""" Statespace Tools Author: Chad Fulton License: Simplified-BSD """ import numpy as np from scipy.linalg import solve_sylvester import pandas as pd from statsmodels.compat.pandas import Appender from statsmodels.tools.data import _is_using_pandas from scipy.linalg.blas import find_best_blas_type from . import (_initialization, _representation, _kalman_filter, _kalman_smoother, _simulation_smoother, _cfa_simulation_smoother, _tools) compatibility_mode = False has_trmm = True prefix_dtype_map = { 's': np.float32, 'd': np.float64, 'c': np.complex64, 'z': np.complex128 } prefix_initialization_map = { 's': _initialization.sInitialization, 'd': _initialization.dInitialization, 'c': _initialization.cInitialization, 'z': _initialization.zInitialization } prefix_statespace_map = { 's': _representation.sStatespace, 'd': _representation.dStatespace, 'c': _representation.cStatespace, 'z': _representation.zStatespace } prefix_kalman_filter_map = { 's': _kalman_filter.sKalmanFilter, 'd': _kalman_filter.dKalmanFilter, 'c': _kalman_filter.cKalmanFilter, 'z': _kalman_filter.zKalmanFilter } prefix_kalman_smoother_map = { 's': _kalman_smoother.sKalmanSmoother, 'd': _kalman_smoother.dKalmanSmoother, 'c': _kalman_smoother.cKalmanSmoother, 'z': _kalman_smoother.zKalmanSmoother } prefix_simulation_smoother_map = { 's': _simulation_smoother.sSimulationSmoother, 'd': _simulation_smoother.dSimulationSmoother, 'c': _simulation_smoother.cSimulationSmoother, 'z': _simulation_smoother.zSimulationSmoother } prefix_cfa_simulation_smoother_map = { 's': _cfa_simulation_smoother.sCFASimulationSmoother, 'd': _cfa_simulation_smoother.dCFASimulationSmoother, 'c': _cfa_simulation_smoother.cCFASimulationSmoother, 'z': _cfa_simulation_smoother.zCFASimulationSmoother } prefix_pacf_map = { 's': _tools._scompute_coefficients_from_multivariate_pacf, 'd': _tools._dcompute_coefficients_from_multivariate_pacf, 'c': _tools._ccompute_coefficients_from_multivariate_pacf, 'z': _tools._zcompute_coefficients_from_multivariate_pacf } prefix_sv_map = { 's': _tools._sconstrain_sv_less_than_one, 'd': _tools._dconstrain_sv_less_than_one, 'c': _tools._cconstrain_sv_less_than_one, 'z': _tools._zconstrain_sv_less_than_one } prefix_reorder_missing_matrix_map = { 's': _tools.sreorder_missing_matrix, 'd': _tools.dreorder_missing_matrix, 'c': _tools.creorder_missing_matrix, 'z': _tools.zreorder_missing_matrix } prefix_reorder_missing_vector_map = { 's': _tools.sreorder_missing_vector, 'd': _tools.dreorder_missing_vector, 'c': _tools.creorder_missing_vector, 'z': _tools.zreorder_missing_vector } prefix_copy_missing_matrix_map = { 's': _tools.scopy_missing_matrix, 'd': _tools.dcopy_missing_matrix, 'c': _tools.ccopy_missing_matrix, 'z': _tools.zcopy_missing_matrix } prefix_copy_missing_vector_map = { 's': _tools.scopy_missing_vector, 'd': _tools.dcopy_missing_vector, 'c': _tools.ccopy_missing_vector, 'z': _tools.zcopy_missing_vector } prefix_copy_index_matrix_map = { 's': _tools.scopy_index_matrix, 'd': _tools.dcopy_index_matrix, 'c': _tools.ccopy_index_matrix, 'z': _tools.zcopy_index_matrix } prefix_copy_index_vector_map = { 's': _tools.scopy_index_vector, 'd': _tools.dcopy_index_vector, 'c': _tools.ccopy_index_vector, 'z': _tools.zcopy_index_vector } prefix_compute_smoothed_state_weights_map = { 's': _tools._scompute_smoothed_state_weights, 'd': _tools._dcompute_smoothed_state_weights, 'c': _tools._ccompute_smoothed_state_weights, 'z': _tools._zcompute_smoothed_state_weights } def set_mode(compatibility=None): if compatibility: raise NotImplementedError('Compatibility mode is only available in' ' statsmodels <= 0.9') def companion_matrix(polynomial): r""" Create a companion matrix Parameters ---------- polynomial : array_like or list If an iterable, interpreted as the coefficients of the polynomial from which to form the companion matrix. Polynomial coefficients are in order of increasing degree, and may be either scalars (as in an AR(p) model) or coefficient matrices (as in a VAR(p) model). If an integer, it is interpreted as the size of a companion matrix of a scalar polynomial, where the polynomial coefficients are initialized to zeros. If a matrix polynomial is passed, :math:`C_0` may be set to the scalar value 1 to indicate an identity matrix (doing so will improve the speed of the companion matrix creation). Returns ------- companion_matrix : ndarray Notes ----- Given coefficients of a lag polynomial of the form: .. math:: c(L) = c_0 + c_1 L + \dots + c_p L^p returns a matrix of the form .. math:: \begin{bmatrix} \phi_1 & 1 & 0 & \cdots & 0 \\ \phi_2 & 0 & 1 & & 0 \\ \vdots & & & \ddots & 0 \\ & & & & 1 \\ \phi_n & 0 & 0 & \cdots & 0 \\ \end{bmatrix} where some or all of the :math:`\phi_i` may be non-zero (if `polynomial` is None, then all are equal to zero). If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_p)`, then the companion matrix is an :math:`n \times n` matrix formed with the elements in the first column defined as :math:`\phi_i = -\frac{c_i}{c_0}, i \in 1, \dots, p`. If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_p)`, each of shape :math:`(m, m)`, then the companion matrix is an :math:`nm \times nm` matrix formed with the elements in the first column defined as :math:`\phi_i = -C_0^{-1} C_i', i \in 1, \dots, p`. It is important to understand the expected signs of the coefficients. A typical AR(p) model is written as: .. math:: y_t = a_1 y_{t-1} + \dots + a_p y_{t-p} + \varepsilon_t This can be rewritten as: .. math:: (1 - a_1 L - \dots - a_p L^p )y_t = \varepsilon_t \\ (1 + c_1 L + \dots + c_p L^p )y_t = \varepsilon_t \\ c(L) y_t = \varepsilon_t The coefficients from this form are defined to be :math:`c_i = - a_i`, and it is the :math:`c_i` coefficients that this function expects to be provided. """ identity_matrix = False if isinstance(polynomial, (int, np.integer)): # GH 5570, allow numpy integer types, but coerce to python int n = int(polynomial) m = 1 polynomial = None else: n = len(polynomial) - 1 if n < 1: raise ValueError("Companion matrix polynomials must include at" " least two terms.") if isinstance(polynomial, (list, tuple)): try: # Note: cannot use polynomial[0] because of the special # behavior associated with matrix polynomials and the constant # 1, see below. m = len(polynomial[1]) except TypeError: m = 1 # Check if we just have a scalar polynomial if m == 1: polynomial = np.asanyarray(polynomial) # Check if 1 was passed as the first argument (indicating an # identity matrix) elif polynomial[0] == 1: polynomial[0] = np.eye(m) identity_matrix = True else: m = 1 polynomial = np.asanyarray(polynomial) matrix = np.zeros((n * m, n * m), dtype=np.asanyarray(polynomial).dtype) idx = np.diag_indices((n - 1) * m) idx = (idx[0], idx[1] + m) matrix[idx] = 1 if polynomial is not None and n > 0: if m == 1: matrix[:, 0] = -polynomial[1:] / polynomial[0] elif identity_matrix: for i in range(n): matrix[i * m:(i + 1) * m, :m] = -polynomial[i+1].T else: inv = np.linalg.inv(polynomial[0]) for i in range(n): matrix[i * m:(i + 1) * m, :m] = -np.dot(inv, polynomial[i+1]).T return matrix def diff(series, k_diff=1, k_seasonal_diff=None, seasonal_periods=1): r""" Difference a series simply and/or seasonally along the zero-th axis. Given a series (denoted :math:`y_t`), performs the differencing operation .. math:: \Delta^d \Delta_s^D y_t where :math:`d =` `diff`, :math:`s =` `seasonal_periods`, :math:`D =` `seasonal\_diff`, and :math:`\Delta` is the difference operator. Parameters ---------- series : array_like The series to be differenced. k_diff : int, optional The number of simple differences to perform. Default is 1. k_seasonal_diff : int or None, optional The number of seasonal differences to perform. Default is no seasonal differencing. seasonal_periods : int, optional The seasonal lag. Default is 1. Unused if there is no seasonal differencing. Returns ------- differenced : ndarray The differenced array. """ pandas = _is_using_pandas(series, None) differenced = np.asanyarray(series) if not pandas else series # Seasonal differencing if k_seasonal_diff is not None: while k_seasonal_diff > 0: if not pandas: differenced = (differenced[seasonal_periods:] - differenced[:-seasonal_periods]) else: sdiffed = differenced.diff(seasonal_periods) differenced = sdiffed[seasonal_periods:] k_seasonal_diff -= 1 # Simple differencing if not pandas: differenced = np.diff(differenced, k_diff, axis=0) else: while k_diff > 0: differenced = differenced.diff()[1:] k_diff -= 1 return differenced def concat(series, axis=0, allow_mix=False): """ Concatenate a set of series. Parameters ---------- series : iterable An iterable of series to be concatenated axis : int, optional The axis along which to concatenate. Default is 1 (columns). allow_mix : bool Whether or not to allow a mix of pandas and non-pandas objects. Default is False. If true, the returned object is an ndarray, and additional pandas metadata (e.g. column names, indices, etc) is lost. Returns ------- concatenated : array or pd.DataFrame The concatenated array. Will be a DataFrame if series are pandas objects. """ is_pandas = np.r_[[_is_using_pandas(s, None) for s in series]] ndim = np.r_[[np.ndim(s) for s in series]] max_ndim = np.max(ndim) if max_ndim > 2: raise ValueError('`tools.concat` does not support arrays with 3 or' ' more dimensions.') # Make sure the iterable is mutable if isinstance(series, tuple): series = list(series) # Standardize ndim for i in range(len(series)): if ndim[i] == 0 and max_ndim == 1: series[i] = np.atleast_1d(series[i]) elif ndim[i] == 0 and max_ndim == 2: series[i] = np.atleast_2d(series[i]) elif ndim[i] == 1 and max_ndim == 2 and is_pandas[i]: name = series[i].name series[i] = series[i].to_frame() series[i].columns = [name] elif ndim[i] == 1 and max_ndim == 2 and not is_pandas[i]: series[i] = np.atleast_2d(series[i]).T if np.all(is_pandas): if isinstance(series[0], pd.DataFrame): base_columns = series[0].columns else: base_columns = pd.Index([series[0].name]) for i in range(1, len(series)): s = series[i] if isinstance(s, pd.DataFrame): # Handle case where we were passed a dataframe and a series # to concatenate, and the series did not have a name. if s.columns.equals(pd.Index([None])): s.columns = base_columns[:1] s_columns = s.columns else: s_columns = pd.Index([s.name]) if axis == 0 and not base_columns.equals(s_columns): raise ValueError('Columns must match to concatenate along' ' rows.') elif axis == 1 and not series[0].index.equals(s.index): raise ValueError('Index must match to concatenate along' ' columns.') concatenated = pd.concat(series, axis=axis) elif np.all(~is_pandas) or allow_mix: concatenated = np.concatenate(series, axis=axis) else: raise ValueError('Attempted to concatenate Pandas objects with' ' non-Pandas objects with `allow_mix=False`.') return concatenated def is_invertible(polynomial, threshold=1 - 1e-10): r""" Determine if a polynomial is invertible. Requires all roots of the polynomial lie inside the unit circle. Parameters ---------- polynomial : array_like or tuple, list Coefficients of a polynomial, in order of increasing degree. For example, `polynomial=[1, -0.5]` corresponds to the polynomial :math:`1 - 0.5x` which has root :math:`2`. If it is a matrix polynomial (in which case the coefficients are coefficient matrices), a tuple or list of matrices should be passed. threshold : number Allowed threshold for `is_invertible` to return True. Default is 1. See Also -------- companion_matrix Notes ----- If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_n)`, then the corresponding polynomial is :math:`c_0 + c_1 L + \dots + c_n L^n`. If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_n)`, then the corresponding polynomial is :math:`C_0 + C_1 L + \dots + C_n L^n`. There are three equivalent methods of determining if the polynomial represented by the coefficients is invertible: The first method factorizes the polynomial into: .. math:: C(L) & = c_0 + c_1 L + \dots + c_n L^n \\ & = constant (1 - \lambda_1 L) (1 - \lambda_2 L) \dots (1 - \lambda_n L) In order for :math:`C(L)` to be invertible, it must be that each factor :math:`(1 - \lambda_i L)` is invertible; the condition is then that :math:`|\lambda_i| < 1`, where :math:`\lambda_i` is a root of the polynomial. The second method factorizes the polynomial into: .. math:: C(L) & = c_0 + c_1 L + \dots + c_n L^n \\ & = constant (L - \zeta_1) (L - \zeta_2) \dots (L - \zeta_3) The condition is now :math:`|\zeta_i| > 1`, where :math:`\zeta_i` is a root of the polynomial with reversed coefficients and :math:`\lambda_i = \frac{1}{\zeta_i}`. Finally, a companion matrix can be formed using the coefficients of the polynomial. Then the eigenvalues of that matrix give the roots of the polynomial. This last method is the one actually used. """ # First method: # np.all(np.abs(np.roots(np.r_[1, params])) < 1) # Second method: # np.all(np.abs(np.roots(np.r_[1, params][::-1])) > 1) # Final method: eigvals = np.linalg.eigvals(companion_matrix(polynomial)) return np.all(np.abs(eigvals) < threshold) def solve_discrete_lyapunov(a, q, complex_step=False): r""" Solves the discrete Lyapunov equation using a bilinear transformation. Notes ----- This is a modification of the version in Scipy (see https://github.com/scipy/scipy/blob/master/scipy/linalg/_solvers.py) which allows passing through the complex numbers in the matrix a (usually the transition matrix) in order to allow complex step differentiation. """ eye = np.eye(a.shape[0], dtype=a.dtype) if not complex_step: aH = a.conj().transpose() aHI_inv = np.linalg.inv(aH + eye) b = np.dot(aH - eye, aHI_inv) c = 2*np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv) return solve_sylvester(b.conj().transpose(), b, -c) else: aH = a.transpose() aHI_inv = np.linalg.inv(aH + eye) b = np.dot(aH - eye, aHI_inv) c = 2*np.dot(np.dot(np.linalg.inv(a + eye), q), aHI_inv) return solve_sylvester(b.transpose(), b, -c) def constrain_stationary_univariate(unconstrained): """ Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation Parameters ---------- unconstrained : ndarray Unconstrained parameters used by the optimizer, to be transformed to stationary coefficients of, e.g., an autoregressive or moving average component. Returns ------- constrained : ndarray Constrained parameters of, e.g., an autoregressive or moving average component, to be transformed to arbitrary parameters used by the optimizer. References ---------- .. [*] Monahan, John F. 1984. "A Note on Enforcing Stationarity in Autoregressive-moving Average Models." Biometrika 71 (2) (August 1): 403-404. """ n = unconstrained.shape[0] y = np.zeros((n, n), dtype=unconstrained.dtype) r = unconstrained/((1 + unconstrained**2)**0.5) for k in range(n): for i in range(k): y[k, i] = y[k - 1, i] + r[k] * y[k - 1, k - i - 1] y[k, k] = r[k] return -y[n - 1, :] def unconstrain_stationary_univariate(constrained): """ Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer Parameters ---------- constrained : ndarray Constrained parameters of, e.g., an autoregressive or moving average component, to be transformed to arbitrary parameters used by the optimizer. Returns ------- unconstrained : ndarray Unconstrained parameters used by the optimizer, to be transformed to stationary coefficients of, e.g., an autoregressive or moving average component. References ---------- .. [*] Monahan, John F. 1984. "A Note on Enforcing Stationarity in Autoregressive-moving Average Models." Biometrika 71 (2) (August 1): 403-404. """ n = constrained.shape[0] y = np.zeros((n, n), dtype=constrained.dtype) y[n-1:] = -constrained for k in range(n-1, 0, -1): for i in range(k): y[k-1, i] = (y[k, i] - y[k, k]*y[k, k-i-1]) / (1 - y[k, k]**2) r = y.diagonal() x = r / ((1 - r**2)**0.5) return x def _constrain_sv_less_than_one_python(unconstrained, order=None, k_endog=None): """ Transform arbitrary matrices to matrices with singular values less than one. Parameters ---------- unconstrained : list Arbitrary matrices. Should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. order : int, optional The order of the autoregression. k_endog : int, optional The dimension of the data vector. Returns ------- constrained : list Partial autocorrelation matrices. Should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. See Also -------- constrain_stationary_multivariate Notes ----- Corresponds to Lemma 2.2 in Ansley and Kohn (1986). See `constrain_stationary_multivariate` for more details. """ from scipy import linalg constrained = [] # P_s, s = 1, ..., p if order is None: order = len(unconstrained) if k_endog is None: k_endog = unconstrained[0].shape[0] eye = np.eye(k_endog) for i in range(order): A = unconstrained[i] B, lower = linalg.cho_factor(eye + np.dot(A, A.T), lower=True) constrained.append(linalg.solve_triangular(B, A, lower=lower)) return constrained def _compute_coefficients_from_multivariate_pacf_python( partial_autocorrelations, error_variance, transform_variance=False, order=None, k_endog=None): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. Parameters ---------- partial_autocorrelations : list Partial autocorrelation matrices. Should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. error_variance : ndarray The variance / covariance matrix of the error term. Should be sized `k_endog` x `k_endog`. This is used as input in the algorithm even if is not transformed by it (when `transform_variance` is False). The error term variance is required input when transformation is used either to force an autoregressive component to be stationary or to force a moving average component to be invertible. transform_variance : bool, optional Whether or not to transform the error variance term. This option is not typically used, and the default is False. order : int, optional The order of the autoregression. k_endog : int, optional The dimension of the data vector. Returns ------- coefficient_matrices : list Transformed coefficient matrices leading to a stationary VAR representation. See Also -------- constrain_stationary_multivariate Notes ----- Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See `constrain_stationary_multivariate` for more details. """ from scipy import linalg if order is None: order = len(partial_autocorrelations) if k_endog is None: k_endog = partial_autocorrelations[0].shape[0] # If we want to keep the provided variance but with the constrained # coefficient matrices, we need to make a copy here, and then after the # main loop we will transform the coefficients to match the passed variance if not transform_variance: initial_variance = error_variance # Need to make the input variance large enough that the recursions # do not lead to zero-matrices due to roundoff error, which would case # exceptions from the Cholesky decompositions. # Note that this will still not always ensure positive definiteness, # and for k_endog, order large enough an exception may still be raised error_variance = np.eye(k_endog) * (order + k_endog)**10 forward_variances = [error_variance] # \Sigma_s backward_variances = [error_variance] # \Sigma_s^*, s = 0, ..., p autocovariances = [error_variance] # \Gamma_s # \phi_{s,k}, s = 1, ..., p # k = 1, ..., s+1 forwards = [] # \phi_{s,k}^* backwards = [] error_variance_factor = linalg.cholesky(error_variance, lower=True) forward_factors = [error_variance_factor] backward_factors = [error_variance_factor] # We fill in the entries as follows: # [1,1] # [2,2], [2,1] # [3,3], [3,1], [3,2] # ... # [p,p], [p,1], ..., [p,p-1] # the last row, correctly ordered, is then used as the coefficients for s in range(order): # s = 0, ..., p-1 prev_forwards = forwards prev_backwards = backwards forwards = [] backwards = [] # Create the "last" (k = s+1) matrix # Note: this is for k = s+1. However, below we then have to fill # in for k = 1, ..., s in order. # P L*^{-1} = x # x L* = P # L*' x' = P' forwards.append( linalg.solve_triangular( backward_factors[s], partial_autocorrelations[s].T, lower=True, trans='T')) forwards[0] = np.dot(forward_factors[s], forwards[0].T) # P' L^{-1} = x # x L = P' # L' x' = P backwards.append( linalg.solve_triangular( forward_factors[s], partial_autocorrelations[s], lower=True, trans='T')) backwards[0] = np.dot(backward_factors[s], backwards[0].T) # Update the variance # Note: if s >= 1, this will be further updated in the for loop # below # Also, this calculation will be re-used in the forward variance tmp = np.dot(forwards[0], backward_variances[s]) autocovariances.append(tmp.copy().T) # Create the remaining k = 1, ..., s matrices, # only has an effect if s >= 1 for k in range(s): forwards.insert(k, prev_forwards[k] - np.dot( forwards[-1], prev_backwards[s-(k+1)])) backwards.insert(k, prev_backwards[k] - np.dot( backwards[-1], prev_forwards[s-(k+1)])) autocovariances[s+1] += np.dot(autocovariances[k+1], prev_forwards[s-(k+1)].T) # Create forward and backwards variances forward_variances.append( forward_variances[s] - np.dot(tmp, forwards[s].T) ) backward_variances.append( backward_variances[s] - np.dot( np.dot(backwards[s], forward_variances[s]), backwards[s].T ) ) # Cholesky factors forward_factors.append( linalg.cholesky(forward_variances[s+1], lower=True) ) backward_factors.append( linalg.cholesky(backward_variances[s+1], lower=True) ) # If we do not want to use the transformed variance, we need to # adjust the constrained matrices, as presented in Lemma 2.3, see above variance = forward_variances[-1] if not transform_variance: # Here, we need to construct T such that: # variance = T * initial_variance * T' # To do that, consider the Cholesky of variance (L) and # input_variance (M) to get: # L L' = T M M' T' = (TM) (TM)' # => L = T M # => L M^{-1} = T initial_variance_factor = np.linalg.cholesky(initial_variance) transformed_variance_factor = np.linalg.cholesky(variance) transform = np.dot(initial_variance_factor, np.linalg.inv(transformed_variance_factor)) inv_transform = np.linalg.inv(transform) for i in range(order): forwards[i] = ( np.dot(np.dot(transform, forwards[i]), inv_transform) ) return forwards, variance def constrain_stationary_multivariate_python(unconstrained, error_variance, transform_variance=False, prefix=None): r""" Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression. Parameters ---------- unconstrained : array or list Arbitrary matrices to be transformed to stationary coefficient matrices of the VAR. If a list, should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. If an array, should be the matrices horizontally concatenated and sized `k_endog` x `k_endog * order`. error_variance : ndarray The variance / covariance matrix of the error term. Should be sized `k_endog` x `k_endog`. This is used as input in the algorithm even if is not transformed by it (when `transform_variance` is False). The error term variance is required input when transformation is used either to force an autoregressive component to be stationary or to force a moving average component to be invertible. transform_variance : bool, optional Whether or not to transform the error variance term. This option is not typically used, and the default is False. prefix : {'s','d','c','z'}, optional The appropriate BLAS prefix to use for the passed datatypes. Only use if absolutely sure that the prefix is correct or an error will result. Returns ------- constrained : array or list Transformed coefficient matrices leading to a stationary VAR representation. Will match the type of the passed `unconstrained` variable (so if a list was passed, a list will be returned). Notes ----- In the notation of [1]_, the arguments `(variance, unconstrained)` are written as :math:`(\Sigma, A_1, \dots, A_p)`, where :math:`p` is the order of the vector autoregression, and is here determined by the length of the `unconstrained` argument. There are two steps in the constraining algorithm. First, :math:`(A_1, \dots, A_p)` are transformed into :math:`(P_1, \dots, P_p)` via Lemma 2.2 of [1]_. Second, :math:`(\Sigma, P_1, \dots, P_p)` are transformed into :math:`(\Sigma, \phi_1, \dots, \phi_p)` via Lemmas 2.1 and 2.3 of [1]_. If `transform_variance=True`, then only Lemma 2.1 is applied in the second step. While this function can be used even in the univariate case, it is much slower, so in that case `constrain_stationary_univariate` is preferred. References ---------- .. [1] Ansley, Craig F., and Robert Kohn. 1986. "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to Enforce Stationarity." Journal of Statistical Computation and Simulation 24 (2): 99-106. .. [*] Ansley, Craig F, and Paul Newbold. 1979. "Multivariate Partial Autocorrelations." In Proceedings of the Business and Economic Statistics Section, 349-53. American Statistical Association """ use_list = type(unconstrained) is list if not use_list: k_endog, order = unconstrained.shape order //= k_endog unconstrained = [ unconstrained[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(order) ] order = len(unconstrained) k_endog = unconstrained[0].shape[0] # Step 1: convert from arbitrary matrices to those with singular values # less than one. sv_constrained = _constrain_sv_less_than_one_python( unconstrained, order, k_endog) # Step 2: convert matrices from our "partial autocorrelation matrix" space # (matrices with singular values less than one) to the space of stationary # coefficient matrices constrained, var = _compute_coefficients_from_multivariate_pacf_python( sv_constrained, error_variance, transform_variance, order, k_endog) if not use_list: constrained = np.concatenate(constrained, axis=1).reshape( k_endog, k_endog * order) return constrained, var @Appender(constrain_stationary_multivariate_python.