"""
ML Estimation of Spatial Error Model
"""

__author__ = "Luc Anselin luc.anselin@asu.edu,\
              Serge Rey srey@asu.edu, \
              Levi Wolf levi.john.wolf@asu.edu"

import numpy as np
import numpy.linalg as la
from scipy import sparse as sp
from scipy.sparse.linalg import splu as SuperLU
import pysal as ps
from .utils import RegressionPropsY, RegressionPropsVM
from . import diagnostics as DIAG
from . import user_output as USER
from . import summary_output as SUMMARY
from . import regimes as REGI
from .w_utils import symmetrize
try:
    from scipy.optimize import minimize_scalar
    minimize_scalar_available = True
except ImportError:
    minimize_scalar_available = False

__all__ = ["ML_Error"]


class BaseML_Error(RegressionPropsY, RegressionPropsVM, REGI.Regimes_Frame):

    """
    ML estimation of the spatial error model (note no consistency
    checks, diagnostics or constants added); Anselin (1988) [Anselin1988]_

    Parameters
    ----------
    y            : array
                   nx1 array for dependent variable
    x            : array
                   Two dimensional array with n rows and one column for each
                   independent (exogenous) variable, excluding the constant
    w            : Sparse matrix
                   Spatial weights sparse matrix
    method       : string
                   if 'full', brute force calculation (full matrix expressions)
                   if 'ord', Ord eigenvalue calculation
                   if 'LU', LU decomposition for sparse matrices
    epsilon      : float
                   tolerance criterion in mimimize_scalar function and inverse_product
    regimes_att  : dictionary
                   Dictionary containing elements to be used in case of a regimes model,
                   i.e. 'x' before regimes, 'regimes' list and 'cols2regi'


    Attributes
    ----------
    betas        : array
                   kx1 array of estimated coefficients
    lam          : float
                   estimate of spatial autoregressive coefficient
    u            : array
                   nx1 array of residuals
    e_filtered   : array
                   spatially filtered residuals
    predy        : array
                   nx1 array of predicted y values
    n            : integer
                   Number of observations
    k            : integer
                   Number of variables for which coefficients are estimated
                   (including the constant, excluding the rho)
    y            : array
                   nx1 array for dependent variable
    x            : array
                   Two dimensional array with n rows and one column for each
                   independent (exogenous) variable, including the constant
    method       : string
                   log Jacobian method
                   if 'full': brute force (full matrix computations)
                   if 'ord' : Ord eigenvalue method
    epsilon      : float
                   tolerance criterion used in minimize_scalar function and inverse_product
    mean_y       : float
                   Mean of dependent variable
    std_y        : float
                   Standard deviation of dependent variable
    vm           : array
                   Variance covariance matrix (k+1 x k+1) - includes lambda
    vm1          : array
                   2x2 array of variance covariance for lambda, sigma
    sig2         : float
                   Sigma squared used in computations
    logll        : float
                   maximized log-likelihood (including constant terms)

