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#' Plug-in bandwidth for local polynomial estimator of a psychometric function
#'
#' The function calculates an estimate of the AMISE optimal bandwidth for
#' a local polynomial estimate of the psychometric function.
#'
#' @usage bandwidth_plugin( r, m, x, link = "logit", guessing = 0,
#' lapsing = 0, K = 2, p = 1, ker = "dnorm" )
#
# INPUT
#
#' @param r number of successes at points x
#' @param m number of trials at points x
#' @param x stimulus levels
#
# OPTIONAL INPUT
#
#' @param link (optional) name of the link function to be used; default is "logit"
#' @param guessing (optional) guessing rate; default is 0
#' @param lapsing (optional) lapsing rate; default is 0
#' @param K (optional) power parameter for Weibull and reverse Weibull link; default is 2
#' @param p (optional) degree of the polynomial; default is 1
#' @param ker (optional) kernel function for weights; default is "dnorm"
#
# OUTPUT
#
#' @returns \verb{h } plug-in bandwidth (ISE optimal on eta-scale)
#' @importFrom stats integrate predict glm binomial deriv coefficients
#' @importFrom PolynomF polynom
#'
#' @examples
#' data("Miranda_Henson")
#' x = Miranda_Henson$x
#' r = Miranda_Henson$r
#' m = Miranda_Henson$m
#' numxfit <- 199; # Number of new points to be generated minus 1
#' xfit <- (max(x)-min(x)) * (0:numxfit) / numxfit + min(x)
#' # Find a plug-in bandwidth
#' bwd <- bandwidth_plugin( r, m, x)
#' pfit <- locglmfit( xfit, r, m, x, bwd )$pfit
#' # Plot the fitted curve
#' plot( x, r / m, xlim = c( 0.1, 1.302 ), ylim = c( 0.0165, 0.965 ), type = "p", pch="*" )
#' lines(xfit, pfit )
#' @export
bandwidth_plugin<-function( r, m, x, link = "logit", guessing = 0,
lapsing = 0, K = 2, p = 1, ker = "dnorm" ) {
#
# The function calculates an estimate of the AMISE optimal bandwidth for
# a local polynomial estimate of the psychometric function.
#
# INPUT
#
# r - number of successes at points x
# m - number of trials at points x
# x - stimulus levels
#
# OPTIONAL INPUT
#
# link - name of the link function to be used; default is "logit"
# guessing - guessing rate; default is 0
# lapsing - lapsing rate; default is 0
# K - power parameter for Weibull and reverse Weibull link; default is 2
# p - degree of the polynomial; default is 1
# ker - kernel function for weights; default is "dnorm"
#
# OUTPUT
#
# h - plug-in bandwidth (ISE optimal on eta-scale)
# INTERNAL FUNCTIONS
# KERNELS
# Epanechnikov
epanechnikov<-function( x ) {
X <- x;
X[which( abs( X ) > 1 )] <- 1;
return( 0.75 * ( 1 - X^2 ) );
}
# triangular
triangular<-function( x ) {
X <- x;
X[which( abs( X ) > 1 )] <- 1;
return( ( 1 - abs( X ) ) );
}
# tri-cube
tricube<-function( x ) {
X <- x;
X[which( abs( X ) > 1 )] <- 1;
return( ( ( 1 - abs( X )^3 )^3 ) );
}
# bi-square
bisquare<-function( x ){
X <- x;
X[which( abs( X ) > 1 )] <- 1;
return( ( (1 - abs( X )^2 )^2) );
}
# uniform
uniform<-function( x ) {
X <- x;
return( ( abs( X ) <= 1 ) / 2 );
}
# EQUIVALENT KERNEL
ker_eqv <- function( x, deg = p, K = ker, S1 = S ) {
K2 <- NULL;
K2 <- get( K );
return( ( ( e_0 %*% solve( S1 ) %*% t( matrix( x, length( x ), deg + 1 )^
t( matrix( 0:deg, deg + 1, length( x ) ) ) ) ) * K2( x ) ) );
}
ker_eqv2 <- function( x, deg = p, K = ker, S1 = S ) {
return( ker_eqv( x, deg, K, S1 )^2 );
}
# MOMENTS OF KERNEL
moments <- function( K, l ) {
getxK <- function( x, power = 0, f = K ) {
fun <- NULL;
fun <- get( K );
return( x^power * fun( x ) );
}
int <- integrate( getxK, -Inf, Inf, power = l );
return( int$value );
}
# VARIANCE FUNCTION FOR BINOMIAL DISTRIBUTION
varfun <- function( x1, fit1 = fit ) {
# estimated mean
mu <- as.numeric( predict( fit1, data.frame( x = x1 ), type = "response" ) );
# offset
epsilon <- .001;
ind <- rep( 0, length( x1 ) );
ind[which( ( mu >= epsilon ) & ( mu <= 1 - epsilon ) )] <- 1;
