MarkovChain: Sparse matrix support

quantecon/graph_tools.py

@@ -263,6 +263,27 @@ def cyclic_components(self):
  return [np.where(self._cyclic_components_proj == k)[0]
  for k in range(self.period)]
 
+ def subgraph(self, nodes):
+ """
+ Return the subgraph consisting of the given nodes and edges
+ between thses nodes.
+
+ Parameters
+ ----------
+ nodes : array_like(int, ndim=1)
+ Array of nodes.
+
+ Returns
+ -------
+ DiGraph
+ A DiGraph representing the subgraph.
+
+ """
+ adj_matrix = self.csgraph[nodes, :][:, nodes]
+
+ weighted = True # To copy the dtype
+ return DiGraph(adj_matrix, weighted=weighted)
+
 
 def _csr_matrix_indices(S):
  """

quantecon/markov/core.py

@@ -86,6 +86,7 @@
 """
 from __future__ import division
 import numpy as np
+from scipy import sparse
 from fractions import gcd
 from .gth_solve import gth_solve
 from ..graph_tools import DiGraph
@@ -103,12 +104,13 @@ class MarkovChain(object):
 
  Parameters
  ----------
- P : array_like(float, ndim=2)
+ P : array_like or scipy sparse matrix (float, ndim=2)
  The transition matrix. Must be of shape n x n.
 
  Attributes
  ----------
- P : see Parameters
+ P : ndarray or scipy.sparse.csr_matrix (float, ndim=2)
+ See Parameters
 
  stationary_distributions : array_like(float, ndim=2)
  Array containing stationary distributions, one for each
@@ -139,32 +141,50 @@ class MarkovChain(object):
  List of numpy arrays containing the cyclic classes. Defined only
  when the Markov chain is irreducible.
 
+ Notes
+ -----
+ In computing stationary distributions, if the input matrix is a
+ sparse matrix, internally it is converted to a dense matrix.
+
  """
 
  def __init__(self, P):
- self.P = np.asarray(P)
+ if sparse.issparse(P): # Sparse matrix
+ self.P = sparse.csr_matrix(P)
+ self.is_sparse = True
+ else: # Dense matrix
+ self.P = np.asarray(P)
+ self.is_sparse = False
 
  # Check Properties
  # Double check that P is a square matrix
  if len(self.P.shape) != 2 or self.P.shape[0] != self.P.shape[1]:
  raise ValueError('P must be a square matrix')
 
+ # The number of states
+ self.n = self.P.shape[0]
+
  # Double check that P is a nonnegative matrix
- if not np.all(self.P >= 0):
+ if not self.is_sparse:
+ data_nonnegative = (self.P >= 0) # ndarray
+ else:
+ data_nonnegative = (self.P.data >= 0) # csr_matrx
+ if not np.all(data_nonnegative):
  raise ValueError('P must be nonnegative')
 
  # Double check that the rows of P sum to one
- if not np.allclose(np.sum(self.P, axis=1), np.ones(self.P.shape[0])):
+ row_sums = self.P.sum(axis=1)
+ if self.is_sparse: # row_sums is np.matrix (ndim=2)
+ row_sums = row_sums.getA1()
+ if not np.allclose(row_sums, np.ones(self.n)):
  raise ValueError('The rows of P must sum to 1')
 
- # The number of states
- self.n = self.P.shape[0]
-
  # To analyze the structure of P as a directed graph
  self._digraph = None
 
  self._stationary_dists = None
- self._cdfs = None
+ self._cdfs = None # For dense matrix
+ self._cdfs1d = None # For sparse matrix
 
  def __repr__(self):
  msg = "Markov chain with transition matrix \nP = \n{0}"
@@ -221,7 +241,7 @@ def period(self):
  # Determine the period, the LCM of the periods of rec_classes
  d = 1
  for rec_class in rec_classes:
- period = DiGraph(self.P[rec_class, :][:, rec_class]).period
+ period = self.digraph.subgraph(rec_class).period
  d = (d * period) // gcd(d, period)
 
  return d
@@ -241,13 +261,23 @@ def _compute_stationary(self):
 
  """
  if self.is_irreducible:
- stationary_dists = gth_solve(self.P).reshape(1, self.n)
+ if not self.is_sparse: # Dense
+ stationary_dists = gth_solve(self.P).reshape(1, self.n)
+ else: # Sparse
+ stationary_dists = \
+ gth_solve(self.P.toarray(),
+ overwrite=True).reshape(1, self.n)
  else:
  rec_classes = self.recurrent_classes
  stationary_dists = np.zeros((len(rec_classes), self.n))
  for i, rec_class in enumerate(rec_classes):
- stationary_dists[i, rec_class] = \
- gth_solve(self.P[rec_class, :][:, rec_class])
+ if not self.is_sparse: # Dense
+ stationary_dists[i, rec_class] = \
+ gth_solve(self.P[rec_class, :][:, rec_class])
+ else: # Sparse
+ stationary_dists[i, rec_class] = \
+ gth_solve(self.P[rec_class, :][:, rec_class].toarray(),
+ overwrite=True)
 
  self._stationary_dists = stationary_dists
 
@@ -259,13 +289,27 @@ def stationary_distributions(self):
 
  @property
  def cdfs(self):
- if self._cdfs is None:
+ if (self._cdfs is None) and not self.is_sparse:
  # See issue #137#issuecomment-96128186
  cdfs = np.empty((self.n, self.n), order='C')
  np.cumsum(self.P, axis=-1, out=cdfs)
  self._cdfs = cdfs
  return self._cdfs
 
