"""
Statespace Tools

Author: Chad Fulton
License: Simplified-BSD
"""
from __future__ import division, absolute_import, print_function

import numpy as np
from scipy.linalg import solve_sylvester
from statsmodels.tools.data import _is_using_pandas
from . import _statespace

has_find_best_blas_type = True
try:
    from scipy.linalg.blas import find_best_blas_type
except ImportError:  # pragma: no cover
    has_find_best_blas_type = False
    # Shim for SciPy 0.11, derived from tag=0.11 scipy.linalg.blas
    _type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z', 'G': 'z'}

    def find_best_blas_type(arrays):
        dtype, index = max(
            [(ar.dtype, i) for i, ar in enumerate(arrays)])
        prefix = _type_conv.get(dtype.char, 'd')
        return prefix, dtype, None

has_trmm = True
try:
    from scipy.linalg.blas import dtrmm
except ImportError:
    has_trmm = False


prefix_dtype_map = {
    's': np.float32, 'd': np.float64, 'c': np.complex64, 'z': np.complex128
}
prefix_statespace_map = {
    's': _statespace.sStatespace, 'd': _statespace.dStatespace,
    'c': _statespace.cStatespace, 'z': _statespace.zStatespace
}
prefix_kalman_filter_map = {
    's': _statespace.sKalmanFilter, 'd': _statespace.dKalmanFilter,
    'c': _statespace.cKalmanFilter, 'z': _statespace.zKalmanFilter
}
if has_trmm:
    prefix_pacf_map = {
        's': _statespace._scompute_coefficients_from_multivariate_pacf,
        'd': _statespace._dcompute_coefficients_from_multivariate_pacf,
        'c': _statespace._ccompute_coefficients_from_multivariate_pacf,
        'z': _statespace._zcompute_coefficients_from_multivariate_pacf
    }
    prefix_sv_map = {
        's': _statespace._sconstrain_sv_less_than_one,
        'd': _statespace._dconstrain_sv_less_than_one,
        'c': _statespace._cconstrain_sv_less_than_one,
        'z': _statespace._zconstrain_sv_less_than_one
    }


def companion_matrix(polynomial):
    r"""
    Create a companion matrix

    Parameters
    ----------
    polynomial : array_like or list
        If an iterable, interpreted as the coefficients of the polynomial from
        which to form the companion matrix. Polynomial coefficients are in
        order of increasing degree, and may be either scalars (as in an AR(p)
        model) or coefficient matrices (as in a VAR(p) model). If an integer,
        it is interpereted as the size of a companion matrix of a scalar
        polynomial, where the polynomial coefficients are initialized to zeros.
        If a matrix polynomial is passed, :math:`C_0` may be set to the scalar
        value 1 to indicate an identity matrix (doing so will improve the speed
        of the companion matrix creation).

    Returns
    -------
    companion_matrix : array

    Notes
    -----
    Given coefficients of a lag polynomial of the form:

    .. math::
        c(L) = c_0 + c_1 L + \dots + c_p L^p

    returns a matrix of the form

    .. math::
        \begin{bmatrix}
            \phi_1 & 1      & 0 & \cdots & 0 \\
            \phi_2 & 0      & 1 &        & 0 \\
            \vdots &        &   & \ddots & 0 \\
                   &        &   &        & 1 \\
            \phi_n & 0      & 0 & \cdots & 0 \\
        \end{bmatrix}

    where some or all of the :math:`\phi_i` may be non-zero (if `polynomial` is
    None, then all are equal to zero).

    If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_p)`,
    then the companion matrix is an :math:`n \times n` matrix formed with the
    elements in the first column defined as
    :math:`\phi_i = -\frac{c_i}{c_0}, i \in 1, \dots, p`.

    If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_p)`,
    each of shape :math:`(m, m)`, then the companion matrix is an
    :math:`nm \times nm` matrix formed with the elements in the first column
    defined as :math:`\phi_i = -C_0^{-1} C_i', i \in 1, \dots, p`.

    It is important to understand the expected signs of the coefficients. A
    typical AR(p) model is written as:

    .. math::
        y_t = a_1 y_{t-1} + \dots + a_p y_{t-p} + \varepsilon_t

    This can be rewritten as:

    .. math::
        (1 - a_1 L - \dots - a_p L^p )y_t = \varepsilon_t \\
        (1 + c_1 L + \dots + c_p L^p )y_t = \varepsilon_t \\
        c(L) y_t = \varepsilon_t

    The coefficients from this form are defined to be :math:`c_i = - a_i`, and
    it is the :math:`c_i` coefficients that this function expects to be
    provided.

    """
    identity_matrix = False
    if isinstance(polynomial, int):
        n = polynomial
        m = 1
        polynomial = None
    else:
        n = len(polynomial) - 1

        if n < 1:
            raise ValueError("Companion matrix polynomials must include at"
                             " least two terms.")

        if isinstance(polynomial, list) or isinstance(polynomial, tuple):
            try:
                # Note: can't use polynomial[0] because of the special behavior
                # associated with matrix polynomials and the constant 1, see
                # below.
                m = len(polynomial[1])
            except TypeError:
                m = 1

            # Check if we just have a scalar polynomial
            if m == 1:
                polynomial = np.asanyarray(polynomial)
            # Check if 1 was passed as the first argument (indicating an
            # identity matrix)
            elif polynomial[0] == 1:
                polynomial[0] = np.eye(m)
                identity_matrix = True
        else:
            m = 1
            polynomial = np.asanyarray(polynomial)

    matrix = np.zeros((n * m, n * m), dtype=np.asanyarray(polynomial).dtype)
    idx = np.diag_indices((n - 1) * m)
    idx = (idx[0], idx[1] + m)
    matrix[idx] = 1
    if polynomial is not None and n > 0:
        if m == 1:
            matrix[:, 0] = -polynomial[1:] / polynomial[0]
        elif identity_matrix:
            for i in range(n):
                matrix[i * m:(i + 1) * m, :m] = -polynomial[i+1].T
        else:
            inv = np.linalg.inv(polynomial[0])
            for i in range(n):
                matrix[i * m:(i + 1) * m, :m] = -np.dot(inv, polynomial[i+1]).T
    return matrix


def diff(series, k_diff=1, k_seasonal_diff=None, k_seasons=1):
    r"""
    Difference a series simply and/or seasonally along the zero-th axis.