# variance
return( ind / ( mu * (1 - mu ) ) );
}
# MAIN PROGRAM
# First 3 arguments are mandatory
if( missing("r") || missing("m") || missing("x") ) {
stop("Check input. First 3 arguments are mandatory");
}
# CHECK ROBUSTNESS OF INPUT PARAMETERS
checkdata<-list();
checkdata[[1]] <- x;
checkdata[[2]] <- r;
checkdata[[3]] <- m;
checkinput( "psychometricdata", checkdata );
rm( checkdata )
pn <- list()
pn[[1]] <- p
pn[[2]] <- x
checkinput( "degreepolynomial", pn );
checkinput( "kernel", ker );
checkinput( 'linkfunction', link );
if( length( guessing ) > 1 ) {
stop( "Guessing rate must be a scalar" );
}
if( length( lapsing ) > 1 ) {
stop( "Lapsing rate must be a scalar" );
}
checkinput( "guessingandlapsing", c( guessing, lapsing ) );
if (link == "weibull" || link == "revweibull"){
checkinput( "exponentk", K );
}
n <- length(r);
# p+Dp - degree of polynomial used for parametric fit
if( ( n - p ) > 5 ) {
Dp <- 3;
}
else {
if( ( n - p ) == 5 ) {
Dp <- 2;
}
else {
if ( ( n - p ) > 2 ) {
Dp = 1;
}
else {
stop( paste( "Not enough data to fit polynomial of degree", p,
sep = " " ) );
}
}
}
# PARAMETRIC ESTIMATOR
# parametric estimator of order p + Dp
# INITIAL
p1 <- p + Dp;
lx <- length( x );
# GLOBAL FIT
# create data frame
glmdata <- data.frame( cbind( r/m ,m , x ) );
names( glmdata ) <- c( "resp", "m", "x" );
# formula
glmformula <- c( "resp ~ x" );
if( p1 > 1 ) {
for( pp in 2:p1 ) {
glmformula <- paste( glmformula, " + I(x^", pp,")", sep = "");
}
}
# fit
if( ( link == "weibull" ) ||
( link == "revweibull" ) ) {
linkfun <- get( paste( link, "_link_private", sep = "" ) );
fit <- glm( glmformula, data = glmdata, weights = m,
family = binomial( linkfun( K, guessing, lapsing ) ) );
}
else {
if( link == "logit" ||
link == "probit" ||
link == "loglog" ||
link == "comploglog" ){
linkfun <- get( paste( link, "_link_private", sep = "" ) );
fit <- glm( glmformula, data = glmdata, weights = m,
family = binomial( linkfun( guessing, lapsing ) ) );
}
else{
linkfun <- get( link );
fit <- glm( glmformula, data = glmdata, weights = m,
family = binomial( linkfun( guessing, lapsing ) ) );
}
}
if (any(!is.finite(fit$coefficients)) ) stop('Result is degenerate: the link function is probably inappropriate')
pol <- polynom( fit$coefficients );
# adjust degenrated values to aviod degenerate results
epsilon <- .001;
ind <- rep( 0, lx );
ind[which( ( r / m >= epsilon ) & ( r / m <= 1 - epsilon ) )] <- 1;
# coefficients for (p+1)th derivative of parametric estimator
for( l in 1:(p+1) ) {
pol <- deriv( pol );
}
tmp_b_p1 <- coefficients( pol );
# calculate the (p+1)th derivative of eta
tmp_eta_p1 <-pol( x ) * ind;
# approximate int(eta^(p+1))^2
int_eta <- sum( tmp_eta_p1^2 ) * m[1];
# EQUIVALENT KERNEL
tmp <- NULL;
for( s in 0:(2 * p) ) {
tmp[s+1] <- moments( ker, s );
}
# create S matrix (all moments)
S <- matrix( 0, p + 1, p + 1 );
for(pp in 1:(p+1) ) {
S[pp,] <- tmp[pp+(0:p)];
}
# indicator vector
e_0 <- diag( 1, 1, p + 1 );
# FUNCTIONS OF THE EQUIVALENT KERNEL
# integral of squared equivalent kernel
k2 <- integrate( ker_eqv2, -Inf, Inf, deg = p, K = ker )$value;
# (p+1)th moment of equivalent kernel
muK_2 <- moments( "ker_eqv", p + 1 )^2;
if( muK_2 == 0 ) {
muK_2 <- .Machine$double.eps;
}
# INTEGRAL OF VARIANCE FUNCTION
int_sg <- integrate( varfun, min( x ) - 1, max( x ) + 1 )$value;
# ensure the above integral is positive which might not be the case for
# near degenerate samples; issue warning
if( int_sg == 0 ) {
int_sg <- .Machine$double.eps;
warning("The estimated value was 0; sample is probably degenerate");
}
# KERNEL DEPENDANT CONSTANT
C <- ( ( factorial( p + 1 )^2 * k2 ) / ( 2 * ( p + 1 ) * muK_2 ) )^
( 1 / ( 2 * p + 3 ) );
# BANDWIDTH
return( C * ( int_sg / int_eta )^( 1 / ( 2 * p + 3 ) ) );
}
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