PARAmetric MAPping : PARAMAP

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

  • The NewMDSX commands in full