rfsrc.RdFast OpenMP parallel computing of random forests (Breiman 2001) for regression, classification, survival analysis (Ishwaran et al. 2008), competing risks (Ishwaran et al. 2012), multivariate (Segal and Xiao 2011), unsupervised (Mantero and Ishwaran 2020), quantile regression (Meinhausen 2006, Zhang et al. 2019, Greenwald-Khanna 2001), and class imbalanced q-classification (O'Brien and Ishwaran 2019). Different splitting rules invoked under deterministic or random splitting (Geurts et al. 2006, Ishwaran 2015) are available for all families. Different types of variable importance (VIMP), holdout VIMP, as well as confidence regions (Ishwaran and Lu 2019) can be calculated for single and grouped variables. Minimal depth variable selection (Ishwaran et al. 2010, 2011). Fast interface for missing data imputation using a variety of different random forest methods (Tang and Ishwaran 2017).
Highlighted new items:
For computational speed, the default VIMP is no longer
"permute" (Breiman-Cutler permutation importance) and has been
switched to "anti" (importance="anti",
importance=TRUE; see below for details). Be aware in some
situations, such as highly imbalanced classification, that
permutation VIMP may perform better. Permutation VIMP is
obtained using importance="permute".
save.memory can be used for big data to save memory;
especially useful for survival and competing risks.
Mahalanobis splitting for multivariate regression with
correlated y-outcomes (splitrule="mahalanobis"). Now allows
for a user specified covariance matrix.
Visualize trees on your Safari or Google Chrome
browser (works for all families). See get.tree.
This is the main entry point to the randomForestSRC
package. For more information about this package and OpenMP parallel
processing, use the command package?randomForestSRC.
rfsrc(formula, data, ntree = 500,
mtry = NULL, ytry = NULL,
nodesize = NULL, nodedepth = NULL,
splitrule = NULL, nsplit = NULL,
importance = c(FALSE, TRUE, "none", "anti", "permute", "random"),
block.size = if (any(is.element(as.character(importance),
c("none", "FALSE")))) NULL else 10,
bootstrap = c("by.root", "none", "by.user"),
samptype = c("swor", "swr"), samp = NULL, membership = FALSE,
sampsize = if (samptype == "swor") function(x){x * .632} else function(x){x},
na.action = c("na.omit", "na.impute"), nimpute = 1,
ntime = 150, cause,
perf.type = NULL,
proximity = FALSE, distance = FALSE, forest.wt = FALSE,
xvar.wt = NULL, yvar.wt = NULL, split.wt = NULL, case.wt = NULL,
case.depth = FALSE,
forest = TRUE,
save.memory = FALSE,
var.used = c(FALSE, "all.trees", "by.tree"),
split.depth = c(FALSE, "all.trees", "by.tree"),
seed = NULL,
do.trace = FALSE,
statistics = FALSE,
...)
## convenient interface for growing a CART tree
rfsrc.cart(formula, data, ntree = 1, mtry = ncol(data), bootstrap = "none", ...)Object of class 'formula' describing the model to fit. Interaction terms are not supported. If missing, unsupervised splitting is implemented.
Data frame containing the y-outcome and x-variables.
Number of trees.
Number of variables to possibly split at each node. Default is number of variables divided by 3 for regression. For all other families (including unsupervised settings), the square root of number of variables. Values are rounded up.
The number of randomly selected pseudo-outcomes for
unsupervised families (see details below). Default is
ytry=1.
Minumum size of terminal node. The defaults are:
survival (15), competing risk (15), regression (5), classification
(1), mixed outcomes (3), unsupervised (3). It is recommended to
experiment with different nodesize values.
Maximum depth to which a tree should be grown. Parameter is ignored by default.
Splitting rule (see below).
Non-negative integer specifying number of random splits for splitting a variable. When zero, all split values are used (deterministic splitting), which can be slower. By default 10 is used.
Method for computing variable importance (VIMP); see
below. Default action is importance="none" but VIMP can
be recovered later using vimp or predict.
Determines how cumulative error rate is calculated.
When NULL, it is calculated once for the entire forest,
resulting in a flat cumulative error rate plot. To view the
cumulative error rate for every nth tree, set block.size to
an integer between 1 and ntree. If importance is
requested, VIMP is calculated in blocks of size block.size,
balancing between ensemble and tree VIMP. The default is 10 trees.
Bootstrap protocol. Default is by.root, which
bootstraps the data by sampling with or without replacement (default
is without replacement; see samptype below). If none,
the data is not bootstrapped (OOB ensembles or prediction error
cannot be returned). If by.user, the bootstrap specified by
samp is used.
Type of bootstrap used when by.root is in
effect. Choices are swor (sampling without replacement;
the default) and swr (sampling with replacement).
Bootstrap specification when by.user is in
effect. Array of dim n x ntree specifying how many
times each record appears inbag in the bootstrap for each tree.
Should terminal node membership and inbag information be returned?
Specifies bootstrap size when by.root is in
effect. For sampling without replacement, it is the requested size
of the sample, defaulting to .632 times the sample size. For
sampling with replacement, it is the sample size. Can also be
specified as a number.
Action taken if the data contains
NAs. Possible values are na.omit or
na.impute. The default, na.omit, removes the entire
record if any entry is NA. Selecting na.impute imputes
the data (see below for details). Also see the function
impute for fast imputation.
Number of iterations of the missing data algorithm.
Performance measures such as out-of-bag (OOB) error rates are
optimistic if nimpute is greater than 1.
Integer value for survival to constrain ensemble
calculations to an ntime grid of time points over the
observed event times. If a vector of values with length greater than
one is supplied, it is assumed these are the time points to be used
(adjusted to match closest observed event times). Setting
ntime to zero (or NULL) uses all observed event times.
Integer value between 1 and J indicating the event
of interest for splitting a node for competing risks, where J
is the number of event types. If not specified, the default is a
composite splitting rule averaging over all event types. Can also
be a vector of non-negative weights of length J specifying
weights for each event (a vector of ones reverts to the default
composite split statistic). Estimates for all event types are
returned regardless of how cause is specified.
