


"""
Markov based methods for spatial dynamics.
"""
__author__ = "Sergio J. Rey <srey@asu.edu"

__all__ = ["Markov", "LISA_Markov", "Spatial_Markov", "kullback",
           "prais", "shorrock", "homogeneity"]

import numpy as np
from pysal.spatial_dynamics.ergodic import fmpt
from pysal.spatial_dynamics.ergodic import steady_state as STEADY_STATE
from scipy import stats
from operator import gt
import pysal

# TT predefine LISA transitions
# TT[i,j] is the transition type from i to j
# i = quadrant in period 0
# j = quadrant in period 1
# uses one offset so first row and col of TT are ignored
TT = np.zeros((5, 5), int)
c = 1
for i in range(1, 5):
    for j in range(1, 5):
        TT[i, j] = c
        c += 1

# MOVE_TYPES is a dictionary that returns the move type of a LISA transition
# filtered on the significance of the LISA end points
# True indicates significant LISA in a particular period
# e.g. a key of (1, 3, True, False) indicates a significant LISA located in
# quadrant 1 in period 0 moved to quadrant 3 in period 1 but was not
# significant in quadrant 3.

MOVE_TYPES = {}
c = 1
cases = (True, False)
sig_keys = [(i, j) for i in cases for j in cases]

for i, sig_key in enumerate(sig_keys):
    c = 1 + i * 16
    for i in range(1, 5):
        for j in range(1, 5):
            key = (i, j, sig_key[0], sig_key[1])
            MOVE_TYPES[key] = c
            c += 1


class Markov(object):
    """
    Classic Markov transition matrices.

    Parameters
    ----------
    class_ids    : array
                   (n, t), one row per observation, one column recording the
                   state of each observation, with as many columns as time
                   periods.
    classes      : array
                   (k, 1), all different classes (bins) of the matrix.

    Attributes
    ----------
    p            : matrix
                   (k, k), transition probability matrix.
    steady_state : matrix
                   (k, 1), ergodic distribution.
    transitions  : matrix
                   (k, k), count of transitions between each state i and j.

    Examples
    --------
    >>> c = [['b','a','c'],['c','c','a'],['c','b','c']]
    >>> c.extend([['a','a','b'], ['a','b','c']])
    >>> c = np.array(c)
    >>> m = Markov(c)
    >>> m.classes.tolist()
    ['a', 'b', 'c']
    >>> m.p
    matrix([[ 0.25      ,  0.5       ,  0.25      ],
            [ 0.33333333,  0.        ,  0.66666667],
            [ 0.33333333,  0.33333333,  0.33333333]])
    >>> m.steady_state
    matrix([[ 0.30769231],
            [ 0.28846154],
            [ 0.40384615]])

    US nominal per capita income 48 states 81 years 1929-2009

    >>> import pysal
    >>> f = pysal.open(pysal.examples.get_path("usjoin.csv"))
    >>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])

    set classes to quintiles for each year

    >>> q5 = np.array([pysal.Quantiles(y).yb for y in pci]).transpose()
    >>> m = Markov(q5)
    >>> m.transitions
    array([[ 729.,   71.,    1.,    0.,    0.],
           [  72.,  567.,   80.,    3.,    0.],
           [   0.,   81.,  631.,   86.,    2.],
           [   0.,    3.,   86.,  573.,   56.],
           [   0.,    0.,    1.,   57.,  741.]])
    >>> m.p
    matrix([[ 0.91011236,  0.0886392 ,  0.00124844,  0.        ,  0.        ],
            [ 0.09972299,  0.78531856,  0.11080332,  0.00415512,  0.        ],
            [ 0.        ,  0.10125   ,  0.78875   ,  0.1075    ,  0.0025    ],
            [ 0.        ,  0.00417827,  0.11977716,  0.79805014,  0.07799443],
            [ 0.        ,  0.        ,  0.00125156,  0.07133917,  0.92740926]])
    >>> m.steady_state
    matrix([[ 0.20774716],
            [ 0.18725774],
            [ 0.20740537],
            [ 0.18821787],
            [ 0.20937187]])

    Relative incomes

    >>> pci = pci.transpose()
    >>> rpci = pci/(pci.mean(axis=0))
    >>> rq = pysal.Quantiles(rpci.flatten()).yb
    >>> rq.shape = (48,81)
    >>> mq = Markov(rq)
    >>> mq.transitions
    array([[ 707.,   58.,    7.,    1.,    0.],
           [  50.,  629.,   80.,    1.,    1.],
           [   4.,   79.,  610.,   73.,    2.],
           [   0.,    7.,   72.,  650.,   37.],
           [   0.,    0.,    0.,   48.,  724.]])
    >>> mq.steady_state
    matrix([[ 0.17957376],
            [ 0.21631443],
            [ 0.21499942],
            [ 0.21134662],
            [ 0.17776576]])

    """
    def __init__(self, class_ids, classes=None):
        if classes is not None:
            self.classes = classes
        else:
            self.classes = np.unique(class_ids)

        n, t = class_ids.shape
        k = len(self.classes)
        js = list(range(t - 1))

        classIds = self.classes.tolist()
        transitions = np.zeros((k, k))
        for state_0 in js:
            state_1 = state_0 + 1
            state_0 = class_ids[:, state_0]
            state_1 = class_ids[:, state_1]
            initial = np.unique(state_0)
            for i in initial:
                ending = state_1[state_0 == i]
                uending = np.unique(ending)
                row = classIds.index(i)
                for j in uending:
                    col = classIds.index(j)
                    transitions[row, col] += sum(ending == j)
        self.transitions = transitions
        row_sum = transitions.sum(axis=1)
        p = np.dot(np.diag(1 / (row_sum + (row_sum == 0))), transitions)
        self.p = np.matrix(p)

    @property
    def steady_state(self):
        if not hasattr(self, '_steady_state'):
            self._steady_state = STEADY_STATE(self.p)
        return self._steady_state


class Spatial_Markov(object):
    """
    Markov transitions conditioned on the value of the spatial lag.

    Parameters
    ----------
    y               : array
                      (n,t), one row per observation, one column per state of
                      each observation, with as many columns as time periods.
    w               : W
                      spatial weights object.
    k               : integer
                      number of classes (quantiles).
    permutations    : int, optional
                      number of permutations for use in randomization based
                      inference (the default is 0).
    fixed           : bool
                      If true, quantiles are taken over the entire n*t
                      pooled series. If false, quantiles are taken each
                      time period over n.
    variable_name   : string
                      name of variable.

