sparseFLMM {sparseFLMM} | R Documentation |
Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data
Description
Estimation of functional linear mixed models (FLMMs) for irregularly or sparsely sampled data based on functional principal component analysis (FPCA). The implemented models are special cases of the general FLMM
Y_i(t_{ij}) = \mu(t_{ij},x_i) + z_i^T U(t_{ij}) + \epsilon_i(t_{ij}), i = 1,...,n, j = 1,...,D_i,
with Y_i(t_{ij})
the value of the response of curve i
at observation point
t_{ij}
, \mu(t_{ij},x_i)
is a mean function, which may depend on covariates
x_i = (x_{i1},\ldots,x_{ip})^T
. z_i
is a covariate vector,
which is multiplied with the vector of functional random effects U(t_{ij})
.
\epsilon_i(t_{ij})
is independent and identically distributed white noise
measurement error with homoscedastic, constant variance. For more details, see references below.
The current implementation can be used to fit four special cases
of the above general FLMM:
a model for independent functional data (e.g. longitudinal data), for which
z_i^T U(t_{ij})
only consists of a smooth curve-specific deviation (smooth error curve)a model for correlated functional data with one functional random intercept (fRI) for one grouping variable in addition to a smooth curve-specific error
a model for correlated functional data with two crossed fRIs for two grouping variables in addition to a smooth curve-specific error
a model for correlated functional data with two nested fRIs for two grouping variables in addition to a smooth curve-specific error.
Usage
sparseFLMM(
curve_info,
use_RI = FALSE,
use_simple = FALSE,
method = "fREML",
use_bam = TRUE,
bs = "ps",
d_grid = 100,
min_grid = 0,
max_grid = 1,
my_grid = NULL,
bf_mean = 8,
bf_covariates = 8,
m_mean = c(2, 3),
covariate = FALSE,
num_covariates,
covariate_form,
interaction,
which_interaction = matrix(NA),
save_model_mean = FALSE,
para_estim_mean = FALSE,
para_estim_mean_nc = 0,
bf_covs,
m_covs,
use_whole = FALSE,
use_tri = FALSE,
use_tri_constr = TRUE,
use_tri_constr_weights = FALSE,
np = TRUE,
mp = TRUE,
use_discrete_cov = FALSE,
para_estim_cov = FALSE,
para_estim_cov_nc = 0,
var_level = 0.95,
N_B = NA,
N_C = NA,
N_E = NA,
use_famm = FALSE,
use_bam_famm = TRUE,
bs_int_famm = list(bs = "ps", k = 8, m = c(2, 3)),
bs_y_famm = list(bs = "ps", k = 8, m = c(2, 3)),
save_model_famm = FALSE,
use_discrete_famm = FALSE,
para_estim_famm = FALSE,
para_estim_famm_nc = 0,
nested = FALSE
)
Arguments
curve_info |
data table in which each row represents a single observation
point.
For models with two crossed functional random intercepts, the data table additionally needs to have columns:
For models with two nested functional random intercepts, the data table additionally needs to have columns: #'
For models with covariates as part of the mean function |
use_RI |
TRUE to specify a model with one functional random intercept
for the first grouping variable ( |
use_simple |
|
method |
estimation method for |
use_bam |
|
bs |
spline basis function type for the estimation of the mean function and
the auto-covariance, see |
d_grid |
pre-specified grid length for equidistant grid on which the mean, the auto-covariance surfaces, the eigenfunctions
and the functional random effects are evaluated. NOTE: the length of the grid can be important for computation time (approx. quadratic influence).
