Title: | Tools for Analyzing Climate Extremes |
---|---|
Description: | Functions for fitting GEV and POT (via point process fitting) models for extremes in climate data, providing return values, return probabilities, and return periods for stationary and nonstationary models. Also provides differences in return values and differences in log return probabilities for contrasts of covariate values. Functions for estimating risk ratios for event attribution analyses, including uncertainty. Under the hood, many of the functions use functions from 'extRemes', including for fitting the statistical models. Details are given in Paciorek, Stone, and Wehner (2018) <doi:10.1016/j.wace.2018.01.002>. |
Authors: | Christopher Paciorek [aut, cre], Harinarayan Krishnan [ctb] |
Maintainer: | Christopher Paciorek <[email protected]> |
License: | BSD_3_clause + file LICENSE |
Version: | 0.3.1 |
Built: | 2025-03-08 03:14:59 UTC |
Source: | https://bitbucket.org/lbl-cascade/climextremes-dev |
Calculates log return period given return value(s) of interest, using model fit from extRemes::fevd
. Standard error is obtained via the delta method. The return period is the average number of blocks expected to occur before the return value is exceeded and is equal to the inverse of the probability of exceeding the return value in a single block. For non-stationary models (those that include covariates for the location, scale, and/or shape parameters, log return periods and standard errors are returned for as many sets of covariates as provided.
calc_logReturnPeriod_fevd(fit, returnValue, covariates = NULL, upper = FALSE)
calc_logReturnPeriod_fevd(fit, returnValue, covariates = NULL, upper = FALSE)
fit |
fitted object from extRemes |
returnValue |
value(s) for which return period is desired |
covariates |
matrix of covariate values, each row a set of covariates for which the log return period is desired |
upper |
logical value indicating whether upper tail or lower tail is being considered |
Results are calculated (and returned) on log scale as delta-method based standard errors are more accurate for the log period. Confidence intervals on the return period scale should be calculated by calculating a confidence interval for the log return period and exponentiating the endpoints of the interval.
Calculates log return probability given return value(s) of interest, using model fit from extRemes::fevd
. Standard error is obtained via the delta method. The return probability is the probability of exceeding the return value in a single block. For non-stationary models (those that include covariates for the location, scale, and/or shape parameters, log probabilities and standard errors are returned for as many sets of covariates as provided.
calc_logReturnProb_fevd( fit, returnValue, covariates = NULL, getSE = TRUE, scaling = 1, upper = FALSE )
calc_logReturnProb_fevd( fit, returnValue, covariates = NULL, getSE = TRUE, scaling = 1, upper = FALSE )
fit |
fitted object from extRemes |
returnValue |
value(s) for which log return probability is desired |
covariates |
matrix of covariate values, each row a set of covariates for which the return probability is desired |
getSE |
logical indicating whether standard error is desired, in addition to the point estimate |
scaling |
if |
upper |
logical value indicating whether upper tail or lower tail is being considered |
Results are calculated (and returned) on log scale as delta-method based standard errors are more accurate for the log probability. Confidence intervals on the probability scale should be calculated by calculating a confidence interval for the log probability and exponentiating the endpoints of the interval.
Calculates difference in log return probabilities for two sets of covariates given return value(s) of interest, using model fit from extRemes::fevd
. Standard error is obtained via the delta method. The return probability is the probability of exceeding the return value in a single block. Differences and standard errors are returned for as many contrasts between covariate sets as provided.
calc_logReturnProbDiff_fevd( fit, returnValue, covariates1, covariates2, getSE = TRUE, scaling = 1, upper = FALSE )
calc_logReturnProbDiff_fevd( fit, returnValue, covariates1, covariates2, getSE = TRUE, scaling = 1, upper = FALSE )
fit |
fitted object from extRemes |
returnValue |
value(s) for which the log return probability difference is desired |
covariates1 |
matrix of covariate values, each row a set of covariates for which the log return probability difference relative to the corresponding row of |
covariates2 |
matrix of covariate values, each row a set of covariates |
getSE |
logical indicating whether standard error is desired, in addition to the point estimate |
scaling |
if |
upper |
logical value indicating whether upper tail or lower tail is being considered |
Results are calculated (and returned) on log scale as delta-method based standard errors are more accurate for the log probability. Confidence intervals for the ratio of return probabilities should be calculated by calculating a confidence interval for the log probability difference and exponentiating the endpoints of the interval.
This is designed to calculate differences in log return probabilities and associated standard errors for different covariate values based on the same model fit. It is not designed for differences based on separate model fits, although it may be possible handle this case by fit two models in a single model specification using dummy variables.
Calculates return value (also known as the return level) given return period(s) of interest, using model fit from extRemes::fevd
. Standard error is obtained via the delta method. The return value is the value for which the expected number of blocks until an event that exceeds that value is equal to the return period. For non-stationary models (those that include covariates for the location, scale, and/or shape parameters, return values and standard errors are returned for as many sets of covariates as provided.
calc_returnValue_fevd(fit, returnPeriod, covariates = NULL)
calc_returnValue_fevd(fit, returnPeriod, covariates = NULL)
fit |
fitted object from extRemes |
returnPeriod |
value(s) for which return value is desired |
covariates |
matrix of covariate values, each row a set of covariates for which the return value is desired |
Calculates difference in return values (also known as return levels) for two sets of covariates given return period(s) of interest, using model fit from extRemes::fevd
. Standard error is obtained via the delta method. The return value is the value for which the expected number of blocks until an event that exceeds that value is equal to the return period. Differences and standard errors are returned for as many contrasts between covariate sets as provided.
calc_returnValueDiff_fevd( fit, returnPeriod, covariates1, covariates2, getSE = TRUE )
calc_returnValueDiff_fevd( fit, returnPeriod, covariates1, covariates2, getSE = TRUE )
fit |
fitted object from extRemes |
returnPeriod |
value(s) for which return value difference is desired |
covariates1 |
matrix of covariate values, each row a set of covariates for which the return value difference relative to the corresponding row of |
covariates2 |
matrix of covariate values, each row a set of covariates |
getSE |
logical indicating whether standard error is desired, in addition to the point estimate |
This is designed to calculate differences in return values and associated standard errors for different covariate values based on the same model fit. It is not designed for differences based on separate model fits, although it may be possible handle this case by fit two models in a single model specification using dummy variables.
Compute risk ratio and uncertainty by fitting binomial models to counts of events relative to possible number of events. The risk ratio is the ratio of the probability of an event under the model fit to the first dataset to the probability under the model fit to the second dataset. Default standard errors are based on the usual MLE asymptotics using a delta-method-based approximation, but standard errors based on the nonparametric bootstrap and on a likelihood ratio procedure can also be computed.
calc_riskRatio_binom( y, n, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL )
calc_riskRatio_binom( y, n, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL )
y |
vector of two values, the number of events in the two scenarios |
n |
vector of two values, the number of samples (possible occurrences of events) in the two scenarios |
ciLevel |
statistical confidence level for confidence intervals; in repeated experimentation, this proportion of confidence intervals should contain the true risk ratio. Note that if only one endpoint of the resulting interval is used, for example the lower bound, then the effective confidence level increases by half of one minus |
ciType |
character vector indicating which type of confidence intervals to compute. See |
bootSE |
logical indicating whether to use the bootstrap to estimate the standard error of the risk ratio |
bootControl |
a list of control parameters for the bootstrapping, used only when at least one bootstrap confidence interval is requested via |
lrtControl |
list containing a single component, |
ciType
can include one or more of the following: 'delta'
, 'koopman'
, 'lrt'
, 'boot_norm'
, 'boot_perc'
, 'boot_basic'
, 'boot_stud'
, 'boot_bca'
. 'delta'
uses the delta method to compute an asymptotic interval based on the standard error of the log risk ratio. 'koopman'
uses the method described in Koopman (1984), following the implementation discussed in Fageland et al. (2015), including the calculation of Nam (1995). 'lrt'
inverts a likelihood-ratio test. Bootstrap-based options are the normal-based interval using the bootstrap standard error ('boot_norm'
), the percentile bootstrap ('boot_perc'
), the basic bootstrap ('boot_basic'
), the bootstrap-t ('boot_stud'
), and the bootstrap BCA method ('boot_bca'
). See Paciorek et al. for more details.
