Package 'climextRemes'

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

Help Index


Calculates log return period and standard error given return value(s) of interest

Description

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.

Usage

calc_logReturnPeriod_fevd(fit, returnValue, covariates = NULL, upper = FALSE)

Arguments

fit

fitted object from extRemes fevd

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

Details

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 and standard error given return value(s) of interest

Description

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.

Usage

calc_logReturnProb_fevd(
  fit,
  returnValue,
  covariates = NULL,
  getSE = TRUE,
  scaling = 1,
  upper = FALSE
)

Arguments

fit

fitted object from extRemes fevd

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 returnValue is scaled for numerics, this allows names of output to be on original scale

upper

logical value indicating whether upper tail or lower tail is being considered

Details

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 log return probability difference for two sets of covariates and standard error of difference given return value(s) of interest

Description

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.

Usage

calc_logReturnProbDiff_fevd(
  fit,
  returnValue,
  covariates1,
  covariates2,
  getSE = TRUE,
  scaling = 1,
  upper = FALSE
)

Arguments

fit

fitted object from extRemes fevd

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 is desired

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 returnValue is scaled for numerics, this allows names of output to be on original scale

upper

logical value indicating whether upper tail or lower tail is being considered

Details

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 and standard error given return period(s) of interest

Description

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.

Usage

calc_returnValue_fevd(fit, returnPeriod, covariates = NULL)

Arguments

fit

fitted object from extRemes fevd

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 return value difference for two sets of covariates and standard error of difference given return period(s) of interest

Description

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.

Usage

calc_returnValueDiff_fevd(
  fit,
  returnPeriod,
  covariates1,
  covariates2,
  getSE = TRUE
)

Arguments

fit

fitted object from extRemes fevd

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 is desired

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

Details

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 based on binomial models for counts of events relative to possible number of events

Description

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.

Usage

calc_riskRatio_binom(
  y,
  n,
  ciLevel = 0.9,
  ciType,
  bootSE,
  bootControl = NULL,
  lrtControl = NULL
)

Arguments

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 ciLevel. For example, a two-sided 0.90 confidence interval corresponds to a one-sided 0.95 confidence interval.

ciType

character vector indicating which type of confidence intervals to compute. See Details.

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 ciType. See Details.

lrtControl

list containing a single component, bounds, which sets the range inside which the algorithm searches for the endpoints of the likelihood ratio-based confidence interval. This avoids numerical issues with endpoints converging to zero and infinity. If an endpoint is not found within the interval, it is set to NA. Used only when 'lrt' is one of the ciType values. Default is (0.01, 100).

Details

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.

Value

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.

Author(s)

Christopher J. Paciorek

References

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.

Examples

# 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 based on generalized extreme value model fit to block maxima or minima

Description

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.

Usage

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
)

Arguments

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 Details for how the values of y1 should be ordered if there are multiple replicates and the values of x1 are identical for all replicates. For better optimization performance, it is recommended that y1 have magnitude around one (see Details), for which one can use scaling1.

y2

a numeric vector of observed maxima or minima values for the second dataset. Analogous to y1.

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 y1 or (if there is more than one replicate) it can optionally equal the number of observations in a single replicate, in which case the values will be assumed to be the same for all replicates.

x2

analogous to x1 but for the second dataset

locationFun1

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1.

locationFun2

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from x2 for the second dataset. x2 must be supplied if this is anything other than NULL or ~1.

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 x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1. logScale1 controls whether this determines the log of the scale or the scale directly.

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 x2 for the second dataset. x2 must be supplied if this is anything other than NULL or ~1. logScale2 controls whether this determines the log of the scale or the scale directly.

shapeFun1

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1.

shapeFun2

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x2 for the first dataset. x2 must be supplied if this is anything other than NULL or ~1.

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 y1. Used (and therefore required) only when using bootstrapping with the resampling by replicates based on the by element of bootControl.

replicateIndex2

numeric vector providing the index of the replicate corresponding to each element of y2. Analogous to replicateIndex1.

