boostmtree (version 1.5.1)

boostmtree: Boosted multivariate trees for longitudinal data

Description

Multivariate extension of Friedman's gradient descent boosting method for modeling continuous or binary longitudinal response using multivariate tree base learners (Pande et al., 2017). Covariate-time interactions are modeled using penalized B-splines (P-splines) with estimated adaptive smoothing parameter.

Usage

boostmtree(x,
            tm,
            id,
            y,
            family = c("Continuous","Binary","Nominal","Ordinal"),
            y_reference = NULL,
            M = 200,
            nu = 0.05,
            na.action = c("na.omit","na.impute")[2],
            K = 5,
            mtry = NULL,
            nknots = 10,
            d = 3,
            pen.ord = 3,
            lambda,
            rho,
            lambda.max = 1e6,
            lambda.iter = 2,
            svd.tol = 1e-6,
            forest.tol = 1e-3,
            verbose = TRUE,
            cv.flag = FALSE,
            eps = 1e-5,
            mod.grad = TRUE,
            NR.iter = 3,
            ...)

Arguments

x

Data frame (or matrix) containing the x-values. Rows must be duplicated to match the number of time points for an individual. That is, if individual i has n[i] outcome y-values, then there must be n[i] duplicate rows of i's x-value.

tm

Vector of time values, one entry for each row in x.

id

Unique subject identifier, one entry for each row in x.

y

Observed y-value, one entry for each row in x.

family

Family of the response variable y. Use any one from {"Continuous", "Binary","Nominal","Ordinal"} based on the scale of y.

y_reference

Set this value, among the unique y values when family == "Nominal". If NULL, lowest value, among unique y values, is used.

M

Number of boosting iterations

nu

Boosting regularization parameter. A value in (0,1].

na.action

Remove missing values (casewise) or impute it. Default is to impute the missign values.

K

Number of terminal nodes used for the multivariate tree learner.

mtry

Number of x variables selected randomly for tree fitting. Default is use all x variables.

nknots

Number of knots used for the B-spline for modeling the time interaction effect.

d

Degree of the piecewise B-spline polynomial (no time effect is fit when d < 1).

pen.ord

Differencing order used to define the penalty with increasing values implying greater smoothness.

lambda

Smoothing (penalty) parameter used for B-splines with increasing values associated with increasing smoothness/penalization. If missing, or non-positive, the value is estimated adaptively using a mixed models approach.

rho

If missing, rho is estimated, else, use the rho value specified in this argument.

lambda.max

Tolerance used for adaptively estimated lambda (caps it). For experts only.

lambda.iter

Number of iterations used to estimate lambda (only applies when lambda is not supplied and adaptive smoothing is employed).

svd.tol

Tolerance value used in the SVD calculation of the penalty matrix. For experts only.

forest.tol

Tolerance used for forest weighted least squares solution. Experimental and for experts only.

verbose

Should verbose output be printed?

cv.flag

Should in-sample cross-validation (CV) be used to determine optimal stopping using out of bag data?

eps

Tolerance value used for determining the optimal M. Applies only if cv.flag = TRUE. For experts only.

mod.grad

Use a modified gradient? See details below.

NR.iter

Number of Newton-Raphson iteration. Applied for family = {Binary","Nominal","Ordinal"}.

...

Further arguments passed to or from other methods.

Value

An object of class (boostmtree, grow) with the following components:

x

The x-values, but with only one row per individual (i.e. duplicated rows are removed). Values sorted on id.

xvar.names

X-variable names.

time

List with each component containing the time points for a given individual. Values sorted on id.

id

Sorted subject identifier.

y

List with each component containing the observed y-values for a given individual. Values sorted on id.

Yorg

For family == "Nominal" or family == "Ordinal", this provides the response in list-format where each element coverted the response into the binary response.

family

Family of y.

ymean

Overall mean of y-values for all individuals. If family = "Binary", ymean = 0.

ysd

Overall standard deviation of y-values for all individuals. If family = "Binary", ysd = 1.

na.action

Remove missing values or impute?

n

Total number of subjects.

ni

Number of repeated measures for each subject.

n.Q

Number of class labels for non-continuous response.