    Attributes
    ----------
    p               : matrix
                      (k, k), transition probability matrix for a-spatial
                      Markov.
    s               : matrix
                      (k, 1), ergodic distribution for a-spatial Markov.
    transitions     : matrix
                      (k, k), counts of transitions between each state i and j
                      for a-spatial Markov.
    T               : matrix
                      (k, k, k), counts of transitions for each conditional
                      Markov.  T[0] is the matrix of transitions for
                      observations with lags in the 0th quantile; T[k-1] is the
                      transitions for the observations with lags in the k-1th.
    P               : matrix
                      (k, k, k), transition probability matrix for spatial
                      Markov first dimension is the conditioned on the lag.
    S               : matrix
                      (k, k), steady state distributions for spatial Markov.
                      Each row is a conditional steady_state.
    F               : matrix
                      (k, k, k),first mean passage times.
                      First dimension is conditioned on the lag.
    shtest          : list
                      (k elements), each element of the list is a tuple for a
                      multinomial difference test between the steady state
                      distribution from a conditional distribution versus the
                      overall steady state distribution: first element of the
                      tuple is the chi2 value, second its p-value and the third
                      the degrees of freedom.
    chi2            : list
                      (k elements), each element of the list is a tuple for a
                      chi-squared test of the difference between the
                      conditional transition matrix against the overall
                      transition matrix: first element of the tuple is the chi2
                      value, second its p-value and the third the degrees of
                      freedom.
    x2              : float
                      sum of the chi2 values for each of the conditional tests.
                      Has an asymptotic chi2 distribution with k(k-1)(k-1)
                      degrees of freedom. Under the null that transition
                      probabilities are spatially homogeneous.
                      (see chi2 above)
    x2_dof          : int
                      degrees of freedom for homogeneity test.
    x2_pvalue       : float
                      pvalue for homogeneity test based on analytic.
                      distribution
    x2_rpvalue      : float
                      (if permutations>0)
                      pseudo p-value for x2 based on random spatial
                      permutations of the rows of the original transitions.
    x2_realizations : array
                      (permutations,1), the values of x2 for the random
                      permutations.
    Q               : float
                      Chi-square test of homogeneity across lag classes based
                      on Bickenbach and Bode (2003) [Bickenbach2003]_.
    Q_p_value       : float
                      p-value for Q.
    LR              : float
                      Likelihood ratio statistic for homogeneity across lag
                      classes based on Bickenback and Bode (2003)
                      [Bickenbach2003]_.
    LR_p_value      : float
                      p-value for LR.
    dof_hom         : int
                      degrees of freedom for LR and Q, corrected for 0 cells.

    Notes
    -----
    Based on  Rey (2001) [Rey2001]_.

    The shtest and chi2 tests should be used with caution as they are based on
    classic theory assuming random transitions. The x2 based test is
    preferable since it simulates the randomness under the null. It is an
    experimental test requiring further analysis.

    This is new

    Examples
    --------
    >>> import pysal as ps
    >>> f = ps.open(ps.examples.get_path("usjoin.csv"))
    >>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
    >>> pci = pci.transpose()
    >>> rpci = pci/(pci.mean(axis=0))
    >>> w = ps.open(ps.examples.get_path("states48.gal")).read()
    >>> w.transform = 'r'
    >>> sm = ps.Spatial_Markov(rpci, w, fixed=True, k=5, variable_name='rpci')
    >>> for p in sm.P:
    ...     print(p)
    ...
    [[ 0.96341463  0.0304878   0.00609756  0.          0.        ]
     [ 0.06040268  0.83221477  0.10738255  0.          0.        ]
     [ 0.          0.14        0.74        0.12        0.        ]
     [ 0.          0.03571429  0.32142857  0.57142857  0.07142857]
     [ 0.          0.          0.          0.16666667  0.83333333]]
    [[ 0.79831933  0.16806723  0.03361345  0.          0.        ]
     [ 0.0754717   0.88207547  0.04245283  0.          0.        ]
     [ 0.00537634  0.06989247  0.8655914   0.05913978  0.        ]
     [ 0.          0.          0.06372549  0.90196078  0.03431373]
     [ 0.          0.          0.          0.19444444  0.80555556]]
    [[ 0.84693878  0.15306122  0.          0.          0.        ]
     [ 0.08133971  0.78947368  0.1291866   0.          0.        ]
     [ 0.00518135  0.0984456   0.79274611  0.0984456   0.00518135]
     [ 0.          0.          0.09411765  0.87058824  0.03529412]
     [ 0.          0.          0.          0.10204082  0.89795918]]
    [[ 0.8852459   0.09836066  0.          0.01639344  0.        ]
     [ 0.03875969  0.81395349  0.13953488  0.          0.00775194]
     [ 0.0049505   0.09405941  0.77722772  0.11881188  0.0049505 ]
     [ 0.          0.02339181  0.12865497  0.75438596  0.09356725]
     [ 0.          0.          0.          0.09661836  0.90338164]]
    [[ 0.33333333  0.66666667  0.          0.          0.        ]
     [ 0.0483871   0.77419355  0.16129032  0.01612903  0.        ]
     [ 0.01149425  0.16091954  0.74712644  0.08045977  0.        ]
     [ 0.          0.01036269  0.06217617  0.89637306  0.03108808]
     [ 0.          0.          0.          0.02352941  0.97647059]]


    The probability of a poor state remaining poor is 0.963 if their
    neighbors are in the 1st quintile and 0.798 if their neighbors are
    in the 2nd quintile. The probability of a rich economy remaining
    rich is 0.976 if their neighbors are in the 5th quintile, but if their
    neighbors are in the 4th quintile this drops to 0.903.

    The Q  and likelihood ratio statistics are both significant indicating
    the dynamics are not homogeneous across the lag classes:

    >>> "%.3f"%sm.LR
    '170.659'
    >>> "%.3f"%sm.Q
    '200.624'
    >>> "%.3f"%sm.LR_p_value
    '0.000'
    >>> "%.3f"%sm.Q_p_value
    '0.000'
    >>> sm.dof_hom
    60

    The long run distribution for states with poor (rich) neighbors has
    0.435 (0.018) of the values in the first quintile, 0.263 (0.200) in
    the second quintile, 0.204 (0.190) in the third, 0.0684 (0.255) in the
    fourth and 0.029 (0.337) in the fifth quintile.

    >>> sm.S
    array([[ 0.43509425,  0.2635327 ,  0.20363044,  0.06841983,  0.02932278],
           [ 0.13391287,  0.33993305,  0.25153036,  0.23343016,  0.04119356],
           [ 0.12124869,  0.21137444,  0.2635101 ,  0.29013417,  0.1137326 ],
           [ 0.0776413 ,  0.19748806,  0.25352636,  0.22480415,  0.24654013],
           [ 0.01776781,  0.19964349,  0.19009833,  0.25524697,  0.3372434 ]])

    States with incomes in the first quintile with neighbors in the
    first quintile return to the first quartile after 2.298 years, after
    leaving the first quintile. They enter the fourth quintile after
    80.810 years after leaving the first quintile, on average.
    Poor states within neighbors in the fourth quintile return to the
    first quintile, on average, after 12.88 years, and would enter the
    fourth quintile after 28.473 years.