Optional character value specifying the metric used for
predicted value, variable importance (VIMP), and error rate.
Defaults to the family metric if not specified.
perf.type="none" turns off performance entirely, useful for
turning off C-error rate calculations for large survival data (which can
be expensive). Values allowed for univariate/multivariate
classification are: perf.type="misclass" (default),
perf.type="brier", and perf.type="gmean".
Proximity of cases as measured by the frequency of
sharing the same terminal node. This is an nxn
matrix, which can be large. Choices are inbag, oob,
all, TRUE, or FALSE. Setting proximity =
TRUE is equivalent to proximity = "inbag".
Distance between cases as measured by the ratio of the
sum of the count of edges from each case to their immediate common
ancestor node to the sum of the count of edges from each case to the
root node. If the cases are co-terminal for a tree, this measure is
zero and reduces to 1 - the proximity measure. This is an
nxn matrix, which can be large. Choices are
inbag, oob, all, TRUE, or FALSE.
Setting distance = TRUE is equivalent to distance =
"inbag".
Calculate the forest weight matrix? Creates an
nxn matrix which can be used for prediction and
constructing customized estimators. Choices are similar to
proximity: inbag, oob, all, TRUE, or
FALSE. The default is TRUE which is equivalent to
inbag.
Vector of non-negative weights (does not have to sum to 1) representing the probability of selecting a variable for splitting. Default is uniform weights.
Vector of non-negative weights (does not have to sum to 1) representing the probability of selecting a response variable in multivariate regression. Used when the number of responses are high-dimensional. See the example below.
Vector of non-negative weights used for multiplying the split statistic for a variable. A large value encourages the node to split on a specific variable. Default is uniform weights.
Vector of non-negative weights (does not have to sum to 1) for sampling cases. Observations with larger weights will be selected with higher probability in the bootstrap (or subsampled) samples. It is generally better to use real weights rather than integers. See the breast data example below illustrating its use for class imbalanced data.
Return a matrix recording the depth at which a case
first splits in a tree. Default is FALSE.
Save key forest values? Used for prediction on new data and required by many of the package functions. Turn this off if you are only interested in training a forest.
Save memory? Default is to store terminal node quantities used for prediction on test data. This yields rapid prediction but can be memory intensive for big data, especially competing risks and survival models. Turn this flag off in those cases.
Return statistics on number of times a variable split?
Default is FALSE. Possible values are all.trees which
returns total number of splits of each variable, and by.tree
which returns a matrix of number a splits for each variable for each
tree.
Records the minimal depth for each variable.
Default is FALSE. Possible values are all.trees which
returns a matrix of the average minimal depth for a variable
(columns) for a specific case (rows), and by.tree which
returns a three-dimensional array recording minimal depth for a
specific case (first dimension) for a variable (second dimension)
for a specific tree (third dimension).
Negative integer specifying seed for the random number generator.
Number of seconds between updates to the user on approximate time to completion.
Should split statistics be returned? Values can be
parsed using stat.split.
Further arguments passed to or from other methods.
Types of forests
There is no need to set the type of forest as the package automagically determines the underlying random forest requested from the type of outcome and the formula supplied. There are several possible scenarios:
Regression forests for continuous outcomes.
Classification forests for factor outcomes.
Multivariate forests for continuous and/or factor outcomes and for mixed (both type) of outcomes.
Unsupervised forests when there is no outcome.
Survival forests for right-censored survival.
Competing risk survival forests for competing risk.
Splitting
Splitting rules are specified by the option splitrule.
For all families, pure random splitting can be invoked by setting
splitrule="random".
For all families, computational speed can be increased using
randomized splitting invoked by the option nsplit.
See Improving Computational Speed.
Available splitting rules
Regression analysis:
splitrule="mse" (default split rule): weighted
mean-squared error splitting (Breiman et al. 1984, Chapter 8.4).
splitrule="quantile.regr": quantile regression splitting via
the "check-loss" function. Requires specifying the target
quantiles. See quantreg.rfsrc for further details.
la.quantile.regr: local adaptive quantile
regression splitting. See quantreg.rfsrc.
Classification analysis:
splitrule="gini" (default splitrule): Gini
index splitting (Breiman et al. 1984, Chapter 4.3).
splitrule="auc": AUC (area under the ROC curve) splitting
for both two-class and multiclass setttings. AUC splitting is
appropriate for imbalanced data. See imbalanced for
more information.
splitrule="entropy": entropy splitting (Breiman et
al. 1984, Chapter 2.5, 4.3).
Survival analysis:
splitrule="logrank" (default splitrule):
log-rank splitting (Segal, 1988; Leblanc and Crowley, 1993).
splitrule="bs.gradient": gradient-based (global non-quantile)
brier score splitting. The time horizon used for the Brier
score is set to the 90th percentile of the observed event times.
This can be over-ridden by the option prob, which must be
a value between 0 and 1 (set to .90 by default).
splitrule="logrankscore": log-rank score splitting (Hothorn and
Lausen, 2003).
Competing risk analysis (for details see Ishwaran et al., 2014):
splitrule="logrankCR" (default splitrule): modified
weighted log-rank splitting rule modeled after Gray's test
(Gray, 1988). Use this to find *all* variables
that are informative and when the goal is long term
prediction.
splitrule="logrank": weighted log-rank splitting where each
event type is treated as the event of interest and all
other events are treated as censored. The split rule is
the weighted value of each of log-rank statistics,
standardized by the variance. Use this to find variables
that affect a *specific* cause of interest and when
the goal is a targeted analysis of a specific cause.
However in order for this to be effective, remember to set
the cause option to the targeted cause of interest.
See examples below.
Multivariate analysis:
Default is the multivariate normalized composite split rule using mean-squared error and Gini index (Tang and Ishwaran, 2017).
splitrule="mahalanobis": Mahalanobis splitting
that adjusts for correlation (also allows for a user specified
covariance matrix, see example below). Only works for
multivariate regression (all outcomes must be real).
Unsupervised analysis: In settings where there is no
outcome, unsupervised splitting that uses pseudo-outcomes is
applied using the default multivariate splitting rule (see
below for details) Also see sidClustering for a
more sophisticated method for unsupervised analysis (Mantero
and Ishwaran, 2020).