See fit_pot
for information on the bootControl
argument.
The primary outputs of this function are as follows: the log of the risk ratio and standard error of that log risk ratio (logRiskRatio
and se_logRiskRatio
) as well the risk ratio itself (riskRatio
). The standard error is based on the usual MLE asymptotics using a delta-method-based approximation. If requested via ciType
, confidence intervals will be returned, as discussed in Details
.
Christopher J. Paciorek
Paciorek, C.J., D.A. Stone, and M.F. Wehner. 2018. Quantifying uncertainty in the attribution of human influence on severe weather. Weather and Climate Extremes 20:69-80. arXiv preprint <https://arxiv.org/abs/1706.03388>.
Koopman, P.A.R. 1984. Confidence intervals for the ratio of two binomial proportions. Biometrics 40: 513-517.
Fagerland, M.W., S. Lydersen, and P. Laake. 2015. Recommended confidence intervals for two independent binomial proportions. Statistical Methods in Medical Research 24: 224-254.
# risk ratio for 40/400 compared to 8/400 events and for # 4/100 compared to 0/100 events calc_riskRatio_binom(c(40, 8), c(400, 400), ciType = c('lrt', 'boot_stud', 'koopman')) # LRT and Koopman methods can estimate lower confidence interval endpoint # even if estimated risk ratio is infinity: calc_riskRatio_binom(c(4,0), c(100, 100), ciType = c('lrt', 'boot_stud', 'koopman'))
# risk ratio for 40/400 compared to 8/400 events and for # 4/100 compared to 0/100 events calc_riskRatio_binom(c(40, 8), c(400, 400), ciType = c('lrt', 'boot_stud', 'koopman')) # LRT and Koopman methods can estimate lower confidence interval endpoint # even if estimated risk ratio is infinity: calc_riskRatio_binom(c(4,0), c(100, 100), ciType = c('lrt', 'boot_stud', 'koopman'))
Compute risk ratio and uncertainty by fitting generalized extreme value model, designed specifically for climate data, to exceedance-only data, using the point process approach. The risk ratio is the ratio of the probability of exceedance of a pre-specified value under the model fit to the first dataset to the probability under the model fit to the second dataset. Default standard errors are based on the usual MLE asymptotics using a delta-method-based approximation, but standard errors based on the nonparametric bootstrap and on a likelihood ratio procedure can also be computed.
calc_riskRatio_gev( returnValue, y1, y2, x1 = NULL, x2 = NULL, locationFun1 = NULL, locationFun2 = NULL, scaleFun1 = NULL, scaleFun2 = NULL, shapeFun1 = NULL, shapeFun2 = NULL, nReplicates1 = 1, nReplicates2 = 1, replicateIndex1 = NULL, replicateIndex2 = NULL, weights1 = NULL, weights2 = NULL, xNew1 = NULL, xNew2 = NULL, maxes = TRUE, scaling1 = 1, scaling2 = 1, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL, optimArgs = NULL, optimControl = NULL, initial1 = NULL, initial2 = NULL, logScale1 = NULL, logScale2 = NULL, getReturnCalcs = FALSE, getParams = FALSE, getFit = FALSE )
calc_riskRatio_gev( returnValue, y1, y2, x1 = NULL, x2 = NULL, locationFun1 = NULL, locationFun2 = NULL, scaleFun1 = NULL, scaleFun2 = NULL, shapeFun1 = NULL, shapeFun2 = NULL, nReplicates1 = 1, nReplicates2 = 1, replicateIndex1 = NULL, replicateIndex2 = NULL, weights1 = NULL, weights2 = NULL, xNew1 = NULL, xNew2 = NULL, maxes = TRUE, scaling1 = 1, scaling2 = 1, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL, optimArgs = NULL, optimControl = NULL, initial1 = NULL, initial2 = NULL, logScale1 = NULL, logScale2 = NULL, getReturnCalcs = FALSE, getParams = FALSE, getFit = FALSE )
returnValue |
numeric value giving the value for which the risk ratio should be calculated, where the resulting period will be the average number of blocks until the value is exceeded and the probability the probability of exceeding the value in any single block. |
y1 |
a numeric vector of observed maxima or minima values for the first dataset. See |
y2 |
a numeric vector of observed maxima or minima values for the second dataset. Analogous to |
x1 |
a data frame, or object that can be converted to a data frame with columns corresponding to covariate/predictor/feature variables and each row containing the values of the variable for the corresponding observed maximum/minimum. The number of rows should either equal the length of |
x2 |
analogous to |
locationFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
locationFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
scaleFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
scaleFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
shapeFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
shapeFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
nReplicates1 |
numeric value indicating the number of replicates for the first dataset. |
nReplicates2 |
numeric value indicating the number of replicates for the second dataset. |
replicateIndex1 |
numeric vector providing the index of the replicate corresponding to each element of |
replicateIndex2 |
numeric vector providing the index of the replicate corresponding to each element of |
weights1 |
a vector providing the weights for each observation in the first dataset. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of observations. When weights vary by replicate, this should be of equal length to |
weights2 |
a vector providing the weights for each observation in the second dataset. Analogous to |
xNew1 |
object of the same form as |
xNew2 |
object of the same form as |
maxes |
logical indicating whether analysis is for block maxima (TRUE) or block minima (FALSE); in the latter case, the function works with the negative of the values, changing the sign of the resulting location parameters |
scaling1 |
positive-valued scalar used to scale the data values of the first dataset for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
scaling2 |
positive-valued scalar used to scale the data values of the second dataset for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
ciLevel |
statistical confidence level for confidence intervals; in repeated experimentation, this proportion of confidence intervals should contain the true risk ratio. Note that if only one endpoint of the resulting interval is used, for example the lower bound, then the effective confidence level increases by half of one minus |
ciType |
character vector indicating which type of confidence intervals to compute. See |
bootSE |
logical indicating whether to use the bootstrap to estimate the standard error of the risk ratio |
bootControl |
a list of control parameters for the bootstrapping. See |
lrtControl |
list containing a single component, |
optimArgs |
a list with named components matching exactly any arguments that the user wishes to pass to |
optimControl |
a list with named components matching exactly any elements that the user wishes to pass as the |
initial1 |
a list with components named |
initial2 |
a list with components named |
logScale1 |
logical indicating whether optimization for the scale parameter should be done on the log scale for the first dataset. By default this is FALSE when the scale is not a function of covariates and TRUE when the scale is a function of covariates (to ensure the scale is positive regardless of the regression coefficients). |
logScale2 |
logical indicating whether optimization for the scale parameter should be done on the log scale for the second dataset. By default this is FALSE when the scale is not a function of covariates and TRUE when the scale is a function of covariates (to ensure the scale is positive regardless of the regression coefficients). |
getReturnCalcs |
logical indicating whether to return the estimated return values/probabilities/periods from the fitted models. |
getParams |
logical indicating whether to return the fitted parameter values and their standard errors for the fitted models; WARNING: parameter values for models with covariates for the scale parameter must interpreted based on the value of |
getFit |
logical indicating whether to return the full fitted models (potentially useful for model evaluation and for understanding optimization problems); note that estimated parameters in the fit object for nonstationary models will not generally match the MLE provided when |
See fit_gev
for more details on fitting the block maxima model for each dataset, including details on blocking and replication. Also see fit_gev
for information on the bootControl
argument.
