NAG FL Interfaceg01kkf (pdf_​gamma_​vector)

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1Purpose

g01kkf returns a number of values of the probability density function (PDF), or its logarithm, for the gamma distribution.

2Specification

Fortran Interface
 Subroutine g01kkf ( ilog, lx, x, la, a, lb, b, pdf,
 Integer, Intent (In) :: ilog, lx, la, lb Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: x(lx), a(la), b(lb) Real (Kind=nag_wp), Intent (Out) :: pdf(*)
#include <nag.h>
 void g01kkf_ (const Integer *ilog, const Integer *lx, const double x[], const Integer *la, const double a[], const Integer *lb, const double b[], double pdf[], Integer ivalid[], Integer *ifail)
The routine may be called by the names g01kkf or nagf_stat_pdf_gamma_vector.

3Description

The gamma distribution with shape parameter ${\alpha }_{i}$ and scale parameter ${\beta }_{i}$ has PDF
 $f (xi,αi,βi) = 1 βi αi Γ(αi) xi αi-1 e -xi / βi if ​ xi ≥ 0 ; αi , βi > 0 f(xi,αi,βi)=0 otherwise.$
If $0.01\le {x}_{i},{\alpha }_{i},{\beta }_{i}\le 100$ then an algorithm based directly on the gamma distribution's PDF is used. For values outside this range, the function is calculated via the Poisson distribution's PDF as described in Loader (2000) (see Section 9).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

4References

Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

5Arguments

1: $\mathbf{ilog}$Integer Input
On entry: the value of ilog determines whether the logarithmic value is returned in pdf.
${\mathbf{ilog}}=0$
$f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)$, the probability density function is returned.
${\mathbf{ilog}}=1$
$\mathrm{log}\left(f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)\right)$, the logarithm of the probability density function is returned.
Constraint: ${\mathbf{ilog}}=0$ or $1$.
2: $\mathbf{lx}$Integer Input
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
3: $\mathbf{x}\left({\mathbf{lx}}\right)$Real (Kind=nag_wp) array Input
On entry: ${x}_{i}$, the values at which the PDF is to be evaluated with ${x}_{i}={\mathbf{x}}\left(j\right)$, , for $i=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
4: $\mathbf{la}$Integer Input
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
5: $\mathbf{a}\left({\mathbf{la}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\alpha }_{i}$, the shape parameter with ${\alpha }_{i}={\mathbf{a}}\left(j\right)$, .
Constraint: ${\mathbf{a}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
6: $\mathbf{lb}$Integer Input
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
7: $\mathbf{b}\left({\mathbf{lb}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\beta }_{i}$, the scale parameter with ${\beta }_{i}={\mathbf{b}}\left(j\right)$, .
Constraint: ${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
8: $\mathbf{pdf}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array pdf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: $f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)$ or $\mathrm{log}\left(f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)\right)$.
9: $\mathbf{ivalid}\left(*\right)$Integer array Output
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${\alpha }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=2$
${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=3$
$\frac{{x}_{i}}{{\beta }_{i}}$ overflows, the value returned should be a reasonable approximation.
10: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of x, a or b was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ilog}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ilog}}=0$ or $1$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lx}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{la}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lb}}>0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

8Parallelism and Performance

g01kkf is not threaded in any implementation.

Due to the lack of a stable link to Loader (2000) paper, we give a brief overview of the method, as applied to the Poisson distribution. The Poisson distribution has a continuous mass function given by,
 $p(x;λ) = λx x! e-λ .$ (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
 $log(p(x;λ)) = x log⁡λ - log(x!) - λ .$
For large $x$ and $\lambda$, $x\mathrm{log}\lambda$ and $\mathrm{log}\left(x!\right)$ are very large, of the same order of magnitude and when calculated have rounding errors. The subtraction of these two terms can, therefore, result in a number, many orders of magnitude smaller and hence we lose accuracy due to subtraction errors. For example for $x=2×{10}^{6}$ and $\lambda =2×{10}^{6}$, $\mathrm{log}\left(x!\right)\approx 2.7×{10}^{7}$ and $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.17326744645834$. But calculated with the method shown later we have $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.1732674441334492$. The difference between these two results suggests a loss of about $7$ significant figures of precision.
Loader introduces an alternative way of expressing (1) based on the saddle point expansion,
 $log(p(x;λ)) = log(p(x;x)) - D(x;λ) ,$ (2)
where $D\left(x;\lambda \right)$, the deviance for the Poisson distribution is given by,
 $D(x;λ) = log(p(x;x)) - log(p(x;λ)) , = λ D0 ( x λ ) ,$ (3)
and
 $D0 (ε) = ε log⁡ε + 1 - ε .$
For $\epsilon$ close to $1$, ${D}_{0}\left(\epsilon \right)$ can be evaluated through the series expansion
 $λ D0 ( x λ ) = (x-λ) 2 x+λ + 2x ∑ j=1 ∞ v 2j+1 2j+1 , where ​ v = x-λ x+λ ,$
otherwise ${D}_{0}\left(\epsilon \right)$ can be evaluated directly. In addition, Loader suggests evaluating $\mathrm{log}\left(x!\right)$ using the Stirling–De Moivre series,
 $log(x!) = 12 log⁡ (2πx) + x log(x) -x + δ(x) ,$ (4)
where the error $\delta \left(x\right)$ is given by
 $δ(x) = 112x - 1 360x3 + 1 1260x5 + O (x−7) .$
Finally $\mathrm{log}\left(p\left(x;\lambda \right)\right)$ can be evaluated by combining equations (1)(4) to get,
 $p (x;λ) = 1 2πx e - δ(x) - λ D0 (x/λ) .$

10Example

This example prints the value of the gamma distribution PDF at six different points ${x}_{i}$ with differing ${\alpha }_{i}$ and ${\beta }_{i}$.

10.1Program Text

Program Text (g01kkfe.f90)

10.2Program Data

Program Data (g01kkfe.d)

10.3Program Results

Program Results (g01kkfe.r)