MatrixUtil.py

"""
Matrix formatting tools.

This module uses numpy arrays and not numpy matrices.
"""

import unittest
from StringIO import StringIO
import random

import numpy as np
from scipy import linalg

import graph
import matrixio
from matrixio import m_to_string
from matrixio import read_matrix


def m_to_mathematica_string(M):
    elements = []
    for row in M:
        s = '{' + ','.join(str(x) for x in row) + '}'
        elements.append(s)
    return '{' + ','.join(s for s in elements) + '}'


#TODO put these global arrays into const data files

# this is a rotation of a contrast matrix of the example tree in "Why neighbor joining works"
g_example_loading_matrix = np.array([
    [0.187795256419, -0.178119343731, 0.107859503348, -0.566026656126, -0.561620966297, 1.71570965693, -2.65637730248],
    [0.187795256419, -0.178119343731, 0.107859503348, -0.566026656126, -0.561620966297, -1.71570965693, 2.65637730248],
    [0.181174971619, -0.156327719297, 0.0923993967909, 1.1689340078, 1.17504869278, 0.0, 0.0],
    [0.0942074702731, 0.512566406759, -0.607716820144, -0.0368806955525, -0.0206980063065, 0.0, 0.0],
    [-0.187795256419, -0.178119343731, -0.107859503348, -0.566026656126, 0.561620966296, 2.65637730248, 1.71570965693],
    [-0.187795256419, -0.178119343731, -0.107859503348, -0.566026656126, 0.561620966296, -2.65637730248, -1.71570965693],
    [-0.181174971619, -0.156327719297, -0.0923993967909, 1.16893400781, -1.17504869278, 0.0, 0.0],
    [-0.0942074702731, 0.512566406759, 0.607716820144, -0.0368806955526, 0.0206980063064, 0.0, 0.0]])

# this is the varimax rotation of the example loading matrix for eps=1e-5
g_example_rotated_matrix = np.array([
    [0.330815132017, -0.256910070995, 0.0801815618365, -0.534892400091, -0.496438433195, -0.000808741457446, -3.16227755675],
    [0.330815132017, -0.256910070995, 0.0801815618365, -0.534892400091, -0.496438433195, 0.000808741457446, 3.16227755675],
    [-0.0786945562714, 0.0117297718391, 0.00988669420354, 1.17928258946, 1.18987853431, 0.0, 0.0],
    [-0.0371881711244, 0.502090370696, -0.612105936609, -0.109497791434, -0.0506611396544, 0.0, 0.0],
    [-0.330815129488, -0.256910072776, -0.0801815614454, -0.534892421301, 0.496438411167, 3.16227755675, -0.000808741457446],
    [-0.330815129488, -0.256910072776, -0.0801815614454, -0.534892421301, 0.496438411167, -3.16227755675, 0.000808741457446],
    [0.0786945522922, 0.0117297739385, -0.00988669475083, 1.17928263783, -1.18987848662, 0.0, 0.0],
    [0.0371881700454, 0.50209037107, 0.612105936374, -0.109497793077, 0.0506611360203, 0.0, 0.0]])

class MatrixError(ValueError): pass

def ndot(*args):
    M = args[0]
    for B in args[1:]:
        M = np.dot(M, B)
    return M

def assert_1d(M):
    if M.ndim != 1:
        raise MatrixError('the array is not 1d')

def assert_2d(M):
    if M.ndim != 2:
        raise MatrixError('the array is not 2d')

def assert_allclose(A, B):
    if not np.allclose(A, B):
        raise MatrixError('the matrices are not close')

def assert_square(M):
    assert_2d(M)
    nrows, ncols = M.shape
    if nrows != ncols:
        raise MatrixError('the matrix is not square')

def assert_sign_symmetric(M):
    assert_square(M)
    S = np.sign(M).astype(int)
    if not np.array_equal(S, S.T):
        raise MatrixError('the matrix is not sign symmetric')

def assert_symmetric(M):
    assert_square(M)
    if not np.allclose(M, M.T):
        raise MatrixError('the matrix is not symmetric')

def assert_nonnegative(M):
    if (M < 0).any():
        raise MatrixError('the matrix has a negative element')

