def test_dimensions(self):
graph = self.graph.copy()
sparse = nx.to_scipy_sparse_matrix(graph)
loopless = graspologic.utils.remove_loops(sparse)
ranked = graspologic.utils.pass_to_ranks(loopless)
lse = LaplacianSpectralEmbed(n_components=100,
n_elbows=None,
svd_seed=1234,
form="R-DAD")
core_response = lse.fit_transform(ranked)
embedding = laplacian_spectral_embedding(self.graph.copy(),
svd_seed=1234)
np.testing.assert_array_almost_equal(core_response,
embedding.embeddings())
def spectral_clustering(adj, n_components=4, method="lse", return_embedding=False):
if method == "ase":
embedder = AdjacencySpectralEmbed(n_components=n_components)
elif method == "lse":
embedder = LaplacianSpectralEmbed(n_components=n_components, form="R-DAD")
latent = embedder.fit_transform(adj)
gc = AutoGMMCluster(min_components=4, max_components=4)
pred_labels = gc.fit_predict(latent)
if return_embedding:
return pred_labels, latent
else:
return pred_labels
import seaborn as sns
def plot_latents(latent_positions, *, title, labels, ax=None):
if ax is None:
ax = plt.gca()
plot = sns.scatterplot(latent_positions[:, 0], latent_positions[:, 1], hue=labels,
palette="Set1", linewidth=0, s=10, ax=ax)
plot.set_title(title, wrap=True);
ax.axes.xaxis.set_visible(False)
ax.axes.yaxis.set_visible(False)
return plot
L = to_laplacian(A, form="R-DAD")
lse = LSE(form="R-DAD", n_components=2)
L_latents = lse.fit_transform(L)
plot_latents(L_latents, title="Latent positions when we\n only embed the Laplacian",
labels=labels);
We'd like to use extra information to more clearly distinguish between the first and second community. We don't have this information in our network: it needs to come from somewhere else.
### Covariates
But we're in luck - we have a set of covariates for each node! These covariates contain the extra information we need that allows us to separate our first and second community. However, with only these extra covariate features, we can no longer distinguish between the last two communities - they contain the same information.
Below is a visualization of our covariates. Remember, each node is associated with its own group of covariates that provide information about that node. We'll organize this information into a matrix, where the $i_{th}$ row contains the covariates associated with node $i$. For simplicity, the only values in this matrix will be 1 and 0.
import numpy as np
from scipy.stats import bernoulli
import seaborn as sns
# currtime = time.time()
# X_transformed = sca.fit_transform(X_train)
# print(f"{time.time() - currtime} elapsed")
#%%
neuron_type_palette = dict(
zip(np.unique(scrna_meta.loc[index_train, "Neuron_type"]), cc.glasbey_light)
)
#%%
from graspologic.embed import LaplacianSpectralEmbed
lse = LaplacianSpectralEmbed(form="R-DAD", n_components=16)
U, V = lse.fit_transform(adj)
Y = np.concatenate((U, V), axis=1)
#%%
embed_map = dict(zip(label_df["neuron_class"], Y))
#%%
neuron_types = scrna_meta.loc[index_train, "Neuron_type"]
Y_expanded = neuron_types.map(embed_map)
Y_expanded = np.stack(Y_expanded.values)
Y_expanded
# %%
def laplacian_spectral_embedding(
graph: Union[nx.Graph, nx.OrderedGraph, nx.DiGraph, nx.OrderedDiGraph],
form: LaplacianFormType = "R-DAD",
dimensions: int = 100,
elbow_cut: Optional[int] = None,
svd_solver_algorithm: SvdAlgorithmType = "randomized",
svd_solver_iterations: int = 5,
svd_seed: Optional[int] = None,
weight_attribute: str = "weight",
regularizer: Optional[numbers.Real] = None,
) -> Embeddings:
"""
Given a directed or undirected networkx graph (*not* multigraph), generate an
Embeddings object.
The laplacian spectral embedding process is similar to the adjacency spectral
embedding process, with the key differentiator being that the LSE process looks
further into the latent space when it captures changes, whereas the ASE process
is egocentric and focused on immediate differentiators in a node's periphery.
All weights will be rescaled based on their relative rank in the graph,
which is beneficial in minimizing anomalous results if some edge weights are
extremely atypical of the rest of the graph.
Parameters
----------
graph : Union[nx.Graph, nx.OrderedGraph, nx.DiGraph, nx.OrderedDiGraph]
An undirected or directed graph. The graph **must**:
- be fully numerically weighted (every edge must have a real, numeric weight
or else it will be treated as an unweighted graph)
- be a basic graph (meaning it should not be a multigraph; if you have a
multigraph you must first decide how you want to handle the weights of the
edges between two nodes, whether summed, averaged, last-wins,
maximum-weight-only, etc)
form : str (default="R-DAD")
Specifies the type of Laplacian normalization to use. Allowed values are:
{ "DAD", "I-DAD", "R-DAD" }. See
:func:`~graspologic.utils.to_laplacian` for more details regarding form.
dimensions : int (default=100)
Dimensions to use for the svd solver.
For undirected graphs, if ``elbow_cut==None``, you will receive an embedding
that has ``nodes`` rows and ``dimensions`` columns.
For directed graphs, if ``elbow_cut==None``, you will receive an embedding that
has ``nodes`` rows and ``2*dimensions`` columns.
If ``elbow_cut`` is specified to be not ``None``, we will cut the embedding at
``elbow_cut`` elbow, but the provided ``dimensions`` will be used in the
creation of the SVD.
elbow_cut : Optional[int] (default=None)
Using a process described by Zhu & Ghodsi in their paper "Automatic
dimensionality selection from the scree plot via the use of profile likelihood",
truncate the dimensionality of the return on the ``elbow_cut``-th elbow.