Q_set

Class labels for the non-continuous response.

y.unq

Unique y values for the non-continous response.

y_reference

Reference value for family == "Nominal".

tm.unq

Unique time points.

gamma

List of length M, with each component containing the boosted tree fitted values.

mu

List with each component containing the estimated mean values for an individual. That is, each component contains the estimated time-profile for an individual. When in-sample cross-validation is requested using cv.flag=TRUE, the estimated mean is cross-validated and evaluated at the optimal number of iterations Mopt. If the family == "Nominal" or family == "Ordinal", mu will have a higher level of list to accommodate binary responses generated from nominal or ordinal response.

Prob_class

For family == "Ordinal", this provides individual probabilty rather than cumulative probabilty.

lambda

Smoothing parameter. Results provided in vector or matrix form, depending on whether family == c("Continuous","Binary") or family == c("Nominal", "Ordinal").

phi

Variance parameter.Results provided in vector or matrix form, depending on whether family == c("Continuous","Binary") or family == c("Nominal", "Ordinal").

rho

Correlation parameter.Results provided in vector or matrix form, depending on whether family == c("Continuous","Binary") or family == c("Nominal", "Ordinal").

baselearner

List of length M containing the base learners.

membership

List of length M, with each component containing the terminal node membership for a given boosting iteration.

X.tm

Design matrix for all the unique time points.

D

Design matrix for each subject.

d

Degree of the piecewise B-spline polynomial.

pen.ord

Penalization difference order.

K

Number of terminal nodes.

M

Number of boosting iterations.

nu

Boosting regularization parameter.

ntree

Number of trees.

cv.flag

Whether in-sample CV is used or not?

err.rate

In-sample standardized estimate of l1-error and RMSE.

rmse

In-sample standardized RMSE at optimized M.

Mopt

The optimized M.

gamma.i.list

Estimate of gamma obtained from in-sample CV if cv.flag = TRUE, else NULL

forest.tol

Forest tolerance value (needed for prediction).

Details

Each individual has observed y-values, over possibly different time points, with possibly differing number of time points. Given y, the time points, and x, the conditional mean time profile of y is estimated using gradient boosting in which the gradient is derived from a criterion function involving a working variance matrix for y specified as an equicorrelation matrix with parameter rho multiplied by a variance parameter phi. Multivariate trees are used for base learners and weighted least squares is used for solving the terminal node optimization problem. This provides solutions to the core parameters of the algorithm. For ancillary parameters, a mixed-model formulation is used to estimate the smoothing parameter associated with the B-splines used for the time-interaction effect, although the user can manually set the smoothing parameter as well. Ancillary parameters rho and phi are estimated using GLS (generalized least squares).

In the original boostmtree algorithm (Pande et al., 2017), the equicorrelation parameter rho is used in two places in the algorithm: (1) for growing trees using the gradient, which depends upon rho; and (2) for solving the terminal node optimization problem which also uses the gradient. However, Pande (2017) observed that setting rho to zero in the gradient used for growing trees improved performance of the algorithm, especially in high dimensions. For this reason the default setting used in this algorithm is to set rho to zero in the gradient for (1). The rho in the gradient for (2) is not touched. The option mod.grad specifies whether a modified gradient is used in the tree growing process and is TRUE by default.

By default, trees are grown from a bootstrap sample of the data -- thus the boosting method employed here is a modified example of stochastic gradient descent boosting (Friedman, 2002). Stochastic descent often improves performance and has the added advantage that out-of-sample data (out-of-bag, OOB) can be used to calculate variable importance (VIMP).

The package implements R-side parallel processing by replacing the R function lapply with mclapply found in the parallel package. You can set the number of cores accessed by mclapply by issuing the command options(mc.cores = x), where x is the number of cores. The options command can also be placed in the users .Rprofile file for convenience. You can, alternatively, initialize the environment variable MC_CORES in your shell environment.

As an example, issuing the following options command uses all available cores for R-side parallel processing:

options(mc.cores=detectCores())

However, be cautious when setting mc.cores. This can create not only high CPU usage but also high RAM usage, especially when using functions partialPlot and predict.

The method can impute the missing observations in x (covariates) using on the fly imputation. Details regarding can be found in the randomForestSRC package. If missing values are present in the tm, id or y, the user should either impute or delete these values before executing the function.