    >>> for f in sm.F:
    ...     print(f)
    ...
    [[   2.29835259   28.95614035   46.14285714   80.80952381  279.42857143]
     [  33.86549708    3.79459555   22.57142857   57.23809524  255.85714286]
     [  43.60233918    9.73684211    4.91085714   34.66666667  233.28571429]
     [  46.62865497   12.76315789    6.25714286   14.61564626  198.61904762]
     [  52.62865497   18.76315789   12.25714286    6.           34.1031746 ]]
    [[   7.46754205    9.70574606   25.76785714   74.53116883  194.23446197]
     [  27.76691978    2.94175577   24.97142857   73.73474026  193.4380334 ]
     [  53.57477715   28.48447637    3.97566318   48.76331169  168.46660482]
     [  72.03631562   46.94601483   18.46153846    4.28393653  119.70329314]
     [  77.17917276   52.08887197   23.6043956     5.14285714   24.27564033]]
    [[   8.24751154    6.53333333   18.38765432   40.70864198  112.76732026]
     [  47.35040872    4.73094099   11.85432099   34.17530864  106.23398693]
     [  69.42288828   24.76666667    3.794921     22.32098765   94.37966594]
     [  83.72288828   39.06666667   14.3           3.44668119   76.36702977]
     [  93.52288828   48.86666667   24.1           9.8           8.79255406]]
    [[  12.87974382   13.34847151   19.83446328   28.47257282   55.82395142]
     [  99.46114206    5.06359731   10.54545198   23.05133495   49.68944423]
     [ 117.76777159   23.03735526    3.94436301   15.0843986    43.57927247]
     [ 127.89752089   32.4393006    14.56853107    4.44831643   31.63099455]
     [ 138.24752089   42.7893006    24.91853107   10.35          4.05613474]]
    [[  56.2815534     1.5          10.57236842   27.02173913  110.54347826]
     [  82.9223301     5.00892857    9.07236842   25.52173913  109.04347826]
     [  97.17718447   19.53125       5.26043557   21.42391304  104.94565217]
     [ 127.1407767    48.74107143   33.29605263    3.91777427   83.52173913]
     [ 169.6407767    91.24107143   75.79605263   42.5           2.96521739]]


    """
    def __init__(self, y, w, k=4, permutations=0, fixed=False,
                 variable_name=None):

        self.y = y
        rows, cols = y.shape
        self.k = k
        self.cols = cols
        npa = np.array
        self.fixed = fixed
        self.variable_name = variable_name
        if fixed:
            yf = y.flatten()
            yb = pysal.Quantiles(yf, k=k).yb
            yb.shape = (rows, cols)
            classes = yb
        else:
            classes = npa([pysal.Quantiles(y[:, i], k=k)
                           .yb for i in np.arange(cols)]).transpose()
        classic = Markov(classes)
        self.classes = classes
        self.p = classic.p
        self.transitions = classic.transitions
        T, P = self._calc(y, w, classes, k=k)
        self.T = T
        self.P = P

        if permutations:
            nrp = np.random.permutation
            counter = 0
            x2_realizations = np.zeros((permutations, 1))
            for perm in range(permutations):
                T, P = self._calc(nrp(y), w, classes, k=k)
                x2 = [chi2(T[i], self.transitions)[0] for i in range(k)]
                x2s = sum(x2)
                x2_realizations[perm] = x2s
                if x2s >= self.x2:
                    counter += 1
            self.x2_rpvalue = (counter + 1.0) / (permutations + 1.)
            self.x2_realizations = x2_realizations

    @property
    def s(self):
        if not hasattr(self, '_s'):
            self._s = STEADY_STATE(self.p)
        return self._s

    @property
    def S(self):
        if not hasattr(self, '_S'):
            S = np.zeros_like(self.p)
            for i, p in enumerate(self.P):
                S[i] = STEADY_STATE(p)
            self._S = np.asarray(S)
        return self._S

    @property
    def F(self):
        if not hasattr(self, '_F'):
            F = np.zeros_like(self.P)
            for i, p in enumerate(self.P):
                F[i] = fmpt(np.asmatrix(p))
            self._F = np.asarray(F)
        return self._F

    # bickenbach and bode tests
    @property
    def ht(self):
        if not hasattr(self, '_ht'):
            self._ht = homogeneity(self.T)
        return self._ht

    @property
    def Q(self):
        if not hasattr(self, '_Q'):
            self._Q = self.ht.Q
        return self._Q

    @property
    def Q_p_value(self):
        self._Q_p_value = self.ht.Q_p_value
        return self._Q_p_value

    @property
    def LR(self):
        self._LR = self.ht.LR
        return self._LR

    @property
    def LR_p_value(self):
        self._LR_p_value = self.ht.LR_p_value
        return self._LR_p_value

    @property
    def dof_hom(self):
        self._dof_hom = self.ht.dof
        return self._dof_hom

    # shtests
    @property
    def shtest(self):
        if not hasattr(self, '_shtest'):
            self._shtest = self._mn_test()
        return self._shtest

    @property
    def chi2(self):
        if not hasattr(self, '_chi2'):
            self._chi2 = self._chi2_test()
        return self._chi2

    @property
    def x2(self):
        if not hasattr(self, '_x2'):
            self._x2 = sum([c[0] for c in self.chi2])
        return self._x2

    @property
    def x2_pvalue(self):
        if not hasattr(self, '_x2_pvalue'):
            self._x2_pvalue = 1 - stats.chi2.cdf(self.x2, self.x2_dof)
        return self._x2_pvalue

    @property
    def x2_dof(self):
        if not hasattr(self, '_x2_dof'):
            k = self.k
            self._x2_dof = k * (k - 1) * (k - 1)
        return self._x2_dof

    def _calc(self, y, w, classes, k):
        ly = pysal.lag_spatial(w, y)
        npa = np.array
        if self.fixed:
            l_classes = pysal.Quantiles(ly.flatten(), k=k).yb
            l_classes.shape = ly.shape
        else:
            l_classes = npa([pysal.Quantiles(
                ly[:, i], k=k).yb for i in np.arange(self.cols)])
            l_classes = l_classes.transpose()
        T = np.zeros((k, k, k))
        n, t = y.shape
        for t1 in range(t - 1):
            t2 = t1 + 1
            for i in range(n):
                T[l_classes[i, t1], classes[i, t1], classes[i, t2]] += 1

        P = np.zeros_like(T)
        for i, mat in enumerate(T):
            row_sum = mat.sum(axis=1)
            row_sum = row_sum + (row_sum == 0)
            p_i = np.matrix(np.diag(1. / row_sum) * np.matrix(mat))
            P[i] = p_i
        return T, P

    def _mn_test(self):
        """
        helper to calculate tests of differences between steady state
        distributions from the conditional and overall distributions.
        """
        n, t = self.y.shape
        n0, n1, n2 = self.T.shape
        rn = list(range(n0))
        mat = [self._ssmnp_test(
            self.s, self.S[i], self.T[i].sum()) for i in rn]
        return mat

    def _ssmnp_test(self, p1, p2, nt):
        """
        Steady state multinomial probability difference test.

        Arguments
        ---------
        p1       :  array
                    (k, 1), first steady state probability distribution.
        p1       :  array
                    (k, 1), second steady state probability distribution.
        nt       :  int
                    number of transitions to base the test on.

