The CRAN Task View on Extreme Value Analysis provides information about R packages that perform various extreme value analyses. The lax package supplements the functionality of nine of these packages, namely eva (Bader and Yan 2020), evd (Stephenson 2002), evir (Pfaff and McNeil 2018), extRemes (Gilleland and Katz 2016), fExtremes (Wuertz, Setz, and Yohan Chalabi 2017), ismev (Heffernan and Stephenson 2018), mev (Belzile et al. 2019), POT (Ribatet and Dutang 2016) and texmex (Southworth, Heffernan, and Metcalfe 2017). Univariate extreme value models, including regression models, are supported.
The chandwich package is used to provide robust sandwich estimation of parameter covariance matrix and loglikelihood adjustment for models fitted by maximum likelihood estimation. This may be useful for cluster correlated data when interest lies in the parameters of the marginal distributions, or for performing inferences that are robust to certain types of model misspecification.
lax works in an object-oriented way, operating on R objects returned from functions in other packages that summarise the fit of an extreme value model. Loglikelihood adjustment and sandwich estimation is performed by an
alogLik S3 method. We demonstrate this using the following running example, which involves fitting a Generalised Extreme Value (GEV) regression model. Firstly, we use the ismev package to produce the fitted GEV regression model object on which
alogLik will operate, and illustrate what can be done with the objects returned from
alogLik. We use the ismev package because it provides an example of a case where we need to refit the model in order to obtain the information that we need to perform adjusted inferences. Then we repeat the adjustment using seven of the other eight packages. The POT package specialises in Generalised Pareto (GP) modelling, for which we use a different example.
This example is based on the analysis presented in Section 5.2 of Chandler and Bate (2007). The data, which are available in the data frame
ow, are a bivariate time series of annual maximum temperatures, recorded in degrees Fahrenheit, at Oxford and Worthing in England, for the period 1901 to 1980. If interest is only in the marginal distributions of high temperatures in Oxford and Worthing, then we might fit a GEV regression model in which some or all of the parameters may vary between Oxford and Worthing. However, we should adjust for the cluster dependence between temperatures recorded during the same year.
gev.fit() function in ismev fits GEV regression models. It allows covariate effects in any of the (location, scale and shape) parameters of the GEV distribution. However, an object returned from
gev.fit() does not provide all the information about a fitted regression model that
alogLik needs, in order to calculate loglikelihood contributions from individual observations: the design matrices are missing. Therefore, lax provides the function
gev_refit, which is a version of
gev.fit that adds this information.
The following code fits a GEV regression model in which the location, scale and shape parameters of the GEV distribution vary between Oxford and Worthing. Then
alogLik is used to provide adjusted standard errors and an adjusted loglikelihood.
library(lax) # Column 4 of ow contains 1 for Oxford and -1 for Worthing <- gev_refit(ow$temp, ow, mul = 4, sigl = 4, shl = 4, show = FALSE, large method = "BFGS") # Adjust the loglikelihood and standard errors <- alogLik(large, cluster = ow$year, cadjust = FALSE) adj_large # MLEs, SEs and adjusted SEs t(summary(adj_large)) #> loc locloc scale scaleloc shape shapeloc #> MLE 81.1700 2.6690 3.7290 0.5312 -0.19890 -0.08835 #> SE 0.3282 0.3282 0.2292 0.2292 0.04937 0.04937 #> adj. SE 0.4036 0.2128 0.2425 0.1911 0.03944 0.03625
This reproduces the values in rows 1, 3 and 4 in Table 2 of Chandler and Bate (2007). The estimation of the ‘meat’ of the sandwich adjustment is performed using the sandwich package. In this example, we need to pass
cadjust = FALSE to
sandwich::meatCL in order that the adjustment is the same as that used in Chandler and Bate (2007). Otherwise,
meatCL makes a finite-cluster bias correction.