Optimization failures:
It is not uncommon for maximization of the log-likelihood to fail for extreme value models. Please see the help information for fit_gev
. Also note that if the probability in the denominator of the risk ratio is near one, one may achieve better numerical performance by swapping the two datasets and computing the risk ratio for the probability under dataset 2 relative to the probability under dataset 1.
ciType
can include one or more of the following: 'delta'
, 'lrt'
, 'boot_norm'
, 'boot_perc'
, 'boot_basic'
, 'boot_stud'
, 'boot_bca'
. 'delta'
uses the delta method to compute an asymptotic interval based on the standard error of the log risk ratio. 'lrt'
inverts a likelihood-ratio test. Bootstrap-based options are the normal-based interval using the bootstrap standard error ('boot_norm'
), the percentile bootstrap ('boot_perc'
), the basic bootstrap ('boot_basic'
), the bootstrap-t ('boot_stud'
), and the bootstrap BCA method ('boot_bca'
). See Paciorek et al. for more details.
See fit_pot
for information on the bootControl
argument.
The primary outputs of this function are as follows: the log of the risk ratio and standard error of that log risk ratio (logRiskRatio
and se_logRiskRatio
) as well the risk ratio itself (riskRatio
). The standard error is based on the usual MLE asymptotics using a delta-method-based approximation. If requested via ciType
, confidence intervals will be returned, as discussed in Details
.
Christopher J. Paciorek
Paciorek, C.J., D.A. Stone, and M.F. Wehner. 2018. Quantifying uncertainty in the attribution of human influence on severe weather. Weather and Climate Extremes 20:69-80. arXiv preprint <https://arxiv.org/abs/1706.03388>.
Jeon S., C.J. Paciorek, and M.F. Wehner. 2016. Quantile-based bias correction and uncertainty quantification of extreme event attribution statements. Weather and Climate Extremes 12: 24-32. <DOI:10.1016/j.wace.2016.02.001>. arXiv preprint: <http://arxiv.org/abs/1602.04139>.
data(Fort, package = 'extRemes') FortMax <- aggregate(Prec ~ year, data = Fort, max) earlyYears <- 1900:1929 lateYears <- 1970:1999 earlyPeriod <- which(FortMax$year %in% earlyYears) latePeriod <- which(FortMax$year %in% lateYears) # contrast late period with early period, assuming a nonstationary fit # within each time period and finding RR based on midpoint of each period ## Not run: out <- calc_riskRatio_gev(returnValue = 3, y1 = FortMax$Prec[earlyPeriod], y2 = FortMax$Prec[latePeriod], x1 = data.frame(years = earlyYears), x2 = data.frame(years = lateYears), locationFun1 = ~years, locationFun2 = ~years, xNew1 = data.frame(years = mean(earlyYears)), xNew2 = data.frame(years = mean(lateYears))) ## End(Not run)
data(Fort, package = 'extRemes') FortMax <- aggregate(Prec ~ year, data = Fort, max) earlyYears <- 1900:1929 lateYears <- 1970:1999 earlyPeriod <- which(FortMax$year %in% earlyYears) latePeriod <- which(FortMax$year %in% lateYears) # contrast late period with early period, assuming a nonstationary fit # within each time period and finding RR based on midpoint of each period ## Not run: out <- calc_riskRatio_gev(returnValue = 3, y1 = FortMax$Prec[earlyPeriod], y2 = FortMax$Prec[latePeriod], x1 = data.frame(years = earlyYears), x2 = data.frame(years = lateYears), locationFun1 = ~years, locationFun2 = ~years, xNew1 = data.frame(years = mean(earlyYears)), xNew2 = data.frame(years = mean(lateYears))) ## End(Not run)
Compute risk ratio and uncertainty by fitting peaks-over-threshold model, designed specifically for climate data, to exceedance-only data, using the point process approach. The risk ratio is the ratio of the probability of exceedance of a pre-specified value under the model fit to the first dataset to the probability under the model fit to the second dataset. Default standard errors are based on the usual MLE asymptotics using a delta-method-based approximation, but standard errors based on the nonparametric bootstrap and on a likelihood ratio procedure can also be computed.
