GeDS (version 0.1.2)

GGeDS: 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, data, family = gaussian(), weights, beta, phi = 0.99,
  min.intknots, max.intknots, q = 2L, Xextr = NULL, show.iters = FALSE,
  stoptype = "SR")

Arguments

formula

a description of the structure of the predictor 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 predictor model. In case the variables are not found in data, they are taken from environment(formula), typically the environment from which GGeDS is called.

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. "gaussian"), the family function itself (e.g. gaussian) or the result of a call to a family function (e.g. gaussian()). See family for details on family functions.

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 NULL by default.

beta

numeric parameter in the interval \([0,1]\) tuning the knot placement in stage A of GeDS. See details below.

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 (i.e. \(N-2\), where \(N\) is the number of observations).

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.

show.iters

logical variable indicating whether or not to print information at each step. By default equal to FALSE.

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 below.

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.

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. (2017).

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. (2017).

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: doi.org/10.1007/s00180-015-0621-7

Dimitrova, D.S., Kaishev, V.K., Lattuada A. and Verrall, R.J. (2017). Geometrically designed, variable knot splines in Generalized (Non-)Linear Models. Available at openaccess.city.ac.uk

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

# NOT RUN {
######################################################################
# 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. (2017)
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))
# 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)


##########################################
# A real data example
# See Dimitrova et al. (2017), 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. (2017), 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)
# }
# NOT RUN {
##########################################
# Life tables
# See Dimitrova et al. (2017), 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)

# }