    Given a series (denoted :math:`y_t`), performs the differencing operation

    .. math::

        \Delta^d \Delta_s^D y_t

    where :math:`d =` `diff`, :math:`s =` `k_seasons`,
    :math:`D =` `seasonal\_diff`, and :math:`\Delta` is the difference
    operator.

    Parameters
    ----------
    series : array_like
        The series to be differenced.
    diff : int, optional
        The number of simple differences to perform. Default is 1.
    seasonal_diff : int or None, optional
        The number of seasonal differences to perform. Default is no seasonal
        differencing.
    k_seasons : int, optional
        The seasonal lag. Default is 1. Unused if there is no seasonal
        differencing.

    Returns
    -------
    differenced : array
        The differenced array.
    """
    pandas = _is_using_pandas(series, None)
    differenced = np.asanyarray(series) if not pandas else series

    # Seasonal differencing
    if k_seasonal_diff is not None:
        while k_seasonal_diff > 0:
            if not pandas:
                differenced = (
                    differenced[k_seasons:] - differenced[:-k_seasons]
                )
            else:
                differenced = differenced.diff(k_seasons)[k_seasons:]
            k_seasonal_diff -= 1

    # Simple differencing
    if not pandas:
        differenced = np.diff(differenced, k_diff, axis=0)
    else:
        while k_diff > 0:
            differenced = differenced.diff()[1:]
            k_diff -= 1
    return differenced


def is_invertible(polynomial, threshold=1.):
    r"""
    Determine if a polynomial is invertible.

    Requires all roots of the polynomial lie inside the unit circle.

    Parameters
    ----------
    polynomial : array_like or tuple, list
        Coefficients of a polynomial, in order of increasing degree.
        For example, `polynomial=[1, -0.5]` corresponds to the polynomial
        :math:`1 - 0.5x` which has root :math:`2`. If it is a matrix
        polynomial (in which case the coefficients are coefficient matrices),
        a tuple or list of matrices should be passed.
    threshold : number
        Allowed threshold for `is_invertible` to return True. Default is 1.

    Notes
    -----

    If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_n)`,
    then the corresponding polynomial is :math:`c_0 + c_1 L + \dots + c_n L^n`.


    If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_n)`,
    then the corresponding polynomial is :math:`C_0 + C_1 L + \dots + C_n L^n`.

    There are three equivalent methods of determining if the polynomial
    represented by the coefficients is invertible:

    The first method factorizes the polynomial into:

    .. math::

        C(L) & = c_0 + c_1 L + \dots + c_n L^n \\
             & = constant (1 - \lambda_1 L)
                 (1 - \lambda_2 L) \dots (1 - \lambda_n L)

    In order for :math:`C(L)` to be invertible, it must be that each factor
    :math:`(1 - \lambda_i L)` is invertible; the condition is then that
    :math:`|\lambda_i| < 1`, where :math:`\lambda_i` is a root of the
    polynomial.

    The second method factorizes the polynomial into:

    .. math::

        C(L) & = c_0 + c_1 L + \dots + c_n L^n \\
             & = constant (L - \zeta_1) (L - \zeta_2) \dots (L - \zeta_3)

    The condition is now :math:`|\zeta_i| > 1`, where :math:`\zeta_i` is a root
    of the polynomial with reversed coefficients and
    :math:`\lambda_i = \frac{1}{\zeta_i}`.

    Finally, a companion matrix can be formed using the coefficients of the
    polynomial. Then the eigenvalues of that matrix give the roots of the
    polynomial. This last method is the one actually used.

    See Also
    --------
    companion_matrix
    """
    # First method:
    # np.all(np.abs(np.roots(np.r_[1, params])) < 1)
    # Second method:
    # np.all(np.abs(np.roots(np.r_[1, params][::-1])) > 1)
    # Final method:
    eigvals = np.linalg.eigvals(companion_matrix(polynomial))
    return np.all(np.abs(eigvals) < threshold)


def solve_discrete_lyapunov(a, q, complex_step=False):
    r"""
    Solves the discrete Lyapunov equation using a bilinear transformation.

    Notes
    -----
    This is a modification of the version in Scipy (see
    https://github.com/scipy/scipy/blob/master/scipy/linalg/_solvers.py)
    which allows passing through the complex numbers in the matrix a
    (usually the transition matrix) in order to allow complex step
    differentiation.
    """
    eye = np.eye(a.shape[0])
    if not complex_step:
        aH = a.conj().transpose()
        aHI_inv = np.linalg.inv(aH + eye)
        b = np.dot(aH - eye, aHI_inv)
        c = 2*np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
        return solve_sylvester(b.conj().transpose(), b, -c)
    else:
        aH = a.transpose()
        aHI_inv = np.linalg.inv(aH + eye)
        b = np.dot(aH - eye, aHI_inv)
        c = 2*np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv)
        return solve_sylvester(b.transpose(), b, -c)


def constrain_stationary_univariate(unconstrained):
    """
    Transform unconstrained parameters used by the optimizer to constrained
    parameters used in likelihood evaluation

    Parameters
    ----------
    unconstrained : array
        Unconstrained parameters used by the optimizer, to be transformed to
        stationary coefficients of, e.g., an autoregressive or moving average
        component.

    Returns
    -------
    constrained : array
        Constrained parameters of, e.g., an autoregressive or moving average
        component, to be transformed to arbitrary parameters used by the
        optimizer.

