'''Recipes for more efficient work with linalg using classes


intended for use for multivariate normal and linear regression
calculations

x  is the data (nobs, nvars)
m  is the moment matrix (x'x) or a covariance matrix Sigma

examples:
x'sigma^{-1}x
z = Px  where P=Sigma^{-1/2}  or P=Sigma^{1/2}

Initially assume positive definite, then add spectral cutoff and
regularization of moment matrix, and extend to PCA

maybe extend to sparse if some examples work out
(transformation matrix P for random effect and for toeplitz)


Author: josef-pktd
Created on 2010-10-20
'''

import numpy as np
from scipy import linalg

from statsmodels.tools.decorators import cache_readonly


class PlainMatrixArray(object):
    '''Class that defines linalg operation on an array

    simplest version as benchmark

    linear algebra recipes for multivariate normal and linear
    regression calculations

    '''
    def __init__(self, data=None, sym=None):
        if data is not None:
            if sym is None:
                self.x = np.asarray(data)
                self.m = np.dot(self.x.T, self.x)
            else:
                raise ValueError('data and sym cannot be both given')
        elif sym is not None:
            self.m = np.asarray(sym)
            self.x = np.eye(*self.m.shape) #default

        else:
            raise ValueError('either data or sym need to be given')

    @cache_readonly
    def minv(self):
        return np.linalg.inv(self.m)

    def m_y(self, y):
        return np.dot(self.m, y)

    def minv_y(self, y):
        return np.dot(self.minv, y)

    @cache_readonly
    def mpinv(self):
        return linalg.pinv(self.m)

    @cache_readonly
    def xpinv(self):
        return linalg.pinv(self.x)

    def yt_m_y(self, y):
        return np.dot(y.T, np.dot(self.m, y))

    def yt_minv_y(self, y):
        return np.dot(y.T, np.dot(self.minv, y))

    #next two are redundant
    def y_m_yt(self, y):
        return np.dot(y, np.dot(self.m, y.T))

    def y_minv_yt(self, y):
        return np.dot(y, np.dot(self.minv, y.T))

    @cache_readonly
    def mdet(self):
        return linalg.det(self.m)

    @cache_readonly
    def mlogdet(self):
        return np.log(linalg.det(self.m))

    @cache_readonly
    def meigh(self):
        evals, evecs = linalg.eigh(self.m)
        sortind = np.argsort(evals)[::-1]
        return evals[sortind], evecs[:,sortind]

    @cache_readonly
    def mhalf(self):
        evals, evecs = self.meigh
        return np.dot(np.diag(evals**0.5), evecs.T)
        #return np.dot(evecs, np.dot(np.diag(evals**0.5), evecs.T))
        #return np.dot(evecs, 1./np.sqrt(evals) * evecs.T))

    @cache_readonly
    def minvhalf(self):
        evals, evecs = self.meigh
        return np.dot(evecs, 1./np.sqrt(evals) * evecs.T)



class SvdArray(PlainMatrixArray):
    '''Class that defines linalg operation on an array

    svd version, where svd is taken on original data array, if
    or when it matters

    no spectral cutoff in first version
    '''

    def __init__(self, data=None, sym=None):
        super(SvdArray, self).__init__(data=data, sym=sym)

        u, s, v = np.linalg.svd(self.x, full_matrices=1)
        self.u, self.s, self.v = u, s, v
        self.sdiag = linalg.diagsvd(s, *x.shape)
        self.sinvdiag = linalg.diagsvd(1./s, *x.shape)

    def _sdiagpow(self, p):
        return linalg.diagsvd(np.power(self.s, p), *x.shape)

    @cache_readonly
    def minv(self):
        sinvv = np.dot(self.sinvdiag, self.v)
        return np.dot(sinvv.T, sinvv)

    @cache_readonly
    def meigh(self):
        evecs = self.v.T
        evals = self.s**2
        return evals, evecs

    @cache_readonly
    def mdet(self):
        return self.meigh[0].prod()

    @cache_readonly
    def mlogdet(self):
        return np.log(self.meigh[0]).sum()

    @cache_readonly
    def mhalf(self):
        return np.dot(np.diag(self.s), self.v)

    @cache_readonly
    def xxthalf(self):
        return np.dot(self.u, self.sdiag)

