DMSCR

 

   more...

Uses Fisher’s linear discriminant analysis method to reduce the number of variables.

Required Arguments

XMEAN — NGROUP by NVAR matrix containing the means of the variables in each group. (Input)

SUMWT — Vector of length NGROUP containing the sum of the weights of the observations in each group. (Input)

COV — NVAR by NVAR matrix containing the pooled within‑groups variance‑covariance matrix Sp. (Input)

NNV — Number of eigenvectors extracted from

 

the standardized between‑groups variance‑covariance matrix. (Output)
Sp is the pooled within‑groups variance‑covariance matrix, and Sb is the between‑groups variance‑covariance matrix. NNV is usually the minimum of NVAR and NGROUP ‑1, but it may be smaller if any row of XMEAN or COV is a linear combination of the other rows.

EVAL — Vector of length NNV containing the eigenvalues extracted from the standardized between‑means variancecovariance matrix, in descending order. (Output)
NNV is less than or equal to the minimum of NVAR and (NGROUP ‑1).

COEF — NVAR by NNV matrix of eigenvectors from the standardized between‑means variance‑covariance matrix. (Output)
The eigenvector coefficients have been standardized such that the canonical scores can be obtained directly by multiplication of the original data by COEF.

CMEAN — NGROUP by NNV matrix of group means of the canonical variables. (Output)

Optional Arguments

NGROUP — Number of groups. (Input)
Default: NGROUP = size (XMEAN,1).

NVAR — Number of variables. (Input)
Default: NVAR = size (XMEAN,2).

LDXMEA — Leading dimension of XMEAN exactly as specified in the dimension statement in the calling program. (Input)
Default: LDXMEA = size (XMEAN,1).

LDCOV — Leading dimension of COV exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCOV = size (COV,1).

LDCOEF — Leading dimension of COEF exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCOEF = size (COEF,1).

LDCMEA — Leading dimension of CMEAN exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCMEA = size (CMEAN,1).

FORTRAN 90 Interface

Generic: CALL DMSCR (XMEAN, SUMWT, COV, NNV, EVAL, COEF, CMEAN [, …])

Specific: The specific interface names are S_DMSCR and D_DMSCR.

FORTRAN 77 Interface

Single: CALL DMSCR (NGROUP, NVAR, XMEAN, LDXMEA, SUMWT, COV, LDCOV, NNV, EVAL, COEF, LDCOEF, CMEAN, LDCMEA)

Double: The double precision name is DDMSCR.

Description

Routine DMSCR is a natural generalization of R.A. Fisher’s linear discrimination procedure for two groups. This method of discrimination obtains those linear combinations of the observed random variables that maximize the between‑groups variation relative to the within groups variation. Denote the first of these linear combinations by

 

where β1 is a column vector of coefficients of length NVAR and x is an observation to be classified. On the basis of one linear combination, the discriminant rule assigns the observation, z, to a group (characterized by the group mean) by minimizing the Euclidean distance between z and the group mean.

To obtain β1 (see, e.g., Tatsuoka 1971, page 158), let Sp denote the pooled within‑groups covariance matrix (Sp is defined and can be computed via routine DSCRM) and let Sb denote the between‑groups covariance matrix defined by

 

where g is the number of groups,

 

is the mean vector for the i-th group of observations, denotes the vector of means over all observations, wi is the sum of the weights times the frequencies as input in SUMWT and as used in the computation of

 

and N is the total number of observations used in computing COV. Then, β1, such that

 

can be computed as the maximum of

 

This yields β1 as the eigenvector associated with the largest eigenvalue from

 

Generally,

 

has rank m, where m = min(g ‑ 1, p) and p = NVAR.

 

has m such eigenvectors, and the matrix COEF is obtained as (β1, β2, …., βm), where each βi is an eigenvector.

The matrix CMEAN is taken as the within‑group means vector of the linear combinations zi defined by the β’s. For each observation x, scores

 

can be computed, because of the restriction on βi, the sample variance of the zi is 1.0. The observation is classified into the group (as specified by the group mean of the zi’s) to which, on the basis of the zi, the Euclidean distance is the least.

