Fmatch will either rotate both of the matrices simultaneously, to a compromise position, or will take one of the two matrices as a target and rotate the other matrix to congruence with the target. The two solution options are known as case I and case II respectively.
Case I, factor matching, simultaneously matches the two solutions is analogous to finding the orientation of r space and orientation of s space and matching the n projections in each of these spaces. The axes of the two spaces are rotated so that the columns of the rotated matrices are as similar as possible. This problem is one of finding eigenroots and eigenvectors and applying those transformations to the two original matrices.
Case II focuses on a fixed target matrix T and attempts to find an orthogonal transformation OT, which, when applied to the original matrix A, produces a matrix A* which is as similar as possible to A. Case II uses the degree of collinearity between corresponding rows of A and A* to measure the congruence of the solution. (The matching procedure is also known as PROCRUSTEAN matching)
The data input for FMATCH includes:
1. PROBLEM LINE
2. FORMAT STATEMENT FOR READING THE TARGET MATRIX
3. FORMAT STATEMENT FOR READING THE "DATA" MATRICES
4. TARGET MATRIX DATA INPUT
5. "DATA" MATRICES ARRANGED IN THE DESIRED ORDER
PROGRAM LIMITATION: FMATCH can handle a target matrix of up to 50 rows and 50 columns. The data matrix must be of the same dimension. Any number of data matrices can be matched to the target matrix in the same analysis.
Line 1: PARAMETERS EXPLANATION (FREE FORMAT)
1. Number of rows in all matrices used as inputs
2. Number of columns in the target matrix
3. Number of columns in the "data" matrices
4. Number of matches to be performed
5. Type of matching:
1 = DATA MATCHED TO TARGET
2 = DATA MATCHED TO TARGET AND VICE VERSA
Line 2: TARGET MATRIX FORMAT
A format statement for the target data matrix:
Line 3: DATA MATRIX FORMAT
A format statement for the data matrix:
Line 4+: Supply data as specified above, first the target matrix, followed by the other matrices.
NOTES1. FMATCH is also known as C-MATCH, or CLIFF MATCH
(The side notes are for explanation and do not belong in the data set):
10 2 2 1 2 (3X, 2F8.0) This is input format statement for data set 1 (3X, 2F8.0) This is input format statement for data set 2 1 -0.0463 0.5558 This is the start of data set 1 2 0.5972 -0.0163 3 0.5724 -0.3562 4 -0.3914 -0.2518 5 -0.4079 0.1736 6 -0.3826 -0.2096 7 -0.4379 0.1296 8 0.4792 0.2585 9 0.4341 -0.0634 10 -0.4169 -0.2203 1 -0.3606 -0.4072 This is the start of data set 2 2 0.4806 -0.2670 3 0.5702 0.3010 4 -0.3474 0.0876 5 -0.4191 -0.1672 6 -0.3724 0.1822 7 -0.3253 0.3763 8 0.5712 -0.1395 9 0.5858 0.0547 10 -0.3830 -0.0210
F M A T C H
FACTOR MATCHING PROGRAM
WRITTEN BY NORMAN CLIFF
PC-MDS VERSION
ANALYSIS TITLE: FMATCH TEST DATA
DATA IS READ FROM FILE: FMATCH.DAT
OUTPUT FILE IS: B:FMATCH.PRN
INPUT PARAMETERS:
DATA IS 10 BY 2
TARGET IS 10 BY 2
NUMBER OF MATCHES: 1
DATA MATCHED TO TARGET AND VICE VERSA
TARGET
ROW: 1 -.046 .556
ROW: 2 .597 -.016