        Returns
        -------
        tuple
                   (3 elements)
                   (chi2 value, pvalue, degrees of freedom)

        """
        p1 = np.array(p1)
        k, c = p1.shape
        p1.shape = (k, )
        o = nt * p2
        e = nt * p1
        d = np.multiply((o - e), (o - e))
        d = d / e
        chi2 = d.sum()
        pvalue = 1 - stats.chi2.cdf(chi2, k - 1)
        return (chi2, pvalue, k - 1)

    def _chi2_test(self):
        """
        helper to calculate tests of differences between the conditional
        transition matrices and the overall transitions matrix.
        """
        n, t = self.y.shape
        n0, n1, n2 = self.T.shape
        rn = list(range(n0))
        mat = [chi2(self.T[i], self.transitions) for i in rn]
        return mat

    def summary(self, file_name=None):
        class_names = ["C%d" % i for i in range(self.k)]
        regime_names = ["LAG%d" % i for i in range(self.k)]
        ht = homogeneity(self.T, class_names=class_names,
                         regime_names=regime_names)
        title = "Spatial Markov Test"
        if self.variable_name:
            title = title + ": " + self.variable_name
        if file_name:
            ht.summary(file_name=file_name, title=title)
        else:
            ht.summary(title=title)


def chi2(T1, T2):
    """
    chi-squared test of difference between two transition matrices.

    Parameters
    ----------
    T1    : matrix
            (k, k), matrix of transitions (counts).
    T2    : matrix
            (k, k), matrix of transitions (counts) to use to form the
            probabilities under the null.

    Returns
    -------
          : tuple
            (3 elements).
            (chi2 value, pvalue, degrees of freedom).

    Examples
    --------
    >>> import pysal
    >>> f = pysal.open(pysal.examples.get_path("usjoin.csv"))
    >>> years = range(1929, 2010)
    >>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
    >>> rpci = pci/(pci.mean(axis=0))
    >>> w = pysal.open(pysal.examples.get_path("states48.gal")).read()
    >>> w.transform='r'
    >>> sm = Spatial_Markov(rpci, w, fixed=True)
    >>> T1 = sm.T[0]
    >>> T1
    array([[ 562.,   22.,    1.,    0.],
           [  12.,  201.,   22.,    0.],
           [   0.,   17.,   97.,    4.],
           [   0.,    0.,    3.,   19.]])
    >>> T2 = sm.transitions
    >>> T2
    array([[ 884.,   77.,    4.,    0.],
           [  68.,  794.,   87.,    3.],
           [   1.,   92.,  815.,   51.],
           [   1.,    0.,   60.,  903.]])
    >>> chi2(T1,T2)
    (23.397284414732951, 0.0053631167048613371, 9)

    Notes
    -----
    Second matrix is used to form the probabilities under the null.
    Marginal sums from first matrix are distributed across these probabilities
    under the null. In other words the observed transitions are taken from T1
    while the expected transitions are formed as follows

    .. math::

            E_{i,j} = \sum_j T1_{i,j} * T2_{i,j}/\sum_j T2_{i,j}

    Degrees of freedom corrected for any rows in either T1 or T2 that have
    zero total transitions.
    """
    rs2 = T2.sum(axis=1)
    rs1 = T1.sum(axis=1)
    rs2nz = rs2 > 0
    rs1nz = rs1 > 0
    dof1 = sum(rs1nz)
    dof2 = sum(rs2nz)
    rs2 = rs2 + (rs2 == 0)
    dof = (dof1 - 1) * (dof2 - 1)
    p = np.diag(1 / rs2) * np.matrix(T2)
    E = np.diag(rs1) * np.matrix(p)
    num = T1 - E
    num = np.multiply(num, num)
    E = E + (E == 0)
    chi2 = num / E
    chi2 = chi2.sum()
    pvalue = 1 - stats.chi2.cdf(chi2, dof)
    return chi2, pvalue, dof


class LISA_Markov(Markov):
    """
    Markov for Local Indicators of Spatial Association

    Parameters
    ----------
    y                  : array
                         (n, t), n cross-sectional units observed over t time
                         periods.
    w                  : W
                         spatial weights object.
    permutations       : int, optional
                         number of permutations used to determine LISA
                         significance (the default is 0).
    significance_level : float, optional
                         significance level (two-sided) for filtering
                         significant LISA endpoints in a transition (the
                         default is 0.05).
    geoda_quads        : bool
                         If True use GeoDa scheme: HH=1, LL=2, LH=3, HL=4.
                         If False use PySAL Scheme: HH=1, LH=2, LL=3, HL=4.
                         (the default is False).

    Attributes
    ----------
    chi_2        : tuple
                   (3 elements)
                   (chi square test statistic, p-value, degrees of freedom) for
                   test that dynamics of y are independent of dynamics of wy.
    classes      : array
                   (4, 1)
                   1=HH, 2=LH, 3=LL, 4=HL (own, lag)
                   1=HH, 2=LL, 3=LH, 4=HL (own, lag) (if geoda_quads=True)
    expected_t   : array
                   (4, 4), expected number of transitions under the null that
                   dynamics of y are independent of dynamics of wy.
    move_types   : matrix
                   (n, t-1), integer values indicating which type of LISA
                   transition occurred (q1 is quadrant in period 1, q2 is
                   quadrant in period 2).

    .. Table:: Move Types

                   ==  ==     ========
                   q1  q2     move_type
                   ==  ==     ========
                   1   1      1
                   1   2      2
                   1   3      3
                   1   4      4
                   2   1      5
                   2   2      6
                   2   3      7
                   2   4      8
                   3   1      9
                   3   2      10
                   3   3      11
                   3   4      12
                   4   1      13
                   4   2      14
                   4   3      15
                   4   4      16
                   ==  ==     ========

    p            : matrix
                   (k, k), transition probability matrix.
    p_values     : matrix
                   (n, t), LISA p-values for each end point (if permutations >
                   0).
    significant_moves : matrix
                        (n, t-1), integer values indicating the type and
                        significance of a LISA transition. st = 1 if
                        significant in period t, else st=0 (if permutations >
                        0).

    .. Table:: Significant Moves

                       ===============  ===================
                       (s1,s2)          move_type
                       ===============  ===================
                       (1,1)            [1, 16]
                       (1,0)            [17, 32]
                       (0,1)            [33, 48]
                       (0,0)            [49, 64]
                       ===============  ===================


                       == ==  ==  ==  =========
                       q1 q2  s1  s2  move_type
                       == ==  ==  ==  =========
                        1  1   1   1   1
                        1  2   1   1   2
                        1  3   1   1   3
                        1  4   1   1   4
                        2  1   1   1   5
                        2  2   1   1   6
                        2  3   1   1   7
                        2  4   1   1   8
                        3  1   1   1   9
                        3  2   1   1   10
                        3  3   1   1   11
                        3  4   1   1   12
                        4  1   1   1   13
                        4  2   1   1   14
                        4  3   1   1   15
                        4  4   1   1   16
                        1  1   1   0   17
                        1  2   1   0   18
                        .  .   .   .    .
                        .  .   .   .    .
                        4  3   1   0   31
                        4  4   1   0   32
                        1  1   0   1   33
                        1  2   0   1   34
                        .  .   .   .    .
                        .  .   .   .    .
                        4  3   0   1   47
                        4  4   0   1   48
                        1  1   0   0   49
                        1  2   0   0   50
                        .  .   .   .    .
                        .  .   .   .    .
                        4  3   0   0   63
                        4  4   0   0   64
                       == ==  ==  ==  =========

    steady_state : matrix
                   (k, 1), ergodic distribution.
    transitions  : matrix
                   (4, 4), count of transitions between each state i and j.
    spillover    : array
                   (n, 1) binary array, locations that were not part of a
                   cluster in period 1 but joined a prexisting cluster in
                   period 2.

