gam.style: Modified generalized additive model plots for interpreting QRNN models

Description

Generalized additive model (GAM)-style effects plots provide a graphical means of interpreting relationships between covariates and conditional quantiles predicted by a QRNN. From Plate et al. (2000): The effect of the ith input variable at a particular input point Delta.i.x is the change in f resulting from changing X1 to x1 from b1 (the baseline value [...]) while keeping the other inputs constant. The effects are plotted as short line segments, centered at (x.i, Delta.i.x), where the slope of the segment is given by the partial derivative. Variables that strongly influence the function value have a large total vertical range of effects. Functions without interactions appear as possibly broken straight lines (linear functions) or curves (nonlinear functions). Interactions show up as vertical spread at a particular horizontal location, that is, a vertical scattering of segments. Interactions are present when the effect of a variable depends on the values of other variables.

Usage

gam.style(x, parms, column, baseline=mean(x[,column]),
         epsilon=1e-5, seg.len=0.02, seg.cols="black",
         plot=TRUE, return.results=FALSE, trim=0,
         ...)

Value

A list with elements:

effects: a matrix of covariate effects.
partials: a matrix of covariate partial derivatives.

Arguments

x: matrix with number of rows equal to the number of samples and number of columns equal to the number of covariate variables.
parms: list returned by qrnn.fit.
column: column of x for which effects plots should be returned.
baseline: value of x[,column] to be used as the baseline for calculation of covariate effects; defaults to mean(x[,column]).
epsilon: step-size used in the finite difference calculation of the partial derivatives.
seg.len: length of effects line segments expressed as a fraction of the range of x[,column].
seg.cols: colors of effects line segments.
plot: if TRUE (the default) then an effects plots for the given model is produced.
return.results: if TRUE then values of effects and partial derivatives are returned.
trim: if plot=TRUE and parms is for a model with n.ensemble > 1, value of trim passed to censored.mean.
...: further arguments to be passed to plot.

References

Cannon, A.J. and I.G. McKendry, 2002. A graphical sensitivity analysis for interpreting statistical climate models: Application to Indian monsoon rainfall prediction by artificial neural networks and multiple linear regression models. International Journal of Climatology, 22:1687-1708.

Plate, T., J. Bert, J. Grace, and P. Band, 2000. Visualizing the function computed by a feedforward neural network. Neural Computation, 12(6): 1337-1354.

Examples

Run this code

## YVR precipitation data with seasonal cycle and NCEP/NCAR Reanalysis
## covariates
data(YVRprecip)

y <- YVRprecip$precip
x <- cbind(sin(2*pi*seq_along(y)/365.25),
           cos(2*pi*seq_along(y)/365.25),
           YVRprecip$ncep)

## Fit QRNN, additive QRNN (QADD), and quantile regression (QREG)
## models for the conditional 75th percentile
set.seed(1)
train <- c(TRUE, rep(FALSE, 49))
w.qrnn <- qrnn.fit(x=x[train,], y=y[train,,drop=FALSE],
                   n.hidden=2, tau=0.75, iter.max=500,
                   n.trials=1, lower=0, penalty=0.01)
w.qadd <- qrnn.fit(x=x[train,], y=y[train,,drop=FALSE],
                   n.hidden=ncol(x), tau=0.75, iter.max=250,
                   n.trials=1, lower=0, additive=TRUE)
w.qreg <- qrnn.fit(x=x[train,], y=y[train,,drop=FALSE],
                   tau=0.75, iter.max=100, n.trials=1,
                   lower=0, Th=linear, Th.prime=linear.prime)

## GAM-style plots for slp, sh700, and z500
for (column in 3:5) {
    gam.style(x[train,], parms=w.qrnn, column=column,
              main="QRNN")
    gam.style(x[train,], parms=w.qadd, column=column,
              main="QADD")
    gam.style(x[train,], parms=w.qreg, column=column,
              main="QREG")
}