mmlt: Multivariate Conditional Transformation Models In tram: Transformation Models

Description

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

Usage

 1 2 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

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.

Value

An object of class mmlt with coef and predict methods.

References

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

Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 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.

tram documentation built on Jan. 14, 2022, 5:07 p.m.