__doc__) def constrain_stationary_multivariate(unconstrained, variance, transform_variance=False, prefix=None): use_list = type(unconstrained) is list if use_list: unconstrained = np.concatenate(unconstrained, axis=1) k_endog, order = unconstrained.shape order //= k_endog if order < 1: raise ValueError('Must have order at least 1') if k_endog < 1: raise ValueError('Must have at least 1 endogenous variable') if prefix is None: prefix, dtype, _ = find_best_blas_type( [unconstrained, variance]) dtype = prefix_dtype_map[prefix] unconstrained = np.asfortranarray(unconstrained, dtype=dtype) variance = np.asfortranarray(variance, dtype=dtype) # Step 1: convert from arbitrary matrices to those with singular values # less than one. # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, # k_endog, prefix) sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) # Step 2: convert matrices from our "partial autocorrelation matrix" # space (matrices with singular values less than one) to the space of # stationary coefficient matrices constrained, variance = prefix_pacf_map[prefix]( sv_constrained, variance, transform_variance, order, k_endog) constrained = np.array(constrained, dtype=dtype) variance = np.array(variance, dtype=dtype) if use_list: constrained = [ constrained[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(order) ] return constrained, variance def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): """ Transform matrices with singular values less than one to arbitrary matrices. Parameters ---------- constrained : list The partial autocorrelation matrices. Should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. order : int, optional The order of the autoregression. k_endog : int, optional The dimension of the data vector. Returns ------- unconstrained : list Unconstrained matrices. A list of length `order`, where each element is an array sized `k_endog` x `k_endog`. See Also -------- unconstrain_stationary_multivariate Notes ----- Corresponds to the inverse of Lemma 2.2 in Ansley and Kohn (1986). See `unconstrain_stationary_multivariate` for more details. """ from scipy import linalg unconstrained = [] # A_s, s = 1, ..., p if order is None: order = len(constrained) if k_endog is None: k_endog = constrained[0].shape[0] eye = np.eye(k_endog) for i in range(order): P = constrained[i] # B^{-1} B^{-1}' = I - P P' B_inv, lower = linalg.cho_factor(eye - np.dot(P, P.T), lower=True) # A = BP # B^{-1} A = P unconstrained.append(linalg.solve_triangular(B_inv, P, lower=lower)) return unconstrained def _compute_multivariate_sample_acovf(endog, maxlag): r""" Computer multivariate sample autocovariances Parameters ---------- endog : array_like Sample data on which to compute sample autocovariances. Shaped `nobs` x `k_endog`. maxlag : int Maximum lag to use when computing the sample autocovariances. Returns ------- sample_autocovariances : list A list of the first `maxlag` sample autocovariance matrices. Each matrix is shaped `k_endog` x `k_endog`. Notes ----- This function computes the forward sample autocovariances: .. math:: \hat \Gamma(s) = \frac{1}{n} \sum_{t=1}^{n-s} (Z_t - \bar Z) (Z_{t+s} - \bar Z)' See page 353 of Wei (1990). This function is primarily implemented for checking the partial autocorrelation functions below, and so is quite slow. References ---------- .. [*] Wei, William. 1990. Time Series Analysis : Univariate and Multivariate Methods. Boston: Pearson. """ # Get the (demeaned) data as an array endog = np.array(endog) if endog.ndim == 1: endog = endog[:, np.newaxis] endog -= np.mean(endog, axis=0) # Dimensions nobs, k_endog = endog.shape sample_autocovariances = [] for s in range(maxlag + 1): sample_autocovariances.append(np.zeros((k_endog, k_endog))) for t in range(nobs - s): sample_autocovariances[s] += np.outer(endog[t], endog[t+s]) sample_autocovariances[s] /= nobs return sample_autocovariances def _compute_multivariate_acovf_from_coefficients( coefficients, error_variance, maxlag=None, forward_autocovariances=False): r""" Compute multivariate autocovariances from vector autoregression coefficient matrices Parameters ---------- coefficients : array or list The coefficients matrices. If a list, should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. If an array, should be the coefficient matrices horizontally concatenated and sized `k_endog` x `k_endog * order`. error_variance : ndarray The variance / covariance matrix of the error term. Should be sized `k_endog` x `k_endog`. maxlag : int, optional The maximum autocovariance to compute. Default is `order`-1. Can be zero, in which case it returns the variance. forward_autocovariances : bool, optional Whether or not to compute forward autocovariances :math:`E(y_t y_{t+j}')`. Default is False, so that backward autocovariances :math:`E(y_t y_{t-j}')` are returned. Returns ------- autocovariances : list A list of the first `maxlag` autocovariance matrices. Each matrix is shaped `k_endog` x `k_endog`. Notes ----- Computes .. math:: \Gamma(j) = E(y_t y_{t-j}') for j = 1, ..., `maxlag`, unless `forward_autocovariances` is specified, in which case it computes: .. math:: E(y_t y_{t+j}') = \Gamma(j)' Coefficients are assumed to be provided from the VAR model: .. math:: y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t Autocovariances are calculated by solving the associated discrete Lyapunov equation of the state space representation of the VAR process. """ from scipy import linalg # Convert coefficients to a list of matrices, for use in # `companion_matrix`; get dimensions if type(coefficients) is list: order = len(coefficients) k_endog = coefficients[0].shape[0] else: k_endog, order = coefficients.shape order //= k_endog coefficients = [ coefficients[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(order) ] if maxlag is None: maxlag = order-1 # Start with VAR(p): w_{t+1} = phi_1 w_t + ... + phi_p w_{t-p+1} + u_{t+1} # Then stack the VAR(p) into a VAR(1) in companion matrix form: # z_{t+1} = F z_t + v_t companion = companion_matrix( [1] + [-np.squeeze(coefficients[i]) for i in range(order)] ).T # Compute the error variance matrix for the stacked form: E v_t v_t' selected_variance = np.zeros(companion.shape) selected_variance[:k_endog, :k_endog] = error_variance # Compute the unconditional variance of z_t: E z_t z_t' stacked_cov = linalg.solve_discrete_lyapunov(companion, selected_variance) # The first (block) row of the variance of z_t gives the first p-1 # autocovariances of w_t: \Gamma_i = E w_t w_t+i with \Gamma_0 = Var(w_t) # Note: these are okay, checked against ArmaProcess autocovariances = [ stacked_cov[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(min(order, maxlag+1)) ] for i in range(maxlag - (order-1)): stacked_cov = np.dot(companion, stacked_cov) autocovariances += [ stacked_cov[:k_endog, -k_endog:] ] if forward_autocovariances: for i in range(len(autocovariances)): autocovariances[i] = autocovariances[i].T return autocovariances def _compute_multivariate_sample_pacf(endog, maxlag): """ Computer multivariate sample partial autocorrelations Parameters ---------- endog : array_like Sample data on which to compute sample autocovariances. Shaped `nobs` x `k_endog`. maxlag : int Maximum lag for which to calculate sample partial autocorrelations. Returns ------- sample_pacf : list A list of the first `maxlag` sample partial autocorrelation matrices. Each matrix is shaped `k_endog` x `k_endog`. """ sample_autocovariances = _compute_multivariate_sample_acovf(endog, maxlag) return _compute_multivariate_pacf_from_autocovariances( sample_autocovariances) def _compute_multivariate_pacf_from_autocovariances(autocovariances, order=None, k_endog=None): """ Compute multivariate partial autocorrelations from autocovariances. Parameters ---------- autocovariances : list Autocorrelations matrices. Should be a list of length `order` + 1, where each element is an array sized `k_endog` x `k_endog`. order : int, optional The order of the autoregression. k_endog : int, optional The dimension of the data vector. Returns ------- pacf : list List of first `order` multivariate partial autocorrelations. See Also -------- unconstrain_stationary_multivariate Notes ----- Note that this computes multivariate partial autocorrelations. Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See `unconstrain_stationary_multivariate` for more details. Computes sample partial autocorrelations if sample autocovariances are given. """ from scipy import linalg if order is None: order = len(autocovariances)-1 if k_endog is None: k_endog = autocovariances[0].shape[0] # Now apply the Ansley and Kohn (1986) algorithm, except that instead of # calculating phi_{s+1, s+1} = L_s P_{s+1} {L_s^*}^{-1} (which requires # the partial autocorrelation P_{s+1} which is what we're trying to # calculate here), we calculate it as in Ansley and Newbold (1979), using # the autocovariances \Gamma_s and the forwards and backwards residual # variances \Sigma_s, \Sigma_s^*: # phi_{s+1, s+1} = [ \Gamma_{s+1}' - \phi_{s,1} \Gamma_s' - ... - # \phi_{s,s} \Gamma_1' ] {\Sigma_s^*}^{-1} # Forward and backward variances forward_variances = [] # \Sigma_s backward_variances = [] # \Sigma_s^*, s = 0, ..., p # \phi_{s,k}, s = 1, ..., p # k = 1, ..., s+1 forwards = [] # \phi_{s,k}^* backwards = [] forward_factors = [] # L_s backward_factors = [] # L_s^*, s = 0, ..., p # Ultimately we want to construct the partial autocorrelation matrices # Note that this is "1-indexed" in the sense that it stores P_1, ... P_p # rather than starting with P_0. partial_autocorrelations = [] # We fill in the entries of phi_{s,k} as follows: # [1,1] # [2,2], [2,1] # [3,3], [3,1], [3,2] # ... # [p,p], [p,1], ..., [p,p-1] # the last row, correctly ordered, should be the same as the coefficient # matrices provided in the argument `constrained` for s in range(order): # s = 0, ..., p-1 prev_forwards = list(forwards) prev_backwards = list(backwards) forwards = [] backwards = [] # Create forward and backwards variances Sigma_s, Sigma*_s forward_variance = autocovariances[0].copy() backward_variance = autocovariances[0].T.copy() for k in range(s): forward_variance -= np.dot(prev_forwards[k], autocovariances[k+1]) backward_variance -= np.dot(prev_backwards[k], autocovariances[k+1].T) forward_variances.append(forward_variance) backward_variances.append(backward_variance) # Cholesky factors forward_factors.append( linalg.cholesky(forward_variances[s], lower=True) ) backward_factors.append( linalg.cholesky(backward_variances[s], lower=True) ) # Create the intermediate sum term if s == 0: # phi_11 = \Gamma_1' \Gamma_0^{-1} # phi_11 \Gamma_0 = \Gamma_1' # \Gamma_0 phi_11' = \Gamma_1 forwards.append(linalg.cho_solve( (forward_factors[0], True), autocovariances[1]).T) # backwards.append(forwards[-1]) # phi_11_star = \Gamma_1 \Gamma_0^{-1} # phi_11_star \Gamma_0 = \Gamma_1 # \Gamma_0 phi_11_star' = \Gamma_1' backwards.append(linalg.cho_solve( (backward_factors[0], True), autocovariances[1].T).T) else: # G := \Gamma_{s+1}' - # \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' tmp_sum = autocovariances[s+1].T.copy() for k in range(s): tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) # Create the "last" (k = s+1) matrix # Note: this is for k = s+1. However, below we then have to # fill in for k = 1, ..., s in order. # phi = G Sigma*^{-1} # phi Sigma* = G # Sigma*' phi' = G' # Sigma* phi' = G' # (because Sigma* is symmetric) forwards.append(linalg.cho_solve( (backward_factors[s], True), tmp_sum.T).T) # phi = G' Sigma^{-1} # phi Sigma = G' # Sigma' phi' = G # Sigma phi' = G # (because Sigma is symmetric) backwards.append(linalg.cho_solve( (forward_factors[s], True), tmp_sum).T) # Create the remaining k = 1, ..., s matrices, # only has an effect if s >= 1 for k in range(s): forwards.insert(k, prev_forwards[k] - np.dot( forwards[-1], prev_backwards[s-(k+1)])) backwards.insert(k, prev_backwards[k] - np.dot( backwards[-1], prev_forwards[s-(k+1)])) # Partial autocorrelation matrix: P_{s+1} # P = L^{-1} phi L* # L P = (phi L*) partial_autocorrelations.append(linalg.solve_triangular( forward_factors[s], np.dot(forwards[s], backward_factors[s]), lower=True)) return partial_autocorrelations def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, order=None, k_endog=None): r""" Transform matrices corresponding to a stationary (or invertible) process to matrices with singular values less than one. Parameters ---------- constrained : array or list The coefficients matrices. If a list, should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. If an array, should be the coefficient matrices horizontally concatenated and sized `k_endog` x `k_endog * order`. error_variance : ndarray The variance / covariance matrix of the error term. Should be sized `k_endog` x `k_endog`. order : int, optional The order of the autoregression. k_endog : int, optional The dimension of the data vector. Returns ------- pacf : list List of first `order` multivariate partial autocorrelations. See Also -------- unconstrain_stationary_multivariate Notes ----- Note that this computes multivariate partial autocorrelations. Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See `unconstrain_stationary_multivariate` for more details. Notes ----- Coefficients are assumed to be provided from the VAR model: .. math:: y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t """ if type(constrained) is list: order = len(constrained) k_endog = constrained[0].shape[0] else: k_endog, order = constrained.shape order //= k_endog # Get autocovariances for the process; these are defined to be # E z_t z_{t-j}' # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' _acovf = _compute_multivariate_acovf_from_coefficients autocovariances = [ autocovariance.T for autocovariance in _acovf(constrained, error_variance, maxlag=order)] return _compute_multivariate_pacf_from_autocovariances(autocovariances) def unconstrain_stationary_multivariate(constrained, error_variance): """ Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer Parameters ---------- constrained : array or list Constrained parameters of, e.g., an autoregressive or moving average component, to be transformed to arbitrary parameters used by the optimizer. If a list, should be a list of length `order`, where each element is an array sized `k_endog` x `k_endog`. If an array, should be the coefficient matrices horizontally concatenated and sized `k_endog` x `k_endog * order`. error_variance : ndarray The variance / covariance matrix of the error term. Should be sized `k_endog` x `k_endog`. This is used as input in the algorithm even if is not transformed by it (when `transform_variance` is False). Returns ------- unconstrained : ndarray Unconstrained parameters used by the optimizer, to be transformed to stationary coefficients of, e.g., an autoregressive or moving average component. Will match the type of the passed `constrained` variable (so if a list was passed, a list will be returned). Notes ----- Uses the list representation internally, even if an array is passed. References ---------- .. [*] Ansley, Craig F., and Robert Kohn. 1986. "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to Enforce Stationarity." Journal of Statistical Computation and Simulation 24 (2): 99-106. """ use_list = type(constrained) is list if not use_list: k_endog, order = constrained.shape order //= k_endog constrained = [ constrained[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(order) ] else: order = len(constrained) k_endog = constrained[0].shape[0] # Step 1: convert matrices from the space of stationary # coefficient matrices to our "partial autocorrelation matrix" space # (matrices with singular values less than one) partial_autocorrelations = _compute_multivariate_pacf_from_coefficients( constrained, error_variance, order, k_endog) # Step 2: convert from arbitrary matrices to those with singular values # less than one. unconstrained = _unconstrain_sv_less_than_one( partial_autocorrelations, order, k_endog) if not use_list: unconstrained = np.concatenate(unconstrained, axis=1) return unconstrained, error_variance def validate_matrix_shape(name, shape, nrows, ncols, nobs): """ Validate the shape of a possibly time-varying matrix, or raise an exception Parameters ---------- name : str The name of the matrix being validated (used in exception messages) shape : array_like The shape of the matrix to be validated. May be of size 2 or (if the matrix is time-varying) 3. nrows : int The expected number of rows. ncols : int The expected