text: """
  State Space Analysis using the Kalman Filter
  
  References
  -----------
  Durbin., J and Koopman, S.J.  `Time Series Analysis by State Space Methods`.
      Oxford, 2001.
  
  Hamilton, J.D.  `Time Series Analysis`.  Princeton, 1994.
  
  Harvey, A.C. `Forecasting, Structural Time Series Models and the Kalman Filter`.
      Cambridge, 1989.
  
  Notes
  -----
  This file follows Hamilton's notation pretty closely.
  The ARMA Model class follows Durbin and Koopman notation.
  Harvey uses Durbin and Koopman notation.
  """
  #Anderson and Moore `Optimal Filtering` provides a more efficient algorithm
  # namely the information filter
  # if the number of series is much greater than the number of states
  # e.g., with a DSGE model.  See also
  # http://www.federalreserve.gov/pubs/oss/oss4/aimindex.html
  # Harvey notes that the square root filter will keep P_t pos. def. but
  # is not strictly needed outside of the engineering (long series)
  
  import numpy as np
  from numpy import dot, identity, kron, log, zeros, pi, exp, eye, issubdtype, ones
  from numpy.linalg import inv, pinv
  from scikits.statsmodels.tools.tools import chain_dot
  try:
      from . import kalman_loglike
      fast_kalman = 1
  except:
      fast_kalman = 0
  #TODO: change to use only Cython when we switch
  
  #Fast filtering and smoothing for multivariate state space models
  # and The Riksbank -- Strid and Walentin (2008)
  # Block Kalman filtering for large-scale DSGE models
  # but this is obviously macro model specific
  
  def _init_diffuse(T,R):
      m = T.shape[1] # number of states
      r = R.shape[1] # should also be the number of states?
      Q_0 = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F'))
      return zeros((m,1)), Q_0.reshape(r,r,order='F')
  
  
  def kalmansmooth(F, A, H, Q, R, y, X, xi10):
      pass
  
  def kalmanfilter(F, A, H, Q, R, y, X, xi10, ntrain, history=False):
      """
      Returns the negative log-likelihood of y conditional on the information set
  
      Assumes that the initial state and all innovations are multivariate
      Gaussian.
  
      Parameters
      -----------
      F : array-like
          The (r x r) array holding the transition matrix for the hidden state.
      A : array-like
          The (nobs x k) array relating the predetermined variables to the
          observed data.
      H : array-like
          The (nobs x r) array relating the hidden state vector to the
          observed data.
      Q : array-like
          (r x r) variance/covariance matrix on the error term in the hidden
          state transition.
      R : array-like
          (nobs x nobs) variance/covariance of the noise in the observation
          equation.
      y : array-like
          The (nobs x 1) array holding the observed data.
      X : array-like
          The (nobs x k) array holding the predetermined variables data.
      xi10 : array-like
          Is the (r x 1) initial prior on the initial state vector.
      ntrain : int
          The number of training periods for the filter.  This is the number of
          observations that do not affect the likelihood.
  
  
      Returns
      -------
      likelihood
          The negative of the log likelihood
      history or priors, history of posterior
          If history is True.
  
      Notes
      -----
      No input checking is done.
      """
  # uses log of Hamilton 13.4.1
      F = np.asarray(F)
      H = np.atleast_2d(np.asarray(H))
      n = H.shape[1]  # remember that H gets transposed
      y = np.asarray(y)
      A = np.asarray(A)
      X = np.asarray(X)
      if y.ndim == 1: # note that Y is in rows for now
          y = y[:,None]
      nobs = y.shape[0]
      xi10 = np.atleast_2d(np.asarray(xi10))
  #    if xi10.ndim == 1:
  #        xi10[:,None]
      if history:
          state_vector = [xi10]
      Q = np.asarray(Q)
      r = xi10.shape[0]
  # Eq. 12.2.21, other version says P0 = Q
  #    p10 = np.dot(np.linalg.inv(np.eye(r**2)-np.kron(F,F)),Q.ravel('F'))
  #    p10 = np.reshape(P0, (r,r), order='F')
  # Assume a fixed, known intial point and set P0 = Q
  #TODO: this looks *slightly * different than Durbin-Koopman exact likelihood
  # initialization p 112 unless I've misunderstood the notational translation.
      p10 = Q
  
