Title:	Random Forests for Longitudinal Data
Version:	0.9
Description:	Random forests are a statistical learning method widely used in many areas of scientific research essentially for its ability to learn complex relationships between input and output variables and also its capacity to handle high-dimensional data. However, current random forests approaches are not flexible enough to handle longitudinal data. In this package, we propose a general approach of random forests for high-dimensional longitudinal data. It includes a flexible stochastic model which allows the covariance structure to vary over time. Furthermore, we introduce a new method which takes intra-individual covariance into consideration to build random forests. The method is fully detailled in Capitaine et.al. (2020) <doi:10.1177/0962280220946080> Random forests for high-dimensional longitudinal data.
License:	GPL-2
Imports:	stats, randomForest, rpart, mvtnorm, latex2exp
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.1.1
Suggests:	testthat
NeedsCompilation:	no
Packaged:	2020-08-25 22:24:05 UTC; capitainelouis
Author:	Louis Capitaine [aut, cre]
Maintainer:	Louis Capitaine <Louis.capitaine@u-bordeaux.fr>
Repository:	CRAN
Date/Publication:	2020-08-31 09:10:07 UTC

Longitudinal data generator

Description

Simulate longitudinal data according to the semi-parametric stochastic mixed-effects model given by:

Y_i(t)=f(X_i(t))+Z_i(t)\beta_i + \omega_i(t)+\epsilon_i

with Y_i(t) the output at time t for the ith individual; X_i(t) the input predictors (fixed effects) at time t for the ith individual; Z_i(t) are the random effects at time t for the ith individual; \omega_i(t) is a Brownian motion with volatility \gamma^2=0.8 at time t for the ith individual; \epsilon_i is the residual error with variance \sigma^2=0.5. The data are simulated according to the simulations in low dimensional in the low dimensional scheme of the paper <doi:10.1177/0962280220946080>

Usage

DataLongGenerator(n = 50, p = 6, G = 6)

Arguments

Value

a list of the following elements:

• Y: vector of the output trajectories.

• X : matrix of the fixed-effects predictors.

• Z: matrix of the random-effects predictors.

• id: vector of the identifiers for each individual.

• time: vector the the time measurements for each individual.

Examples

oldpar <- par()
oldopt <- options()
data <- DataLongGenerator(n=17, p=6,G=6) # Generate the data
# Let's see the output :
w <- which(data$id==1)
plot(data$time[w],data$Y[w],type="l",ylim=c(min(data$Y),max(data$Y)), col="grey")
for (i in unique(data$id)){
  w <- which(data$id==i)
  lines(data$time[w],data$Y[w], col='grey')
}
# Let's see the fixed effects predictors:
par(mfrow=c(2,3), mar=c(2,3,3,2))
for (i in 1:ncol(data$X)){
  w <- which(data$id==1)
  plot(data$time[w],data$X[w,i], col="grey",ylim=c(min(data$X[,i]),
  max(data$X[,i])),xlim=c(1,max(data$time)),main=latex2exp::TeX(paste0("$X^{(",i,")}$")))
  for (k in unique(data$id)){
    w <- which(data$id==k)
    lines(data$time[w],data$X[w,i], col="grey")
  }
}
par(oldpar)
options(oldopt)

(S)MERF algorithm

Description

(S)MERF is an adaptation of the random forest regression method to longitudinal data introduced by Hajjem et. al. (2014) <doi:10.1080/00949655.2012.741599>. The model has been improved by Capitaine et. al. (2020) <doi:10.1177/0962280220946080> with the addition of a stochastic process. The algorithm will estimate the parameters of the following semi-parametric stochastic mixed-effects model:

Y_i(t)=f(X_i(t))+Z_i(t)\beta_i + \omega_i(t)+\epsilon_i

with Y_i(t) the output at time t for the ith individual; X_i(t) the input predictors (fixed effects) at time t for the ith individual; Z_i(t) are the random effects at time t for the ith individual; \omega_i(t) is the stochastic process at time t for the ith individual which model the serial correlations of the output measurements; \epsilon_i is the residual error.

Usage

MERF(
  X,
  Y,
  id,
  Z,
  iter = 100,
  mtry = ceiling(ncol(X)/3),
  ntree = 500,
  time,
  sto,
  delta = 0.001
)

Arguments

Value

A fitted (S)MERF model which is a list of the following elements:

• forest: Random forest obtained at the last iteration.