Custom splitting: All families except unsupervised are
available for user defined custom splitting. Some basic
C-programming skills are required. The harness for defining
these rules is in splitCustom.c. In this file we
give examples of how to code rules for regression,
classification, survival, and competing risk. Each family
can support up to sixteen custom split rules. Specifying
splitrule="custom" or splitrule="custom1" will
trigger the first split rule for the family defined by the
training data set. Multivariate families will need a custom
split rule for both regression and classification. In the
examples, we demonstrate how the user is presented with the
node specific membership. The task is then to define a
split statistic based on that membership. Take note of the
instructions in splitCustom.c on how to
register the custom split rules. It is suggested
that the existing custom split rules be kept in place for
reference and that the user proceed to develop
splitrule="custom2" and so on. The package must be
recompiled and installed for the custom split rules to
become available.
Improving computational speed
See the function rfsrc.fast for a fast
implementation of rfsrc. Key methods for
increasing speed are as follows:
Nodesize
Increasing nodesize has the greatest effect in speeding
calculations. In some big data settings this can also lead to
better prediction performance.
Save memory
Use option save.memory="TRUE" for big data competing risk
and survival models. By default the package stores terminal
node quantities to be used in prediction for test data but this
can be memory intensive for big data.
Block size
Make sure block.size="NULL" (or set to number of trees)
so that the cumulative error is calculated only once.
Turn off performace
The C-error rate calculation can be very expensive for big
survival data. Set perf.type="none" to turn this off and
all other performance calculations (then consider using the
function get.brier.survival as a fast way to get survival
performance).
perf.type="none" applies to all other families as well.
Randomized splitting rules
Set nsplit to a small non-zero integer value. Then a
maximum of nsplit split points are chosen randomly for
each of the candidate splitting variables when splitting a tree
node, thus significantly reducing computational costs.
For more details about randomized splitting see Loh and Shih
(1997), Dietterich (2000), and Lin and Jeon (2006). Geurts et
al. (2006) introduced extremely randomized trees using the
extra-trees algorithm. This algorithm corresponds to
nsplit=1. In our experience however this may be too low
for general use (Ishwaran, 2015).
For completely randomized (pure random) splitting use
splitrule="random". In pure splitting, nodes are split
by randomly selecting a variable and randomly selecting its
split point (Cutler and Zhao, 2001).
Subsampling
Reduce the size of the bootstrap using sampsize and
samptype. See rfsrc.fast for a fast forest
implementation using subsampling.
Unique time points
Setting ntime to a reasonably small value such as 50
constrains survival ensemble calculations to a restricted grid
of time points and significantly improves computational times.
Large number of variables
Try filtering variables ahead of time. Make sure not to request
VIMP (variable importance can always be recovered later using
vimp or predict). Also if variable
selection is desired, but is too slow, consider using
max.subtree which calculates minimal depth, a measure
of the depth that a variable splits, and yields fast variable
selection (Ishwaran, 2010).
Prediction Error
Prediction error is calculated using OOB data. The metric used is
mean-squared-error for regression, and misclassification error for
classification. A normalized Brier score (relative to a coin-toss)
and the AUC (area under the ROC curve) is also provided upon
printing a classification forest. Performance for Brier score can
be specified using perf.type="brier". G-mean performance is
also available, see the function imbalanced for more
details.
For survival, prediction error is measured by the C-error rate defined as 1-C, where C is Harrell's (Harrell et al., 1982) concordance index. The C-error is between 0 and 1, and measures error in ranking (classifying) two random individuals in terms of survival. A value of 0.5 is no better than random guessing. A value of 0 is perfect.
When bootstrapping is by none, a coherent OOB subset is not
available to assess prediction error. Thus, all outputs dependent
on this are suppressed. In such cases, prediction error is only
available via classical cross-validation (the user will need to use
the predict.rfsrc function).
Variable Importance (VIMP)
VIMP is calculated using OOB data in several ways.
importance="permute" yields permutation VIMP (Breiman-Cutler
importance) by permuting OOB cases. importance="random" uses
random left/right assignments whenever a split is encountered for
the target variable. The default importance="anti"
(equivalent to importance=TRUE) assigns cases to the anti
(opposite) split.
VIMP depends upon block.size, an integer value between 1 and
ntree, specifying number of trees in a block used for VIMP.
When block.size=1, VIMP is calculated for each tree. When
block.size="ntree", VIMP is calculated for the entire forest
by comparing the perturbed OOB forest ensemble (using all trees) to
the unperturbed OOB forest ensemble (using all trees). This yields
ensemble VIMP, which does not measure the tree average effect of a
variable, but rather its overall forest effect.
A useful compromise between tree VIMP and ensemble VIMP can be
obtained by setting block.size to a value between 1 and
ntree. Smaller values generally gives better accuracy,
however computational times will be higher because VIMP is
calculated over more blocks. However, see imbalanced for
imbalanced classification data where larger block.size
often works better (O'Brien and Ishwaran, 2019).
See vimp for a user interface for extracting VIMP and
subsampling for calculating confidence intervals for VIMP.
Also see holdout.vimp for holdout VIMP, which calculates
importance by holding out variables. This is more conservative, but
with good false discovery properties.
For classification, VIMP is returned as a matrix with J+1 columns where J is the number of classes. The first column "all" is the unconditional VIMP, while the remaining columns are conditional VIMP calculated using only OOB cases with the class label.
Multivariate Forests
Multivariate forests can be specified in two ways:
rfsrc(Multivar(y1, y2, ..., yd) ~ . , my.data, ...)
rfsrc(cbind(y1, y2, ..., yd) ~ . , my.data, ...)
By default, a multivariate normalized composite splitting rule is used to split nodes (for multivariate regression, users have the option to use Mahalanobis splitting).
The nature of the outcomes informs the code as to what type of multivariate forest is grown; i.e. whether it is real-valued, categorical, or a combination of both (mixed). Performance measures (when requested) are returned for all outcomes.