number of columns. nobs : int The number of observations (used to validate the last dimension of a time-varying matrix) Raises ------ ValueError If the matrix is not of the desired shape. """ ndim = len(shape) # Enforce dimension if ndim not in [2, 3]: raise ValueError('Invalid value for %s matrix. Requires a' ' 2- or 3-dimensional array, got %d dimensions' % (name, ndim)) # Enforce the shape of the matrix if not shape[0] == nrows: raise ValueError('Invalid dimensions for %s matrix: requires %d' ' rows, got %d' % (name, nrows, shape[0])) if not shape[1] == ncols: raise ValueError('Invalid dimensions for %s matrix: requires %d' ' columns, got %d' % (name, ncols, shape[1])) # If we do not yet know `nobs`, do not allow time-varying arrays if nobs is None and not (ndim == 2 or shape[-1] == 1): raise ValueError('Invalid dimensions for %s matrix: time-varying' ' matrices cannot be given unless `nobs` is specified' ' (implicitly when a dataset is bound or else set' ' explicity)' % name) # Enforce time-varying array size if ndim == 3 and nobs is not None and shape[-1] not in [1, nobs]: raise ValueError('Invalid dimensions for time-varying %s' ' matrix. Requires shape (*,*,%d), got %s' % (name, nobs, str(shape))) def validate_vector_shape(name, shape, nrows, nobs): """ Validate the shape of a possibly time-varying vector, or raise an exception Parameters ---------- name : str The name of the vector being validated (used in exception messages) shape : array_like The shape of the vector to be validated. May be of size 1 or (if the vector is time-varying) 2. nrows : int The expected number of rows (elements of the vector). nobs : int The number of observations (used to validate the last dimension of a time-varying vector) Raises ------ ValueError If the vector is not of the desired shape. """ ndim = len(shape) # Enforce dimension if ndim not in [1, 2]: raise ValueError('Invalid value for %s vector. Requires a' ' 1- or 2-dimensional array, got %d dimensions' % (name, ndim)) # Enforce the shape of the vector if not shape[0] == nrows: raise ValueError('Invalid dimensions for %s vector: requires %d' ' rows, got %d' % (name, nrows, shape[0])) # If we do not yet know `nobs`, do not allow time-varying arrays if nobs is None and not (ndim == 1 or shape[-1] == 1): raise ValueError('Invalid dimensions for %s vector: time-varying' ' vectors cannot be given unless `nobs` is specified' ' (implicitly when a dataset is bound or else set' ' explicity)' % name) # Enforce time-varying array size if ndim == 2 and shape[1] not in [1, nobs]: raise ValueError('Invalid dimensions for time-varying %s' ' vector. Requires shape (*,%d), got %s' % (name, nobs, str(shape))) def reorder_missing_matrix(matrix, missing, reorder_rows=False, reorder_cols=False, is_diagonal=False, inplace=False, prefix=None): """ Reorder the rows or columns of a time-varying matrix where all non-missing values are in the upper left corner of the matrix. Parameters ---------- matrix : array_like The matrix to be reordered. Must have shape (n, m, nobs). missing : array_like of bool The vector of missing indices. Must have shape (k, nobs) where `k = n` if `reorder_rows is True` and `k = m` if `reorder_cols is True`. reorder_rows : bool, optional Whether or not the rows of the matrix should be re-ordered. Default is False. reorder_cols : bool, optional Whether or not the columns of the matrix should be re-ordered. Default is False. is_diagonal : bool, optional Whether or not the matrix is diagonal. If this is True, must also have `n = m`. Default is False. inplace : bool, optional Whether or not to reorder the matrix in-place. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- reordered_matrix : array_like The reordered matrix. """ if prefix is None: prefix = find_best_blas_type((matrix,))[0] reorder = prefix_reorder_missing_matrix_map[prefix] if not inplace: matrix = np.copy(matrix, order='F') reorder(matrix, np.asfortranarray(missing), reorder_rows, reorder_cols, is_diagonal) return matrix def reorder_missing_vector(vector, missing, inplace=False, prefix=None): """ Reorder the elements of a time-varying vector where all non-missing values are in the first elements of the vector. Parameters ---------- vector : array_like The vector to be reordered. Must have shape (n, nobs). missing : array_like of bool The vector of missing indices. Must have shape (n, nobs). inplace : bool, optional Whether or not to reorder the matrix in-place. Default is False. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- reordered_vector : array_like The reordered vector. """ if prefix is None: prefix = find_best_blas_type((vector,))[0] reorder = prefix_reorder_missing_vector_map[prefix] if not inplace: vector = np.copy(vector, order='F') reorder(vector, np.asfortranarray(missing)) return vector def copy_missing_matrix(A, B, missing, missing_rows=False, missing_cols=False, is_diagonal=False, inplace=False, prefix=None): """ Copy the rows or columns of a time-varying matrix where all non-missing values are in the upper left corner of the matrix. Parameters ---------- A : array_like The matrix from which to copy. Must have shape (n, m, nobs) or (n, m, 1). B : array_like The matrix to copy to. Must have shape (n, m, nobs). missing : array_like of bool The vector of missing indices. Must have shape (k, nobs) where `k = n` if `reorder_rows is True` and `k = m` if `reorder_cols is True`. missing_rows : bool, optional Whether or not the rows of the matrix are a missing dimension. Default is False. missing_cols : bool, optional Whether or not the columns of the matrix are a missing dimension. Default is False. is_diagonal : bool, optional Whether or not the matrix is diagonal. If this is True, must also have `n = m`. Default is False. inplace : bool, optional Whether or not to copy to B in-place. Default is False. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- copied_matrix : array_like The matrix B with the non-missing submatrix of A copied onto it. """ if prefix is None: prefix = find_best_blas_type((A, B))[0] copy = prefix_copy_missing_matrix_map[prefix] if not inplace: B = np.copy(B, order='F') # We may have been given an F-contiguous memoryview; in that case, we do # not want to alter it or convert it to a numpy array try: if not A.is_f_contig(): raise ValueError() except (AttributeError, ValueError): A = np.asfortranarray(A) copy(A, B, np.asfortranarray(missing), missing_rows, missing_cols, is_diagonal) return B def copy_missing_vector(a, b, missing, inplace=False, prefix=None): """ Reorder the elements of a time-varying vector where all non-missing values are in the first elements of the vector. Parameters ---------- a : array_like The vector from which to copy. Must have shape (n, nobs) or (n, 1). b : array_like The vector to copy to. Must have shape (n, nobs). missing : array_like of bool The vector of missing indices. Must have shape (n, nobs). inplace : bool, optional Whether or not to copy to b in-place. Default is False. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- copied_vector : array_like The vector b with the non-missing subvector of b copied onto it. """ if prefix is None: prefix = find_best_blas_type((a, b))[0] copy = prefix_copy_missing_vector_map[prefix] if not inplace: b = np.copy(b, order='F') # We may have been given an F-contiguous memoryview; in that case, we do # not want to alter it or convert it to a numpy array try: if not a.is_f_contig(): raise ValueError() except (AttributeError, ValueError): a = np.asfortranarray(a) copy(a, b, np.asfortranarray(missing)) return b def copy_index_matrix(A, B, index, index_rows=False, index_cols=False, is_diagonal=False, inplace=False, prefix=None): """ Copy the rows or columns of a time-varying matrix where all non-index values are in the upper left corner of the matrix. Parameters ---------- A : array_like The matrix from which to copy. Must have shape (n, m, nobs) or (n, m, 1). B : array_like The matrix to copy to. Must have shape (n, m, nobs). index : array_like of bool The vector of index indices. Must have shape (k, nobs) where `k = n` if `reorder_rows is True` and `k = m` if `reorder_cols is True`. index_rows : bool, optional Whether or not the rows of the matrix are a index dimension. Default is False. index_cols : bool, optional Whether or not the columns of the matrix are a index dimension. Default is False. is_diagonal : bool, optional Whether or not the matrix is diagonal. If this is True, must also have `n = m`. Default is False. inplace : bool, optional Whether or not to copy to B in-place. Default is False. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- copied_matrix : array_like The matrix B with the non-index submatrix of A copied onto it. """ if prefix is None: prefix = find_best_blas_type((A, B))[0] copy = prefix_copy_index_matrix_map[prefix] if not inplace: B = np.copy(B, order='F') # We may have been given an F-contiguous memoryview; in that case, we do # not want to alter it or convert it to a numpy array try: if not A.is_f_contig(): raise ValueError() except (AttributeError, ValueError): A = np.asfortranarray(A) copy(A, B, np.asfortranarray(index), index_rows, index_cols, is_diagonal) return B def copy_index_vector(a, b, index, inplace=False, prefix=None): """ Reorder the elements of a time-varying vector where all non-index values are in the first elements of the vector. Parameters ---------- a : array_like The vector from which to copy. Must have shape (n, nobs) or (n, 1). b : array_like The vector to copy to. Must have shape (n, nobs). index : array_like of bool The vector of index indices. Must have shape (n, nobs). inplace : bool, optional Whether or not to copy to b in-place. Default is False. prefix : {'s', 'd', 'c', 'z'}, optional The Fortran prefix of the vector. Default is to automatically detect the dtype. This parameter should only be used with caution. Returns ------- copied_vector : array_like The vector b with the non-index subvector of b copied onto it. """ if prefix is None: prefix = find_best_blas_type((a, b))[0] copy = prefix_copy_index_vector_map[prefix] if not inplace: b = np.copy(b, order='F') # We may have been given an F-contiguous memoryview; in that case, we do # not want to alter it or convert it to a numpy array try: if not a.is_f_contig(): raise ValueError() except (AttributeError, ValueError): a = np.asfortranarray(a) copy(a, b, np.asfortranarray(index)) return b def prepare_exog(exog): k_exog = 0 if exog is not None: exog_is_using_pandas = _is_using_pandas(exog, None) if not exog_is_using_pandas: exog = np.asarray(exog) # Make sure we have 2-dimensional array if exog.ndim == 1: if not exog_is_using_pandas: exog = exog[:, None] else: exog = pd.DataFrame(exog) k_exog = exog.shape[1] return (k_exog, exog) def prepare_trend_spec(trend): # Trend if trend is None or trend == 'n': polynomial_trend = np.ones(0) elif trend == 'c': polynomial_trend = np.r_[1] elif trend == 't': polynomial_trend = np.r_[0, 1] elif trend == 'ct': polynomial_trend = np.r_[1, 1] elif trend == 'ctt': # TODO deprecate ctt? polynomial_trend = np.r_[1, 1, 1] else: trend = np.array(trend) if trend.ndim > 0: polynomial_trend = (trend > 0).astype(int) else: raise ValueError( "Valid trend inputs are 'c' (constant), 't' (linear trend in " "time), 'ct' (both), 'ctt' (both with trend squared) or an " "interable defining a polynomial, e.g., [1, 1, 0, 1] is `a + " f"b*t + ct**3`. Received {trend}" ) # Note: k_trend is not the degree of the trend polynomial, because e.g. # k_trend = 1 corresponds to the degree zero polynomial (with only a # constant term). k_trend = int(np.sum(polynomial_trend)) return polynomial_trend, k_trend def prepare_trend_data(polynomial_trend, k_trend, nobs, offset=1): # Cache the arrays for calculating the intercept from the trend # components time_trend = np.arange(offset, nobs + offset) trend_data = np.zeros((nobs, k_trend)) i = 0 for k in polynomial_trend.nonzero()[0]: if k == 0: trend_data[:, i] = np.ones(nobs,) else: trend_data[:, i] = time_trend**k i += 1 return trend_data def _safe_cond(a): """Compute condition while protecting from LinAlgError""" try: return np.linalg.cond(a) except np.linalg.LinAlgError: if np.any(np.isnan(a)): return np.nan else: return np.inf def _compute_smoothed_state_weights(ssm, compute_t=None, compute_j=None, compute_prior_weights=None, scale=1.0): # Get references to the Cython objects _model = ssm._statespace _kfilter = ssm._kalman_filter _smoother = ssm._kalman_smoother # Determine the appropriate function for the dtype func = prefix_compute_smoothed_state_weights_map[ssm.prefix] # Handle compute_t and compute_j indexes if compute_t is None: compute_t = np.arange(ssm.nobs) if compute_j is None: compute_j = np.arange(ssm.nobs) compute_t = np.unique(np.atleast_1d(compute_t).astype(np.int32)) compute_t.sort() compute_j = np.unique(np.atleast_1d(compute_j).astype(np.int32)) compute_j.sort() # Default setting for computing the prior weights if compute_prior_weights is None: compute_prior_weights = compute_j[0] == 0 # Validate that compute_prior_weights is valid if compute_prior_weights and compute_j[0] != 0: raise ValueError('If `compute_prior_weights` is set to True, then' ' `compute_j` must include the time period 0.') # Compute the weights weights, state_intercept_weights, prior_weights, _ = func( _smoother, _kfilter, _model, compute_t, compute_j, scale, bool(compute_prior_weights)) # Re-order missing entries correctly and transpose to the appropriate # shape t0 = min(compute_t[0], compute_j[0]) missing = np.isnan(ssm.endog[:, t0:]) if np.any(missing): shape = weights.shape # Transpose m, p, t, j, -> t, m, p, j so that we can use the # `reorder_missing_matrix` function weights = np.asfortranarray(weights.transpose(2, 0, 1, 3).reshape( shape[2] * shape[0], shape[1], shape[3], order='C')) missing = np.asfortranarray(missing.astype(np.int32)) reorder_missing_matrix(weights, missing, reorder_cols=True, inplace=True) # Transpose t, m, p, j -> t, j, m, p, weights = (weights.reshape(shape[2], shape[0], shape[1], shape[3]) .transpose(0, 3, 1, 2)) else: # Transpose m, p, t, j -> t, j, m, p weights = weights.transpose(2, 3, 0, 1) # Transpose m, l, t, j -> t, j, m, l state_intercept_weights = state_intercept_weights.transpose(2, 3, 0, 1) # Transpose m, l, t -> t, m, l prior_weights = prior_weights.transpose(2, 0, 1) # Subset to the actual computed t, j elements ix_tj = np.ix_(compute_t - t0, compute_j - t0) weights = weights[ix_tj] state_intercept_weights = state_intercept_weights[ix_tj] if compute_prior_weights: prior_weights = prior_weights[compute_t - t0] return weights, state_intercept_weights, prior_weights def compute_smoothed_state_weights(results, compute_t=None, compute_j=None, compute_prior_weights=None, resmooth=None): r""" Construct the weights of observations and the prior on the smoothed state Parameters ---------- results : MLEResults object Results object from fitting a state space model. compute_t : array_like, optional An explicit list of periods `t` of the smoothed state vector to compute weights for (see the Returns section for more details about the dimension `t`). Default is to compute weights for all periods `t`. However, if weights for only a few time points are desired, then performance can be improved by specifying this argument. compute_j : array_like, optional An explicit list of periods `j` of observations to compute weights for (see the Returns section for more details about the dimension `j`). Default is to compute weights for all periods `j`. However, if weights for only a few time points are desired, then performance can be improved by specifying this argument. compute_prior_weights : bool, optional Whether or not to compute the weight matrices associated with the prior mean (also called the "initial state"). Note that doing so requires that period 0 is in the periods defined in `compute_j`. Default is True if 0 is in `compute_j` (or if the `compute_j` argument is not passed) and False otherwise. resmooth : bool, optional Whether or not to re-perform filtering and smoothing prior to constructing the weights. Default is to resmooth if the smoothed_state vector is different between the given results object and the underlying smoother. Caution is adviced when changing this setting. See the Notes section below for more details. Returns ------- weights : array_like Weight matrices that can be used to construct the smoothed state from the observations. The returned matrix is always shaped `(nobs, nobs, k_states, k_endog)`, and entries that are not computed are set to NaNs. (Entries will not be computed if they are not included in `compute_t` and `compute_j`, or if they correspond to missing observations, or if they are for periods in which the exact diffuse Kalman filter is operative). The `(t, j, m, p)`-th element of this matrix contains the weight of the `p`-th