Functions to compute and plot summary statistics of prediction errors to (cross-)validate fitted spatial linear models by the criteria proposed by Gneiting et al. (2007) for assessing probabilistic forecasts.
validate.predictions(data, pred, se.pred,
statistic = c("crps", "pit", "mc", "bs", "st"), ncutoff = NULL)
# S3 method for cv.georob
plot(x,
type = c("sc", "lgn.sc", "ta", "qq", "hist.pit", "ecdf.pit", "mc", "bs"),
smooth = TRUE, span = 2/3, ncutoff = NULL, add = FALSE,
col, pch, lty, main, xlab, ylab, ...)
# S3 method for cv.georob
print(x, digits = max(3, getOption("digits") - 3), ...)
# S3 method for cv.georob
summary(object, se = FALSE, ...)a numeric vector with observations about a response (mandatory argument).
a numeric vector with predictions for the response (mandatory argument).
a numeric vector with prediction standard errors (mandatory argument).
character keyword defining what statistic of the prediction errors should be computed. Possible values are (see Details):
"crps": continuous ranked probability score (default),
"pit": probability integral transform,
"mc": average predictive distribution (marginal
calibration),
"bs": Brier score,
"st": mean and dispersion statistics of (standardized)
prediction errors.
positive integer (\(N\)) giving the number of quantiles,
for which CDFs are evaluated (type = "mc"), or the number of
thresholds for which the Brier score is computed (type = "bs"),
see Details (default: min(500, length(data))).
objects of class cv.georob.
positive integer indicating the number of decimal digits to print.
character keyword defining what type of plot is created by the
plot.cv.georob. Possible values are:
"sc": a scatter-plot of the (possibly log-transformed) response
vs. the respective predictions (default).
"lgn.sc": a scatter-plot of the untransformed response
against back- transformed predictions of the log-transformed response.
"ta": Tukey-Anscombe plot (plot of standardized prediction
errors vs. predictions).
"qq": normal QQ plot of standardized prediction errors.
"hist.pit": histogram of probability integral transform, see
Details.
"ecdf.pit": empirical CDF of probability integral
transform, see Details.
"mc": a marginal calibration plot, see Details,
"bs": a plot of Brier score vs. threshold, see
Details.
control whether scatter plots of data vs. predictions
should be smoothed by loess.smooth.
smoothness parameter for loess (see loess.smooth).
logical controlling whether the current high-level plot is
added to an existing graphics without cleaning the frame before (default:
FALSE).
title and axes labels of plot.
color, symbol and line type.
logical controlling if the standard errors of the averaged
continuous ranked probability score and of the mean and dispersion
statistics of the prediction errors (see Details) are computed
from the respective values computed for the \(K\) cross-validation
subsets (default: FALSE).
additional arguments passed to the methods.
Depending on the argument statistic, the function
validate.predictions returns
a numeric vector of PIT values if statistic is equal to "pit",
a named numeric vector with summary statistics of the
(standardized) prediction errors if statistic is equal to "st". The
following statistics are computed:
me |
mean prediction error |
mede |
median prediction error |
rmse |
root mean squared prediction error |
made |
median absolute prediction error |
qne |
Qn dispersion measure of prediction errors
(see Qn) |
msse |
mean squared standardized prediction error |
medsse |
median squared standardized prediction error |
a data frame if statistic is equal to "mc" or
"bs" with the components (see Details):
z |
the sorted unique (cross-)validation
observations (or a subset of size
ncutoff of them) |
ghat |
the empirical CDF of the (cross-)validation observations \(\widehat{G}_n(y)\) |
fbar |
the average predictive distribution \(\bar{F}_n(y)\) |
bs |
the Brier score \(\mathrm{BS}(y)\) |
validate.predictions computes the items required to evaluate (and
plot) the diagnostic criteria proposed by Gneiting et al. (2007) for
assessing the calibration and the sharpness of
probabilistic predictions of (cross-)validation data. To this aim,
validate.predictions uses the assumption that the prediction
errors
\(Y(\mbox{\boldmath$s$\unboldmath})-\widehat{Y}(\mbox{\boldmath$s$\unboldmath})\)
follow normal distributions with zero mean and standard deviations equal
to the Kriging standard errors. This assumption is an approximation if
the errors \(\varepsilon\) come from a long-tailed
distribution. Furthermore, for the time being, the Kriging variance of
the response \(Y\) is approximated by adding the estimated
nugget \(\widehat{\tau}^2\) to the Kriging variance of the
signal \(Z\). This approximation likely underestimates the mean
squared prediction error of the response if the errors come from a
long-tailed distribution. Hence, for robust Kriging, the standard errors of
the (cross-)validation errors are likely too small.
