| GGeDS {GeDS} | R Documentation |
Generalized Geometrically Designed Spline regression estimation
Description
GGeDS constructs a Geometrically Designed (univariate) variable knots
spline regression model for the predictor in the context of Generalized
(Non-)Linear Models, referred to as a GeDS model, for a response with a
pre-specified distribution from the Exponential Family.
Usage
GGeDS(
formula,
family = gaussian(),
data,
weights,
beta,
phi = 0.99,
min.intknots,
max.intknots,
q = 2L,
Xextr = NULL,
Yextr = NULL,
show.iters = FALSE,
stoptype = "SR",
higher_order = TRUE
)
Arguments
formula |
a description of the structure of the predictor model to be
fitted, including the dependent and independent variables. See
|
family |
a description of the error distribution and link function to be
used in the model. This can be a character string naming a family function
(e.g. |
data |
an optional data frame, list or environment containing the
variables of the predictor model. In case the variables are not found in
|
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 is |
beta |
numeric parameter in the interval |
phi |
numeric parameter in the interval |
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 (i.e.
|
q |
numeric parameter which allows to fine-tune the stopping rule of stage A of GeDS, by default equal to 2. See details below. |
Xextr |
numeric vector of 2 elements representing the left-most and right-most limits of the interval embedding the observations of the 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. By default equal to |
stoptype |
a character string indicating the type of GeDS stopping rule
to be used. It should be either one of |
higher_order |
a logical that defines whether to compute the higher
order fits (quadratic and cubic) after stage A is run. Default is
|
Details
The GGeDS function extends the GeDS methodology, recently developed
by Kaishev et al. (2016) and implemented in the NGeDS function
for the Normal case, to the more general GNM (GLM) context, allowing for the
response to have any distribution from the Exponential Family. Under the
GeDS-GNM 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 procedure. In stage A, a linear
variable-knot spline is fitted to the data applying iteratively re-weighted
least squares (see IRLSfit 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.
A detailed description of the underlying algorithm can be found in
Dimitrova et al. (2023).
As noted in formula, the argument formula
allows the user to specify predictor 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 GGeDS
only one independent variable is allowed for the spline component and
arbitrary many independent variables for the parametric component of the
predictor. 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
'working' residuals (see IRLSfit) 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 (see
Kaishev et al., 2016, for further details).
The default values of beta are beta = 0.5 if the response is
assumed to be Gaussian, beta = 0.2 if it is Poisson (or Quasipoisson),
while if it is Binomial, Quasibinomial or Gamma beta = 0.1, which
reflect our experience of running GeDS for different underlying functional
dependencies.
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.
Value
A 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.
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: 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: doi:10.1016/j.amc.2022.127493
See Also
NGeDS; GeDS-Class; S3 methods such as
coef.GeDS, deviance.GeDS,
knots.GeDS, print.GeDS and
predict.GeDS; Integrate and Derive;
PPolyRep.
Examples
######################################################################
# Generate a data sample for the response variable Y and the covariate X
# assuming Poisson distributed error and log link function
# See section 4.1 in Dimitrova et al. (2023)
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 <- exp(f_1(X))
#############
## POISSON ##
#############
# Generate Poisson distributed Y according to the mean model
Y <- rpois(N, means)
# Fit a Poisson GeDS regression using GGeDS
(Gmod <- GGeDS(Y ~ f(X), beta = 0.2, phi = 0.995, family = poisson(),
Xextr = c(-2,2)))
# Plot the quadratic and cubic GeDS fits
plot(X,log(Y),xlab = "x", ylab = expression(f[1](x)))
lines(Gmod, n = 3, col = "red")
lines(Gmod, n = 4, col = "blue", lty = 2)
legend("topleft", c("Quadratic", "Cubic"),
col = c("red", "blue"), lty = c(1,2))
# Generate GeDS prediction at X=0, first on the response scale and then on
# the predictor scale
predict(Gmod, n = 3, newdata = data.frame(X = 0))
predict(Gmod, n = 3, newdata = data.frame(X = 0), type = "link")
