The MRSCAL
algorithm is a metric counterpart to MINISSA. Its aim
is to position a set of stimulus objects as a set of points in a space of
minimum dimensionality in much the same way as MINISSA, except that the distances in this space will be
a linear (or optionally a logarithmic) function of the dissimilarities between
the stimuli.
Data: 2-way, 1-mode
dis/similarities
Transform: Linear or Logarithmic Model:
Euclidean (and other Minkowski) Distance
Output
from MRSCAL may be input to PINDIS.
MRSCAL
accepts as input the lower triangle (without diagonal) or a full symmetric
matrix of (dis)similarities. The program will also accept negative values such
as product moment correlations and covariances.
The
user may provide a starting configuration by means of the command READ
CONFIG, using an associated INPUT FORMAT specification if the input
is not free format. A coordinate for each point on each dimension is input. This
may be done either by stimuli (rows) by dimensions (columns) or dimensions
(rows) by stimuli (columns). In this latter case MATFORM(1) should set in
the PARAMETER command.
If this
is not done, however, then the program constructs an initial configuration from
the original data by the Lingoes-Roskam (1973) procedure which is a good initial
approximation to a solution and has desirable geometrical
properties.
The
user may choose the way in which the distance between the points in the
configuration is defined using the MINKOWSKI parameter. The default value
2 provides for the Euclidean metric, but the user may specify any non-negative
value for the parameter. Commonly used values include 1, the so-called
'city-block' metric where the distance between the two points is the sum of the
differences between their co-ordinates on the axes of the space, and infinity
(in MRSCAL approximated by a large number(>25)), the so-called 'dominance'
metric when the largest difference on any one axis will dominate all others.
(Users are warned that high MINKOWSKI values are liable to produce
program failure due to overflow).
Linear
and logarithmic transformations
The most common use of MRSCAL is to
find a linear transformation of the data which best fits a configuration of
points in the chosen dimensionality. The program will also, however, perform an
analysis using logarithmic transformations of the data values. In this case the
Shepard diagram will show a smooth exponential curve. The user must specify
which transformation is required. If no PARAMETERS command is read and/or
no specification of the transformation given, then no analysis will be
performed.
Dimensionality
As
a general rule solutions should be computed in a number of dimensionalities.
Since a perfect fit will be obtained in n-2 dimensions the trial
dimensionalities should always be in dimensionalities less than n-3. As a
practical guide to the choice of trial dimensionalities it is recommended that
the data compression ratio (defined by Young as the product of stimuli x dimensions divided
by the number of input data elements) should be greater than 2.
A
further guide involves examining the plot of stress by dimensionality. Since MU
is a measure of goodness of fit the plot will show an ascending function. The
'appropriate' dimensionality, is that at which the graph shows an 'elbow', i.e.
where the addition of extra dimensions is otiose.
PARAMETERS
Keyword
Default Value
Function
DATA
TYPE
0
0: Lower-triangle matrix
of similarities
(high values mean high
similarities
between
points).
1: Lower-triangle matrix of
dissimilarities
(high values mean high
dissimilarities
between
points).
2: Full-symmetric
matrix
of similarities
(high values mean high
similarities
between
points).
3: Full-symmetric matrix of
dissimilarities
(high values mean high
dissimilarities
between points).
LINEAR TRANSFORMATION
0 0: Linear transformation is
not
performed.
1:
Linear transformation is performed.
LOG
TRANSFORMATION 0
0: Logarithmic transformation is
not
performed.
1:
Logarithmic transformation is
performed.
CRITERION
0.00001 Sets the criterion value for
terminating
the
iterations.
MINKOWSKI 2
Sets the Minkowski metric
for
the
analysis.
MATFORM 0
(RELEVANT ONLY WHEN 'READ
CONFIG'
IS
USED)
0:
The input configuration is
entered:
stimuli (rows) by
dimensions
(columns)
1: The
input configuration is
entered:
dimensions
(rows) by
stimuli
(columns)
N.B.
Either LINEAR TRANSFORMATION or LOG TRANSFORMATION must be specified
NOTES
1. N OF
SUBJECTS is not valid with MRSCAL.
2. N OF STIMULI may be replaced by N OF
POINTS
3. a) The program expects input to be in the form of the lower
>
triangle of a matrix of real (F-type) numbers.
b) The INPUT FORMAT, if
specified, should read the longest,
i.e. last, row of this matrix.
4.
LABELS followed by a series of labels (<= 65 char), each on a
separate
line, optionally identifies the stimuli, in order and
without
omissions.
PRINT
options (to main output
file)
Option
Form
Description
INITIAL
p x r matrix Initial configuration,
either
generated
by the program
or provided by the
user
(p = no. of
stimuli, r = no. of
dimensions)
FINAL p
x r matrix Final configuration,
rotated
to
principal
components.
DISTANCES triangle
Solution distances between
points
with diagonal calculated
according to
MINKOWSKI
parameter
FITTING Lower
triangle Fitting values : the
disparities
with diagonal (DHAT)
values.
RESIDUALS Lower
triangle The difference between the
distances
with diagonal and the
disparities.
By
default, only the final configuration and the final STRESS values are output.
PLOT
options (to main output
file)
Option
Description
INITIAL
Up to r(r-1)/2 plots of the
initial
configuration (r=no. of
dimensions)
FINAL
Up to r(r-1)/2 plots of the
final
configuration.
SHEPARD The
Shepard diagram of
distances
plotted against data. Fitted
values
are shown by *, actual
data/distance
pairs by
0.
STRESS
Plot of STRESS values by
iteration,
with a summary plot of
stress
by
dimensions.
POINT
Histogram of point contributions
to
STRESS.
RESIDUALS
Histogram of residual values (logged).
By
default, only the Shepard diagram and the final configuration are
plotted.
PROGRAM
LIMITS
Maximum no.
stimuli = 300
Maximum dimensions = 8
See also