def assert_positive(M):
    if (M <= 0).any():
        raise MatrixError('the matrix has a non-positive element')

def assert_hollow(M):
    assert_square(M)
    if np.diag(M).any():
        raise MatrixError('the matrix is not hollow')

def assert_predistance(M):
    assert_symmetric(M)
    assert_nonnegative(M)

def assert_weighted_adjacency(M):
    assert_symmetric(M)
    assert_nonnegative(M)

def assert_symmetric_irreducible(M):
    assert_symmetric(M)
    V = set(range(len(M)))
    E = set()
    for i in range(len(M)):
        for j in range(i):
            if M[i, j]:
                edge = frozenset((i,j))
                E.add(edge)
    nd = graph.g_to_nd(V, E)
    V_component, D_component = graph.nd_to_dag_component(nd, 0)
    if V_component != V:
        raise MatrixError('the matrix is not irreducible')

def is_symmetric_irreducible(M):
    try:
        assert_symmetric_irreducible(M)
        return True
    except MatrixError as e:
        return False


#FIXME do something about this
def assert_detailed_balance(M):
    pass

def assert_rate_matrix(M):
    """
    @param M: a numpy array
    """
    assert_square(M)
    n = M.shape[0]
    if not (np.diag(M) < 0).all():
        raise MatrixError('diagonal elements of a rate matrix should be less than zero')
    if ((M - np.diag(M)) < 0).any():
        raise MatrixError('off diagonal elements of a rate matrix should be nonnegative')
    if not np.allclose(np.sum(M, 1), np.zeros(n)):
        raise MatrixError('rows of a rate matrix should each sum to zero')

def assert_distribution(v):
    if not np.allclose(np.sum(v), 1):
        raise MatrixError('sum of entries should be 1')
    if not np.allclose(v - np.abs(v), 0):
        raise MatrixError('each entry should be nonnegative')

def assert_transition_matrix(M):
    """
    @param M: a numpy array
    """
    assert_square(M)
    n = M.shape[0]
    if (M < 0).any():
        raise MatrixError('each element of a transition matrix should be nonnegative')
    if not np.allclose(np.sum(M, 1), np.ones(n)):
        raise MatrixError('rows of a transition matrix should each sum to 1.0')

def row_major_to_dict(matrix, ordered_row_labels, ordered_column_labels):
    """
    Convert between matrix formats.
    @param matrix: a matrix represented a row major list of lists
    @param ordered_row_labels: the row labels
    @param ordered_column_labels: the column labels
    @return: a matrix represented as a dictionary of (row, column) label pairs
    """
    # do some sanity checking
    nrows = len(ordered_row_labels)
    ncols = len(ordered_column_labels)
    assert matrix
    assert len(matrix) == nrows
    assert len(matrix[0]) == ncols
    # create the matrix
    dict_matrix = {}
    for row_label, row in zip(ordered_row_labels, matrix):
        for column_label, element in zip(ordered_column_labels, row):
            key = (row_label, column_label)
            dict_matrix[key] = element
    return dict_matrix

def dict_to_row_major(matrix, ordered_row_labels, ordered_column_labels):
    """
    Convert between matrix formats.
    @param matrix: a matrix represented as a dictionary of (row, column) label pairs
    @param ordered_row_labels: the row labels
    @param ordered_column_labels: the column labels
    @return: a row major matrix without label information
    """
    # do some sanity checking
    nrows = len(ordered_row_labels)
    ncols = len(ordered_column_labels)
    assert len(matrix) == nrows * ncols
    # create the matrix
    row_major_matrix = []
    for a in ordered_row_labels:
        row = []
        for b in ordered_column_labels:
            row.append(matrix[(a, b)])
        row_major_matrix.append(row)
    return row_major_matrix

def m_to_matlab_string(matrix):
    """
    @param matrix: a list of lists
    @return: a single line string suitable for copying and pasting into matlab
    """
    return '[' + '; '.join(' '.join(str(x) for x in row) for row in matrix) + ']'