# Apply some of the other available methods, e.g.
# knots, coefficients and deviance extractions for the
# quadratic GeDS fit
knots(Gmod)
coef(Gmod)
deviance(Gmod)
# the same but for the cubic GeDS fit
knots(Gmod, n = 4)
coef(Gmod, n = 4)
deviance(Gmod, n = 4)
###########
## GAMMA ##
###########
# Generate Gamma distributed Y according to the mean model
Y <- rgamma(N, shape = means, rate = 0.1)
# Fit a Gamma GeDS regression using GGeDS
Gmod <- GGeDS(Y ~ f(X), beta = 0.1, phi = 0.995, family = Gamma(log),
Xextr = c(-2,2))
plot(Gmod, f = function(x) exp(f_1(x))/0.1)
##############
## BINOMIAL ##
##############
# Generate Binomial distributed Y according to the mean model
eta <- f_1(X) - 4
means <- exp(eta)/(1+exp(eta))
Y <- rbinom(N, size = 50, prob = means) / 50
# Fit a Binomial GeDS regression using GGeDS
Gmod <- GGeDS(Y ~ f(X), beta = 0.1, phi = 0.995, family = "binomial",
Xextr = c(-2,2))
plot(Gmod, f = function(x) exp(f_1(x) - 4)/(1 + exp(f_1(x) - 4)))
##########################################
# A real data example
# See Dimitrova et al. (2023), Section 4.2
data("coalMining")
(Gmod2 <- GGeDS(formula = accidents ~ f(years), beta = 0.1, phi = 0.98,
family = poisson(), data = coalMining))
(Gmod3 <- GGeDS(formula = accidents ~ f(years), beta = 0.1, phi = 0.985,
family = poisson(), data = coalMining))
plot(coalMining$years, coalMining$accidents, type = "h", xlab = "Years",
ylab = "Accidents")
lines(Gmod2, tr = exp, n = 4, col = "red")
lines(Gmod3, tr = exp, n = 4, col = "blue", lty = 2)
legend("topright", c("phi = 0.98","phi = 0.985"), col = c("red", "blue"),
lty=c(1, 2))
## Not run:
##########################################
# The same regression in the example of GeDS
# but assuming Gamma and Poisson responses
# See Dimitrova et al. (2023), Section 4.2
data('BaFe2As2')
(Gmod4 <- GGeDS(intensity ~ f(angle), data = BaFe2As2, beta = 0.6, phi = 0.995, q = 3,
family = Gamma(log), stoptype = "RD"))
plot(Gmod4)
(Gmod5 <- GGeDS(intensity ~ f(angle), data = BaFe2As2, beta = 0.1, phi = 0.995, q = 3,
family = poisson(), stoptype = "SR"))
plot(Gmod5)
## End(Not run)
##########################################
# Life tables
# See Dimitrova et al. (2023), Section 4.2
data(EWmortality)
attach(EWmortality)
(M1 <- GGeDS(formula = Deaths ~ f(Age) + offset(log(Exposure)),
family = poisson(), phi = 0.99, beta = 0.1, q = 3,
stoptype = "LR"))
Exposure_init <- Exposure + 0.5 * Deaths
Rate <- Deaths / Exposure_init
(M2 <- GGeDS(formula = Rate ~ f(Age), weights = Exposure_init,
family = quasibinomial(), phi = 0.99, beta = 0.1,
q = 3, stoptype = "LR"))
op <- par(mfrow=c(2,2))
plot(Age, Deaths/Exposure, ylab = expression(mu[x]), xlab = "Age")
lines(M1, n = 3, tr = exp, lwd = 1, col = "red")
plot(Age, Rate, ylab = expression(q[x]), xlab = "Age")
lines(M2, n = 3, tr = quasibinomial()$linkinv, lwd = 1, col = "red")
plot(Age, log(Deaths/Exposure), ylab = expression(log(mu[x])), xlab = "Age")
lines(M1, n = 3, lwd = 1, col = "red")
plot(Age, quasibinomial()$linkfun(Rate), ylab = expression(logit(q[x])), xlab = "Age")
lines(M2, n = 3, lwd = 1, col = "red")
par(op)
#########################################
# bivariate example
set.seed(123)
doublesin <- function(x) {
# Adjusting the output to ensure it's positive
exp(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)
# Calculate lambda for Poisson distribution
lambda <- doublesin(cbind(X,Y))
# Generate Z from Poisson distribution
Z <- rpois(400, lambda)
data <- data.frame(X, Y, Z)
# Fit a Poisson GeDS regression using GGeDS
BivGeDS <- GGeDS(Z ~ f(X,Y), beta = 0.2, phi = 0.995, family = "poisson",
Xextr = c(0, 3), Yextr = c(0, 3))
# MSEs w.r.t data
mean((Z-BivGeDS$Linear$Predicted)^2)
mean((Z-BivGeDS$Quadratic$Predicted)^2)
mean((Z-BivGeDS$Cubic$Predicted)^2)
# MSEs w.r.t true function
f_XY <- apply(cbind(X, Y), 1, function(row) doublesin(matrix(row, ncol = 2)))
mean((f_XY - BivGeDS$Linear$Predicted)^2)
mean((f_XY - BivGeDS$Quadratic$Predicted)^2)
mean((f_XY - BivGeDS$Cubic$Predicted)^2)
# Surface plot of the generating function (doublesin)
plot(BivGeDS, f = doublesin)
# Surface plot of the fitted model
plot(BivGeDS)