NAG CL Interface
g03gac (gaussian_​mixture)

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1 Purpose

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

2 Specification

#include <nag.h>
void  g03gac (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)
The function may be called by the names: g03gac or nag_mv_gaussian_mixture.

3 Description

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.

4 References

Hartigan J A (1975) Clustering Algorithms Wiley

5 Arguments

1: n Integer Input
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 Integer Input
On entry: the total number of variables in array x.
Constraint: m1.
3: x[n×pdx] const double Input
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 Integer Input
On entry: the stride separating matrix column elements in the array x.
Constraint: pdxm.
5: isx[m] const Integer Input
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 Integer Input
On entry: p, the number of variables included in the calculations.
Constraint: 1nvarm.
7: ng Integer Input
On entry: k, the number of groups in the mixture model.
Constraint: ng1.
8: popt Nag_Boolean Input
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] double Input/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 Integer Input
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 Integer Input
On entry: if riter>0, membership probabilities are rounded to 0.0 or 1.0 after the completion of the first riter iterations.
Suggested value: 0
13: w[ng] double Output
On exit: wj, the mixing probability for the jth group.
14: g[nvar×ng] double Output
On exit: g[(i-1)×ng+j-1] gives the estimated mean of the ith variable in the jth group.
15: sopt Nag_VarCovar Input
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] double Output
Note: the dimension, dim, of the array s must be at least a×b×c.
where S(i,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, S(i,j,k) gives the (i,j)th element of the kth group, with a=b=nvar and c=ng.
If sopt=Nag_PooledCovar, S(i,j,1) gives the (i,j)th element of the pooled covariance, with a=b=nvar and c=1.
If sopt=Nag_GroupVar, S(j,k,1) gives the jth variance in the kth group, with a=nvar, b=ng and c=1.
If sopt=Nag_PooledVar, S(j,1,1) gives the jth pooled variance., with a=nvar and b=c=1
If sopt=Nag_OverallVar, S(1,1,1) gives the overall variance, with a=b=c=1.
17: f[n×ng] double Output
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 double Input
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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g03gac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

9 Further Comments


10 Example

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).

10.1 Program Text

Program Text (g03gace.c)

10.2 Program Data

Program Data (g03gace.d)

10.3 Program Results

Program Results (g03gace.r)