def list_to_matrix(arr, f):
    """
    Apply a function to pairwise elements of a list.
    @param arr: a list of values, pairs of which will be passed to f
    @param f: a function that takes two arguments
    @return: a symmetric row major matrix
    """
    #FIXME this function is used but is stupidly named
    M = []
    for a in arr:
        row = [f(a, b) for b in arr]
        M.append(row)
    return M

def get_principal_submatrix(M, indices):
    """
    @param M: a square numpy array
    @param indices: indices of M to be included in the output matrix
    @return: a square numpy array no bigger than M
    """
    assert_square(M)
    n_small = len(indices)
    R = np.zeros((n_small, n_small))
    for i_small, i_big in enumerate(indices):
        for j_small, j_big in enumerate(indices):
            R[i_small][j_small] = M[i_big][j_big]
    return R

def double_centered_slow(M):
    """
    This is a very explicit implementation.
    The form of.
    @param M: a numpy array
    @return: a doubly centered numpy array
    """
    assert_square(M)
    n = len(M)
    e = np.ones(n)
    I = np.eye(n)
    P = np.outer(e, e) / np.inner(e, e)
    H = I - P
    return ndot(H, M, H)

def double_centered(M):
    """
    This is a somewhat less explicit but more efficient implementation.
    @param M: a numpy array
    @return: a doubly centered numpy array
    """
    assert_square(M)
    n = len(M)
    r = np.mean(M, 0)
    c = np.mean(M, 1)
    m = np.mean(M)
    e = np.ones(n)
    HMH = M - np.outer(e, r) - np.outer(c, e) + m
    return HMH

def varimax(M, eps=1e-5, itermax=200):
    """
    @param M: a numpy matrix to rotate
    @param eps: use smaller values to give a more accurate rotation
    @param itermax: do at most this many iterations
    @return: a rotation matrix R such that MR rotates M to satisfy the varimax criterion
    """
    nrows, ncols = M.shape
    R = np.eye(ncols)
    d = 0
    for i in range(itermax):
        MR = np.dot(M, R)
        D = np.diag(np.sum(MR**2, 0))/nrows
        U, S, V_T = np.linalg.svd(np.dot(M.T, MR**3 - np.dot(MR, D)))
        R = np.dot(U, V_T)
        d_next = np.sum(S)
        if d_next < d * (1 + eps):
            break
        d = d_next
    return R

def get_best_reflection(A, B):
    """
    Find best {-1, +1} multiples of columns of A to approximate B.
    That is, minimize the frobenius norm of AS - B
    where S is a diagonal matrix with {-1, +1} diagonal entries.
    The returned array has the diagonal elements of S.
    A motivation for this function is for the rows of matrix A
    to represent new points up to reflection across each axis,
    whereas rows of matrix B represent old points.
    If v is the output of this function then, using numpy notation,
    A*v gives the canonically reflected points of A
    that best map onto the points of B.
    @param A: a matrix
    @param B: a matrix
    @return: a 1d array of {-1, +1} entries
    """
    assert_2d(A)
    assert_2d(B)
    if A.shape != B.shape:
        raise MatrixError('the shapes of A and B should be the same')
    nrows, ncols = A.shape
    best_signs = np.ones(ncols)
    for i in range(ncols):
        pos_error = np.linalg.norm(A.T[i] - B.T[i])
        neg_error = np.linalg.norm(-A.T[i] - B.T[i])
        if neg_error < pos_error:
            best_signs[i] = -1
    return best_signs

def sample_pos_sym_matrix(n):
    """
    Note that this matrix is symmetric and is entrywise positive.
    Its definiteness is not specified.
    If you care much about the particular distribution of the entries,
    then you should probably use your own function instead of this one.
    @param n: number of rows and of columns
    @return: a numpy array
    """
    M = np.random.exponential(1, (n, n))
    return M + M.T

def get_stationary_distribution(P):
    """
    This uses a direct solver.
    It does not use power iteration or an eigendecomposition.
    @param P: a transition matrix
    @return: the equilibrium (stationary) distribution
    """
    assert_transition_matrix(P)
    nstates = len(P)
    b = np.zeros(nstates)
    A = P.T - np.eye(nstates)
    A[0] = np.ones(nstates)
    b[0] = 1
    v = linalg.solve(A, b, overwrite_a=True, overwrite_b=True)
    assert_distribution(v)
    # take extra care to clean up entries that are
    # technically negative but negligibly tiny.
    v = (v + np.abs(v)) / 2
    v = v / np.sum(v)
    assert_distribution(v)
    return v 

