# smile.mds

Multidimensional Scaling

### isomds

`(isomds proximity k)`

`(isomds proximity k tol max-iter)`

Kruskal's nonmetric MDS. In non-metric MDS, only the rank order of entries in the proximity matrix (not the actual dissimilarities) is assumed to contain the significant information. Hence, the distances of the final configuration should as far as possible be in the same rank order as the original data. Note that a perfect ordinal re-scaling of the data into distances is usually not possible. The relationship is typically found using isotonic regression. `proximity` is the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric. `k` is the dimension of the projection. `tol` is the tolerance for stopping iterations. `max-iter` is the maximum number of iterations.

### mds

`(mds proximity k)`

`(mds proximity k positive)`

Classical multidimensional scaling, also known as principal coordinates analysis. Given a matrix of dissimilarities (e.g. pairwise distances), MDS finds a set of points in low dimensional space that well-approximates the dissimilarities in A. We are not restricted to using a Euclidean distance metric. However, when Euclidean distances are used MDS is equivalent to PCA. `proximity` is 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. `k` is the dimension of the projection. If `positive` is true, estimate an appropriate constant to be added to all the dissimilarities, apart from the self-dissimilarities, that makes the learning matrix positive semi-definite. The other formulation of the additive constant problem is as follows. If the proximity is measured in an interval scale, where there is no natural origin, then there is not a sympathy of the dissimilarities to the distances in the Euclidean space used to represent the objects. In this case, we can estimate a constant `c` such that proximity + c may be taken as ratio data, and also possibly to minimize the dimensionality of the Euclidean space required for representing the objects.

### sammon

`(sammon proximity k)`

`(sammon proximity k lambda tol step-tol max-iter)`

Sammon's mapping. 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. `proximity the nonnegative proximity matrix of dissimilarities. The diagonal should be zero and all other elements should be positive and symmetric. `k` is the dimension of the projection. `lambda` is the initial value of the step size constant in diagonal Newton method. `tol` is the tolerance for stopping iterations. `step-tol` is the tolerance on step size. `max-iter` is the maximum number of iterations.