Helper functions get.mv.formula,
get.mv.predicted, get.mv.error can be used for
defining the multivariate forest formula and extracting predicted
values (all outcomes) and VIMP (all variables, all outcomes;
assuming importance was requested in the call). The latter two
functions also work for univariate (regular) forests. Both
functions return standardized values (dividing by the variance for
regression, or multiplying by 100, otherwise) using option
standardize="TRUE".
Unsupervised Forests and sidClustering
See sidClustering sidClustering for a more sophisticated
method for unsupervised analysis.
Otherwise a more direct (but naive) way to proceed is to use the unsupervised splitting rule. The following are equivalent ways to grow an unsupervised forest via unsupervised splitting:
rfsrc(data = my.data)
rfsrc(Unsupervised() ~ ., data = my.data)
In unsupervised mode, features take turns acting as target
y-outcomes and x-variables for splitting. Specifically, mtry
x-variables are randomly selected for splitting the node. Then for
each mtry feature, ytry variables are selected
from the remaining features to act as the target pseduo-outcomes.
Splitting uses the multivariate normalized composite splitting rule.
The default value of ytry is 1 but can be increased. As
illustration, the following equivalent unsupervised calls set
mtry=10 and ytry=5:
rfsrc(data = my.data, ytry = 5, mtry = 10)
rfsrc(Unsupervised(5) ~ ., my.data, mtry = 10)
Note that all performance values (error rates, VIMP, prediction) are turned off in unsupervised mode.
Survival, Competing Risks
Survival settings require a time and censoring variable which
should be identifed in the formula as the outcome using the standard
Surv formula specification. A typical formula call looks like:
Surv(my.time, my.status) ~ .
where my.time and my.status are the variables names for
the event time and status variable in the users data set.
For survival forests (Ishwaran et al. 2008), the censoring variable must be coded as a non-negative integer with 0 reserved for censoring and (usually) 1=death (event).
For competing risk forests (Ishwaran et al., 2013), the implementation is similar to survival, but with the following caveats:
Censoring must be coded as a non-negative integer, where 0
indicates right-censoring, and non-zero values indicate different
event types. While 0,1,2,..,J is standard, and recommended,
events can be coded non-sequentially, although 0 must always be used
for censoring.
Setting the splitting rule to logrankscore will result
in a survival analysis in which all events are treated as if they
are the same type (indeed, they will coerced as such).
Generally, competing risks requires a larger nodesize than
survival settings.
Missing data imputation
na.action="na.impute" imputes missing data (both x and
y-variables) using the missing data algorithm of Ishwaran et
al. (2008). But also see the impute for an
alternate way to do fast and accurate imputation.
The missing data algorithm can be iterated by setting nimpute
to a positive integer greater than 1. When iterated, at the
completion of each iteration, missing data is imputed using OOB
non-missing terminal node data which is then used as input to grow a
new forest. A side effect of iteration is that missing values in
the returned objects xvar, yvar are replaced by
imputed values. In other words the incoming data is overlaid with
the missing data. Also, performance measures such as error rates
and VIMP become optimistically biased.
Records in which all outcome and x-variable information are missing are removed from the forest analysis. Variables having all missing values are also removed.
Allowable data types and factors
Data types must be real valued, integer, factor or logical --
however all except factors are coerced and treated as if real
valued. For ordered x-variable factors, splits are similar to real
valued variables. For unordered factors, a split will move a subset
of the levels in the parent node to the left daughter, and the
complementary subset to the right daughter. All possible
complementary pairs are considered and apply to factors with an
unlimited number of levels. However, there is an optimization check
to ensure number of splits attempted is not greater than number of
cases in a node or the value of nsplit.
For coherence, an immutable map is applied to each factor that ensures factor levels in the training data are consistent with the factor levels in any subsequent test data. This map is applied to each factor before and after the native C library is executed. Because of this, if all x-variables all factors, then computational time will be long in high dimensional problems. Consider converting factors to real if this is the case.
An object of class (rfsrc, grow) with the following
components:
The original call to rfsrc.
The family used in the analysis.
Sample size of the data (depends upon NA's, see na.action).
Number of trees grown.
Number of variables randomly selected for splitting at each node.
Minimum size of terminal nodes.
Maximum depth allowed for a tree.
Splitting rule used.
Number of randomly selected split points.
y-outcome values.
A character vector of the y-outcome names.
Data frame of x-variables.
A character vector of the x-variable names.
Vector of non-negative weights specifying the probability used to select a variable for splitting a node.
Vector of non-negative weights specifying multiplier by which the split statistic for a covariate is adjusted.
Vector of weights used for the composite competing risk splitting rule.
Number of terminal nodes for each tree in the
forest. Vector of length ntree. A value of zero indicates
a rejected tree (can occur when imputing missing data).
Values of one indicate tree stumps.
Proximity matrix recording the frequency of pairs of data points occur within the same terminal node.
If forest=TRUE, the forest object is returned.
This object is used for prediction with new test data
sets and is required for other R-wrappers.
Forest weight matrix.
Matrix recording terminal node membership where each column records node mebership for a case for a tree (rows).
Splitting rule used.
Matrix recording inbag membership where each column contains the number of times that a case appears in the bootstrap sample for a tree (rows).
Count of the number of times a variable is used in growing the forest.
Vector of indices for cases with missing values.
Data frame of the imputed data. The first column(s) are reserved for the y-outcomes, after which the x-variables are listed.
Matrix (i,j) or array (i,j,k) recording the minimal depth for variable j for case i, either averaged over the forest, or by tree k.
Split statistics returned when
statistics=TRUE which can be parsed using stat.split.
Tree cumulative OOB error rate.
When importance=TRUE, vector of the
cumulative error rate for each ensemble block comprised of
block.size trees. So with block.size = 10, entries are
the cumulative error rate for the first 10 trees, the first 20 trees,
30 trees, and so on. As another exmple, if block.size = 1,
entries are the error rate for each tree.
Variable importance (VIMP) for each x-variable.
In-bag predicted value.
OOB predicted value.
for classification settings, additionally ++++++++
In-bag predicted class labels.
OOB predicted class labels.
for multivariate settings, additionally ++++++++
List containing performance values for multivariate regression outcomes (applies only in multivariate settings).
