| Type: | Package |
| Title: | Hierarchical Sufficient Dimension Reduction |
| Version: | 0.1 |
| Description: | Provides semiparametric sufficient dimension reduction for central mean subspaces for heterogeneous data defined by combinations of binary factors (such as chronic conditions). Subspaces are estimated to be hierarchically nested to respect the structure of subpopulations with overlapping characteristics. This package is an implementation of the proposed methodology of Huling and Yu (2021) <doi:10.1111/biom.13546>. |
| BugReports: | https://github.com/jaredhuling/hierSDR/issues |
| License: | GPL-2 |
| Encoding: | UTF-8 |
| Depends: | R (≥ 3.2.0), MASS, Matrix, locfit, lbfgs |
| Imports: | numDeriv, optimx |
| RoxygenNote: | 7.1.1 |
| NeedsCompilation: | no |
| Packaged: | 2021-09-23 16:46:15 UTC; huling |
| Author: | Jared Huling [aut, cre] |
| Maintainer: | Jared Huling <jaredhuling@gmail.com> |
| Repository: | CRAN |
| Date/Publication: | 2021-09-23 18:20:02 UTC |
Angle between two subspaces
Description
Measures angle between two subspaces. Smallest value is 0, largest is 90 from http://www4.stat.ncsu.edu/~li/software/GroupDR.R http://lexinli.biostat.berkeley.edu/softwares/dr/GroupDR.R
Usage
angle(B1, B2)
Arguments
B1 |
first matrix |
B2 |
second matrix |
Value
scalar value of the angle between B1 and B2
Examples
## case where any relation between b1 and b2 is random
b1 <- matrix(rnorm(10 * 2), ncol = 2)
b2 <- matrix(rnorm(10 * 2), ncol = 2)
angle(b1, b2)
## angle here should be small
b1 <- matrix(rnorm(10 * 2), ncol = 2)
b2 <- b1 + matrix(rnorm(10 * 2, sd = 0.2), ncol = 2)
angle(b1, b2)
Main hierarchical SDR fitting function
Description
fits hierarchical SDR models
Usage
hier.phd.nt(
x,
y,
z,
z.combinations,
d,
weights = rep(1L, NROW(y)),
constrain.none.subpop = TRUE,
pooled = FALSE,
...
)
Arguments
x |
an n x p matrix of covariates, where each row is an observation and each column is a predictor |
y |
vector of responses of length n |
z |
an n x C matrix of binary indicators, where each column is a binary variable indicating the presence
of a binary variable which acts as a stratifying variable. Each combination of all columns of |
z.combinations |
a matrix of dimensions 2^C x C with each row indicating a different combination of the possible
values in |
d |
an integer vector of length 2^C of structural dimensions. Specified in the same order as the rows in
|
weights |
vector of observation weights |
constrain.none.subpop |
should the "none" subpopulation be constrained to be contained in every other subpopulation's
dimension reduction subspace? Recommended to set to |
pooled |
should the estimator be a pooled estimator? |
... |
not used |
Value
A list with the following elements
beta a list of estimated sufficient dimension reduction matrices, one for each subpopulation
directions a list of estimated sufficient dimension reduction directions (i.e. the reduced dimension predictors/variables), one for each subpopulation. These have number of rows equal to the sample size for the subpopulation and number of columns equal to the specified dimensions of the reduced dimension spaces.
y.list a list of vectors of responses for each subpopulation
z.combinations the
z.combinationsspecified as an inputcov list of variance covariance matrices for the covariates for each subpopulation
sqrt.inv.cov list of inverse square roots of the variance covariance matrices for the covariates for each subpopulation. These are used for scaling
Examples
library(hierSDR)
Main hierarchical sufficient dimension reduction fitting function
Description
fits hierarchically nested sufficient dimension reduction models
Usage
hier.sphd(
x,
y,
z,
z.combinations,
d,
weights = rep(1L, NROW(y)),
maxit = 250L,
tol = 1e-09,
h = NULL,
opt.method = c("lbfgs2", "lbfgs.x", "bfgs.x", "bfgs", "lbfgs", "spg", "ucminf", "CG",
"nlm", "nlminb", "newuoa"),
init.method = c("random", "phd"),
vic = TRUE,
grassmann = TRUE,
nn = NULL,
nn.try = c(0.15, 0.25, 0.5, 0.75, 0.9, 0.95),
n.random = 100L,
optimize.nn = FALSE,
separate.nn = FALSE,
constrain.none.subpop = TRUE,
verbose = TRUE,
degree = 2,
pooled = FALSE,
maxk = 5000,
...