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 y. Likelihood contribution of each observation is multiplied by the corresponding weight.

weights2

a vector providing the weights for each observation in the second dataset. Analogous to weights1.

xNew1

object of the same form as x1, providing covariate/predictor/feature values for which one desires log risk ratios.

xNew2

object of the same form as x2, providing covariate/predictor/feature values for which log risk ratios are desired. Must provide the same number of covariate sets as xNew1 as the risk ratio is based on contrasting return probabilities under xNew1 and xNew2.

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 ciLevel. For example, a two-sided 0.90 confidence interval corresponds to a one-sided 0.95 confidence interval.

ciType

character vector indicating which type of confidence intervals to compute. See Details.

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 Details.

lrtControl

list containing a single component, bounds, which sets the range inside which the algorithm searches for the endpoints of the likelihood ratio-based confidence interval. This avoids numerical issues with endpoints converging to zero and infinity. If an endpoint is not found within the interval, it is set to NA.

optimArgs

a list with named components matching exactly any arguments that the user wishes to pass to optim. See help(optim) for details. Of particular note, 'method' can be used to choose the optimization method used for maximizing the log-likelihood to fit the model and 'control=list(maxit=VALUE)' for a user-chosen VALUE can be used to increase the number of iterations if the optimization is converging slowly.

optimControl

a list with named components matching exactly any elements that the user wishes to pass as the control list to R's optim function. See help(optim) for details. Primarily provided for the Python interface because control can also be passed as part of optimArgs.

initial1

a list with components named 'location', 'scale', and 'shape' providing initial parameter values for the first dataset, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling1, logScale1 and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale1 = TRUE, initial value(s) for 'scale' should be specified on the log scale.

initial2

a list with components named 'location', 'scale', and 'shape' providing initial parameter values for the second dataset, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling2, logScale2 and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale2 = TRUE, initial value(s) for 'scale' should be specified on the log scale.

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 logScale.

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 getParams is TRUE because covariates are normalized before fitting and the fit object is based on the normalized covariates. Similarly, parameters will not match if scaling is not 1.

Details

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.

Value

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.

Author(s)

Christopher J. Paciorek

References

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>.

Examples

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 based on peaks-over-threshold models fit to exceedances over a threshold

Description

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.

Usage

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
)

Arguments

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 y1 have magnitude around one (see Details), for which one can use scaling1.

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 x1 but for the second dataset

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 threshold1 but for the second dataset

locationFun1

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1.

locationFun2

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from x2 for the second dataset. x2 must be supplied if this is anything other than NULL or ~1.

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 x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1. logScale1 controls whether this determines the log of the scale or the scale directly.

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 x2 for the second dataset. x2 must be supplied if this is anything other than NULL or ~1. logScale2 controls whether this determines the log of the scale or the scale directly.

shapeFun1

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x1 for the first dataset. x1 must be supplied if this is anything other than NULL or ~1.

shapeFun2

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x2 for the first dataset. x2 must be supplied if this is anything other than NULL or ~1.

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 returnPeriod and returnValue.

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 returnPeriod and returnValue.

blockIndex1

numeric vector providing the index of the block corresponding to each element of y1. Used only when x1 is provided to match exceedances to the covariate/predictor/feature value for the exceedance or when using bootstrapping with the resampling based on blocks based on the by element of bootControl. If firstBlock1 is not equal to one, then blockIndex1 need not have one as its smallest possible value.

blockIndex2

numeric vector providing the index of the block corresponding to each element of y2. Analogous to blockIndex1.

firstBlock1

single numeric value indicating the numeric value of the first possible block of blockIndex1. For example the values in blockIndex1 might indicate the year of each exceedance with the first year of data being 1969, in which case firstBlock1 would be 1969. Note that the first block may not have any exceedances so it may not be represented in blockIndex1. Used only to adjust blockIndex1 so that the block indices start at one and therefore correspond to the rows of x1.

firstBlock2

single numeric value indicating the numeric value of the first possible block of blockIndex2. Analogous to firstBlock1.

index1

numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of y1. For example if there are 10 original observations and the third, fourth, and seventh values are exceedances, then index1 would be the vector 3,4,7. Used only when declustering is provided to determine which exceedances occur sequentially or within a contiguous set of values of a given length. The actual values are arbitrary; only the lags between the values are used.

index2

numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of y2. Analogous to index1.