      loglikelihood = 0
      for i in range(nobs):
          HTPHR = np.atleast_1d(np.squeeze(chain_dot(H.T,p10,H)+R))
  #        print HTPHR
  #        print HTPHR.ndim
  #        print HTPHR.shape
          if HTPHR.ndim == 1:
              HTPHRinv = 1./HTPHR
          else:
              HTPHRinv = np.linalg.inv(HTPHR) # correct
  #        print A.T
  #        print X
  #        print H.T
  #        print xi10
  #        print y[i]
          part1 = y[i] - np.dot(A.T,X) - np.dot(H.T,xi10) # correct
          if i >= ntrain: # zero-index, but ntrain isn't
              HTPHRdet = np.linalg.det(np.atleast_2d(HTPHR)) # correct
              part2 = -.5*chain_dot(part1.T,HTPHRinv,part1) # correct
  #TODO: Need to test with ill-conditioned problem.
              loglike_interm = (-n/2.) * np.log(2*np.pi) - .5*\
                          np.log(HTPHRdet) + part2
              loglikelihood += loglike_interm
  
          # 13.2.15 Update current state xi_t based on y
          xi11 = xi10 + chain_dot(p10, H, HTPHRinv, part1)
          # 13.2.16 MSE of that state
          p11 = p10 - chain_dot(p10, H, HTPHRinv, H.T, p10)
          # 13.2.17 Update forecast about xi_{t+1} based on our F
          xi10 = np.dot(F,xi11)
          if history:
              state_vector.append(xi10)
          # 13.2.21 Update the MSE of the forecast
          p10 = chain_dot(F,p11,F.T) + Q
      if not history:
          return -loglikelihood
      else:
          return -loglikelihood, np.asarray(state_vector[:-1])
  
  #TODO: this works if it gets refactored, but it's not quite as accurate
  # as KalmanFilter
  #    def loglike_exact(self, params):
  #        """
  #        Exact likelihood for ARMA process.
  #
  #        Notes
  #        -----
  #        Computes the exact likelihood for an ARMA process by modifying the
  #        conditional sum of squares likelihood as suggested by Shephard (1997)
  #        "The relationship between the conditional sum of squares and the exact
  #        likelihood for autoregressive moving average models."
  #        """
  #        p = self.p
  #        q = self.q
  #        k = self.k
  #        y = self.endog.copy()
  #        nobs = self.nobs
  #        if self.transparams:
  #            newparams = self._transparams(params)
  #        else:
  #            newparams = params
  #        if k > 0:
  #            y -= dot(self.exog, newparams[:k])
  #        if p != 0:
  #            arcoefs = newparams[k:k+p][::-1]
  #            T = KalmanFilter.T(arcoefs)
  #        else:
  #            arcoefs = 0
  #        if q != 0:
  #            macoefs = newparams[k+p:k+p+q][::-1]
  #        else:
  #            macoefs = 0
  #        errors = [0] * q # psuedo-errors
  #        rerrors = [1] * q # error correction term
  #        # create pseudo-error and error correction series iteratively
  #        for i in range(p,len(y)):
  #            errors.append(y[i]-sum(arcoefs*y[i-p:i])-\
  #                                sum(macoefs*errors[i-q:i]))
  #            rerrors.append(-sum(macoefs*rerrors[i-q:i]))
  #        errors = np.asarray(errors)
  #        rerrors = np.asarray(rerrors)
  #
  #        # compute bayesian expected mean and variance of initial errors
  #        one_sumrt2 = 1 + np.sum(rerrors**2)
  #        sum_errors2 = np.sum(errors**2)
  #        mup = -np.sum(errors * rerrors)/one_sumrt2
  #
  #        # concentrating out the ML estimator of "true" sigma2 gives
  #        sigma2 = 1./(2*nobs)  * (sum_errors2 - mup**2*(one_sumrt2))
  #
  #        # which gives a variance of the initial errors of
  #        sigma2p = sigma2/one_sumrt2
  #
  #        llf = -(nobs-p)/2. * np.log(2*pi*sigma2) - 1./(2*sigma2)*sum_errors2 \
  #                + 1./2*log(one_sumrt2) + 1./(2*sigma2) * mup**2*one_sumrt2
  #        Z_mat = KalmanFilter.Z(r)
  #        R_mat = KalmanFilter.R(newparams, r, k, q, p)
  #        T_mat = KalmanFilter.T(newparams, r, k, p)
  #        # initial state and its variance
  #        alpha = zeros((m,1))
  #        Q_0 = dot(inv(identity(m**2)-kron(T_mat,T_mat)),
  #                dot(R_mat,R_mat.T).ravel('F'))
  #        Q_0 = Q_0.reshape(r,r,order='F')
  #        P = Q_0
  #        v = zeros((nobs,1))
  #        F = zeros((nobs,1))
  #        B = array([T_mat, 0], dtype=object)
  #
  #
  #        for i in xrange(int(nobs)):
  #            v_mat = (y[i],0) - dot(z_mat,B)
  #
  #        B_0 = (T,0)
  #        v_t = (y_t,0) - z*B_t
  #        llf = -nobs/2.*np.log(2*pi*sigma2) - 1/(2.*sigma2)*se_n - \
  #            1/2.*logdet(Sigma_a) + 1/(2*sigma2)*s_n_prime*sigma_a*s_n
  #        return llf
  #
  