Defaults to |
min_grid |
minimum value of equidistant grid (should approx. correspond to minimum value of time interval). Defaults to |
max_grid |
maximum value of equidistant grid (should approx. correspond to maximum value of time interval). Defaults to |
my_grid |
optional evaluation grid, which can be specified and used instead of |
bf_mean |
basis dimension (number of basis functions) used for the functional intercept
|
bf_covariates |
basis dimension (number of basis functions) used for the functional effects
of covariates in the mean estimation via |
m_mean |
order of the penalty for this term in |
covariate |
|
num_covariates |
number of covariates that are included in the model. NOTE: not number of effects in case interactions of covariates are specified. |
covariate_form |
vector with entries for each covariate that specify the form in which the
respective covariate enters the mean function. Possible forms are |
interaction |
|
which_interaction |
symmetric matrix that specifies which interactions should be considered in case |
save_model_mean |
|
para_estim_mean |
|
para_estim_mean_nc |
number of cores for parallelization of mean estimation (only possible using |
bf_covs |
vector of marginal basis dimensions (number of basis functions) used for covariance estimation via |
m_covs |
list of marginal orders of the penalty for |
use_whole |
|
use_tri |
|
use_tri_constr |
|
use_tri_constr_weights |
|
np |
|
mp |
|
use_discrete_cov |
|
para_estim_cov |
|
para_estim_cov_nc |
number of cores (if |
var_level |
pre-specified level of explained variance used for the choice of the number of the functional principal
components (FPCs). Alternatively, a specific number of FPCs can be specified (see below). Defaults to |
N_B |
number of components for B (fRI for first grouping variable) to keep, overrides |
N_C |
number of components for C (fRI for second grouping variable) to keep, overrides |
N_E |
number of components for E (smooth error) to keep, overrides |
use_famm |
|
use_bam_famm |
|
bs_int_famm |
specification of the estimation of the functional intercept |
bs_y_famm |
specification of the estimation of the covariates effects (as part of the mean function), see |
save_model_famm |
|
use_discrete_famm |
|
para_estim_famm |
|
para_estim_famm_nc |
number of cores (if |
nested |
|
Details
The code can handle irregularly and possibly sparsely sampled
data. Of course, it can also be used to analyze regular grid data,
but as it is especially designed for the irregular case and there may
be a more efficient way to analyze regular grid data.
The mean function is of the form
\mu(t_{ij},x_i) = f_0(t_{ij}) +
\sum_{k=1}^r f_k(t_{ij},x_{ik}),
where f_0(t_{ij})
is a functional
intercept.
Currently implemented are effects of dummy-coded and metric covariates which act as
varying-coefficients of the
form f_k(t_{ij})*x_{ik}
and smooth effects of metric covariates (smooth in t and in the covariate)
of the form f(t_{ij}, x_{ik})
. NOTE: metric covariates should be centered such that the global functional intercept can be interpreted as global mean function and
the effect can be interpreted as difference from the global mean. Interaction effects of dummy-coded
covariates acting as varying coefficients are possible.
The estimation consists of four main steps:
estimation of the smooth mean function (including covariate effects) under independence assumption using splines.
estimation of the smooth auto-covariances of the functional random effects. A fast bivariate symmetric smoother implemented in the smooth class 'symm' can be used to speed up estimation (see below).
eigen decomposition of the estimated auto-covariances, which are evaluated on a pre-specified equidistant grid. This yields estimated eigenvalues and eigenfunctions, which are rescaled to ensure orthonormality with respect to the L2-scalar product.
prediction of the functional principal component weights (scores) yielding predictions for the functional random effects.
The estimation of the mean function and auto-covariance functions is based on package mgcv.
The functional principal component weights (scores) are predicted as best (linear)
unbiased predictors. In addition, this implementation allows to embed the model
in the general framework of functional additive mixed models (FAMM) based on package refund, which allows for the construction of
point-wise confidence bands for covariate effects (in the mean function) conditional on the FPCA.
Note that the estimation as FAMM may be computationally expensive as the model
is re-estimated in a mixed model framework.