Notwithstanding these difficulties and imperfections, validate.predictions computes
the probability integral transform (PIT),
$$\mathrm{PIT}_i = F_i(y_i),$$
where \(F_i(y_i)\) denotes the (plug-in) predictive CDF evaluated at \(y_i\), the value of the \(i\)th (cross-)validation datum,
the average predictive CDF (plug-in)
$$\bar{F}_n(y)=1/n \sum_{i=1}^n F_i(y),$$
where \(n\) is the number of (cross-)validation observations and the \(F_i\) are evaluated at \(N\) quantiles equal to the set of distinct \(y_i\) (or a subset of size \(N\) of them),
the Brier Score (plug-in)
$$\mathrm{BS}(y) = 1/n \sum_{i=1}^n \left(F_i(y) - I(y_i \leq y) \right)^2,$$
where \(I(x)\) is the indicator function for the event \(x\), and the Brier score is again evaluated at the unique values of the (cross-)validation observations (or a subset of size \(N\) of them),
the averaged continuous ranked probability score, CRPS, a strictly proper scoring criterion to rank predictions, which is related to the Brier score by
$$\mathrm{CRPS} = \int_{-\infty}^\infty \mathrm{BS}(y) \,dy.$$
Gneiting et al. (2007) proposed the following plots to validate probabilistic predictions:
A histogram (or a plot of the empirical CDF) of the PIT values. For ideal predictions, with observed coverages of prediction intervals matching nominal coverages, the PIT values have an uniform distribution.
Plots of \(\bar{F}_n(y)\) and of the empirical CDF of the data, say \(\widehat{G}_n(y)\), and of their difference, \(\bar{F}_n(y)-\widehat{G}_n(y)\) vs \(y\). The forecasts are said to be marginally calibrated if \(\bar{F}_n(y)\) and \(\widehat{G}_n(y)\) match.
A plot of \(\mathrm{BS}(y)\) vs. \(y\). Probabilistic predictions are said to be sharp if the area under this curve, which equals CRPS, is minimized.
The plot method for class cv.georob allows to create
these plots, along with scatter-plots of observations and predictions,
Tukey-Anscombe and normal QQ plots of the standardized prediction
errors.
summary.cv.georob computes the mean and dispersion statistics
of the (standardized) prediction errors (by a call to
validate.prediction with argument statistic = "st", see
Value) and the averaged continuous ranked probability score
(crps). If present in the cv.georob object, the error
statistics are also computed for the errors of the unbiasedly
back-transformed predictions of a log-transformed response. If se
is TRUE then these statistics are evaluated separately for the
\(K\) cross-validation subsets and the standard errors of the means of
these statistics are returned as well.
The print method for class cv.georob returns the mean
and dispersion statistics of the (standardized) prediction errors.
Gneiting, T., Balabdaoui, F. and Raftery, A. E.(2007) Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society Series B 69, 243--268.
georob for (robust) fitting of spatial linear models;
cv.georob for assessing the goodness of a fit by georob.
# NOT RUN {
data(meuse)
r.logzn <- georob(log(zinc) ~ sqrt(dist), data = meuse, locations = ~ x + y,
variogram.model = "RMexp",
param = c(variance = 0.15, nugget = 0.05, scale = 200),
tuning.psi = 1)
r.logzn.cv.1 <- cv(r.logzn, seed = 1, lgn = TRUE)
r.logzn.cv.2 <- cv(r.logzn, formula = .~. + ffreq, seed = 1, lgn = TRUE)
summary(r.logzn.cv.1, se = TRUE)
summary(r.logzn.cv.2, se = TRUE)
op <- par(mfrow = c(2,2))
plot(r.logzn.cv.1, type = "lgn.sc")
plot(r.logzn.cv.2, type = "lgn.sc", add = TRUE, col = "red")
abline(0, 1, lty= "dotted")
plot(r.logzn.cv.1, type = "ta")
plot(r.logzn.cv.2, type = "ta", add = TRUE, col = "red")
abline(h=0, lty= "dotted")
plot(r.logzn.cv.2, type = "mc", add = TRUE, col = "red")
plot(r.logzn.cv.1, type = "bs")
plot(r.logzn.cv.2, type = "bs", add = TRUE, col = "red")
legend("topright", lty = 1, col = c("black", "red"), bty = "n",
legend = c("log(Zn) ~ sqrt(dist)", "log(Zn) ~ sqrt(dist) + ffreq"))
par(op)
# }
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