calc_riskRatio_pot( returnValue, y1, y2, x1 = NULL, x2 = NULL, threshold1, threshold2, locationFun1 = NULL, locationFun2 = NULL, scaleFun1 = NULL, scaleFun2 = NULL, shapeFun1 = NULL, shapeFun2 = NULL, nBlocks1 = nrow(x1), nBlocks2 = nrow(x2), blockIndex1 = NULL, blockIndex2 = NULL, firstBlock1 = 1, firstBlock2 = 1, index1 = NULL, index2 = NULL, nReplicates1 = 1, nReplicates2 = 1, replicateIndex1 = NULL, replicateIndex2 = NULL, weights1 = NULL, weights2 = NULL, proportionMissing1 = NULL, proportionMissing2 = NULL, xNew1 = NULL, xNew2 = NULL, declustering = NULL, upperTail = TRUE, scaling1 = 1, scaling2 = 1, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL, optimArgs = NULL, optimControl = NULL, initial1 = NULL, initial2 = NULL, logScale1 = NULL, logScale2 = NULL, getReturnCalcs = FALSE, getParams = FALSE, getFit = FALSE )
calc_riskRatio_pot( returnValue, y1, y2, x1 = NULL, x2 = NULL, threshold1, threshold2, locationFun1 = NULL, locationFun2 = NULL, scaleFun1 = NULL, scaleFun2 = NULL, shapeFun1 = NULL, shapeFun2 = NULL, nBlocks1 = nrow(x1), nBlocks2 = nrow(x2), blockIndex1 = NULL, blockIndex2 = NULL, firstBlock1 = 1, firstBlock2 = 1, index1 = NULL, index2 = NULL, nReplicates1 = 1, nReplicates2 = 1, replicateIndex1 = NULL, replicateIndex2 = NULL, weights1 = NULL, weights2 = NULL, proportionMissing1 = NULL, proportionMissing2 = NULL, xNew1 = NULL, xNew2 = NULL, declustering = NULL, upperTail = TRUE, scaling1 = 1, scaling2 = 1, ciLevel = 0.9, ciType, bootSE, bootControl = NULL, lrtControl = NULL, optimArgs = NULL, optimControl = NULL, initial1 = NULL, initial2 = NULL, logScale1 = NULL, logScale2 = NULL, getReturnCalcs = FALSE, getParams = FALSE, getFit = FALSE )
returnValue |
numeric value giving the value for which the risk ratio should be calculated, where the resulting period will be the average number of blocks until the value is exceeded and the probability the probability of exceeding the value in any single block. |
y1 |
a numeric vector of exceedance values for the first dataset (values of the outcome variable above the threshold). For better optimization performance, it is recommended that the |
y2 |
a numeric vector of exceedance values for the second dataset (values of the outcome variable above the threshold). |
x1 |
a data frame, or object that can be converted to a data frame with columns corresponding to covariate/predictor/feature variables and each row containing the values of the variable for a block (e.g., often a year with climate data) for the first dataset. The number of rows must equal the number of blocks. |
x2 |
analogous to |
threshold1 |
a single numeric value for constant threshold or a numeric vector with length equal to the number of blocks, indicating the threshold for each block for the first dataset. |
threshold2 |
analogous to |
locationFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
locationFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
scaleFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
scaleFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
shapeFun1 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
shapeFun2 |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
nBlocks1 |
number of blocks (e.g., a block will often be a year with climate data) in first dataset; note this value determines the interpretation of return values/periods/probabilities; see |
nBlocks2 |
number of blocks (e.g., a block will often be a year with climate data) in second dataset; note this value determines the interpretation of return values/periods/probabilities; see |
blockIndex1 |
numeric vector providing the index of the block corresponding to each element of |
blockIndex2 |
numeric vector providing the index of the block corresponding to each element of |
firstBlock1 |
single numeric value indicating the numeric value of the first possible block of |
firstBlock2 |
single numeric value indicating the numeric value of the first possible block of |
index1 |
numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of |
index2 |
numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of |
nReplicates1 |
numeric value indicating the number of replicates for the first dataset. |
nReplicates2 |
numeric value indicating the number of replicates for the second dataset. |
replicateIndex1 |
numeric vector providing the index of the replicate corresponding to each element of |
replicateIndex2 |
numeric vector providing the index of the replicate corresponding to each element of |
weights1 |
a vector or matrix providing the weights by block for the first dataset. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of blocks. When weights vary by replicate, a matrix with rows corresponding to blocks and columns to replicates. Likelihood contribution of each block is multiplied by the corresponding weight. |
weights2 |
a vector or matrix providing the weights by block for the second dataset. Analogous to |
proportionMissing1 |
a numeric value, vector or matrix indicating the proportion of missing values in the original first dataset before exceedances were selected. When the proportion missing is the same for all blocks and replicates, a single value. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of blocks. When weights vary by replicate, a matrix with rows corresponding to blocks and columns to replicates. |
proportionMissing2 |
a numeric value, vector or matrix indicating the proportion of missing values in the original second dataset before exceedances were selected. Analogous to |
xNew1 |
object of the same form as |
xNew2 |
object of the same form as |
declustering |
one of |
upperTail |
logical indicating whether one is working with exceedances over a high threshold (TRUE) or exceedances under a low threshold (FALSE); in the latter case, the function works with the negative of the values and the threshold, changing the sign of the resulting location parameters. |
scaling1 |
positive-valued scalar used to scale the data values of the first dataset for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
scaling2 |
positive-valued scalar used to scale the data values of the second dataset for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
ciLevel |
statistical confidence level for confidence intervals; in repeated experimentation, this proportion of confidence intervals should contain the true risk ratio. Note that if only one endpoint of the resulting interval is used, for example the lower bound, then the effective confidence level increases by half of one minus |
ciType |
character vector indicating which type of confidence intervals to compute. See |
bootSE |
logical indicating whether to use the bootstrap to estimate the standard error of the risk ratio |
bootControl |
a list of control parameters for the bootstrapping. See |
lrtControl |
list containing a single component, |
optimArgs |
a list with named components matching exactly any arguments that the user wishes to pass to |
optimControl |
a list with named components matching exactly any elements that the user wishes to pass as the |
initial1 |
a list with components named |
initial2 |
a list with components named |
logScale1 |
logical indicating whether optimization for the scale parameter should be done on the log scale for the first dataset. By default this is |
logScale2 |
logical indicating whether optimization for the scale parameter should be done on the log scale for the second dataset. By default this is |
getReturnCalcs |
logical indicating whether to return the estimated return values/probabilities/periods from the fitted models. |
getParams |
logical indicating whether to return the fitted parameter values and their standard errors for the fitted models; WARNING: parameter values for models with covariates for the scale parameter must interpreted based on the value of |
getFit |
logical indicating whether to return the full fitted models (potentially useful for model evaluation and for understanding optimization problems); note that estimated parameters in the fit object for nonstationary models will not generally match the MLE provided when |
See fit_pot
for more details on fitting the peaks-over-threshold model for each dataset, including details on blocking and replication. Also see fit_pot
for information on the bootControl
argument.
Optimization failures:
It is not uncommon for maximization of the log-likelihood to fail for extreme value models. Please see the help information for fit_pot
. Also note that if the probability in the denominator of the risk ratio is near one, one may achieve better numerical performance by swapping the two datasets and computing the risk ratio for the probability under dataset 2 relative to the probability under dataset 1.
ciType
can include one or more of the following: 'delta'
, 'lrt'
, 'boot_norm'
, 'boot_perc'
, 'boot_basic'
, 'boot_stud'
, 'boot_bca'
. 'delta'
uses the delta method to compute an asymptotic interval based on the standard error of the log risk ratio. 'lrt'
inverts a likelihood-ratio test. Bootstrap-based options are the normal-based interval using the bootstrap standard error ('boot_norm'
), the percentile bootstrap ('boot_perc'
), the basic bootstrap ('boot_basic'
), the bootstrap-t ('boot_stud'
), and the bootstrap BCA method ('boot_bca'
). See Paciorek et al. for more details.
See fit_pot
for information on the bootControl
argument.
The primary outputs of this function are as follows: the log of the risk ratio and standard error of that log risk ratio (logRiskRatio
and se_logRiskRatio
) as well the risk ratio itself (riskRatio
). The standard error is based on the usual MLE asymptotics using a delta-method-based approximation. If requested via ciType
, confidence intervals will be returned, as discussed in Details
.
Christopher J. Paciorek
Paciorek, C.J., D.A. Stone, and M.F. Wehner. 2018. Quantifying uncertainty in the attribution of human influence on severe weather. Weather and Climate Extremes 20:69-80. arXiv preprint <https://arxiv.org/abs/1706.03388>.