• random_effects : Predictions of random effects for different trajectories.

• id_btilde: Identifiers of individuals associated with the predictions random_effects.

• var_random_effects: Estimation of the variance covariance matrix of random effects.

• sigma_sto: Estimation of the volatility parameter of the stochastic process.

• sigma: Estimation of the residual variance parameter.

• time: The vector of the measurement times associated with the trajectories in Y,Z and X.

• sto: Stochastic process used in the model.

• Vraisemblance: Log-likelihood of the different iterations.

• id: Vector of the identifiers for the different trajectories.

• OOB: OOB error of the fitted random forest at each iteration.

Examples

set.seed(123)
data <- DataLongGenerator(n=20) # Generate the data composed by n=20 individuals.
# Train a SMERF model on the generated data. Should take ~ 50 seconds
# The data are generated with a Brownian motion,
# so we use the parameter sto="BM" to specify a Brownian motion as stochastic process
smerf <- MERF(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,mtry=2,ntree=500,sto="BM")
smerf$forest # is the fitted random forest (obtained at the last iteration).
smerf$random_effects # are the predicted random effects for each individual.
smerf$omega # are the predicted stochastic processes.
plot(smerf$Vraisemblance) # evolution of the log-likelihood.
smerf$OOB # OOB error at each iteration.

(S)MERT algorithm

Description

(S)MERT is an adaptation of the random forest regression method to longitudinal data introduced by Hajjem et. al. (2011) <doi:10.1016/j.spl.2010.12.003>. The model has been improved by Capitaine et. al. (2020) <doi:10.1177/0962280220946080> with the addition of a stochastic process. The algorithm will estimate the parameters of the following semi-parametric stochastic mixed-effects model:

Y_i(t)=f(X_i(t))+Z_i(t)\beta_i + \omega_i(t)+\epsilon_i

with Y_i(t) the output at time t for the ith individual; X_i(t) the input predictors (fixed effects) at time t for the ith individual; Z_i(t) are the random effects at time t for the ith individual; \omega_i(t) is the stochastic process at time t for the ith individual which model the serial correlations of the output measurements; \epsilon_i is the residual error.

Usage

MERT(X, Y, id, Z, iter = 100, time, sto, delta = 0.001)

Arguments

Value

A fitted (S)MERF model which is a list of the following elements:

• forest: Tree obtained at the last iteration.

• random_effects : Predictions of random effects for different trajectories.

• id_btilde: Identifiers of individuals associated with the predictions random_effects.

• var_random_effects: Estimation of the variance covariance matrix of random effects.

• sigma_sto: Estimation of the volatility parameter of the stochastic process.

• sigma: Estimation of the residual variance parameter.

• time: The vector of the measurement times associated with the trajectories in Y,Z and X.

• sto: Stochastic process used in the model.

• Vraisemblance: Log-likelihood of the different iterations.

• id: Vector of the identifiers for the different trajectories.

Examples

set.seed(123)
data <- DataLongGenerator(n=20) # Generate the data composed by n=20 individuals.
# Train a SMERF model on the generated data. Should take ~ 50 secondes
# The data are generated with a Brownian motion,
# so we use the parameter sto="BM" to specify a Brownian motion as stochastic process
smert <- MERF(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,sto="BM")
smert$forest # is the fitted random forest (obtained at the last iteration).
smert$random_effects # are the predicted random effects for each individual.
smert$omega # are the predicted stochastic processes.
plot(smert$Vraisemblance) #evolution of the log-likelihood.

Title

Description

Title

Usage

Moy(id, Btilde, sigmahat, Phi, Y, Z)

Title

Description

Title

Usage

Moy_exp(id, Btilde, sigmahat, Phi, Y, Z, alpha, time, sigma2)

Title

Description

Title

Usage

Moy_fbm(id, Btilde, sigmahat, Phi, Y, Z, H, time, sigma2)

Title

Description

Title

Usage

Moy_sto(id, Btilde, sigmahat, Phi, Y, Z, sto, time, sigma2)

(S)REEMforest algorithm

Description

(S)REEMforest algorithm

Usage

REEMforest(
  X,
  Y,
  id,
  Z,
  iter = 100,
  mtry,
  ntree = 500,
  time,
  sto,
  delta = 0.001
)

Arguments

Details

(S)REEMforest is an adaptation of the random forest regression method to longitudinal data introduced by Capitaine et. al. (2020) <doi:10.1177/0962280220946080>. The algorithm will estimate the parameters of the following semi-parametric stochastic mixed-effects model:

Y_i(t)=f(X_i(t))+Z_i(t)\beta_i + \omega_i(t)+\epsilon_i

with Y_i(t) the output at time t for the ith individual; X_i(t) the input predictors (fixed effects) at time t for the ith individual; Z_i(t) are the random effects at time t for the ith individual; \omega_i(t) is the stochastic process at time t for the ith individual which model the serial correlations of the output measurements; \epsilon_i is the residual error.

