NAG Library Routine Document

G03DCF

INTEGER	NVAR, NG, NIG(NG), LDGMN, NOBS, M, ISX(M), LDX, LDP, IAG(NOBS), IFAIL
REAL (KIND=nag_wp)	GMN(LDGMN,NVAR), GC((NG+1)NVAR(NVAR+1)/2), DET(NG), X(LDX,M), PRIOR(NG), P(LDP,NG), ATI(LDP,), WK(2NVAR)
LOGICAL	ATIQ
CHARACTER(1)	TYP, EQUAL, PRIORS

3 Description

Discriminant analysis is concerned with the allocation of observations to groups using information from other observations whose group membership is known,

X_{t}

; these are called the training set. Consider

p

variables observed on

n_{g}

populations or groups. Let

{\bar{x}}_{j}

be the sample mean and

S_{j}

the within-group variance-covariance matrix for the

j

th group; these are calculated from a training set of

n

observations with

n_{j}

observations in the

j

th group, and let

x_{k}

be the

k

th observation from the set of observations to be allocated to the

n_{g}

groups. The observation can be allocated to a group according to a selected rule. The allocation rule or discriminant function will be based on the distance of the observation from an estimate of the location of the groups, usually the group means. A measure of the distance of the observation from the

j

th group mean is given by the Mahalanobis distance,

D_{k j}

D_{k j}^{2} = {(x_{k} - {\bar{x}}_{j})}^{T} S_{j}^{- 1} (x_{k} - {\bar{x}}_{j}) .

(1)

If the pooled estimate of the variance-covariance matrix

S

is used rather than the within-group variance-covariance matrices, then the distance is:

D_{k j}^{2} = {(x_{k} - {\bar{x}}_{j})}^{T} S^{- 1} (x_{k} - {\bar{x}}_{j}) .

(2)

Instead of using the variance-covariance matrices

S

and

S_{j}

, G03DCF uses the upper triangular matrices

R

and

R_{j}

supplied by G03DAF such that

S = R^{T} R

and

S_{j} = R_{j}^{T} R_{j}

D_{k j}^{2}

can then be calculated as

z^{T} z

where

{R^{T}}_{j} z = (x_{k} - x_{j})

R^{T} z = (x_{k} - x)

as appropriate.

In addition to the distances, a set of prior probabilities of group membership,

π_{j}

, for

j = 1, 2, \dots, n_{g}

, may be used, with

\sum π_{j} = 1

. The prior probabilities reflect your view as to the likelihood of the observations coming from the different groups. Two common cases for prior probabilities are

π_{1} = π_{2} = \dots = π_{n_{g}}

, that is, equal prior probabilities, and

π_{j} = n_{j} / n

, for

j = 1, 2, \dots, n_{g}

, that is, prior probabilities proportional to the number of observations in the groups in the training set.

G03DCF uses one of four allocation rules. In all four rules the

p

variables are assumed to follow a multivariate Normal distribution with mean

μ_{j}

and variance-covariance matrix

Σ_{j}

if the observation comes from the

j

th group. The different rules depend on whether or not the within-group variance-covariance matrices are assumed equal, i.e.,

Σ_{1} = Σ_{2} = \dots = Σ_{n_{g}}

, and whether a predictive or estimative approach is used. If

p (x_{k} ∣ μ_{j}, Σ_{j})

is the probability of observing the observation

x_{k}

from group

j

, then the posterior probability of belonging to group

j

is:

p (j ∣ x_{k}, μ_{j}, Σ_{j}) \propto p (x_{k} ∣ μ_{j}, Σ_{j}) π_{j} .

(3)

In the estimative approach, the parameters

μ_{j}

and

Σ_{j}

in (3) are replaced by their estimates calculated from

X_{t}

. In the predictive approach, a non-informative prior distribution is used for the parameters and a posterior distribution for the parameters,

p (μ_{j}, Σ_{j} ∣ X_{t})

, is found. A predictive distribution is then obtained by integrating

p (j ∣ x_{k}, μ_{j}, Σ_{j}) p (μ_{j}, Σ_{j} ∣ X)

over the parameter space. This predictive distribution then replaces

p (x_{k} ∣ μ_{j}, Σ_{j})

in (3). See Aitchison and Dunsmore (1975), Aitchison et al. (1977) and Moran and Murphy (1979) for further details.