confint method calculates approximate (95%) confidence intervals for the parameters, based on the adjusted loglikelihood. Chandler and Bate (2007) consider three types of loglikelihood adjustment: one vertical and two horizontal. The type of adjustment is selected by the argument
type. The default is
type = "vertical" and there is an option to perform no adjustment.
confint(adj_large) #> Waiting for profiling to be done... #> 2.5 % 97.5 % #> loc 80.3729001 81.9602100 #> locloc 2.2437419 3.0831279 #> scale 3.2991896 4.2598984 #> scaleloc 0.1611912 0.9433844 #> shape -0.2741177 -0.1157069 #> shapeloc -0.1652089 -0.0200268 confint(adj_large, type = "none") #> Waiting for profiling to be done... #> 2.5 % 97.5 % #> loc 80.52440863 81.813160870 #> locloc 2.01969229 3.308352990 #> scale 3.32435238 4.236272757 #> scaleloc 0.09100223 1.020139218 #> shape -0.28957175 -0.094326264 #> shapeloc -0.18850187 0.008628523
conf_region function in the chandwich package can be used to produce confidence regions for pairs of parameters. Here, we consider the ‘central’ (midway between Oxford and Worthing) scale and shape intercept parameters: \(\sigma_0\) and \(\xi_0\) in Chandler and Bate (2007).
library(chandwich) <- c("scale", "shape") which_pars <- conf_region(adj_large, which_pars = which_pars, type = "none") gev_none #> Waiting for profiling to be done... <- conf_region(adj_large, which_pars = which_pars) gev_vertical #> Waiting for profiling to be done... plot(gev_none, gev_vertical, lwd = 2, xlim = c(3.1, 4.5), ylim = c(-0.35, -0.05), xlab = expression(sigma), ylab = expression(xi))
alogLik also has an
anova method, which can be used to perform (adjusted) loglikelihood ratio tests of nested models. To illustrate this we fit, and adjust, a smaller model, in which Oxford and Worthing have a common GEV shape parameter, and then compare this model to the larger one.
<- gev_refit(ow$temp, ow, mul = 4, sigl = 4, show = FALSE, small method = "BFGS") <- alogLik(small, cluster = ow$year, cadjust = FALSE) adj_small summary(adj_small) #> MLE SE adj. SE #> loc 81.1200 0.3260 0.40640 #> locloc 2.4540 0.3047 0.20370 #> scale 3.7230 0.2232 0.24020 #> scaleloc 0.3537 0.2096 0.16840 #> shape -0.1845 0.0488 0.04028 anova(adj_large, adj_small) #> Analysis of (Adjusted) Deviance Table #> #> Model.Df Df ALRTS Pr(>ALRTS) #> adj_large 6 #> adj_small 5 1 6.3572 0.01169 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 anova(adj_large, adj_small, type = "none") #> Analysis of (Adjusted) Deviance Table #> #> Model.Df Df ALRTS Pr(>ALRTS) #> adj_large 6 #> adj_small 5 1 3.1876 0.0742 . #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
We see that the adjustment of the loglikelihood for clustering makes enough of a difference to matter: if we perform a test at the 5% significance level then we choose the larger model when we adjust but the smaller model if we do not.
library(texmex, quietly = TRUE) # Note: phi = log(scale) <- evm(temp, ow, gev, mu = ~ loc, phi = ~ loc, xi = ~loc) evm_fit <- alogLik(evm_fit, cluster = ow$year) adj_evm_fit summary(adj_evm_fit) #> MLE SE adj. SE #> mu: (Intercept) 81.17000 0.32820 0.40620 #> mu: loc 2.66800 0.32820 0.21420 #> phi: (Intercept) 1.30600 0.06091 0.06531 #> phi: loc 0.14330 0.06091 0.05106 #> xi: (Intercept) -0.19900 0.04937 0.03968 #> xi: loc -0.08821 0.04937 0.03647
fgev() function in evd fits GEV regression models, but it only allows covariate effects in the location parameter.
