J. J. Chang and J. D. Carroll
Bell Laboratories
Murray Hill, New Jersey 07974
The PROFIT program relates a stimulus space to several sets of independently determined physical attributes called properties.
PROFIT starts with coordinate input describing a configuration of stimuli (N) in a specified number of dimensions (K). This input is normally derived from a multidimensional scaling procedure such as KYST or INDSCAL. Also input is the set of independently determined physical measures called properties, on which each stimuli is ranked.
PROFIT will find for each property a vector, or direction, in the K dimensional space such that the projections of the N stimulus points on that vector correspond optimally to the values input for the property vectors.
Two different approaches are employed in finding the vectors, including a procedure based on linear regression (see Miller, Shepard and Chang, 1964) and, a procedure based on optimizing an index of nonlinear correlation, an algorithm devised by J. D. Carroll (1964).
The first procedure is equivalent to a multiple linear regression analysis, in which each property plays the role of a dependent variable, while the stimulus coordinates of the multidimensional space are treated as independent variables. The second procedure can be viewed as providing a variety of nonlinear regression analyses, in which a very general nonlinear relation is assumed between the dependent variables and a linear combination of the independent variables.
Various plotting facilities are provided by PROFIT, allowing for plotting of best fitting property vectors against one another and against the coordinates of the multidimensional space.
The particular problem this program deals with can be simply stated as follows:
Suppose we have the coordinates of N stimulus points in K dimensional space, determined, say, from a multidimensional scaling procedure, and several sets of physical measures, which we call properties. Each property consists of N independently determined measures, one value for each stimulus point in the psychological space. The program will then find for each property a vector in the K dimensional space such that the projections of the N points on the vector correspond optimally with the given property values. 'Optimal correspondence' is defined in terms of optimizing either linear correlation of a general measure of nonlinear correlation, proposed by Carroll, based on notions of 'continuity' or 'smoothness'.
The approach to this problem devised by Carroll (1964) in the nonlinear regression procedure is quite different from the linear case (Miller, Shepard and Chang, 1964). However, the input and output of these two analyses are similar. For this reason they are combined into one computer program.
The input to the program consists of the coordinates of stimulus points in a specified number of dimensions and a set of properties.
The program provides output to the screen and to a user specified disk file.
The mathematics for the linear and nonlinear regression procedures will be briefly stated here. The reader is referred to the references for detailed description.
Property fitting by linear regression.
We define the following variables:
X || Xij || is the matrix of coordinates of N objects in K dimensional space.
( i = 1-----N, and j = 1-----K)
The origin of the K dimensional space is assumed to be at the centroid of the N objects. This means that the columns of the X matrix sum to zero. In practice this is assured by initially subtracting out the column means.
T (Tj) is the column vector of direction cosines of the fitted vector. (j = 1-----K)
H (Hi) is the row vector of projections of N points on the fitted vector. (i = 1-----N)
We want to find T and H such that | P - H |2 is minimum where H = XT; that is, we seek the least squares solution for T. The solution, by well-known results, is given by:
T = (XTX)-1 XTP while H = XT = X(XTX)-1 XTP
Property fitting by nonlinear regression
A general index of nonlinear correlation (x) between an independent variable P and a dependent variable X was defined by Carroll (1964) as:
with f a monotone decreasing function, and
In the current application the independent variable P corresponds to one property and the dependent variable X refers to the projections of the N objects, on a vector in the K dimensional space, while in most applications Wij = 1/(Pi - Pj)2 + CONSTANT
X is an inverse measure of nonlinear correlation. Therefore, the procedure applied in the program aims to minimize x. Such a procedure entails finding the direction cosines (with respect to the initial reference axes) of a vector such that the projections of stimuli on that vector are as closely related as possible to the corresponding property values.
As an initial step we must transform the initially given coordinates to a new 'orthonormal' set (i.e., such that XTX = I). We do this by transforming to 'principal axes' and then standardizing each principle axis to unit length.
To find the direction cosines (T) of the fitted vector and the projections (H) of stimuli on the vector, we first construct a symmetric matrix XTAX, where the elements of A are defined by:
The smallest characteristic root of XTAX is the minimum value of x and the corresponding characteristic vector gives the direction cosines (T) of the new vector. As before, H = XT. Once we have found the direction cosines with respect to the orthonormal coordinates, we must apply a transformation to find direction cosines with respect to the original coordinate system.