    Examples
    --------
    >>> import numpy as np
    >>> import pysal as ps
    >>> np.set_printoptions(suppress=True) #prevent scientific format
    >>> db = ps.open(ps.examples.get_path("south.dbf"),'r')
    >>> y_name = "HR90"
    >>> y = np.array(db.by_col(y_name))
    >>> y.shape = (len(y),1)
    >>> x_names = ["RD90","PS90","UE90","DV90"]
    >>> x = np.array([db.by_col(var) for var in x_names]).T
    >>> x = np.hstack((np.ones((len(y),1)),x))
    >>> ww = ps.open(ps.examples.get_path("south_q.gal"))
    >>> w = ww.read()
    >>> ww.close()
    >>> w.transform = 'r'
    >>> mlerr = BaseML_Error(y,x,w) #doctest: +SKIP
    >>> "{0:.6f}".format(mlerr.lam) #doctest: +SKIP
    '0.299078'
    >>> np.around(mlerr.betas, decimals=4) #doctest: +SKIP
    array([[ 6.1492],
           [ 4.4024],
           [ 1.7784],
           [-0.3781],
           [ 0.4858],
           [ 0.2991]])
    >>> "{0:.6f}".format(mlerr.mean_y) #doctest: +SKIP
    '9.549293'
    >>> "{0:.6f}".format(mlerr.std_y) #doctest: +SKIP
    '7.038851'
    >>> np.diag(mlerr.vm) #doctest: +SKIP
    array([ 1.06476526,  0.05548248,  0.04544514,  0.00614425,  0.01481356,
            0.00143001])
    >>> "{0:.6f}".format(mlerr.sig2[0][0]) #doctest: +SKIP
    '32.406854'
    >>> "{0:.6f}".format(mlerr.logll) #doctest: +SKIP
    '-4471.407067'
    >>> mlerr1 = BaseML_Error(y,x,w,method='ord') #doctest: +SKIP
    >>> "{0:.6f}".format(mlerr1.lam) #doctest: +SKIP
    '0.299078'
    >>> np.around(mlerr1.betas, decimals=4) #doctest: +SKIP
    array([[ 6.1492],
           [ 4.4024],
           [ 1.7784],
           [-0.3781],
           [ 0.4858],
           [ 0.2991]])
    >>> "{0:.6f}".format(mlerr1.mean_y) #doctest: +SKIP
    '9.549293'
    >>> "{0:.6f}".format(mlerr1.std_y) #doctest: +SKIP
    '7.038851'
    >>> np.around(np.diag(mlerr1.vm), decimals=4) #doctest: +SKIP
    array([ 1.0648,  0.0555,  0.0454,  0.0061,  0.0148,  0.0014])
    >>> "{0:.4f}".format(mlerr1.sig2[0][0]) #doctest: +SKIP
    '32.4069'
    >>> "{0:.4f}".format(mlerr1.logll) #doctest: +SKIP
    '-4471.4071'

    """

    def __init__(self, y, x, w, method='full', epsilon=0.0000001, regimes_att=None):
        # set up main regression variables and spatial filters
        self.y = y
        if regimes_att:
            self.x = x.toarray()
        else:
            self.x = x
        self.n, self.k = self.x.shape
        self.method = method
        self.epsilon = epsilon

        #W = w.full()[0] #wait to build pending what is needed
        #Wsp = w.sparse

        ylag = ps.lag_spatial(w, self.y)
        xlag = self.get_x_lag(w, regimes_att)

        # call minimizer using concentrated log-likelihood to get lambda
        methodML = method.upper()
        if methodML in ['FULL', 'LU', 'ORD']:
            if methodML == 'FULL':  
                W = w.full()[0]      # need dense here
                res = minimize_scalar(err_c_loglik, 0.0, bounds=(-1.0, 1.0),
                                      args=(self.n, self.y, ylag, self.x,
                                            xlag, W), method='bounded',
                                      tol=epsilon)
            elif methodML == 'LU':
                I = sp.identity(w.n)
                Wsp = w.sparse   # need sparse here
                res = minimize_scalar(err_c_loglik_sp, 0.0, bounds=(-1.0,1.0),
                                      args=(self.n, self.y, ylag, 
                                            self.x, xlag, I, Wsp),
                                      method='bounded', tol=epsilon)
            elif methodML == 'ORD':
                # check on symmetry structure
                if w.asymmetry(intrinsic=False) == []:
                    ww = symmetrize(w)
                    WW = ww.todense()
                    evals = la.eigvalsh(WW)
                else:
                    W = w.full()[0]      # need dense here
                    evals = la.eigvals(W)
                res = minimize_scalar(
                    err_c_loglik_ord, 0.0, bounds=(-1.0, 1.0),
                    args=(self.n, self.y, ylag, self.x,
                          xlag, evals), method='bounded',
                    tol=epsilon)
        else:
            raise Exception("{0} is an unsupported method".format(method))

        self.lam = res.x

        # compute full log-likelihood, including constants
        ln2pi = np.log(2.0 * np.pi)
        llik = -res.fun - self.n / 2.0 * ln2pi - self.n / 2.0

        self.logll = llik

        # b, residuals and predicted values

        ys = self.y - self.lam * ylag
        xs = self.x - self.lam * xlag
        xsxs = np.dot(xs.T, xs)
        xsxsi = np.linalg.inv(xsxs)
        xsys = np.dot(xs.T, ys)
        b = np.dot(xsxsi, xsys)

        self.betas = np.vstack((b, self.lam))

        self.u = y - np.dot(self.x, b)
        self.predy = self.y - self.u

        # residual variance

        self.e_filtered = self.u - self.lam * ps.lag_spatial(w, self.u)
        self.sig2 = np.dot(self.e_filtered.T, self.e_filtered) / self.n

