# Source code for filterpy.kalman.sigma_points

    # -*- coding: utf-8 -*-
    # pylint: disable=invalid-name, too-many-instance-attributes

    """Copyright 2015 Roger R Labbe Jr.

    FilterPy library.
    http://github.com/rlabbe/filterpy

    Documentation at:
    https://filterpy.readthedocs.org

    Supporting book at:
    https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

    This is licensed under an MIT license. See the readme.MD file
    for more information.
    """

    from __future__ import division
    import numpy as np
    from scipy.linalg import cholesky
    from filterpy.common import pretty_str

    class MerweScaledSigmaPoints(object):

        """
        Generates sigma points and weights according to Van der Merwe's
        2004 dissertation[1] for the UnscentedKalmanFilter class.. It
        parametizes the sigma points using alpha, beta, kappa terms, and
        is the version seen in most publications.

        Unless you know better, this should be your default choice.

        Parameters
        ----------

        n : int
            Dimensionality of the state. 2n+1 weights will be generated.

        alpha : float
            Determins the spread of the sigma points around the mean.
            Usually a small positive value (1e-3) according to [3].

        beta : float
            Incorporates prior knowledge of the distribution of the mean. For
            Gaussian x beta=2 is optimal, according to [3].

        kappa : float, default=0.0
            Secondary scaling parameter usually set to 0 according to [4],
            or to 3-n according to [5].

        sqrt_method : function(ndarray), default=scipy.linalg.cholesky
            Defines how we compute the square root of a matrix, which has
            no unique answer. Cholesky is the default choice due to its
            speed. Typically your alternative choice will be
            scipy.linalg.sqrtm. Different choices affect how the sigma points
            are arranged relative to the eigenvectors of the covariance matrix.
            Usually this will not matter to you; if so the default cholesky()
            yields maximal performance. As of van der Merwe's dissertation of
            2004 [6] this was not a well reseached area so I have no advice
            to give you.

            If your method returns a triangular matrix it must be upper
            triangular. Do not use numpy.linalg.cholesky - for historical
            reasons it returns a lower triangular matrix. The SciPy version
            does the right thing.

        subtract : callable (x, y), optional
            Function that computes the difference between x and y.
            You will have to supply this if your state variable cannot support
            subtraction, such as angles (359-1 degreees is 2, not 358). x and y
            are state vectors, not scalars.

        Attributes
        ----------

        Wm : np.array
            weight for each sigma point for the mean

        Wc : np.array
            weight for each sigma point for the covariance

        Examples
        --------

        See my book Kalman and Bayesian Filters in Python
        https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

        References
        ----------

        .. [1] R. Van der Merwe "Sigma-Point Kalman Filters for Probabilitic
               Inference in Dynamic State-Space Models" (Doctoral dissertation)

        """

        def __init__(self, n, alpha, beta, kappa, sqrt_method=None, subtract=None):
            #pylint: disable=too-many-arguments

            self.n = n
            self.alpha = alpha
            self.beta = beta
            self.kappa = kappa
            if sqrt_method is None:
                self.sqrt = cholesky
            else:
                self.sqrt = sqrt_method

            if subtract is None:
                self.subtract = np.subtract
            else:
                self.subtract = subtract

            self._compute_weights()

        def num_sigmas(self):
            """ Number of sigma points for each variable in the state x"""
            return 2*self.n + 1

        def sigma_points(self, x, P):
            """ Computes the sigma points for an unscented Kalman filter
            given the mean (x) and covariance(P) of the filter.
            Returns tuple of the sigma points and weights.

            Works with both scalar and array inputs:
            sigma_points (5, 9, 2) # mean 5, covariance 9
            sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

            Parameters
            ----------

            x : An array-like object of the means of length n
                Can be a scalar if 1D.
                examples: 1, [1,2], np.array([1,2])

            P : scalar, or np.array
               Covariance of the filter. If scalar, is treated as eye(n)*P.

            Returns
            -------

            sigmas : np.array, of size (n, 2n+1)
                Two dimensional array of sigma points. Each column contains all of
                the sigmas for one dimension in the problem space.

                Ordered by Xi_0, Xi_{1..n}, Xi_{n+1..2n}
            """

            if self.n != np.size(x):
                raise ValueError("expected size(x) {}, but size is {}".format(
                    self.n, np.size(x)))

            n = self.n

            if np.isscalar(x):
                x = np.asarray([x])

            if  np.isscalar(P):
                P = np.eye(n)*P
            else:
                P = np.atleast_2d(P)

            lambda_ = self.alpha**2 * (n + self.kappa) - n
            U = self.sqrt((lambda_ + n)*P)

            sigmas = np.zeros((2*n+1, n))
            sigmas[0] = x
            for k in range(n):
                # pylint: disable=bad-whitespace
                sigmas[k+1]   = self.subtract(x, -U[k])
                sigmas[n+k+1] = self.subtract(x, U[k])

            return sigmas

        def _compute_weights(self):
            """ Computes the weights for the scaled unscented Kalman filter.

