tram (version 0.6-2)

mmlt: Multivariate Conditional Transformation Models

Description

A proof-of-concept implementation of multivariate conditional transformation models

Usage

mmlt(..., formula = ~ 1, data, theta = NULL, 
     control.outer = list(trace = FALSE), scale = FALSE, dofit = TRUE)

Arguments

…

marginal transformation models, one for each response

formula

a model formula describing a model for the dependency structure via the lambda parameters. The default is set to ~ 1 for constant lambdas.

data

a data.frame

theta

an optional vector of starting values

control.outer

a list controlling auglag

scale

logical; parameters are not scaled prior to optimisation by default

dofit

logical; parameters are fitted by default, otherwise a list with log-likelihood and score function is returned

Value

An object of class mmlt with coef and predict methods.

Details

The function implements multivariate conditional transformation models as described by Klein et al (2020). The response is assumed absolutely continuous at the moment, discrete versions will be added later.

Below is a simple example for an unconditional bivariate distribution. See demo("undernutrition", package = "tram") for a conditional three-variate example.

References

Nadja Klein, Torsten Hothorn, Luisa Barbanti, Thomas Kneib (2020), Multivariate Conditional Transformation Models. Scandinavian Journal of Statistics, 10.1111/sjos.12501.

Examples

# NOT RUN {
  data("cars")

  ### fit unconditional bivariate distribution of speed and distance to stop
  ## fit unconditional marginal transformation models
  m_speed <- BoxCox(speed ~ 1, data = cars, support = ss <- c(4, 25), 
                    add = c(-5, 5))
  m_dist <- BoxCox(dist ~ 1, data = cars, support = sd <- c(0, 120), 
                   add = c(-5, 5))

  ## fit multivariate unconditional transformation model
  m_speed_dist <- mmlt(m_speed, m_dist, formula = ~ 1, data = cars)

  ## lambda defining the Cholesky of the precision matrix,
  ## with standard error
  coef(m_speed_dist, newdata = cars[1,], type = "Lambda")
  sqrt(vcov(m_speed_dist)["dist.sped.(Intercept)", 
                          "dist.sped.(Intercept)"])

  ## linear correlation, ie Pearson correlation of speed and dist after
  ## transformation to bivariate normality
  (r <- coef(m_speed_dist, newdata = cars[1,], type = "Corr"))
  
  ## Spearman's rho (rank correlation), can be computed easily 
  ## for Gaussian copula as
  (rs <- 6 * asin(r / 2) / pi)

  ## evaluate joint and marginal densities (needs to be more user-friendly)
  nd <- expand.grid(c(nd_s <- mkgrid(m_speed, 100), nd_d <- mkgrid(m_dist, 100)))
  nd$hs <- predict(m_speed_dist, newdata = nd, marginal = 1L)
  nd$hps <- predict(m_speed_dist, newdata = nd, marginal = 1L, 
                    deriv = c("speed" = 1))
  nd$hd <- predict(m_speed_dist, newdata = nd, marginal = 2L)
  nd$hpd <- predict(m_speed_dist, newdata = nd, marginal = 2L, 
                    deriv = c("dist" = 1))

  ## joint density
  nd$d <- with(nd, 
               dnorm(hs) * 
               dnorm(coef(m_speed_dist)["dist.sped.(Intercept)"] * hs + hd) * 
               hps * hpd)

  ## compute marginal densities
  nd_s <- as.data.frame(nd_s)
  nd_s$d <- predict(m_speed_dist, newdata = nd_s, type = "density")
  nd_d <- as.data.frame(nd_d)
  nd_d$d <- predict(m_speed_dist, newdata = nd_d, marginal = 2L, 
                    type = "density")

  ## plot bivariate and marginal distribution
  col1 <- rgb(.1, .1, .1, .9)
  col2 <- rgb(.1, .1, .1, .5)
  w <- c(.8, .2)
  layout(matrix(c(2, 1, 4, 3), nrow = 2), width = w, height = rev(w))
  par(mai = c(1, 1, 0, 0) * par("mai"))
  sp <- unique(nd$speed)
  di <- unique(nd$dist)
  d <- matrix(nd$d, nrow = length(sp))
  contour(sp, di, d, xlab = "Speed (in mph)", ylab = "Distance (in ft)", xlim = ss, ylim = sd,
          col = col1)
  points(cars$speed, cars$dist, pch = 19, col = col2)
  mai <- par("mai")
  par(mai = c(0, 1, 0, 1) * mai)
  plot(d ~ speed, data = nd_s, xlim = ss, type = "n", axes = FALSE, 
       xlab = "", ylab = "")
  polygon(nd_s$speed, nd_s$d, col = col2, border = FALSE)
  par(mai = c(1, 0, 1, 0) * mai)
  plot(dist ~ d, data = nd_d, ylim = sd, type = "n", axes = FALSE, 
       xlab = "", ylab = "")
  polygon(nd_d$d, nd_d$dist, col = col2, border = FALSE)

  ### NOTE: marginal densities are NOT normal, nor is the joint
  ### distribution. The non-normal shape comes from the data-driven 
  ### transformation of both variables to joint normality in this model.

# }