Since the nonlinear regression procedure embraces a more sophisticated approach, there is some complication in interpreting the nonlinear correlation index kappa. Hence, a considerable portion of the program has been devoted to nonlinear regression. For this reason we shall describe the procedures adopted respectively by nonlinear and linear regression in succession.
1) Compute the first four moments of x, as well as skewness and kurtosis. These moments are computed by equations derived by von Neumann (1941), and are appropriate, under the usual normality and independence assumptions, when only a single 'dependent variable' is involved. In our case, where more than one 'dependent variable' is involved, and we optimize x, the distribution is no longer valid. However, the Z squared measure discussed below should be at least approximately valid.
2) Compute Z and Z squared (ZSQ), where Z is a standardized index of nonlinear correlation and ZSQ might be taken as an approximate chi square measure with K (number of dimensions) degrees of freedom. Z and ZSQ are defined by:
In these equations µx and x are, the mean and standard deviation of x, as computed above. The third and fourth moments (as well as skewness' and 'kurtosis') are computed only to assess how the basic distribution of x departs from normality. If this distribution were normal, skewness and kurtosis would be, respectively, zero and three.
3) Construct the symmetric matrix XTAX and find its smallest characteristic root (x). The corresponding characteristic vector (T) defines direction cosines of the vector on which the projections of N points have the maximum nonlinear correlation (of 'x') with the given property values.
4) Compute the projections (H) of the N points on the vector defined by T.
Step 3 - When all vectors have been found, the program proceeds to compute the cosine of angles between all pairs of vectors.
Step 4 - For each pair of vectors, plot the vectors and the projections of N points on the plane formed by the pair.
Step 5 - The configuration X is then transformed back to the original coordinates.
Step 6 - Compute the direction cosines of all vectors with respect to the original coordinates.
Step 7 - Plot the projections of all vectors onto planes defined by pairs of coordinates in the original space, together with the corresponding projections of stimulus points.
2. Linear Regression
Step 1 - Normalize X by columns, i.e., subtract the respective column means. Each column represents one dimension. Then compute the matrix XMAT, where XMAT = (XTX)-1XT
Step 2 - For each property the following operations are carried out:
1) Compute the direction cosines (the Tj's of the fitted vector.
2) Compute the projections of the N points (the Hi's) on the fitted vector.
3) Compute the product moment correlation (RHO) between the projections of N points on the fitted vector and the corresponding property values.
Proceed on to steps 3, 4, and 7 as described in nonlinear regression.
It is essential that the input lines be arranged in the order given below. In general, each input set refers to one input image. However, when a matrix or an array of numbers belonging to a single variable is referred to, several lines may be used.
Input Set #1: Parameters - All parameters are entered FREE format.
| LANA | 1 Do linear regression only. |
| 2 Do nonlinear regression only. | |
| 3 Do both linear and nonlinear regression treated as two independent analyses. |
N Number of stimulus points. Maximum N is 500.
K Number of dimensions. Maximum K is 10.
M Number of properties. Maximum M is 60
| IRX | 0 The stimulus matrix is entered N by K. Namely each row consists of the coordinates of one stimulus point. Each row must begin on a new line. |
| 1 The stimulus matrix is entered K by N. Each row, representing one dimension, begins on a new line and may continue to several lines if necessary. | |
| IWGT | This parameter is relevant only when doing nonlinear regression analysis. A detailed description of IWGT is in section V. |
| IPLOT | Option on plots. |
| 0 The program will not generate any plot. | |
| 1 Plot projections of all prop. in original space. | |
| 2 Plot projections of all properties in the original space and for each property plot the given property values against the projections of stimuli on its fitted vector. | |
| 3 In addition to the above, plots showing the projections of stimulus points on a plane formed by pairs of fitted vectors. | |
| BC0 | BC0 is also relevant only in nonlinear regression analysis. See section V for a discussion of BC0. |
| Input set #2 | A FORMAT for reading in the stimulus matrix (use floating point format). |
| Input set #3 | Stimulus Matrix (X) - This matrix may be entered either K by N or N by K. Option IRX indicates which way it is entered. Each row of the matrix must begin on a new line. |
| Input set #4 | A format for reading in the properties (floating point format). |
| Input set #5 | Property matrix - Each property must be preceded by a line containing its property name. The property matrix must be arranged in the following order: |
| 1st property name (restricted to one line) | |
| 1st property values - read in according to format given above. May use several lines if necessary | |
| Mth property name | |
| Mth property value | |
| Input set #6 | Labels for Stumulus matrix (property labels have already been given) |
IWGT concerns the available options on the weight function and BC0 is a constant associated with
it. At present there are only three options available:
(1) When IWGT = 0 (This is considered the
normal case), Wij = 1/[(Pi - Pj)2 + CONS], where
The reason for the addition of the constant is evident in the case where Pi = Pj. Wij becomes undefined unless this constant is given some non-zero value. It is also felt that the inclusion of this term may improve the measure in many cases. In this case of noisy data, there is some reason to believe BC0 should be about equal to the ratio of 'error' variance to 'true' variance.