Jeon S., C.J. Paciorek, and M.F. Wehner. 2016. Quantile-based bias correction and uncertainty quantification of extreme event attribution statements. Weather and Climate Extremes 12: 24-32. <DOI:10.1016/j.wace.2016.02.001>. arXiv preprint: <http://arxiv.org/abs/1602.04139>.
data(Fort, package = 'extRemes') threshold <- 0.395 ord <- order(Fort$year, Fort$month, Fort$day) Fort <- Fort[ord, ] ind <- Fort$Prec > threshold FortExc <- Fort[ind, ] earlyYears <- 1900:1929 lateYears <- 1970:1999 earlyPeriod <- which(FortExc$year %in% earlyYears) latePeriod <- which(FortExc$year %in% lateYears) # contrast late period with early period, assuming a nonstationary fit # within each time period and finding RR based on midpoint of each period ## Not run: out <- calc_riskRatio_pot(returnValue = 3, y1 = FortExc$Prec[earlyPeriod], y2 = FortExc$Prec[latePeriod], x1 = data.frame(years = earlyYears), x2 = data.frame(years = lateYears), threshold1 = threshold, threshold2 = threshold, locationFun1 = ~years, locationFun2 = ~years, xNew1 = data.frame(years = mean(earlyYears)), xNew2 = data.frame(years = mean(lateYears)), blockIndex1 = FortExc$year[earlyPeriod], blockIndex2 = FortExc$year[latePeriod], firstBlock1 = earlyYears[1], firstBlock2 = lateYears[1]) ## End(Not run)
data(Fort, package = 'extRemes') threshold <- 0.395 ord <- order(Fort$year, Fort$month, Fort$day) Fort <- Fort[ord, ] ind <- Fort$Prec > threshold FortExc <- Fort[ind, ] earlyYears <- 1900:1929 lateYears <- 1970:1999 earlyPeriod <- which(FortExc$year %in% earlyYears) latePeriod <- which(FortExc$year %in% lateYears) # contrast late period with early period, assuming a nonstationary fit # within each time period and finding RR based on midpoint of each period ## Not run: out <- calc_riskRatio_pot(returnValue = 3, y1 = FortExc$Prec[earlyPeriod], y2 = FortExc$Prec[latePeriod], x1 = data.frame(years = earlyYears), x2 = data.frame(years = lateYears), threshold1 = threshold, threshold2 = threshold, locationFun1 = ~years, locationFun2 = ~years, xNew1 = data.frame(years = mean(earlyYears)), xNew2 = data.frame(years = mean(lateYears)), blockIndex1 = FortExc$year[earlyPeriod], blockIndex2 = FortExc$year[latePeriod], firstBlock1 = earlyYears[1], firstBlock2 = lateYears[1]) ## End(Not run)
Fit a generalized extreme value model, designed specifically for climate data. It includes options for variable weights (useful for local likelihood), as well as for bootstrapping to estimate uncertainties. Results can be returned in terms of parameter values, return values, return periods, return probabilities, and differences in either return values or log return probabilities (i.e., log risk ratios).
fit_gev( y, x = NULL, locationFun = NULL, scaleFun = NULL, shapeFun = NULL, nReplicates = 1, replicateIndex = NULL, weights = NULL, returnPeriod = NULL, returnValue = NULL, getParams = FALSE, getFit = FALSE, xNew = NULL, xContrast = NULL, maxes = TRUE, scaling = 1, bootSE = FALSE, bootControl = NULL, optimArgs = NULL, optimControl = NULL, missingFlag = NULL, initial = NULL, logScale = NULL, .normalizeX = TRUE, .getInputs = FALSE, .allowNoInt = TRUE )
fit_gev( y, x = NULL, locationFun = NULL, scaleFun = NULL, shapeFun = NULL, nReplicates = 1, replicateIndex = NULL, weights = NULL, returnPeriod = NULL, returnValue = NULL, getParams = FALSE, getFit = FALSE, xNew = NULL, xContrast = NULL, maxes = TRUE, scaling = 1, bootSE = FALSE, bootControl = NULL, optimArgs = NULL, optimControl = NULL, missingFlag = NULL, initial = NULL, logScale = NULL, .normalizeX = TRUE, .getInputs = FALSE, .allowNoInt = TRUE )
y |
a numeric vector of observed maxima or minima values. See |
x |
a data frame, or object that can be converted to a data frame with columns corresponding to covariate/predictor/feature variables and each row containing the values of the variable for the corresponding observed maximum/minimum. The number of rows should either equal the length of |
locationFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
scaleFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
shapeFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
nReplicates |
numeric value indicating the number of replicates. |
replicateIndex |
numeric vector providing the index of the replicate corresponding to each element of |
weights |
a vector providing the weights for each observation. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of observations. When weights vary by replicate, this should be of equal length to |
returnPeriod |
numeric value giving the number of blocks for which return values should be calculated. For example a returnPeriod of 20 corresponds to the value of an event that occurs with probability 1/20 in any block and therefore occurs on average every 20 blocks. Often blocks will correspond to years. |
returnValue |
numeric value giving the value for which return probabilities/periods should be calculated, where the period would be the average number of blocks until the value is exceeded and the probability the probability of exceeding the value in any single block. |
getParams |
logical indicating whether to return the fitted parameter values and their standard errors; WARNING: parameter values for models with covariates for the scale parameter must interpreted based on the value of |
getFit |
logical indicating whether to return the full fitted model (potentially useful for model evaluation and for understanding optimization problems); note that estimated parameters in the fit object for nonstationary models will not generally match the MLE provided when |
xNew |
object of the same form as |
xContrast |
object of the same form and dimensions as |
maxes |
logical indicating whether analysis is for block maxima (TRUE) or block minima (FALSE); in the latter case, the function works with the negative of the values, changing the sign of the resulting location parameters |
scaling |
positive-valued scalar used to scale the data values for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
bootSE |
logical indicating whether to use the bootstrap to estimate standard errors. |
bootControl |
a list of control parameters for the bootstrapping. See |
optimArgs |
a list with named components matching exactly any arguments that the user wishes to pass to R's |
optimControl |
a list with named components matching exactly any elements that the user wishes to pass as the |
missingFlag |
value to be interpreted as missing values (instead of |
initial |
a list with components named |
logScale |
logical indicating whether optimization for the scale parameter should be done on the log scale. By default this is |
.normalizeX |
logical indicating whether to normalize |
.getInputs |
logical indicating whether to return intermediate objects used in fitting. Defaults to |
.allowNoInt |
logical indicating whether no-intercept models are allowed. Defaults to |
This function allows one to fit stationary or nonstationary block maxima/minima models using the generalized extreme value distribution. The function can return parameter estimates, return value/level for a given return period (number of blocks), and return probabilities/periods for a given return value/level. The function provides standard errors based on the usual MLE asymptotics, with delta-method-based standard errors for functionals of the parameters, but also standard errors based on the nonparametric bootstrap, either resampling by block or by replicate or both.
Replicates:
Replicates are repeated datasets, each with the same structure, including the same number of block maxima/minima. The additional observations in multiple replicates could simply be treated as additional blocks without replication (see next paragraph), but when the covariate values and weights are the same across replicates, it is simpler to make use of nReplicates
and replicateIndex
.
When using multiple replicates (e.g., multiple members of a climate model initial condition ensemble), the standard input format is to append observations for additional replicates to the y
argument and indicate the replicate ID for each value via replicateIndex
, which would be of the form 1,1,1,...2,2,2,...3,3,3,... etc. The values for each replicate should be grouped together and in the same order within replicate so that x
can be correctly matched to the y
values when x
is only supplied for the first replicate. In other words, y
should first contain all the values for the first replicate, then all the values for the second replicate in the same block order as for the first replicate, and so forth. Note that if y
is provided as a matrix with the number of rows equal to the number of observations in each replicate and the columns corresponding to replicates, this ordering will occur naturally.