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 y1. Used for three purposes: (1) when using bootstrapping with the resampling based on replicates based on the by element of bootControl, (2) to avoid treating values in different replicates as potentially being sequential or within a short interval when removing values based on declustering, and (3) to match outcomes to weights or proportionMissing when either vary by replicate.

replicateIndex2

numeric vector providing the index of the replicate corresponding to each element of y2. Analogous to replicateIndex1.

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 weights1.

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 proportionMissing1.

xNew1

object of the same form as x1, providing covariate/predictor/feature values for which log risk ratios are desired.

xNew2

object of the same form as x2, providing covariate/predictor/feature values for which log risk ratios are desired. Must provide the same number of covariate sets as xNew1 as the risk ratio is based on contrasting return probabilities under xNew1 and xNew2.

declustering

one of NULL, "noruns", or a number. If 'noruns' is specified, only the maximum (or minimum if upperTail = FALSE) value within a set of exceedances corresponding to successive indices is included. If a number, this should indicate the size of the interval (which will be used with the index argument) within which to allow only the largest (or smallest if upperTail = FALSE) value.

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 ciLevel. For example, a two-sided 0.90 confidence interval corresponds to a one-sided 0.95 confidence interval.

ciType

character vector indicating which type of confidence intervals to compute. See Details.

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 Details.

lrtControl

list containing a single component, bounds, which sets the range inside which the algorithm searches for the endpoints of the likelihood ratio-based confidence interval. This avoids numerical issues with endpoints converging to zero and infinity. If an endpoint is not found within the interval, it is set to NA.

optimArgs

a list with named components matching exactly any arguments that the user wishes to pass to optim. See help(optim) for details. Of particular note, 'method' can be used to choose the optimization method used for maximizing the log-likelihood to fit the model and 'control=list(maxit=VALUE)' for a user-chosen VALUE can be used to increase the number of iterations if the optimization is converging slowly.

optimControl

a list with named components matching exactly any elements that the user wishes to pass as the control argument to R's optim function. See help(optim) for details. Primarily provided for the Python interface because control can also be passed as part of optimArgs.

initial1

a list with components named 'location', 'scale', and 'shape' providing initial parameter values for the first dataset, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling1, logScale1 and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale1 = TRUE, initial value(s) for 'scale' should be specified on the log scale.

initial2

a list with components named 'location', 'scale', and 'shape' providing initial parameter values for the second dataset, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling2, logScale2 and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale2 = TRUE, initial value(s) for 'scale' should be specified on the log scale.

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 logScale.

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 getParams is TRUE because covariates are normalized before fitting and the fit object is based on the normalized covariates. Similarly, parameters will not match if scaling is not 1.

Details

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.

Value

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.

Author(s)

Christopher J. Paciorek

References

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>.

Examples

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 to block maxima or minima

Description

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).

Usage

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
)

Arguments

y

a numeric vector of observed maxima or minima values. See Details for how the values of y should be ordered if there are multiple replicates and the values of x are identical for all replicates. For better optimization performance, it is recommended that the y have magnitude around one (see Details), for which one can use scaling.

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 y or (if there is more than one replicate) it can optionally equal the number of observations in a single replicate, in which case the values will be assumed to be the same for all replicates.

locationFun

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the location parameter using columns from x. x must be supplied if this is anything other than NULL or ~1.

