classical mds python
from __future__ import division
import numpy as np
def cmdscale(D):
"""
Classical multidimensional scaling (MDS)
Parameters
----------
D : (n, n) array
Symmetric distance matrix.
Returns
-------
Y : (n, p) array
Configuration matrix. Each column represents a dimension. Only the
p dimensions corresponding to positive eigenvalues of B are returned.
Note that each dimension is only determined up to an overall sign,
corresponding to a reflection.
e : (n,) array
Eigenvalues of B.
"""
# Number of points
n = len(D)
# Centering matrix
H = np.eye(n) - np.ones((n, n))/n
# YY^T
B = -H.dot(D**2).dot(H)/2
# Diagonalize
evals, evecs = np.linalg.eigh(B)
# Sort by eigenvalue in descending order
idx = np.argsort(evals)[::-1]
evals = evals[idx]
evecs = evecs[:,idx]
# Compute the coordinates using positive-eigenvalued components only
w, = np.where(evals > 0)
L = np.diag(np.sqrt(evals[w]))
V = evecs[:,w]
Y = V.dot(L)
return Y, evals
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