However, if one has different covariate values for different replicates, then one needs to treat the additional replicates as providing additional blocks, with only a single replicate (and nReplicates
set to 1). The covariate values can then be included as additional rows in x
. Similarly, if there is a varying number of replicates by block, then all block-replicate pairs should be treated as individual blocks with a corresponding row in x
(and nReplicates
set to 1).
bootControl
arguments:
The bootControl
argument is a list (or dictionary when calling from Python) that can supply any of the following components:
seed. Value of the random number seed as a single value, or in the form of .Random.seed
, to set before doing resampling. Defaults to 1
.
n. Number of bootstrap samples. Defaults to 250
.
by. Character string, one of 'block'
, 'replicate'
, or 'joint'
, indicating the basis for the resampling. If 'block'
, resampled datasets will consist of blocks drawn at random from the original set of blocks; if there are replicates, each replicate will occur once for every resampled block. If 'replicate'
, resampled datasets will consist of replicates drawn at random from the original set of replicates; all blocks from a replicate will occur in each resampled replicate. Note that this preserves any dependence across blocks rather than assuming independence between blocks. If 'joint'
resampled datasets will consist of block-replicate pairs drawn at random from the original set of block-replicate pairs. Defaults to 'block'
.
getSample. Logical/boolean indicating whether the user wants the full bootstrap sample of parameter estimates and/or return value/period/probability information returned for use in subsequent calculations; if FALSE (the default), only the bootstrap-based estimated standard errors are returned.
Optimization failures:
It is not uncommon for maximization of the log-likelihood to fail for extreme value models. Users should carefully check the info
element of the return object to ensure that the optimization converged. For better optimization performance, it is recommended that the observations be scaled to have magnitude around one (e.g., converting precipitation from mm to cm). When there is a convergence failure, one can try a different optimization method, more iterations, or different starting values – see optimArgs
and initial
. In particular, the Nelder-Mead method is used; users may want to try the BFGS method by setting optimArgs = list(method = 'BFGS')
(or optimArgs = {'method': 'BFGS'}
when calling from Python).
When using the bootstrap, users should check that the number of convergence failures when fitting to the boostrapped datasets is small, as it is not clear how to interpret the bootstrap results when there are convergence failures for some bootstrapped datasets.
The primary outputs of this function are as follows, depending on what is requested via returnPeriod
, returnValue
, getParams
and xContrast
:
when returnPeriod
is given: for the period given in returnPeriod
the return value(s) (returnValue
) and its corresponding asymptotic standard error (se_returnValue
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_returnValue_boot
). For nonstationary models, these correspond to the covariate values given in x
.
when returnValue
is given: for the value given in returnValue
, the log exceedance probability (logReturnProb
) and the corresponding asymptotic standard error (se_logReturnProb
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_logReturnProb_boot
). This exceedance probability is the probability of exceedance for a single block. Also returned are the log return period (logReturnPeriod
) and its corresponding asymptotic standard error (se_logReturnPeriod
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_logReturnPeriod_boot
). For nonstationary models, these correspond to the covariate values given in x
. Note that results are on the log scale as probabilities and return times are likely to be closer to normally distributed on the log scale and therefore standard errors are more naturally given on this scale. Confidence intervals for return probabilities/periods can be obtained by exponentiating the interval obtained from plus/minus twice the standard error of the log probabilities/periods.
when getParams=TRUE
: the MLE for the model parameters (mle
) and corresponding asymptotic standard error (se_mle
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_mle_boot
).
when xContrast
is specified for nonstationary models: the difference in return values (returnValueDiff
) and its corresponding asymptotic standard error (se_returnValueDiff
) and, when bootSE=TRUE
, bootstrapped standard error (se_returnValueDiff_boot
). These differences correspond to the differences when contrasting each row in x
with the corresponding row in xContrast
. Also returned are the difference in log return probabilities (i.e., the log risk ratio) (logReturnProbDiff
) and its corresponding asymptotic standard error (se_logReturnProbDiff
) and, when bootSE=TRUE
, bootstrapped standard error (se_logReturnProbDiff_boot
).
Christopher J. Paciorek
Coles, S. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer.
Paciorek, C.J., D.A. Stone, and M.F. Wehner. 2018. Quantifying uncertainty in the attribution of human influence on severe weather. Weather and Climate Extremes 20:69-80. arXiv preprint <https://arxiv.org/abs/1706.03388>.
data(Fort, package = 'extRemes') FortMax <- aggregate(Prec ~ year, data = Fort, max) # stationary fit out <- fit_gev(FortMax$Prec, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, bootSE = FALSE) # nonstationary fit with location linear in year out <- fit_gev(FortMax$Prec, x = data.frame(years = FortMax$year), locationFun = ~years, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(FortMax$year)), bootSE = FALSE)
data(Fort, package = 'extRemes') FortMax <- aggregate(Prec ~ year, data = Fort, max) # stationary fit out <- fit_gev(FortMax$Prec, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, bootSE = FALSE) # nonstationary fit with location linear in year out <- fit_gev(FortMax$Prec, x = data.frame(years = FortMax$year), locationFun = ~years, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(FortMax$year)), bootSE = FALSE)
Fit a peaks-over-threshold model, designed specifically for climate data, to exceedance-only data, using the point process approach. Any covariates/predictors/features assumed to vary only between and not within blocks of observations. It includes options for variable weights (useful for local likelihood) and variable proportions of missing data, as well as for bootstrapping to estimate uncertainties. Results can be returned in terms of parameter values, return values, return periods, return probabilities, and differences in either return values or log return probabilities (i.e., log risk ratios).