    References
    ----------
    .. [1] Monahan, John F. 1984.
       "A Note on Enforcing Stationarity in
       Autoregressive-moving Average Models."
       Biometrika 71 (2) (August 1): 403-404.
    """

    n = unconstrained.shape[0]
    y = np.zeros((n, n), dtype=unconstrained.dtype)
    r = unconstrained/((1 + unconstrained**2)**0.5)
    for k in range(n):
        for i in range(k):
            y[k, i] = y[k - 1, i] + r[k] * y[k - 1, k - i - 1]
        y[k, k] = r[k]
    return -y[n - 1, :]


def unconstrain_stationary_univariate(constrained):
    """
    Transform constrained parameters used in likelihood evaluation
    to unconstrained parameters used by the optimizer

    Parameters
    ----------
    constrained : array
        Constrained parameters of, e.g., an autoregressive or moving average
        component, to be transformed to arbitrary parameters used by the
        optimizer.

    Returns
    -------
    unconstrained : array
        Unconstrained parameters used by the optimizer, to be transformed to
        stationary coefficients of, e.g., an autoregressive or moving average
        component.

    References
    ----------
    .. [1] Monahan, John F. 1984.
       "A Note on Enforcing Stationarity in
       Autoregressive-moving Average Models."
       Biometrika 71 (2) (August 1): 403-404.
    """
    n = constrained.shape[0]
    y = np.zeros((n, n), dtype=constrained.dtype)
    y[n-1:] = -constrained
    for k in range(n-1, 0, -1):
        for i in range(k):
            y[k-1, i] = (y[k, i] - y[k, k]*y[k, k-i-1]) / (1 - y[k, k]**2)
    r = y.diagonal()
    x = r / ((1 - r**2)**0.5)
    return x


def _constrain_sv_less_than_one_python(unconstrained, order=None,
                                       k_endog=None):
    """
    Transform arbitrary matrices to matrices with singular values less than
    one.

    Parameters
    ----------
    unconstrained : list
        Arbitrary matrices. Should be a list of length `order`, where each
        element is an array sized `k_endog` x `k_endog`.
    order : integer, optional
        The order of the autoregression.
    k_endog : integer, optional
        The dimension of the data vector.

    Returns
    -------
    constrained : list
        Partial autocorrelation matrices. Should be a list of length
        `order`, where each element is an array sized `k_endog` x `k_endog`.

    Notes
    -----
    Corresponds to Lemma 2.2 in Ansley and Kohn (1986). See
    `constrain_stationary_multivariate` for more details.

    There is a Cython implementation of this function that can be much faster,
    but which requires SciPy 0.14.0 or greater. See
    `constrain_stationary_multivariate` for details.

    """

    from scipy import linalg

    constrained = []  # P_s,  s = 1, ..., p
    if order is None:
        order = len(unconstrained)
    if k_endog is None:
        k_endog = unconstrained[0].shape[0]

    eye = np.eye(k_endog)
    for i in range(order):
        A = unconstrained[i]
        B, lower = linalg.cho_factor(eye + np.dot(A, A.T), lower=True)
        constrained.append(linalg.solve_triangular(B, A, lower=lower))
    return constrained


def _compute_coefficients_from_multivariate_pacf_python(
        partial_autocorrelations, error_variance, transform_variance=False,
        order=None, k_endog=None):
    """
    Transform matrices with singular values less than one to matrices
    corresponding to a stationary (or invertible) process.

    Parameters
    ----------
    partial_autocorrelations : list
        Partial autocorrelation matrices. Should be a list of length `order`,
        where each element is an array sized `k_endog` x `k_endog`.
    error_variance : array
        The variance / covariance matrix of the error term. Should be sized
        `k_endog` x `k_endog`. This is used as input in the algorithm even if
        is not transformed by it (when `transform_variance` is False). The
        error term variance is required input when transformation is used
        either to force an autoregressive component to be stationary or to
        force a moving average component to be invertible.
    transform_variance : boolean, optional
        Whether or not to transform the error variance term. This option is
        not typically used, and the default is False.
    order : integer, optional
        The order of the autoregression.
    k_endog : integer, optional
        The dimension of the data vector.

    Returns
    -------
    coefficient_matrices : list
        Transformed coefficient matrices leading to a stationary VAR
        representation.

    Notes
    -----
    Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See
    `constrain_stationary_multivariate` for more details.

    There is a Cython implementation of this function that can be much faster,
    but which requires SciPy 0.14.0 or greater. See
    `constrain_stationary_multivariate` for details.
    """
    from scipy import linalg

    if order is None:
        order = len(partial_autocorrelations)
    if k_endog is None:
        k_endog = partial_autocorrelations[0].shape[0]

    # If we want to keep the provided variance but with the constrained
    # coefficient matrices, we need to make a copy here, and then after the
    # main loop we will transform the coefficients to match the passed variance
    if not transform_variance:
        initial_variance = error_variance
        # Need to make the input variance large enough that the recursions
        # don't lead to zero-matrices due to roundoff error, which would case
        # exceptions from the Cholesky decompositions.
        # Note that this will still not always ensure positive definiteness,
        # and for k_endog, order large enough an exception may still be raised
        error_variance = np.eye(k_endog) * (order + k_endog)**10

    forward_variances = [error_variance]   # \Sigma_s
    backward_variances = [error_variance]  # \Sigma_s^*,  s = 0, ..., p
    autocovariances = [error_variance]     # \Gamma_s
    # \phi_{s,k}, s = 1, ..., p
    #             k = 1, ..., s+1
    forwards = []
    # \phi_{s,k}^*
    backwards = []

    error_variance_factor = linalg.cholesky(error_variance, lower=True)

    forward_factors = [error_variance_factor]
    backward_factors = [error_variance_factor]

