...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/non_central_chi_squared.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class non_central_chi_squared_distribution; typedef non_central_chi_squared_distribution<> non_central_chi_squared; template <class RealType, class Policy> class non_central_chi_squared_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor: non_central_chi_squared_distribution(RealType v, RealType lambda); // Accessor to degrees of freedom parameter v: RealType degrees_of_freedom()const; // Accessor to non centrality parameter lambda: RealType non_centrality()const; // Parameter finders: static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p); template <class A, class B, class C> static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c); static RealType find_non_centrality(RealType v, RealType x, RealType p); template <class A, class B, class C> static RealType find_non_centrality(const complemented3_type<A,B,C>& c); }; }} // namespaces
The noncentral chisquared distribution is a generalization of the Chi Squared Distribution. If X_{i} are ν independent, normally distributed random variables with means μ_{i} and variances σ_{i}^{2}, then the random variable
is distributed according to the noncentral chisquared distribution.
The noncentral chisquared distribution has two parameters: ν which specifies the number of degrees of freedom (i.e. the number of X_{i}), and λ which is related to the mean of the random variables X_{i} by:
(Note that some references define λ as one half of the above sum).
This leads to a PDF of:
where f(x;k) is the central chisquared distribution PDF, and I_{v}(x) is a modified Bessel function of the first kind.
The following graph illustrates how the distribution changes for different values of λ:
non_central_chi_squared_distribution(RealType v, RealType lambda);
Constructs a ChiSquared distribution with v degrees of freedom and noncentrality parameter lambda.
Requires v > 0 and lambda >= 0, otherwise calls domain_error.
RealType degrees_of_freedom()const;
Returns the parameter v from which this object was constructed.
RealType non_centrality()const;
Returns the parameter lambda from which this object was constructed.
static RealType find_degrees_of_freedom(RealType lambda, RealType x, RealType p);
This function returns the number of degrees of freedom v
such that: cdf(non_central_chi_squared<RealType, Policy>(v, lambda), x) == p
template <class A, class B, class C> static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c);
When called with argument boost::math::complement(lambda, x,
q)
this function returns the number of degrees of freedom v
such that:
cdf(complement(non_central_chi_squared<RealType, Policy>(v, lambda), x)) == q
.
static RealType find_non_centrality(RealType v, RealType x, RealType p);
This function returns the non centrality parameter lambda such that:
cdf(non_central_chi_squared<RealType, Policy>(v, lambda), x) == p
template <class A, class B, class C> static RealType find_non_centrality(const complemented3_type<A,B,C>& c);
When called with argument boost::math::complement(v,
x,
q)
this function returns the non centrality parameter lambda
such that:
cdf(complement(non_central_chi_squared<RealType, Policy>(v, lambda), x)) == q
.
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random variable is [0, +∞].
There is a worked example for the noncentral chisquared distribution.
The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types. The failures in the comparison to the R Math library, seem to be mostly in the corner cases when the probablity would be very small. Unless otherwise specified any floatingpoint type that is narrower than the one shown will have effectively zero error.
Table 5.6. Error rates for non central chi squared CDF
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Non Central Chi Squared, medium parameters 
Max = 0.99ε (Mean = 0.0544ε) 
Max = 46.5ε (Mean = 10.3ε) 
Max = 115ε (Mean = 13.9ε) 
Max = 48.9ε (Mean = 10ε) 
Non Central Chi Squared, large parameters 
Max = 1.07ε (Mean = 0.102ε) 
Max = 3.07e+03ε (Mean = 336ε) 
Max = 6.17e+03ε (Mean = 677ε) 
Max = 9.79e+03ε (Mean = 723ε) 
Table 5.7. Error rates for non central chi squared CDF complement
GNU C++ version 7.1.0 
GNU C++ version 7.1.0 
Sun compiler version 0x5150 
Microsoft Visual C++ version 14.1 


Non Central Chi Squared, medium parameters 
Max = 0.96ε (Mean = 0.0635ε) 
Max = 107ε (Mean = 17.2ε) 
Max = 171ε (Mean = 22.8ε) 
Max = 98.6ε (Mean = 15.8ε) 
Non Central Chi Squared, large parameters 
Max = 2.11ε (Mean = 0.278ε) 
Max = 5.02e+03ε (Mean = 630ε) 
Max = 5.1e+03ε (Mean = 577ε) 
Max = 5.43e+03ε (Mean = 705ε) 
Error rates for the quantile functions are broadly similar. Special mention
should go to the mode
function:
there is no closed form for this function, so it is evaluated numerically
by finding the maxima of the PDF: in principal this can not produce an
accuracy greater than the square root of the machine epsilon.