  
  class StateSpaceModel(object):
      """
      Generic StateSpaceModel class. Meant to be a base class.
  
      This class lays out the methods that are to be defined by any child
      class.
  
      Parameters
      ----------
      endog : array-like
          An `nobs` x `p` array of observations
      exog : array-like, optional
          An `nobs` x `k` array of exogenous variables.
      **kwargs
          Anything provided to the constructor will be attached as an
          attribute.
  
      Notes
      -----
      The state space model is assumed to be of the form
  
      y[t] = Z[t].dot(alpha[t]) + epsilon[t]
      alpha[t+1] = T[t].dot(alpha[t]) + R[t].dot(eta[t])
  
      where
  
      epsilon[t] ~ N(0, H[t])
      eta[t] ~ N(0, Q[t])
      alpha[0] ~ N(a[0], P[0])
  
      Where y is the `p` x 1 observations vector, and alpha is the `m` x 1
      state vector.
  
      References
      -----------
      Durbin, J. and S.J. Koopman. 2001. `Time Series Analysis by State Space
          Methods.` Oxford.
      """
      def __init__(self, endog, exog=None, **kwargs):
          dict.__init__(self, kwargs)
          self.__dict__ = self
  
          endog = np.asarray(endog)
          if endog.ndim == 1:
              endog = endog[:,None]
          self.endog = endog
          p = endog.shape[1]
          self.p = nobs
          self.nobs = endog.shape[0]
          if exog:
              self.exog = exog
  
      def T(self, params):
          pass
  
      def R(self, params):
          pass
  
      def Z(self, params):
          pass
  
      def H(self, params):
          pass
  
      def Q(self, params):
          pass
  
      def _univariatefilter(self, params, init_state, init_var):
          """
          Implements the Kalman Filter recursions. Optimized for univariate case.
          """
          y = self.endog
          nobs = self.nobs
  
          R = self.R
          T = self.T
          Z = self.Z
          H = self.H
          Q = self.Q
          if not init_state and not init_var:
              alpha, P = _init_diffuse(T,R)
          #NOTE: stopped here
  
      def _univariatefilter_update(self):
          pass
          # does the KF but calls _update after each loop to update the matrices
          # for time-varying coefficients
  
      def kalmanfilter(self, params, init_state=None, init_var=None):
          """
          Runs the Kalman Filter
          """
          # determine if
          if self.p == 1:
              return _univariatefilter(init_state, init_var)
          else:
              raise ValueError("No multivariate filter written yet")
  
  
      def _updateloglike(self, params, xi10, ntrain, penalty, upperbounds, lowerbounds,
              F,A,H,Q,R, history):
          """
          """
          paramsorig = params
          # are the bounds binding?
          if penalty:
              params = np.min((np.max((lowerbounds, params), axis=0),upperbounds),
                  axis=0)
          #TODO: does it make sense for all of these to be allowed to be None?
          if F != None and callable(F):
              F = F(params)
          elif F == None:
              F = 0
          if A != None and callable(A):
              A = A(params)
          elif A == None:
              A = 0
          if H != None and callable(H):
              H = H(params)
          elif H == None:
              H = 0
          print callable(Q)
          if Q != None and callable(Q):
              Q = Q(params)
          elif Q == None:
              Q = 0
          if R != None and callable(R):
              R = R(params)
          elif R == None:
              R = 0
          X = self.exog
          if X == None:
              X = 0
          y = self.endog
          loglike = kalmanfilter(F,A,H,Q,R,y,X, xi10, ntrain, history)
          # use a quadratic penalty function to move away from bounds
          if penalty:
              loglike += penalty * np.sum((paramsorig-params)**2)
          return loglike
  