    Examples
    --------
    >>> import pysal as ps
    >>> import numpy as np
    >>> f = ps.open(ps.examples.get_path("usjoin.csv"))
    >>> years = range(1929, 2010)
    >>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
    >>> w = ps.open(ps.examples.get_path("states48.gal")).read()
    >>> lm = ps.LISA_Markov(pci,w)
    >>> lm.classes
    array([1, 2, 3, 4])
    >>> lm.steady_state
    matrix([[ 0.28561505],
            [ 0.14190226],
            [ 0.40493672],
            [ 0.16754598]])
    >>> lm.transitions
    array([[  1.08700000e+03,   4.40000000e+01,   4.00000000e+00,
              3.40000000e+01],
           [  4.10000000e+01,   4.70000000e+02,   3.60000000e+01,
              1.00000000e+00],
           [  5.00000000e+00,   3.40000000e+01,   1.42200000e+03,
              3.90000000e+01],
           [  3.00000000e+01,   1.00000000e+00,   4.00000000e+01,
              5.52000000e+02]])
    >>> lm.p
    matrix([[ 0.92985458,  0.03763901,  0.00342173,  0.02908469],
            [ 0.07481752,  0.85766423,  0.06569343,  0.00182482],
            [ 0.00333333,  0.02266667,  0.948     ,  0.026     ],
            [ 0.04815409,  0.00160514,  0.06420546,  0.88603531]])
    >>> lm.move_types[0,:3]
    array([11, 11, 11])
    >>> lm.move_types[0,-3:]
    array([11, 11, 11])

    Now consider only moves with one, or both, of the LISA end points being
    significant

    >>> np.random.seed(10)
    >>> lm_random = pysal.LISA_Markov(pci, w, permutations=99)
    >>> lm_random.significant_moves[0, :3]
    array([11, 11, 11])
    >>> lm_random.significant_moves[0,-3:]
    array([59, 43, 27])


    Any value less than 49 indicates at least one of the LISA end points was
    significant. So for example, the first spatial unit experienced a
    transition of type 11 (LL, LL)  during the first three and last tree
    intervals (according to lm.move_types), however, the last three of these
    transitions involved insignificant LISAS in both the start and ending year
    of each transition.

    Test whether the moves of y are independent of the moves of wy

    >>> "Chi2: %8.3f, p: %5.2f, dof: %d" % lm.chi_2
    'Chi2: 1058.208, p:  0.00, dof: 9'

    Actual transitions of LISAs

    >>> lm.transitions
    array([[  1.08700000e+03,   4.40000000e+01,   4.00000000e+00,
              3.40000000e+01],
           [  4.10000000e+01,   4.70000000e+02,   3.60000000e+01,
              1.00000000e+00],
           [  5.00000000e+00,   3.40000000e+01,   1.42200000e+03,
              3.90000000e+01],
           [  3.00000000e+01,   1.00000000e+00,   4.00000000e+01,
              5.52000000e+02]])

    Expected transitions of LISAs under the null y and wy are moving
    independently of one another

    >>> lm.expected_t
    array([[  1.12328098e+03,   1.15377356e+01,   3.47522158e-01,
              3.38337644e+01],
           [  3.50272664e+00,   5.28473882e+02,   1.59178880e+01,
              1.05503814e-01],
           [  1.53878082e-01,   2.32163556e+01,   1.46690710e+03,
              9.72266513e+00],
           [  9.60775143e+00,   9.86856346e-02,   6.23537392e+00,
              6.07058189e+02]])

    If the LISA classes are to be defined according to GeoDa, the `geoda_quad`
    option has to be set to true

    >>> lm.q[0:5,0]
    array([3, 2, 3, 1, 4])
    >>> lm = ps.LISA_Markov(pci,w, geoda_quads=True)
    >>> lm.q[0:5,0]
    array([2, 3, 2, 1, 4])

    """
    def __init__(self, y, w, permutations=0,
                 significance_level=0.05, geoda_quads=False):
        y = y.transpose()
        pml = pysal.Moran_Local
        gq = geoda_quads
        ml = ([pml(yi, w, permutations=permutations, geoda_quads=gq)
               for yi in y])
        q = np.array([mli.q for mli in ml]).transpose()
        classes = np.arange(1, 5)  # no guarantee all 4 quadrants are visited
        Markov.__init__(self, q, classes)
        self.q = q
        self.w = w
        n, k = q.shape
        k -= 1
        self.significance_level = significance_level
        move_types = np.zeros((n, k), int)
        sm = np.zeros((n, k), int)
        self.significance_level = significance_level
        if permutations > 0:
            p = np.array([mli.p_z_sim for mli in ml]).transpose()
            self.p_values = p
            pb = p <= significance_level
        else:
            pb = np.zeros_like(y.T)
        for t in range(k):
            origin = q[:, t]
            dest = q[:, t + 1]
            p_origin = pb[:, t]
            p_dest = pb[:, t + 1]
            for r in range(n):
                move_types[r, t] = TT[origin[r], dest[r]]
                key = (origin[r], dest[r], p_origin[r], p_dest[r])
                sm[r, t] = MOVE_TYPES[key]
        if permutations > 0:
            self.significant_moves = sm
        self.move_types = move_types

        # null of own and lag moves being independent

        ybar = y.mean(axis=0)
        r = y / ybar
        ylag = np.array([pysal.lag_spatial(w, yt) for yt in y])
        rlag = ylag / ybar
        rc = r < 1.
        rlagc = rlag < 1.
        markov_y = pysal.Markov(rc)
        markov_ylag = pysal.Markov(rlagc)
        A = np.matrix([[1, 0, 0, 0],
                       [0, 0, 1, 0],
                       [0, 0, 0, 1],
                       [0, 1, 0, 0]])

        kp = A * np.kron(markov_y.p, markov_ylag.p) * A.T
        trans = self.transitions.sum(axis=1)
        t1 = np.diag(trans) * kp
        t2 = self.transitions
        t1 = t1.getA()
        self.chi_2 = pysal.spatial_dynamics.markov.chi2(t2, t1)
        self.expected_t = t1
        self.permutations = permutations

    def spillover(self, quadrant=1, neighbors_on=False):
        """
        Detect spillover locations for diffusion in LISA Markov.

        Parameters
        ----------
        quadrant     : int
                       which quadrant in the scatterplot should form the core
                       of a cluster.
        neighbors_on : binary
                       If false, then only the 1st order neighbors of a core
                       location are included in the cluster.
                       If true, neighbors of cluster core 1st order neighbors
                       are included in the cluster.

