Source code for statsmodels.tsa.statespace.kalman_filter

"""
State Space Representation and Kalman Filter

Author: Chad Fulton
License: Simplified-BSD
"""
from __future__ import division, absolute_import, print_function

from warnings import warn
from contextlib import contextmanager

import numpy as np
from .representation import OptionWrapper, Representation, FrozenRepresentation
from .tools import (validate_vector_shape, validate_matrix_shape,
                    reorder_missing_matrix, reorder_missing_vector)
from . import tools
from statsmodels.tools.sm_exceptions import ValueWarning

# Define constants
FILTER_CONVENTIONAL = 0x01     # Durbin and Koopman (2012), Chapter 4
FILTER_EXACT_INITIAL = 0x02    # ibid., Chapter 5.6
FILTER_AUGMENTED = 0x04        # ibid., Chapter 5.7
FILTER_SQUARE_ROOT = 0x08      # ibid., Chapter 6.3
FILTER_UNIVARIATE = 0x10       # ibid., Chapter 6.4
FILTER_COLLAPSED = 0x20        # ibid., Chapter 6.5
FILTER_EXTENDED = 0x40         # ibid., Chapter 10.2
FILTER_UNSCENTED = 0x80        # ibid., Chapter 10.3
FILTER_CONCENTRATED = 0x100    # Harvey (1989), Chapter 3.4

INVERT_UNIVARIATE = 0x01
SOLVE_LU = 0x02
INVERT_LU = 0x04
SOLVE_CHOLESKY = 0x08
INVERT_CHOLESKY = 0x10

STABILITY_FORCE_SYMMETRY = 0x01

MEMORY_STORE_ALL = 0
MEMORY_NO_FORECAST = 0x01
MEMORY_NO_PREDICTED = 0x02
MEMORY_NO_FILTERED = 0x04
MEMORY_NO_LIKELIHOOD = 0x08
MEMORY_NO_GAIN = 0x10
MEMORY_NO_SMOOTHING = 0x20
MEMORY_NO_STD_FORECAST = 0x40
MEMORY_CONSERVE = (
    MEMORY_NO_FORECAST | MEMORY_NO_PREDICTED | MEMORY_NO_FILTERED |
    MEMORY_NO_LIKELIHOOD | MEMORY_NO_GAIN | MEMORY_NO_SMOOTHING |
    MEMORY_NO_STD_FORECAST
)

