NAG Library Function Document

nag_mv_gaussian_mixture (g03gac)


    1  Purpose
    7  Accuracy


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


#include <nag.h>
#include <nagg03.h>
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)


A Normal (Gaussian) mixture model is a weighted sum of k group Normal densities given by,
p xw,μ,Σ = j=1 k wj g xμj,Σj ,  xp  
Optionally, the (co)variance structure may be pooled (common to all groups) or calculated for each group, and may be full or diagonal.


Hartigan J A (1975) Clustering Algorithms Wiley


1:     n IntegerInput
On entry: n, the number of objects. There must be more objects than parameters in the model.
  • if sopt=Nag_GroupCovar, n>ng× nvar×nvar+nvar;
  • if sopt=Nag_PooledCovar, n>nvar× ng+nvar ;
  • if sopt=Nag_GroupVar, n>2×ng×nvar ;
  • if sopt=Nag_PooledVar, n>nvar× ng+1 ;
  • if sopt=Nag_OverallVar, n>nvar×ng+1 .
2:     m IntegerInput
On entry: the total number of variables in array x.
Constraint: m1.
3:     x[n×pdx] const doubleInput
On entry: x[i-1×pdx+j-1] must contain the value of the jth variable for the ith object, for i=1,2,,n and j=1,2,,m.
4:     pdx IntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: pdxm.
5:     isx[m] const IntegerInput
On entry: if nvar=m all available variables are included in the model and isx is not referenced; otherwise the jth variable will be included in the analysis if isx[j-1]=1 and excluded if isx[j-1]=0, for j=1,2,,m.
Constraint: if nvarm, isx[j-1]=1 for nvar values of j and isx[j-1]=0 for the remaining m-nvar values of j, for j=1,2,,m.
6:     nvar IntegerInput
On entry: p, the number of variables included in the calculations.
Constraint: 1nvarm.
7:     ng IntegerInput
On entry: k, the number of groups in the mixture model.
Constraint: ng1.
8:     popt Nag_BooleanInput
On entry: if popt=Nag_TRUE, the initial membership probabilities in prob are set internally; otherwise these probabilities must be supplied.
9:     prob[n×tdprob] doubleInput/Output
On entry: if poptNag_TRUE, prob[i-1×tdprob+j-1] is the probability that the ith object belongs to the jth group. (These probabilities are normalised internally.)
On exit: prob[i-1×tdprob+j-1] is the probability of membership of the ith object to the jth group for the fitted model.
10:   tdprob IntegerInput
On entry: the stride separating matrix column elements in the array prob.
Constraint: tdprobng.
11:   niter Integer *Input/Output
On entry: the maximum number of iterations.
Suggested value: 15
On exit: the number of completed iterations.
Constraint: niter1.
12:   riter IntegerInput
On entry: if riter>0, membership probabilities are rounded to 0.0 or 1.0 after the completion of every riter iterations.
Suggested value: 5
13:   w[ng] doubleOutput
On exit: wj, the mixing probability for the jth group.
14:   g[nvar×ng] doubleOutput
On exit: g[i-1×ng+j-1] gives the estimated mean of the ith variable in the jth group.
15:   sopt Nag_VarCovarInput
On entry: determines the (co)variance structure:
Groupwise covariance matrices.
Pooled covariance matrix.
Groupwise variances.
Pooled variances.
Overall variance.
Constraint: sopt=Nag_GroupCovar, Nag_PooledCovar, Nag_GroupVar, Nag_PooledVar or Nag_OverallVar.
16:   s[dim] doubleOutput
Note: the dimension, dim, of the array s must be at least a×b×c.
Where Si,j,k appears in this document, it refers to the array element s[k-1×a×b+j-1×a+i-1].
On exit: if sopt=Nag_GroupCovar, Si,j,k gives the i,jth element of the kth group, with a=b=nvar and c=ng.
If sopt=Nag_PooledCovar, Si,j,1 gives the i,jth element of the pooled covariance, with a=b=nvar and c=1.
If sopt=Nag_GroupVar, Sj,k,1 gives the jth variance in the kth group, with a=nvar, b=ng and c=1.
If sopt=Nag_PooledVar, Sj,1,1 gives the jth pooled variance., with a=nvar and b=c=1
If sopt=Nag_OverallVar, S1,1,1 gives the overall variance, with a=b=c=1.
17:   f[n×ng] doubleOutput
On exit: f[i-1×ng+j-1] gives the p-variate Normal (Gaussian) density of the ith object in the jth group.
18:   tol doubleInput
On entry: iterations cease the first time an improvement in log-likelihood is less than tol. If tol0 a value of 10-3 is used.
19:   loglik double *Output
On exit: the log-likelihood for the fitted mixture model.
20:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, pdx=value and n=value.
Constraint: pdxn.
On entry, tdprob=value and n=value.
Constraint: tdprobn.
On entry, argument value had an illegal value.
An iteration cannot continue due to an empty group, try a different initial allocation.
On entry, m=value.
Constraint: m1.
On entry, ng=value.
Constraint: ng1.
On entry, niter=value.
Constraint: niter1.
On entry, nvar=value and m=value.
Constraint: 1nvarm.
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.
A covariance matrix is not positive definite, try a different initial allocation.
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.
On entry, n=value and p=value.
Constraint: n>p, the number of parameters, i.e., too few objects have been supplied for the model.
On entry, row value of supplied prob does not sum to 1.
On entry, nvarm and isx is invalid.


Not applicable.

Parallelism 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.

Further Comments



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).

Program Text

Program Text (g03gace.c)

Program Data

Program Data (g03gace.d)

Program Results

Program Results (g03gace.r)

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