Predict matrix method for (skew-)symmetric bivariate smooths.

Description

Predict matrix method for (skew-)symmetric bivariate smooths.

Usage

## S3 method for class 'symm.smooth'
Predict.matrix(object, data)

Arguments

Author(s)

Jona Cederbaum, Almond Stoecker

See Also

Predict.matrix and smoothCon for details on constructors.

Phonetics acoustic data (complete)

Description

The data are part of a large study on consonant assimilation, which is the phenomenon that the articulation of two consonants becomes phonetically more alike when they appear subsequently in fluent speech. The data set contains the audio signals of nine different speakers which repeated the same sixteen German target words each five times. The target words are bisyllabic noun-noun compound words which contained the two abutting consonants of interest, s and sh, in either order. Consonant assimilation is accompanied by a complex interplay of language-specific, perceptual and articulatory factors. The aim in the study was to investigate the assimilation of the two consonants as a function of their order (either first s, then sh or vice-versa), syllable stress (stressed or unstressed) and vowel context, i.e. which vowels are immediately adjacent to the target consonants of interest. The vowels are either of the form ia or ai. For more details, see references below.

Format

A data frame with 24830 rows and 11 variables

Details

The variables are as follows:

• subject_long: unique identification number for each speaker.

• word_long: unique identification number for each target word.

• combi_long: number of the repetition of the combination of the corresponding speaker and target word.

• y_vec: the response values for each observation point

• n_long: unique identification number for each curve.

• t: the observations point locations.

• covariate.1: (order of the consonants, reference category first /s/ then /sh/).

• covariate.2: (stress of the final syllable of the first compound, reference category 'stressed').

• covariate.3: (stress of the initial syllable of the second compound, reference category 'stressed').

• covariate.4: (vowel context, reference category ia).

• word_names_long: names of the target words.

References

Pouplier, Marianne and Hoole, Philip (2016): Articulatory and Acoustic Characteristics of German Fricative Clusters, Phonetica, 73(1), 52–78.

Cederbaum, Pouplier, Hoole, Greven (2016): Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data. Statistical Modelling, 16(1), 67-88.

Phonetics acoustic data (subset)

Description

A small subset of the phonetics acoustic data set acoustic with observations from two speakers and two items only. This will not produce meaningful results but can be used as a toy data set when testing the code. The variables are as in the full data set, see acoustic.

Format

A data frame with 656 rows and 11 variables

References

Pouplier, Marianne and Hoole, Philip (2016): Articulatory and Acoustic Characteristics of German Fricative Clusters, Phonetica, 73(1), 52–78.

Cederbaum, Pouplier, Hoole, Greven (2016): Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data. Statistical Modelling, 16(1), 67-88.

Construct symmetry constraint matrix for bivariate symmetric smoothing.

Description

This function can be used to construct a symmetry constraint matrix that imposes a (skew-)symmetry constraint on (cyclic) spline coefficients in symmetric bivariate smoothing problems and is especially designed for constructing objects of the class "symm.smooth", see smooth.construct.symm.smooth.spec.

Usage

make_summation_matrix(F, skew = FALSE, cyclic.degree = 0)

Arguments

Details

Imposing a symmetry constraint to the spline coefficients in order to obtain a reduced coefficient vector is equivalent to right multiplication of the bivariate design matrix with the symmetry constraint matrix obtained with function make_summation_matrix. The penalty matrix of the bivariate smooth needs to be adjusted to the reduced coefficient vector by left and right multiplication with the symmetry constraint matrix. This function is used in the constructor function smooth.construct.symm.smooth.spec.

Value

A basis transformation matrix of dimension F^2 \times G with G<F^2 depending on the specified constraint.

Author(s)

Jona Cederbaum, Almond Stoecker

References

Cederbaum, Scheipl, Greven (2016): Fast symmetric additive covariance smoothing. Submitted on arXiv.

See Also

smooth.construct and smoothCon for details on constructors

Symmetric bivariate smooths constructor

Description

The symm class is a smooth class that is appropriate for symmetric bivariate smooths, e.g. of covariance functions, using tensor-product smooths in a gam formula. A constraint matrix is constructed (see make_summation_matrix) to impose a (skew-)symmetry constraint on the (cyclic) spline coefficients, which considerably reduces the number of coefficients that have to be estimated.

Usage

## S3 method for class 'symm.smooth.spec'
smooth.construct(object, data, knots)

Arguments

Details

By default a symmetric bivariate B-spline smooth g is specified, in the sense that g(s, t) = g(t, s). By setting s(..., bs = "symm", xt = list(skew = TRUE)), a skew-symmetric (or anti-smmetric) smooth with g(s, t) = -g(t, s) can be specified instead. In both cases, the smooth can also be constraint to be cyclic with the property g(s, t) = g(s + c, t) = g(s, t + c) for some fixed constant c via specifying xt = list(cyclic = TRUE). Note that this does not correspond to specifying a tensor-product smooth from cyclic marginal B-splines as given by the cp-smooth. In the cyclic case, it is recommended to explicitly specify the range of the domain of the smooth via the knots argument, as this determines the period and often deviates from the observed range.

The underlying procedure is the following: First, the marginal spline design matrices and the corresponding marginal difference penalties are built. Second, the tensor product of the marginal design matrices is built and the bivariate penalty matrix is set up. Third, the constraint matrix is applied to the tensor product design matrix and to the penalty matrix.

Value

An object of class "symm.smooth". See smooth.construct for the elements it will contain.

Author(s)

Jona Cederbaum, Almond Stoecker

References

Cederbaum, Scheipl, Greven (2016): Fast symmetric additive covariance smoothing. Submitted on arXiv.

