def indiv_space_for_sparse(X, joint_scores, joint_rank, signal_rank,
sv_threshold):
# compute a rank R1 SVD of I
# if the estimated individual rank is less than R1 we are done
# otherwise compute a rank R2 SVD of I
# keep going until we find the individual rank
# TODO: this could use lots of optimizing
X_orthog = col_proj_orthog(X, joint_scores)
# start with a low rank SVD
max_rank = min(X.shape) - joint_rank # saves computation
current_rank = min(int(1.2 * signal_rank),
max_rank) # 1.2 is somewhat arbitrary
U, D, V = svd_wrapper(X_orthog, current_rank)
indiv_rank = sum(D > sv_threshold)
if indiv_rank == current_rank: # SVD rank is still too low
found_indiv_rank = False
for t in range(3):
# current guess at an upper bound for the individual rank
additional_rank = signal_rank
current_rank = current_rank + additional_rank
current_rank = min(current_rank, max_rank)
# compute additional additional_rank SVD components
# TODO: possibly use svds_additional to speed up calculation
# U, D, V = svds_additional(I, scores, sv, loadings, additional_rank)
U, D, V = svd_wrapper(X_orthog, current_rank)
indiv_rank = sum(D > sv_threshold)
# we are done if the individual rank estimate is less
# than the current_rank or if the current_rank is equal to the maximal rank
if (indiv_rank < current_rank) or (current_rank == max_rank):
found_indiv_rank = True
break
if not found_indiv_rank:
warnings.warn('individual rank estimate probably too low')
return U[:, 0:indiv_rank], D[0:indiv_rank], V[:, 0:indiv_rank], indiv_rank
def _get_rand_sample(num_obs, signal_ranks):
M = [None for _ in range(len(signal_ranks))]
for k in range(len(signal_ranks)):
# sample random orthonormal basis
Z = np.random.normal(size=(num_obs, signal_ranks[k]))
M[k] = np.linalg.qr(Z)[0]
# compute largest sing val of random joint matrix
M = np.bmat(M)
_, svs, __ = svd_wrapper(M, rank=1)
return svs.item() ** 2
def fit(self, X):
"""
Computes the PCA decomposition of X.
Parameters
----------
X: {array-like, sparse matrix}, shape (n_samples, n_features)
Fit PCA with data matrix X. If X is a pd.DataFrame, the observation
and feature names will be extracted from its index/columns.
Note X can be either dense or sparse.
"""
self.shape_, obs_names, var_names, self.n_components, \
= _arg_checker(X, self.n_components)
# possibly mean center X
X, self.m_ = centering(X, self.center)
# compute SVD
U, D, V = svd_wrapper(X, self.n_components)
# compute variance explained
if self.n_components == min(X.shape):
self.frob_norm_ = np.sqrt(sum(D**2))
else:
self.frob_norm_ = _safe_frob_norm(X)
self.var_expl_prop_ = D**2 / self.frob_norm_**2
self.var_expl_cum_ = np.cumsum(self.var_expl_prop_)
if self.n_components is None:
self.n_components = self.scores_.shape[1]
self.scores_, self.svals_, self.loadings_ = \
svd2pd(U, D, V, obs_names=obs_names, var_names=var_names)
return self
def fit(self, blocks, precomp_init_svd=None):
"""
Fits the AJIVE decomposition.
Parameters
----------
blocks: list, dict
The data matrices. If dict, will name blocks by keys, otherwise
blocks are named by 0, 1, ...K. Data matrices must have observations
on the rows and have the same number of observations i.e. the
kth data matrix is shape (n_samples, n_features[k]).
precomp_init_svd: {list, dict, None}, optional
Precomputed initial SVD. Must have one entry for each data block.
The SVD should be a 3 tuple (scores, svals, loadings), see output
of jive.utils.svd_wrapper for formatting details.
