public class SammonMapping
extends java.lang.Object
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 interpoint distances we can minimize the difference between corresponding interpoint 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.
Modifier and Type  Field and Description 

double[][] 
coordinates
The coordinates.

double 
stress
The final stress achieved.

Constructor and Description 

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

Modifier and Type  Method and Description 

static SammonMapping 
of(double[][] proximity)
Fits Sammon's mapping with default k = 2, lambda = 0.2, tolerance = 1E4 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.

public final double stress
public final double[][] coordinates
public SammonMapping(double stress, double[][] coordinates)
stress
 the objective function value.coordinates
 the principal coordinatespublic static SammonMapping of(double[][] proximity)
proximity
 the nonnegative proximity matrix of dissimilarities. The
diagonal should be zero and all other elements should be positive and symmetric.public static SammonMapping of(double[][] proximity, int k)
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.public static SammonMapping of(double[][] proximity, java.util.Properties prop)
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.public static SammonMapping of(double[][] proximity, int k, double lambda, double tol, double stepTol, int maxIter)
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.public static SammonMapping of(double[][] proximity, double[][] init, double lambda, double tol, double stepTol, int maxIter)
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.