The four special cases of the general FLMM (two nested fRIs, two crossed fRIs, one fRI, independent curves) are implemented as follows:
In the special case with two nested fRIs, three random processes B, C, and E are considered, where B is the fRI for the first grouping variable (e. g. patient in a random controlled trial), C denotes the fRI for the second grouping variable (e.g. individual specific effect in the follow-up) and E denotes the smooth error. For this special case, arguments
use_RI
anduse_simple
are both set toFALSE
and argumentnested
is set toTRUE
. Note that this implementation only allows for a simple before/after study design.In the special case with two crossed fRIs, three random processes B, C, and E are considered, where B is the fRI for the first grouping variable (e.g. speakers in the phonetics example below), C denotes the fRI for the second grouping variable (e.g. target words in the phonetics example below) and E denotes the smooth error. For this special case, arguments
use_RI
anduse_simple
are both set toFALSE
.In the special case with only one fRI, only B and E are considered and the number of levels for the second grouping variable is to zero. For this special case, argument
use_RI
is set toTRUE
and argumentuse_simple
is set toFALSE
.The special case with independent curves is internally seen as a special case of the model with one fRI for the first grouping variable, with the number of levels for this grouping variable corresponding to the number of curves. Thus, for each level of the first grouping variable there is one curve. Therefore, for the special case of independent curves, the estimation returns an estimate for the auto-covariance of B (instead of E) and all corresponding results are indicated with
'_B'
, although they correspond to the smooth error. For this special case, argumentsuse_RI
anduse_simple
are both set toTRUE
.
Value
The function returns a list of two elements: time_all
and results
.
time_all
contains the total system.time() for calling function sparseFLMM()
.
results
is a list of all estimates, including:
-
mean_hat
: includes the components of the estimated mean function.-
mean_pred
contains effects of dummy covariates or metric covariates with a linear effect (varying coefficients). -
mean_pred_smooth
contains effects of metric covariates with a smooth effect. -
intercept
is the estimated intercept, which is part off_0(t_{ij})
.
-
For each auto-covariance smoothing alternative X
(use_whole
, use_tri
,
use_tri_constr
, use_tri_constr_weights
):
-
cov_hat_X
: includes-
sigmasq
: the estimated error variance -
sigmasq_int
: the integral of the estimated error variance over the domain -
grid_mat_B/C/E
: the estimated auto-covariance(s) evaluated on the pre-specified grid -
sp
: the smoothing parameter(s) for smoothing the auto-covariance(s) -
time_cov_estim
: the time for the smoothing the auto-covariance(s) only -
time_cov_pred_grid
: the time for evaluating the estimated auto-covariance(s) on the pre-specified grid.
-
-
time_cov_X
: the total system.time() for the auto-covariance estimation -
fpc_hat_X
: including-
phi_B/C/E_hat_grid
: the estimated rescaled eigenfunctions evaluated on the pre-specified grid -
nu_B/C/E_hat
: the estimated rescaled eigenvalues -
N_B/C/E
: the estimated truncation numbers, i.e., number of FPCs which are chosen -
total_var
: the estimated total variance -
var_explained
: the estimated explained variance -
xi_B/C/E_hat
: the predicted FPC weights (scores).
-
-
time_fpc_X
: the total system.time() for the eigen decompositions and prediction on the FPC weights (scores) Ifuse_famm = TRUE
, the listresults
additionally contains:-
fpc_famm_hat_X
: including-
intercept
: the estimated intercept, which is part off_0(t_{ij})
-
residuals
: the residuals of the FAMM estimation -
xi_B/C/E_hat_famm
: the predicted basis weights -
famm_predict_B/C/E
: the predicted functional processes evaluated on the pre-specified grid -
famm_cb_mean
: the re-estimated functional interceptf_0(t_{ij})
-
famm_cb_covariate.1
,famm_cb_covariate.1
, etc: possible re-estimated covariate effects -
famm_cb_inter_1_2
,famm_cb_inter_1_3
, etc: possible interaction effects -
time_fpc_famm_X
: the total system.time() for the FAMM estimation.
-
The unique identification numbers for the levels of the grouping variables and curves are renumbered for convenience during estimation from 1 in ascending order. The original identification numbers are returned in the list
results
:-
n_orig
: curve levels as they entered the estimation -
subject_orig
: levels of the first grouping variable as they entered the estimation -
word_orig
: levels of the second grouping variable (if existent) as they entered the estimation -
my_grid
: pre-specified grid.
-
Author(s)
Jona Cederbaum
References
Cederbaum, Pouplier, Hoole, Greven (2016): Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data. Statistical Modelling, 16(1), 67-88.
