File: morph.Rd

package info (click to toggle)
r-cran-mcmc 0.9-7-1
 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128 \name{morph} \encoding{UTF-8} \alias{morph} \alias{morph.identity} \title{Variable Transformation} \description{ Utility functions for variable transformation. } \usage{ morph(b, r, p, center) morph.identity() } \arguments{ \item{b}{Positive real number. May be missing.} \item{r}{Non-negative real number. May be missing. If \code{p} is specified, \code{r} defaults to 0.} \item{p}{Real number strictly greater than 2. May be missing. If \code{r} is specified, \code{p} defaults to 3.} \item{center}{Real scalar or vector. May be missing. If \code{center} is a vector it should be the same length of the state of the Markov chain, \code{center} defaults to 0} } \section{Warning}{ The equations for the returned \code{transform} function (see below) do not have a general analytical solution when \code{p} is not equal to 3. This implementation uses numerical approximation to calculate \code{transform} when \code{p} is not equal to 3. If computation speed is a factor, it is advisable to use \code{p=3}. This is not a factor when using \code{\link{morph.metrop}}, as \code{transform} is only called once during setup, and not at all while running the Markov chain. } \details{ The \code{morph} function facilitates using variable transformations by providing functions to (using \eqn{X} for the original random variable with the pdf \eqn{f_X}{f.X}, and \eqn{Y} for the transformed random variable with the pdf \eqn{f_Y}{f.Y}): \itemize{ \item Calculate the log unnormalized probability density for \eqn{Y} induced by the transformation. \item Transform an arbitrary function of \eqn{X} to a function of \eqn{Y}. \item Transform values of \eqn{X} to values of \eqn{Y}. \item Transform values of \eqn{Y} to values of \eqn{X} (the inverse transformation). } for a select few transformations. \code{morph.identity} implements the identity transformation, \eqn{Y=X}. The parameters \code{r}, \code{p}, \code{b} and \code{center} specify the transformation function. In all cases, \code{center} gives the center of the transformation, which is the value \eqn{c} in the equation \deqn{Y = f(X - c).} If no parameters are specified, the identity transformation, \eqn{Y=X}, is used. The parameters \code{r}, \code{p} and \code{b} specify a function \eqn{g}, which is a monotonically increasing bijection from the non-negative reals to the non-negative reals. Then \deqn{f(X) = g\bigl(|X|\bigr) \frac{X}{|X|}}{f(X) = g(|X|) * X / |X|} where \eqn{|X|} represents the Euclidean norm of the vector \eqn{X}. The inverse function is given by \deqn{f^{-1}(Y) = g^{-1}\bigl(|Y|\bigr) \frac{Y}{|Y|}.}{f^{-1}(Y) = g^{-1}(|Y|) * Y / |Y|.} The parameters \code{r} and \code{p} are used to define the function \deqn{g_1(x) = x + (x-r)^p I(x > r)}{g1(x) = x + (x-r)^p * I(x > r)} where \eqn{I( \cdot )}{I(•)} is the indicator function. We require that \code{r} is non-negative and \code{p} is strictly greater than 2. The parameter \code{b} is used to define the function \deqn{g_2(x) = \bigl(e^{bx} - e / 3\bigr) I(x > \frac{1}{b}) + \bigl(x^3 b^3 e / 6 + x b e / 2\bigr) I(x \leq \frac{1}{b})}{ g2(x) = (exp(b * x) - exp(1) / 3) * I(x > 1 / b) + (x^3 * b^3 exp(1) / 6 + x * b * exp(1) / 2) * I(x <= 1 / b).} We require that \eqn{b} is positive. The parameters \code{r}, \code{p} and \code{b} specify \eqn{f^{-1}} in the following manner: \itemize{ \item If one or both of \code{r} and \code{p} is specified, and \code{b} is not specified, then \deqn{f^{-1}(X) = g_1(|X|) \frac{X}{|X|}.}{f^{-1}(X) = g1(|X|) * X / |X|.} If only \code{r} is specified, \code{p = 3} is used. If only \code{p} is specified, \code{r = 0} is used. \item If only \code{b} is specified, then \deqn{f^{-1}(X) = g_2(|X|) \frac{X}{|X|}.}{f^{-1}(X) = g2(|X|) * X / |X|.} \item If one or both of \code{r} and \code{p} is specified, and \code{b} is also specified, then \deqn{f^{-1}(X) = g_2(g_1(|X|)) \frac{X}{|X|}.}{f^{-1}(X) = g2(g1(|X|)) * X / |X|.} } } \value{ a list containing the functions \itemize{ \item \code{outfun(f)}, a function that operates on functions. \code{outfun(f)} returns the function \code{function(state, ...) f(inverse(state), ...)}. \item \code{inverse}, the inverse transformation function. \item \code{transform}, the transformation function. \item \code{lud}, a function that operates on functions. As input, \code{lud} takes a function that calculates a log unnormalized probability density, and returns a function that calculates the log unnormalized density by transforming a random variable using the \code{transform} function. \code{lud(f) = function(state, ...) f(inverse(state), ...) + log.jacobian(state)}, where \code{log.jacobian} represents the function that calculate the log Jacobian of the transformation. \code{log.jacobian} is not returned. } } \examples{ # use an exponential transformation, centered at 100. b1 <- morph(b=1, center=100) # original log unnormalized density is from a t distribution with 3 # degrees of freedom, centered at 100. lud.transformed <- b1\$lud(function(x) dt(x - 100, df=3, log=TRUE)) d.transformed <- Vectorize(function(x) exp(lud.transformed(x))) \dontrun{ curve(d.transformed, from=-3, to=3, ylab="Induced Density") } } \seealso{ \code{\link{morph.metrop}} } \keyword{misc}