept = FALSE), lqs.control))
            } else stop("'init' method is unknown")
            coef <- temp$coefficients
            resid <- temp$residuals
        } else {
            if(is.list(init)) coef <- init$coef
            else coef <- init
            resid <- drop(y - x %*% coef)
        }
    } else if(method == "MM") {
        scale.est <- "MM"
        temp <- do.call("lqs",
                        c(list(x, y, intercept = FALSE, method = "S",
                               k0 = 1.548), lqs.control))
        coef <- temp$coefficients
        resid <- temp$residuals
        psi <- psi.bisquare
        if(length(arguments <- list(...)))
            if(match("c", names(arguments), nomatch = 0L)) {
                c0 <- arguments$c
                if (c0 > 1.548) formals(psi)$c <- c0
                else
                    warning("'c' must be at least 1.548 and has been ignored")
            }
        scale <- temp$scale
    } else stop("'method' is unknown")

    done <- FALSE
    conv <- NULL
    n1 <- (if(is.null(wt)) nrow(x) else sum(wt)) - ncol(x)
    theta <- 2*pnorm(k2) - 1
    gamma <- theta + k2^2 * (1 - theta) - 2 * k2 * dnorm(k2)
    ## At this point the residuals are weighted for inv.var and
    ## unweighted for case weights.  Only Huber handles case weights
    ## correctly.
    if(scale.est != "MM")
        scale <- if(is.null(wt)) mad(resid, 0) else wmad(resid, wt)
    for(iiter in 1L:maxit) {
        if(!is.null(test.vec)) testpv <- get(test.vec)
        if(scale.est != "MM") {
            scale <- if(scale.est == "MAD")
                if(is.null(wt)) median(abs(resid))/0.6745 else wmad(resid, wt)
            else if(is.null(wt))
                sqrt(sum(pmin(resid^2, (k2 * scale)^2))/(n1*gamma))
            else sqrt(sum(wt*pmin(resid^2, (k2 * scale)^2))/(n1*gamma))
            if(scale == 0) {
                done <- TRUE
                break
            }
        }
        w <- psi(resid/scale)
        if(!is.null(wt)) w <- w * weights
        temp <- lm.wfit(x, y, w, method="qr")
        coef <- temp$coefficients
        resid <- temp$residuals
        if(!is.null(test.vec)) convi <- irls.delta(testpv, get(test.vec))
        else convi <- irls.rrxwr(x, w, resid)
        conv <- c(conv, convi)
        done <- (convi <= acc)
        if(done) break
    }
    if(!done)
        warning(gettextf("'rlm' failed to converge in %d steps", maxit),
                domain = NA)
    fitted <- drop(xx %*% coef)
    ## fix up call to refer to the generic, but leave arg name as `formula'
    cl <- match.call()
    cl[[1L]] <- as.name("rlm")
    fit <- list(coefficients = coef, residuals = yy - fitted, wresid = resid,
                effects = temp$effects,
                rank = temp$rank, fitted.values = fitted,
                assign = temp$assign,  qr = temp$qr, df.residual = NA, w = w,
                s = scale, psi = psi, k2 = k2,
                weights = if(!missing(weights)) weights,
                conv = conv, converged = done, x = xx, call = cl)
    class(fit) <- c("rlm", "lm")
    fit
}

print.rlm <- function(x, ...)
{
    if(!is.null(cl <- x$call)) {
        cat("Call:\n")
        dput(cl, control=NULL)
    }
    if(x$converged)
        cat("Converged in", length(x$conv), "iterations\n")
    else cat("Ran", length(x$conv), "iterations without convergence\n")
    coef <- x$coefficients
    cat("\nCoefficients:\n")
    print(coef, ...)
    nobs <- length(x$residuals)
    rdf <- nobs - length(coef)
    cat("\nDegrees of freedom:", nobs, "total;", rdf, "residual\n")
    if(nzchar(mess <- naprint(x$na.action))) cat("  (", mess, ")\n", sep="")
    cat("Scale estimate:", format(signif(x$s,3)), "\n")
    invisible(x)
}

