tests/dontrun/test-derivatives.R

In mlysy/hlm: Inference for a Heteroscedastic Linear Model

#--- derivaties of the heteroscedastic model ------------------------------

sim_Z <- function(n, p) matrix(rnorm(n*p), n, p)
sim_gam <- function(p) rnorm(p)

# random data
p <- sample(3:5, 1)
n <- sample(10:15,1)
Z <- sim_Z(n, p)
gam <- sim_gam(p)
sig <- exp(.5 * c(Z %*% gam))
y <- rnorm(n, sd = sig)

# likelihood function

# unsimplified
ll_un <- function(gam) {
  sum(dnorm(y, sd = exp(.5 * c(Z %*% gam)), log = TRUE))
}

# simplified
ll_si <- function(gam) {
  zg <- c(Z %*% gam)
  - .5 * sum(y^2/exp(zg) + zg)
}

# ok
replicate(10, {
  gam <- sim_gam(p)
  ll_un(gam) - ll_si(gam)
})

# gradient
require(numDeriv)

ll_grad <- function(gam) {
  zg <- c(Z %*% gam)
  .5 * colSums((y^2/exp(zg) - 1) * Z)
}

# ok
ans <- replicate(10, {
  gam <- sim_gam(p)
  ll_grad(gam) - numDeriv::grad(ll_un, x = gam)
})
range(ans)

# hessian

ll_hess <- function(gam) {
  zg <- c(Z %*% gam)
  - .5 * crossprod(Z, (y^2/exp(zg)) * Z)
}

ans <- replicate(10, {
  gam <- sim_gam(p)
  ll_hess(gam) - numDeriv::hessian(ll_un, x = gam)
})
range(ans)


#--- initialize algorithm -------------------------------------------------

# expectation of log-chi-square distribution

mean(log(rchisq(1e5, df = 1)))
digamma(.5) + log(2)

debug(glm.fit)

glm.fit(x = Z, y = y^2, family = Gamma(link = "log"))