library(evd, quietly = TRUE) <- fgev(ow$temp, nsloc = ow[, "loc"]) fgev_fit <- alogLik(fgev_fit, cluster = ow$year) adj_fgev_fit summary(adj_fgev_fit) #> MLE SE adj. SE #> loc 81.1600 0.32980 0.40830 #> loctrend 2.5040 0.31080 0.19910 #> scale 3.7900 0.22810 0.25230 #> shape -0.2097 0.04765 0.04063
library(extRemes, quietly = TRUE) <- fevd(temp, ow, location.fun = ~ ow$loc, scale.fun = ~ ow$loc, fevd_fit shape.fun = ~ ow$loc) <- alogLik(fevd_fit, cluster = ow$year) adj_fevd_fit summary(adj_fevd_fit) #> MLE SE adj. SE #> mu0 81.17000 0.32820 0.40620 #> mu1 2.66800 0.32820 0.21420 #> sigma0 3.72900 0.22930 0.24410 #> sigma1 0.53090 0.22930 0.19230 #> xi0 -0.19890 0.04938 0.03969 #> xi1 -0.08828 0.04938 0.03648
library(eva, quietly = TRUE) <- gevrFit(ow$temp, information = "observed", gevr_fit locvars = ow, locform = ~ ow$loc, scalevars = ow, scaleform = ~ ow$loc, shapevars = ow, shapeform = ~ ow$loc) <- alogLik(gevr_fit, cluster = ow$year) adj_gevr_fit summary(adj_gevr_fit) #> MLE SE adj. SE #> Location (Intercept) 81.1700 0.32810 0.40620 #> Location ow$loc 2.6680 0.32810 0.21420 #> Scale (Intercept) 3.7290 0.22920 0.24400 #> Scale ow$loc 0.5307 0.22920 0.19230 #> Shape (Intercept) -0.1989 0.04936 0.03968 #> Shape ow$loc -0.0882 0.04936 0.03647
gev() function in evir only fits stationary GEV models.
library(evir, quietly = TRUE) <- gev(ow$temp) gev_fit <- alogLik(gev_fit) adj_gev_fit summary(adj_gev_fit) #> MLE SE adj. SE #> xi -0.1801 0.06051 0.0509 #> sigma 4.3980 0.28180 0.2558 #> mu 80.7800 0.39240 0.4128
gevFit() function in fExtremes only fits stationary GEV models.
library(fExtremes, quietly = TRUE) <- gevFit(ow$temp) gevFit_fit <- alogLik(gevFit_fit) adj_gevFit_fit summary(adj_gevFit_fit) #> MLE SE adj. SE #> xi -0.1802 0.06047 0.05087 #> mu 80.7900 0.39240 0.41290 #> beta 4.3980 0.28170 0.25570
fit.gev() function in mev only fits stationary GEV models.
library(mev, quietly = TRUE) <- fit.gev(ow$temp) gfit <- alogLik(gfit) adj_gfit summary(adj_gfit) #> MLE SE adj. SE #> loc 80.7800 0.3924 0.41290 #> scale 4.3980 0.2818 0.25580 #> shape -0.1801 0.0605 0.05089
Among other things, the
fitgpd() function in the POT package can fit a GP distribution to threshold excesses using maximum likelihood estimation. We illustrate
alogLik using an example from the
fitgpd documentation. There is no cluster dependence here. However, there may be interest in using a sandwich estimator of covariance if we are concerned about model misspecification. In this case, where we simulate from the correct model, we expect the adjustment to make little difference, and so it proves.
library(POT, quietly = TRUE) set.seed(24082019) <- POT::rgpd(200, 1, 2, 0.25) x <- fitgpd(x, 1, "mle") fit <- alogLik(fit) adj_fit summary(adj_fit) #> MLE SE adj. SE #> scale 1.8670 0.20930 0.19260 #> shape 0.2573 0.08887 0.07121