  #        r = self.r
  #        n = self.n
  #        F = np.diagonal(np.ones(r-1), k=-1) # think this will be wrong for VAR
                                              # cf. 13.1.22 but think VAR
  #        F[0] = params[:p] # assumes first p start_params are coeffs
                                  # of obs. vector, needs to be nxp for VAR?
  #        self.F = F
  #        cholQ = np.diag(start_params[p:]) # fails for bivariate
                                                          # MA(1) section
                                                          # 13.4.2
  #        Q = np.dot(cholQ,cholQ.T)
  #        self.Q = Q
  #        HT = np.zeros((n,r))
  #        xi10 = self.xi10
  #        y = self.endog
  #        ntrain = self.ntrain
   #       loglike = kalmanfilter(F,H,y,xi10,Q,ntrain)
  
      def fit_kalman(self, start_params, xi10, ntrain=1, F=None, A=None, H=None,
              Q=None,
              R=None, method="bfgs", penalty=True, upperbounds=None,
              lowerbounds=None):
          """
          Parameters
          ----------
          method : str
              Only "bfgs" is currently accepted.
          start_params : array-like
              The first guess on all parameters to be estimated.  This can
              be in any order as long as the F,A,H,Q, and R functions handle
              the parameters appropriately.
          xi10 : arry-like
              The (r x 1) vector of initial states.  See notes.
          F,A,H,Q,R : functions or array-like, optional
              If functions, they should take start_params (or the current
              value of params during iteration and return the F,A,H,Q,R matrices).
              See notes.  If they are constant then can be given as array-like
              objects.  If not included in the state-space representation then
              can be left as None.  See example in class docstring.
          penalty : bool,
              Whether or not to include a penalty for solutions that violate
              the bounds given by `lowerbounds` and `upperbounds`.
          lowerbounds : array-like
              Lower bounds on the parameter solutions.  Expected to be in the
              same order as `start_params`.
          upperbounds : array-like
              Upper bounds on the parameter solutions.  Expected to be in the
              same order as `start_params`
          """
          y = self.endog
          ntrain = ntrain
          _updateloglike = self._updateloglike
          params = start_params
          if method.lower() == 'bfgs':
              (params, llf, score, cov_params, func_calls, grad_calls,
              warnflag) = optimize.fmin_bfgs(_updateloglike, params,
                      args = (xi10, ntrain, penalty, upperbounds, lowerbounds,
                          F,A,H,Q,R, False), gtol= 1e-8, epsilon=1e-5,
                          full_output=1)
              #TODO: provide more options to user for optimize
          # Getting history would require one more call to _updatelikelihood
          self.params = params
          self.llf = llf
          self.gradient = score
          self.cov_params = cov_params # how to interpret this?
          self.warnflag = warnflag
  
  def updatematrices(params, y, xi10, ntrain, penalty, upperbound, lowerbound):
      """
      TODO: change API, update names
  
      This isn't general.  Copy of Luca's matlab example.
      """
      paramsorig = params
      # are the bounds binding?
      params = np.min((np.max((lowerbound,params),axis=0),upperbound), axis=0)
      rho = params[0]
      sigma1 = params[1]
      sigma2 = params[2]
  
      F = np.array([[rho, 0],[0,0]])
      cholQ = np.array([[sigma1,0],[0,sigma2]])
      H = np.ones((2,1))
      q = np.dot(cholQ,cholQ.T)
      loglike = kalmanfilter(F,0,H,q,0, y, 0, xi10, ntrain)
      loglike = loglike + penalty*np.sum((paramsorig-params)**2)
      return loglike
  
  class KalmanFilter(object):
      """
      Kalman Filter code intended for use with the ARMA model.
  
      Notes
      -----
      The notation for the state-space form follows Durbin and Koopman (2001).
  
