This package provides a set of tools for analysing and estimating extreme value distributions.
It defines two types, BlockMaxima and PeakOverThreshold, which can be used to filter a
collection of values into a collection of maxima.
Given a collection of maxima produced by either model above, one can start estimating heavy-tail distributions and plotting classical extreme value statistics.
Get the latest stable release with Julia's package manager:
] add ExtremeStatsGiven a collection of values xs (e.g. time series), one can define two maxima models:
using ExtremeStats
# maximum values within blocks of size 50
bm = BlockMaxima(xs, 50)
# maximum values above a threshold of 100.0
pm = PeakOverThreshold(xs, 100.0)For the block maxima model, the values xs need to represent a measurement over time,
whereas the peak over threshold model does not assume any ordering in the data. Both
models are lazy, and the maxima are only returned via a collect call.
Generalized extreme value (GEV) and generalized Pareto (GP) distributions from the Distributions.jl
package can be fit to maxima via constrained optimization (maximum likelihood + extreme value index constraints):
using Distributions
# fit GEV to block maxima
fit(GeneralizedExtremeValue, bm)
# fit GP to peak over threshold
fit(GeneralizedPareto, pm)A few plots are defined as Makie.jl recipes:
# choose a backend
import GLMakie
# mean excess plot
excessplot(xs)
# Pareto quantile plot
paretoplot(xs)
# return level plot
returnplot(xs)Please refer to our visual regression tests to learn more about axes conventions.
The following statistics are exported for convenience:
# return statistics
returnlevels(xs)
# mean excess with previous k values
meanexcess(xs, k)The book An Introduction to Statistical Modeling of Extreme Values by Stuart Coles gives a practical introduction to the theory. Most other books I've encountered are too theoretical or expose topics that are somewhat disconnected.