Generalised additive model on the flattened representation

One additive model per response, a smooth function of every bin-by-channel column of the representation, fitted as mgcv's gam(y ~ s(x1) + s(x2) + ..., method = "GCV.Cp") fits it and as biomod2 fits GAM. Each column enters as a thin plate regression spline of k basis functions (Wood 2003): a cubic radial function centred on each of the column's distinct values, reduced to its k - 2 directions of greatest eigenvalue, together with the linear function, which the penalty on the spline's squared second derivative leaves free. Each smooth sums to zero over the units, beside one intercept.

Usage

additive(data = NULL, k = 10L, gamma = 1, max_knots = 2000L, threads = 1L)

Arguments

data

A representation the learner is pinned to, or NULL to run across every representation of the run.

k

The basis dimension of a column's smooth, the constant included, at least 3.

gamma

The criterion's charge per effective degree of freedom.

max_knots

The most distinct values of a column the radial functions are centred on.

threads

Columns and responses worked at once. The model is the same on any number.

Value

A learner().

Details

The coefficients maximise the penalised likelihood, by penalised iteratively reweighted least squares, and the smoothing parameters, one per column, minimise the unbiased risk estimator under presence-absence and a count head and the generalised cross-validation score under a squared-error head, by Newton's method with the exact derivatives (Wood 2008). gamma multiplies the charge each effective degree of freedom adds to that criterion, as mgcv's gamma does; above one it gives smoother fits. Over columns as alike as neighbouring weeks the criterion can have more than one local minimum; the one the search settles in then depends on where it starts, and mgcv, starting from a rule of its own parametrisation, can settle in another.

The defaults are mgcv's, which biomod2 passes unchanged: k = 10 and a criterion charge of one. Above max_knots distinct values, the radial functions are centred on that many of them, drawn as mgcv draws them, so that the basis is the one mgcv fits. A column of fewer than k distinct values takes as many basis functions as it holds values, a column of two enters linearly, and a column of one is left out; mgcv refuses the first and the last of those. A column whose linear part the columns before it already span keeps its penalised part and loses the linear one.

The case weights are the response head's, positive_weights() under presence-absence, and enter the likelihood as mgcv's prior weights. The model holds at most as many coefficients as there are units, as mgcv's does: one for the intercept and k - 1 per column. Under the shipped presence-absence head those weights are on, so a default additive() is mgcv's specification fitted under them; a head registered without weights fits it unweighted.

The basis, the fit and the choice of the smoothing parameters live in the core the Python package calls, pinned against mgcv in the fixtures, so the two languages fit the same model. A response holding one value is predicted its mean and named in unfitted; one whose smoothing parameter search stopped short of its tolerance is named in stopped.

Examples

additive()
additive(data = grain("season"), k = 5)