"""
blocks, self.init_signal_ranks, self.indiv_ranks, precomp_init_svd,\
self.center, obs_names, var_names, self.shapes_ = \
arg_checker(blocks,
self.init_signal_ranks,
self.joint_rank,
self.indiv_ranks,
precomp_init_svd,
self.center)
block_names = list(blocks.keys())
num_obs = list(blocks.values())[0].shape[0]
# center blocks
self.centers_ = {}
for bn in block_names:
blocks[bn], self.centers_[bn] = centering(blocks[bn],
method=self.center[bn])
################################################################
# step 1: initial signal space extraction by SVD on each block #
################################################################
init_signal_svd = {}
self.sv_threshold_ = {}
for bn in block_names:
# compute rank init_signal_ranks[bn] + 1 SVD of the data block
if precomp_init_svd[bn] is None:
# signal rank + 1 to get individual rank sv threshold
U, D, V = svd_wrapper(blocks[bn],
self.init_signal_ranks[bn] + 1)
else:
U = precomp_init_svd[bn]['scores']
D = precomp_init_svd[bn]['svals']
V = precomp_init_svd[bn]['loadings']
# The SV threshold is halfway between the init_signal_ranks[bn]th
# and init_signal_ranks[bn] + 1 st singular value. Recall that
# python is zero indexed.
self.sv_threshold_[bn] = (D[self.init_signal_ranks[bn] - 1] \
+ D[self.init_signal_ranks[bn]]) / 2
init_signal_svd[bn] = {
'scores': U[:, 0:self.init_signal_ranks[bn]],
'svals': D[0:self.init_signal_ranks[bn]],
'loadings': V[:, 0:self.init_signal_ranks[bn]]
}
##################################
# step 2: joint space estimation #
##################################
# this step estimates the joint rank and computes the common
# joint space basis
# SVD of joint signal matrix
joint_scores_matrix = np.bmat(
[init_signal_svd[bn]['scores'] for bn in block_names])
joint_scores, joint_svals, joint_loadings = svd_wrapper(
joint_scores_matrix)
self.all_joint_svals_ = deepcopy(joint_svals)
# estimate joint rank using wedin bound and random direction if a
# joint rank estimate has not already been provided
# TODO: maybe make this into an object or function
if self.joint_rank is None:
# if the random sv samples are not already provided compute them
if self.random_sv_samples_ is None:
self.random_sv_samples_ = \
sample_randdir(num_obs,
signal_ranks=list(self.init_signal_ranks.values()),
R=self.n_randdir_samples,
n_jobs=self.n_jobs)
# if the wedin samples are not already provided compute them
if self.wedin_samples_ is None:
self.wedin_samples_ = {}
for bn in block_names:
self.wedin_samples_[bn] = \
get_wedin_samples(X=blocks[bn],
U=init_signal_svd[bn]['scores'],
D=init_signal_svd[bn]['svals'],
V=init_signal_svd[bn]['loadings'],
rank=self.init_signal_ranks[bn],
R=self.n_wedin_samples,
n_jobs=self.n_jobs)
self.wedin_sv_samples_ = len(blocks) - \
np.array([sum(self.wedin_samples_[bn][i] ** 2 for bn in block_names)
for i in range(self.n_wedin_samples)])
# given the wedin and random bound samples, compute the joint rank
# SV cutoff
self.wedin_cutoff_ = np.percentile(self.wedin_sv_samples_,
self.wedin_percentile)
self.rand_cutoff_ = np.percentile(self.random_sv_samples_,
self.randdir_percentile)
self.svalsq_cutoff_ = max(self.wedin_cutoff_, self.rand_cutoff_)
self.joint_rank_wedin_est_ = sum(
joint_svals**2 > self.svalsq_cutoff_)
self.joint_rank = deepcopy(self.joint_rank_wedin_est_)
# check identifiability constraint and possibly remove some
# joint components
if self.reconsider_joint_components:
joint_scores, joint_svals, joint_loadings, self.joint_rank = \
reconsider_joint_components(blocks, self.sv_threshold_,
joint_scores, joint_svals, joint_loadings,
self.joint_rank)
