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 3.7 in How to Use the NAG Library and its Documentation).

6

Error 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, $pdx = 〈value〉$ 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.
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, $n = 〈value〉$ 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

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.

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 Document

nag_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

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