(2) When IWGT = 1
Wij = 1 - if | Pi - Pj | < BC0
0 - otherwise
If the property values correspond to the integer values 1 through N (though not necessarily in that order) and BC0 = 1.0, Wij will equal 1 if i and j are adjacent and 0 otherwise. In this case x is equivalent to von Neuman's N, (the ratio of the 'mean square successive difference' to the variance) as defined in von Neuman (3). Let
Where Wij has been defined in option 2, (with Wii = 1) .
Then we redefine Wij = Wij/ NiNj .
This option has been introduced because it allows, under certain conditions, the equivalence of 1-x and N2, the so-called 'correlation ratio'. To attain this equivalence, it is necessary only that objects in the same group have property values that differ by no more than BC0, while objects in different groups differ by more than BC0. If this option is used, and the conditions obtain making x equivalent to N2, the non-linear regression procedure itself becomes equivalent to finding the best linear discriminant in a discriminant function analysis. There is reason to believe, however, that this option is sensible in more general situations.
1) The normalized configuration of N stimulus points in K dimensional space. (printed as a K by N matrix)
2) For each property the original values on properties and the projections on fitted vectors.
3) For the linear case, the maximum correlation between each given property and its fitted vector. For the nonlinear case, the smallest characteristic root associated with each property, as well as Z, ZSQ and the four moments of x.
4) Direction cosines of the fitted vector in normalized space.
5) Cosines of angles between fitted vectors.
6) For nonlinear case only--the direction cosines of fitted vectors in the original space.
1. Carroll, J. Douglas and Jih-Jie Chang (1964). "A General Index of Nonlinear Correlation and Its Application to the Problem of Relating Physical and Psychological Dimensions." American Psychologist, 1964, 19, 540.
2. Miller, J. E., Shepard, R. N. and Chang, J.-J. "An Analytical Approach to the Interpretation of Multidimensional Scaling Solutions." Paper presented at 1964 meeting of A.P.A. Abstract in American Psychologist, 1964, 19, 579-580.
3. Von Neuman, J. (1941). "Distribution of the Ratio of the Mean Square Successive Difference to the Variance." Am. Math. Statist., 12, 367-395.
1. The original version of this paper has been modified for the PC-MDS Version of PROFIT by Scott M. Smith.
SAMPLE PROFIT DATA SET
3 20 4 4 0 0 2 0.0
(2X,4F7.3)
1 .309 -.476 1.126 .75
2 .588 -.294 -2.251 .25
3 .809 .476 2.694 -.85
4 .951 .294 -1.864 -.45
5 1.000 -.294 2.118 .85
6 .951 -.476 -.352 -.15
7 .809 .294 .699 -.25
8 .588 -.476 -2.135 -.95
9 .309 .000 1.597 .65
10 .000 -.294 -2.006 .45
11 -.309 .476 -1.131 -.65
12 -.588 -.294 -.638 .55
13 -.809 .000 3.330 .95
14 -.951 -.476 -1.342 .15
15 -1.000 .000 -.036 .35
16 -.951 .476 .313 .05
17 -.809 .000 -.896 -.75
18 -.588 .294 -1.707 -.55
19 -.309 .294 -1.534 -.35
20 .000 .476 4.033 -.05
(20F3.1)
SIN FUNCTION
.1 .2 .3 .4 .5 .6 .7 .8 .91.01.11.21.31.41.51.61.71.81.92.0
SIN.COS
1.8 .6 .2 .4 .91.71.1 .72. 1.61.21.9 .5 .81.01.31.5 .11.4 .3
EXPONENTIAL
1.5 .11.8 .41.71.11.4 .21.6 .3 .81.01.9 .71.21.3 .9 .5 .62.0
LINEAR
1.81.3 .2 .61.9 .9 .8 .11.71.5 .41.62.01.21.41.1 .3 .5 .71.0
O1
O2
O3
O4
O5
O6
O7
O8
O9
1T
T1
T2
T3
T4
T5
T6
T7
T8
T9
2T
SAMPLE PROFIT DATA SET
1 11 2 12 0 0 2 0.0
(2X,10F7.3)
-0.383 0.672 PARAMETER LINE IS FOLLOWED BY FORMAT
0.429 -0.708 STATEMENT AND CONFIGURATION MATRIX.