The observation is allocated to the group with the highest posterior probability. Denoting the posterior probabilities,

p (j ∣ x_{k}, μ_{j}, Σ_{j})

, by

q_{j}

, the four allocation rules are:

(i)

Estimative with equal variance-covariance matrices – Linear Discrimination

\log q_{j} \propto - \frac{1}{2} D_{k j}^{2} + \log π_{j}

(ii)

Estimative with unequal variance-covariance matrices – Quadratic Discrimination

\log q_{j} \propto - \frac{1}{2} D_{k j}^{2} + \log π_{j} - \frac{1}{2} \log |S_{j}|

(iii)

Predictive with equal variance-covariance matrices

q_{j}^{- 1} \propto {((n_{j} + 1) / n_{j})}^{p / 2} {\{1 + [n_{j} / ((n - n_{g}) (n_{j} + 1))] D_{k j}^{2}\}}^{(n + 1 - n_{g}) / 2}

(iv)

Predictive with unequal variance-covariance matrices

q_{j}^{- 1} \propto C {\{((n_{j}^{2} - 1) / n_{j}) |S_{j}|\}}^{p / 2} {\{1 + (n_{j} / (n_{j}^{2} - 1)) D_{k j}^{2}\}}^{n_{j} / 2},

where

C = \frac{Γ (\frac{1}{2} (n_{j} - p))}{Γ (\frac{1}{2} n_{j})} .

In the above the appropriate value of

D_{k j}^{2}

from (1) or (2) is used. The values of the

q_{j}

are standardized so that,

\sum_{j = 1}^{n_{g}} q_{j} = 1 .

Moran and Murphy (1979) show the similarity between the predictive methods and methods based upon likelihood ratio tests.

In addition to allocating the observation to a group, G03DCF computes an atypicality index,

I_{j} (x_{k})

. The predictive atypicality index is returned, irrespective of the value of the parameter TYP. This represents the probability of obtaining an observation more typical of group

j

than the observed

x_{k}

(see Aitchison and Dunsmore (1975) and Aitchison et al. (1977)). The atypicality index is computed for unequal within-group variance-covariance matrices as:

I_{j} (x_{k}) = P (B \leq z : \frac{1}{2} p, \frac{1}{2} (n_{j} - p))

where

P (B \leq β : a, b)

is the lower tail probability from a beta distribution and

z = D_{k j}^{2} / (D_{k j}^{2} + (n_{j}^{2} - 1) / n_{j}),

and for equal within-group variance-covariance matrices as:

I_{j} (x_{k}) = P (B \leq z : \frac{1}{2} p, \frac{1}{2} (n - n_{g} - p + 1)),

with

z = D_{k j}^{2} / (D_{k j}^{2} + (n - n_{g}) (n_{j} + 1) / n_{j}) .

I_{j} (x_{k})

is close to

1

for all groups it indicates that the observation may come from a grouping not represented in the training set. Moran and Murphy (1979) provide a frequentist interpretation of

I_{j} (x_{k})

4 References

Aitchison J and Dunsmore I R (1975) Statistical Prediction Analysis Cambridge

Aitchison J, Habbema J D F and Kay J W (1977) A critical comparison of two methods of statistical discrimination Appl. Statist. 26 15–25

Kendall M G and Stuart A (1976) The Advanced Theory of Statistics (Volume 3) (3rd Edition) Griffin

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

Moran M A and Murphy B J (1979) A closer look at two alternative methods of statistical discrimination Appl. Statist. 28 223–232

Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill

5 Parameters

1: $TYP$ – CHARACTER(1)Input

On entry: whether the estimative or predictive approach is used.

$TYP ='E'$: The estimative approach is used.
$TYP ='P'$: The predictive approach is used.

Constraint:

TYP ='E'

'P'

2: $EQUAL$ – CHARACTER(1)Input

On entry: indicates whether or not the within-group variance-covariance matrices are assumed to be equal and the pooled variance-covariance matrix used.