element of the observation vector at time `j` in constructing the `m`-th element of the smoothed state vector at time `t`. prior_weights : array_like Weight matrices that describe the impact of the prior (also called the initialization) on the smoothed state vector. The returned matrix is always shaped `(nobs, k_states, k_states)`. If prior weights are not computed, then all entries will be set to NaNs. The `(t, m, l)`-th element of this matrix contains the weight of the `l`-th element of the prior mean (also called the "initial state") in constructing the `m`-th element of the smoothed state vector at time `t`. Notes ----- In [1]_, Chapter 4.8, it is shown how the smoothed state vector can be written as a weighted vector sum of observations: .. math:: \hat \alpha_t = \sum_{j=1}^n \omega_{jt}^{\hat \alpha} y_j One output of this function is the weights :math:`\omega_{jt}^{\hat \alpha}`. Note that the description in [1]_ assumes that the prior mean (or "initial state") is fixed to be zero. More generally, the smoothed state vector will also depend partly on the prior. The second output of this function are the weights of the prior mean. There are two important technical notes about the computations used here: 1. In the univariate approach to multivariate filtering (see e.g. Chapter 6.4 of [1]_), all observations are introduced one at a time, including those from the same time period. As a result, the weight of each observation can be different than when all observations from the same time point are introduced together, as in the typical multivariate filtering approach. Here, we always compute weights as in the multivariate filtering approach, and we handle singular forecast error covariance matrices by using a pseudo-inverse. 2. Constructing observation weights for periods in which the exact diffuse filter (see e.g. Chapter 5 of [1]_) is operative is not done here, and so the corresponding entries in the returned weight matrices will always be set equal to zeros. While handling these periods may be implemented in the future, one option for constructing these weights is to use an approximate (instead of exact) diffuse initialization for this purpose. Finally, one note about implementation: to compute the weights, we use attributes of the underlying filtering and smoothing Cython objects directly. However, these objects are not frozen with the result computation, and we cannot guarantee that their attributes have not changed since `res` was created. As a result, by default we re-run the filter and smoother to ensure that the attributes there actually correspond to the `res` object. This can be overridden by the user for a small performance boost if they are sure that the attributes have not changed; see the `resmooth` argument. References ---------- .. [1] Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press. """ # Get the python model object mod = results.model # Always update the parameters to be consistent with `res` mod.update(results.params) # By default, resmooth if it appears the results have changed; check is # based on the smoothed state vector if resmooth is None: resmooth = np.any(results.smoothed_state != mod.ssm._kalman_smoother.smoothed_state) # Resmooth if necessary, otherwise at least update the Cython model if resmooth: mod.ssm.smooth(conserve_memory=0, update_representation=False, update_filter=False, update_smoother=False) else: mod.ssm._initialize_representation() return _compute_smoothed_state_weights( mod.ssm, compute_t=compute_t, compute_j=compute_j, compute_prior_weights=compute_prior_weights, scale=results.filter_results.scale) def get_impact_dates(previous_model, updated_model, impact_date=None, start=None, end=None, periods=None): """ Compute start/end periods and an index, often for impacts of data updates Parameters ---------- previous_model : MLEModel Model used to compute default start/end periods if None are given. In the case of computing impacts of data updates, this would be the model estimated with the previous dataset. Otherwise, can be the same as `updated_model`. updated_model : MLEModel Model used to compute the index. In the case of computing impacts of data updates, this would be the model estimated with the updated dataset. Otherwise, can be the same as `previous_model`. impact_date : {int, str, datetime}, optional Specific individual impact date. Cannot be used in combination with `start`, `end`, or `periods`. start : {int, str, datetime}, optional Starting point of the impact dates. If given, one of `end` or `periods` must also be given. If a negative integer, will be computed relative to the dates in the `updated_model` index. Cannot be used in combination with `impact_date`. end : {int, str, datetime}, optional Ending point of the impact dates. If given, one of `start` or `periods` must also be given. If a negative integer, will be computed relative to the dates in the `updated_model` index. Cannot be used in combination with `impact_date`. periods : int, optional Number of impact date periods. If given, one of `start` or `end` must also be given. Cannot be used in combination with `impact_date`. Returns ------- start : int Integer location of the first included impact dates. end : int Integer location of the last included impact dates (i.e. this integer location is included in the returned `index`). index : pd.Index Index associated with `start` and `end`, as computed from the `updated_model`'s index. Notes ----- This function is typically used as a helper for standardizing start and end periods for a date range where the most sensible default values are based on some initial dataset (here contained in the `previous_model`), while index-related operations (especially relative start/end dates given via negative integers) are most sensibly computed from an updated dataset (here contained in the `updated_model`). """ # There doesn't seem to be any universal default that both (a) make # sense for all data update combinations, and (b) work with both # time-invariant and time-varying models. So we require that the user # specify exactly two of start, end, periods. if impact_date is not None: if not (start is None and end is None and periods is None): raise ValueError('Cannot use the `impact_date` argument in' ' combination with `start`, `end`, or' ' `periods`.') start = impact_date periods = 1 if start is None and end is None and periods is None: start = previous_model.nobs - 1 end = previous_model.nobs - 1 if int(start is None) + int(end is None) + int(periods is None) != 1: raise ValueError('Of the three parameters: start, end, and' ' periods, exactly two must be specified') # If we have the `periods` object, we need to convert `start`/`end` to # integers so that we can compute the other one. That's because # _get_prediction_index doesn't support a `periods` argument elif start is not None and periods is not None: start, _, _, _ = updated_model._get_prediction_index(start, start) end = start + (periods - 1) elif end is not None and periods is not None: _, end, _, _ = updated_model._get_prediction_index(end, end) start = end - (periods - 1) elif start is not None and end is not None: pass # Get the integer-based start, end and the prediction index start, end, out_of_sample, prediction_index = ( updated_model._get_prediction_index(start, end)) end = end + out_of_sample return start, end, prediction_index def _atleast_1d(*arys): """ Version of `np.atleast_1d`, copied from https://github.com/numpy/numpy/blob/master/numpy/core/shape_base.py, with the following modifications: 1. It allows for `None` arguments, and passes them directly through """ res = [] for ary in arys: if ary is None: result = None else: ary = np.asanyarray(ary) if ary.ndim == 0: result = ary.reshape(1) else: result = ary res.append(result) if len(res) == 1: return res[0] else: return res def _atleast_2d(*arys): """ Version of `np.atleast_2d`, copied from https://github.com/numpy/numpy/blob/master/numpy/core/shape_base.py, with the following modifications: 1. It allows for `None` arguments, and passes them directly through 2. Instead of creating new axis at the beginning, it creates it at the end """ res = [] for ary in arys: if ary is None: result = None else: ary = np.asanyarray(ary) if ary.ndim == 0: result = ary.reshape(1, 1) elif ary.ndim == 1: result = ary[:, np.newaxis] else: result = ary res.append(result) if len(res) == 1: return res[0] else: return res