            """

            n = self.n
            lambda_ = self.alpha**2 * (n +self.kappa) - n

            c = .5 / (n + lambda_)
            self.Wc = np.full(2*n + 1, c)
            self.Wm = np.full(2*n + 1, c)
            self.Wc[0] = lambda_ / (n + lambda_) + (1 - self.alpha**2 + self.beta)
            self.Wm[0] = lambda_ / (n + lambda_)

        def __repr__(self):

            return '\n'.join([
                'MerweScaledSigmaPoints object',
                pretty_str('n', self.n),
                pretty_str('alpha', self.alpha),
                pretty_str('beta', self.beta),
                pretty_str('kappa', self.kappa),
                pretty_str('Wm', self.Wm),
                pretty_str('Wc', self.Wc),
                pretty_str('subtract', self.subtract),
                pretty_str('sqrt', self.sqrt)
                ])

    class JulierSigmaPoints(object):
        """
        Generates sigma points and weights according to Simon J. Julier
        and Jeffery K. Uhlmann's original paper[1]. It parametizes the sigma
        points using kappa.

        Parameters
        ----------

        n : int
            Dimensionality of the state. 2n+1 weights will be generated.

        kappa : float, default=0.
            Scaling factor that can reduce high order errors. kappa=0 gives
            the standard unscented filter. According to [Julier], if you set
            kappa to 3-dim_x for a Gaussian x you will minimize the fourth
            order errors in x and P.

        sqrt_method : function(ndarray), default=scipy.linalg.cholesky
            Defines how we compute the square root of a matrix, which has
            no unique answer. Cholesky is the default choice due to its
            speed. Typically your alternative choice will be
            scipy.linalg.sqrtm. Different choices affect how the sigma points
            are arranged relative to the eigenvectors of the covariance matrix.
            Usually this will not matter to you; if so the default cholesky()
            yields maximal performance. As of van der Merwe's dissertation of
            2004 [6] this was not a well reseached area so I have no advice
            to give you.

            If your method returns a triangular matrix it must be upper
            triangular. Do not use numpy.linalg.cholesky - for historical
            reasons it returns a lower triangular matrix. The SciPy version
            does the right thing.

        subtract : callable (x, y), optional
            Function that computes the difference between x and y.
            You will have to supply this if your state variable cannot support
            subtraction, such as angles (359-1 degreees is 2, not 358). x and y

        Attributes
        ----------

        Wm : np.array
            weight for each sigma point for the mean

        Wc : np.array
            weight for each sigma point for the covariance

        References
        ----------

        .. [1] Julier, Simon J.; Uhlmann, Jeffrey "A New Extension of the Kalman
            Filter to Nonlinear Systems". Proc. SPIE 3068, Signal Processing,
            Sensor Fusion, and Target Recognition VI, 182 (July 28, 1997)
       """

        def __init__(self, n, kappa=0., sqrt_method=None, subtract=None):

            self.n = n
            self.kappa = kappa
            if sqrt_method is None:
                self.sqrt = cholesky
            else:
                self.sqrt = sqrt_method

            if subtract is None:
                self.subtract = np.subtract
            else:
                self.subtract = subtract

            self._compute_weights()

        def num_sigmas(self):
            """ Number of sigma points for each variable in the state x"""
            return 2*self.n + 1

        def sigma_points(self, x, P):
            r""" Computes the sigma points for an unscented Kalman filter
            given the mean (x) and covariance(P) of the filter.
            kappa is an arbitrary constant. Returns sigma points.

            Works with both scalar and array inputs:
            sigma_points (5, 9, 2) # mean 5, covariance 9
            sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

            Parameters
            ----------

            x : array-like object of the means of length n
                Can be a scalar if 1D.
                examples: 1, [1,2], np.array([1,2])

            P : scalar, or np.array
               Covariance of the filter. If scalar, is treated as eye(n)*P.

            kappa : float
                Scaling factor.

            Returns
            -------

            sigmas : np.array, of size (n, 2n+1)
                2D array of sigma points :math:`\chi`. Each column contains all of
                the sigmas for one dimension in the problem space. They
                are ordered as:

                .. math::
                    :nowrap:

                    \begin{eqnarray}
                      \chi[0]    = &x \\
                      \chi[1..n] = &x + [\sqrt{(n+\kappa)P}]_k \\
                      \chi[n+1..2n] = &x - [\sqrt{(n+\kappa)P}]_k
                    \end{eqnarray}

            """

            if self.n != np.size(x):
                raise ValueError("expected size(x) {}, but size is {}".format(
                    self.n, np.size(x)))

            n = self.n

            if np.isscalar(x):
                x = np.asarray([x])

            n = np.size(x)  # dimension of problem

            if np.isscalar(P):
                P = np.eye(n) * P
            else:
                P = np.atleast_2d(P)

            sigmas = np.zeros((2*n+1, n))

