NGeDS: Geometrically Designed Spline regression estimation

Description

NGeDS constructs a Geometrically Designed variable knots spline regression model referred to as a GeDS model, for a response having a Normal distribution.

Usage

NGeDS(
  formula,
  data,
  weights,
  beta = 0.5,
  phi = 0.99,
  min.intknots = 0,
  max.intknots = 500,
  q = 2,
  Xextr = NULL,
  Yextr = NULL,
  show.iters = FALSE,
  stoptype = "RD",
  higher_order = TRUE,
  intknots_init = NULL,
  fit_init = NULL,
  only_pred = FALSE
)

Value

GeDS-Class object, i.e. a list of items that summarizes the main details of the fitted GeDS regression. See GeDS-Class

for details. Some S3 methods are available in order to make these objects tractable, such as coef, deviance, knots, predict and print as well as S4 methods for lines and plot.

Arguments

formula: a description of the structure of the model to be fitted, including the dependent and independent variables. See formula for details.
data: an optional data frame, list or environment containing the variables of the model. If not found in data, the variables are taken from environment(formula), typically the environment from which NGeDS is called.
weights: an optional vector of `prior weights' to be put on the observations in the fitting process in case the user requires weighted GeDS fitting. It should be NULL or a numeric vector of the same length as the response variable in the argument formula.
beta: numeric parameter in the interval \([0,1]\) tuning the knot placement in stage A of GeDS. See details.
phi: numeric parameter in the interval \([0,1]\) specifying the threshold for the stopping rule (model selector) in stage A of GeDS. See also stoptype and details below.
min.intknots: optional parameter allowing the user to set a minimum number of internal knots required. By default equal to zero.
max.intknots: optional parameter allowing the user to set a maximum number of internal knots to be added by the GeDS estimation algorithm. By default equal to the number of knots for the saturated GeDS model.
q: numeric parameter which allows to fine-tune the stopping rule of stage A of GeDS, by default equal to 2. See details.
Xextr: numeric vector of 2 elements representing the left-most and right-most limits of the interval embedding the observations of the first independent variable. See details.
Yextr: numeric vector of 2 elements representing the left-most and right-most limits of the interval embedding the observations of the second independent variable (if the bivariate GeDS is run). See details.
show.iters: logical variable indicating whether or not to print information at each step.
stoptype: a character string indicating the type of GeDS stopping rule to be used. It should be either one of "SR", "RD" or "LR", partial match allowed. See details.
higher_order: a logical that defines whether to compute the higher order fits (quadratic and cubic) after stage A is run. Default is TRUE.
intknots_init: vector of starting internal knots. Default is NULL.
fit_init: A list containing fitted values pred, along with corresponding intknots and coef, representing the initial fit from which to begin Stage A GeDS iteration (i.e. departing from step 2).
only_pred: logical, if TRUE only predictions are computed.

Details

The NGeDS function implements the GeDS methodology, recently developed by Kaishev et al. (2016) and extended in the GGeDS function for the more general GNM, (GLM) context, allowing for the response to have any distribution from the Exponential Family. Under the GeDS approach the (non-)linear predictor is viewed as a spline with variable knots which are estimated along with the regression coefficients and the order of the spline, using a two stage algorithm. In stage A, a linear variable-knot spline is fitted to the data applying iteratively least squares regression (see lm function). In stage B, a Schoenberg variation diminishing spline approximation to the fit from stage A is constructed, thus simultaneously producing spline fits of order 2, 3 and 4, all of which are included in the output, a GeDS-Class object.

As noted in formula, the argument formula allows the user to specify models with two components, a spline regression (non-parametric) component involving part of the independent variables identified through the function f and an optional parametric component involving the remaining independent variables. For NGeDS one or two independent variables are allowed for the spline component and arbitrary many independent variables for the parametric component. Failure to specify the independent variable for the spline regression component through the function f will return an error. See formula.

Within the argument formula, similarly as in other R functions, it is possible to specify one or more offset variables, i.e. known terms with fixed regression coefficients equal to 1. These terms should be identified via the function offset.

The parameter beta tunes the placement of a new knot in stage A of the algorithm. Once a current second-order spline is fitted to the data the regression residuals are computed and grouped by their sign. A new knot is placed at a location defined by the group for which a certain measure attains its maximum. The latter measure is defined as a weighted linear combination of the range of each group and the mean of the absolute residuals within it. The parameter beta determines the weights in this measure correspondingly as beta and 1 - beta. The higher it is, the more weight is put to the mean of the residuals and the less to the range of their corresponding x-values. The default value of beta is 0.5.