class TestMatrixUtil(unittest.TestCase):

    def test_sign_symmetric_true(self):
        M = np.array([
                [0.0, -0.5],
                [-0.5, 0.0]])
        assert_sign_symmetric(M)

    def test_sign_symmetric_false(self):
        M = np.array([
                [0.0, 0.5],
                [-0.5, 0.0]])
        self.assertRaises(MatrixError, assert_sign_symmetric, M)

    def test_symmetric_irreducible_true(self):
        M = np.array([
                [1, 1, 0],
                [1, 1, 1],
                [0, 1, 1]])
        assert_symmetric_irreducible(M)

    def test_symmetric_irreducible_false_reducible(self):
        M = np.array([
                [1, 0, 0],
                [0, 1, 1],
                [0, 1, 1]])
        self.assertRaises(MatrixError, assert_symmetric_irreducible, M)

    def test_symmetric_irreducible_false_asymmetric(self):
        M = np.array([
                [0, 1],
                [0, 0]])
        self.assertRaises(MatrixError, assert_symmetric_irreducible, M)

    def test_dict_to_row_major(self):
        d = {
                (0, 0) : 1, (0, 1) : 2, (0, 2) : 3,
                (1, 0) : 4, (1, 1) : 5, (1, 2) : 6
                }
        expected = [[1, 2, 3], [4, 5, 6]]
        row_labels = [0, 1]
        column_labels = [0, 1, 2]
        observed = dict_to_row_major(d, row_labels, column_labels)
        self.assertEquals(expected, observed)

    def test_read_square_matrix_bad(self):
        s = '1 2 3 \n 4 5 6'
        M = np.array(matrixio.read_matrix(StringIO(s)))
        self.assertRaises(MatrixError, assert_square, M)

    def test_read_square_matrix_good(self):
        s = '1 2 3 \n 4 5 6 \n 7 8 9'
        row_major_observed = matrixio.read_matrix(StringIO(s))
        row_major_expected = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
        self.assertEquals(row_major_observed, row_major_expected)

    def test_read_rate_matrix(self):
        arr = [[-3, 1, 1, 1], [1, -3, 1, 1], [1, 1, -3, 1], [1, 1, 1, -3]]
        s = '\n'.join('\t'.join(str(x) for x in row) for row in arr)
        sio = StringIO(s)
        M = np.array(matrixio.read_matrix(sio))
        assert_rate_matrix(M)

    def test_m_to_matlab_string(self):
        observed = m_to_matlab_string([[1, 2, 3], [4, 5, 6]])
        expected = '[1 2 3; 4 5 6]'
        self.assertEquals(expected, observed)

    def test_varimax(self):
        M = g_example_loading_matrix
        MR = np.dot(M, varimax(M))
        self.assertTrue(np.allclose(MR, g_example_rotated_matrix))

    def test_double_centered_a(self):
        M = np.array([[1.0, 2.0], [3.0, 4.0]])
        expected = np.array([[0, 0], [0, 0]])
        observed = double_centered(M)
        observed_slow = double_centered_slow(M)
        self.assertTrue(np.allclose(observed, expected))
        self.assertTrue(np.allclose(observed_slow, expected))

    def test_double_centered_b(self):
        M = np.array([[1.0, 2.0], [1.0, 1.0]])
        expected = np.array([[-0.25, 0.25], [0.25, -0.25]])
        observed = double_centered(M)
        observed_slow = double_centered_slow(M)
        self.assertTrue(np.allclose(observed, expected))
        self.assertTrue(np.allclose(observed_slow, expected))

    def test_sample_pos_sym_matrix(self):
        n = 5
        M = sample_pos_sym_matrix(n)
        assert_symmetric(M)
        assert_positive(M)


if __name__ == '__main__':
    unittest.main()