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 x. x must be supplied if this is anything other than NULL or ~1. logScale controls whether this determines the log of the scale or the scale directly.

shapeFun

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x. x must be supplied if this is anything other than NULL or ~1.

nReplicates

numeric value indicating the number of replicates.

replicateIndex

numeric vector providing the index of the replicate corresponding to each element of y. Used (and therefore required) only when using bootstrapping with the resampling by replicates based on the by element of bootControl.

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 y. Likelihood contribution of each observation is multiplied by the corresponding weight.

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 logScale.

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 getParams is TRUE because covariates are normalized before fitting and the fit object is based on the normalized covariates. Similarly, parameters will not match if scaling is not 1.

xNew

object of the same form as x, providing covariate/predictor/feature values for which return values/periods/probabilities are desired.

xContrast

object of the same form and dimensions as xNew, providing covariate/predictor/feature values for which to calculate the differences of the return values and/or log return probabilities relative to the values in xNew. This provides a way to estimate the difference in return value or log return probabilities (i.e., log risk ratios).

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 Details.

optimArgs

a list with named components matching exactly any arguments that the user wishes to pass to R's optim function. See help(optim) for details. Of particular note, 'method' can be used to choose the optimization method used for maximizing the log-likelihood to fit the model and control=list(maxit=VALUE) for a user-specified VALUE can be used to increase the number of iterations if the optimization is converging slowly.

optimControl

a list with named components matching exactly any elements that the user wishes to pass as the control argument to R's optim function. See help(optim) for details. Primarily provided for the Python interface because control can also be passed as part of optimArgs.

missingFlag

value to be interpreted as missing values (instead of NA), intended for use in other languages (e.g., Python) calling this function

initial

a list with components named 'location', 'scale', and 'shape' providing initial parameter values, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling, logScale, and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale = TRUE, initial value(s) for 'scale' should be specified on the log scale.

logScale

logical indicating whether optimization for the scale parameter should be done on the log scale. 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).

.normalizeX

logical indicating whether to normalize x values for better numerical performance; default is TRUE.

.getInputs

logical indicating whether to return intermediate objects used in fitting. Defaults to FALSE and intended for internal use only

.allowNoInt

logical indicating whether no-intercept models are allowed. Defaults to TRUE and provided primarily to enable backwards compatibility with versions <= 0.2.2.

Details

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.

Value

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).

Author(s)

Christopher J. Paciorek

References

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>.

Examples

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 to exceedances over a threshold

Description

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).

Usage

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
)

Arguments

y

a numeric vector of exceedance values (values of the outcome variable above the threshold). For better optimization performance, it is recommended that the y have magnitude around one (see Details), for which one can use scaling.

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 x. x must be supplied if this is anything other than NULL or ~1.

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 x. x must be supplied if this is anything other than NULL or ~1. logScale controls whether this determines the log of the scale or the scale directly.

shapeFun

formula, vector of character strings, or indices describing a linear model (i.e., regression function) for the shape parameter using columns from x. x must be supplied if this is anything other than NULL or ~1.

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 returnPeriod and returnValue.

blockIndex

numeric vector providing the index of the block corresponding to each element of y. Used only when x is provided to match exceedances to the covariate/predictor/feature value for the exceedance or when using bootstrapping with the resampling based on blocks based on the by element of bootControl. If firstBlock is not equal to one, then blockIndex need not have one as its smallest possible value.

firstBlock

single numeric value indicating the numeric value of the first possible block of blockIndex. For example the values in blockIndex might indicate the year of each exceedance with the first year of data being 1969, in which case firstBlock would be 1969. Note that the first block may not have any exceedances so it may not be represented in blockIndex. Used only to adjust blockIndex so that the block indices start at one and therefore correspond to the rows of x.

index

numeric vector providing the integer-valued index (e.g., julian day for daily climate data) corresponding to each element of y. For example if there are 10 original observations and the third, fourth, and seventh values are exceedances, then index would be the vector 3,4,7. Used only when declustering is provided to determine which exceedances occur sequentially or within a contiguous set of values of a given length. The actual values are arbitrary; only the lags between the values are used.

nReplicates

numeric value indicating the number of replicates.

replicateIndex

numeric vector providing the index of the replicate corresponding to each element of y. Used for three purposes: (1) when using bootstrapping with the resampling based on replicates based on the by element of bootControl, (2) to avoid treating values in different replicates as potentially being sequential or within a short interval when removing values based on declustering, and (3) to match outcomes to weights or proportionMissing when either vary by replicate.