    # We fill in the entries as follows:
    # [1,1]
    # [2,2], [2,1]
    # [3,3], [3,1], [3,2]
    # ...
    # [p,p], [p,1], ..., [p,p-1]
    # the last row, correctly ordered, is then used as the coefficients
    for s in range(order):  # s = 0, ..., p-1
        prev_forwards = forwards
        prev_backwards = backwards
        forwards = []
        backwards = []

        # Create the "last" (k = s+1) matrix
        # Note: this is for k = s+1. However, below we then have to fill
        # in for k = 1, ..., s in order.
        # P L*^{-1} = x
        # x L* = P
        # L*' x' = P'
        forwards.append(
            linalg.solve_triangular(
                backward_factors[s], partial_autocorrelations[s].T,
                lower=True, trans='T'))
        forwards[0] = np.dot(forward_factors[s], forwards[0].T)

        # P' L^{-1} = x
        # x L = P'
        # L' x' = P
        backwards.append(
            linalg.solve_triangular(
                forward_factors[s], partial_autocorrelations[s],
                lower=True, trans='T'))
        backwards[0] = np.dot(backward_factors[s], backwards[0].T)

        # Update the variance
        # Note: if s >= 1, this will be further updated in the for loop
        # below
        # Also, this calculation will be re-used in the forward variance
        tmp = np.dot(forwards[0], backward_variances[s])
        autocovariances.append(tmp.copy().T)

        # Create the remaining k = 1, ..., s matrices,
        # only has an effect if s >= 1
        for k in range(s):
            forwards.insert(k, prev_forwards[k] - np.dot(
                forwards[-1], prev_backwards[s-(k+1)]))

            backwards.insert(k, prev_backwards[k] - np.dot(
                backwards[-1], prev_forwards[s-(k+1)]))

            autocovariances[s+1] += np.dot(autocovariances[k+1],
                                           prev_forwards[s-(k+1)].T)

        # Create forward and backwards variances
        forward_variances.append(
            forward_variances[s] - np.dot(tmp, forwards[s].T)
        )
        backward_variances.append(
            backward_variances[s] -
            np.dot(
                np.dot(backwards[s], forward_variances[s]),
                backwards[s].T
            )
        )

        # Cholesky factors
        forward_factors.append(
            linalg.cholesky(forward_variances[s+1], lower=True)
        )
        backward_factors.append(
            linalg.cholesky(backward_variances[s+1], lower=True)
        )

    # If we do not want to use the transformed variance, we need to
    # adjust the constrained matrices, as presented in Lemma 2.3, see above
    variance = forward_variances[-1]
    if not transform_variance:
        # Here, we need to construct T such that:
        # variance = T * initial_variance * T'
        # To do that, consider the Cholesky of variance (L) and
        # input_variance (M) to get:
        # L L' = T M M' T' = (TM) (TM)'
        # => L = T M
        # => L M^{-1} = T
        initial_variance_factor = np.linalg.cholesky(initial_variance)
        transformed_variance_factor = np.linalg.cholesky(variance)
        transform = np.dot(initial_variance_factor,
                           np.linalg.inv(transformed_variance_factor))
        inv_transform = np.linalg.inv(transform)

        for i in range(order):
            forwards[i] = (
                np.dot(np.dot(transform, forwards[i]), inv_transform)
            )

    return forwards, variance


def constrain_stationary_multivariate_python(unconstrained, error_variance,
                                             transform_variance=False,
                                             prefix=None):
    """
    Transform unconstrained parameters used by the optimizer to constrained
    parameters used in likelihood evaluation for a vector autoregression.

    Parameters
    ----------
    unconstrained : array or list
        Arbitrary matrices to be transformed to stationary coefficient matrices
        of the VAR. If a list, should be a list of length `order`, where each
        element is an array sized `k_endog` x `k_endog`. If an array, should be
        the matrices horizontally concatenated and sized
        `k_endog` x `k_endog * order`.
    error_variance : array
        The variance / covariance matrix of the error term. Should be sized
        `k_endog` x `k_endog`. This is used as input in the algorithm even if
        is not transformed by it (when `transform_variance` is False). The
        error term variance is required input when transformation is used
        either to force an autoregressive component to be stationary or to
        force a moving average component to be invertible.
    transform_variance : boolean, optional
        Whether or not to transform the error variance term. This option is
        not typically used, and the default is False.
    prefix : {'s','d','c','z'}, optional
        The appropriate BLAS prefix to use for the passed datatypes. Only
        use if absolutely sure that the prefix is correct or an error will
        result.

    Returns
    -------
    constrained : array or list
        Transformed coefficient matrices leading to a stationary VAR
        representation. Will match the type of the passed `unconstrained`
        variable (so if a list was passed, a list will be returned).

    Notes
    -----
    In the notation of [1]_, the arguments `(variance, unconstrained)` are
    written as :math:`(\Sigma, A_1, \dots, A_p)`, where :math:`p` is the order
    of the vector autoregression, and is here determined by the length of
    the `unconstrained` argument.

    There are two steps in the constraining algorithm.

    First, :math:`(A_1, \dots, A_p)` are transformed into
    :math:`(P_1, \dots, P_p)` via Lemma 2.2 of [1]_.

    Second, :math:`(\Sigma, P_1, \dots, P_p)` are transformed into
    :math:`(\Sigma, \phi_1, \dots, \phi_p)` via Lemmas 2.1 and 2.3 of [1]_.

    If `transform_variance=True`, then only Lemma 2.1 is applied in the second
    step.

    While this function can be used even in the univariate case, it is much
    slower, so in that case `constrain_stationary_univariate` is preferred.