ROW: 3 .572 -.356
ROW: 4 -.391 -.252
ROW: 5 -.408 .174
ROW: 6 -.383 -.210
ROW: 7 -.438 .130
ROW: 8 .479 .259
ROW: 9 .434 -.063
ROW:10 -.417 -.220
EIGEN ROOTS
1 2
1.93603 .70970
EIGENVECTORS IN COLUMNS
ROW: 1 1.000 .000
ROW: 2 .000 1.000
TRANSLATED TO ZERO MEAN AND ROTATED TO PRINCIPAL AXES
ROW: 1 -.046 .556
ROW: 2 .597 -.016
ROW: 3 .572 -.356
ROW: 4 -.391 -.252
ROW: 5 -.408 .174
ROW: 6 -.383 -.210
ROW: 7 -.438 .130
ROW: 8 .479 .259
ROW: 9 .434 -.063
ROW:10 -.417 -.220
NORMALIZED TARGET
ROW: 1 -.091 1.093
ROW: 2 1.174 -.032
ROW: 3 1.126 -.700
ROW: 4 -.770 -.495
ROW: 5 -.802 .341
ROW: 6 -.752 -.412
ROW: 7 -.861 .255
ROW: 8 .942 .508
ROW: 9 .854 -.125
ROW:10 -.820 -.433
DATA
ROW: 1 -.361 -.407
ROW: 2 .481 -.267
ROW: 3 .570 .301
ROW: 4 -.347 .088
ROW: 5 -.419 -.167
ROW: 6 -.372 .182
ROW: 7 -.325 .376
ROW: 8 .571 -.140
ROW: 9 .586 .055
ROW:10 -.383 -.021
EIGEN ROOTS
1 2
2.04309 .56102
EIGENVECTORS IN COLUMNS
ROW: 1 1.000 .000
ROW: 2 .000 1.000
TRANSLATED TO ZERO MEAN AND ROTATED TO PRINCIPAL AXES
ROW: 1 -.361 -.407
ROW: 2 .481 -.267
ROW: 3 .570 .301
ROW: 4 -.347 .088
ROW: 5 -.419 -.167
ROW: 6 -.372 .182
ROW: 7 -.325 .376
ROW: 8 .571 -.139
ROW: 9 .586 .055
ROW:10 -.383 -.021
NORMALIZED DATA
ROW: 1 -.718 -.811
ROW: 2 .957 -.532
ROW: 3 1.136 .600
ROW: 4 -.692 .174
ROW: 5 -.835 -.333
ROW: 6 -.742 .363
ROW: 7 -.648 .750
ROW: 8 1.138 -.278
ROW: 9 1.167 .109
ROW:10 -.763 -.042
MATCH: 1
SIMULTANEOUS TRANSFORMATIONS
TRANSFORMATION MATRIX - TARGET TO DATA
ORTHOGONALITY OF THIS TRANSFORMATION IS QUESTIONABLE
1 .998 .067
2 -.067 .998
TRANSFORMED TARGET
1 -.164 1.084
2 1.174 .046
3 1.170 -.624
4 -.735 -.545
5 -.823 .287
6 -.723 -.461
7 -.876 .197
8 .906 .570
9 .860 -.068
10 -.789 -.487
TRANSFORMATION MATRIX - DATA TO TARGET
ORTHOGONALITY OF THIS TRANSFORMATION IS QUESTIONABLE
1 .996 -.091
2 -.091 -.996
TRANSFORMED DATA MATRIX
1 -.641 .873
2 1.002 .442
3 1.076 -.701
4 -.705 -.111
5 -.801 .408
6 -.772 -.294
7 -.714 -.687
8 1.159 .173
9 1.152 -.215
10 -.756 .111
*****IDENTIFICATION KEY FOR PLOTS WITH IDENTIFIED POINTS*****
PT # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
CHAR 1 2 3 4 5 6 7 8 9 A B C D E F
PT # 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
CHAR G H I J K L M N O P Q R S T U
PT # 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
CHAR V W X Y Z + / = * & $ @ % ? <
PT # 46 47 48 49 50
CHAR ( ) " # @
POINT NUMBERS ABOVE 50 IDENTIFIED AS >, MULTIPLE POINTS IDENTIFIED AS #
THE FIRST 10 POINTS ARE TARGET---THE LAST 10 ARE DATA
+....+....+....+....+....+....+....+....+....+....+....+....+
1.17+ | +
. B 1| .
. | .
.96+ | +
. | .
. | .
.74+ | +
. | .
. | I .
.52+ | 8 +
. | .
. 5 | .
.30+ F | +
. # | .
. | .
.08+-------------------------0-----------------------------------+
. | C .
. | J 2 .
-.14+ | 9 +
. | .
. | .
-.35+ | +
. A 6G | .
. K4 | .
-.57+ E | +
. | D .
. | 3 .
+....+....+....+....+....+....+....+....+....+....+....+....+
-1.0 -.8 -.6 -.4 -.2 .0 .1 .3 .5 .7 .9 1.1 1.3
THE GOODNESS OF FIT IS .889 -1. IS WORST, +1. IS PERFECT
CORRELATION MATRIX
ROW: 1 .972 .000
ROW: 2 .000 .654
THE CORRELATION OF THE DISTANCE VECTORS IS .856