Package 'FGLMtrunc'

Fit a truncated Functional Generalized Linear Model

Description

Fit a truncated functional linear or logistic regression model using nested group lasso penalty. The solution path is computed efficiently using active set algorithm with warm start. Optimal tuning parameters ($\lambda_s, \lambda_t$) are chosen by Bayesian information criterion (BIC).

Usage

fglm_trunc(
  Y,
  X.curves,
  S = NULL,
  grid = NULL,
  family = c("gaussian", "binomial"),
  degree = 3,
  nbasis = NULL,
  knots = NULL,
  nlambda.s = 10,
  lambda.s.seq = NULL,
  precision = 1e-05,
  parallel = FALSE
)

Arguments

Y	n-by-1 vector of response. Each row is an observed scalar response, which is continous for family="gaussian" and binary (i.e. 0 and 1) for family="binomal".
X.curves	n-by-p matrix of functional predictors. Each row is an observation vector at p finite points on [0,T] for some T>0.
S	(optional) n-by-s matrix of scalar predictors. Binary variable should be coded as numeric rather than factor.
grid	A sequence of p points at which X is recorded, including both boundaries 0 and T. If not specified, an equally spaced sequence of length p between 0 and 1 will be used.
family	Choice of exponential family for the model. The function then uses corresponding canonical link function to fit model.
degree	Degree of the piecewise polynomial. Default 3 for cubic splines.
nbasis	Number of B-spline basis. If knots is unspecified, the function choose nbasis - degree - 1 internal knots at suitable quantiles of grid. If knots is specified, the value of nbasis will be ignored.
knots	k internal breakpoints that define that spline.
nlambda.s	(optional) Length of sequence of smoothing regularization parameters. Default 10.
lambda.s.seq	(optional) Sequence of smoothing regularization parameters.
precision	(optional) Error tolerance of the optimization. Default 1e-5.
parallel	(optional) If TRUE, use parallel foreach to fit each value of lambda.s.seq. Must register parallel before hand, such as doMC or others.

Details

Details on spline estimator

For an order q B-splines (q = degree + 1 since an intercept is used) with k internal knots 0 < t_1 <...< t_k < T, the number of B-spline basis equals q + k. Without truncation ($\lambda$_t=0), the function returns smoothing estimate that is equivalent to the method of Cardot and Sarda (2005), and optimal smoothing parameter is chosen by Generalized Cross Validation (GCV).

Details on family

The model can work with Gaussian or Bernoulli responses. If family="gaussian", identity link is used. If family="binomial", logit link is used.

Details on scalar predictors

FGLMtrunc allows using scalar predictors together with functional predictors. If scalar predictors are used, their estimated coefficients are included in alpha form fitted model.

Value

A list with components:

grid	The grid sequence used.
knots	The knots sequence used.
degree	The degree of the piecewise polynomial used.
eta.0	Estimate of B-spline coefficients $\eta$ without truncation penalty.
beta.0	Estimate of functional parameter $\beta$ without truncation penalty.
eta.truncated	Estimate of B-spline coefficients $\eta$ with truncation penalty.
beta.truncated	Estimate of functional parameter $\beta$ with truncation penalty.
lambda.s0	Optimal smoothing regularization parameter without truncation chosen by GCV.
lambda.s	Optimal smoothing regularization parameter with truncation chosen by BIC.
lambda.t	Optimal truncation regularization parameter chosen by BIC.
trunc.point	Truncation point $\delta$ where $\beta(t)$ = 0 for $t \ge \delta$.
alpha	Intercept (and coefficients of scalar predictors if used) of truncated model.
scalar.pred	Logical variable indicating whether any scalar predictor was used.
call	Function call of fitted model.
family	Choice of exponential family used.

References

Xi Liu, Afshin A. Divani, and Alexander Petersen. "Truncated estimation in functional generalized linear regression models" (2022). Computational Statistics & Data Analysis.

Hervé Cardot and Pacal Sarda. "Estimation in generalized linear models for functional data via penalized likelihood" (2005). Journal of Multivariate Analysis.

See Also

bSpline from splines2 R package for usage of B-spline basis.

Examples

# Gaussian response
data(LinearExample)
Y_linear = LinearExample$Y
Xcurves_linear = LinearExample$X.curves
fit1 = fglm_trunc(Y_linear, Xcurves_linear, nbasis = 20, nlambda.s = 1)
print(fit1)
plot(fit1)

---

Simulated data for functional linear regression.

Description

Randomly generated data with Gaussian responses for functional linear regression example follows Case I from Liu et. al. (2022).

Usage

data(LinearExample)

Format

List containing the following elements:

X.curves

200 by 101 matrix of functional predictors.

Y

200 by 1 numeric vector of Gaussian responses.

beta.true

The true functional parameter $\beta$.

References

Xi Liu, Afshin A. Divani, and Alexander Petersen. "Truncated estimation in functional generalized linear regression models" (2022). Computational Statistics & Data Analysis.

---

Simulated data for functional logistic regression.

Description

Randomly generated data with Bernoulli responses for functional logistic regression example follows Case I from Liu et. al. (2022).

Usage

data(LogisticExample)

Format

List containing the following elements:

X.curves

200 by 101 matrix of functional predictors.

Y

200 by 1 numeric vector of Bernoulli responses.

beta.true

The true functional parameter $\beta$.

References

Xi Liu, Afshin A. Divani, and Alexander Petersen. "Truncated estimation in functional generalized linear regression models" (2022). Computational Statistics & Data Analysis.

---

Plot functional parameters $\beta$ from a FGLMtrunc object

Description

Plot functional parameters $\beta$ as a function of $t$ for a fitted FGLMtrunc object.

Usage

## S3 method for class 'FGLMtrunc'
plot(x, include_smooth = TRUE, ...)

Arguments

x	fitted FGLMtrunc object
include_smooth	If TRUE, smoothing estimate without truncation of $\beta$ is plotted.
...	additional plot arguments

Value

No return value.

---

Make predictions from FGLMtrunc fitted model

Description

This function returns truncated estimate of linear predictors, fitted values, and functional parameter $\beta$ for a fitted FGLMtrunc object.

Usage

## S3 method for class 'FGLMtrunc'
predict(
  object,
  newX.curves,
  newS = NULL,
  type = c("link", "response", "coefficients"),
  ...
)

Arguments

object	fitted FGLMtrunc object
newX.curves	Matrix of new values for functional predictors X.curves.
newS	Matrix of new values for scalar predictors S.
type	Type of prediction. For logistic regression (family = "binomial"), type="link" gives the linear predictors, which is log-odds, and type="response" gives the predicted probabilities. For linear regression (family = "gaussian"), both type="link" and type="response" give fitted values. For both linear regression and logistic regression, type="coefficients" gives truncated estimate of functional parameter $\beta$.
...	additional predict arguments (Not applicable for FGLMtrunc)

Value

Predictions depends on chosen type.

See Also

predict.glm.

---

Print a FGLMtrunc object

Description

Print a summary of truncation point of the fitted FGLMtrunc model.

Usage

## S3 method for class 'FGLMtrunc'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x	fitted FGLMtrunc object
digits	significant digits in printout
...	additional print arguments

Details

Truncation point estimate of $\delta$ is printed.

Value

The fitted object is silently return.

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