JuliaEarth / ExtremeStats.jl

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.

Installation

Get the latest stable release with Julia's package manager:

] add ExtremeStats

Usage

Given a collection of values xs (e.g. time series), one can retrieve its maxima:

using ExtremeStats

# find maxima with blocks of size 50
bm = BlockMaxima(xs, 50)

# get values above a threshold of 100.
pm = PeakOverThreshold(xs, 100.)

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.

Plotting

A few plot recipes are defined for maxima as well as for the original values xs:

using Plots

# mean excess plot
excessplot(xs)

# Pareto quantile plot
paretoplot(xs)

# return level plot
returnplot(xs)

Fitting

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)

Statistics

A few statistics are defined:

# return statistics
returnlevels(xs)

# mean excess with previous k values
meanexcess(xs, k)

References

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.