statistics.h File Reference

Statistical functions. More...

#include "../core/constants.h"
#include "../core/real_analysis.h"
#include "../core/special.h"
#include "../calculus/integral.h"
#include "../calculus/gauss.h"
#include "../core/dataset.h"

Go to the source code of this file.

Namespaces
namespace	theoretica
	Main namespace of the library which contains all functions and objects.
namespace	theoretica::stats
	Statistical functions.

Functions
template<typename Dataset >
real	theoretica::stats::mean (const Dataset &X)
	Compute the mean of a dataset.
template<typename Dataset >
real	theoretica::stats::range (const Dataset &X)
	Computes the range of a data set, defined as \(x_{max} - {x_min}\).
template<typename Dataset >
real	theoretica::stats::semidispersion (const Dataset &X)
	Computes the maximum semidispersion of a data set defined as \((x_{max} - {x_min}) / 2\).
template<typename Dataset >
real	theoretica::stats::propagate_sum (const Dataset &sigma)
	Propagate the error over a sum of random variables under quadrature, as \(\sqrt{\sum_{i = 1}^n \sigma_i^2}\), where each \(\sigma_i\) corresponds to the standard deviation of a variable.
template<typename Dataset1 , typename Dataset2 >
real	theoretica::stats::propagate_product (const Dataset1 &sigma, const Dataset2 &mean)
	Propagate the error over a product of random variables under quadrature, as \(\sqrt{\sum_{i = 1}} (\sigma_i / \mu_i)^2}\), where each \(\sigma_i\) corresponds to the standard deviation of a variable.
template<typename Dataset >
real	theoretica::stats::total_sum_squares (const Dataset &X)
	Compute the total sum of squares (TSS) of a given dataset as \(sum(square(x_i - x_{mean}))\) using Welford's one-pass method.
template<typename Dataset >
real	theoretica::stats::variance (const Dataset &X, unsigned int constraints=1)
	Compute the variance given a dataset and the number of constraints.
template<typename Dataset >
void	theoretica::stats::moments2 (const Dataset &X, real &out_mean, real &out_variance, unsigned int constraints=1)
	Compute the mean and the variance of a dataset in a single pass, using Welford's method, with the given number of constraints (defaults to 1 for Bessel's correction).
template<typename Dataset >
real	theoretica::stats::stdev (const Dataset &data, unsigned int constraints=1)
	Compute the standard deviation given a dataset and the number of constraints.
template<typename Dataset >
real	theoretica::stats::stdom (const Dataset &X)
	Compute the standard deviation of the mean given a dataset.
template<typename Dataset >
real	theoretica::stats::standard_relative_error (const Dataset &X)
	Compute the relative error on a dataset using estimates of its mean and standard deviation, with the given number of constraints (defaults to 1 for Bessel's correction).
template<typename Dataset1 , typename Dataset2 >
real	theoretica::stats::covariance (const Dataset1 &X, const Dataset2 &Y, unsigned int constraints=1)
	Compute the covariance between two datasets with the given number of constraints.
template<typename Dataset1 , typename Dataset2 >
real	theoretica::stats::correlation_coefficient (const Dataset1 &X, const Dataset2 &Y)
	Compute Pearson's correlation coefficient R between two datasets.
template<typename Dataset >
real	theoretica::stats::autocorrelation (const Dataset &X, unsigned int n=1)
	Compute the lag-n autocorrelation of a dataset as \(\).
template<typename Dataset >
real	theoretica::stats::absolute_deviation (const Dataset &X)
	Compute the mean absolute deviation of a dataset as \(\frac{\sum_{i = 1}^n |x_i - \hat \mu|}{n}\).
template<typename Dataset >
real	theoretica::stats::skewness (const Dataset &X)
	Compute the skewness of a dataset as \(\frac{\sum_{i=1}^n (\frac{x_i - \hat \mu}{\hat \sigma})^3}{n}\).
template<typename Dataset >
real	theoretica::stats::kurtosis (const Dataset &X)
	Compute the normalized kurtosis of a dataset as \(\frac{\sum_{i=1}^n (\frac{x_i - \hat \mu}{\hat \sigma})^4}{n} - 3\).
template<typename RealFunction >
real	theoretica::stats::gaussian_expectation (RealFunction g, real mean, real sigma)
	Compute the expectation value of a given function with respect to a Gaussian distribution with the given parameters.
real	theoretica::stats::z_score (real x, real mean, real sigma)
	Compute the Z-score of an observed value with respect to a Gaussian distribution with the given parameters.
template<typename Dataset >
Dataset	theoretica::stats::normalize_z_score (const Dataset &X)
	Normalize a data set using Z-score normalization.
template<typename Dataset1 , typename Dataset2 , typename Dataset3 >
real	theoretica::stats::chi_square (const Dataset1 &O, const Dataset2 &E, const Dataset3 &sigma)
	Compute the chi-square from the set of observed quantities, expected quantities and errors.
real	theoretica::stats::pvalue_chi_squared (real chi_sqr, unsigned int ndf)
	Compute the (right-tailed) p-value associated to a computed Chi-square value as the integral of the Chi-squared distribution from the given value to infinity (right-tailed).
template<typename Dataset1 , typename Dataset2 , typename Dataset3 >
real	theoretica::stats::chi_square_linear (const Dataset1 &X, const Dataset2 &Y, const Dataset3 &sigma, real intercept, real slope)
	Compute the chi-square on a linear regression, as the sum of the squares of the residuals divided by the standard deviation.
template<typename Dataset1 , typename Dataset2 , typename Dataset3 >
real	theoretica::stats::reduced_chi_square_linear (const Dataset1 &X, const Dataset2 &Y, const Dataset3 &sigma, real intercept, real slope)
	Compute the reduced chi-squared on a linear regression, computed as the usual chi-square (computed by chi_square_linear) divided by the number of degrees of freedom of the model ( \(N - 2\)).

Detailed Description

Statistical functions.