    References
    ----------
    .. [1] Ansley, Craig F., and Robert Kohn. 1986.
       "A Note on Reparameterizing a Vector Autoregressive Moving Average Model
       to Enforce Stationarity."
       Journal of Statistical Computation and Simulation 24 (2): 99-106.
    .. [2] Ansley, Craig F, and Paul Newbold. 1979.
       "Multivariate Partial Autocorrelations."
       In Proceedings of the Business and Economic Statistics Section, 349-53.
       American Statistical Association

    """

    use_list = type(unconstrained) == list
    if not use_list:
        k_endog, order = unconstrained.shape
        order //= k_endog

        unconstrained = [
            unconstrained[:k_endog, i*k_endog:(i+1)*k_endog]
            for i in range(order)
        ]

    order = len(unconstrained)
    k_endog = unconstrained[0].shape[0]

    # Step 1: convert from arbitrary matrices to those with singular values
    # less than one.
    sv_constrained = _constrain_sv_less_than_one_python(
        unconstrained, order, k_endog)

    # Step 2: convert matrices from our "partial autocorrelation matrix" space
    # (matrices with singular values less than one) to the space of stationary
    # coefficient matrices
    constrained, var = _compute_coefficients_from_multivariate_pacf_python(
        sv_constrained, error_variance, transform_variance, order, k_endog)

    if not use_list:
        constrained = np.concatenate(constrained, axis=1).reshape(
            k_endog, k_endog * order)

    return constrained, var


# Conditionally use the Cython versions of the multivariate constraint if
# possible (i.e. if Scipy >= 0.14.0 is available.)
if has_trmm:

    def constrain_stationary_multivariate(unconstrained, variance,
                                          transform_variance=False,
                                          prefix=None):

        use_list = type(unconstrained) == list
        if use_list:
            unconstrained = np.concatenate(unconstrained, axis=1)

        k_endog, order = unconstrained.shape
        order //= k_endog

        if order < 1:
            raise ValueError('Must have order at least 1')
        if k_endog < 1:
            raise ValueError('Must have at least 1 endogenous variable')

        if prefix is None:
            prefix, dtype, _ = find_best_blas_type(
                [unconstrained, variance])
        dtype = prefix_dtype_map[prefix]

        unconstrained = np.asfortranarray(unconstrained, dtype=dtype)
        variance = np.asfortranarray(variance, dtype=dtype)

        # Step 1: convert from arbitrary matrices to those with singular values
        # less than one.
        # sv_constrained = _constrain_sv_less_than_one(unconstrained, order,
        #                                              k_endog, prefix)
        sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog)

        # Step 2: convert matrices from our "partial autocorrelation matrix"
        # space (matrices with singular values less than one) to the space of
        # stationary coefficient matrices
        constrained, variance = prefix_pacf_map[prefix](
            sv_constrained, variance, transform_variance, order, k_endog)

        constrained = np.array(constrained, dtype=dtype)
        variance = np.array(variance, dtype=dtype)

        if use_list:
            constrained = [
                constrained[:k_endog, i*k_endog:(i+1)*k_endog]
                for i in range(order)
            ]

        return constrained, variance
    constrain_stationary_multivariate.__doc__ = (
        constrain_stationary_multivariate_python.__doc__)

else:
    constrain_stationary_multivariate = (
        constrain_stationary_multivariate_python)


def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None):
    """
    Transform matrices with singular values less than one to arbitrary
    matrices.

    Parameters
    ----------
    constrained : list
        The partial autocorrelation matrices. Should be a list of length
        `order`, where each element is an array sized `k_endog` x `k_endog`.
    order : integer, optional
        The order of the autoregression.
    k_endog : integer, optional
        The dimension of the data vector.

    Returns
    -------
    unconstrained : list
        Unconstrained matrices. A list of length `order`, where each element is
        an array sized `k_endog` x `k_endog`.

    Notes
    -----
    Corresponds to the inverse of Lemma 2.2 in Ansley and Kohn (1986). See
    `unconstrain_stationary_multivariate` for more details.

    """
    from scipy import linalg

    unconstrained = []  # A_s,  s = 1, ..., p
    if order is None:
        order = len(constrained)
    if k_endog is None:
        k_endog = constrained[0].shape[0]

    eye = np.eye(k_endog)
    for i in range(order):
        P = constrained[i]
        # B^{-1} B^{-1}' = I - P P'
        B_inv, lower = linalg.cho_factor(eye - np.dot(P, P.T), lower=True)
        # A = BP
        # B^{-1} A = P
        unconstrained.append(linalg.solve_triangular(B_inv, P, lower=lower))
    return unconstrained


def _compute_multivariate_sample_acovf(endog, maxlag):
    """
    Computer multivariate sample autocovariances

    Parameters
    ----------
    endog : array_like
        Sample data on which to compute sample autocovariances. Shaped
        `nobs` x `k_endog`.

    Returns
    -------
    sample_autocovariances : list
        A list of the first `maxlag` sample autocovariance matrices. Each
        matrix is shaped `k_endog` x `k_endog`.

    Notes
    -----
    This function computes the forward sample autocovariances:

    .. math::

        \hat \Gamma(s) = \frac{1}{n} \sum_{t=1}^{n-s}
        (Z_t - \bar Z) (Z_{t+s} - \bar Z)'

    See page 353 of Wei (1990). This function is primarily implemented for
    checking the partial autocorrelation functions below, and so is quite slow.

