NAG Library Function Document

1Purpose

nag_mv_gaussian_mixture (g03gac) performs a mixture of Normals (Gaussians) for a given (co)variance structure.

2Specification

 #include #include
 void nag_mv_gaussian_mixture (Integer n, Integer m, const double x[], Integer pdx, const Integer isx[], Integer nvar, Integer ng, Nag_Boolean popt, double prob[], Integer tdprob, Integer *niter, Integer riter, double w[], double g[], Nag_VarCovar sopt, double s[], double f[], double tol, double *loglik, NagError *fail)

3Description

A Normal (Gaussian) mixture model is a weighted sum of $k$ group Normal densities given by,
 $p x∣w,μ,Σ = ∑ j=1 k wj g x∣μj,Σj , x∈ℝp$
where:
• $x$ is a $p$-dimensional object of interest;
• ${w}_{j}$ is the mixture weight for the $j$th group and $\sum _{\mathit{j}=1}^{k}{w}_{j}=1$;
• ${\mu }_{j}$ is a $p$-dimensional vector of means for the $j$th group;
• ${\Sigma }_{j}$ is the covariance structure for the $j$th group;
• $g\left(·\right)$ is the $p$-variate Normal density:
 $g x∣μj,Σj = 1 2π p/2 Σj 1/2 exp - 12 x-μj Σ j -1 x-μj T .$
Optionally, the (co)variance structure may be pooled (common to all groups) or calculated for each group, and may be full or diagonal.