summary.rlm <- function(object, method = c("XtX", "XtWX"),
                        correlation = FALSE, ...)
{
    method <- match.arg(method)
    s <- object$s
    coef <- object$coefficients
    ptotal <- length(coef)
    wresid <- object$wresid
    res <- object$residuals
    n <- length(wresid)
    if(any(na <- is.na(coef))) coef <- coef[!na]
    cnames <- names(coef)
    p <- length(coef)
    rinv <- diag(p)
    dimnames(rinv) <- list(cnames, cnames)
    wts <- if(length(object$weights)) object$weights else rep(1, n)
    if(length(object$call$wt.method) && object$call$wt.method == "case") {
        rdf <- sum(wts) - p
        w <- object$psi(wresid/s)
        S <- sum(wts * (wresid*w)^2)/rdf
        psiprime <- object$psi(wresid/s, deriv=1)
        m1 <- sum(wts*psiprime)
        m2 <- sum(wts*psiprime^2)
        nn <- sum(wts)
        mn <- m1/nn
        kappa <- 1 + p*(m2 - m1^2/nn)/(nn-1)/(nn*mn^2)
        stddev <- sqrt(S)*(kappa/mn)
    } else {
        res <- res * sqrt(wts)
        rdf <- n - p
        w <- object$psi(wresid/s)
        S <- sum((wresid*w)^2)/rdf
        psiprime <- object$psi(wresid/s, deriv=1)
        mn <- mean(psiprime)
        kappa <- 1 + p*var(psiprime)/(n*mn^2)
        stddev <- sqrt(S)*(kappa/mn)
    }
    X <- if(length(object$weights)) object$x * sqrt(object$weights) else object$x
    if(method == "XtWX")  {
        mn <- sum(wts*w)/sum(wts)
        X <- X * sqrt(w/mn)
    }
    R <- qr(X)$qr
    R <- R[1L:p, 1L:p, drop = FALSE]
    R[lower.tri(R)] <- 0
    rinv <- solve(R, rinv)
    dimnames(rinv) <- list(cnames, cnames)
    rowlen <- (rinv^2 %*% rep(1, p))^0.5
    names(rowlen) <- cnames
    if(correlation) {
        correl <- rinv * array(1/rowlen, c(p, p))
        correl <- correl %*% t(correl)
    } else correl <- NULL
    coef <- array(coef, c(p, 3L))
    dimnames(coef) <- list(cnames, c("Value", "Std. Error", "t value"))
    coef[, 2] <- rowlen %o% stddev
    coef[, 3] <- coef[, 1]/coef[, 2]
    object <- object[c("call", "na.action")]
    object$residuals <- res
    object$coefficients <- coef
    object$sigma <- s
    object$stddev <- stddev
    object$df <- c(p, rdf, ptotal)
    object$r.squared <- NA
    object$cov.unscaled <- rinv %*% t(rinv)
    object$correlation <- correl
    object$terms <- NA
    class(object) <- "summary.rlm"
    object
}

print.summary.rlm <-
function(x, digits = max(3, .Options$digits - 3), ...)
{
    cat("\nCall: ")
    dput(x$call, control=NULL)
    resid <- x$residuals
    df <- x$df
    rdf <- df[2L]
    cat(if(!is.null(x$weights) && diff(range(x$weights))) "Weighted ",
        "Residuals:\n", sep="")
    if(rdf > 5L) {
        if(length(dim(resid)) == 2L) {
            rq <- apply(t(resid), 1L, quantile)
            dimnames(rq) <- list(c("Min", "1Q", "Median", "3Q", "Max"),
                                 colnames(resid))
        } else {
            rq <- quantile(resid)
            names(rq) <- c("Min", "1Q", "Median", "3Q", "Max")
        }
        print(rq, digits = digits, ...)
    } else if(rdf > 0L) {
        print(resid, digits = digits, ...)
    }
    if(nsingular <- df[3L] - df[1L])
        cat("\nCoefficients: (", nsingular,
            " not defined because of singularities)\n", sep = "")
    else cat("\nCoefficients:\n")
    print(format(round(x$coefficients, digits = digits)), quote = FALSE, ...)
    cat("\nResidual standard error:", format(signif(x$sigma, digits)),
        "on", rdf, "degrees of freedom\n")
    if(nzchar(mess <- naprint(x$na.action))) cat("  (", mess, ")\n", sep="")
    if(!is.null(correl <- x$correlation)) {
        p <- dim(correl)[2L]
        if(p > 1L) {
            cat("\nCorrelation of Coefficients:\n")
            ll <- lower.tri(correl)
            correl[ll] <- format(round(correl[ll], digits))
            correl[!ll] <- ""
            print(correl[-1L, -p, drop = FALSE], quote = FALSE, digits = digits, ...)
        }
    }
    invisible(x)
}