$EQUAL ='E'$: The within-group variance-covariance matrices are assumed equal and the matrix $R$ stored in the first $p (p + 1) / 2$ elements of GC is used.
$EQUAL ='U'$: The within-group variance-covariance matrices are assumed to be unequal and the matrices $R_{i}$ , for $i = 1, 2, \dots, n_{g}$ , stored in the remainder of GC are used.

Constraint:

EQUAL ='E'

'U'

3: $PRIORS$ – CHARACTER(1)Input

On entry: indicates the form of the prior probabilities to be used.

$PRIORS ='E'$: Equal prior probabilities are used.
$PRIORS ='P'$: Prior probabilities proportional to the group sizes in the training set, $n_{j}$ , are used.
$PRIORS ='I'$: The prior probabilities are input in PRIOR.

Constraint:

PRIORS ='E'

'I'

'P'

4: $NVAR$ – INTEGERInput

On entry:

p

, the number of variables in the variance-covariance matrices.

Constraint:

NVAR \geq 1

5: $NG$ – INTEGERInput

On entry: the number of groups,

n_{g}

Constraint:

NG \geq 2

6: $NIG (NG)$ – INTEGER arrayInput

On entry: the number of observations in each group in the training set,

n_{j}

Constraints:

if $EQUAL ='E'$ , $NIG (j) > 0$ and $\sum_{j = 1}^{n_{g}} NIG (j) > NG + NVAR$ , for $j = 1, 2, \dots, n_{g}$ ;
if $EQUAL ='U'$ , $NIG (j) > NVAR$ , for $j = 1, 2, \dots, n_{g}$ .

7: $GMN (LDGMN, NVAR)$ – REAL (KIND=nag_wp) arrayInput

On entry: the

j

th row of GMN contains the means of the

p

variables for the

j

th group, for

j = 1, 2, \dots, n_{j}

. These are returned by G03DAF.

8: $LDGMN$ – INTEGERInput

On entry: the first dimension of the array GMN as declared in the (sub)program from which G03DCF is called.

Constraint:

LDGMN \geq NG

9: $GC ((NG + 1) \times NVAR \times (NVAR + 1) / 2)$ – REAL (KIND=nag_wp) arrayInput

On entry: the first

p (p + 1) / 2

elements of GC should contain the upper triangular matrix

R

and the next

n_{g}

blocks of

p (p + 1) / 2

elements should contain the upper triangular matrices

R_{j}

All matrices must be stored packed by column. These matrices are returned by G03DAF. If

EQUAL ='E'

only the first

p (p + 1) / 2

elements are referenced, if

EQUAL ='U'

only the elements

p (p + 1) / 2 + 1

(n_{g} + 1) p (p + 1) / 2

are referenced.

Constraints:

if $EQUAL ='E'$ , the diagonal elements of $R$ must be $\neq 0.0$ ;
if $EQUAL ='U'$ , the diagonal elements of the $R_{j}$ must be $\neq 0.0$ , for $j = 1, 2, \dots, n_{g}$ .

10: $DET (NG)$ – REAL (KIND=nag_wp) arrayInput

On entry: if

EQUAL ='U'

. the logarithms of the determinants of the within-group variance-covariance matrices as returned by G03DAF. Otherwise DET is not referenced.

11: $NOBS$ – INTEGERInput

On entry: the number of observations in X which are to be allocated.

Constraint:

NOBS \geq 1

12: $M$ – INTEGERInput

On entry: the number of variables in the data array X.

Constraint:

M \geq NVAR

13: $ISX (M)$ – INTEGER arrayInput

On entry:

ISX (l)

indicates if the

l

th variable in X is to be included in the distance calculations.

ISX (l) > 0

, the

l

th variable is included, for

l = 1, 2, \dots, M

; otherwise the

l

th variable is not referenced.

Constraint:

ISX (l) > 0

for NVAR values of

l

14: $X (LDX, M)$ – REAL (KIND=nag_wp) arrayInput

On entry:

X (k, l)

must contain the

k

th observation for the

l

th variable, for

k = 1, 2, \dots, NOBS

and

l = 1, 2, \dots, M

15: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G03DCF is called.

Constraint:

LDX \geq NOBS

16: $PRIOR (NG)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

PRIORS ='I'

, the prior probabilities for the

n_{g}

groups.