List containing performance values for multivariate categorical (factor) outcomes (applies only in multivariate settings).
for survival settings, additionally ++++++++
In-bag survival function.
OOB survival function.
In-bag cumulative hazard function (CHF).
OOB CHF.
Ordered unique death times.
Number of deaths.
for competing risks, additionally ++++++++
In-bag cause-specific cumulative hazard function (CSCHF) for each event.
OOB CSCHF.
In-bag cumulative incidence function (CIF) for each event.
OOB CIF.
Values returned depend heavily on the family. In particular,
predicted values from the forest (predicted and
predicted.oob) are as follows:
For regression, a vector of predicted y-outcomes.
For classification, a matrix with columns containing the estimated class probability for each class. Performance values and VIMP for classification are reported as a matrix with J+1 columns where J is the number of classes. The first column "all" is the unconditional value for performance (VIMP), while the remaining columns are performance (VIMP) conditioned on cases corresponding to that class label.
For survival, a vector of mortality values (Ishwaran et al.,
2008) representing estimated risk for each individual calibrated
to the scale of the number of events (as a specific example, if
i has a mortality value of 100, then if all individuals had
the same x-values as i, we would expect an average of 100
events). Also returned are matrices containing
the CHF and survival function. Each row corresponds to an
individual's ensemble CHF or survival function evaluated at each
time point in time.interest.
For competing risks, a matrix with one column for each event
recording the expected number of life years lost due to the event
specific cause up to the maximum follow up (Ishwaran et al.,
2013). Also returned are the cause-specific cumulative hazard
function (CSCHF) and the cumulative incidence function (CIF) for
each event type. These are encoded as a three-dimensional array,
with the third dimension used for the event type, each time point
in time.interest making up the second dimension (columns),
and the case (individual) being the first dimension (rows).
For multivariate families, predicted values (and other
performance values such as VIMP and error rates) are stored in the
lists regrOutput and clasOutput which can be
extracted using functions get.mv.error,
get.mv.predicted and get.mv.vimp.
Breiman L., Friedman J.H., Olshen R.A. and Stone C.J. (1984). Classification and Regression Trees, Belmont, California.
Breiman L. (2001). Random forests, Machine Learning, 45:5-32.
Cutler A. and Zhao G. (2001). PERT-Perfect random tree ensembles. Comp. Sci. Statist., 33: 490-497.
Dietterich, T. G. (2000). An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization. Machine Learning, 40, 139-157.
Gray R.J. (1988). A class of k-sample tests for comparing the cumulative incidence of a competing risk, Ann. Statist., 16: 1141-1154.
Geurts, P., Ernst, D. and Wehenkel, L., (2006). Extremely randomized trees. Machine learning, 63(1):3-42.
Greenwald M. and Khanna S. (2001). Space-efficient online computation of quantile summaries. Proceedings of ACM SIGMOD, 30(2):58-66.
Harrell et al. F.E. (1982). Evaluating the yield of medical tests, J. Amer. Med. Assoc., 247:2543-2546.
Hothorn T. and Lausen B. (2003). On the exact distribution of maximally selected rank statistics, Comp. Statist. Data Anal., 43:121-137.
Ishwaran H. (2007). Variable importance in binary regression trees and forests, Electronic J. Statist., 1:519-537.
Ishwaran H. and Kogalur U.B. (2007). Random survival forests for R, Rnews, 7(2):25-31.
Ishwaran H., Kogalur U.B., Blackstone E.H. and Lauer M.S. (2008). Random survival forests, Ann. App. Statist., 2:841-860.
Ishwaran H., Kogalur U.B., Gorodeski E.Z, Minn A.J. and Lauer M.S. (2010). High-dimensional variable selection for survival data. J. Amer. Statist. Assoc., 105:205-217.
Ishwaran H., Kogalur U.B., Chen X. and Minn A.J. (2011). Random survival forests for high-dimensional data. Stat. Anal. Data Mining, 4:115-132
Ishwaran H., Gerds T.A., Kogalur U.B., Moore R.D., Gange S.J. and Lau B.M. (2014). Random survival forests for competing risks. Biostatistics, 15(4):757-773.
Ishwaran H. and Malley J.D. (2014). Synthetic learning machines. BioData Mining, 7:28.
Ishwaran H. (2015). The effect of splitting on random forests. Machine Learning, 99:75-118.
Lin, Y. and Jeon, Y. (2006). Random forests and adaptive nearest neighbors. J. Amer. Statist. Assoc., 101(474), 578-590.
Lu M., Sadiq S., Feaster D.J. and Ishwaran H. (2018). Estimating individual treatment effect in observational data using random forest methods. J. Comp. Graph. Statist, 27(1), 209-219
Ishwaran H. and Lu M. (2019). Standard errors and confidence intervals for variable importance in random forest regression, classification, and survival. Statistics in Medicine, 38, 558-582.
LeBlanc M. and Crowley J. (1993). Survival trees by goodness of split, J. Amer. Statist. Assoc., 88:457-467.
Loh W.-Y and Shih Y.-S (1997). Split selection methods for classification trees, Statist. Sinica, 7:815-840.
Mantero A. and Ishwaran H. (2021). Unsupervised random forests. Statistical Analysis and Data Mining, 14(2):144-167.
Meinshausen N. (2006) Quantile regression forests, Journal of Machine Learning Research, 7:983-999.
Mogensen, U.B, Ishwaran H. and Gerds T.A. (2012). Evaluating random forests for survival analysis using prediction error curves, J. Statist. Software, 50(11): 1-23.
O'Brien R. and Ishwaran H. (2019). A random forests quantile classifier for class imbalanced data. Pattern Recognition, 90, 232-249
Segal M.R. (1988). Regression trees for censored data, Biometrics, 44:35-47.
Segal M.R. and Xiao Y. Multivariate random forests. (2011). Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery. 1(1):80-87.
Tang F. and Ishwaran H. (2017). Random forest missing data algorithms. Statistical Analysis and Data Mining, 10:363-377.
Zhang H., Zimmerman J., Nettleton D. and Nordman D.J. (2019). Random forest prediction intervals. The American Statistician. 4:1-5.
