# NAG CL Interfaceg03gac (gaussian_​mixture)

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## 1Purpose

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

## 2Specification

 #include
 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.

## 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.
Hartigan J A (1975) Clustering Algorithms Wiley

## 5Arguments

1: $\mathbf{n}$Integer Input
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: $\mathbf{m}$Integer Input
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 double Input
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: $\mathbf{pdx}$Integer Input
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 Integer Input
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: $\mathbf{nvar}$Integer Input
On entry: $p$, the number of variables included in the calculations.
Constraint: $1\le {\mathbf{nvar}}\le {\mathbf{m}}$.
7: $\mathbf{ng}$Integer Input
On entry: $k$, the number of groups in the mixture model.
Constraint: ${\mathbf{ng}}\ge 1$.
8: $\mathbf{popt}$Nag_Boolean Input
On entry: if ${\mathbf{popt}}=\mathrm{Nag_TRUE}$, the initial membership probabilities in prob are set internally; otherwise these probabilities must be supplied.
9: $\mathbf{prob}\left[{\mathbf{n}}×{\mathbf{tdprob}}\right]$double Input/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: $\mathbf{tdprob}$Integer Input
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: $\mathbf{riter}$Integer Input
On entry: if ${\mathbf{riter}}>0$, membership probabilities are rounded to $0.0$ or $1.0$ after the completion of the first riter iterations.
Suggested value: $0$
13: $\mathbf{w}\left[{\mathbf{ng}}\right]$double Output
On exit: ${w}_{j}$, the mixing probability for the $j$th group.
14: $\mathbf{g}\left[{\mathbf{nvar}}×{\mathbf{ng}}\right]$double Output
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_VarCovar Input
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: $\mathbf{s}\left[\mathit{dim}\right]$double Output
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: $\mathbf{f}\left[{\mathbf{n}}×{\mathbf{ng}}\right]$double Output
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: $\mathbf{tol}$double Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

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

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.

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)