There are two sets of test data used to verify this implementation: firstly we can compare with published data, for example with Table 6 of "SelfValidating Computations of Probabilities for Selected Central and Noncentral Univariate Probability Functions", Morgan C. Wang and William J. Kennedy, Journal of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994), pp. 878887. Secondly, we have tables of test data, computed with this implementation and using interval arithmetic  this data should be accurate to at least 50 decimal digits  and is the used for our accuracy tests.
The CDF and its complement are evaluated as follows:
First we determine which of the two values (the CDF or its complement) is likely to be the smaller: for this we can use the relation due to Temme (see "Asymptotic and Numerical Aspects of the Noncentral ChiSquare Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No. 5, 5563, 1993) that:
F(ν,λ;ν+λ) ≈ 0.5
and so compute the CDF when the random variable is less than ν+λ, and its complement when the random variable is greater than ν+λ. If necessary the computed result is then subtracted from 1 to give the desired result (the CDF or its complement).
For small values of the non centrality parameter, the CDF is computed using the method of Ding (see "Algorithm AS 275: Computing the NonCentral #2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol. 41, No. 2. (1992), pp. 478482). This uses the following series representation:
which requires just one call to gamma_p_derivative with the subsequent terms being computed by recursion as shown above.
For larger values of the noncentrality parameter, Ding's method can take an unreasonable number of terms before convergence is achieved. Furthermore, the largest term is not the first term, so in extreme cases the first term may be zero, leading to a zero result, even though the true value may be nonzero.
Therefore, when the noncentrality parameter is greater than 200, the method due to Krishnamoorthy (see "Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient", Denise Benton and K. Krishnamoorthy, Computational Statistics & Data Analysis, 43, (2003), 249267) is used.
This method uses the well known sum:
Where P_{a}(x) is the incomplete gamma function.
The method starts at the λth term, which is where the Poisson weighting function achieves its maximum value, although this is not necessarily the largest overall term. Subsequent terms are calculated via the normal recurrence relations for the incomplete gamma function, and iteration proceeds both forwards and backwards until sufficient precision has been achieved. It should be noted that recurrence in the forwards direction of P_{a}(x) is numerically unstable. However, since we always start after the largest term in the series, numeric instability is introduced more slowly than the series converges.
Computation of the complement of the CDF uses an extension of Krishnamoorthy's method, given that:
we can again start at the λ'th term and proceed in both directions from there until the required precision is achieved. This time it is backwards recursion on the incomplete gamma function Q_{a}(x) which is unstable. However, as long as we start well before the largest term, this is not an issue in practice.
The PDF is computed directly using the relation:
Where f(x; v) is the PDF of the central Chi Squared Distribution and I_{v}(x) is a modified Bessel function, see cyl_bessel_i. For small values of the noncentrality parameter the relation in terms of cyl_bessel_i is used. However, this method fails for large values of the noncentrality parameter, so in that case the infinite sum is evaluated using the method of Benton and Krishnamoorthy, and the usual recurrence relations for successive terms.
The quantile functions are computed by numeric inversion of the CDF. An improve starting quess is from Thomas Luu, Fast and accurate parallel computation of quantile functions for random number generation, Doctorial Thesis, 2016.
There is no closed form for the mode of the noncentral chisquared distribution: it is computed numerically by finding the maximum of the PDF. Likewise, the median is computed numerically via the quantile.
The remaining nonmember functions use the following formulas:
Some analytic properties of noncentral distributions (particularly unimodality, and monotonicity of their modes) are surveyed and summarized by:
Andrea van Aubel & Wolfgang Gawronski, Applied Mathematics and Computation, 141 (2003) 312.