# \donttest{
##------------------------------------------------------------
## survival analysis
##------------------------------------------------------------
## veteran data
## randomized trial of two treatment regimens for lung cancer
data(veteran, package = "randomForestSRC")
v.obj <- rfsrc(Surv(time, status) ~ ., data = veteran, block.size = 1)
## plot tree number 3
plot(get.tree(v.obj, 3))
## print results of trained forest
print(v.obj)
## plot results of trained forest
plot(v.obj)
## plot survival curves for first 10 individuals -- direct way
matplot(v.obj$time.interest, 100 * t(v.obj$survival.oob[1:10, ]),
xlab = "Time", ylab = "Survival", type = "l", lty = 1)
## plot survival curves for first 10 individuals
## using function "plot.survival"
plot.survival(v.obj, subset = 1:10)
## obtain Brier score using KM and RSF censoring distribution estimators
bs.km <- get.brier.survival(v.obj, cens.model = "km")$brier.score
bs.rsf <- get.brier.survival(v.obj, cens.model = "rfsrc")$brier.score
## plot the brier score
plot(bs.km, type = "s", col = 2)
lines(bs.rsf, type ="s", col = 4)
legend("topright", legend = c("cens.model = km", "cens.model = rfsrc"), fill = c(2,4))
## plot CRPS (continuous rank probability score) as function of time
## here's how to calculate the CRPS for every time point
trapz <- randomForestSRC:::trapz
time <- v.obj$time.interest
crps.km <- sapply(1:length(time), function(j) {
trapz(time[1:j], bs.km[1:j, 2] / diff(range(time[1:j])))
})
crps.rsf <- sapply(1:length(time), function(j) {
trapz(time[1:j], bs.rsf[1:j, 2] / diff(range(time[1:j])))
})
plot(time, crps.km, ylab = "CRPS", type = "s", col = 2)
lines(time, crps.rsf, type ="s", col = 4)
legend("bottomright", legend=c("cens.model = km", "cens.model = rfsrc"), fill=c(2,4))
## fast nodesize optimization for veteran data
## optimal nodesize in survival is larger than other families
## see the function "tune" for more examples
tune.nodesize(Surv(time,status) ~ ., veteran)
## Primary biliary cirrhosis (PBC) of the liver
data(pbc, package = "randomForestSRC")
pbc.obj <- rfsrc(Surv(days, status) ~ ., pbc)
print(pbc.obj)
## save.memory example for survival
## growing many deep trees creates memory issue without this option!
data(pbc, package = "randomForestSRC")
print(rfsrc(Surv(days, status) ~ ., pbc, splitrule = "random",
ntree = 25000, nodesize = 1, save.memory = TRUE))
##------------------------------------------------------------
## trees can be plotted for any family
## see get.tree for details and more examples
##------------------------------------------------------------
## survival where factors have many levels
data(veteran, package = "randomForestSRC")
vd <- veteran
vd$celltype=factor(vd$celltype)
vd$diagtime=factor(vd$diagtime)
vd.obj <- rfsrc(Surv(time,status)~., vd, ntree = 100, nodesize = 5)
plot(get.tree(vd.obj, 3))
## classification
iris.obj <- rfsrc(Species ~., data = iris)
plot(get.tree(iris.obj, 25, class.type = "bayes"))
plot(get.tree(iris.obj, 25, target = "setosa"))
plot(get.tree(iris.obj, 25, target = "versicolor"))
plot(get.tree(iris.obj, 25, target = "virginica"))
## ------------------------------------------------------------
## simple example of VIMP using iris classification
## ------------------------------------------------------------
## directly from trained forest
print(rfsrc(Species~.,iris,importance=TRUE)$importance)
## VIMP (and performance) use misclassification error by default
## but brier prediction error can be requested
print(rfsrc(Species~.,iris,importance=TRUE,perf.type="brier")$importance)
## example using vimp function (see vimp help file for details)
iris.obj <- rfsrc(Species ~., data = iris)
print(vimp(iris.obj)$importance)
print(vimp(iris.obj,perf.type="brier")$importance)
## example using hold out vimp (see holdout.vimp help file for details)
print(holdout.vimp(Species~.,iris)$importance)
print(holdout.vimp(Species~.,iris,perf.type="brier")$importance)
## ------------------------------------------------------------
## confidence interval for vimp using subsampling
## compare with holdout vimp
## ------------------------------------------------------------
## new York air quality measurements
o <- rfsrc(Ozone ~ ., data = airquality)
so <- subsample(o)
plot(so)
## compare with holdout vimp
print(holdout.vimp(Ozone ~ ., data = airquality)$importance)
##------------------------------------------------------------
## example of imputation in survival analysis
##------------------------------------------------------------
data(pbc, package = "randomForestSRC")
pbc.obj2 <- rfsrc(Surv(days, status) ~ ., pbc, na.action = "na.impute")
## same as above but iterate the missing data algorithm
pbc.obj3 <- rfsrc(Surv(days, status) ~ ., pbc,
na.action = "na.impute", nimpute = 3)
## fast way to impute data (no inference is done)
## see impute for more details
pbc.imp <- impute(Surv(days, status) ~ ., pbc, splitrule = "random")
##------------------------------------------------------------
## compare RF-SRC to Cox regression
## Illustrates C-error and Brier score measures of performance
## assumes "pec" and "survival" libraries are loaded
##------------------------------------------------------------
if (library("survival", logical.return = TRUE)
& library("pec", logical.return = TRUE)
& library("prodlim", logical.return = TRUE))
{
##prediction function required for pec
predictSurvProb.rfsrc <- function(object, newdata, times, ...){
ptemp <- predict(object,newdata=newdata,...)$survival
pos <- sindex(jump.times = object$time.interest, eval.times = times)
p <- cbind(1,ptemp)[, pos + 1]
if (NROW(p) != NROW(newdata) || NCOL(p) != length(times))
stop("Prediction failed")
p
}
## data, formula specifications
data(pbc, package = "randomForestSRC")
pbc.na <- na.omit(pbc) ##remove NA's
surv.f <- as.formula(Surv(days, status) ~ .)
