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Estimation of functional regression models for partially observed functional data

po_fit.Rd

The po_fit function fits functional regression models for partially observed functional data, where each curve is only observed over part of the common domain.

Usage

po_fit(formula, data, family = stats::gaussian(), offset = NULL)

Arguments

formula

a formula object with at least one ffpo term.

data

a list object containing the response variable and the covariates as the components of the list.

family

a family object specifying the distribution from which the data originates. The default distribution is gaussian.

offset

an offset vector. The default value is NULL.

Value

An object of class po_fit. It is a list containing the following items:

  • An item named fit of class sop. See sop.fit.

  • An item named Beta which is a list with one data.frame per functional term, each containing the grid (t), the estimated functional coefficient (beta), its standard error (se) and the lower and upper limits of the pointwise confidence interval (lower, upper).

  • An item named intercept which is the estimated intercept of the model.

  • An item named theta which is the basis coefficient vector of the estimated functional coefficient.

  • An item named covar_theta which is the covariance matrix of the basis coefficients, used to build the pointwise confidence intervals.

  • An item named M which holds the observed domain information for each functional term.

  • An item named ffpo_evals which is the result of the evaluations of the ffpo terms in the formula.

See also

ffpo

Examples

# PARTIALLY OBSERVED FUNCTIONAL DATA EXAMPLE

# set seed for reproducibility
set.seed(123)

# generate example data with partially observed curves
sim  <- data_generator_po_1d(n = 50, grid_points = 60)
X    <- sim$noisy_curves_miss
y    <- sim$response
grid <- sim$grid
mp   <- sim$missing_points

# Fit the model using an 'ffpo' term for the partially observed covariate.
# 'nbasis' sets the number of basis functions for the data reconstruction
# and for the functional coefficient, respectively.
fit  <- po_fit(
  response ~ ffpo(X = X, missing_points = mp, grid = grid, nbasis = c(20, 20)),
  data = list(response = y, X = X, grid = grid, missing_points = mp)
)

# Inspect the structure of the returned object
str(fit, max.level = 1)
#> List of 7
#>  $ fit        :List of 15
#>   ..- attr(*, "class")= chr "sop"
#>  $ Beta       :List of 1
#>  $ intercept  : num 0.123
#>  $ theta      : num [1:20, 1] -0.03136 -0.01256 0.00602 0.02258 0.03373 ...
#>  $ covar_theta: num [1:20, 1:20] 8.72e-04 4.35e-04 1.23e-04 -1.74e-05 -3.27e-05 ...
#>  $ M          :List of 1
#>  $ ffpo_evals :List of 1
#>  - attr(*, "class")= chr "po_fit"
#>  - attr(*, "N")= int 50

# The estimated intercept and the basis coefficients can be accessed directly
fit$intercept
#> [1] 0.1229534
fit$theta
#>               [,1]
#>  [1,] -0.031363841
#>  [2,] -0.012559819
#>  [3,]  0.006023635
#>  [4,]  0.022584795
#>  [5,]  0.033734844
#>  [6,]  0.036421863
#>  [7,]  0.033038168
#>  [8,]  0.026912137
#>  [9,]  0.015454328
#> [10,] -0.002773189
#> [11,] -0.023712627
#> [12,] -0.042205336
#> [13,] -0.057212897
#> [14,] -0.068095604
#> [15,] -0.073799895
#> [16,] -0.072385012
#> [17,] -0.062012250
#> [18,] -0.044650855
#> [19,] -0.025073615
#> [20,] -0.005355858

# A summary of the underlying fit can be obtained using the summary function
summary(fit)
#> 
#> Family: gaussian 
#> Link function: identity 
#> 
#> 
#> Formula:
#> NULL
#> 
#> 
#> Fixed terms: 
#> [1] 0.1229534 0.1145223 0.0503246
#> 
#> 
#> Estimated degrees of freedom:
#> Total edf     Total      <NA> 
#>    3.6034    3.6034    6.6034 
#> 
#> R-sq.(adj) =  0.732   Deviance explained = 81.9%  n = 50
#> 
#> Number of iterations: 1