      The observation equations is
  
      .. math:: y_{t} = Z_{t}\\alpha_{t} + \\epsilon_{t}
  
      The state equation is
  
      .. math:: \\alpha_{t+1} = T_{t}\\alpha_{t} + R_{t}\\eta_{t}
  
      For the present purposed \epsilon_{t} is assumed to always be zero.
      """
  
      @classmethod
      def T(cls, params, r, k, p): # F in Hamilton
          """
          The coefficient matrix for the state vector in the state equation.
  
          Its dimension is r+k x r+k.
  
          Parameters
          ----------
          r : int
              In the context of the ARMA model r is max(p,q+1) where p is the
              AR order and q is the MA order.
          k : int
              The number of exogenous variables in the ARMA model, including
              the constant if appropriate.
          p : int
              The AR coefficient in an ARMA model.
  
          References
          ----------
          Durbin and Koopman Section 3.7.
          """
          arr = zeros((r,r), dtype=params.dtype) # allows for complex-step
                                                    # derivative
          params_padded = zeros(r, dtype=params.dtype) # handle zero coefficients if necessary
          #NOTE: squeeze added for cg optimizer
          params_padded[:p] = params[k:p+k]
          arr[:,0] = params_padded   # first p params are AR coeffs w/ short params
          arr[:-1,1:] = eye(r-1)
          return arr
  
      @classmethod
      def R(cls, params, r, k, q, p): # R is H in Hamilton
          """
          The coefficient matrix for the state vector in the observation equation.
  
          Its dimension is r+k x 1.
  
          Parameters
          ----------
          r : int
              In the context of the ARMA model r is max(p,q+1) where p is the
              AR order and q is the MA order.
          k : int
              The number of exogenous variables in the ARMA model, including
              the constant if appropriate.
          q : int
              The MA order in an ARMA model.
          p : int
              The AR order in an ARMA model.
  
          References
          ----------
          Durbin and Koopman Section 3.7.
          """
          arr = zeros((r,1), dtype=params.dtype) # this allows zero coefficients
                                                    # dtype allows for compl. der.
          arr[1:q+1,:] = params[p+k:p+k+q][:,None]
          arr[0] = 1.0
          return arr
  
      @classmethod
      def Z(cls, r):
          """
          Returns the Z selector matrix in the observation equation.
  
          Parameters
          ----------
          r : int
              In the context of the ARMA model r is max(p,q+1) where p is the
              AR order and q is the MA order.
  
          Notes
          -----
          Currently only returns a 1 x r vector [1,0,0,...0].  Will need to
          be generalized when the Kalman Filter becomes more flexible.
          """
          arr = zeros((1,r))
          arr[:,0] = 1.
          return arr
  
      @classmethod
      def loglike(cls, params, arma_model):
          """
          The loglikelihood for an ARMA model using the Kalman Filter recursions.
  
          Parameters
          ----------
          params : array
              The coefficients of the ARMA model, assumed to be in the order of
              trend variables and `k` exogenous coefficients, the `p` AR
              coefficients, then the `q` MA coefficients.
          arma_model : `scikits.statsmodels.tsa.arima.ARMA` instance
              A reference to the ARMA model instance.
  
          Notes
          -----
          This works for both real valued and complex valued parameters. The
          complex values being used to compute the numerical derivative. If
          available will use a Cython version of the Kalman Filter.
          """
          #TODO: see section 3.4.6 in Harvey for computing the derivatives in the
          # recursion itself.
          #TODO: this won't work for time-varying parameters
          paramsdtype = params.dtype
          y = arma_model.endog.copy().astype(paramsdtype)
          k = arma_model.k_exog + arma_model.k_trend
          nobs = arma_model.nobs
          k_ar = arma_model.k_ar
          k_ma = arma_model.k_ma
          k_lags = arma_model.k_lags
  
          if arma_model.transparams:
              newparams = arma_model._transparams(params)
          else:
              newparams = params  # don't need a copy if not modified.
  
          if k > 0:
              y -= dot(arma_model.exog, newparams[:k])
  