By default this value is ``None`` but can be used to reduce the dimensionality
of the returned tensors.
svd_solver_algorithm : str (default="randomized")
allowed values: {'randomized', 'full', 'truncated'}
SVD solver to use:
- 'randomized'
Computes randomized svd using
:func:`sklearn.utils.extmath.randomized_svd`
- 'full'
Computes full svd using :func:`scipy.linalg.svd`
Does not support ``graph`` input of type scipy.sparse.csr_matrix
- 'truncated'
Computes truncated svd using :func:`scipy.sparse.linalg.svds`
svd_solver_iterations : int (default=5)
Number of iterations for randomized SVD solver. Not used by 'full' or
'truncated'. The default is larger than the default in randomized_svd
to handle sparse matrices that may have large slowly decaying spectrum.
svd_seed : Optional[int] (default=None)
Used to seed the PRNG used in the ``randomized`` svd solver algorithm.
weight_attribute : str (default="weight")
The edge dictionary key that contains the weight of the edge.
regularizer : Optional[numbers.Real] (default=None)
Only used when form="R-DAD". Must be None or nonnegative.
Constant to be added to the diagonal of degree matrix. If None, average
node degree is added. If int or float, must be >= 0.
Returns
-------
Embeddings
Raises
------
beartype.roar.BeartypeCallHintParamViolation if parameters do not match type hints
ValueError if values are not within appropriate ranges or allowed values
See Also
--------
graspologic.pipeline.embed.Embeddings
graspologic.embed.LaplacianSpectralEmbed
graspologic.embed.select_svd
graspologic.utils.to_laplacian
Notes
-----
The singular value decomposition:
.. math:: A = U \Sigma V^T
is used to find an orthonormal basis for a matrix, which in our case is the
Laplacian matrix of the graph. These basis vectors (in the matrices U or V) are
ordered according to the amount of variance they explain in the original matrix.
By selecting a subset of these basis vectors (through our choice of dimensionality
reduction) we can find a lower dimensional space in which to represent the graph.
References
----------
.. [1] Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E. "A
Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs,"
Journal of the American Statistical Association, Vol. 107(499), 2012.
.. [2] Von Luxburg, Ulrike. "A tutorial on spectral clustering," Statistics
and computing, Vol. 17(4), pp. 395-416, 2007.
.. [3] Rohe, Karl, Sourav Chatterjee, and Bin Yu. "Spectral clustering and
the high-dimensional stochastic blockmodel," The Annals of Statistics,
Vol. 39(4), pp. 1878-1915, 2011.
.. [4] Zhu, M. and Ghodsi, A. (2006). Automatic dimensionality selection from the
scree plot via the use of profile likelihood. Computational Statistics & Data
Analysis, 51(2), pp.918-930.
"""
check_argument(form in __FORMS,
f"form must be one of the values in {','.join(__FORMS)}")
check_argument(dimensions >= 1, "dimensions must be positive")
check_argument(elbow_cut is None or elbow_cut >= 1,
"elbow_cut must be positive")
check_argument(
svd_solver_algorithm in __SVD_SOLVER_TYPES,
f"svd_solver_algorithm must be one of the values in {','.join(__SVD_SOLVER_TYPES)}",
)
check_argument(svd_solver_iterations >= 1,
"svd_solver_iterations must be positive")
check_argument(
svd_seed is None or 0 <= svd_seed <= 2**32 - 1,
"svd_seed must be a nonnegative, 32-bit integer",
)
check_argument(
regularizer is None or float(regularizer) >= 0,
"regularizer must be nonnegative",
)
check_argument(
not graph.is_multigraph(),
"Multigraphs are not supported; you must determine how to represent at most "
"one edge between any two nodes, and handle the corresponding weights "
"accordingly",
)
used_weight_attribute: Optional[str] = weight_attribute
if not is_real_weighted(graph, weight_attribute=weight_attribute):
warnings.warn(
f"Graphs with edges that do not have a real numeric weight set for every "
f"{weight_attribute} attribute on every edge are treated as an unweighted "
f"graph - which presumes all weights are `1.0`. If this is incorrect, "
f"please add a '{weight_attribute}' attribute to every edge with a real, "
f"numeric value (e.g. an integer or a float) and call this function again."
)
used_weight_attribute = None # this supercedes what the user said, because
# not all of the weights are real numbers, if they exist at all
# this weight=1.0 treatment actually happens in nx.to_scipy_sparse_matrix()
node_labels = np.array(list(graph.nodes()))
graph_as_csr = nx.to_scipy_sparse_matrix(graph,
weight=used_weight_attribute,
nodelist=node_labels)
if not is_fully_connected(graph):
warnings.warn("More than one connected component detected")
graph_sans_loops = remove_loops(graph_as_csr)
ranked_graph = pass_to_ranks(graph_sans_loops)
embedder = LaplacianSpectralEmbed(
form=form,
n_components=dimensions,
n_elbows=None, # in the short term, we do our own elbow finding
algorithm=svd_solver_algorithm,
n_iter=svd_solver_iterations,
svd_seed=svd_seed,
concat=False,
)
results = embedder.fit_transform(ranked_graph)
results_arr: np.ndarray
if elbow_cut is None:
if isinstance(results, tuple) or graph.is_directed():
results_arr = np.concatenate(results, axis=1)
else:
results_arr = results
else:
column_index = _index_of_elbow(embedder.singular_values_, elbow_cut)
if isinstance(results, tuple):
left, right = results
left = left[:, :column_index]
right = right[:, :column_index]
results_arr = np.concatenate((left, right), axis=1)
else:
results_arr = results[:, :column_index]
embeddings = Embeddings(node_labels, results_arr)
return embeddings