    References
    ----------
    .. [1] Wei, William. 1990.
        Time Series Analysis : Univariate and Multivariate Methods.
       Boston: Pearson.
    """
    # Get the (demeaned) data as an array
    endog = np.array(endog)
    if endog.ndim == 1:
        endog = endog[:, np.newaxis]
    endog -= np.mean(endog, axis=0)

    # Dimensions
    nobs, k_endog = endog.shape

    sample_autocovariances = []
    for s in range(maxlag + 1):
        sample_autocovariances.append(np.zeros((k_endog, k_endog)))
        for t in range(nobs - s):
            sample_autocovariances[s] += np.outer(endog[t], endog[t+s])
        sample_autocovariances[s] /= nobs

    return sample_autocovariances


def _compute_multivariate_acovf_from_coefficients(
        coefficients, error_variance, maxlag=None,
        forward_autocovariances=False):
    """
    Compute multivariate autocovariances from vector autoregression coefficient
    matrices

    Parameters
    ----------
    coefficients : array or list
        The coefficients matrices. If a list, should be a list of length
        `order`, where each element is an array sized `k_endog` x `k_endog`. If
        an array, should be the coefficient matrices horizontally concatenated
        and sized `k_endog` x `k_endog * order`.
    error_variance : array
        The variance / covariance matrix of the error term. Should be sized
        `k_endog` x `k_endog`.
    maxlag : integer, optional
        The maximum autocovariance to compute. Default is `order`-1. Can be
        zero, in which case it returns the variance.
    forward_autocovariances : boolean, optional
        Whether or not to compute forward autocovariances
        :math:`E(y_t y_{t+j}')`. Default is False, so that backward
        autocovariances :math:`E(y_t y_{t-j}')` are returned.

    Returns
    -------
    autocovariances : list
        A list of the first `maxlag` autocovariance matrices. Each matrix is
        shaped `k_endog` x `k_endog`.

    Notes
    -----
    Computes

    ..math::

        \Gamma(j) = E(y_t y_{t-j}')

    for j = 1, ..., `maxlag`, unless `forward_autocovariances` is specified,
    in which case it computes:

    ..math::

        E(y_t y_{t+j}') = \Gamma(j)'

    Coefficients are assumed to be provided from the VAR model:

    .. math::
        y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t

    Autocovariances are calculated by solving the associated discrete Lyapunov
    equation of the state space representation of the VAR process.

    """
    from scipy import linalg

    # Convert coefficients to a list of matrices, for use in
    # `companion_matrix`; get dimensions
    if type(coefficients) == list:
        order = len(coefficients)
        k_endog = coefficients[0].shape[0]
    else:
        k_endog, order = coefficients.shape
        order //= k_endog

        coefficients = [
            coefficients[:k_endog, i*k_endog:(i+1)*k_endog]
            for i in range(order)
        ]

    if maxlag is None:
        maxlag = order-1

    # Start with VAR(p): w_{t+1} = phi_1 w_t + ... + phi_p w_{t-p+1} + u_{t+1}
    # Then stack the VAR(p) into a VAR(1) in companion matrix form:
    # z_{t+1} = F z_t + v_t
    companion = companion_matrix(
        [1] + [-coefficients[i] for i in range(order)]
    ).T

    # Compute the error variance matrix for the stacked form: E v_t v_t'
    selected_variance = np.zeros(companion.shape)
    selected_variance[:k_endog, :k_endog] = error_variance

    # Compute the unconditional variance of z_t: E z_t z_t'
    stacked_cov = linalg.solve_discrete_lyapunov(companion, selected_variance)

    # The first (block) row of the variance of z_t gives the first p-1
    # autocovariances of w_t: \Gamma_i = E w_t w_t+i with \Gamma_0 = Var(w_t)
    # Note: these are okay, checked against ArmaProcess
    autocovariances = [
        stacked_cov[:k_endog, i*k_endog:(i+1)*k_endog]
        for i in range(min(order, maxlag+1))
    ]

    for i in range(maxlag - (order-1)):
        stacked_cov = np.dot(companion, stacked_cov)
        autocovariances += [
            stacked_cov[:k_endog, -k_endog:]
        ]

    if forward_autocovariances:
        for i in range(len(autocovariances)):
            autocovariances[i] = autocovariances[i].T

    return autocovariances


def _compute_multivariate_sample_pacf(endog, maxlag):
    """
    Computer multivariate sample partial autocorrelations

    Parameters
    ----------
    endog : array_like
        Sample data on which to compute sample autocovariances. Shaped
        `nobs` x `k_endog`.
    maxlag : integer
        Maximum lag for which to calculate sample partial autocorrelations.

    Returns
    -------
    sample_pacf : list
        A list of the first `maxlag` sample partial autocorrelation matrices.
        Each matrix is shaped `k_endog` x `k_endog`.

    """
    sample_autocovariances = _compute_multivariate_sample_acovf(endog, maxlag)

    return _compute_multivariate_pacf_from_autocovariances(
        sample_autocovariances)


def _compute_multivariate_pacf_from_autocovariances(autocovariances,
                                                    order=None, k_endog=None):
    """
    Compute multivariate partial autocorrelations from autocovariances.

    Parameters
    ----------
    autocovariances : list
        Autocorrelations matrices. Should be a list of length `order` + 1,
        where each element is an array sized `k_endog` x `k_endog`.
    order : integer, optional
        The order of the autoregression.
    k_endog : integer, optional
        The dimension of the data vector.

    Returns
    -------
    pacf : list
        List of first `order` multivariate partial autocorrelations.

    Notes
    -----
    Note that this computes multivariate partial autocorrelations.

    Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See
    `unconstrain_stationary_multivariate` for more details.

    Notes
    -----
    Computes sample partial autocorrelations if sample autocovariances are
    given.