Finally note cv.flag can be used for an in-sample cross-validated estimate of prediction error. This is used to determine the optimized number of boosting iterations Mopt. The final mu predictor is evaluated at this value and is cross-validated. The prediction error returned via err.rate is standardized by the overall standard deviation of y.

References

Friedman J.H. (2001). Greedy function approximation: a gradient boosting machine, Ann. of Statist., 5:1189-1232.

Friedman J.H. (2002). Stochastic gradient boosting. Comp. Statist. Data Anal., 38(4):367--378.

Pande A., Li L., Rajeswaran J., Ehrlinger J., Kogalur U.B., Blackstone E.H., Ishwaran H. (2017). Boosted multivariate trees for longitudinal data, Machine Learning, 106(2): 277--305.

Pande A. (2017). Boosting for longitudinal data. Ph.D. Dissertation, Miller School of Medicine, University of Miami.

See Also

marginalPlot partialPlot, plot.boostmtree, predict.boostmtree, print.boostmtree, simLong, vimpPlot

Examples

# NOT RUN {
##------------------------------------------------------------
## synthetic example (Response y is continuous)
## 0.8 correlation, quadratic time with quadratic interaction
##-------------------------------------------------------------
#simulate the data (use a small sample size for illustration)
dta <- simLong(n = 50, N = 5, rho =.80, model = 2,family = "Continuous")$dtaL

#basic boosting call (M set to a small value for illustration)
boost.grow <- boostmtree(dta$features, dta$time, dta$id, dta$y,family = "Continuous",M = 20)

#print results
print(boost.grow)

#plot.results
plot(boost.grow)

##------------------------------------------------------------
## synthetic example (Response y is binary)
## 0.8 correlation, quadratic time with quadratic interaction
##-------------------------------------------------------------
#simulate the data (use a small sample size for illustration)
dta <- simLong(n = 50, N = 5, rho =.80, model = 2, family = "Binary")$dtaL

#basic boosting call (M set to a small value for illustration)
boost.grow <- boostmtree(dta$features, dta$time, dta$id, dta$y,family = "Binary", M = 20)

#print results
print(boost.grow)

#plot.results
plot(boost.grow)

# }
# NOT RUN {
##------------------------------------------------------------
## Same synthetic example as above with continuous response
## but with in-sample cross-validation estimate for RMSE
##-------------------------------------------------------------
dta <- simLong(n = 50, N = 5, rho =.80, model = 2,family = "Continuous")$dtaL
boost.cv.grow <- boostmtree(dta$features, dta$time, dta$id, dta$y,
                 family = "Continuous", M = 300, cv.flag = TRUE)
plot(boost.cv.grow)
print(boost.cv.grow)

##----------------------------------------------------------------------------
## spirometry data (Response is continuous)
##----------------------------------------------------------------------------
data(spirometry, package = "boostmtree")

#boosting call: cubic B-splines with 15 knots
spr.obj <- boostmtree(spirometry$features, spirometry$time, spirometry$id, spirometry$y,
                        family = "Continuous",M = 100, nu = .025, nknots = 15)
plot(spr.obj)


##----------------------------------------------------------------------------
## Atrial Fibrillation data (Response is binary)
##----------------------------------------------------------------------------
data(AF, package = "boostmtree")

#boosting call: cubic B-splines with 15 knots
AF.obj <- boostmtree(AF$feature, AF$time, AF$id, AF$y,
                        family = "Binary",M = 100, nu = .025, nknots = 15)
plot(AF.obj)


##----------------------------------------------------------------------------
## sneaky way to use boostmtree for (univariate) regression: boston housing
##----------------------------------------------------------------------------

if (library("mlbench", logical.return = TRUE)) {

  ## assemble the data
  data(BostonHousing)
  x <- BostonHousing; x$medv <- NULL
  y <- BostonHousing$medv
  trn <- sample(1:nrow(x), size = nrow(x) * (2 / 3), replace = FALSE)

  ## run boosting in univariate mode
  o <- boostmtree(x = x[trn,], y = y[trn],family = "Continuous")
  o.p <- predict(o, x = x[-trn, ], y = y[-trn])
  print(o)
  plot(o.p)

  ## run boosting in univariate mode to obtain RMSE and vimp
  o.cv <- boostmtree(x = x, y = y, M = 100,family = "Continuous",cv.flag = TRUE)
  print(o.cv)
  plot(o.cv)
}

# }