          # system matrices
          Z_mat = KalmanFilter.Z(k_lags)
          m = Z_mat.shape[1] # r
          R_mat = KalmanFilter.R(newparams, k_lags, k, k_ma, k_ar)
          T_mat = KalmanFilter.T(newparams, k_lags, k, k_ar)
  
          if fast_kalman:
              if issubdtype(paramsdtype, float):
                  loglike, sigma2 =  kalman_loglike.kalman_loglike_double(y, k,
                                          k_ar, k_ma, k_lags, int(nobs), Z_mat,
                                          R_mat, T_mat)
              elif issubdtype(paramsdtype, complex):
                  loglike, sigma2 =  kalman_loglike.kalman_loglike_complex(y, k,
                                          k_ar, k_ma, k_lags, int(nobs), Z_mat,
                                          R_mat, T_mat)
              else:
                  raise TypeError("This dtype %s is not supported\n\
  Please files a bug report." % paramsdtype)
          else:
              # initial state and its variance
              alpha = zeros((m,1)) # if constant (I-T)**-1 * c
              Q_0 = dot(inv(identity(m**2)-kron(T_mat,T_mat)),
                                  dot(R_mat,R_mat.T).ravel('F'))
              #TODO: above is only valid if Eigenvalues of T_mat are inside the
              # unit circle, if not then Q_0 = kappa * eye(m**2)
             # w/ kappa some large value say 1e7, but DK recommends not doing this
              # for a diffuse prior
              # Note that we enforce stationarity
              Q_0 = Q_0.reshape(k_lags,k_lags,order='F')
              P = Q_0
              sigma2 = 0
              loglikelihood = 0
              v = zeros((nobs,1), dtype=paramsdtype)
              F = ones((nobs,1), dtype=paramsdtype)
              #NOTE: can only do quick recursions if Z is time-invariant
              #so could have recursions for pure ARMA vs ARMAX
  #            for i in xrange(int(nobs)):
              F_mat = 0
              i = 0
              while not F_mat == 1 and i < nobs:
                  # Predict
                  v_mat = y[i] - dot(Z_mat,alpha) # one-step forecast error
                  v[i] = v_mat
                  F_mat = dot(dot(Z_mat, P), Z_mat.T)
                  F[i] = F_mat
                  Finv = 1./F_mat # always scalar for univariate series
                  K = dot(dot(dot(T_mat,P),Z_mat.T),Finv) # Kalman Gain Matrix
                  # update state
                  alpha = dot(T_mat, alpha) + dot(K,v_mat)
                  L = T_mat - dot(K,Z_mat)
                  P = dot(dot(T_mat, P), L.T) + dot(R_mat, R_mat.T)
                  loglikelihood += log(F_mat)
                  i += 1
              for i in xrange(i,int(nobs)):
                  v_mat = y[i] - dot(Z_mat, alpha)
                  v[i] = v_mat
                  alpha = dot(T_mat, alpha) + dot(K, v_mat)
              sigma2 = 1./nobs * np.sum(v**2 / F)
              loglike = -.5 *(loglikelihood + nobs*log(sigma2))
              loglike -= nobs/2. * (log(2*pi) + 1)
          arma_model.sigma2 = sigma2
          return loglike.item() # return a scalar not a 0d array
  
  
  if __name__ == "__main__":
      import numpy as np
      from scipy.linalg import block_diag
      import numpy as np
      # Make our observations as in 13.1.13
      np.random.seed(54321)
      nobs = 600
      y = np.zeros(nobs)
      rho = [.5, -.25, .35, .25]
      sigma = 2.0 # std dev. or noise
      for i in range(4,nobs):
          y[i] = np.dot(rho,y[i-4:i][::-1]) + np.random.normal(scale=sigma)
      y = y[100:]
  
      # make an MA(2) observation equation as in example 13.3
      # y = mu + [1 theta][e_t e_t-1]'
      mu = 2.
      theta = .8
      rho = np.array([1, theta])
      np.random.randn(54321)
      e = np.random.randn(101)
      y = mu + rho[0]*e[1:]+rho[1]*e[:-1]
      # might need to add an axis
      r = len(rho)
      x = np.ones_like(y)
  
      # For now, assume that F,Q,A,H, and R are known
      F = np.array([[0,0],[1,0]])
      Q = np.array([[1,0],[0,0]])
      A = np.array([mu])
      H = rho[:,None]
      R = 0
  
      # remember that the goal is to solve recursively for the
      # state vector, xi, given the data, y (in this case)
      # we can also get a MSE matrix, P, associated with *each* observation
  
      # given that our errors are ~ NID(0,variance)
      # the starting E[e(1),e(0)] = [0,0]
      xi0 = np.array([[0],[0]])
      # with variance = 1 we know that
  #    P0 = np.eye(2)  # really P_{1|0}
  