glm.fit <- function (x, y, weights = rep(1, nobs), start = NULL, etastart = NULL,
    mustart = NULL, offset = rep(0, nobs), family = gaussian(),
    control = list(), intercept = TRUE, singular.ok = TRUE)
{
    control <- do.call("glm.control", control)
    x <- as.matrix(x)
    xnames <- dimnames(x)[[2L]]
    ynames <- if (is.matrix(y)) rownames(y) else names(y)
    conv <- FALSE
    nobs <- NROW(y)
    nvars <- ncol(x)
    EMPTY <- nvars == 0
    if (is.null(weights)) weights <- rep.int(1, nobs)
    if (is.null(offset)) offset <- rep.int(0, nobs)
    variance <- family$variance
    linkinv <- family$linkinv
    if (!is.function(variance) || !is.function(linkinv))
        stop("'family' argument seems not to be a valid family object",
            call. = FALSE)
    dev.resids <- family$dev.resids
    aic <- family$aic
    mu.eta <- family$mu.eta
    unless.null <- function(x, if.null) if (is.null(x)) if.null else x
    valideta <- unless.null(family$valideta, function(eta) TRUE)
    validmu <- unless.null(family$validmu, function(mu) TRUE)
    if (is.null(mustart)) {
        eval(family$initialize)
    } else {
        mukeep <- mustart
        eval(family$initialize)
        mustart <- mukeep
    }
    if (EMPTY) {
        eta <- rep.int(0, nobs) + offset
        if (!valideta(eta))
            stop("invalid linear predictor values in empty model",
                call. = FALSE)
        mu <- linkinv(eta)
        if (!validmu(mu))
            stop("invalid fitted means in empty model", call. = FALSE)
        dev <- sum(dev.resids(y, mu, weights))
        w <- sqrt((weights * mu.eta(eta)^2)/variance(mu))
        residuals <- (y - mu)/mu.eta(eta)
        good <- rep_len(TRUE, length(residuals))
        boundary <- conv <- TRUE
        coef <- numeric()
        iter <- 0L
    }
    else {
        coefold <- NULL
        eta <- if (!is.null(etastart)) {
                 etastart
               } else if (!is.null(start)) {
                 if (length(start) != nvars)
                   stop(gettextf("length of 'start' should equal %d and correspond to initial coefs for %s",
                                 nvars, paste(deparse(xnames), collapse = ", ")),
                        domain = NA)
                 else {
                   coefold <- start
                   offset + as.vector(if (NCOL(x) == 1L)
                                        x * start
                                      else x %*% start)
                 }
               } else family$linkfun(mustart)
        mu <- linkinv(eta)
        if (!(validmu(mu) && valideta(eta)))
            stop("cannot find valid starting values: please specify some",
                call. = FALSE)
        devold <- sum(dev.resids(y, mu, weights))
        boundary <- conv <- FALSE
        for (iter in 1L:control$maxit) {
            good <- weights > 0
            varmu <- variance(mu)[good]
            if (anyNA(varmu))
                stop("NAs in V(mu)")
            if (any(varmu == 0))
                stop("0s in V(mu)")
            mu.eta.val <- mu.eta(eta)
            if (any(is.na(mu.eta.val[good])))
                stop("NAs in d(mu)/d(eta)")
            good <- (weights > 0) & (mu.eta.val != 0)
            if (all(!good)) {
                conv <- FALSE
                warning(gettextf("no observations informative at iteration %d",
                  iter), domain = NA)
                break
            }
            z <- (eta - offset)[good] + (y - mu)[good]/mu.eta.val[good]
            w <- sqrt((weights[good] * mu.eta.val[good]^2)/variance(mu)[good])
            fit <- .Call(C_Cdqrls, x[good, , drop = FALSE] *
                w, z * w, min(1e-07, control$epsilon/1000), check = FALSE)
            if (any(!is.finite(fit$coefficients))) {
                conv <- FALSE
                warning(gettextf("non-finite coefficients at iteration %d",
                  iter), domain = NA)
                break
            }
            if (nobs < fit$rank)
                stop(sprintf(ngettext(nobs, "X matrix has rank %d, but only %d observation",
                  "X matrix has rank %d, but only %d observations"),
                  fit$rank, nobs), domain = NA)
            if (!singular.ok && fit$rank < nvars)
                stop("singular fit encountered")
            start[fit$pivot] <- fit$coefficients
            eta <- drop(x %*% start)
            mu <- linkinv(eta <- eta + offset)
            dev <- sum(dev.resids(y, mu, weights))
            if (control$trace)
                cat("Deviance = ", dev, " Iterations - ", iter,
                  "\n", sep = "")
            boundary <- FALSE
            if (!is.finite(dev)) {
                if (is.null(coefold))
                  stop("no valid set of coefficients has been found: please supply starting values",
                    call. = FALSE)
                warning("step size truncated due to divergence",
                  call. = FALSE)
                ii <- 1
                while (!is.finite(dev)) {
                  if (ii > control$maxit)
                    stop("inner loop 1; cannot correct step size",
                      call. = FALSE)
                  ii <- ii + 1
                  start <- (start + coefold)/2
                  eta <- drop(x %*% start)
                  mu <- linkinv(eta <- eta + offset)
                  dev <- sum(dev.resids(y, mu, weights))
                }
                boundary <- TRUE
                if (control$trace)
                  cat("Step halved: new deviance = ", dev, "\n",
                    sep = "")
            }
            if (!(valideta(eta) && validmu(mu))) {
                if (is.null(coefold))
                  stop("no valid set of coefficients has been found: please supply starting values",
                    call. = FALSE)
                warning("step size truncated: out of bounds",
                  call. = FALSE)
                ii <- 1
                while (!(valideta(eta) && validmu(mu))) {
                  if (ii > control$maxit)
                    stop("inner loop 2; cannot correct step size",
                      call. = FALSE)
                  ii <- ii + 1
                  start <- (start + coefold)/2
                  eta <- drop(x %*% start)
                  mu <- linkinv(eta <- eta + offset)
                }
                boundary <- TRUE
                dev <- sum(dev.resids(y, mu, weights))
                if (control$trace)
                  cat("Step halved: new deviance = ", dev, "\n",
                    sep = "")
            }
            if (abs(dev - devold)/(0.1 + abs(dev)) < control$epsilon) {
                conv <- TRUE
                coef <- start
                break
            }
            else {
                devold <- dev
                coef <- coefold <- start
            }
        }
        if (!conv)
            warning("glm.fit: algorithm did not converge", call. = FALSE)
        if (boundary)
            warning("glm.fit: algorithm stopped at boundary value",
                call. = FALSE)
        eps <- 10 * .Machine$double.eps
        if (family$family == "binomial") {
            if (any(mu > 1 - eps) || any(mu < eps))
                warning("glm.fit: fitted probabilities numerically 0 or 1 occurred",
                  call. = FALSE)
        }
        if (family$family == "poisson") {
            if (any(mu < eps))
                warning("glm.fit: fitted rates numerically 0 occurred",
                  call. = FALSE)
        }
        if (fit$rank < nvars)
            coef[fit$pivot][seq.int(fit$rank + 1, nvars)] <- NA
        xxnames <- xnames[fit$pivot]
        residuals <- (y - mu)/mu.eta(eta)
        fit$qr <- as.matrix(fit$qr)
        nr <- min(sum(good), nvars)
        if (nr < nvars) {
            Rmat <- diag(nvars)
            Rmat[1L:nr, 1L:nvars] <- fit$qr[1L:nr, 1L:nvars]
        }
        else Rmat <- fit$qr[1L:nvars, 1L:nvars]
        Rmat <- as.matrix(Rmat)
        Rmat[row(Rmat) > col(Rmat)] <- 0
        names(coef) <- xnames
        colnames(fit$qr) <- xxnames
        dimnames(Rmat) <- list(xxnames, xxnames)
    }
    names(residuals) <- ynames
    names(mu) <- ynames
    names(eta) <- ynames
    wt <- rep.int(0, nobs)
    wt[good] <- w^2
    names(wt) <- ynames
    names(weights) <- ynames
    names(y) <- ynames
    if (!EMPTY)
        names(fit$effects) <- c(xxnames[seq_len(fit$rank)], rep.int("",
            sum(good) - fit$rank))
    wtdmu <- if (intercept)
        sum(weights * y)/sum(weights)
    else linkinv(offset)
    nulldev <- sum(dev.resids(y, wtdmu, weights))
    n.ok <- nobs - sum(weights == 0)
    nulldf <- n.ok - as.integer(intercept)
    rank <- if (EMPTY)
        0
    else fit$rank
    resdf <- n.ok - rank
    aic.model <- aic(y, n, mu, weights, dev) + 2 * rank
    list(coefficients = coef, residuals = residuals, fitted.values = mu,
        effects = if (!EMPTY) fit$effects, R = if (!EMPTY) Rmat,
        rank = rank, qr = if (!EMPTY) structure(fit[c("qr", "rank",
            "qraux", "pivot", "tol")], class = "qr"), family = family,
        linear.predictors = eta, deviance = dev, aic = aic.model,
        null.deviance = nulldev, iter = iter, weights = wt, prior.weights = weights,
        df.residual = resdf, df.null = nulldf, y = y, converged = conv,
        boundary = boundary)
}