Cederbaum, Scheipl, Greven (2016): Fast symmetric additive covariance smoothing. Submitted on arXiv.
Scheipl, F., Staicu, A.-M. and Greven, S. (2015): Functional Additive Mixed Models, Journal of Computational and Graphical Statistics, 24(2), 477-501.
See Also
Note that sparseFLMM
calls bam
or gam
directly.
For functional linear mixed models with complex correlation structures
for data sampled on equal grids based on functional principal component analysis,
see function denseFLMM
in package denseFLMM
.
Examples
## Not run:
# subset of acoustic data (very small subset, no meaningful results can be expected and
# FAMM estimation does not work for this subset example. For FAMM estimation, see below.)
data("acoustic_subset")
acoustic_results <- sparseFLMM(curve_info = acoustic_subset, use_RI = FALSE, use_simple = FALSE,
method = "fREML", use_bam = TRUE, bs = "ps", d_grid = 100, min_grid = 0,
max_grid = 1, my_grid = NULL, bf_mean = 8, bf_covariates = 8, m_mean = c(2,3),
covariate = TRUE, num_covariates = 4, covariate_form = rep("by", 4),
interaction = TRUE,
which_interaction = matrix(c(FALSE, TRUE, TRUE, TRUE, TRUE,
FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, TRUE, FALSE,
FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE),
byrow = TRUE, nrow = 4, ncol = 4),
save_model_mean = FALSE, para_estim_mean = FALSE, para_estim_mean_nc = 0,
bf_covs = c(5, 5, 5), m_covs = list(c(2, 3), c(2, 3), c(2, 3)),
use_whole = FALSE, use_tri = FALSE, use_tri_constr = TRUE,
use_tri_constr_weights = FALSE, np = TRUE, mp = TRUE,
use_discrete_cov = FALSE,
para_estim_cov = FALSE, para_estim_cov_nc = 5,
var_level = 0.95, N_B = NA, N_C = NA, N_E = NA,
use_famm = FALSE, use_bam_famm = TRUE,
bs_int_famm = list(bs = "ps", k = 8, m = c(2, 3)),
bs_y_famm = list(bs = "ps", k = 8, m = c(2, 3)),
save_model_famm = FALSE, use_discrete_famm = FALSE,
para_estim_famm = FALSE, para_estim_famm_nc = 0)
## End(Not run)
## Not run:
# whole data set with estimation in the FAMM framework
data("acoustic")
acoustic_results <- sparseFLMM(curve_info = acoustic, use_RI = FALSE, use_simple = FALSE,
method = "fREML", use_bam = TRUE, bs = "ps", d_grid = 100, min_grid = 0,
max_grid = 1, my_grid = NULL, bf_mean = 8, bf_covariates = 8, m_mean = c(2,3),
covariate = TRUE, num_covariates = 4, covariate_form = rep("by", 4),
interaction = TRUE,
which_interaction = matrix(c(FALSE, TRUE, TRUE, TRUE, TRUE,
FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, TRUE, FALSE,
FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE, FALSE),
byrow = TRUE, nrow = 4, ncol = 4),
save_model_mean = FALSE, para_estim_mean = FALSE, para_estim_mean_nc = 0,
bf_covs = c(5, 5, 5), m_covs = list(c(2, 3), c(2, 3), c(2, 3)),
use_whole = FALSE, use_tri = FALSE, use_tri_constr = TRUE,
use_tri_constr_weights = FALSE, np = TRUE, mp = TRUE,
use_discrete_cov = FALSE,
para_estim_cov = TRUE, para_estim_cov_nc = 5,
var_level = 0.95, N_B = NA, N_C = NA, N_E = NA,
use_famm = TRUE, use_bam_famm = TRUE,
bs_int_famm = list(bs = "ps", k = 8, m = c(2, 3)),
bs_y_famm = list(bs = "ps", k = 8, m = c(2, 3)),
save_model_famm = FALSE, use_discrete_famm = FALSE,
para_estim_famm = TRUE, para_estim_famm_nc = 5)
## End(Not run)