  # Using the note below
      P0 = np.dot(np.linalg.inv(np.eye(r**2)-np.kron(F,F)),Q.ravel('F'))
      P0 = np.reshape(P0, (r,r), order='F')
  
      # more generally, if the eigenvalues for F are in the unit circle
      # (watch out for rounding error in LAPACK!) then
      # the DGP of the state vector is var/cov stationary, we know that
      # xi0 = 0
      # Furthermore, we could start with
      # vec(P0) = np.dot(np.linalg.inv(np.eye(r**2) - np.kron(F,F)),vec(Q))
      # where vec(X) = np.ravel(X, order='F') with a possible [:,np.newaxis]
      # if you really want a "2-d" array
      # a fortran (row-) ordered raveled array
      # If instead, some eigenvalues are on or outside the unit circle
      # xi0 can be replaced with a best guess and then
      # P0 is a positive definite matrix repr the confidence in the guess
      # larger diagonal elements signify less confidence
  
  
      # we also know that y1 = mu
      # and MSE(y1) = variance*(1+theta**2) = np.dot(np.dot(H.T,P0),H)
  
      state_vector = [xi0]
      forecast_vector = [mu]
      MSE_state = [P0]    # will be a list of matrices
      MSE_forecast = []
      # must be numerical shortcuts for some of this...
      # this should be general enough to be reused
      for i in range(len(y)-1):
          # update the state vector
          sv = state_vector[i]
          P = MSE_state[i]
          HTPHR = np.dot(np.dot(H.T,P),H)+R
          if np.ndim(HTPHR) < 2: # we have a scalar
              HTPHRinv = 1./HTPHR
          else:
              HTPHRinv = np.linalg.inv(HTPHR)
          FPH = np.dot(np.dot(F,P),H)
          gain_matrix = np.dot(FPH,HTPHRinv)  # correct
          new_sv = np.dot(F,sv)
          new_sv += np.dot(gain_matrix,y[i] - np.dot(A.T,x[i]) -
                  np.dot(H.T,sv))
          state_vector.append(new_sv)
          # update the MSE of the state vector forecast using 13.2.28
          new_MSEf = np.dot(np.dot(F - np.dot(gain_matrix,H.T),P),F.T - np.dot(H,
              gain_matrix.T)) + np.dot(np.dot(gain_matrix,R),gain_matrix.T) + Q
          MSE_state.append(new_MSEf)
          # update the in sample forecast of y
          forecast_vector.append(np.dot(A.T,x[i+1]) + np.dot(H.T,new_sv))
          # update the MSE of the forecast
          MSE_forecast.append(np.dot(np.dot(H.T,new_MSEf),H) + R)
      MSE_forecast = np.array(MSE_forecast).squeeze()
      MSE_state = np.array(MSE_state)
      forecast_vector = np.array(forecast_vector)
      state_vector = np.array(state_vector).squeeze()
  
  ##########
  #    Luca's example
      # choose parameters governing the signal extraction problem
      rho = .9
      sigma1 = 1
      sigma2 = 1
      nobs = 100
  
  # get the state space representation (Hamilton's notation)\
      F = np.array([[rho, 0],[0, 0]])
      cholQ = np.array([[sigma1, 0],[0,sigma2]])
      H = np.ones((2,1))
  
  # generate random data
      np.random.seed(12345)
      xihistory = np.zeros((2,nobs))
      for i in range(1,nobs):
          xihistory[:,i] = np.dot(F,xihistory[:,i-1]) + \
                  np.dot(cholQ,np.random.randn(2,1)).squeeze()
                  # this makes an ARMA process?
                  # check notes, do the math
      y = np.dot(H.T, xihistory)
      y = y.T
  
      params = np.array([rho, sigma1, sigma2])
      penalty = 1e5
      upperbounds = np.array([.999, 100, 100])
      lowerbounds = np.array([-.999, .001, .001])
      xi10 = xihistory[:,0]
      ntrain = 1
      bounds = zip(lowerbounds,upperbounds) # if you use fmin_l_bfgs_b
  #    results = optimize.fmin_bfgs(updatematrices, params,
  #        args=(y,xi10,ntrain,penalty,upperbounds,lowerbounds),
  #        gtol = 1e-8, epsilon=1e-10)
  #    array([ 0.83111567,  1.2695249 ,  0.61436685])
  