    """
    from scipy import linalg

    if order is None:
        order = len(autocovariances)-1
    if k_endog is None:
        k_endog = autocovariances[0].shape[0]

    # Now apply the Ansley and Kohn (1986) algorithm, except that instead of
    # calculating phi_{s+1, s+1} = L_s P_{s+1} {L_s^*}^{-1} (which requires
    # the partial autocorrelation P_{s+1} which is what we're trying to
    # calculate here), we calculate it as in Ansley and Newbold (1979), using
    # the autocovariances \Gamma_s and the forwards and backwards residual
    # variances \Sigma_s, \Sigma_s^*:
    # phi_{s+1, s+1} = [ \Gamma_{s+1}' - \phi_{s,1} \Gamma_s' - ... -
    #                    \phi_{s,s} \Gamma_1' ] {\Sigma_s^*}^{-1}

    # Forward and backward variances
    forward_variances = []   # \Sigma_s
    backward_variances = []  # \Sigma_s^*,  s = 0, ..., p
    # \phi_{s,k}, s = 1, ..., p
    #             k = 1, ..., s+1
    forwards = []
    # \phi_{s,k}^*
    backwards = []

    forward_factors = []   # L_s
    backward_factors = []  # L_s^*,  s = 0, ..., p

    # Ultimately we want to construct the partial autocorrelation matrices
    # Note that this is "1-indexed" in the sense that it stores P_1, ... P_p
    # rather than starting with P_0.
    partial_autocorrelations = []

    # We fill in the entries of phi_{s,k} as follows:
    # [1,1]
    # [2,2], [2,1]
    # [3,3], [3,1], [3,2]
    # ...
    # [p,p], [p,1], ..., [p,p-1]
    # the last row, correctly ordered, should be the same as the coefficient
    # matrices provided in the argument `constrained`
    for s in range(order):  # s = 0, ..., p-1
        prev_forwards = list(forwards)
        prev_backwards = list(backwards)
        forwards = []
        backwards = []

        # Create forward and backwards variances Sigma_s, Sigma*_s
        forward_variance = autocovariances[0].copy()
        backward_variance = autocovariances[0].T.copy()

        for k in range(s):
            forward_variance -= np.dot(prev_forwards[k],
                                       autocovariances[k+1])
            backward_variance -= np.dot(prev_backwards[k],
                                        autocovariances[k+1].T)

        forward_variances.append(forward_variance)
        backward_variances.append(backward_variance)

        # Cholesky factors
        forward_factors.append(
            linalg.cholesky(forward_variances[s], lower=True)
        )
        backward_factors.append(
            linalg.cholesky(backward_variances[s], lower=True)
        )

        # Create the intermediate sum term
        if s == 0:
            # phi_11 = \Gamma_1' \Gamma_0^{-1}
            # phi_11 \Gamma_0 = \Gamma_1'
            # \Gamma_0 phi_11' = \Gamma_1
            forwards.append(linalg.cho_solve(
                (forward_factors[0], True), autocovariances[1]).T)
            # backwards.append(forwards[-1])
            # phi_11_star = \Gamma_1 \Gamma_0^{-1}
            # phi_11_star \Gamma_0 = \Gamma_1
            # \Gamma_0 phi_11_star' = \Gamma_1'
            backwards.append(linalg.cho_solve(
                (backward_factors[0], True), autocovariances[1].T).T)
        else:
            # G := \Gamma_{s+1}' -
            #      \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1'
            tmp_sum = autocovariances[s+1].T.copy()

            for k in range(s):
                tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T)

            # Create the "last" (k = s+1) matrix
            # Note: this is for k = s+1. However, below we then have to
            # fill in for k = 1, ..., s in order.
            # phi = G Sigma*^{-1}
            # phi Sigma* = G
            # Sigma*' phi' = G'
            # Sigma* phi' = G'
            # (because Sigma* is symmetric)
            forwards.append(linalg.cho_solve(
                (backward_factors[s], True), tmp_sum.T).T)

            # phi = G' Sigma^{-1}
            # phi Sigma = G'
            # Sigma' phi' = G
            # Sigma phi' = G
            # (because Sigma is symmetric)
            backwards.append(linalg.cho_solve(
                (forward_factors[s], True), tmp_sum).T)

        # Create the remaining k = 1, ..., s matrices,
        # only has an effect if s >= 1
        for k in range(s):
            forwards.insert(k, prev_forwards[k] - np.dot(
                forwards[-1], prev_backwards[s-(k+1)]))
            backwards.insert(k, prev_backwards[k] - np.dot(
                backwards[-1], prev_forwards[s-(k+1)]))

        # Partial autocorrelation matrix: P_{s+1}
        # P = L^{-1} phi L*
        # L P = (phi L*)
        partial_autocorrelations.append(linalg.solve_triangular(
            forward_factors[s], np.dot(forwards[s], backward_factors[s]),
            lower=True))

    return partial_autocorrelations


def _compute_multivariate_pacf_from_coefficients(constrained, error_variance,
                                                 order=None, k_endog=None):
    """
    Transform matrices corresponding to a stationary (or invertible) process
    to matrices with singular values less than one.

    Parameters
    ----------
    constrained : array or list
        The coefficients matrices. If a list, should be a list of length
        `order`, where each element is an array sized `k_endog` x `k_endog`. If
        an array, should be the coefficient matrices horizontally concatenated
        and sized `k_endog` x `k_endog * order`.
    error_variance : array
        The variance / covariance matrix of the error term. Should be sized
        `k_endog` x `k_endog`.
    order : integer, optional
        The order of the autoregression.
    k_endog : integer, optional
        The dimension of the data vector.

    Returns
    -------
    pacf : list
        List of first `order` multivariate partial autocorrelations.

    Notes
    -----
    Note that this computes multivariate partial autocorrelations.

    Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See
    `unconstrain_stationary_multivariate` for more details.