        Returns
        -------
        results      : dictionary
                       two keys - values pairs:
                       'components' - array (n, t)
                       values are integer ids (starting at 1) indicating which
                       component/cluster observation i in period t belonged to.
                       'spillover' - array (n, t-1)
                       binary values indicating if the location was a
                       spill-over location that became a new member of a
                       previously existing cluster.

        Examples
        --------
        >>> f = pysal.open(pysal.examples.get_path("usjoin.csv"))
        >>> years = range(1929, 2010)
        >>> pci = np.array([f.by_col[str(y)] for y in years]).transpose()
        >>> w = pysal.open(pysal.examples.get_path("states48.gal")).read()
        >>> np.random.seed(10)
        >>> lm_random = pysal.LISA_Markov(pci, w, permutations=99)
        >>> r = lm_random.spillover()
        >>> r['components'][:,12]
        array([ 0.,  0.,  0.,  2.,  0.,  1.,  1.,  0.,  0.,  2.,  0.,  0.,  0.,
                0.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  2.,
                1.,  1.,  0.,  1.,  0.,  0.,  1.,  0.,  2.,  1.,  1.,  0.,  0.,
                0.,  0.,  0.,  1.,  0.,  2.,  1.,  0.,  0.])
        >>> r['components'][:,14]
        array([ 0.,  2.,  0.,  2.,  0.,  1.,  1.,  0.,  0.,  2.,  0.,  0.,  0.,
                0.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  2.,
                0.,  1.,  0.,  1.,  0.,  0.,  1.,  0.,  2.,  1.,  1.,  0.,  0.,
                0.,  0.,  0.,  1.,  0.,  2.,  1.,  0.,  0.])
        >>> r['spill_over'][:,12]
        array([ 0.,  1.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
                0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,
                0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
                0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  1.])

        Including neighbors of core neighbors

        >>> rn = lm_random.spillover(neighbors_on=True)
        >>> rn['components'][:,12]
        array([ 0.,  2.,  0.,  2.,  0.,  1.,  1.,  0.,  0.,  2.,  0.,  1.,  0.,
                0.,  1.,  0.,  1.,  1.,  1.,  1.,  0.,  0.,  0.,  2.,  0.,  2.,
                1.,  1.,  0.,  1.,  0.,  0.,  1.,  0.,  2.,  1.,  1.,  0.,  0.,
                0.,  0.,  2.,  1.,  1.,  2.,  1.,  0.,  2.])
        >>> rn["components"][:,13]
        array([ 0.,  2.,  0.,  2.,  2.,  1.,  1.,  0.,  0.,  2.,  0.,  1.,  0.,
                2.,  1.,  0.,  1.,  1.,  1.,  1.,  0.,  0.,  0.,  2.,  2.,  2.,
                1.,  1.,  2.,  1.,  0.,  2.,  1.,  2.,  2.,  1.,  1.,  0.,  2.,
                0.,  2.,  2.,  1.,  1.,  2.,  1.,  0.,  2.])
        >>> rn["spill_over"][:,12]
        array([ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
                1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,
                0.,  0.,  1.,  0.,  0.,  1.,  0.,  1.,  0.,  0.,  0.,  0.,  1.,
                0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])

        """
        n, k = self.q.shape
        if self.permutations:
            spill_over = np.zeros((n, k - 1))
            components = np.zeros((n, k))
            i2id = {}  # handle string keys
            for key in list(self.w.neighbors.keys()):
                idx = self.w.id2i[key]
                i2id[idx] = key
            sig_lisas = (self.q == quadrant) \
                * (self.p_values <= self.significance_level)
            sig_ids = [np.nonzero(
                sig_lisas[:, i])[0].tolist() for i in range(k)]

            neighbors = self.w.neighbors
            for t in range(k - 1):
                s1 = sig_ids[t]
                s2 = sig_ids[t + 1]
                g1 = pysal.region.components.Graph(undirected=True)
                for i in s1:
                    for neighbor in neighbors[i2id[i]]:
                        g1.add_edge(i2id[i], neighbor, 1.0)
                        if neighbors_on:
                            for nn in neighbors[neighbor]:
                                g1.add_edge(neighbor, nn, 1.0)
                components1 = g1.connected_components(op=gt)
                components1 = [list(c.nodes) for c in components1]
                g2 = pysal.region.components.Graph(undirected=True)
                for i in s2:
                    for neighbor in neighbors[i2id[i]]:
                        g2.add_edge(i2id[i], neighbor, 1.0)
                        if neighbors_on:
                            for nn in neighbors[neighbor]:
                                g2.add_edge(neighbor, nn, 1.0)
                components2 = g2.connected_components(op=gt)
                components2 = [list(c.nodes) for c in components2]
                c2 = []
                c1 = []
                for c in components2:
                    c2.extend(c)
                for c in components1:
                    c1.extend(c)

                new_ids = [j for j in c2 if j not in c1]
                spill_ids = []
                for j in new_ids:
                    # find j's component in period 2
                    cj = [c for c in components2 if j in c][0]
                    # for members of j's component in period 2, check if they
                    # belonged to any components in period 1
                    for i in cj:
                        if i in c1:
                            spill_ids.append(j)
                            break
                for spill_id in spill_ids:
                    id = self.w.id2i[spill_id]
                    spill_over[id, t] = 1
                for c, component in enumerate(components1):
                    for i in component:
                        ii = self.w.id2i[i]
                        components[ii, t] = c + 1
            results = {}
            results['components'] = components
            results['spill_over'] = spill_over
            return results

        else:
            return None


def kullback(F):
    """
    Kullback information based test of Markov Homogeneity.

    Parameters
    ----------
    F : array
        (s, r, r), values are transitions (not probabilities) for
        s strata, r initial states, r terminal states.

    Returns
    -------
    Results : dictionary
              (key - value)

              Conditional homogeneity - (float) test statistic for homogeneity
              of transition probabilities across strata.

              Conditional homogeneity pvalue - (float) p-value for test
              statistic.

              Conditional homogeneity dof - (int) degrees of freedom =
              r(s-1)(r-1).

    Notes
    -----
    Based on  Kullback, Kupperman and Ku (1962) [Kullback1962]_.
    Example below is taken from Table 9.2 .

