Detailed Description

Resampling algorithms for bias-corrected estimates of functions of averages.

Many physical observables of interest are non-linear functions of statistical averages, for which the naive estimate \( f(\overline{x}_1, \overline{x}_2, \ldots) \) carries a bias of order \( 1/N \) that has to be corrected.

triqs::stat::jackknife performs the delete-one jackknife resampling on one or more sample ranges and returns a tuple containing:

the bias-corrected estimate \( f_{\mathrm{corr}} = N \overline{f} - (N - 1) \overline{f}_{\mathrm{JK}} \),
the jackknife error estimate \( s_f \),
the jackknife function estimate \( \overline{f}_{\mathrm{JK}} \), and
the naive estimate \( \overline{f} = f(\overline{x}_1, \overline{x}_2, \ldots) \).

The MPI-aware variant triqs::stat::jackknife_mpi performs the same calculation on samples distributed across multiple ranks; results are made available on all ranks. Both functions accept any callable and any number of triqs::stat::StatCompatibleRange inputs.

Functions
template<typename F, StatCompatibleRange R, StatCompatibleRange... Rs> requires (not std::same_as<std::remove_cvref_t<F>, mpi::communicator>)
auto	triqs::stat::jackknife (F &&f, R &&rg, Rs &&...rgs)
	Perform jackknife resampling.
template<typename F, StatCompatibleRange R, StatCompatibleRange... Rs>
auto	triqs::stat::jackknife_mpi (mpi::communicator comm, F &&f, R &&rg, Rs &&...rgs)
	Perform jackknife resampling with MPI support.

Function Documentation

◆ jackknife()

template<typename F, StatCompatibleRange R, StatCompatibleRange... Rs>
requires (not std::same_as<std::remove_cvref_t<F>, mpi::communicator>)

auto triqs::stat::jackknife	(	F &&	f,
		R &&	rg,
		Rs &&...	rgs )

#include <triqs/stat/jackknife.hpp>

Perform jackknife resampling.

Calculate the value and error of a general function \( f \) of the averages of sampled observables using jackknife resampling.

Let \( \left( X_1, X_2, \ldots \right) \) denote the different observables and let \( \left\{ x_j^{(i)} : i = 1, \ldots, N \right\} \) be the set of their corresponding samples. We define the deleted averages

\[ \overline{x}_j(i) = \frac{1}{N - 1} \sum_{k \neq i} x_j^{(k)} = \frac{N \overline{x}_j - x_j^{(i)}}{N - 1} \; , \]

and their corresponding function evaluations

\[ \overline{f}(i) = f\left( \overline{x}_1(i), \overline{x}_2(i), \ldots \right) \; . \]

We can then calculate the following quantities:

naive function estimate: \( \overline{f} = f(\overline{x}_1, \overline{x}_2 \ldots) \),
jackknife function estimate: \( \overline{f}_{\mathrm{JK}} = \frac{1}{N} \sum_{i = 1}^N \overline{f}(i) \),
bias corrected function estimate: \( f_{\mathrm{corr}} = N \overline{f} - (N - 1) \overline{f}_{\mathrm{JK}} \) and
jackknife error estimate: \( s_f^2 = \frac{N - 1}{N} \sum_{i = 1}^N \left( \overline{f}(i) - \overline{f}_{\mathrm{JK}} \right)^2 \).

Template Parameters

F	Callable type.
R	triqs::stat::StatCompatibleRange type.
Rs	triqs::stat::StatCompatibleRange types.

Parameters

f	Callable to be applied to the averages of the sampled observables.
rg	Range containing the samples of one observable.
rgs	Range objects containing the samples of the other observables.

Returns: std::tuple containing the bias corrected function estimate, the jackknife error estimate, the jackknife function estimate and the naive function estimate.

Definition at line 120 of file jackknife.hpp.

◆ jackknife_mpi()

template<typename F, StatCompatibleRange R, StatCompatibleRange... Rs>

auto triqs::stat::jackknife_mpi	(	mpi::communicator	comm,
		F &&	f,
		R &&	rg,
		Rs &&...	rgs )

#include <triqs/stat/jackknife.hpp>

Perform jackknife resampling with MPI support.

See triqs::stat::jackknife for more details.