fit_pot( y, x = NULL, threshold, locationFun = NULL, scaleFun = NULL, shapeFun = NULL, nBlocks = nrow(x), blockIndex = NULL, firstBlock = 1, index = NULL, nReplicates = 1, replicateIndex = NULL, weights = NULL, proportionMissing = NULL, returnPeriod = NULL, returnValue = NULL, getParams = FALSE, getFit = FALSE, xNew = NULL, xContrast = NULL, declustering = NULL, upperTail = TRUE, scaling = 1, bootSE = FALSE, bootControl = NULL, optimArgs = NULL, optimControl = NULL, initial = NULL, logScale = NULL, .normalizeX = TRUE, .getInputs = FALSE, .allowNoInt = TRUE )
fit_pot( y, x = NULL, threshold, locationFun = NULL, scaleFun = NULL, shapeFun = NULL, nBlocks = nrow(x), blockIndex = NULL, firstBlock = 1, index = NULL, nReplicates = 1, replicateIndex = NULL, weights = NULL, proportionMissing = NULL, returnPeriod = NULL, returnValue = NULL, getParams = FALSE, getFit = FALSE, xNew = NULL, xContrast = NULL, declustering = NULL, upperTail = TRUE, scaling = 1, bootSE = FALSE, bootControl = NULL, optimArgs = NULL, optimControl = NULL, initial = NULL, logScale = NULL, .normalizeX = TRUE, .getInputs = FALSE, .allowNoInt = TRUE )
y |
a numeric vector of exceedance values (values of the outcome variable above the threshold). For better optimization performance, it is recommended that the |
x |
a data frame, or object that can be converted to a data frame with columns corresponding to covariate/predictor/feature variables and each row containing the values of the variable for a block (e.g., often a year with climate data). The number of rows must equal the number of blocks. |
threshold |
a single numeric value for constant threshold or a numeric vector with length equal to the number of blocks, indicating the threshold for each block. |
locationFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from |
scaleFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the (potentially transformed) scale parameter using columns from |
shapeFun |
formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from |
nBlocks |
number of blocks (e.g., a block will often be a year with climate data); note this value determines the interpretation of return values/periods/probabilities; see |
blockIndex |
numeric vector providing the index of the block corresponding to each element of |
firstBlock |
single numeric value indicating the numeric value of the first possible block of |
index |
numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of |
nReplicates |
numeric value indicating the number of replicates. |
replicateIndex |
numeric vector providing the index of the replicate corresponding to each element of |
weights |
a vector or matrix providing the weights by block. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of blocks. When weights vary by replicate, a matrix with rows corresponding to blocks and columns to replicates. Likelihood contribution of each block is multiplied by the corresponding weight. |
proportionMissing |
a numeric value, vector or matrix indicating the proportion of missing values in the original dataset before exceedances were selected. When the proportion missing is the same for all blocks and replicates, a single value. When there is only one replicate or the weights do not vary by replicate, a vector of length equal to the number of blocks. When weights vary by replicate, a matrix with rows corresponding to blocks and columns to replicates. |
returnPeriod |
numeric value giving the number of blocks for which return values should be calculated. For example a |
returnValue |
numeric value giving the value for which return probabilities/periods should be calculated, where the resulting period will be the average number of blocks until the value is exceeded and the probability the probability of exceeding the value in any single block. |
getParams |
logical indicating whether to return the fitted parameter values and their standard errors; WARNING: parameter values for models with covariates for the scale parameter must interpreted based on the value of |
getFit |
logical indicating whether to return the full fitted model (potentially useful for model evaluation and for understanding optimization problems); note that estimated parameters in the fit object for nonstationary models will not generally match the MLE provided when |
xNew |
object of the same form as |
xContrast |
object of the same form and dimensions as |
declustering |
one of |
upperTail |
logical indicating whether one is working with exceedances over a high threshold (TRUE) or exceedances under a low threshold (FALSE); in the latter case, the function works with the negative of the values and the threshold, changing the sign of the resulting location parameters. |
scaling |
positive-valued scalar used to scale the data values for more robust optimization performance. When multiplied by the values, it should produce values with magnitude around 1. |
bootSE |
logical indicating whether to use the bootstrap to estimate standard errors. |
bootControl |
a list of control parameters for the bootstrapping. See ‘Details’. |
optimArgs |
a list with named components matching exactly any arguments that the user wishes to pass to R's |
optimControl |
a list with named components matching exactly any elements that the user wishes to pass as the |
initial |
a list with components named |
logScale |
logical indicating whether optimization for the scale parameter should be done on the log scale. By default this is |
.normalizeX |
logical indicating whether to normalize |
.getInputs |
logical indicating whether to return intermediate objects used in fitting. Defaults to |
.allowNoInt |
logical indicating whether no-intercept models are allowed. Defaults to |
This function allows one to fit stationary or nonstationary peaks-over-threshold models using the point process approach. The function can return parameter estimates (which are asymptotically equivalent to GEV model parameters for block maxima data), return value/level for a given return period (number of blocks), and return probabilities/periods for a given return value/level. The function provides standard errors based on the usual MLE asymptotics, with delta-method-based standard errors for functionals of the parameters, but also standard errors based on the nonparametric bootstrap, either resampling by block or by replicate or both.
Analyzing aggregated observations, such as yearly averages:
Aggregated average or summed data such as yearly or seasonal averages can be fit using this function. The best way to do this is to specify nBlocks
to be the number of observations (i.e., the length of the observation period, not the number of exceedances). Then any return probabilities can be interpreted as the probabilities for a single block (e.g., a year). If instead nBlocks
were one (i.e., a single block) then probabilities would be interpreted as the probability of occurrence in a multi-year block.
Blocks and replicates:
Note that blocks and replicates are related concepts. Blocks are the grouping of values such that return values and return periods are calculated based on the equivalent block maxima (or minima) generalized extreme value model. For example if a block is a year, the return period is the average number of years before the given value is seen, while the return value when returnPeriod
is, say, 20, is the value exceeded on average once in 20 years. A given dataset will generally have multiple blocks. In some cases a block may contain only a single value, such as when analyzing yearly sums or averages.
Replicates are repeated datasets, each with the same structure, including the same number of blocks. The additional blocks in multiple replicates could simply be treated as additional blocks without replication, but when the predictor variables and weights are the same across replicates, it is simpler to make use of nReplicates
and replicateIndex
(see next paragraph). A given replicate might only contain a single block, such as with an ensemble of short climate model runs that are run only for the length of a single block (e.g., a single year). In this case nBlocks
should be set to one.
When using multiple replicates (e.g., multiple members of a climate model initial condition ensemble), the standard input format is to append outcome values for additional replicates to the y
argument and indicate the replicate ID for each exceedance in replicateIndex
. However, if one has different covariate values or thresholds for different replicates, then one needs to treat the additional replicates as providing additional blocks, with only a single replicate. The covariate values can then be included as additional rows in x
, with nBlocks
reflecting the product of the number of blocks per replicate and the number of replicates and nReplicates
set to 1. In this situation, if declustering
is specified, make sure to set index
such that the values for separate replicates do not overlap with each other, to avoid treating exceedances from different replicates as being sequential or from a contiguous set of values. Similarly, if there is a varying number of replicates by block, then all block-replicate pairs should be treated as individual blocks with a corresponding row in x
.
bootControl
arguments:
The bootControl
argument is a list (or dictionary when calling from Python) that can supply any of the following components:
seed. Value of the random number seed as a single value, or in the form of .Random.seed
, to set before doing resampling. Defaults to 1
.
n. Number of bootstrap samples. Defaults to 250
.
by. Character string, one of 'block'
, 'replicate'
, or 'joint'
, indicating the basis for the resampling. If 'block'
, resampled datasets will consist of blocks drawn at random from the original set of blocks; if there are replicates, each replicate will occur once for every resampled block. If 'replicate'
, resampled datasets will consist of replicates drawn at random from the original set of replicates; all blocks from a replicate will occur in each resampled replicate. Note that this preserves any dependence across blocks rather than assuming independence between blocks. If 'joint'
resampled datasets will consist of block-replicate pairs drawn at random from the original set of block-replicate pairs. Defaults to 'block'
.
getSample. Logical/boolean indicating whether the user wants the full bootstrap sample of parameter estimates and/or return value/period/probability information provided for use in subsequent calculations; if FALSE (the default), only the bootstrap-based estimated standard errors are returned.
Optimization failures:
It is not uncommon for maximization of the log-likelihood to fail for extreme value models. Users should carefully check the info
element of the return object to ensure that the optimization converged. For better optimization performance, it is recommended that the observations be scaled to have magnitude around one (e.g., converting precipitation from mm to cm). When there is a convergence failure, one can try a different optimization method, more iterations, or different starting values – see optimArgs
and initial
. In particular, the Nelder-Mead method is used; users may want to try the BFGS method by setting optimArgs = list(method = 'BFGS')
(or optimArgs = {'method': 'BFGS'}
if calling from Python).
When using the bootstrap, users should check that the number of convergence failures when fitting to the boostrapped datasets is small, as it is not clear how to interpret the bootstrap results when there are convergence failures for some bootstrapped datasets.