  
      F = lambda x : np.array([[x[0],0],[0,0]])
      def Q(x):
          cholQ = np.array([[x[1],0],[0,x[2]]])
          return np.dot(cholQ,cholQ.T)
      H = np.ones((2,1))
  #    ssm_model = StateSpaceModel(y)  # need to pass in Xi10!
  #    ssm_model.fit_kalman(start_params=params, xi10=xi10, F=F, Q=Q, H=H,
  #            upperbounds=upperbounds, lowerbounds=lowerbounds)
  # why does the above take 3 times as many iterations than direct max?
  
      # compare directly to matlab output
      from scipy import io
  #    y_matlab = io.loadmat('./kalman_y.mat')['y'].reshape(-1,1)
  #    ssm_model2 = StateSpaceModel(y_matlab)
  #    ssm_model2.fit_kalman(start_params=params, xi10=xi10, F=F, Q=Q, H=H,
  #            upperbounds=upperbounds, lowerbounds=lowerbounds)
  
  # matlab output
  #    thetaunc = np.array([0.7833, 1.1688, 0.5584])
  #    np.testing.assert_almost_equal(ssm_model2.params, thetaunc, 4)
      # maybe add a line search check to make sure we didn't get stuck in a local
      # max for more complicated ssm?
  
  
  
  # Examples from Durbin and Koopman
      import zipfile
      try:
          dk = zipfile.ZipFile('/home/skipper/statsmodels/statsmodels-skipper/scikits/statsmodels/sandbox/tsa/DK-data.zip')
      except:
          raise IOError("Install DK-data.zip from http://www.ssfpack.com/DKbook.html or specify its correct local path.")
      nile = dk.open('Nile.dat').readlines()
      nile = [float(_.strip()) for _ in nile[1:]]
      nile = np.asarray(nile)
  #    v = np.zeros_like(nile)
  #    a = np.zeros_like(nile)
  #    F = np.zeros_like(nile)
  #    P = np.zeros_like(nile)
  #    P[0] = 10.**7
  #    sigma2e = 15099.
  #    sigma2n = 1469.1
  #    for i in range(len(nile)):
  #        v[i] = nile[i] - a[i] # Kalman filter residual
  #        F[i] = P[i] + sigma2e # the variance of the Kalman filter residual
  #        K = P[i]/F[i]
  #        a[i+1] = a[i] + K*v[i]
  #        P[i+1] = P[i]*(1.-K) + sigma2n
  
      nile_ssm = StateSpaceModel(nile)
      R = lambda params : np.array(params[0])
      Q = lambda params : np.array(params[1])
  #    nile_ssm.fit_kalman(start_params=[1.0,1.0], xi10=0, F=[1.], H=[1.],
  #                Q=Q, R=R, penalty=False, ntrain=0)
  
  # p. 162 univariate structural time series example
      seatbelt = dk.open('Seatbelt.dat').readlines()
      seatbelt = [map(float,_.split()) for _ in seatbelt[2:]]
      sb_ssm = StateSpaceModel(seatbelt)
      s = 12 # monthly data
  # s p.
      H = np.zeros((s+1,1)) # Z in DK, H' in Hamilton
      H[::2] = 1.
      lambdaj = np.r_[1:6:6j]
      lambdaj *= 2*np.pi/s
      T = np.zeros((s+1,s+1))
      C = lambda j : np.array([[np.cos(j), np.sin(j)],[-np.sin(j), np.cos(j)]])
      Cj = [C(j) for j in lambdaj] + [-1]
  #NOTE: the above is for handling seasonality
  #TODO: it is just a rotation matrix.  See if Robert's link has a better way
  #http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=5F5145BE25D61F87478B25AD1493C8F4?doi=10.1.1.110.5134&rep=rep1&type=pdf&ei=QcetSefqF4GEsQPnx4jSBA&sig2=HjJILSBPFgJTfuifbvKrxw&usg=AFQjCNFbABIxusr-NEbgrinhtR6buvjaYA
      from scipy import linalg
      F = linalg.block_diag(*Cj) # T in DK, F in Hamilton
      R = np.eye(s-1)
      sigma2_omega = 1.
      Q = np.eye(s-1) * sigma2_omega
  