Constraint: if

PRIORS ='I'

PRIOR (j) > 0.0

and

|1 - \sum_{j = 1}^{n_{g}} PRIOR (j)| \leq 10 \times machine precision

, for

j = 1, 2, \dots, n_{g}

On exit: if

PRIORS ='P'

, the computed prior probabilities in proportion to group sizes for the

n_{g}

groups.

PRIORS ='I'

, the input prior probabilities will be unchanged.

PRIORS ='E'

, PRIOR is not set.

17: $P (LDP, NG)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

P (k, j)

contains the posterior probability

p_{k j}

for allocating the

k

th observation to the

j

th group, for

k = 1, 2, \dots, NOBS

and

j = 1, 2, \dots, n_{g}

18: $LDP$ – INTEGERInput

On entry: the first dimension of the arrays P and ATI as declared in the (sub)program from which G03DCF is called.

Constraint:

LDP \geq NOBS

19: $IAG (NOBS)$ – INTEGER arrayOutput

On exit: the groups to which the observations have been allocated.

20: $ATIQ$ – LOGICALInput

On entry: ATIQ must be .TRUE. if atypicality indices are required. If ATIQ is .FALSE. the array ATI is not set.

21: $ATI (LDP *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array ATI must be at least

NG

ATIQ = .TRUE.

, and at least

1

otherwise.

On exit: if ATIQ is .TRUE.,

ATI (k, j)

will contain the predictive atypicality index for the

k

th observation with respect to the

j

th group, for

k = 1, 2, \dots, NOBS

and

j = 1, 2, \dots, n_{g}

If ATIQ is .FALSE., ATI is not set.

22: $WK (2 \times NVAR)$ – REAL (KIND=nag_wp) arrayWorkspace

23: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NVAR < 1$ ,
or	$NG < 2$ ,
or	$NOBS < 1$ ,
or	$M < NVAR$ ,
or	$LDGMN < NG$ ,
or	$LDX < NOBS$ ,
or	$LDP < NOBS$ ,
or	$TYP \neq'E'$ or ‘P’,
or	$EQUAL \neq'E'$ or ‘U’,
or	$PRIORS \neq'E'$ , ‘I’ or ‘P’.

$IFAIL = 2$

On entry,	the number of variables indicated by ISX is not equal to NVAR,
or	$EQUAL ='E'$ and $NIG (j) \leq 0$ , for some $j$ ,
or	$EQUAL ='E'$ and $\sum_{j = 1}^{n_{g}} NIG (j) \leq NG + NVAR$ ,
or	$EQUAL ='U'$ and $NIG (j) \leq NVAR$ for some $j$ .

$IFAIL = 3$

On entry,	$PRIORS ='I'$ and $PRIOR (j) \leq 0.0$ for some $j$ ,
or	$PRIORS ='I'$ and $\sum_{j = 1}^{n_{g}} PRIOR (j)$ is not within $10 \times machine precision$ of $1$ .

$IFAIL = 4$

On entry,	$EQUAL ='E'$ and a diagonal element of $R$ is zero,
or	$EQUAL ='U'$ and a diagonal element of $R_{j}$ for some $j$ is zero.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The accuracy of the returned posterior probabilities will depend on the accuracy of the input

R

R_{j}

matrices. The atypicality index should be accurate to four significant places.

8 Parallelism and Performance

G03DCF is not threaded by NAG in any implementation.

G03DCF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The distances

D_{k j}^{2}

can be computed using G03DBF if other forms of discrimination are required.

10 Example

The data, taken from Aitchison and Dunsmore (1975), is concerned with the diagnosis of three ‘types’ of Cushing's syndrome. The variables are the logarithms of the urinary excretion rates (mg/24hr) of two steroid metabolites. Observations for a total of

21

patients are input and the group means and

R

matrices are computed by G03DAF. A further six observations of unknown type are input and allocations made using the predictive approach and under the assumption that the within-group covariance matrices are not equal. The posterior probabilities of group membership,

q_{j}

, and the atypicality index are printed along with the allocated group. The atypicality index shows that observations

5

and

6

do not seem to be typical of the three types present in the initial

21

observations.

NAG Library Routine DocumentG03DCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G03DCF