    Examples
    --------
    >>> s1 = np.array([
    ...         [ 22, 11, 24,  2,  2,  7],
    ...         [ 5, 23, 15,  3, 42,  6],
    ...         [ 4, 21, 190, 25, 20, 34],
    ...         [0, 2, 14, 56, 14, 28],
    ...         [32, 15, 20, 10, 56, 14],
    ...         [5, 22, 31, 18, 13, 134]
    ...     ])
    >>> s2 = np.array([
    ...     [3, 6, 9, 3, 0, 8],
    ...     [1, 9, 3, 12, 27, 5],
    ...     [2, 9, 208, 32, 5, 18],
    ...     [0, 14, 32, 108, 40, 40],
    ...     [22, 14, 9, 26, 224, 14],
    ...     [1, 5, 13, 53, 13, 116]
    ...     ])
    >>>
    >>> F = np.array([s1, s2])
    >>> res = kullback(F)
    >>> "%8.3f"%res['Conditional homogeneity']
    ' 160.961'
    >>> "%d"%res['Conditional homogeneity dof']
    '30'
    >>> "%3.1f"%res['Conditional homogeneity pvalue']
    '0.0'

    """

    F1 = F == 0
    F1 = F + F1
    FLF = F * np.log(F1)
    T1 = 2 * FLF.sum()

    FdJK = F.sum(axis=0)
    FdJK1 = FdJK + (FdJK == 0)
    FdJKLFdJK = FdJK * np.log(FdJK1)
    T2 = 2 * FdJKLFdJK.sum()

    FdJd = F.sum(axis=0).sum(axis=1)
    FdJd1 = FdJd + (FdJd == 0)
    T3 = 2 * (FdJd * np.log(FdJd1)).sum()

    FIJd = F[:, :].sum(axis=1)
    FIJd1 = FIJd + (FIJd == 0)
    T4 = 2 * (FIJd * np.log(FIJd1)).sum()

    T6 = F.sum()
    T6 = 2 * T6 * np.log(T6)

    s, r, r1 = F.shape
    chom = T1 - T4 - T2 + T3
    cdof = r * (s - 1) * (r - 1)
    results = {}
    results['Conditional homogeneity'] = chom
    results['Conditional homogeneity dof'] = cdof
    results['Conditional homogeneity pvalue'] = 1 - stats.chi2.cdf(chom, cdof)
    return results


def prais(pmat):
    """
    Prais conditional mobility measure.

    Parameters
    ----------
    pmat : matrix
           (k, k), Markov probability transition matrix.

    Returns
    -------
    pr   : matrix
           (1, k), conditional mobility measures for each of the k classes.

    Notes
    -----
    Prais' conditional mobility measure for a class is defined as:

    .. math::

            pr_i = 1 -  p_{i,i}

    Examples
    --------
    >>> import numpy as np
    >>> import pysal
    >>> f = pysal.open(pysal.examples.get_path("usjoin.csv"))
    >>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
    >>> q5 = np.array([pysal.Quantiles(y).yb for y in pci]).transpose()
    >>> m = pysal.Markov(q5)
    >>> m.transitions
    array([[ 729.,   71.,    1.,    0.,    0.],
           [  72.,  567.,   80.,    3.,    0.],
           [   0.,   81.,  631.,   86.,    2.],
           [   0.,    3.,   86.,  573.,   56.],
           [   0.,    0.,    1.,   57.,  741.]])
    >>> m.p
    matrix([[ 0.91011236,  0.0886392 ,  0.00124844,  0.        ,  0.        ],
            [ 0.09972299,  0.78531856,  0.11080332,  0.00415512,  0.        ],
            [ 0.        ,  0.10125   ,  0.78875   ,  0.1075    ,  0.0025    ],
            [ 0.        ,  0.00417827,  0.11977716,  0.79805014,  0.07799443],
            [ 0.        ,  0.        ,  0.00125156,  0.07133917,  0.92740926]])
    >>> pysal.spatial_dynamics.markov.prais(m.p)
    matrix([[ 0.08988764,  0.21468144,  0.21125   ,  0.20194986,  0.07259074]])

    """
    pr = (pmat.sum(axis=1) - np.diag(pmat))[0]
    return pr


def shorrock(pmat):
    """
    Shorrock's mobility measure.

    Parameters
    ----------
    pmat : matrix
           (k, k), Markov probability transition matrix.

    Returns
    -------
    sh   : float
           Shorrock mobility measure.

    Notes
    -----
    Shorock's mobility measure is defined as

    .. math::

         sh = (k  - \sum_{j=1}^{k} p_{j,j})/(k - 1)

    Examples
    --------
    >>> import numpy as np
    >>> import pysal
    >>> f = pysal.open(pysal.examples.get_path("usjoin.csv"))
    >>> pci = np.array([f.by_col[str(y)] for y in range(1929,2010)])
    >>> q5 = np.array([pysal.Quantiles(y).yb for y in pci]).transpose()
    >>> m = pysal.Markov(q5)
    >>> m.transitions
    array([[ 729.,   71.,    1.,    0.,    0.],
           [  72.,  567.,   80.,    3.,    0.],
           [   0.,   81.,  631.,   86.,    2.],
           [   0.,    3.,   86.,  573.,   56.],
           [   0.,    0.,    1.,   57.,  741.]])
    >>> m.p
    matrix([[ 0.91011236,  0.0886392 ,  0.00124844,  0.        ,  0.        ],
            [ 0.09972299,  0.78531856,  0.11080332,  0.00415512,  0.        ],
            [ 0.        ,  0.10125   ,  0.78875   ,  0.1075    ,  0.0025    ],
            [ 0.        ,  0.00417827,  0.11977716,  0.79805014,  0.07799443],
            [ 0.        ,  0.        ,  0.00125156,  0.07133917,  0.92740926]])
    >>> pysal.spatial_dynamics.markov.shorrock(m.p)
    0.19758992000997844

    """
    t = np.trace(pmat)
    k = pmat.shape[1]
    sh = (k - t) / (k - 1)
    return sh


def homogeneity(transition_matrices, regime_names=[], class_names=[],
                title="Markov Homogeneity Test"):
    """
    Test for homogeneity of Markov transition probabilities across regimes.

    Parameters
    ----------
    transition_matrices : list
                          of transition matrices for regimes, all matrices must
                          have same size (r, c). r is the number of rows in the
                          transition matrix and c is the number of columns in
                          the transition matrix.
    regime_names        : sequence
                          Labels for the regimes.
    class_names         : sequence
                          Labels for the classes/states of the Markov chain.
    title               : string
                          name of test.

    Returns
    -------
                        : implicit
                          an instance of Homogeneity_Results.
    """

    return Homogeneity_Results(transition_matrices, regime_names=regime_names,
                               class_names=class_names, title=title)


class Homogeneity_Results:
    """
    Wrapper class to present homogeneity results.

    Parameters
    ----------
    transition_matrices : list
                          of transition matrices for regimes, all matrices must
                          have same size (r, c). r is the number of rows in
                          the transition matrix and c is the number of columns
                          in the transition matrix.
    regime_names        : sequence
                          Labels for the regimes.
    class_names         : sequence
                          Labels for the classes/states of the Markov chain.
    title               : string
                          Title of the table.

    Attributes
    -----------

    Notes
    -----
    Degrees of freedom adjustment follow the approach in Bickenbach and Bode
    (2003) [Bickenbach2003]_.

    Examples
    --------
    See Spatial_Markov above.