    @cache_readonly
    def xxtinvhalf(self):
        return np.dot(self.u, self.sinvdiag)


class CholArray(PlainMatrixArray):
    '''Class that defines linalg operation on an array

    cholesky version, where svd is taken on original data array, if
    or when it matters

    plan: use cholesky factor and cholesky solve
    nothing implemented yet
    '''

    def __init__(self, data=None, sym=None):
        super(SvdArray, self).__init__(data=data, sym=sym)


    def yt_minv_y(self, y):
        '''xSigmainvx
        does not use stored cholesky yet
        '''
        return np.dot(x,linalg.cho_solve(linalg.cho_factor(self.m),x))
        #same as
        #lower = False   #if cholesky(sigma) is used, default is upper
        #np.dot(x,linalg.cho_solve((self.cholsigma, lower),x))



def testcompare(m1, m2):
    from numpy.testing import assert_almost_equal, assert_approx_equal
    decimal = 12

    #inv
    assert_almost_equal(m1.minv, m2.minv, decimal=decimal)

    #matrix half and invhalf
    #fix sign in test, should this be standardized
    s1 = np.sign(m1.mhalf.sum(1))[:,None]
    s2 = np.sign(m2.mhalf.sum(1))[:,None]
    scorr = s1/s2
    assert_almost_equal(m1.mhalf, m2.mhalf * scorr, decimal=decimal)
    assert_almost_equal(m1.minvhalf, m2.minvhalf, decimal=decimal)

    #eigenvalues, eigenvectors
    evals1, evecs1 = m1.meigh
    evals2, evecs2 = m2.meigh
    assert_almost_equal(evals1, evals2, decimal=decimal)
    #normalization can be different: evecs in columns
    s1 = np.sign(evecs1.sum(0))
    s2 = np.sign(evecs2.sum(0))
    scorr = s1/s2
    assert_almost_equal(evecs1, evecs2 * scorr, decimal=decimal)

    #determinant
    assert_approx_equal(m1.mdet, m2.mdet, significant=13)
    assert_approx_equal(m1.mlogdet, m2.mlogdet, significant=13)

####### helper function for interactive work
def tiny2zero(x, eps = 1e-15):
    '''replace abs values smaller than eps by zero, makes copy
    '''
    mask = np.abs(x.copy()) <  eps
    x[mask] = 0
    return x

def maxabs(x):
    return np.max(np.abs(x))


if __name__ == '__main__':


    n = 5
    y = np.arange(n)
    x = np.random.randn(100,n)
    autocov = 2*0.8**np.arange(n) +0.01 * np.random.randn(n)
    sigma = linalg.toeplitz(autocov)

    mat = PlainMatrixArray(sym=sigma)
    print(tiny2zero(mat.mhalf))
    mih = mat.minvhalf
    print(tiny2zero(mih)) #for nicer printing

    mat2 = PlainMatrixArray(data=x)
    print(maxabs(mat2.yt_minv_y(np.dot(x.T, x)) - mat2.m))
    print(tiny2zero(mat2.minv_y(mat2.m)))

    mat3 = SvdArray(data=x)
    print(mat3.meigh[0])
    print(mat2.meigh[0])

    testcompare(mat2, mat3)

    '''
    m = np.dot(x.T, x)

    u,s,v = np.linalg.svd(x, full_matrices=1)
    Sig = linalg.diagsvd(s,*x.shape)

    >>> np.max(np.abs(np.dot(u, np.dot(Sig, v)) - x))
    3.1086244689504383e-015
    >>> np.max(np.abs(np.dot(u.T, u) - np.eye(100)))
    3.3306690738754696e-016
    >>> np.max(np.abs(np.dot(v.T, v) - np.eye(5)))
    6.6613381477509392e-016
    >>> np.max(np.abs(np.dot(Sig.T, Sig) - np.diag(s**2)))
    5.6843418860808015e-014

    >>> evals,evecs = linalg.eigh(np.dot(x.T, x))
    >>> evals[::-1]
    array([ 123.36404464,  112.17036442,  102.04198468,   76.60832278,
             74.70484487])

    >>> s**2
    array([ 123.36404464,  112.17036442,  102.04198468,   76.60832278,
             74.70484487])

    >>> np.max(np.abs(np.dot(v.T, np.dot(np.diag(s**2), v)) - m))
    1.1368683772161603e-013

    >>> us = np.dot(u, Sig)
    >>> np.max(np.abs(np.dot(us, us.T) - np.dot(x, x.T)))
    1.0658141036401503e-014

    >>> sv = np.dot(Sig, v)
    >>> np.max(np.abs(np.dot(sv.T, sv) - np.dot(x.T, x)))
    1.1368683772161603e-013


    '''