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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def expectation(self, observables, statistics, lag_multiple=1, observables_mean_free=False, statistics_mean_free=False):
 r"""Compute future expectation of observable or covariance using the approximated Koopman operator. Parameters observables : np.ndarray((input_dimension, n_observables)) Coefficients that express one or multiple observables in the basis of the input features. statistics : np.ndarray((input_dimension, n_statistics)), optional Coefficients that express one or multiple statistics in the basis of the input features. This parameter can be None. In that case, this method returns the future expectation value of the observable(s). lag_multiple : int If > 1, extrapolate to a multiple of the estimator's lag time by assuming Markovianity of the approximated Koopman operator. observables_mean_free : bool, default=False If true, coefficients in `observables` refer to the input features with feature means removed. If false, coefficients in `observables` refer to the unmodified input features. statistics_mean_free : bool, default=False If true, coefficients in `statistics` refer to the input features with feature means removed. If false, coefficients in `statistics` refer to the unmodified input features. Notes ----- A "future expectation" of a observable g is the average of g computed over a time window that has the same total length as the input data from which the Koopman operator was estimated but is shifted by lag_multiple*tau time steps into the future (where tau is the lag time). It is computed with the equation: .. math:: \mathbb{E}[g]_{\rho_{n}}=\mathbf{q}^{T}\mathbf{P}^{n-1}\mathbf{e}_{1} where .. math:: P_{ij}=\sigma_{i}\langle\psi_{i},\phi_{j}\rangle_{\rho_{1}} and .. math:: q_{i}=\langle g,\phi_{i}\rangle_{\rho_{1}} and :math:`\mathbf{e}_{1}` is the first canonical unit vector. A model prediction of time-lagged covariances between the observable f and the statistic g at a lag-time of lag_multiple*tau is computed with the equation: .. math:: \mathrm{cov}[g,\,f;n\tau]=\mathbf{q}^{T}\mathbf{P}^{n-1}\boldsymbol{\Sigma}\mathbf{r} where :math:`r_{i}=\langle\psi_{i},f\rangle_{\rho_{0}}` and :math:`\boldsymbol{\Sigma}=\mathrm{diag(\boldsymbol{\sigma})}` . """

        # TODO: implement the case lag_multiple=0

        dim = self.dimension()

        S = np.diag(np.concatenate(([1.0], self.singular_values[0:dim])))
        V = self.V[:, 0:dim]
        U = self.U[:, 0:dim]
        m_0 = self.mean_0
        m_t = self.mean_t

        assert lag_multiple >= 1, 'lag_multiple = 0 not implemented'

        if lag_multiple == 1:
            P = S
        else:
            p = np.zeros((dim + 1, dim + 1))
            p[0, 0] = 1.0
            p[1:, 0] = U.T.dot(m_t - m_0)
            p[1:, 1:] = U.T.dot(self.Ctt).dot(V)
            P = np.linalg.matrix_power(S.dot(p), lag_multiple - 1).dot(S)

        Q = np.zeros((observables.shape[1], dim + 1))
        if not observables_mean_free:
            Q[:, 0] = observables.T.dot(m_t)
        Q[:, 1:] = observables.T.dot(self.Ctt).dot(V)

        if statistics is not None:
            # compute covariance
            R = np.zeros((statistics.shape[1], dim + 1))
            if not statistics_mean_free:
                R[:, 0] = statistics.T.dot(m_0)
            R[:, 1:] = statistics.T.dot(self.C00).dot(U)

        if statistics is not None:
            # compute lagged covariance
            return Q.dot(P).dot(R.T)
            # TODO: discuss whether we want to return this or the transpose
            # TODO: from MSMs one might expect to first index to refer to the statistics, here it is the other way round
        else:
            # compute future expectation
            return Q.dot(P)[:, 0]