Note that the linear combinations zi have meaning even when discrimination is not desired. The linear combination of the observed variables that most separates the g groups is z1; z2, giving the second highest such separation orthogonal to the first, and so on. Thus, a plot of the mean vectors of the first two variables gives a good two‑dimensional summarization of the relationships between the groups.

Comments

1. Workspace may be explicitly provided, if desired, by use of D2SCR/DD2SCR. The reference is:

CALL D2SCR (NGROUP, NVAR, XMEAN, LDXMEA, SUMWT, COV, LDCOV, NNV, EVAL, COEF, LDCOEF, CMEAN, LDCMEA, BCOV, EVAL2, EVEC, WKR, WK)

The additional arguments are as follows:

BCOV — Work array of length NVAR * NVAR.

EVAL2 — Work array of length NVAR.

EVEC — Work array of length NVAR * NVAR.

WKR — Work array of length NVAR * NVAR.

WK — Work array of length 2 * NVAR.

2. IMSL routine DSCRM may be used to calculate the input arrays for this routine from the original data.

Example

The following example illustrates a typical sequence. Fisher’s iris data is used. (See routine GDATA, Chapter 19, “Utilities”). Routine DSCRM is first used to perform a discriminant analysis based on all the variables. COV, XMEAN, and NI are obtained from DSCRM. Function DMSCR, which uses these arrays, is then called.

 

USE IMSL_LIBRARIES

 

IMPLICIT NONE

INTEGER IGRP, IMTH, IPRINT, LDCLAS, LDCMEA, LDCO, LDCOEF, &

LDCOV, LDD2, LDPROB, LDX, LDXMEA, NCOL, &

NGROUP, NROW, NVAR

PARAMETER (IGRP=1, IMTH=3, IPRINT=0, LDCOV=4, NCOL=5, NGROUP=3, &

NROW=150, NVAR=4, LDCLAS=NGROUP, LDCMEA=NGROUP, &

LDCO=NGROUP, LDCOEF=NVAR, LDD2=NGROUP, LDPROB=NROW, &

LDX=NROW, LDXMEA=NGROUP)

!

INTEGER ICLASS(NROW), IND(4), NI(NGROUP), NNV, NOBS, NOUT, &

NRMISS, NV

REAL CLASS(LDCLAS,NGROUP), CMEAN(LDCMEA,NGROUP-1), &

CO(LDCO,NVAR+1), COEF(LDCOEF,NGROUP-1), &

COV(LDCOV,LDCOV,1), D2(LDD2,NGROUP), EVAL(NGROUP-1), &

PRIOR(3), PROB(LDPROB,NGROUP), REAL, &

STAT(6+2*NGROUP), SUMWT(NGROUP), X(LDX,5), &

XMEAN(LDXMEA,NVAR)

INTRINSIC REAL

!

DATA IND/2, 3, 4, 5/, PRIOR/0.3333333, 0.3333333, 0.3333333/

!

CALL GDATA (3, X, NOBS, NV)

!

CALL DSCRM (NROW, NVAR, X, NGROUP, COV(1:,1:,1), CO, ICLASS, &

PROB, CLASS, D2, STAT, IND=IND, IGRP=IGRP, IMTH=IMTH, &

PRIOR=PRIOR, XMEAN=XMEAN)

!

SUMWT(1) = STAT(6+NGROUP)

SUMWT(2) = STAT(7+NGROUP)

SUMWT(3) = STAT(8+NGROUP)

!

CALL DMSCR (XMEAN, SUMWT, COV(1:,1:,1), NNV, EVAL, COEF, CMEAN)

 

CALL UMACH (2, NOUT)

WRITE (NOUT,'('' NNV = '',I1)') NNV

CALL WRRRN ('EVAL', EVAL, 1, NNV, 1)

CALL WRRRN ('COEF', COEF)

CALL WRRRN ('CMEAN', CMEAN)

END

Output

 

NNV = 2

 

EVAL

1 2

32.19 0.29

 

COEF

1 2

1 -0.829 0.024

2 -1.534 2.165

3 2.201 -0.932

4 2.810 2.839

 

CMEAN

1 2

1 -5.502 6.877

2 3.930 5.934

3 7.888 7.174