The argument stoptype allows to choose between three alternative stopping rules for the knot selection in stage A of GeDS, the "RD", that stands for Ratio of Deviances, the "SR", that stands for Smoothed Ratio of deviances and the "LR", that stands for Likelihood Ratio. The latter is based on the difference of deviances rather than on their ratio as in the case of "RD" and "SR". Therefore "LR" can be viewed as a log likelihood ratio test performed at each iteration of the knot placement. In each of these cases the corresponding stopping criterion is compared with a threshold value phi (see below).

The argument phi provides a threshold value required for the stopping rule to exit the knot placement in stage A of GeDS. The higher the value of phi, the more knots are added under the "RD" and "SR" stopping rules contrary to the case of the stopping rule "LR" where the lower phi is, more knots are included in the spline regression. Further details for each of the three alternative stopping rules can be found in Dimitrova et al. (2023).

The argument q is an input parameter that allows to fine-tune the stopping rule in stage A. It identifies the number of consecutive iterations over which the deviance should exhibit stable convergence so as the knot placement in stage A is terminated. More precisely, under any of the rules "RD", "SR", or "LR", the deviance at the current iteration is compared to the deviance computed q iterations before, i.e., before selecting the last q knots. Setting a higher q will lead to more knots being added before exiting stage A of GeDS.

References

Kaishev, V.K., Dimitrova, D.S., Haberman, S. and Verrall, R.J. (2016). Geometrically designed, variable knot regression splines. Computational Statistics, 31, 1079--1105.
DOI: tools:::Rd_expr_doi("10.1007/s00180-015-0621-7")

Dimitrova, D. S., Kaishev, V. K., Lattuada, A. and Verrall, R. J. (2023). Geometrically designed variable knot splines in generalized (non-)linear models. Applied Mathematics and Computation, 436.
DOI: tools:::Rd_expr_doi("10.1016/j.amc.2022.127493")

Examples

Run this code


###################################################
# Generate a data sample for the response variable
# Y and the single covariate X
set.seed(123)
N <- 500
f_1 <- function(x) (10*x/(1+100*x^2))*4+4
X <- sort(runif(N, min = -2, max = 2))
# Specify a model for the mean of Y to include only a component
# non-linear in X, defined by the function f_1
means <- f_1(X)
# Add (Normal) noise to the mean of Y
Y <- rnorm(N, means, sd = 0.1)

# Fit a Normal GeDS regression using NGeDS
(Gmod <- NGeDS(Y ~ f(X), beta = 0.6, phi = 0.995, Xextr = c(-2,2)))

# Apply some of the available methods, e.g.
# coefficients, knots and deviance extractions for the
# quadratic GeDS fit
# Note that the first call to the function knots returns
# also the left and right limits of the interval containing
# the data
coef(Gmod, n = 3)
knots(Gmod, n = 3)
knots(Gmod, n = 3, options = "internal")
deviance(Gmod, n = 3)

# Add a covariate, Z, that enters linearly
Z <- runif(N)
Y2 <- Y + 2*Z + 1
# Re-fit the data using NGeDS
(Gmod2 <- NGeDS(Y2 ~ f(X) + Z, beta = 0.6, phi = 0.995, Xextr = c(-2,2)))
coef(Gmod2, n = 3)
coef(Gmod2, onlySpline = FALSE, n = 3)

if (FALSE) {
##########################################
# Real data example
# See Kaishev et al. (2016), section 4.2
data('BaFe2As2')
(Gmod2 <- NGeDS(intensity ~ f(angle), data = BaFe2As2, beta = 0.6, phi = 0.99, q = 3))
plot(Gmod2)
}

#########################################
# bivariate example
# See Dimitrova et al. (2023), section 5

# Generate a data sample for the response variable
# Z and the covariates X and Y assuming Normal noise
set.seed(123)
doublesin <- function(x){
 sin(2*x[,1])*sin(2*x[,2])
}

X <- (round(runif(400, min = 0, max = 3),2))
Y <- (round(runif(400, min = 0, max = 3),2))
Z <- doublesin(cbind(X,Y))
Z <- Z+rnorm(400, 0, sd = 0.1)
# Fit a two dimensional GeDS model using NGeDS
(BivGeDS <- NGeDS(Z ~ f(X, Y), phi = 0.9))

# Extract quadratic coefficients/knots/deviance
coef(BivGeDS, n = 3)
knots(BivGeDS, n = 3)
deviance(BivGeDS, n = 3)

# Surface plot of the generating function (doublesin)
plot(BivGeDS, f = doublesin)
# Surface plot of the fitted model
plot(BivGeDS)

Run the code above in your browser using DataLab