import numpy
from scipy import linalg

# Todo:
#   - complete docstring (explain arguments, pseudo code)
#   - Consider case with discount = 1.0
#   - reevaluation when not at the end of a minibatch

class PcaOnlineEstimator(object):
  """Online estimation of the leading eigen values/vectors of the covariance
  of some samples.

  Maintains a moving (with discount) low rank (n_eigen) estimate of the
  covariance matrix of some observations. New observations are accumulated
  until the batch is complete, at which point the low rank estimate is 
  reevaluated.

  Example:

    pca_esti = pca_online_estimator.PcaOnlineEstimator(dimension_of_the_samples)

    for i in range(number_of_samples):
      pca_esti.observe(samples[i])

    [eigvals, eigvecs] = pca_esti.getLeadingEigen()

  """


  def __init__(self, n_dim, n_eigen = 10, minibatch_size = 25, gamma = 0.999, regularizer = 1e-6, centering = True):
    # dimension of the observations
    self.n_dim = n_dim 
    # rank of the low-rank estimate
    self.n_eigen = n_eigen
    # how many observations between reevaluations of the low rank estimate
    self.minibatch_size = minibatch_size
    # the discount factor in the moving estimate
    self.gamma = gamma
    # regularizer of the covariance estimate
    self.regularizer = regularizer
    # wether we center the observations or not: obtain leading eigen of
    # covariance (centering = True) vs second moment (centering = False)
    self.centering = centering

    # Total number of observations: to compute the normalizer for the mean and
    # the covariance.
    self.n_observations = 0
    # Index in the current minibatch
    self.minibatch_index = 0

    # Matrix containing on its *rows*:
    # - the current unnormalized eigen vector estimates
    # - the observations since the last reevaluation
    self.Xt = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_dim])

    # The discounted sum of the observations.
    self.x_sum = numpy.zeros([self.n_dim])

    # The Gram matrix of the observations, ie Xt Xt' (since Xt is rowwise)
    self.G = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])
    for i in range(self.n_eigen):
      self.G[i,i] = self.regularizer

    # I don't think it's worth "allocating" these 3 next (though they need to be
    # declared). I don't know how to do in place operations... 

    # Hold the results of the eigendecomposition of the Gram matrix G
    # (eigen vectors on columns of V).
    self.d = numpy.zeros([self.n_eigen + self.minibatch_size])
    self.V = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])

    # Holds the unnormalized eigenvectors of the covariance matrix before
    # they're copied back to Xt.
    self.Ut = numpy.zeros([self.n_eigen, self.n_dim])


  def observe(self, x):
    assert(numpy.size(x) == self.n_dim)

    self.n_observations += 1

    # Add the *non-centered* observation to Xt.
    row = self.n_eigen + self.minibatch_index
    self.Xt[row] = x

    # Update the discounted sum of the observations.
    self.x_sum *= self.gamma
    self.x_sum += x

    # To get the mean, we must normalize the sum by:
    # /gamma^(n_observations-1) + /gamma^(n_observations-2) + ... + 1
    normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma);
    #print "normalizer: ", normalizer

    # Now center the observation.
    # We will lose the first observation as it is the only one in the mean.
    if self.centering:
      self.Xt[row] -= self.x_sum / normalizer 

    # Multiply the observation by the discount compensator. Basically
    # we make this observation look "younger" than the previous ones. The actual
    # discount is applied in the reevaluation (and when solving the equations in
    # the case of TONGA) by multiplying every direction with the same aging factor.
    rn = pow(self.gamma, -0.5*(self.minibatch_index+1));
    self.Xt[row] *= rn

    # Update the Gram Matrix.
    # The column.
    self.G[:row+1,row] = numpy.dot( self.Xt[:row+1,:], self.Xt[row,:].transpose() )
    # The symetric row.
    # There are row+1 values, but the diag doesn't need to get copied.
    self.G[row,:row] = self.G[:row,row].transpose()

    self.minibatch_index += 1

    if self.minibatch_index == self.minibatch_size:
      self.reevaluate()


  def reevaluate(self):
    # TODO do the modifications to handle when this is not true.
    assert(self.minibatch_index == self.minibatch_size);

    # Regularize - not necessary but in case
    for i in range(self.n_eigen + self.minibatch_size):
      self.G[i,i] += self.regularizer

    # The Gram matrix is up to date. Get its low rank eigendecomposition.
    # NOTE: the eigenvalues are in ASCENDING order and the vectors are on
    # the columns. 
    # With scipy 0.7, you can ask for only some eigenvalues (the n_eigen top
    # ones) but it doesn't look loke it for scipy 0.6.
    self.d, self.V = linalg.eigh(self.G) #, overwrite_a=True)

    # Convert the n_eigen LAST eigenvectors of the Gram matrix contained in V
    # into *unnormalized* eigenvectors U of the covariance (unnormalized wrt 
    # the eigen values, not the moving average).
    self.Ut = numpy.dot(self.V[:,-self.n_eigen:].transpose(), self.Xt)

    # Take into account the discount factor.
    # Here, minibatch index is minibatch_size. We age everyone. Because of the
    # previous multiplications to make some observations "younger" we multiply
    # everyone by the same factor.
    # TODO VERIFY THIS!
    rn = pow(self.gamma, -0.5*(self.minibatch_index+1))
    inv_rn2 = 1.0/(rn*rn)
    self.Ut *= 1.0/rn
    self.d *= inv_rn2;

    #print "*** Reevaluate! ***"
    #normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma)
    #print "normalizer: ", normalizer
    #print self.d / normalizer
    #print self.Ut # unnormalized eigen vectors (wrt eigenvalues AND moving average).

    # Update Xt, G and minibatch_index
    self.Xt[:self.n_eigen,:] = self.Ut

    for i in range(self.n_eigen):
      self.G[i,i] = self.d[-self.n_eigen+i]

    self.minibatch_index = 0

  # Returns a copy of the current estimate of the eigen values and vectors
  # (normalized vectors on rows), normalized by the discounted number of observations.
  def getLeadingEigen(self):
    # We subtract self.minibatch_index in case this call is not right after a reevaluate call.
    normalizer = (1.0 - pow(self.gamma, self.n_observations - self.minibatch_index)) /(1.0 - self.gamma)

    eigvals = self.d[-self.n_eigen:] / normalizer
    eigvecs = numpy.zeros([self.n_eigen, self.n_dim])
    for i in range(self.n_eigen):
      eigvecs[i] = self.Ut[-self.n_eigen+i] / numpy.sqrt(numpy.dot(self.Ut[-self.n_eigen+i], self.Ut[-self.n_eigen+i]))

    return [eigvals, eigvecs]