        # variance-covariance matrix betas

        varb = self.sig2 * xsxsi

        # variance-covariance matrix lambda, sigma

        a = -self.lam * W
        np.fill_diagonal(a, 1.0)
        ai = la.inv(a)
        wai = np.dot(W, ai)
        tr1 = np.trace(wai)

        wai2 = np.dot(wai, wai)
        tr2 = np.trace(wai2)

        waiTwai = np.dot(wai.T, wai)
        tr3 = np.trace(waiTwai)

        v1 = np.vstack((tr2 + tr3,
                        tr1 / self.sig2))
        v2 = np.vstack((tr1 / self.sig2,
                        self.n / (2.0 * self.sig2 ** 2)))

        v = np.hstack((v1, v2))

        self.vm1 = np.linalg.inv(v)

        # create variance matrix for beta, lambda
        vv = np.hstack((varb, np.zeros((self.k, 1))))
        vv1 = np.hstack(
            (np.zeros((1, self.k)), self.vm1[0, 0] * np.ones((1, 1))))

        self.vm = np.vstack((vv, vv1))


    def get_x_lag(self, w, regimes_att):
        if regimes_att:
            xlag = ps.lag_spatial(w, regimes_att['x'])
            xlag = REGI.Regimes_Frame.__init__(self, xlag,
                                               regimes_att['regimes'], constant_regi=None, cols2regi=regimes_att['cols2regi'])[0]
            xlag = xlag.toarray()
        else:
            xlag = ps.lag_spatial(w, self.x)
        return xlag


class ML_Error(BaseML_Error):

    """
    ML estimation of the spatial lag model with all results and diagnostics;
    Anselin (1988) [Anselin1988]_

    Parameters
    ----------
    y            : array
                   nx1 array for dependent variable
    x            : array
                   Two dimensional array with n rows and one column for each
                   independent (exogenous) variable, excluding the constant
    w            : Sparse matrix
                   Spatial weights sparse matrix
    method       : string
                   if 'full', brute force calculation (full matrix expressions)
                   if 'ord', Ord eigenvalue method
                   if 'LU', LU sparse matrix decomposition
    epsilon      : float
                   tolerance criterion in mimimize_scalar function and inverse_product
    spat_diag    : boolean
                   if True, include spatial diagnostics
    vm           : boolean
                   if True, include variance-covariance matrix in summary
                   results
    name_y       : string
                   Name of dependent variable for use in output
    name_x       : list of strings
                   Names of independent variables for use in output
    name_w       : string
                   Name of weights matrix for use in output
    name_ds      : string
                   Name of dataset for use in output

    Attributes
    ----------
    betas        : array
                   (k+1)x1 array of estimated coefficients (rho first)
    lam          : float
                   estimate of spatial autoregressive coefficient
    u            : array
                   nx1 array of residuals
    e_filtered   : array
                   nx1 array of spatially filtered residuals
    predy        : array
                   nx1 array of predicted y values
    n            : integer
                   Number of observations
    k            : integer
                   Number of variables for which coefficients are estimated
                   (including the constant, excluding lambda)
    y            : array
                   nx1 array for dependent variable
    x            : array
                   Two dimensional array with n rows and one column for each
                   independent (exogenous) variable, including the constant
    method       : string
                   log Jacobian method
                   if 'full': brute force (full matrix computations)
    epsilon      : float
                   tolerance criterion used in minimize_scalar function and inverse_product
    mean_y       : float
                   Mean of dependent variable
    std_y        : float
                   Standard deviation of dependent variable
    varb         : array
                   Variance covariance matrix (k+1 x k+1) - includes var(lambda)
    vm1          : array
                   variance covariance matrix for lambda, sigma (2 x 2)
    sig2         : float
                   Sigma squared used in computations
    logll        : float
                   maximized log-likelihood (including constant terms)
    pr2          : float
                   Pseudo R squared (squared correlation between y and ypred)
    utu          : float
                   Sum of squared residuals
    std_err      : array
                   1xk array of standard errors of the betas
    z_stat       : list of tuples
                   z statistic; each tuple contains the pair (statistic,
                   p-value), where each is a float
    name_y       : string
                   Name of dependent variable for use in output
    name_x       : list of strings
                   Names of independent variables for use in output
    name_w       : string
                   Name of weights matrix for use in output
    name_ds      : string
                   Name of dataset for use in output
    title        : string
                   Name of the regression method used