0.157 -0.661 CONFIGURATION FROM KYST OR INDSCAL
-1.240 0.218
0.238 0.605
-1.385 -0.187
0.470 0.810
-0.970 -0.500
0.792 0.612
0.893 -0.288
0.997 -0.572
(2X,11F4.1) FORMAT STATEMENT FOR READING PROPERTIES
SOPHISTICATED
1 0.3 2.0 2.2-1.0 1.2-2.7 1.4-2.5 1.6 3.0 2.8
EXCITING
2 1.5-1.0 -.8-2.0 2.0-1.8 2.2-1.8 2.5 1.4 1.1
STRONG
3 1.7 1.2 1.5-1.4 1.8-1.6 1.0-1.5 .5 1.4 1.2
UNCONVENTIONAL
4 1.2 .5 1.0-1.4 1.3-2.0 1.5-1.6 2.5 1.8 2.2
BOLD
5 .5-0.2 .3-1.8 2.2-2.0 .9-1.5 2.5 1.2 1.8
COMPLEX
6 -1.6 0.3 .8-2.4 -.1-2.5 1.2-1.8 .6 1.5 1.9
SPORTY
7 .9-1.8-2.6 -.3 1.5 0.2 0.5 0.0 2.0-2.4-2.9
FRESH
8 .5-2.0-1.5 .3 .7-0.5 1.6 -.2 1.8 2.0 1.3
SWIFT
9 2.6 1.5 1.8-0.4 2.2-2.5 2.0 2.1 1.8 .2 1.1
ELEGANT
10-1.5 .5 .8-2.0-0.8-2.8 1.7-2.5 1.3 2.3 2.7
RELIABLE
11 .8 .4 1.3 .2 .9 0.0 1.9-1.7-0.8-1.0 2.1
MASCULINE
12 1.2 -.5 .4 -.8 1.3-1.2 1.0-1.5 0.9-2.0 2.2
MUSTANG
SEVILLE
LINCOLN
ESCORT
CORVETTE
CHEVETTE
300ZX
ALLIANCE
PORSCHE944
XJ6
500SEL
SAMPLE OUTPUT
P R O F I T |INPUT DATA: AUTOMOBILE DATA 11 BRANDS
PROPERTY FITTING ANALYSIS | 1 11 2 12 0 0 2 0.0
PROGRAM WRITTEN BY DR. J. D. CARROLL AND JIH JIE CHANG | (2X,10F7.3)
PC-MDS VERSION | -0.383 0.672
| 0.429 -0.708 CONFIG. MATRIX FROM
ANALYSIS TITLE: CAR DATA KYST CONFIGURATION | 0.157 -0.661 THE KYST PROGRAM
DATA IS READ FROM FILE: CARPROFI.DAT | -1.240 0.218
OUTPUT FILE IS: CARPROFI.PRN | 0.238 0.605
| -1.385 -0.187
| 0.470 0.810
LANA (REGRESSION OPTION): 1 | -0.970 -0.500
N NO. OF STIMULI (100 MAX) 11 | 0.792 0.612
K NO. OF DIMENSIONS (10 MAX) 2 | 0.893 -0.288
M NO. OF PROPERTIES (20 MAX) 12 | 0.997 -0.572
IRX 0 = N X K INPUT; 1 = K X N INPUT 0 |(2X,11F4.1) FORMAT STATEMENT FOR READING
IWT 0 = RATIO OF ERROR VAR. TO TRUE VAR. (USUAL OPTION) 0 | SOPHISTICATED PROPERTIES
1 = RATIO OF MEAN SQ. SUCCESSIVE DIFFERENCE TO VARIANCE |1 0.3 2.0 2.2-1.0 1.2-2.7 1.4-2.5 1.6 3.0 2.8
IPLOT 0 = PROPERTIES ONLY 2 | EXCITING
1 = PLOT PROPERTIES AND FUNCTIONS |2 1.5-1.0 -.8-2.0 2.0-1.8 2.2-1.8 2.5 1.4 1.1