)
Arguments
x |
an n x p matrix of covariates, where each row is an observation and each column is a predictor |
y |
vector of responses of length n |
z |
an n x C matrix of binary indicators, where each column is a binary variable indicating the presence
of a binary variable which acts as a stratifying variable. Each combination of all columns of |
z.combinations |
a matrix of dimensions 2^C x C with each row indicating a different combination of the possible
values in |
d |
an integer vector of length 2^C of structural dimensions. Specified in the same order as the rows in
|
weights |
vector of observation weights |
maxit |
maximum number of iterations for optimization routines |
tol |
convergence tolerance for optimization routines. Defaults to |
h |
bandwidth parameter. By default, a reasonable choice is selected automatically |
opt.method |
optimization method to use. Available choices are
|
init.method |
method for parameter initialization. Either |
vic |
logical value of whether or not to compute the VIC criterion for dimension determination |
grassmann |
logical value of whether or not to enforce parameters to be on the Grassmann manifold |
nn |
nearest neighbor parameter for |
nn.try |
vector of nearest neighbor parameters for |
n.random |
integer number of random initializations for parameters to try |
optimize.nn |
should |
separate.nn |
should each subpopulation have its own |
constrain.none.subpop |
should the "none" subpopulation be constrained to be contained in every other subpopulation's
dimension reduction subspace? Recommended to set to |
verbose |
should results be printed along the way? |
degree |
degree of kernel to use |
pooled |
should the estimator be a pooled estimator? |
maxk |
maxk parameter for |
... |
extra arguments passed to |
Value
A list with the following elements
beta a list of estimated sufficient dimension reduction matrices, one for each subpopulation
beta.init a list of the initial sufficient dimension reduction matrices, one for each subpopulation – do not use, just for the sake of comparisons
directions a list of estimated sufficient dimension reduction directions (i.e. the reduced dimension predictors/variables), one for each subpopulation. These have number of rows equal to the sample size for the subpopulation and number of columns equal to the specified dimensions of the reduced dimension spaces.
y.list a list of vectors of responses for each subpopulation
z.combinations the
z.combinationsspecified as an inputcov list of variance covariance matrices for the covariates for each subpopulation
sqrt.inv.cov list of inverse square roots of the variance covariance matrices for the covariates for each subpopulation. These are used for scaling
solver.obj object returned by the solver/optimization function
value value of the objective function at the solution
value.init value of the objective function at the initial beta (
beta.init) usedvic.est.eqn the average (unpenalized) VIC value across the r different input values. This assesses model fit
vic.eqns the individual (unpenalized) VIC values across the r input values. Not used.
vic the penalized VIC value. This is used for dimension selection, with dimensions chosen by the set of dimensions that minimize this penalized vic value that trades off model complexity and model fit
Examples
library(hierSDR)
set.seed(123)
dat <- simulate_data(nobs = 200, nvars = 6,
x.type = "some_categorical",
sd.y = 1, model = 2)
x <- dat$x ## covariates
z <- dat$z ## factor indicators
y <- dat$y ## response
dat$beta ## true coefficients that generate the subspaces
dat$z.combinations ## what combinations of z represent different subpops
## correct structural dimensions:
dat$d.correct
## fit hier SPHD model:
hiermod <- hier.sphd(x, y, z, dat$z.combinations, d = dat$d.correct,
verbose = FALSE, maxit = 250, maxk = 8200)
## validated inf criterion for choosing dimensions (the smaller the better)
hiermod$vic
cbind(hiermod$beta[[4]], NA, dat$beta[[4]])
## angles between estimated and true subspaces for each population:
mapply(function(x,y) angle(x,y), hiermod$beta, dat$beta)
## projection difference norm between estimated and true subspaces for each population:
mapply(function(x,y) projnorm(x,y), hiermod$beta, dat$beta)
PHD SDR fitting function
Description
fits SDR models (PHD approach)
Usage
phd(x, y, d = 5L)
Arguments
x |
an n x p matrix of covariates, where each row is an observation and each column is a predictor |
y |
vector of responses of length n |
d |
an integer representing the structural dimension |
Value
A list with the following elements
beta.hat estimated sufficient dimension reduction matrix
eta.hat coefficients on the scale of the scaled covariates
cov variance covariance matric for the covariates
sqrt.inv.cov inverse square root of the variance covariance matrix for the covariates. Used for scaling
M matrix from principal Hessian directions
eigenvalues eigenvalues of the M matrix
Plotting hierarchical SDR models
Description
Plots hier.sdr objects
Usage
## S3 method for class 'hier_sdr_fit'
plot(x, ...)