    Examples
    --------

    >>> import numpy as np
    >>> import pysal as ps
    >>> np.set_printoptions(suppress=True)  #prevent scientific format
    >>> db = ps.open(ps.examples.get_path("south.dbf"),'r')
    >>> ds_name = "south.dbf"
    >>> y_name = "HR90"
    >>> y = np.array(db.by_col(y_name))
    >>> y.shape = (len(y),1)
    >>> x_names = ["RD90","PS90","UE90","DV90"]
    >>> x = np.array([db.by_col(var) for var in x_names]).T
    >>> ww = ps.open(ps.examples.get_path("south_q.gal"))
    >>> w = ww.read()
    >>> ww.close()
    >>> w_name = "south_q.gal"
    >>> w.transform = 'r'
    >>> mlerr = ML_Error(y,x,w,name_y=y_name,name_x=x_names,\
               name_w=w_name,name_ds=ds_name) #doctest: +SKIP
    >>> np.around(mlerr.betas, decimals=4) #doctest: +SKIP
    array([[ 6.1492],
           [ 4.4024],
           [ 1.7784],
           [-0.3781],
           [ 0.4858],
           [ 0.2991]])
    >>> "{0:.4f}".format(mlerr.lam) #doctest: +SKIP
    '0.2991'
    >>> "{0:.4f}".format(mlerr.mean_y) #doctest: +SKIP
    '9.5493'
    >>> "{0:.4f}".format(mlerr.std_y) #doctest: +SKIP
    '7.0389'
    >>> np.around(np.diag(mlerr.vm), decimals=4) #doctest: +SKIP
    array([ 1.0648,  0.0555,  0.0454,  0.0061,  0.0148,  0.0014])
    >>> np.around(mlerr.sig2, decimals=4) #doctest: +SKIP
    array([[ 32.4069]])
    >>> "{0:.4f}".format(mlerr.logll) #doctest: +SKIP
    '-4471.4071'
    >>> "{0:.4f}".format(mlerr.aic) #doctest: +SKIP
    '8952.8141'
    >>> "{0:.4f}".format(mlerr.schwarz) #doctest: +SKIP
    '8979.0779'
    >>> "{0:.4f}".format(mlerr.pr2) #doctest: +SKIP
    '0.3058'
    >>> "{0:.4f}".format(mlerr.utu) #doctest: +SKIP
    '48534.9148'
    >>> np.around(mlerr.std_err, decimals=4) #doctest: +SKIP
    array([ 1.0319,  0.2355,  0.2132,  0.0784,  0.1217,  0.0378])
    >>> np.around(mlerr.z_stat, decimals=4) #doctest: +SKIP
    array([[  5.9593,   0.    ],
           [ 18.6902,   0.    ],
           [  8.3422,   0.    ],
           [ -4.8233,   0.    ],
           [  3.9913,   0.0001],
           [  7.9089,   0.    ]])
    >>> mlerr.name_y #doctest: +SKIP
    'HR90'
    >>> mlerr.name_x #doctest: +SKIP
    ['CONSTANT', 'RD90', 'PS90', 'UE90', 'DV90', 'lambda']
    >>> mlerr.name_w #doctest: +SKIP
    'south_q.gal'
    >>> mlerr.name_ds #doctest: +SKIP
    'south.dbf'
    >>> mlerr.title #doctest: +SKIP
    'MAXIMUM LIKELIHOOD SPATIAL ERROR (METHOD = FULL)'


