smile.mds

Class SammonMapping

java.lang.Object

smile.mds.SammonMapping

public class SammonMapping
extends java.lang.Object

The Sammon's mapping is an iterative technique for making interpoint distances in the low-dimensional projection as close as possible to the interpoint distances in the high-dimensional object. Two points close together in the high-dimensional space should appear close together in the projection, while two points far apart in the high dimensional space should appear far apart in the projection. The Sammon's mapping is a special case of metric least-square multidimensional scaling.

Ideally when we project from a high dimensional space to a low dimensional space the image would be geometrically congruent to the original figure. This is called an isometric projection. Unfortunately it is rarely possible to isometrically project objects down into lower dimensional spaces. Instead of trying to achieve equality between corresponding inter-point distances we can minimize the difference between corresponding inter-point distances. This is one goal of the Sammon's mapping algorithm. A second goal of the Sammon's mapping algorithm is to preserve the topology as best as possible by giving greater emphasize to smaller interpoint distances. The Sammon's mapping algorithm has the advantage that whenever it is possible to isometrically project an object into a lower dimensional space it will be isometrically projected into the lower dimensional space. But whenever an object cannot be projected down isometrically the Sammon's mapping projects it down to reduce the distortion in interpoint distances and to limit the change in the topology of the object.

The projection cannot be solved in a closed form and may be found by an iterative algorithm such as gradient descent suggested by Sammon. Kohonen also provides a heuristic that is simple and works reasonably well.

See Also:

MDS, KPCA

Field Summary

Fields

Modifier and Type	Field and Description
double[][]	coordinates The coordinates.
double	stress The final stress achieved.

Constructor Summary

Constructors

Constructor and Description
SammonMapping(double stress, double[][] coordinates) Constructor.

Method Summary

Modifier and Type	Method and Description
static SammonMapping	of(double[][] proximity) Fits Sammon's mapping with default k = 2, lambda = 0.2, tolerance = 1E-4 and maxIter = 100.
static SammonMapping	of(double[][] proximity, double[][] init, double lambda, double tol, double stepTol, int maxIter) Fits Sammon's mapping.
static SammonMapping	of(double[][] proximity, int k) Fits Sammon's mapping.
static SammonMapping	of(double[][] proximity, int k, double lambda, double tol, double stepTol, int maxIter) Fits Sammon's mapping.
static SammonMapping	of(double[][] proximity, java.util.Properties prop) Fits Sammon's mapping.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

stress

public final double stress

The final stress achieved.

coordinates

public final double[][] coordinates

The coordinates.

Constructor Detail

SammonMapping

public SammonMapping(double stress,
                     double[][] coordinates)

Constructor.

Parameters:

stress - the objective function value.

coordinates - the principal coordinates

Method Detail

of

public static SammonMapping of(double[][] proximity)

Fits Sammon's mapping with default k = 2, lambda = 0.2, tolerance = 1E-4 and maxIter = 100.

Parameters:

proximity - the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric.

of

public static SammonMapping of(double[][] proximity,
                               int k)

Fits Sammon's mapping.

Parameters:

proximity - the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric.

k - the dimension of the projection.

of

public static SammonMapping of(double[][] proximity,
                               java.util.Properties prop)

Fits Sammon's mapping.

Parameters:

proximity - the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric. For pairwise distances matrix, it should be just the plain distance, not squared.

of

public static SammonMapping of(double[][] proximity,
                               int k,
                               double lambda,
                               double tol,
                               double stepTol,
                               int maxIter)

Fits Sammon's mapping.

Parameters:

proximity - the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric.

k - the dimension of the projection.

lambda - initial value of the step size constant in diagonal Newton method.

tol - the tolerance on objective function for stopping iterations.

stepTol - the tolerance on step size.

maxIter - maximum number of iterations.

of

public static SammonMapping of(double[][] proximity,
                               double[][] init,
                               double lambda,
                               double tol,
                               double stepTol,
                               int maxIter)

Fits Sammon's mapping.

Parameters:

proximity - the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric.

init - the initial projected coordinates, of which the column size is the projection dimension. It will be modified.

lambda - initial value of the step size constant in diagonal Newton method.

tol - the tolerance for stopping iterations.

stepTol - the tolerance on step size.

maxIter - maximum number of iterations.