2 = DO ALL PLOTS | STRONG
BCO (FLOATING POINT NUMBER FOR NON LINEAR REG.) 0. |3 1.7 1.2 1.5-1.4 1.8-1.6 1.0-1.5 .5 1.4 1.2
| UNCONVENTIONAL
CAR DATA KYST CONFIGURATION |4 1.2 .5 1.0-1.4 1.3-2.0 1.5-1.6 2.5 1.8 2.2
LINEAR REGRESSION | BOLD
|5 .5-0.2 .3-1.8 2.2-2.0 .9-1.5 2.5 1.2 1.8
NORMALIZED CONFIGURATION | COMPLEX
1 -.3828 .4292 .1572 -1.2398 .2382 |6 -1.6 0.3 .8-2.4 -.1-2.5 1.2-1.8 .6 1.5 1.9
-1.3848 .4702 -.9698 .7922 .8932 | SPORTY
.9972 |7 .9-1.8-2.6 -.3 1.5 0.2 0.5 0.0 2.0-2.4-2.9
2 .6719 -.7081 -.6611 .2179 .6049 | FRESH
-.1871 .8099 -.5001 .6119 -.2881 |8 .5-2.0-1.5 .3 .7-0.5 1.6 -.2 1.8 2.0 1.3
-.5721 | SWIFT
|9 2.6 1.5 1.8-0.4 2.2-2.5 2.0 2.1 1.8 .2 1.1
COVARIANCE MATRIX | ELEGANT
1 7.4484 -.0093 |10-1.5 .5 .8-2.0-0.8-2.8 1.7-2.5 1.3 2.3 2.7
2 -.0093 3.5291 | RELIABLE
|11 .8 .4 1.3 .2 .9 0.0 1.9-1.7-0.8-1.0 2.1
X*(X''X) INVERSE | MASCULINE
1 -.0512 .0574 .0209 -.1664 .0322 -.1860 |12 1.2 -.5 .4 -.8 1.3-1.2 1.0-1.5 0.9-2.0 2.2
-.1304 .1066 .1198 .1337 |
2 .1903 -.2005 -.1873 .0613 .1715 -.0535 |
-.1420 .1737 -.0813 -.1618 | X*(X''X) INVERSE = COMPUTATION MATRIX USED
|STIMULUS RATINGS ON THE FITTED STIMULUS PROPERTY
PROPERTY 1 |
INTERMEDIATE SUMS BEFORE SQUARING: 2.1715 -.2093 |
|
SSQ = 4.75913 XL = 2.18154 |
ORIGINAL VALUES ON PROPERTY 1 |
.3000 2.0000 2.2000 -1.0000 1.2000 |
-2.7000 1.4000 -2.5000 1.6000 3.0000 |
2.8000 |
|
PROJECTIONS ON FITTED VECTORS |PROJECTIONS OF THE 11 STIMULI ON PROPERTY VECTOR 1
-.4455 .4951 .2199 -1.2550 .1790 |
-1.3605 .3903 -.9174 .7298 .9167 |
1.0475 |
|
|
PLOT OF ORIGINAL (X-AXIS) VERSUS OBTAINED (Y-AXIS) FOR PROPERTY VECTOR|PLOT FOR PROPERTY 1:
|PLOT OF ORIGINAL RATINGS VS FITTED RATINGS (DERIVED
+.........+.........+.........+.........+.........+.........+ |BY REGRESSION TO MINIMIZE THE SUM OF SQUARES
1.168+ +|DIFFERENCES BETWEEN THE ORIGINAL AND FITTED RATINGS).
. B .|THE CLOSER THE PLOT APPEARS TO AN UPWARD SLOPING
. .|STRAIGHT LINE, THE BETTER THE FIT.