pec.f <- as.formula(Hist(days,status) ~ 1)
## run cox/rfsrc models
## for illustration we use a small number of trees
cox.obj <- coxph(surv.f, data = pbc.na, x = TRUE)
rfsrc.obj <- rfsrc(surv.f, pbc.na, ntree = 150)
## compute bootstrap cross-validation estimate of expected Brier score
## see Mogensen, Ishwaran and Gerds (2012) Journal of Statistical Software
set.seed(17743)
prederror.pbc <- pec(list(cox.obj,rfsrc.obj), data = pbc.na, formula = pec.f,
splitMethod = "bootcv", B = 50)
print(prederror.pbc)
plot(prederror.pbc)
## compute out-of-bag C-error for cox regression and compare to rfsrc
rfsrc.obj <- rfsrc(surv.f, pbc.na)
cat("out-of-bag Cox Analysis ...", "\n")
cox.err <- sapply(1:100, function(b) {
if (b%%10 == 0) cat("cox bootstrap:", b, "\n")
train <- sample(1:nrow(pbc.na), nrow(pbc.na), replace = TRUE)
cox.obj <- tryCatch({coxph(surv.f, pbc.na[train, ])}, error=function(ex){NULL})
if (!is.null(cox.obj)) {
get.cindex(pbc.na$days[-train], pbc.na$status[-train], predict(cox.obj, pbc.na[-train, ]))
} else NA
})
cat("\n\tOOB error rates\n\n")
cat("\tRSF : ", rfsrc.obj$err.rate[rfsrc.obj$ntree], "\n")
cat("\tCox regression : ", mean(cox.err, na.rm = TRUE), "\n")
}
##------------------------------------------------------------
## competing risks
##------------------------------------------------------------
## WIHS analysis
## cumulative incidence function (CIF) for HAART and AIDS stratified by IDU
data(wihs, package = "randomForestSRC")
wihs.obj <- rfsrc(Surv(time, status) ~ ., wihs, nsplit = 3, ntree = 100)
plot.competing.risk(wihs.obj)
cif <- wihs.obj$cif.oob
Time <- wihs.obj$time.interest
idu <- wihs$idu
cif.haart <- cbind(apply(cif[,,1][idu == 0,], 2, mean),
apply(cif[,,1][idu == 1,], 2, mean))
cif.aids <- cbind(apply(cif[,,2][idu == 0,], 2, mean),
apply(cif[,,2][idu == 1,], 2, mean))
matplot(Time, cbind(cif.haart, cif.aids), type = "l",
lty = c(1,2,1,2), col = c(4, 4, 2, 2), lwd = 3,
ylab = "Cumulative Incidence")
legend("topleft",
legend = c("HAART (Non-IDU)", "HAART (IDU)", "AIDS (Non-IDU)", "AIDS (IDU)"),
lty = c(1,2,1,2), col = c(4, 4, 2, 2), lwd = 3, cex = 1.5)
## illustrates the various splitting rules
## illustrates event specific and non-event specific variable selection
if (library("survival", logical.return = TRUE)) {
## use the pbc data from the survival package
## events are transplant (1) and death (2)
data(pbc, package = "survival")
pbc$id <- NULL
## modified Gray's weighted log-rank splitting
## (equivalent to cause=c(1,1) and splitrule="logrankCR")
pbc.cr <- rfsrc(Surv(time, status) ~ ., pbc)
## log-rank cause-1 specific splitting and targeted VIMP for cause 1
pbc.log1 <- rfsrc(Surv(time, status) ~ ., pbc,
splitrule = "logrankCR", cause = c(1,0), importance = TRUE)
## log-rank cause-2 specific splitting and targeted VIMP for cause 2
pbc.log2 <- rfsrc(Surv(time, status) ~ ., pbc,
splitrule = "logrankCR", cause = c(0,1), importance = TRUE)
## extract VIMP from the log-rank forests: event-specific
## extract minimal depth from the Gray log-rank forest: non-event specific
var.perf <- data.frame(md = max.subtree(pbc.cr)$order[, 1],
vimp1 = 100 * pbc.log1$importance[ ,1],
vimp2 = 100 * pbc.log2$importance[ ,2])
print(var.perf[order(var.perf$md), ], digits = 2)
}
## ------------------------------------------------------------
## regression analysis
## ------------------------------------------------------------
## new York air quality measurements
airq.obj <- rfsrc(Ozone ~ ., data = airquality, na.action = "na.impute")
# partial plot of variables (see plot.variable for more details)
plot.variable(airq.obj, partial = TRUE, smooth.lines = TRUE)
## motor trend cars
mtcars.obj <- rfsrc(mpg ~ ., data = mtcars)
## ------------------------------------------------------------
## regression with custom bootstrap
## ------------------------------------------------------------
ntree <- 25
n <- nrow(mtcars)
s.size <- n / 2
swr <- TRUE
samp <- randomForestSRC:::make.sample(ntree, n, s.size, swr)
o <- rfsrc(mpg ~ ., mtcars, bootstrap = "by.user", samp = samp)
## ------------------------------------------------------------
## classification analysis
## ------------------------------------------------------------
## iris data
iris.obj <- rfsrc(Species ~., data = iris)
## wisconsin prognostic breast cancer data
data(breast, package = "randomForestSRC")
breast.obj <- rfsrc(status ~ ., data = breast, block.size=1)
plot(breast.obj)
## ------------------------------------------------------------
## big data set, reduce number of variables using simple method
## ------------------------------------------------------------
## use Iowa housing data set
data(housing, package = "randomForestSRC")
## original data contains lots of missing data, use fast imputation
## however see impute for other methods
housing2 <- impute(data = housing, fast = TRUE)
## run shallow trees to find variables that split any tree
xvar.used <- rfsrc(SalePrice ~., housing2, ntree = 250, nodedepth = 4,
var.used="all.trees", mtry = Inf, nsplit = 100)$var.used
## now fit forest using filtered variables
xvar.keep <- names(xvar.used)[xvar.used >= 1]
o <- rfsrc(SalePrice~., housing2[, c("SalePrice", xvar.keep)])
print(o)