+ @property
+ def cdfs1d(self):
+ if (self._cdfs1d is None) and self.is_sparse:
+ data = self.P.data
+ indices = self.P.indices
+ indptr = self.P.indptr
+
+ cdfs1d = np.empty(self.P.nnz, order='C')
+ for i in range(self.n):
+ cdfs1d[indptr[i]:indptr[i+1]] = \
+ data[indptr[i]:indptr[i+1]].cumsum()
+ self._cdfs1d = cdfs1d
+ return self._cdfs1d
+
  def simulate(self, ts_length, init=None, num_reps=None, random_state=None):
  """
  Simulate time series of state transitions.
@@ -332,7 +376,15 @@ def simulate(self, ts_length, init=None, num_reps=None, random_state=None):
  random_values = random_state.random_sample(size=(k, ts_length-1))
 
  # Generate sample paths and store in X
- _generate_sample_paths(self.cdfs, init_states, random_values, out=X)
+ if not self.is_sparse: # Dense
+ _generate_sample_paths(
+ self.cdfs, init_states, random_values, out=X
+ )
+ else: # Sparse
+ _generate_sample_paths_sparse(
+ self.cdfs1d, self.P.indices, self.P.indptr, init_states,
+ random_values, out=X
+ )
 
  if dim == 1:
  return X[0]
@@ -377,6 +429,55 @@ def _generate_sample_paths(P_cdfs, init_states, random_values, out):
  _generate_sample_paths = jit(nopython=True)(_generate_sample_paths)
 
 
+def _generate_sample_paths_sparse(P_cdfs1d, indices, indptr, init_states,
+ random_values, out):
+ """
+ For sparse matrix.
+
+ Generate num_reps sample paths of length ts_length, where num_reps =
+ out.shape[0] and ts_length = out.shape[1].
+
+ Parameters
+ ----------
+ P_cdfs1d : ndarray(float, ndim=1)
+ 1D array containing the CDFs of the state transition.
+
+ indices : ndarray(int, ndim=1)
+ CSR format index array.
+
+ indptr : ndarray(int, ndim=1)
+ CSR format index pointer array.
+
+ init_states : array_like(int, ndim=1)
+ Array containing the initial states. Its length must be equal to
+ num_reps.
+
+ random_values : ndarray(float, ndim=2)
+ Array containing random values from [0, 1). Its shape must be
+ equal to (num_reps, ts_length-1)
+
+ out : ndarray(int, ndim=2)
+ Array to store the sample paths.
+
+ Notes
+ -----
+ This routine is jit-complied if the module Numba is vailable.
+
+ """
+ num_reps, ts_length = out.shape
+
+ for i in range(num_reps):
+ out[i, 0] = init_states[i]
+ for t in range(ts_length-1):
+ k = searchsorted(P_cdfs1d[indptr[out[i, t]]:indptr[out[i, t]+1]],
+ random_values[i, t])
+ out[i, t+1] = indices[indptr[out[i, t]]+k]
+
+if numba_installed:
+ _generate_sample_paths_sparse = \
+ jit(nopython=True)(_generate_sample_paths_sparse)
+
+
 def mc_compute_stationary(P):
  """
  Computes stationary distributions of P, one for each recurrent

quantecon/markov/gth_solve.py

@@ -18,7 +18,7 @@
  xrange = range
 
 
-def gth_solve(A):
+def gth_solve(A, overwrite=False):
  r"""
  This routine computes the stationary distribution of an irreducible
  Markov transition matrix (stochastic matrix) or transition rate
@@ -51,6 +51,9 @@ def gth_solve(A):
  x : numpy.ndarray(float, ndim=1)
  Stationary distribution of `A`.
 
+ overwrite : bool, optional(default=False)
+ Whether to overwrite `A`.
+
  References
  ----------
  .. [1] W. K. Grassmann, M. I. Taksar and D. P. Heyman, "Regenerative
@@ -61,7 +64,7 @@ def gth_solve(A):
  Simulation, Princeton University Press, 2009.
 
  """
- A1 = np.array(A, dtype=float, copy=True, order='C')
+ A1 = np.array(A, dtype=float, copy=not overwrite, order='C')
  # `order='C'` is for use with Numba <= 0.18.2
  # See issue github.com/numba/numba/issues/1103
 

quantecon/markov/random.py

@@ -9,13 +9,14 @@
 import numpy as np
 import scipy.sparse
 
-from ..util import check_random_state, numba_installed, jit
+from ..util import check_random_state
 from ..random import (
  probvec, sample_without_replacement
 )
 
 from .core import MarkovChain
 
+
 def random_markov_chain(n, k=None, sparse=False, random_state=None):
  """
  Return a randomly sampled MarkovChain instance with n states, where
@@ -32,7 +33,7 @@ def random_markov_chain(n, k=None, sparse=False, random_state=None):
 
  sparse : bool, optional(default=False)
  Whether to store the transition probability matrix in sparse
- matrix form. (Sparse format is not yet implemented.)
+ matrix form.
 
  random_state : scalar(int) or np.random.RandomState,
  optional(default=None)
@@ -59,8 +60,6 @@ def random_markov_chain(n, k=None, sparse=False, random_state=None):
  [ 0.56227226, 0. , 0.43772774]])
 
  """
- if sparse:
- raise NotImplementedError
  P = random_stochastic_matrix(n, k, sparse, format='csr',
  random_state=random_state)
  mc = MarkovChain(P)