4References

Hartigan J A (1975) Clustering Algorithms Wiley

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of objects. There must be more objects than parameters in the model.
Constraints:
• if ${\mathbf{sopt}}=\mathrm{Nag_GroupCovar}$, ${\mathbf{n}}>{\mathbf{ng}}×\left({\mathbf{nvar}}×{\mathbf{nvar}}+{\mathbf{nvar}}\right)$;
• if ${\mathbf{sopt}}=\mathrm{Nag_PooledCovar}$, ${\mathbf{n}}>{\mathbf{nvar}}×\left({\mathbf{ng}}+{\mathbf{nvar}}\right)$;
• if ${\mathbf{sopt}}=\mathrm{Nag_GroupVar}$, ${\mathbf{n}}>2×{\mathbf{ng}}×{\mathbf{nvar}}$;
• if ${\mathbf{sopt}}=\mathrm{Nag_PooledVar}$, ${\mathbf{n}}>{\mathbf{nvar}}×\left({\mathbf{ng}}+1\right)$;
• if ${\mathbf{sopt}}=\mathrm{Nag_OverallVar}$, ${\mathbf{n}}>{\mathbf{nvar}}×{\mathbf{ng}}+1$.
2:    $\mathbf{m}$IntegerInput
On entry: the total number of variables in array x.
Constraint: ${\mathbf{m}}\ge 1$.
3:    $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{pdx}}\right]$const doubleInput
On entry: ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{pdx}}+\mathit{j}-1\right]$ must contain the value of the $\mathit{j}$th variable for the $\mathit{i}$th object, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
4:    $\mathbf{pdx}$IntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
5:    $\mathbf{isx}\left[{\mathbf{m}}\right]$const IntegerInput
On entry: if ${\mathbf{nvar}}={\mathbf{m}}$ all available variables are included in the model and isx is not referenced; otherwise the $j$th variable will be included in the analysis if ${\mathbf{isx}}\left[\mathit{j}-1\right]=1$ and excluded if ${\mathbf{isx}}\left[\mathit{j}-1\right]=0$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
Constraint: if ${\mathbf{nvar}}\ne {\mathbf{m}}$, ${\mathbf{isx}}\left[\mathit{j}-1\right]=1$ for nvar values of $\mathit{j}$ and ${\mathbf{isx}}\left[\mathit{j}-1\right]=0$ for the remaining ${\mathbf{m}}-{\mathbf{nvar}}$ values of $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
6:    $\mathbf{nvar}$IntegerInput
On entry: $p$, the number of variables included in the calculations.
Constraint: $1\le {\mathbf{nvar}}\le {\mathbf{m}}$.
7:    $\mathbf{ng}$IntegerInput
On entry: $k$, the number of groups in the mixture model.
Constraint: ${\mathbf{ng}}\ge 1$.
8:    $\mathbf{popt}$Nag_BooleanInput
On entry: if ${\mathbf{popt}}=\mathrm{Nag_TRUE}$, the initial membership probabilities in prob are set internally; otherwise these probabilities must be supplied.
9:    $\mathbf{prob}\left[{\mathbf{n}}×{\mathbf{tdprob}}\right]$doubleInput/Output
On entry: if ${\mathbf{popt}}\ne \mathrm{Nag_TRUE}$, ${\mathbf{prob}}\left[\left(i-1\right)×{\mathbf{tdprob}}+j-1\right]$ is the probability that the $i$th object belongs to the $j$th group. (These probabilities are normalised internally.)
On exit: ${\mathbf{prob}}\left[\left(i-1\right)×{\mathbf{tdprob}}+j-1\right]$ is the probability of membership of the $i$th object to the $j$th group for the fitted model.
10:  $\mathbf{tdprob}$IntegerInput
On entry: the stride separating matrix column elements in the array prob.
Constraint: ${\mathbf{tdprob}}\ge {\mathbf{ng}}$.
11:  $\mathbf{niter}$Integer *Input/Output
On entry: the maximum number of iterations.
Suggested value: $15$
On exit: the number of completed iterations.
Constraint: ${\mathbf{niter}}\ge 1$.
12:  $\mathbf{riter}$IntegerInput
On entry: if ${\mathbf{riter}}>0$, membership probabilities are rounded to $0.0$ or $1.0$ after the completion of every riter iterations.
Suggested value: $5$
13:  $\mathbf{w}\left[{\mathbf{ng}}\right]$doubleOutput
On exit: ${w}_{j}$, the mixing probability for the $j$th group.
14:  $\mathbf{g}\left[{\mathbf{nvar}}×{\mathbf{ng}}\right]$doubleOutput
On exit: ${\mathbf{g}}\left[\left(i-1\right)×{\mathbf{ng}}+j-1\right]$ gives the estimated mean of the $i$th variable in the $j$th group.
15:  $\mathbf{sopt}$Nag_VarCovarInput
On entry: determines the (co)variance structure:
${\mathbf{sopt}}=\mathrm{Nag_GroupCovar}$
Groupwise covariance matrices.
${\mathbf{sopt}}=\mathrm{Nag_PooledCovar}$
Pooled covariance matrix.
${\mathbf{sopt}}=\mathrm{Nag_GroupVar}$
Groupwise variances.
${\mathbf{sopt}}=\mathrm{Nag_PooledVar}$
Pooled variances.
${\mathbf{sopt}}=\mathrm{Nag_OverallVar}$
Overall variance.
Constraint: ${\mathbf{sopt}}=\mathrm{Nag_GroupCovar}$, $\mathrm{Nag_PooledCovar}$, $\mathrm{Nag_GroupVar}$, $\mathrm{Nag_PooledVar}$ or $\mathrm{Nag_OverallVar}$.
16:  $\mathbf{s}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array s must be at least $\mathit{a}×\mathit{b}×\mathit{c}$.
Where ${\mathbf{S}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{s}}\left[\left(k-1\right)×\mathit{a}×\mathit{b}+\left(j-1\right)×\mathit{a}+i-1\right]$.
On exit: if ${\mathbf{sopt}}=\mathrm{Nag_GroupCovar}$, ${\mathbf{S}}\left(i,j,k\right)$ gives the $\left(i,j\right)$th element of the $k$th group, with $a=b={\mathbf{nvar}}$ and $c={\mathbf{ng}}$.
If ${\mathbf{sopt}}=\mathrm{Nag_PooledCovar}$, ${\mathbf{S}}\left(i,j,1\right)$ gives the $\left(i,j\right)$th element of the pooled covariance, with $a=b={\mathbf{nvar}}$ and $c=1$.
If ${\mathbf{sopt}}=\mathrm{Nag_GroupVar}$, ${\mathbf{S}}\left(j,k,1\right)$ gives the $j$th variance in the $k$th group, with $a={\mathbf{nvar}}$, $b={\mathbf{ng}}$ and $c=1$.
If ${\mathbf{sopt}}=\mathrm{Nag_PooledVar}$, ${\mathbf{S}}\left(j,1,1\right)$ gives the $j$th pooled variance., with $a={\mathbf{nvar}}$ and $b=c=1$
If ${\mathbf{sopt}}=\mathrm{Nag_OverallVar}$, ${\mathbf{S}}\left(1,1,1\right)$ gives the overall variance, with $a=b=c=1$.
17:  $\mathbf{f}\left[{\mathbf{n}}×{\mathbf{ng}}\right]$doubleOutput
On exit: ${\mathbf{f}}\left[\left(i-1\right)×{\mathbf{ng}}+j-1\right]$ gives the $p$-variate Normal (Gaussian) density of the $i$th object in the $j$th group.
18:  $\mathbf{tol}$doubleInput
On entry: iterations cease the first time an improvement in log-likelihood is less than tol. If ${\mathbf{tol}}\le 0$ a value of ${10}^{-3}$ is used.
19:  $\mathbf{loglik}$double *Output
On exit: the log-likelihood for the fitted mixture model.
20:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{tdprob}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tdprob}}\ge {\mathbf{n}}$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CLUSTER_EMPTY
An iteration cannot continue due to an empty group, try a different initial allocation.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{ng}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ng}}\ge 1$.
On entry, ${\mathbf{niter}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{niter}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{nvar}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{nvar}}\le {\mathbf{m}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF
A covariance matrix is not positive definite, try a different initial allocation.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_OBSERVATIONS
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and $p=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>p$, the number of parameters, i.e., too few objects have been supplied for the model.
NE_PROBABILITY
On entry, row $〈\mathit{\text{value}}〉$ of supplied prob does not sum to $1$.
NE_VAR_INCL_INDICATED
On entry, ${\mathbf{nvar}}\ne {\mathbf{m}}$ and isx is invalid.

Not applicable.

8Parallelism and Performance

nag_mv_gaussian_mixture (g03gac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_mv_gaussian_mixture (g03gac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

10Example

This example fits a Gaussian mixture model with pooled covariance structure to New Haven schools test data, see Table 5.1 (p. 118) in Hartigan (1975).

10.1Program Text

Program Text (g03gace.c)

10.2Program Data

Program Data (g03gace.d)

10.3Program Results

Program Results (g03gace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017