psi.huber <- function(u, k = 1.345, deriv=0)
{
    if(!deriv) return(pmin(1, k / abs(u)))
    abs(u) <= k
}

psi.hampel <- function(u, a = 2, b = 4, c = 8, deriv=0)
{
    U <- pmin(abs(u) + 1e-50, c)
    if(!deriv) return(ifelse(U <= a, U, ifelse(U <= b, a, a*(c-U)/(c-b) ))/U)
    ifelse(abs(u) <= c, ifelse(U <= a, 1, ifelse(U <= b, 0, -a/(c-b))), 0)
}

psi.bisquare <- function(u, c = 4.685, deriv=0)
{
    if(!deriv) return((1  - pmin(1, abs(u/c))^2)^2)
    t <- (u/c)^2
    ifelse(t < 1, (1 - t)*(1 - 5*t), 0)
}

se.contrast.rlm <-
    function(object, contrast.obj,
             coef = contr.helmert(ncol(contrast))[, 1L],
             data = NULL, ...)
{
    contrast.weight.aov <- function(object, contrast)
    {
        asgn <- object$assign[object$qr$pivot[1L:object$rank]]
        uasgn <- unique(asgn)
        nterms <- length(uasgn)
        nmeffect <- c("(Intercept)",
                      attr(object$terms, "term.labels"))[1L + uasgn]
        effects <- as.matrix(qr.qty(object$qr, contrast))
        res <- matrix(0, nrow = nterms, ncol = ncol(effects),
                      dimnames = list(nmeffect, colnames(contrast)))
        for(i in seq(nterms)) {
            select <- (asgn == uasgn[i])
            res[i,] <- colSums(effects[seq_along(asgn)[select], , drop = FALSE]^2)
        }
        res
    }
    if(is.null(data)) contrast.obj <- eval(contrast.obj)
    else contrast.obj <- eval(substitute(contrast.obj), data, parent.frame())
    if(!is.matrix(contrast.obj)) { # so a list
        if(!missing(coef)) {
            if(sum(coef) != 0)
                stop("'coef' must define a contrast, i.e., sum to 0")
            if(length(coef) != length(contrast.obj))
                stop("'coef' must have same length as 'contrast.obj'")
        }
        contrast <-
            sapply(contrast.obj, function(x)
               {
                   if(!is.logical(x))
                       stop(gettextf("each element of '%s' must be logical",
                                     substitute(contrasts.list)),
                            domain = NA)
                   x/sum(x)
               })
        if(!length(contrast) || all(is.na(contrast)))
            stop("the contrast defined is empty (has no TRUE elements)")
        contrast <- contrast %*% coef
    } else {
        contrast <- contrast.obj
        if(any(abs(colSums(contrast)) > 1e-8))
            stop("columns of 'contrast.obj' must define a contrast (sum to zero)")
        if(!length(colnames(contrast)))
            colnames(contrast) <- paste("Contrast", seq(ncol(contrast)))
    }
    weights <- contrast.weight.aov(object, contrast)
    summary(object)$stddev *
        if(!is.matrix(contrast.obj)) sqrt(sum(weights)) else sqrt(colSums(weights))
}

predict.rlm <- function (object, newdata = NULL, scale = NULL, ...)
{
    ## problems with using predict.lm are the scale and
    ## the QR decomp which has been done on down-weighted values.
    object$qr <- qr(sqrt(object$weights) * object$x)
    NextMethod(object, scale = object$s, ...)
}

vcov.rlm <- function (object, ...)
{
    so <- summary(object, corr = FALSE)
    so$stddev^2 * so$cov.unscaled
}