plbpsm_models: Specifying generalized geospatial additive models

In funstatpackages/GgAM: Generalized geoAdditive Models

Description

This page is intended to provide some more information on how to specify GGAMs. A GGAM is a GLM in which the linear predictor depends, in part, on a sum of smooth functions of predictors and (possibly) linear functionals of smooth functions of (possibly dummy) predictors. Specifically let y_i denote an independent random variable with mean mu_i and an exponential family distribution, or failing that a known mean variance relationship suitable for use of quasi-likelihood methods. Then the the linear predictor of a GGAM has a structure something like

g(mu_i)=X_i b + f_1(x_1i,x_2i) + f_2(x_3i) + L_i f_3(x_4) + ...

where g is a known smooth monotonic ‘link’ function, X_i b is the parametric part of the linear predictor, the x_j are predictor variables, the u_j and f_j are smooth functions. There may of course be multiple linear functional terms, or none.

The key idea here is that the dependence of the response on the predictors can be represented as a parametric sub-model plus the sum of some (functionals of) smooth functions of one or more of the predictor variables. Thus the model is quite flexible relative to strictly parametric linear or generalized linear models, but still has much more structure than the completely general model that says that the response is just some smooth function of all the covariates.

The following sections discuss specifying model structures for gam. Specification of the distribution and link function is done using the family argument to plbpsm and works in the same way as for glm. This page therefore concentrates on the model formula for plbpsm.

Value

A plbpsm object has the following elements:

aic	AIC of the fitted model: bear in mind that the degrees of freedom used to calculate this are the effective degrees of freedom of the model, and the likelihood is evaluated at the maximum of the penalized likelihood in most cases, not at the MLE.
assign	Array whose elements indicate which model term (listed in pterms) each parameter relates to: applies only to non-smooth terms.
call	the matched call (allows update to be used with plbpsm objects, for example).
cmX	column means of the model matrix (with elements corresponding to smooths set to zero ) — useful for componentwise CI calculation.
coefficients_linear	the linear (parametric) coefficients of the fitted model.
coefficients_nonlinear	the nonlinear (bivariate spline) coefficients of the fitted model.
data	the original supplied data argument (for class "gam" compatibility).
deviance	model deviance (not penalized deviance).
df.null	null degrees of freedom.
df.residual	effective residual degrees of freedom of the model.
edf	estimated degrees of freedom.
family	family object specifying distribution and link used.
fitted.values	fitted model predictions of expected value for each datum.
formula	the model formula.
gcv_opt	The minimized smoothing penalty parameter selection score: GCV.
cv_opt	The minimized smoothing penalty parameter selection score: CV.
linear.predictors	fitted model prediction of link function of expected value for each datum.
criterion	One of "GCV" or CV, depending on the fitting criterion used.
min.edf	Minimum possible degrees of freedom for whole model.
model	model frame containing all variables needed in original model fit.
na.action	The na.action used in fitting.
nsdf	number of parametric, non-smooth, model terms including the intercept.
null.deviance	deviance for single parameter model.
method	One of ALASSO, SCAD, depending on the variable selection method.
pred.formula	one sided formula containing variables needed for prediction, used by predict.plbpsm
pterms	terms object for strictly parametric part of model.
rank	apparent rank of fitted model.
residuals	the working residuals for the fitted model.
scale	defualt=1.
se_beta	estimated standard error for parametric covariates.
sse	Sum of squared error of the estimation.
basis_info	list of smooth objects, containing the basis information for each term in the model formula in the order in which they appear. These smooth objects are what gets returned by the BasisCon objects.
terms	terms object of model model frame.
var.summary	A named list of summary information on the predictor variables
Ve	estimated covariance matrix for the parameter estimators. Particularly useful for testing whether terms are zero. Not so useful for CI's as smooths are usually biased.
weights	final weights used in grplsfit.
y	response data.

Models with simple smooth terms

Consider the example model.

g(mu_i) = b_0 + b_1 x_1i + b_2 x_2i + u(x_3i) + f(x_4i,x_5i)

where the response variables y_i has expectation mu_i and g is a link function.

The plbpsm formula for this would be
y ~ x1 + x2 + u(x3) + f(x4,x5).
This would use the default basis for the smooths (a thin plate regression spline basis for each), with default choices of degree of polynomials for both smooths. Full details of how to control smooths are given in s and u.

The above assumes that x_4 and x_5 are naturally on similar scales (e.g. they might be co-ordinates), so that isotropic smoothing is appropriate.

funstatpackages/GgAM documentation built on Nov. 4, 2019, 12:59 p.m.