    """

    def __init__(self, transition_matrices, regime_names=[], class_names=[],
                 title="Markov Homogeneity Test"):
        self._homogeneity(transition_matrices)
        self.regime_names = regime_names
        self.class_names = class_names
        self.title = title

    def _homogeneity(self, transition_matrices):
        # form null transition probability matrix
        M = np.array(transition_matrices)
        m, r, k = M.shape
        self.k = k
        B = np.zeros((r, m))
        T = M.sum(axis=0)
        self.t_total = T.sum()
        n_i = T.sum(axis=1)
        A_i = (T > 0).sum(axis=1)
        A_im = np.zeros((r, m))
        p_ij = np.dot(np.diag(1./(n_i + (n_i == 0)*1.)), T)
        den = p_ij + 1. * (p_ij == 0)
        b_i = np.zeros_like(A_i)
        p_ijm = np.zeros_like(M)
        # get dimensions
        m, n_rows, n_cols = M.shape
        m = 0
        Q = 0.0
        LR = 0.0
        lr_table = np.zeros_like(M)
        q_table = np.zeros_like(M)

        for nijm in M:
            nim = nijm.sum(axis=1)
            B[:, m] = 1.*(nim > 0)
            b_i = b_i + 1. * (nim > 0)
            p_ijm[m] = np.dot(np.diag(1./(nim + (nim == 0)*1.)), nijm)
            num = (p_ijm[m]-p_ij)**2
            ratio = num / den
            qijm = np.dot(np.diag(nim), ratio)
            q_table[m] = qijm
            Q = Q + qijm.sum()
            # only use nonzero pijm in lr test
            mask = (nijm > 0) * (p_ij > 0)
            A_im[:, m] = (nijm > 0).sum(axis=1)
            unmask = 1.0 * (mask == 0)
            ratio = (mask * p_ijm[m] + unmask) / (mask * p_ij + unmask)
            lr = nijm * np.log(ratio)
            LR = LR + lr.sum()
            lr_table[m] = 2 * lr
            m += 1
        # b_i is the number of regimes that have non-zero observations in row i
        # A_i is the number of non-zero elements in row i of the aggregated
        # transition matrix
        self.dof = int(((b_i-1) * (A_i-1)).sum())
        self.Q = Q
        self.Q_p_value = 1 - stats.chi2.cdf(self.Q, self.dof)
        self.LR = LR * 2.
        self.LR_p_value = 1 - stats.chi2.cdf(self.LR, self.dof)
        self.A = A_i
        self.A_im = A_im
        self.B = B
        self.b_i = b_i
        self.LR_table = lr_table
        self.Q_table = q_table
        self.m = m
        self.p_h0 = p_ij
        self.p_h1 = p_ijm

    def summary(self, file_name=None, title="Markov Homogeneity Test"):
        regime_names = ["%d" % i for i in range(self.m)]
        if self.regime_names:
            regime_names = self.regime_names
        cols = ["P(%s)" % str(regime) for regime in regime_names]
        if not self.class_names:
            self.class_names = list(range(self.k))

        max_col = max([len(col) for col in cols])
        col_width = max([5, max_col])  # probabilities have 5 chars
        n_tabs = self.k
        width = n_tabs * 4 + (self.k+1)*col_width
        lead = "-" * width
        head = title.center(width)
        contents = [lead, head, lead]
        l = "Number of regimes: %d" % int(self.m)
        k = "Number of classes: %d" % int(self.k)
        r = "Regime names: "
        r += ", ".join(regime_names)
        t = "Number of transitions: %d" % int(self.t_total)
        contents.append(k)
        contents.append(t)
        contents.append(l)
        contents.append(r)
        contents.append(lead)
        h = "%7s %20s %20s" % ('Test', 'LR', 'Chi-2')
        contents.append(h)
        stat = "%7s %20.3f %20.3f" % ('Stat.', self.LR, self.Q)
        contents.append(stat)
        stat = "%7s %20d %20d" % ('DOF', self.dof, self.dof)
        contents.append(stat)
        stat = "%7s %20.3f %20.3f" % ('p-value', self.LR_p_value,
                                      self.Q_p_value)
        contents.append(stat)
        print("\n".join(contents))
        print(lead)

        cols = ["P(%s)" % str(regime) for regime in self.regime_names]
        if not self.class_names:
            self.class_names = list(range(self.k))
        cols.extend(["%s" % str(cname) for cname in self.class_names])

        max_col = max([len(col) for col in cols])
        col_width = max([5, max_col])  # probabilities have 5 chars
        p0 = []
        line0 = ['{s: <{w}}'.format(s="P(H0)", w=col_width)]
        line0.extend((['{s: >{w}}'.format(s=cname, w=col_width) for cname in
                       self.class_names]))
        print("    ".join(line0))
        p0.append("&".join(line0))
        for i, row in enumerate(self.p_h0):
            line = ["%*s" % (col_width, str(self.class_names[i]))]
            line.extend(["%*.3f" % (col_width, v) for v in row])
            print("    ".join(line))
            p0.append("&".join(line))
        pmats = [p0]

        print(lead)
        for r, p1 in enumerate(self.p_h1):
            p0 = []
            line0 = ['{s: <{w}}'.format(s="P(%s)" %
                                        regime_names[r], w=col_width)]
            line0.extend((['{s: >{w}}'.format(s=cname, w=col_width) for cname
                           in self.class_names]))
            print("    ".join(line0))
            p0.append("&".join(line0))
            for i, row in enumerate(p1):
                line = ["%*s" % (col_width, str(self.class_names[i]))]
                line.extend(["%*.3f" % (col_width, v) for v in row])
                print("    ".join(line))
                p0.append("&".join(line))
            pmats.append(p0)
            print(lead)

        if file_name:
            k = self.k
            ks = str(k+1)
            with open(file_name, 'w') as f:
                c = []
                fmt = "r"*(k+1)
                s = "\\begin{tabular}{|%s|}\\hline\n" % fmt
                s += "\\multicolumn{%s}{|c|}{%s}" % (ks, title)
                c.append(s)
                s = "Number of classes: %d" % int(self.k)
                c.append("\\hline\\multicolumn{%s}{|l|}{%s}" % (ks, s))
                s = "Number of transitions: %d" % int(self.t_total)
                c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
                s = "Number of regimes: %d" % int(self.m)
                c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
                s = "Regime names: "
                s += ", ".join(regime_names)
                c.append("\\multicolumn{%s}{|l|}{%s}" % (ks, s))
                s = "\\hline\\multicolumn{2}{|l}{%s}" % ("Test")
                s += "&\\multicolumn{2}{r}{LR}&\\multicolumn{2}{r|}{Q}"
                c.append(s)
                s = "Stat."
                s = "\\multicolumn{2}{|l}{%s}" % (s)
                s += "&\\multicolumn{2}{r}{%.3f}" % self.LR
                s += "&\\multicolumn{2}{r|}{%.3f}" % self.Q
                c.append(s)
                s = "\\multicolumn{2}{|l}{%s}" % ("DOF")
                s += "&\\multicolumn{2}{r}{%d}" % int(self.dof)
                s += "&\\multicolumn{2}{r|}{%d}" % int(self.dof)
                c.append(s)
                s = "\\multicolumn{2}{|l}{%s}" % ("p-value")
                s += "&\\multicolumn{2}{r}{%.3f}" % self.LR_p_value
                s += "&\\multicolumn{2}{r|}{%.3f}" % self.Q_p_value
                c.append(s)
                s1 = "\\\\\n".join(c)
                s1 += "\\\\\n"
                c = []
                for mat in pmats:
                    c.append("\\hline\n")
                    for row in mat:
                        c.append(row+"\\\\\n")
                c.append("\\hline\n")
                c.append("\\end{tabular}")
                s2 = "".join(c)
                f.write(s1+s2)