    """

    def __init__(self, y, x, w, method='full', epsilon=0.0000001,
                 spat_diag=False, vm=False, name_y=None, name_x=None,
                 name_w=None, name_ds=None):
        n = USER.check_arrays(y, x)
        USER.check_y(y, n)
        USER.check_weights(w, y, w_required=True)
        x_constant = USER.check_constant(x)
        method = method.upper()
        BaseML_Error.__init__(self, y=y, x=x_constant,
                              w=w, method=method, epsilon=epsilon)
        self.title = "MAXIMUM LIKELIHOOD SPATIAL ERROR" + \
            " (METHOD = " + method + ")"
        self.name_ds = USER.set_name_ds(name_ds)
        self.name_y = USER.set_name_y(name_y)
        self.name_x = USER.set_name_x(name_x, x)
        self.name_x.append('lambda')
        self.name_w = USER.set_name_w(name_w, w)
        self.aic = DIAG.akaike(reg=self)
        self.schwarz = DIAG.schwarz(reg=self)
        SUMMARY.ML_Error(reg=self, w=w, vm=vm, spat_diag=spat_diag)


def err_c_loglik(lam, n, y, ylag, x, xlag, W):
    # concentrated log-lik for error model, no constants, brute force
    ys = y - lam * ylag
    xs = x - lam * xlag
    ysys = np.dot(ys.T, ys)
    xsxs = np.dot(xs.T, xs)
    xsxsi = np.linalg.inv(xsxs)
    xsys = np.dot(xs.T, ys)
    x1 = np.dot(xsxsi, xsys)
    x2 = np.dot(xsys.T, x1)
    ee = ysys - x2
    sig2 = ee[0][0] / n
    nlsig2 = (n / 2.0) * np.log(sig2)
    a = -lam * W
    np.fill_diagonal(a, 1.0)
    jacob = np.log(np.linalg.det(a))
    # this is the negative of the concentrated log lik for minimization
    clik = nlsig2 - jacob
    return clik

def err_c_loglik_sp(lam, n, y, ylag, x, xlag, I, Wsp):
    # concentrated log-lik for error model, no constants, LU
    if isinstance(lam, np.ndarray):
        if lam.shape == (1,1):
            lam = lam[0][0] #why does the interior value change?
    ys = y - lam * ylag
    xs = x - lam * xlag
    ysys = np.dot(ys.T, ys)
    xsxs = np.dot(xs.T, xs)
    xsxsi = np.linalg.inv(xsxs)
    xsys = np.dot(xs.T, ys)
    x1 = np.dot(xsxsi, xsys)
    x2 = np.dot(xsys.T, x1)
    ee = ysys - x2
    sig2 = ee[0][0] / n
    nlsig2 = (n / 2.0) * np.log(sig2)
    a = I - lam * Wsp
    LU = SuperLU(a.tocsc())
    jacob = np.sum(np.log(np.abs(LU.U.diagonal()))) 
    # this is the negative of the concentrated log lik for minimization
    clik = nlsig2 - jacob
    return clik


def err_c_loglik_ord(lam, n, y, ylag, x, xlag, evals):
    # concentrated log-lik for error model, no constants, eigenvalues
    ys = y - lam * ylag
    xs = x - lam * xlag
    ysys = np.dot(ys.T, ys)
    xsxs = np.dot(xs.T, xs)
    xsxsi = np.linalg.inv(xsxs)
    xsys = np.dot(xs.T, ys)
    x1 = np.dot(xsxsi, xsys)
    x2 = np.dot(xsys.T, x1)
    ee = ysys - x2
    sig2 = ee[0][0] / n
    nlsig2 = (n / 2.0) * np.log(sig2)
    revals = lam * evals
    jacob = np.log(1 - revals).sum()
    if isinstance(jacob, complex):
        jacob = jacob.real
    # this is the negative of the concentrated log lik for minimization
    clik = nlsig2 - jacob
    return clik


def _test():
    import doctest
    start_suppress = np.get_printoptions()['suppress']
    np.set_printoptions(suppress=True)
    doctest.testmod()
    np.set_printoptions(suppress=start_suppress)
