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100 lines
3.4 KiB
ReStructuredText
100 lines
3.4 KiB
ReStructuredText
Tools for statistical analysis: binning and jackknife
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======================================================
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Introduction
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-------------
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Given the statistical samples :math:`\lbrace x_i\rbrace _{i=0\dots N-1}` and :math:`\lbrace y_i\rbrace _{i=0\dots N-1}` of random variables :math:`X` and :math:`Y`, one often wants to compute the estimate of the following observables:
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:math:`\langle X \rangle`, :math:`\langle X\rangle/\langle Y \rangle`, :math:`\langle X \rangle^2`, or in general :math:`f(\langle X \rangle , \langle Y \rangle, \dots)`
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as well as the estimate of the errors:
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:math:`\Delta\langle X \rangle`, :math:`\Delta\langle X\rangle /\langle Y \rangle`, :math:`\Delta\langle X\rangle ^2` or :math:`\Delta f(\langle X \rangle , \langle Y \rangle, \dots)`
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The estimate of the expectation values is the empirical average :
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:math:`\langle X \rangle \approx \frac{1}{N} \sum_{i=0}^{N-1} x_i`
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If the samples are independent from each other and :math:`f` is a linear function of its variables (e.g :math:`f=Id`):
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:math:`(\Delta \langle X \rangle)^2 \approx \frac{\frac{N-1}{N} \sigma^2({x})}{N}`
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where :math:`\sigma^2({x})` is the empirical variance of the sample.
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In the general case, however,
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- the samples are correlated (with a characteristic correlation time): one needs to :doc:`bin <binning>` the series to obtain a reliable estimate of the error bar
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- :math:`f` is non-linear in its arguments: one needs to :doc:`jackknife <jackknife>` the series
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This library allows one to reliably compute the estimates of :math:`f(\langle X \rangle , \langle Y \rangle, \dots)` and its error bar :math:`\Delta f(\langle X \rangle , \langle Y \rangle, \dots)` in the general case.
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Synopsis
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---------
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`average_and_error` takes an object with the **Observable** concept (see below) and returns a struct with two members `val` and `error`:
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- `val` is the estimate of the expectation value of the random variable for a given sample of it
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- `error` is the estimate of the error on this expectation value for the given sample
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Concepts
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---------
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TimeSeries
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~~~~~~~~~~~
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An object has the concept of a TimeSeries if it has the following member functions:
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+-------------+-------------------+
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| Return type | Name |
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+=============+===================+
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| value_type | operator[](int i) |
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+-------------+-------------------+
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| int | size() |
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+-------------+-------------------+
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and the following member type:
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+-------------+------------------------------------------+
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| Name | Property |
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+=============+==========================================+
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| value_type | belong to an algebra (has +,- operators) |
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+-------------+------------------------------------------+
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Observable
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~~~~~~~~~~~
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An object has the concept of an observable if it is a TimeSeries and has, additionally, the following member function:
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+-------------+-----------------+
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| Return type | Name |
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+=============+=================+
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| observable& | operator<<(T x) |
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+-------------+-----------------+
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where `T` belongs to an algebra.
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Example
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--------
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.. compileblock::
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#include <triqs/clef.hpp>
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#include <triqs/statistics.hpp>
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using namespace triqs::statistics;
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int main(){
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observable<double> X;
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X<<1.0;
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X<<-1.0;
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X<<.5;
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X<<.0;
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std::cout << average_and_error(X) << std::endl;
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std::cout << average_and_error(X*X) << std::endl;
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return 0;
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}
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.. toctree::
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binning
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jackknife
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autocorrelation_time
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ising2d
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