## ------------------------------------------------------------
## imbalanced classification data
## see the "imbalanced" function for further details
##
## (a) use balanced random forests with undersampling of the majority class
## Specifically let n0, n1 be sample sizes for majority, minority
## cases. We sample 2 x n1 cases with majority, minority cases chosen
## with probabilities n1/n, n0/n where n=n0+n1
##
## (b) balanced random forests using "imbalanced"
##
## (c) q-classifier (RFQ) using "imbalanced"
##
## ------------------------------------------------------------
## Wisconsin breast cancer example
data(breast, package = "randomForestSRC")
breast <- na.omit(breast)
## balanced random forests - brute force
y <- breast$status
obdirect <- rfsrc(status ~ ., data = breast, nsplit = 10,
case.wt = randomForestSRC:::make.wt(y),
sampsize = randomForestSRC:::make.size(y))
print(obdirect)
print(get.imbalanced.performance(obdirect))
## balanced random forests - using "imbalanced"
ob <- imbalanced(status ~ ., data = breast, method = "brf")
print(ob)
print(get.imbalanced.performance(ob))
## q-classifier (RFQ) - using "imbalanced"
oq <- imbalanced(status ~ ., data = breast)
print(oq)
print(get.imbalanced.performance(oq))
## q-classifier (RFQ) - with auc splitting
oqauc <- imbalanced(status ~ ., data = breast, splitrule = "auc")
print(oqauc)
print(get.imbalanced.performance(oqauc))
## ------------------------------------------------------------
## unsupervised analysis
## ------------------------------------------------------------
## two equivalent ways to implement unsupervised forests
mtcars.unspv <- rfsrc(Unsupervised() ~., data = mtcars)
mtcars2.unspv <- rfsrc(data = mtcars)
## illustration of sidClustering for the mtcars data
## see sidClustering for more details
mtcars.sid <- sidClustering(mtcars, k = 1:10)
print(split(mtcars, mtcars.sid$cl[, 3]))
print(split(mtcars, mtcars.sid$cl[, 10]))
## ------------------------------------------------------------
## bivariate regression using Mahalanobis splitting
## also illustrates user specified covariance matrix
## ------------------------------------------------------------
if (library("mlbench", logical.return = TRUE)) {
## load boston housing data, specify the bivariate regression
data(BostonHousing)
f <- formula("Multivar(lstat, nox) ~.")
## Mahalanobis splitting
bh.mreg <- rfsrc(f, BostonHousing, importance = TRUE, splitrule = "mahal")
## performance error and vimp
vmp <- get.mv.vimp(bh.mreg)
pred <- get.mv.predicted(bh.mreg)
## standardized error and vimp
err.std <- get.mv.error(bh.mreg, standardize = TRUE)
vmp.std <- get.mv.vimp(bh.mreg, standardize = TRUE)
## same analysis, but with user specified covariance matrix
sigma <- cov(BostonHousing[, c("lstat","nox")])
bh.mreg2 <- rfsrc(f, BostonHousing, splitrule = "mahal", sigma = sigma)
}
## ------------------------------------------------------------
## multivariate mixed forests (nutrigenomic study)
## study effects of diet, lipids and gene expression for mice
## diet, genotype and lipids used as the multivariate y
## genes used for the x features
## ------------------------------------------------------------
## load the data (data is a list)
data(nutrigenomic, package = "randomForestSRC")
## assemble the multivariate y data
ydta <- data.frame(diet = nutrigenomic$diet,
genotype = nutrigenomic$genotype,
nutrigenomic$lipids)
## multivariate mixed forest call
## uses "get.mv.formula" for conveniently setting formula
mv.obj <- rfsrc(get.mv.formula(colnames(ydta)),
data.frame(ydta, nutrigenomic$genes),
importance=TRUE, nsplit = 10)
## print results for diet and genotype y values
print(mv.obj, outcome.target = "diet")
print(mv.obj, outcome.target = "genotype")
## extract standardized VIMP
svimp <- get.mv.vimp(mv.obj, standardize = TRUE)
## plot standardized VIMP for diet, genotype and lipid for each gene
boxplot(t(svimp), col = "bisque", cex.axis = .7, las = 2,
outline = FALSE,
ylab = "standardized VIMP",
main = "diet/genotype/lipid VIMP for each gene")
## ------------------------------------------------------------
## illustrates yvar.wt which sets the probability of selecting
## the response variables in multivariate regression
## ------------------------------------------------------------
## use mtcars: add fake responses
mult.mtcars <- cbind(mtcars, mtcars$mpg, mtcars$mpg)
names(mult.mtcars) = c(names(mtcars), "mpg2", "mpg3")
## noise up the fake responses
mult.mtcars$mpg2 <- sample(mtcars$mpg)
mult.mtcars$mpg3 <- sample(mtcars$mpg)
formula = as.formula(Multivar(mpg, mpg2, mpg3) ~ .)
## select 2 of the 3 responses randomly at each split with an associated weight vector.
## choose the noisy y responses which should degrade performance
yvar.wt = c(0.000001, 0.5, 0.5)
ytry = 2
mult.grow <- rfsrc(formula = formula, data = mult.mtcars, ytry = ytry, yvar.wt = yvar.wt)
print(mult.grow)
print(get.mv.error(mult.grow))
## Also, compare the following two results, as they should be similar:
yvar.wt = c(1.0, 00000.1, 00000.1)
ytry = 1
result1 = rfsrc(formula = formula, data = mult.mtcars, ytry = ytry, yvar.wt = yvar.wt)
result2 = rfsrc(mpg ~ ., mtcars)
print(get.mv.error(result1))
print(get.mv.error(result2))
## ------------------------------------------------------------
## custom splitting using the pre-coded examples
## ------------------------------------------------------------
## motor trend cars
mtcars.obj <- rfsrc(mpg ~ ., data = mtcars, splitrule = "custom")
## iris analysis
iris.obj <- rfsrc(Species ~., data = iris, splitrule = "custom1")
## WIHS analysis
wihs.obj <- rfsrc(Surv(time, status) ~ ., wihs, nsplit = 3,
ntree = 100, splitrule = "custom1")
# }