.874+ A +|
. 9 .|
. .|
.579+ +|
. 2 .|
. 7 .|
.285+ +|
. 5 3 .|
. .|
-.009+ +|
. .|
. .|
-.304+ +|
. 1 .|
. .|
-.598+ +|
. .|
. .|
-.892+ 8 +|
. .|
. .|
-1.187+ +|
. 4 .|
. 6 .|
-1.481+ +|
+.........+.........+.........+.........+.........+.........+ |
-2.99-2.46-1.94-1.42 -.89 -.37 .15 .67 1.20 1.72 2.24 2.76 3.29|
|
CORRELATION BETWEEN ORIGINAL AND FITTED VECTORS FOR PROPERTY 1 IS: |CORRELATION BETWEEN THE PROJECTIONS OF THE
R = .939 , RSQ = .882 |N POINTS ON THE FITTED VECTOR AND THE
|ORIGINAL RATINGS ON THE PROPERTY VECTOR.
|
PROPERTY 2 |
. |
. |OUTPUT IS THE SAME FOR PROPERTIES 2 - 12
. |
. |
PROPERTY 12 |
|
TABLE 1. THE MAXIMUM CORRELATION BETWEEN THE PROPERTY |EACH CORRELATION REPRESENTS THE RELATIONSHIP BETWEEN
AND THE PROJECTIONS ON FITTED VECTOR |THE PROPERTIES OF THE N POINTS ON THE FITTED VECTORS
|AND THE ORIGINAL RATINGS ON THE PROPERTY VECTOR.
RHO PROPERTY |TABLE 1 IS TAKEN FROM THE CORRELATION REPORTED AFTER
1 .9391 SOPHISTICATED |EACH PROPERTY PLOT.
2 .9686 EXCITING |
3 .8081 STRONG |
4 .9702 UNCONVENTIONAL |
5 .9447 BOLD |
6 .9674 COMPLEX |
7 .8716 SPORTY |
8 .7059 FRESH |
9 .5272 SWIFT |
10 .9475 ELEGANT |
11 .3334 RELIABLE |
12 .6054 MASCULINE |
|
TABLE 2. DIRECTION COSINES OF FITTED VECTORS |
IN NORMALIZED SPACE |
|
DIMENSION |
|
VECTOR 1 2 |CORRELATIONS BETWEEN THE PROPERTY VECTOR AND THE
1 .9954 -.0959 |UNDERLYING DIMENSIONAL SPACE. THE SUM OF
2 .6378 .7702 |SQUARED VALUES ACROSS ROWS EQUALS 1.0.
3 .9661 .2582 |
4 .9280 .3726 |
5 .8693 .4943 |
6 .9825 -.1864 |
7 -.2346 .9721 |
8 .4404 .8978 |
9 .7923 .6101 |
10 .9906 -.1367 |
11 .8375 .5464 |
12 .6288 .7776 |
|
TABLE 3. COSINE OF ANGLES BETWEEN VECTORS |
|
VECTOR: 1 2 3 4 5 6 7 8 9 |C,D,E,F,G,H... CORRELATIONS BETWEEN VECTORS
11 |
2 .561 |
3 .937 .815 |
4 .888 .879 .993 |
5 .818 .935 .967 .991 |
6 .996 .483 .901 .842 .762 |
7 -.327 .599 .024 .145 .276 -.412 |
8 .352 .972 .657 .743 .827 .265 .769 |
9 .730 .975 .923 .963 .990 .665 .407 .897 |
10 .999 .526 .922 .868 .794 .999 -.365 .314 .701|
11 .781 .955 .950 .981 .998 .721 .335 .859 .997|
12 .551 1.000 .808 .873 .931 .473 .608 .975 .973|
|
PLOT FOR FIRST TWO DIMENSIONS OF STIMULUS POINTS AND DIRECTION COSINES |
FITTED PROPERTY VECTORS |
|PLOT OF STIMULUS POINTS AND
+.........+.........+.........+.........+.........+.........+ |DIRECTIONAL COSINES OF FITTED
1.000+ I | +|VECTORS. THE DIRECTIONAL COSINES DEFINING THE VECTOR
. | J .|LOCATIONS IN TERMS OF THE X AND Y AXES ARE FOUND ON
. | .|THE UNIT CIRCLE.
.778+ | 7 # +|
. 1 | .|POINTS = 1 TO B, VECTORS = C TO N
. | 5 # .|
.556+ | M +|
. | G .|
. | F .|
.333+ | +|
. | E .|
. | .|
.111+ | +|
. | .|
.------------------------------0------------------------------.|
-.111+ | # +|
. | H .|
. | A .|
-.333+ | +|
. | .|
. 8 | .|
-.556+ | B +|
. | 3 .|
. | 2 .|
-.778+ | +|
. | .|
. | .|
-1.000+ | +|
+.........+.........+.........+.........+.........+.........+ |
-1.20-1.00 -.80 -.60 -.40 -.20 .00 .20 .40 .60 .80 1.00 1.20|
|
*****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 |