    Notes
    -----

    Coefficients are assumed to be provided from the VAR model:

    .. math::
        y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t
    """

    if type(constrained) == list:
        order = len(constrained)
        k_endog = constrained[0].shape[0]
    else:
        k_endog, order = constrained.shape
        order //= k_endog

    # Get autocovariances for the process; these are defined to be
    # E z_t z_{t-j}'
    # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')'
    _acovf = _compute_multivariate_acovf_from_coefficients

    autocovariances = [
        autocovariance.T for autocovariance in
        _acovf(constrained, error_variance, maxlag=order)]

    return _compute_multivariate_pacf_from_autocovariances(autocovariances)


def unconstrain_stationary_multivariate(constrained, error_variance):
    """
    Transform constrained parameters used in likelihood evaluation
    to unconstrained parameters used by the optimizer

    Parameters
    ----------
    constrained : array or list
        Constrained parameters of, e.g., an autoregressive or moving average
        component, to be transformed to arbitrary parameters used by the
        optimizer. If a list, should be a list of length `order`, where each
        element is an array sized `k_endog` x `k_endog`. If an array, should be
        the coefficient matrices horizontally concatenated and sized
        `k_endog` x `k_endog * order`.
    error_variance : array
        The variance / covariance matrix of the error term. Should be sized
        `k_endog` x `k_endog`. This is used as input in the algorithm even if
        is not transformed by it (when `transform_variance` is False).

    Returns
    -------
    unconstrained : array
        Unconstrained parameters used by the optimizer, to be transformed to
        stationary coefficients of, e.g., an autoregressive or moving average
        component. Will match the type of the passed `constrained`
        variable (so if a list was passed, a list will be returned).

    Notes
    -----
    Uses the list representation internally, even if an array is passed.

    References
    ----------
    .. [1] Ansley, Craig F., and Robert Kohn. 1986.
       "A Note on Reparameterizing a Vector Autoregressive Moving Average Model
       to Enforce Stationarity."
       Journal of Statistical Computation and Simulation 24 (2): 99-106.

    """

    from scipy import linalg

    use_list = type(constrained) == list
    if not use_list:
        k_endog, order = constrained.shape
        order //= k_endog

        constrained = [
            constrained[:k_endog, i*k_endog:(i+1)*k_endog]
            for i in range(order)
        ]
    else:
        order = len(constrained)
        k_endog = constrained[0].shape[0]

    # Step 1: convert matrices from the space of stationary
    # coefficient matrices to our "partial autocorrelation matrix" space
    # (matrices with singular values less than one)
    partial_autocorrelations = _compute_multivariate_pacf_from_coefficients(
        constrained, error_variance, order, k_endog)

    # Step 2: convert from arbitrary matrices to those with singular values
    # less than one.
    unconstrained = _unconstrain_sv_less_than_one(
        partial_autocorrelations, order, k_endog)

    if not use_list:
        unconstrained = np.concatenate(unconstrained, axis=1)

    return unconstrained, error_variance


def validate_matrix_shape(name, shape, nrows, ncols, nobs):
    """
    Validate the shape of a possibly time-varying matrix, or raise an exception

    Parameters
    ----------
    name : str
        The name of the matrix being validated (used in exception messages)
    shape : array_like
        The shape of the matrix to be validated. May be of size 2 or (if
        the matrix is time-varying) 3.
    nrows : int
        The expected number of rows.
    ncols : int
        The expected number of columns.
    nobs : int
        The number of observations (used to validate the last dimension of a
        time-varying matrix)

    Raises
    ------
    ValueError
        If the matrix is not of the desired shape.
    """
    ndim = len(shape)

    # Enforce dimension
    if ndim not in [2, 3]:
        raise ValueError('Invalid value for %s matrix. Requires a'
                         ' 2- or 3-dimensional array, got %d dimensions' %
                         (name, ndim))
    # Enforce the shape of the matrix
    if not shape[0] == nrows:
        raise ValueError('Invalid dimensions for %s matrix: requires %d'
                         ' rows, got %d' % (name, nrows, shape[0]))
    if not shape[1] == ncols:
        raise ValueError('Invalid dimensions for %s matrix: requires %d'
                         ' columns, got %d' % (name, ncols, shape[1]))

    # If we don't yet know `nobs`, don't allow time-varying arrays
    if nobs is None and not (ndim == 2 or shape[-1] == 1):
        raise ValueError('Invalid dimensions for %s matrix: time-varying'
                         ' matrices cannot be given unless `nobs` is specified'
                         ' (implicitly when a dataset is bound or else set'
                         ' explicity)' % name)

    # Enforce time-varying array size
    if ndim == 3 and nobs is not None and not shape[-1] in [1, nobs]:
        raise ValueError('Invalid dimensions for time-varying %s'
                         ' matrix. Requires shape (*,*,%d), got %s' %
                         (name, nobs, str(shape)))


def validate_vector_shape(name, shape, nrows, nobs):
    """
    Validate the shape of a possibly time-varying vector, or raise an exception

    Parameters
    ----------
    name : str
        The name of the vector being validated (used in exception messages)
    shape : array_like
        The shape of the vector to be validated. May be of size 1 or (if
        the vector is time-varying) 2.
    nrows : int
        The expected number of rows (elements of the vector).
    nobs : int
        The number of observations (used to validate the last dimension of a
        time-varying vector)

    Raises
    ------
    ValueError
        If the vector is not of the desired shape.
    """
    ndim = len(shape)
    # Enforce dimension
    if ndim not in [1, 2]:
        raise ValueError('Invalid value for %s vector. Requires a'
                         ' 1- or 2-dimensional array, got %d dimensions' %
                         (name, ndim))
    # Enforce the shape of the vector
    if not shape[0] == nrows:
        raise ValueError('Invalid dimensions for %s vector: requires %d'
                         ' rows, got %d' % (name, nrows, shape[0]))

    # If we don't yet know `nobs`, don't allow time-varying arrays
    if nobs is None and not (ndim == 1 or shape[-1] == 1):
        raise ValueError('Invalid dimensions for %s vector: time-varying'
                         ' vectors cannot be given unless `nobs` is specified'
                         ' (implicitly when a dataset is bound or else set'
                         ' explicity)' % name)

    # Enforce time-varying array size
    if ndim == 2 and not shape[1] in [1, nobs]:
        raise ValueError('Invalid dimensions for time-varying %s'
                         ' vector. Requires shape (*,%d), got %s' %
                         (name, nobs, str(shape)))
