def torgerson(distances, n_components=2, eigen_solver="auto"):
"""
Perform classical mds (Torgerson-Gower scaling).
..note ::
If the distances are euclidean then this is equivalent to projecting
the original data points to the first `n` principal components.
Parameters
----------
distances : (N, N) ndarray
Input distance (dissimilarity) matrix.
n_components : int
Number of components to return
eigen_solver : str
One of `lapack`, `arpack` or `'auto'`. The later chooses between the
former based on the input.
See Also
--------
`cmdscale` in R
"""
distances = np.asarray(distances)
assert distances.shape[0] == distances.shape[1]
N = distances.shape[0]
# O ^ 2
D_sq = distances**2
# double center the D_sq
rsum = np.sum(D_sq, axis=1, keepdims=True)
csum = np.sum(D_sq, axis=0, keepdims=True)
total = np.sum(csum)
D_sq -= rsum / N
D_sq -= csum / N
D_sq += total / (N**2)
B = np.multiply(D_sq, -0.5, out=D_sq)
if eigen_solver == 'auto':
if N > 200 and n_components < 10: # arbitrary - follow skl KernelPCA
eigen_solver = 'arpack'
else:
eigen_solver = 'lapack'
if eigen_solver == "arpack":
v0 = np.random.RandomState(0xD06).uniform(-1, 1, B.shape[0])
w, v = arpack_eigh(B, k=n_components, v0=v0)
assert np.all(np.diff(w) >= 0), "w was not in ascending order"
U, L = v[:, ::-1], w[::-1]
elif eigen_solver == "lapack": # lapack (d|s)syevr
w, v = lapack_eigh(B,
overwrite_a=True,
eigvals=(max(N - n_components, 0), N - 1))
assert np.all(np.diff(w) >= 0), "w was not in ascending order"
U, L = v[:, ::-1], w[::-1]
else:
raise ValueError(eigen_solver)
assert L.shape == (min(n_components, N), )
assert U.shape == (N, min(n_components, N))
# Warn for (sufficiently) negative eig values ...
neg = L < -5 * np.finfo(L.dtype).eps
if np.any(neg):
warnings.warn(
("{} of the {} eigenvalues were negative.".format(
np.sum(neg), L.size)),
UserWarning,
stacklevel=2,
)
# ... and clamp them all to 0
L[L < 0] = 0
if n_components > N:
U = np.hstack((U, np.zeros((N, n_components - N))))
L = np.hstack((L, np.zeros((n_components - N))))
U = U[:, :n_components]
L = L[:n_components]
return U * np.sqrt(L.reshape((1, n_components)))
def torgerson(distances, n_components=2, eigen_solver="auto"):
"""
Perform classical mds (Torgerson-Gower scaling).
..note ::
If the distances are euclidean then this is equivalent to projecting
the original data points to the first `n` principal components.
Parameters
----------
distances : (N, N) ndarray
Input distance (dissimilarity) matrix.
n_components : int
Number of components to return
eigen_solver : str
One of `lapack`, `arpack` or `'auto'`. The later chooses between the
former based on the input.
See Also
--------
`cmdscale` in R
"""
distances = np.asarray(distances)
assert distances.shape[0] == distances.shape[1]
N = distances.shape[0]
# O ^ 2
D_sq = distances ** 2
# double center the D_sq
rsum = np.sum(D_sq, axis=1, keepdims=True)
csum = np.sum(D_sq, axis=0, keepdims=True)
total = np.sum(csum)
D_sq -= rsum / N
D_sq -= csum / N
D_sq += total / (N ** 2)
B = np.multiply(D_sq, -0.5, out=D_sq)
if eigen_solver == 'auto':
if N > 200 and n_components < 10: # arbitrary - follow skl KernelPCA
eigen_solver = 'arpack'
else:
eigen_solver = 'lapack'
if eigen_solver == "arpack":
v0 = np.random.RandomState(0xD06).uniform(-1, 1, B.shape[0])
w, v = arpack_eigh(B, k=n_components, v0=v0)
assert np.all(np.diff(w) >= 0), "w was not in ascending order"
U, L = v[:, ::-1], w[::-1]
elif eigen_solver == "lapack": # lapack (d|s)syevr
w, v = lapack_eigh(B, overwrite_a=True,
eigvals=(max(N - n_components, 0), N - 1))
assert np.all(np.diff(w) >= 0), "w was not in ascending order"
U, L = v[:, ::-1], w[::-1]
else:
raise ValueError(eigen_solver)
assert L.shape == (min(n_components, N),)
assert U.shape == (N, min(n_components, N))
# Warn for (sufficiently) negative eig values ...
neg = L < -5 * np.finfo(L.dtype).eps
if np.any(neg):
warnings.warn(
("{} of the {} eigenvalues were negative."
.format(np.sum(neg), L.size)),
UserWarning, stacklevel=2,
)
# ... and clamp them all to 0
L[L < 0] = 0
if n_components > N:
U = np.hstack((U, np.zeros((N, n_components - N))))
L = np.hstack((L, np.zeros((n_components - N))))
U = U[:, :n_components]
L = L[:n_components]
return U * np.sqrt(L.reshape((1, n_components)))