Value

A fitted (S)REEMforest model which is a list of the following elements:

• forest: Random forest obtained at the last iteration.

• random_effects : Predictions of random effects for different trajectories.

• id_btilde: Identifiers of individuals associated with the predictions random_effects.

• var_random_effects: Estimation of the variance covariance matrix of random effects.

• sigma_sto: Estimation of the volatility parameter of the stochastic process.

• sigma: Estimation of the residual variance parameter.

• time: The vector of the measurement times associated with the trajectories in Y,Z and X.

• sto: Stochastic process used in the model.

• Vraisemblance: Log-likelihood of the different iterations.

• id: Vector of the identifiers for the different trajectories.

• OOB: OOB error of the fitted random forest at each iteration.

Examples

set.seed(123)
data <- DataLongGenerator(n=20) # Generate the data composed by n=20 individuals.
# Train a SREEMforest model on the generated data. Should take ~ 50 secondes
# The data are generated with a Brownian motion
#  so we use the parameter sto="BM" to specify a Brownian motion as stochastic process
SREEMF <- REEMforest(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,mtry=2,ntree=500,sto="BM")
SREEMF$forest # is the fitted random forest (obtained at the last iteration).
SREEMF$random_effects # are the predicted random effects for each individual.
SREEMF$omega # are the predicted stochastic processes.
plot(SREEMF$Vraisemblance) #evolution of the log-likelihood.
SREEMF$OOB # OOB error at each iteration.

(S)REEMtree algorithm

Description

(S)REEMtree is an adaptation of the random forest regression method to longitudinal data introduced by Sela and Simonoff. (2012) <doi:10.1007/s10994-011-5258-3>. The algorithm will estimate the parameters of the following semi-parametric stochastic mixed-effects model:

Y_i(t)=f(X_i(t))+Z_i(t)\beta_i + \omega_i(t)+\epsilon_i

with Y_i(t) the output at time t for the ith individual; X_i(t) the input predictors (fixed effects) at time t for the ith individual; Z_i(t) are the random effects at time t for the ith individual; \omega_i(t) is the stochastic process at time t for the ith individual which model the serial correlations of the output measurements; \epsilon_i is the residual error.

Usage

REEMtree(X, Y, id, Z, iter = 10, time, sto, delta = 0.001)

Arguments

Value

A fitted (S)MERF model which is a list of the following elements:

• forest: Tree obtained at the last iteration.

• random_effects : Predictions of random effects for different trajectories.

• id_btilde: Identifiers of individuals associated with the predictions random_effects.

• var_random_effects: Estimation of the variance covariance matrix of random effects.

• sigma_sto: Estimation of the volatility parameter of the stochastic process.

• sigma: Estimation of the residual variance parameter.

• time: The vector of the measurement times associated with the trajectories in Y,Z and X.

• sto: Stochastic process used in the model.

• Vraisemblance: Log-likelihood of the different iterations.

• id: Vector of the identifiers for the different trajectories.

Examples

set.seed(123)
data <- DataLongGenerator(n=20) # Generate the data composed by n=20 individuals.
# Train a SREEMtree model on the generated data.
# The data are generated with a Brownian motion,
# so we use the parameter sto="BM" to specify a Brownian motion as stochastic process
X.fixed.effects <- as.data.frame(data$X)
sreemt <- REEMtree(X=X.fixed.effects,Y=data$Y,Z=data$Z,id=data$id,time=data$time,
sto="BM", delta=0.0001)
sreemt$forest # is the fitted random forest (obtained at the last iteration).
sreemt$random_effects # are the predicted random effects for each individual.
sreemt$omega # are the predicted stochastic processes.
plot(sreemt$Vraisemblance) #evolution of the log-likelihood.

