..
   Generated automatically by cpp2rst

.. highlight:: c
.. role:: red
.. role:: green
.. role:: param


.. _triqs__stat__jackknife:

triqs::stat::jackknife
======================

*#include <triqs/stat.hpp>*



**Synopsis**

 .. rst-class:: cppsynopsis

    1. | :green:`template<typename F, typename A>`
       | auto :red:`jackknife` (F && :param:`f`, A const &... :param:`a`)

    2. | :green:`template<typename F, typename T>`
       | auto :red:`jackknife` (F && :param:`f`, accumulator<T> const &... :param:`a`)

Calculate the value and error of a general function :math:`f` of the average of sampled observables :math:`f\left(\langle \mathbf{a} \rangle\right)`, using jackknife resampling.



 **1)**   Directly pass data-series in vector like objects



 **2)**   Pass :ref:`accumulators <triqs__stat__accumulator>`, where the jacknife acts on the :ref:`linear binned data <accumulator_linear_bins>`





Template parameters
^^^^^^^^^^^^^^^^^^^

 * :param:`F` return type of function :param:`f` which acts on data

 * :param:`A` vector-like object, defining size() and []

 * :param:`T` type of data stored in the accumulators


Parameters
^^^^^^^^^^

 * :param:`a` one or multiple series with data: :math:`\mathbf{a} = \{a_1, a_2, a_3, \ldots\}`
   Pre-condition: if more than one series is passed, the series have to be equal in size

 * :param:`f` a function which acts on the :math:`i^\mathrm{th}` elements of the series in :param:`a`:


.. math::
		\left(a_1[i], a_2[i],a_3[i],\ldots\right) \to f\left(a_1[i],a_2[i],a_3[i],\ldots\right)

..


Returns
^^^^^^^

std::tuple with four statistical estimators :math:`\left(f_\mathrm{J}^{*}, \Delta{f}_\mathrm{J}, f_\mathrm{J}, f_\mathrm{direct}\right)`, defined below.

 Jackknife resampling takes :math:`N` data points :math:`\mathbf{a}[i]` and creates :math:`N` samples ("jackknifed data"), which we denote :math:`\tilde{\mathbf{a}}[i]`. We calculate three statistical estimators for :math:`f\left(\langle \mathbf{a} \rangle\right)`:
   * The function :math:`f` applied to observed mean of the data


.. math::
		f_\mathrm{direct} = f\left(\bar{\mathbf{a}}\right),\quad \bar{\mathbf{a}} = \frac{1}{N}\sum_{i=0}^{N}\mathbf{a}[i]

..

   * The jacknife estimate defined as


.. math::
		f_\mathrm{J} = \frac{1}{N}\sum_{i=0}^N f(\tilde{\mathbf{a}}[i])

..

   * The jacknife estimate, with bias correction to remove :math:`O(1/N)` effects


.. math::
		f_\mathrm{J}^{*} = N f_\mathrm{direct} - (N - 1) f_\mathrm{J}

..

 Additionally, an estimate in the errror of :math:`f\left(\langle \mathbf{a} \rangle\right)` is given by the jacknife as


.. math::
		\Delta{f}_J = \sqrt{N-1} \cdot \sigma_f

..

 where :math:`\sigma_f` is the standard deviation of :math:`\left\{f(\tilde{\mathbf{a}}[0]), f(\tilde{\mathbf{a}}[1]), \ldots, f(\tilde{\mathbf{a}}[N])\right\}`.