dummy-link

BinningAnalysis

Statistical standard error estimation tools for correlated data

Readme

logo

Build Status Community
DOI

This package provides tools to estimate standard errors and autocorrelation times of correlated time series. A typical example is a Markov chain obtained in a Metropolis Monte Carlo simulation.

Binning tools:

  • Logarithmic Binning
    • Size complexity: O(log(N))
    • Time complexity: O(N)
  • Full Binning (all bin sizes that work out evenly)

Statistical resampling methods:

  • Jackknife resampling.


As per usual, you can install the registered package with

] add BinningAnalysis

Note that there is BinningAnalysisPlots.jl which defines some Plots.jl recipes for LogBinner and FullBinner to facilitate visualizing the error convergence.

Binning tools

Logarithmic Binning

B = LogBinner()
# As per default, 2^32-1 ≈ 4 billion values can be added to the binner. This value can be
# tuned with the `capacity` keyword argument.

push!(B, 4.2)
append!(B, [1,2,3]) # multiple values at once

x  = mean(B)
Δx = std_error(B) # standard error of the mean
tau_x = tau(B) # autocorrelation time

# Alternatively you can provide a time series already in the constructor
x = rand(100)
B = LogBinner(x)

Δx = std_error(B)

Full Binning

B = FullBinner() # <: AbstractVector (lightweight wrapper)

push!(B, 2.0)
append!(B, [1,2,3])

x  = mean(B)
Δx = std_error(B) # standard error of the mean

# Alternatively you can provide a time series already in the constructor
x = rand(100)
F = FullBinner(x)

push!(F, 2.0) # will modify x as F is just a thin wrapper

Δx = std_error(F)

Incremental Binning

# Averages pushed values more and more, starting with no averaging
# Averaging includes 2x more values for every blocksize averages saved
B = IncrementBinner(0.0, blocksize=50)

for x in rand(10_000)
    push!(B, x)
end

# Returns the effective indices for the values saved
# I.e. [1, 2, ...49, 50, 51.5, 53.5, ..., 146.5, 148.5, 151.5, ...]
xs = indices(B)
# Returns the averaged values saved
ys = values(B)

Resampling methods

Jackknife

x = rand(100)

xmean, Δx = jackknife(identity, x) # jackknife estimates for mean and standard error of 

# in this example
# isapprox(Δx, std(x)/sqrt(length(x))) == true

x_inv_mean, Δx_inv = jackknife(identity, 1 ./ x) # # jackknife estimates for mean and standard error of <1/x>

# Multiple time series
x = rand(100)
y = rand(100)

# The inputs of the function `g` must be provided as arguments in `jackknife`.
g(x, y, xy) = x * y / xy  #  / 
g_mean, Δg = jackknife(g, x, y, x .* y)

Error Propagator

ep = ErrorPropagator(N_args=1)
# Essentially a LogBinner that can hold multiple variables. Errors can be derived
# for functions which depend on these variables. The memory overhead of this
# type is O(N_args^2 log(N_samples)), making it much cheaper than jackknife for
# few variables

push!(ep, rand())
append!(ep, rand(99))

# Mean and error of the (first) input
xmean = mean(ep, 1)
Δx = std_error(ep, 1)

# To compute the mean and error of a function we need its gradient
f(x) = x.^2
dfdx(x) = 2x
y = mean(ep, f)[1]
Δy = std_error(ep, dfdx)[1]

# Error propagator with multiple variables:
ep = ErrorPropagator(N_args=3)

# Multiple time series
x = rand(100)
y = rand(100)
append!(ep, x, y, x.*y)

# means and standard error of inputs:
xs = means(ep)
Δxs = std_errors(ep)

# mean and error of a function dependant on x, y and xy
# Note that this function takes a vector input
g(v) = v[1] * v[2] / v[3]  #  / 
dgdx(v) = [v[2]/v[3], v[1]/v[3], -v[1]*v[2]/v[3]^2]
g_mean = mean(ep, g)
Δg = std_error(ep, dgdx)

Convenience wrapper

If you want to calculate the standard error of an existing time series there you can use the convenience wrapper std_error(x[; method=:log]). It takes a keyword argument method, which can be :log, :full, or :jackknife.

ts = rand(1000);
std_error(ts) # default is logarithmic binning
std_error(ts, method=:full)

Supported types

All statistical tools should work with number-like (<: Number) and array-like (<: AbstractArray) elements. Regarding complex numbers, we follow base Julia and define var(x) = var(real(x)) + var(imag(x)).

If you observe unexpected behavior please file an issue!

References

First Commit

01/29/2019

Last Touched

13 days ago

Commits

202 commits