Title

Description

Title

Usage

bay(bhat, Bhat, Z, id, sigmahat)

Title

Description

Title

Usage

bay.exp(bhat, Bhat, Z, id, sigmahat, time, sigma2, alpha)

Arguments

Title

Description

Title

Usage

bay.fbm(bhat, Bhat, Z, id, sigmahat, time, sigma2, h)

Arguments

Title

Description

Title

Usage

bay_sto(bhat, Bhat, Z, id, sigmahat, time, sigma2, sto)

Title

Description

Title

Usage

cov.exp(time, alpha)

Arguments

Title

Description

Title

Usage

cov.fbm(time, H)

Arguments

Title

Description

Title

Usage

gam_exp(Y, sigm, id, Z, Btilde, time, sigma2, omega, id_omega, alpha)

Arguments

Title

Description

Title

Usage

gam_fbm(Y, sigm, id, Z, Btilde, time, sigma2, omega, id_omega, h)

Arguments

Title

Description

Title

Usage

gam_sto(sigma, id, Z, Btilde, time, sigma2, sto, omega)

Title

Description

Title

Usage

logV(Y, f, Z, time, id, B, gamma, sigma, sto)

Title

Description

Title

Usage

logV.exp(Y, fhat, Z, time, id, B, sigma2, sigmahat, H)

Arguments

Title

Description

Title

Usage

logV.fbm(Y, fhat, Z, time, id, B, sigma2, sigmahat, H)

Arguments

Title

Description

Title

Usage

opti.FBM(X, Y, id, Z, iter, mtry, ntree, time)

Arguments

Title

Description

Title

Usage

opti.FBMreem(X, Y, id, Z, iter, mtry, ntree, time)

Arguments

Title

Description

Title

Usage

opti.exp(X, Y, id, Z, iter, mtry, ntree, time)

Arguments

blabla

Description

blabla

Usage

## S3 method for class 'exp'
predict(omega, time.app, time.test, alpha)

Title

Description

Title

Usage

## S3 method for class 'fbm'
predict(omega, time.app, time.test, h)

Predict with longitudinal trees and random forests.

Description

Predict with longitudinal trees and random forests.

Usage

## S3 method for class 'longituRF'
predict(object, X, Z, id, time, ...)

Arguments

Value

vector of the predicted output for the new observations.

Examples

set.seed(123)
data <- DataLongGenerator(n=20) # Generate the data composed by n=20 individuals.
REEMF <- REEMforest(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,mtry=2,ntree=500,sto="BM")
# Then we predict on the learning sample :
pred.REEMF <- predict(REEMF, X=data$X,Z=data$Z,id=data$id, time=data$time)
# Let's have a look at the predictions
# the predictions are in red while the real output trajectories are in blue:
par(mfrow=c(4,5),mar=c(2,2,2,2))
for (i in unique(data$id)){
  w <- which(data$id==i)
  plot(data$time[w],data$Y[w],type="l",col="blue")
  lines(data$time[w],pred.REEMF[w], col="red")
}
# Train error :
mean((pred.REEMF-data$Y)^2)

# The same function can be used with a fitted SMERF model:
smerf <-MERF(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,mtry=2,ntree=500,sto="BM")
pred.smerf <- predict(smerf, X=data$X,Z=data$Z,id=data$id, time=data$time)
# Train error :
mean((pred.smerf-data$Y)^2)
# This function can be used even on a MERF model (when no stochastic process is specified)
merf <-MERF(X=data$X,Y=data$Y,Z=data$Z,id=data$id,time=data$time,mtry=2,ntree=500,sto="none")
pred.merf <- predict(merf, X=data$X,Z=data$Z,id=data$id, time=data$time)
# Train error :
mean((pred.merf-data$Y)^2)

Title

Description

Title

Usage

## S3 method for class 'sto'
predict(omega, time.app, time.test, sto)

Title

Description

Title

Usage

sig(sigma, id, Z, epsilon, Btilde)

Title

Description

Title

Usage

sig.exp(Y, sigma, id, Z, epsilon, Btilde, time, sigma2, alpha)

Arguments

Title

Description

Title

Usage

sig.fbm(Y, sigma, id, Z, epsilon, Btilde, time, sigma2, h)

Arguments

Title

Description

Title

Usage

sig_sto(sigma, id, Z, epsilon, Btilde, time, sigma2, sto)

Title

Description

Title

Usage

sto_analysis(sto, time)