The primary outputs of this function are as follows, depending on what is requested via returnPeriod
, returnValue
, getParams
and xContrast
:
when returnPeriod
is given: for the period given in returnPeriod
the return value(s) (returnValue
) and its corresponding asymptotic standard error (se_returnValue
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_returnValue_boot
). For nonstationary models, these correspond to the covariate values given in x
.
when returnValue
is given: for the value given in returnValue
, the log exceedance probability (logReturnProb
) and the corresponding asymptotic standard error (se_logReturnProb
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_logReturnProb_boot
). This exceedance probability is the probability of exceedance for a single block. Also returned are the log return period (logReturnPeriod
) and its corresponding asymptotic standard error (se_logReturnPeriod
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_logReturnPeriod_boot
). For nonstationary models, these correspond to the covariate values given in x
. Note that results are on the log scale as probabilities and return times are likely to be closer to normally distributed on the log scale and therefore standard errors are more naturally given on this scale. Confidence intervals for return probabilities/periods can be obtained by exponentiating the interval obtained from plus/minus twice the standard error of the log probabilities/periods.
when getParams=TRUE
: the MLE for the model parameters (mle
) and corresponding asymptotic standard error (se_mle
) and, when bootSE=TRUE
, also the bootstrapped standard error (se_mle_boot
).
when xContrast
is specified for nonstationary models: the difference in return values (returnValueDiff
) and its corresponding asymptotic standard error (se_returnValueDiff
) and, when bootSE=TRUE
, bootstrapped standard error (se_returnValueDiff_boot
). These differences correspond to the differences when contrasting each row in x
with the corresponding row in xContrast
. Also returned are the difference in log return probabilities (i.e., the log risk ratio) (logReturnProbDiff
) and its corresponding asymptotic standard error (se_logReturnProbDiff
) and, when bootSE=TRUE
, bootstrapped standard error (se_logReturnProbDiff_boot
).
Christopher J. Paciorek
Coles, S. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer.
Paciorek, C.J., D.A. Stone, and M.F. Wehner. 2018. Quantifying uncertainty in the attribution of human influence on severe weather. Weather and Climate Extremes 20:69-80. arXiv preprint <https://arxiv.org/abs/1706.03388>.
# setup Fort precipitation data data(Fort, package = 'extRemes') firstBlock <- min(Fort$year) years <- min(Fort$year):max(Fort$year) nYears <- length(years) threshold <- 0.395 ord <- order(Fort$year, Fort$month, Fort$day) Fort <- Fort[ord, ] ind <- Fort$Prec > threshold FortExc <- Fort[ind, ] # stationary fit out <- fit_pot(FortExc$Prec, threshold = threshold, nBlocks = nYears, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, bootSE = FALSE) # fit with location linear in year out <- fit_pot(FortExc$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = FortExc$year, firstBlock = firstBlock, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE) # with declustering index <- seq_len(nrow(Fort)) out <- fit_pot(FortExc$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = FortExc$year, firstBlock = firstBlock, index = index[ind], returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), declustering = 'noruns', bootSE = FALSE) # with replicates; for illustration here, I'll just duplicate the Fort data Fort2x <- rbind(FortExc, FortExc) n <- nrow(FortExc) out <- fit_pot(Fort2x$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = Fort2x$year, firstBlock = firstBlock, nReplicates = 2, replicateIndex = c(rep(1, n), rep(2, n)), returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE) # analysis of seasonal total precipitation FortSummer <- Fort[Fort$month %in% 6:8, ] # summer precipitation FortSummerSum <- aggregate(Prec ~ year, data = FortSummer, sum) threshold <- quantile(FortSummerSum$Prec, 0.8) FortSummerSumExc <- FortSummerSum[FortSummerSum$Prec > threshold, ] # each year (single observation) treated as a block, so return probability # can be interpreted as probability of exceeding a value in a single year out <- fit_pot(FortSummerSumExc$Prec, x = data.frame(years = years), threshold = threshold, nBlocks = nYears, blockIndex = FortSummerSumExc$year, firstBlock = firstBlock, locationFun = ~years, returnPeriod = 20, returnValue = 10, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE)
# setup Fort precipitation data data(Fort, package = 'extRemes') firstBlock <- min(Fort$year) years <- min(Fort$year):max(Fort$year) nYears <- length(years) threshold <- 0.395 ord <- order(Fort$year, Fort$month, Fort$day) Fort <- Fort[ord, ] ind <- Fort$Prec > threshold FortExc <- Fort[ind, ] # stationary fit out <- fit_pot(FortExc$Prec, threshold = threshold, nBlocks = nYears, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, bootSE = FALSE) # fit with location linear in year out <- fit_pot(FortExc$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = FortExc$year, firstBlock = firstBlock, returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE) # with declustering index <- seq_len(nrow(Fort)) out <- fit_pot(FortExc$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = FortExc$year, firstBlock = firstBlock, index = index[ind], returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), declustering = 'noruns', bootSE = FALSE) # with replicates; for illustration here, I'll just duplicate the Fort data Fort2x <- rbind(FortExc, FortExc) n <- nrow(FortExc) out <- fit_pot(Fort2x$Prec, x = data.frame(years = years), threshold = threshold, locationFun = ~years, nBlocks = nYears, blockIndex = Fort2x$year, firstBlock = firstBlock, nReplicates = 2, replicateIndex = c(rep(1, n), rep(2, n)), returnPeriod = 20, returnValue = 3.5, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE) # analysis of seasonal total precipitation FortSummer <- Fort[Fort$month %in% 6:8, ] # summer precipitation FortSummerSum <- aggregate(Prec ~ year, data = FortSummer, sum) threshold <- quantile(FortSummerSum$Prec, 0.8) FortSummerSumExc <- FortSummerSum[FortSummerSum$Prec > threshold, ] # each year (single observation) treated as a block, so return probability # can be interpreted as probability of exceeding a value in a single year out <- fit_pot(FortSummerSumExc$Prec, x = data.frame(years = years), threshold = threshold, nBlocks = nYears, blockIndex = FortSummerSumExc$year, firstBlock = firstBlock, locationFun = ~years, returnPeriod = 20, returnValue = 10, getParams = TRUE, xNew = data.frame(years = range(Fort$year)), bootSE = FALSE)
Normalize a vector by subtracting off central point and dividing by range
normalize(vec, shift = NULL, lower = NULL, upper = NULL)
normalize(vec, shift = NULL, lower = NULL, upper = NULL)
vec |
vector of values |
shift |
optional central point (if not provided, uses the mean of |
lower |
optional lower end point of range (if not provided uses min of |
upper |
optional upper end point of range (if not provided uses max of |
Remove runs, i.e., consecutive exceedances, from a vector of values and associated indices (days); for use in declustering
remove_runs(y, index, upperTail = TRUE)
remove_runs(y, index, upperTail = TRUE)
y |
vector of values |
index |
vector of indices, one per value, that indicate which elements of |
upperTail |
logical indicating whether values of |
Remove multiple exceedances within non-overlapping blocks of fixed lengths, for use in declustering
screen_within_block(y, index, blockLength = 10)
screen_within_block(y, index, blockLength = 10)
y |
vector of values |
index |
vector of indices, one per value, that indicate which elements of |
blockLength |
length of block within which to remove all but the most extreme value |