[triqs/statistics] Tools for statistical analysis: binning, jackknife and autocorrelation time


Given the statistical samples \(\lbrace x_i\rbrace _{i=0\dots N-1}\) and \(\lbrace y_i\rbrace _{i=0\dots N-1}\) of random variables \(X\) and \(Y\), one often wants to compute the estimate of the following observables:

\(\langle X \rangle\), \(\langle X\rangle/\langle Y \rangle\), \(\langle X \rangle^2\), or in general \(f(\langle X \rangle , \langle Y \rangle, \dots)\)

as well as the estimate of the errors:

\(\Delta\langle X \rangle\), \(\Delta\langle X\rangle /\langle Y \rangle\), \(\Delta\langle X\rangle ^2\) or \(\Delta f(\langle X \rangle , \langle Y \rangle, \dots)\)

The estimate of the expectation values is the empirical average :

\(\langle X \rangle \approx \frac{1}{N} \sum_{i=0}^{N-1} x_i\)

If the samples are independent from each other and \(f\) is a linear function of its variables (e.g \(f=Id\)):

\((\Delta \langle X \rangle)^2 \approx \frac{\frac{N-1}{N} \sigma^2({x})}{N}\)

where \(\sigma^2({x})\) is the empirical variance of the sample.

In the general case, however,

  • the samples are correlated (with a characteristic correlation time): one needs to bin the series to obtain a reliable estimate of the error bar
  • \(f\) is non-linear in its arguments: one needs to jackknife the series

This library allows one to reliably compute the estimates of \(f(\langle X \rangle , \langle Y \rangle, \dots)\) and its error bar \(\Delta f(\langle X \rangle , \langle Y \rangle, \dots)\) in the general case.


average_and_error takes an object with the Observable concept (see below) and returns a struct with two members val and error:
  • val is the estimate of the expectation value of the random variable for a given sample of it
  • error is the estimate of the error on this expectation value for the given sample



An object has the concept of a TimeSeries if it has the following member functions:

Return type Name
value_type operator[](int i)
int size()

and the following member type:

Name Property
value_type belong to an algebra (has +,-,* operators)


An object has the concept of an observable if it is a TimeSeries and has, additionally, the following member function:

Return type Name
observable& operator<<(T x)

where T belongs to an algebra.


#include <triqs/clef.hpp>
#include <triqs/statistics.hpp>
using namespace triqs::statistics;
int main() {
  observable<double> X;
  X << 1.0;
  X << -1.0;
  X << .5;
  X << .0;
  std::cout << average_and_error(X) << std::endl;
  std::cout << average_and_error(X * X) << std::endl;
  return 0;


histogram is a lightweight object used to represent and to accumulate a histogram of a real random variable.

#include <triqs/statistics.hpp>
using namespace triqs::statistics;

int main() {

  // Histogram with 21 bins over [0;10] range
  histogram hist{0, 10, 21};

  // General information about histogram
  std::cout << "Number of bins = " << hist.size() << std::endl;
  auto limits = hist.limits();
  std::cout << "Histogram range [" << limits.first << ";" << limits.second << "]" << std::endl;

  // Accumulate some value
  for (double x : {-10.0, -0.05, 1.1, 2.0, 2.2, 2.9, 3.4, 5.0, 9.0, 10.0, 10.5, 12.1, 32.2}) hist << x;

  // Print accumulated histogram
  std::cout << "Histogram:\n" << hist << std::endl;

  // Accumulated and lost samples
  std::cout << "Accumulated data points: " << hist.n_data_pts() << std::endl;
  std::cout << "Lost data points: " << hist.n_lost_pts() << std::endl;

  // Direct access to histogram data
  std::cout << "Histogram data: " << hist.data() << std::endl;

  // Make normalized histogram (PDF)
  std::cout << "PDF:\n" << pdf(hist) << std::endl;

  // Make integrated and normalized histogram (CDF)
  std::cout << "CDF:\n" << cdf(hist) << std::endl;

  return 0;