            # implements U'*U = (n+kappa)*P. Returns lower triangular matrix.
            # Take transpose so we can access with U[i]
            U = self.sqrt((n + self.kappa) * P)

            sigmas[0] = x
            for k in range(n):
                # pylint: disable=bad-whitespace
                sigmas[k+1]   = self.subtract(x, -U[k])
                sigmas[n+k+1] = self.subtract(x, U[k])
            return sigmas

        def _compute_weights(self):
            """ Computes the weights for the unscented Kalman filter. In this
            formulation the weights for the mean and covariance are the same.
            """

            n = self.n
            k = self.kappa

            self.Wm = np.full(2*n+1, .5 / (n + k))
            self.Wm[0] = k / (n+k)
            self.Wc = self.Wm

        def __repr__(self):

            return '\n'.join([
                'JulierSigmaPoints object',
                pretty_str('n', self.n),
                pretty_str('kappa', self.kappa),
                pretty_str('Wm', self.Wm),
                pretty_str('Wc', self.Wc),
                pretty_str('subtract', self.subtract),
                pretty_str('sqrt', self.sqrt)
                ])

    class SimplexSigmaPoints(object):

        """
        Generates sigma points and weights according to the simplex
        method presented in [1].

        Parameters
        ----------

        n : int
            Dimensionality of the state. n+1 weights will be generated.

        sqrt_method : function(ndarray), default=scipy.linalg.cholesky
            Defines how we compute the square root of a matrix, which has
            no unique answer. Cholesky is the default choice due to its
            speed. Typically your alternative choice will be
            scipy.linalg.sqrtm

            If your method returns a triangular matrix it must be upper
            triangular. Do not use numpy.linalg.cholesky - for historical
            reasons it returns a lower triangular matrix. The SciPy version
            does the right thing.

        subtract : callable (x, y), optional
            Function that computes the difference between x and y.
            You will have to supply this if your state variable cannot support
            subtraction, such as angles (359-1 degreees is 2, not 358). x and y
            are state vectors, not scalars.

        Attributes
        ----------

        Wm : np.array
            weight for each sigma point for the mean

        Wc : np.array
            weight for each sigma point for the covariance

        References
        ----------

        .. [1] Phillippe Moireau and Dominique Chapelle "Reduced-Order
               Unscented Kalman Filtering with Application to Parameter
               Identification in Large-Dimensional Systems"
               DOI: 10.1051/cocv/2010006
        """

        def __init__(self, n, alpha=1, sqrt_method=None, subtract=None):
            self.n = n
            self.alpha = alpha
            if sqrt_method is None:
                self.sqrt = cholesky
            else:
                self.sqrt = sqrt_method

            if subtract is None:
                self.subtract = np.subtract
            else:
                self.subtract = subtract

            self._compute_weights()

        def num_sigmas(self):
            """ Number of sigma points for each variable in the state x"""
            return self.n + 1

        def sigma_points(self, x, P):
            """
            Computes the implex sigma points for an unscented Kalman filter
            given the mean (x) and covariance(P) of the filter.
            Returns tuple of the sigma points and weights.

            Works with both scalar and array inputs:
            sigma_points (5, 9, 2) # mean 5, covariance 9
            sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I

            Parameters
            ----------

            x : An array-like object of the means of length n
                Can be a scalar if 1D.
                examples: 1, [1,2], np.array([1,2])

            P : scalar, or np.array
               Covariance of the filter. If scalar, is treated as eye(n)*P.

            Returns
            -------

            sigmas : np.array, of size (n, n+1)
                Two dimensional array of sigma points. Each column contains all of
                the sigmas for one dimension in the problem space.

                Ordered by Xi_0, Xi_{1..n}
            """

            if self.n != np.size(x):
                raise ValueError("expected size(x) {}, but size is {}".format(
                    self.n, np.size(x)))

            n = self.n

            if np.isscalar(x):
                x = np.asarray([x])
            x = x.reshape(-1, 1)

            if np.isscalar(P):
                P = np.eye(n) * P
            else:
                P = np.atleast_2d(P)

            U = self.sqrt(P)

            lambda_ = n / (n + 1)
            Istar = np.array([[-1/np.sqrt(2*lambda_), 1/np.sqrt(2*lambda_)]])
            for d in range(2, n+1):
                row = np.ones((1, Istar.shape[1] + 1)) * 1. / np.sqrt(lambda_*d*(d + 1))
                row[0, -1] = -d / np.sqrt(lambda_ * d * (d + 1))
                Istar = np.r_[np.c_[Istar, np.zeros((Istar.shape[0]))], row]

            I = np.sqrt(n)*Istar
            scaled_unitary = U.dot(I)

            sigmas = self.subtract(x, -scaled_unitary)
            return sigmas.T

        def _compute_weights(self):
            """ Computes the weights for the scaled unscented Kalman filter. """

            n = self.n
            c = 1. / (n + 1)
            self.Wm = np.full(n + 1, c)
            self.Wc = self.Wm

        def __repr__(self):
            return '\n'.join([
                'SimplexSigmaPoints object',
                pretty_str('n', self.n),
                pretty_str('alpha', self.alpha),
                pretty_str('Wm', self.Wm),
                pretty_str('Wc', self.Wc),
                pretty_str('subtract', self.subtract),
                pretty_str('sqrt', self.sqrt)
                ])