provides
internal analysis of either a matrix (of
co-ordinates or profiles) or a square symmetric matrix of (dis)similarity
coefficients by means of a distance model, which maximises continuity or local
monotonicity. It is particularly efficient in reducing dramatically the number
of dimensions necessary to represent higher-dimensional data (but at the cost
that the highest valued dissimilarities are relatively poorly represented).
DATA:
are EITHER two-way, two-mode row-conditional profile or co-ordinate data OR a
two-way-one mode lower-triangular matrix of dis/similarity or
covariance/correlation measures.
TRANSFORMATION:
"smoothness" or continuity function (kappa)
MODEL Euclidean distance (but weighted to preserve small distances).
Data
may be input to PARAMAP
1. as a matrix of distances, or
2. as a matrix of coordinates (or alternatively a set of profiles). In the case
of profile data, only row points are represented in the solution.
The type of data input is described by DATA TYPE in the PARAMETERS
command.
The default option DATA TYPE (0) allows the user to input a matrix of
coordinates for p points in r dimensions, or of a set of profiles . This is
converted by the program to a set of squared distances before proceeding. The
input matrix might be an actual matrix of coordinates or profile data for N
subjects on p variables. If this is the case, since these are treated as
coordinates, there should be good grounds for regarding the data as being at
least interval level.
NOTES
1. What we refer to as stimuli in the list of input commands are the entities
actually represented in the configuration, and it is the number of these
entities which is given in N OF STIMULI. These may be associated with
identifiers, using the optional LABELS command.
2. The number of dimensions on which the stimuli are measured is given by N
OF SUBJECTS.
3. These may be replaced, as appropriate, by N OF ROWS and N
OF COLUMNS.
The weighting factors
The generalised index of continuity, K* ("KAPPA star") contains three
factors A, B and C which control the weighting assigned to various elements in
the formula. The basis of the index of continuity is the sum of the ratios of
the data distances to the solution distances. This sum is normalised by the sum
of the solution distances. Each of these elements is weighted by being raised
to a specific power. These powers are the values A, B and C. A is the exponent
associated with the data distances, B with the solution distances and C with
the normalising factor. There are two constraints on the possible values of A,
B and C. The first is that C must be negative, and the second that B + C - A
should equal zero if similarity transformations are required, as will normally
be the case. The default options allow for the values A(1), B(2), C(-1) which
reduces the general index K* to the index K as used in PROFIT.
Users may wish to vary these values. The crucial consideration would seem to be
the ratio between the weights assigned to the data values and to the solution
values (A and B respectively). In general, B should be greater than or equal to
A.
The CRITERION parameter
At step 4 of the algorithm PARAMAP performs a number of tests to determine
whether the iterative process should proceed. One of these is to decide whether
the index of continuity has reached a minimum value. This value is set by the
user by means of the CRITERION parameter. The default value CRITERION
(0) asks the program to try for a perfectly smooth functional relationship
between data and solution. It is, of course, likely that the process will
terminate before KAPPA reaches zero if a minimum is found. The user may specify
non-negative values of CRITERION, reasonably between 0.05 and 0.1, to
make exploratory analyses of a data set.
Normalisation
If a rectangular matrix is input, the user may choose to normalise the matrix
before the distances are computed. There are two options: If the distances are
to be calculated from the matrix without normalisation then NORMALISE (0),
the default option, is appropriate. If the rows of the matrix are to be
normalised, then NORMALISE (1) should be specified in the PARAMETERS
command.
The initial configuration
The user may choose to input an initial configuration of points which represent
a guess at the possible solution configuration. In this case a configuration
containing the stimulus points in the required dimensionalities is input, with
stimuli as rows and dimensions as columns. If solutions are to be obtained in
more than one dimensionality then a configuration for each dimensionality
should be input. These should follow a READ CONFIG command, the configurations
following each other without a break. The lowest dimensionality should come
first (the INPUT FORMAT, if specified, should be suitable for reading
one row of the longest matrix, i.e. the highest dimensionality. By default,
free format input will be assumed).
Alternatively, the program will generate a random configuration of points to
provide the starting configuration. Different starting configurations should be
tried if relatively high values of KAPPA occur. This is done by specifying
different values for the RANDOM "seed" in the PARAMETERS
command.
INPUT
COMMANDS
Keyword
Function
N OF STIMULI [number]
Number of stimuli in the analysis
N OF SUBJECTS [number]
Number of subjects/cases
DIMENSIONS [number]
Dimensions for analysis
LABELS [followed
by a series
Optionally identify the stimuli.
of
labels (<=
65 chars There should
be as many labels
each
on a separate line] as there are
stimuli.
PARAMETERS
Keyword Default Value Function
DATA TYPE 0 0:
Input matrix is a rectangular matrix
of stimulus coordinates.
1:
Input matrix is lower-triangle
covariance matrix with diagonal.
2:
Input matrix is a lower triangle
matrix
of squared interpoint
distances
without diagonal.
3:
Input matrix is lower triangle matrix
of
correlation coefficients without
diagonal.
4:
Input matrix is lower triangle matrix
of
interpoint distances without
diagonal.
MATFORM 0 (Relevant
only when DATA TYPE(0) is specified)
0:
The input matrix is punched
stimuli
(rows) by dimensions (columns).
1:
The input matrix is punched
dimensions
(rows) by stimuli (columns).
NORMALISE 1
0: No normalisation
1: Row effects removed from data.
RANDOM 12345 Enter
any odd five digit integer, to set the
pseudo-random
number generator seed.
A 1
'a' of the KAPPA formula.
B 2 'b'
of the KAPPA formula.
C -1
'c' of the KAPPA formula.
CRITERION 0 Sets
the criterion for the terminating
value
for KAPPA.
PRINT options
Option Form Description
INITIAL p x r The
coordinates at the initial
configuration
are output.
FINAL p
x r The
coordinates of the stimuli in
the
solution configuration are output.
DISTANCES lower triangle The squared distances in the solution
are
output.
HISTORY An
iteration-by-iteration history of
the
algorithm is output.
By default the initial and final configurations and the final value of KAPPA
are printed.
PLOT options
Option Description
INITIAL The initial
configuration is
plotted.
r(r-1)/2 two-way plots are
produced.
FINAL The
solution configuration in the
form
of r(r-1)/2 plots is produced.
FUNCTIONS r x r plots of the functions
required
to translate the r
dimensions
at X into the r
dimensions
of Y.
SHEPARD A plot of initial distance
values
against
the fitted values is produced.
KAPPA A
histogram showing the value of KAPPA
at
each iteration is produced.
By default only the FINAL configuration is plotted.
PUNCH options (to secondary output file)
Option Description
SPSS The
following are output in a fixed
format:
I
= stimulus index
J
= subject index
DATA
= corresponding (squared) data
distance
DISTANCE
= corresponding (squared)
solution
distance
RESIDUAL
= corresponding residual value
FINAL The
coordinates of the stimuli in the
final
configuration are output in a fixed
format.
KAPPA The
values for KAPPA at each iteration
are
output.
By default, no secondary output file is produced.
PROGRAM LIMITS
Maximum no. of subjects (data dimensions) = 100
Maximum no. of stimuli = 60
Maximum no. of (solution) dimensions = 5
See also