See Also

smooth.construct and smoothCon for details on constructors

Examples

require(sparseFLMM)

# (skew-)symmetric smooths ---------------------------------------

# generate random surface 
dat1 <- data.frame(arg1 = 1:50)
dat2 <- expand.grid(arg1 = 1:50, arg2 = 1:50)

Bskew <- Predict.matrix(
  smooth.construct( 
    s(arg1, arg2, bs = "symm", xt = list(skew = TRUE)),
    data = dat2, knots = NULL ),
  data = dat2 )
Bsymm <- Predict.matrix(
  smooth.construct( 
    s(arg1, arg2, bs = "symm", xt = list(skew = FALSE)),
    data = dat2, knots = NULL ),
  data = dat2 )

set.seed(934811)
dat2$yskew <- c(Bskew %*% rnorm(ncol(Bskew)))
dat2$ysymm <- c(Bsymm %*% rnorm(ncol(Bsymm)))

# fit sum of skew-symmetric and symmetric parts with corresponding smooths
modpa <- gam( I(yskew + ysymm) ~ s(arg1, arg2, bs = "symm", xt = list(skew = TRUE)) + 
                s(arg1, arg2, bs = "symm", xt = list(skew = FALSE)), data = dat2)
# predict surfaces
preds <- predict(modpa, type = "terms")
dat1 <- as.list(dat1)
dat1$arg2 <- dat1$arg1
dat1$predskew <- matrix(preds[,1], nrow = length(dat1$arg1))
dat1$predsymm <- matrix(preds[,2], nrow = length(dat1$arg1))

cols <- hcl.colors(12, "RdBu")
opar <- par(mfcol = c(2,2))
# symm part (intercept missing)
with(dat1, image(arg1, arg2, predsymm, asp = 1,
                 main = "Symmetric part of y",
                 col = cols))
with(dat1, image(arg1, arg2, asp = 1, 
                 main = "Fit via symm.smooth",
                 matrix(dat2$ysymm, nrow = length(arg1)), 
                                    col = cols))
# skew-symm part
with(dat1, image(arg1, arg2, predskew, asp = 1,
                 main = "Skew-symmetric part of y",
                 col = cols))
with(dat1, image(arg1, arg2, asp = 1, 
                 main = "Fit via symm.smooth",
                 matrix(dat2$yskew, nrow = length(arg1)), 
                 col = cols))
par(opar)

stopifnot(all.equal(dat1$predskew, - t(dat1$predskew)))
stopifnot(all.equal(dat1$predsymm, t(dat1$predsymm)))




# cyclic (skew-)symmetric splines ---------------------------------------

# fit the above example with cyclic smooths
modpac <- gam( I(yskew + ysymm) ~ s(arg1, arg2, bs = "symm", 
                                   xt = list(skew = TRUE, cyclic = TRUE)) + 
                s(arg1, arg2, bs = "symm", xt = list(skew = FALSE, cyclic = TRUE)),
              knots = list(arg1 = c(1, 50), arg2 = c(1,50)), 
              # specify arg range to specify 'wavelength'! 
              data = dat2)
plot(modpac, asp = 1, se = FALSE, pages = 1)

predsc <- predict(modpac, type = "terms")
dat1$predskewc <- matrix(predsc[,1], nrow = length(dat1$arg1))
dat1$predsymmc <- matrix(predsc[,2], nrow = length(dat1$arg1))

# check cyclic margins
opar <- par(mfrow = c(1,2))
with(dat1, matplot(arg1, predsymmc[, c(1,10, 40)], t = "l",
                   main = "symmetric smooth"))
abline(h = dat1$predsymmc[1, c(1,10, 40)], col = "darkgrey")
abline(v = c(1,50), col = "darkgrey")

with(dat1, matplot(arg1, predskewc[, c(1,10, 40)], t = "l",
                   main = "skew-symmetric smooth"))
abline(h = dat1$predskewc[1, c(1,10, 40)], col = "darkgrey")
abline(v = c(1,50), col = "darkgrey")
par(opar)



# 1D point symmetric B-splines --------------------------------------------

# generate toy data
dat <- data.frame( x = 1:100 )
ps_obj <- with(dat, s(x, bs = "ps"))
B <- Predict.matrix(smooth.construct(ps_obj, dat, NULL), dat)               
set.seed(3904)
dat$y <- B %*% rnorm(ncol(B))
plot(dat, t = "l")

# fit skew-symmetric spline
mod0 <- gam( y ~ s(x, bs = "symm", xt = list(skew = TRUE)), 
             knots = list(x = c(0,100)), # specify x range to determine inversion point 
             dat = dat )
lines(dat$x, predict(mod0), col = "cornflowerblue", lty = "dashed")

# or a symmetric spline to first part only
mod1 <- gam( y ~ s(x, bs = "symm"), 
             knots = list(x=c(0,50)), 
                          dat = dat[1:50, ])
lines(dat[1:50, ]$x, predict(mod1), col = "darkred", lty = "dashed")

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

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:

1. estimation of the smooth mean function (including covariate effects) under independence assumption using splines.

2. 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).

3. 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.

4. 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 and use_simple are both set to FALSE and argument nested is set to TRUE. 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 and use_simple are both set to FALSE.

• 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 to TRUE and argument use_simple is set to FALSE.

• 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, arguments use_RI and use_simple are both set to TRUE.

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 of f_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) If use_famm = TRUE, the list results additionally contains:

  • fpc_famm_hat_X: including

    • intercept: the estimated intercept, which is part of f_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 intercept f_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)