Arguments
x |
fitted object returned by |
... |
not used |
Value
No return value, called for side effects
See Also
hier.sphd for function which fits hierarchical SDR model
Examples
library(hierSDR)
Norm of difference of projections
Description
Measures distance between two subspaces
Usage
projnorm(B1, B2)
Arguments
B1 |
first matrix |
B2 |
second matrix |
Value
scalar value of the projection difference norm between B1 and B2
Examples
b1 <- matrix(rnorm(10 * 2), ncol = 2)
b2 <- matrix(rnorm(10 * 2), ncol = 2)
projnorm(b1, b2)
## angle here should be smalls
b1 <- matrix(rnorm(10 * 2), ncol = 2)
b2 <- b1 + matrix(rnorm(10 * 2, sd = 0.2), ncol = 2)
projnorm(b1, b2)
Semiparametric PHD SDR fitting function
Description
fits semiparametric SDR models (PHD approach)
Usage
semi.phd(
x,
y,
d = 5L,
maxit = 100L,
h = NULL,
opt.method = c("lbfgs.x", "bfgs", "lbfgs2", "bfgs.x", "lbfgs", "spg", "ucminf", "CG",
"nlm", "nlminb", "newuoa"),
nn = 0.95,
init.method = c("random", "phd"),
optimize.nn = FALSE,
verbose = TRUE,
n.samples = 100,
degree = 2,
vic = TRUE,
...
)
Arguments
x |
an n x p matrix of covariates, where each row is an observation and each column is a predictor |
y |
vector of responses of length n |
d |
an integer representing the structural dimension |
maxit |
maximum number of iterations |
h |
bandwidth parameter. By default, a reasonable choice is selected automatically |
opt.method |
optimization method to use. Available choices are
|
nn |
nearest neighbor parameter for |
init.method |
method for parameter initialization. Either |
optimize.nn |
should |
verbose |
should results be printed along the way? |
n.samples |
number of samples for the random initialization method |
degree |
degree of kernel to use |
vic |
logical value of whether or not to compute the VIC criterion for dimension determination |
... |
extra arguments passed to |
Value
A list with the following elements
beta estimated sufficient dimension reduction matrix
beta.init initial sufficient dimension reduction matrix – do not use, just for the sake of comparisons
cov variance covariance matric for the covariates
sqrt.inv.cov inverse square root of the variance covariance matrix for the covariates. Used for scaling
solver.obj object returned by the solver/optimization function
vic the penalized VIC value. This is used for dimension selection, with dimension chosen to minimize this penalized vic value that trades off model complexity and model fit
Simulate data with hierarchical subspaces
Description
Simulates data with hierarchical subspaces. Data are generated with two factors that induce heterogeneity
Usage
simulate_data(
nobs,
nvars,
x.type = c("continuous", "some_categorical"),
sd.y = 1,
rho = 0.5,
model = c("1", "2", "3")
)
Arguments
nobs |
positive integer for the sample size per subpopulation |
nvars |
positive integer for the dimension |
x.type |
variable type for covariates, either |
sd.y |
standard deviation of responsee |
rho |
correlation parameter for AR-1 covariance structure for continuous covariates |
model |
model number used, either "1", "2", or "3", each corresponds to a different outcome model setting |
Value
A list with the following elements
x a matrix of covariates with number of rows equal to the total sample size and columns equal to the number of variables
z a matrix with number of rows equal to the total sample size and columns as dummy variables indicating presence of a stratifying factor
y a vector of all responses
beta a list of the true sufficient dimension reduction matrices, one for each subpopulation
z.combinations all possible combinations of the stratifying factors
zsnr scalar the observed signal-to-noise ratio for the response
d.correct the true dimensions of the dimension reduction spaces
Examples
library(hierSDR)
set.seed(123)
dat <- simulate_data(nobs = 100, nvars = 6,
x.type = "some_categorical",
sd.y = 1, model = 2)
x <- dat$x ## covariates
z <- dat$z ## factor indicators
y <- dat$y ## response
dat$beta ## true coefficients that generate the subspaces
dat$snr ## signal-to-noise ratio
str(x)
str(z)
dat$z.combinations ## what combinations of z represent different subpops
## correct structural dimensions:
dat$d.correct