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 returnPeriod equal to 20 will result in calculation of 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 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 logScale.

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 getParams is TRUE because covariates are normalized before fitting and the fit object is based on the normalized covariates. Similarly, parameters will not match if scaling is not 1.

xNew

object of the same form as x, providing covariate/predictor/feature values for which return values/periods/probabilities are desired.

xContrast

object of the same form and dimensions as xNew, providing covariate/predictor/feature values for which to calculate the differences of the return values and/or log return probabilities relative to the values in xNew. This provides a way to estimate differences in return value or log return probabilities (i.e., log risk ratios).

declustering

one of NULL, "noruns", or a number. If 'noruns' is specified, only the maximum (or minimum if upperTail = FALSE) value within a set of exceedances corresponding to successive indices is included. If a number, this should indicate the size of the interval (which will be used with the index argument) within which to allow only the largest (or smallest if upperTail = FALSE) value.

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 optim function. See help(optim) for details. Of particular note, 'method' can be used to choose the optimization method used for maximizing the log-likelihood to fit the model and control = list(maxit=VALUE) for a user-specified VALUE can be used to increase the number of iterations if the optimization is converging slowly.

optimControl

a list with named components matching exactly any elements that the user wishes to pass as the control argument to R's optim function. See help(optim) for details. Primarily provided for the Python interface because control can also be passed as part of optimArgs.

initial

a list with components named 'location', 'scale', and 'shape' providing initial parameter values, intended for use in speeding up or enabling optimization when the default initial values are resulting in failure of the optimization; note that use of scaling, logScale and .normalizeX = TRUE cause numerical changes in some of the parameters. For example with logScale = TRUE, initial value(s) for 'scale' should be specified on the log scale.

logScale

logical indicating whether optimization for the scale parameter should be done on the log scale. 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).

.normalizeX

logical indicating whether to normalize x values for better numerical performance; default is TRUE.

.getInputs

logical indicating whether to return intermediate objects used in fitting. Defaults to FALSE and intended for internal use only

.allowNoInt

logical indicating whether no-intercept models are allowed. Defaults to TRUE and provided primarily to enable backwards compatibility with versions <= 0.2.2.

Details

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.

Value

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).

Author(s)

Christopher J. Paciorek

References

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>.

Examples

# 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

Description

Normalize a vector by subtracting off central point and dividing by range

Usage

normalize(vec, shift = NULL, lower = NULL, upper = NULL)

Arguments

vec

vector of values

shift

optional central point (if not provided, uses the mean of vec)

lower

optional lower end point of range (if not provided uses min of vec)

upper

optional upper end point of range (if not provided uses max of vec)


Remove consecutive exceedances from a vector

Description

Remove runs, i.e., consecutive exceedances, from a vector of values and associated indices (days); for use in declustering

Usage

remove_runs(y, index, upperTail = TRUE)

Arguments

y

vector of values

index

vector of indices, one per value, that indicate which elements of y are consecutive

upperTail

logical indicating whether values of y are upper (right) tail values (TRUE) or lower (left) tail values. Defaults to FALSE.


Remove multiple exceedances within non-overlapping blocks of fixed length

Description

Remove multiple exceedances within non-overlapping blocks of fixed lengths, for use in declustering

Usage

screen_within_block(y, index, blockLength = 10)

Arguments

y

vector of values

index

vector of indices, one per value, that indicate which elements of y are consecutive

blockLength

length of block within which to remove all but the most extreme value