# TODO: include center?
# TODO: comp_names, var_names
# The common joint space has now been estimated
self.common = PCA.from_precomputed(
scores=joint_scores[:, 0:self.joint_rank],
svals=joint_svals[0:self.joint_rank],
loadings=joint_loadings[:, 0:self.joint_rank],
obs_names=obs_names)
self.common.set_comp_names(base='common',
zero_index=self.zero_index_names)
#######################################
# step 3: compute final decomposition #
#######################################
# this step computes the block specific estimates
block_specific = {bn: {} for bn in block_names}
for bn in block_names:
X = blocks[bn]
########################################
# step 3.1: block specific joint space #
########################################
# project X onto the joint space then compute SVD
if self.joint_rank != 0:
if issparse(X): # lazy evaluation for sparse matrices
J = col_proj(X, joint_scores)
U, D, V = svd_wrapper(J, self.joint_rank)
J = None # kill J matrix to save memory
else:
J = np.array(
np.dot(joint_scores, np.dot(joint_scores.T, X)))
U, D, V = svd_wrapper(J, self.joint_rank)
if not self.store_full:
J = None # kill J matrix to save memory
else:
U, D, V = None, None, None
if self.store_full:
J = np.zeros(shape=blocks[bn].shape)
else:
J = None
block_specific[bn]['joint'] = {
'full': J,
'scores': U,
'svals': D,
'loadings': V,
'rank': self.joint_rank
}
#############################################
# step 3.2: block specific individual space #
#############################################
# project X onto the orthogonal complement of the joint space,
# estimate the individual rank, then compute SVD
if issparse(X): # lazy evaluation for sparse matrices
U, D, V, indiv_rank = indiv_space_for_sparse(
X, joint_scores, self.joint_rank,
self.init_signal_ranks[bn], self.sv_threshold_[bn])
I = None
else:
# project X columns onto orthogonal complement of joint_scores
if self.joint_rank == 0:
X_orthog = X
else:
X_orthog = X - np.dot(joint_scores,
np.dot(joint_scores.T, X))
# estimate individual rank using sv threshold, then compute SVD
if self.indiv_ranks[bn] is None:
max_rank = min(
X.shape) - self.joint_rank # saves computation
U, D, V = svd_wrapper(X_orthog, max_rank)
rank = sum(D > self.sv_threshold_[bn])
if rank == 0:
U, D, V = None, None, None
else:
U = U[:, 0:rank]
D = D[0:rank]
V = V[:, 0:rank]
self.indiv_ranks[bn] = rank
else: # indiv_rank has been provided by the user
rank = self.indiv_ranks[bn]
if rank == 0:
U, D, V = None, None, None
else:
U, D, V = svd_wrapper(X_orthog, rank)
if self.store_full:
if rank == 0:
I = np.zeros(shape=blocks[bn].shape)
else:
I = np.array(np.dot(U, np.dot(np.diag(D), V.T)))
else:
I = None # Kill I matrix to save memory
block_specific[bn]['individual'] = {
'full': I,
'scores': U,
'svals': D,
'loadings': V,
'rank': rank
}
###################################
# step 3.3: estimate noise matrix #
###################################
if self.store_full and not issparse(X):
E = X - (J + I)
else:
E = None
block_specific[bn]['noise'] = E
# save block specific estimates
self.blocks = {}
for bn in block_specific.keys():
self.blocks[bn] = \
BlockSpecificResults(joint=block_specific[bn]['joint'],
individual=block_specific[bn]['individual'],
noise=block_specific[bn]['noise'],
CNS=joint_scores,
block_name=bn,
obs_names=obs_names,
var_names=var_names[bn],
m=self.centers_[bn],
shape=blocks[bn].shape,
zero_index_names=self.zero_index_names,
init_signal_svd=init_signal_svd[bn],
X=blocks[bn])
return self
