NAG Library Function Document

nag_mv_gaussian_mixture (g03gac)

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)

3 Description

A Normal (Gaussian) mixture model is a weighted sum of

k

group Normal densities given by,

p (x ∣ w, μ, Σ) = \sum_{j = 1}^{k} w_{j} g (x ∣ μ_{j}, Σ_{j}), x \in ℝ^{p}

where:

$x$ is a $p$ -dimensional object of interest;
$w_{j}$ is the mixture weight for the $j$ th group and $\sum_{j = 1}^{k} w_{j} = 1$ ;
$μ_{j}$ is a $p$ -dimensional vector of means for the $j$ th group;
$Σ_{j}$ is the covariance structure for the $j$ th group;
$g (\cdot)$ is the $p$ -variate Normal density:
$g (x ∣ μ_{j}, Σ_{j}) = \frac{1}{{(2 π)}^{p / 2} {|Σ_{j}|}^{1 / 2}} \exp [- \frac{1}{2} (x - μ_{j}) Σ_{j}^{- 1} {(x - μ_{j})}^{T}] .$

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 – IntegerInput

On entry:

n

, the number of objects. There must be more objects than parameters in the model.

Constraints:

if $sopt = Nag_GroupCovar$ , $n > ng \times (nvar \times nvar + nvar)$ ;
if $sopt = Nag_PooledCovar$ , $n > nvar \times (ng + nvar)$ ;
if $sopt = Nag_GroupVar$ , $n > 2 \times ng \times nvar$ ;
if $sopt = Nag_PooledVar$ , $n > nvar \times (ng + 1)$ ;
if $sopt = Nag_OverallVar$ , $n > nvar \times ng + 1$ .

2: m – IntegerInput

On entry: the total number of variables in array x.

Constraint:

m \geq 1

3: x[ $n \times pdx$ ] – const doubleInput

On entry:

x [(i - 1) \times pdx + j - 1]

must contain the value of the

j

th variable for the

i

th object, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: pdx – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

pdx \geq m

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

j

th variable will be included in the analysis if

isx [j - 1] = 1

and excluded if

isx [j - 1] = 0

, for

j = 1, 2, \dots, m

Constraint: if

nvar \neq m

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, \dots, m

6: nvar – IntegerInput

On entry:

p

, the number of variables included in the calculations.

Constraint:

1 \leq nvar \leq m

7: ng – IntegerInput

On entry:

k

, the number of groups in the mixture model.

Constraint:

ng \geq 1

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[ $n \times tdprob$ ] – doubleInput/Output

On entry: if

popt \neq Nag_TRUE

prob [(i - 1) \times tdprob + j - 1]

is the probability that the

i

th object belongs to the

j

th group. (These probabilities are normalised internally.)

On exit:

prob [(i - 1) \times tdprob + j - 1]

is the probability of membership of the

i

th object to the

j

th group for the fitted model.

10: tdprob – IntegerInput

On entry: the stride separating matrix column elements in the array prob.

Constraint:

tdprob \geq ng

11: niter – Integer *Input/Output

On entry: the maximum number of iterations.

Suggested value:

15

On exit: the number of completed iterations.

Constraint:

niter \geq 1

12: riter – IntegerInput

On entry: if

riter > 0

, membership probabilities are rounded to

0.0

1.0

after the completion of every riter iterations.

Suggested value:

5

13: w[ng] – doubleOutput

On exit:

w_{j}

, the mixing probability for the

j

th group.

14: g[ $nvar \times ng$ ] – doubleOutput

On exit:

g [(i - 1) \times ng + j - 1]

gives the estimated mean of the

i

th variable in the

j

th group.

15: sopt – Nag_VarCovarInput

On entry: determines the (co)variance structure:

$sopt = Nag_GroupCovar$: Groupwise covariance matrices.
$sopt = Nag_PooledCovar$: Pooled covariance matrix.
$sopt = Nag_GroupVar$: Groupwise variances.
$sopt = Nag_PooledVar$: Pooled variances.
$sopt = Nag_OverallVar$: Overall variance.

Constraint:

sopt = Nag_GroupCovar

Nag_PooledCovar

Nag_GroupVar

Nag_PooledVar

Nag_OverallVar

16: s[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array s must be at least

a \times b \times c

Where

S (i, j, k)

appears in this document, it refers to the array element

s [(k - 1) \times a \times b + (j - 1) \times a + i - 1]

On exit: if

sopt = Nag_GroupCovar

S (i, j, k)

gives the

(i, j)

th element of the

k

th group, with

a = b = nvar

and

c = ng

sopt = Nag_PooledCovar

S (i, j, 1)

gives the

(i, j)

th element of the pooled covariance, with

a = b = nvar

and

c = 1

sopt = Nag_GroupVar

S (j, k, 1)

gives the

j

th variance in the

k

th group, with

a = nvar

b = ng

and

c = 1

sopt = Nag_PooledVar

S (j, 1, 1)

gives the

j

th pooled variance., with

a = nvar

and

b = c = 1

sopt = Nag_OverallVar

S (1, 1, 1)

gives the overall variance, with

a = b = c = 1

17: f[ $n \times ng$ ] – doubleOutput

On exit:

f [(i - 1) \times ng + j - 1]

gives the

p

-variate Normal (Gaussian) density of the

i

th object in the

j

th group.

18: tol – doubleInput

On entry: iterations cease the first time an improvement in log-likelihood is less than tol. If

tol \leq 0

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_SIZE: On entry, $pdx = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdx \geq n$ .
On entry, $tdprob = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $tdprob \geq n$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ 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, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $ng = ⟨value⟩$ .
Constraint: $ng \geq 1$ .
On entry, $niter = ⟨value⟩$ .
Constraint: $niter \geq 1$ .
NE_INT_2: On entry, $nvar = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $1 \leq nvar \leq m$ .
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.
NE_MAT_NOT_POS_DEF: A covariance matrix is not positive definite, try a different initial allocation.
NE_OBSERVATIONS: 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.
NE_PROBABILITY: On entry, row $⟨value⟩$ of supplied prob does not sum to $1$ .
NE_VAR_INCL_INDICATED: On entry, $nvar \neq m$ and isx is invalid.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

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).

NAG Library Function Documentnag_mv_gaussian_mixture (g03gac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_mv_gaussian_mixture (g03gac)