TIMING_INIT_PREDICTED = 0
TIMING_INIT_FILTERED = 1


[docs]class KalmanFilter(Representation): r""" State space representation of a time series process, with Kalman filter Parameters ---------- k_endog : array_like or integer The observed time-series process :math:`y` if array like or the number of variables in the process if an integer. k_states : int The dimension of the unobserved state process. k_posdef : int, optional The dimension of a guaranteed positive definite covariance matrix describing the shocks in the transition equation. Must be less than or equal to `k_states`. Default is `k_states`. loglikelihood_burn : int, optional The number of initial periods during which the loglikelihood is not recorded. Default is 0. tolerance : float, optional The tolerance at which the Kalman filter determines convergence to steady-state. Default is 1e-19. results_class : class, optional Default results class to use to save filtering output. Default is `FilterResults`. If specified, class must extend from `FilterResults`. **kwargs Keyword arguments may be used to provide values for the filter, inversion, and stability methods. See `set_filter_method`, `set_inversion_method`, and `set_stability_method`. Keyword arguments may be used to provide default values for state space matrices. See `Representation` for more details. Notes ----- There are several types of options available for controlling the Kalman filter operation. All options are internally held as bitmasks, but can be manipulated by setting class attributes, which act like boolean flags. For more information, see the `set_*` class method documentation. The options are: filter_method The filtering method controls aspects of which Kalman filtering approach will be used. inversion_method The Kalman filter may contain one matrix inversion: that of the forecast error covariance matrix. The inversion method controls how and if that inverse is performed. stability_method The Kalman filter is a recursive algorithm that may in some cases suffer issues with numerical stability. The stability method controls what, if any, measures are taken to promote stability. conserve_memory By default, the Kalman filter computes a number of intermediate matrices at each iteration. The memory conservation options control which of those matrices are stored. filter_timing By default, the Kalman filter follows Durbin and Koopman, 2012, in initializing the filter with predicted values. Kim and Nelson, 1999, instead initialize the filter with filtered values, which is essentially just a different timing convention. The `filter_method` and `inversion_method` options intentionally allow the possibility that multiple methods will be indicated. In the case that multiple methods are selected, the underlying Kalman filter will attempt to select the optional method given the input data. For example, it may be that INVERT_UNIVARIATE and SOLVE_CHOLESKY are indicated (this is in fact the default case). In this case, if the endogenous vector is 1-dimensional (`k_endog` = 1), then INVERT_UNIVARIATE is used and inversion reduces to simple division, and if it has a larger dimension, the Cholesky decomposition along with linear solving (rather than explicit matrix inversion) is used. If only SOLVE_CHOLESKY had been set, then the Cholesky decomposition method would *always* be used, even in the case of 1-dimensional data. See Also -------- FilterResults statsmodels.tsa.statespace.representation.Representation """ filter_methods = [ 'filter_conventional', 'filter_exact_initial', 'filter_augmented', 'filter_square_root', 'filter_univariate', 'filter_collapsed', 'filter_extended', 'filter_unscented', 'filter_concentrated' ] filter_conventional = OptionWrapper('filter_method', FILTER_CONVENTIONAL) """ (bool) Flag for conventional Kalman filtering. """ filter_exact_initial = OptionWrapper('filter_method', FILTER_EXACT_INITIAL) """ (bool) Flag for exact initial Kalman filtering. Not implemented. """ filter_augmented = OptionWrapper('filter_method', FILTER_AUGMENTED) """ (bool) Flag for augmented Kalman filtering. Not implemented. """ filter_square_root = OptionWrapper('filter_method', FILTER_SQUARE_ROOT) """ (bool) Flag for square-root Kalman filtering. Not implemented. """ filter_univariate = OptionWrapper('filter_method', FILTER_UNIVARIATE) """ (bool) Flag for univariate filtering of multivariate observation vector. """ filter_collapsed = OptionWrapper('filter_method', FILTER_COLLAPSED) """ (bool) Flag for Kalman filtering with collapsed observation vector. """ filter_extended = OptionWrapper('filter_method', FILTER_EXTENDED) """ (bool) Flag for extended Kalman filtering. Not implemented. """ filter_unscented = OptionWrapper('filter_method', FILTER_UNSCENTED) """ (bool) Flag for unscented Kalman filtering. Not implemented. """ filter_concentrated = OptionWrapper('filter_method', FILTER_CONCENTRATED) """ (bool) Flag for Kalman filtering with concentrated log-likelihood. """ inversion_methods = [ 'invert_univariate', 'solve_lu', 'invert_lu', 'solve_cholesky', 'invert_cholesky' ] invert_univariate = OptionWrapper('inversion_method', INVERT_UNIVARIATE) """ (bool) Flag for univariate inversion method (recommended). """ solve_lu = OptionWrapper('inversion_method', SOLVE_LU) """ (bool) Flag for LU and linear solver inversion method. """ invert_lu = OptionWrapper('inversion_method', INVERT_LU) """ (bool) Flag for LU inversion method. """ solve_cholesky = OptionWrapper('inversion_method', SOLVE_CHOLESKY) """ (bool) Flag for Cholesky and linear solver inversion method (recommended). """ invert_cholesky = OptionWrapper('inversion_method', INVERT_CHOLESKY) """ (bool) Flag for Cholesky inversion method. """ stability_methods = ['stability_force_symmetry'] stability_force_symmetry = ( OptionWrapper('stability_method', STABILITY_FORCE_SYMMETRY) ) """ (bool) Flag for enforcing covariance matrix symmetry """ memory_options = [ 'memory_store_all', 'memory_no_forecast', 'memory_no_predicted', 'memory_no_filtered', 'memory_no_likelihood', 'memory_no_gain', 'memory_no_smoothing', 'memory_no_std_forecast', 'memory_conserve' ] memory_store_all = OptionWrapper('conserve_memory', MEMORY_STORE_ALL) """ (bool) Flag for storing all intermediate results in memory (default). """ memory_no_forecast = OptionWrapper('conserve_memory', MEMORY_NO_FORECAST) """ (bool) Flag to prevent storing forecasts. """ memory_no_predicted = OptionWrapper('conserve_memory', MEMORY_NO_PREDICTED) """ (bool) Flag to prevent storing predicted state and covariance matrices. """ memory_no_filtered = OptionWrapper('conserve_memory', MEMORY_NO_FILTERED) """ (bool) Flag to prevent storing filtered state and covariance matrices. """ memory_no_likelihood = ( OptionWrapper('conserve_memory', MEMORY_NO_LIKELIHOOD) ) """ (bool) Flag to prevent storing likelihood values for each observation. """ memory_no_gain = OptionWrapper('conserve_memory', MEMORY_NO_GAIN) """ (bool) Flag to prevent storing the Kalman gain matrices. """ memory_no_smoothing = OptionWrapper('conserve_memory', MEMORY_NO_SMOOTHING) """ (bool) Flag to prevent storing likelihood values for each observation. """ memory_no_std_forecast = ( OptionWrapper('conserve_memory', MEMORY_NO_STD_FORECAST)) """ (bool) Flag to prevent storing standardized forecast errors. """ memory_conserve = OptionWrapper('conserve_memory', MEMORY_CONSERVE) """ (bool) Flag to conserve the maximum amount of memory. """ timing_options = [ 'timing_init_predicted', 'timing_init_filtered' ] timing_init_predicted = OptionWrapper('filter_timing', TIMING_INIT_PREDICTED) """ (bool) Flag for the default timing convention (Durbin and Koopman, 2012). """ timing_init_filtered = OptionWrapper('filter_timing', TIMING_INIT_FILTERED) """ (bool) Flag for the alternate timing convention (Kim and Nelson, 2012). """ # Default filter options filter_method = FILTER_CONVENTIONAL """ (int) Filtering method bitmask. """ inversion_method = INVERT_UNIVARIATE | SOLVE_CHOLESKY """ (int) Inversion method bitmask. """ stability_method = STABILITY_FORCE_SYMMETRY """ (int) Stability method bitmask. """ conserve_memory = MEMORY_STORE_ALL """ (int) Memory conservation bitmask. """ filter_timing = TIMING_INIT_PREDICTED """ (int) Filter timing. """ def __init__(self, k_endog, k_states, k_posdef=None, loglikelihood_burn=0, tolerance=1e-19, results_class=None, kalman_filter_classes=None, **kwargs): super(KalmanFilter, self).__init__( k_endog, k_states, k_posdef, **kwargs ) # Setup the underlying Kalman filter storage self._kalman_filters = {} # Filter options self.loglikelihood_burn = loglikelihood_burn self.results_class = ( results_class if results_class is not None else FilterResults ) # Options self.prefix_kalman_filter_map = ( kalman_filter_classes if kalman_filter_classes is not None else tools.prefix_kalman_filter_map.copy()) self.set_filter_method(**kwargs) self.set_inversion_method(**kwargs) self.set_stability_method(**kwargs) self.set_conserve_memory(**kwargs) self.set_filter_timing(**kwargs) self.tolerance = tolerance # Internal flags # The _scale internal flag is used because we may want to # use a fixed scale, in which case we want the flag to the Cython # Kalman filter to indicate that the scale should not be concentrated # out, so that self.filter_concentrated = False, but we still want to # alert the results object that we are viewing the model as one in # which the scale had been concentrated out for e.g. degree of freedom # computations. # This value should always be None, except within the fixed_scale # context, and should not be modified by users or anywhere else. self._scale = None @property def _kalman_filter(self): prefix = self.prefix if prefix in self._kalman_filters: return self._kalman_filters[prefix] return None def _initialize_filter(self, filter_method=None, inversion_method=None, stability_method=None, conserve_memory=None, tolerance=None, filter_timing=None, loglikelihood_burn=None): if filter_method is None: filter_method = self.filter_method if inversion_method is None: inversion_method = self.inversion_method if stability_method is None: stability_method = self.stability_method if conserve_memory is None: conserve_memory = self.conserve_memory if loglikelihood_burn is None: loglikelihood_burn = self.loglikelihood_burn if filter_timing is None: filter_timing = self.filter_timing if tolerance is None: tolerance = self.tolerance # Make sure we have endog if self.endog is None: raise RuntimeError('Must bind a dataset to the model before' ' filtering or smoothing.') # Initialize the representation matrices prefix, dtype, create_statespace = self._initialize_representation() # Determine if we need to (re-)create the filter # (definitely need to recreate if we recreated the _statespace object) create_filter = create_statespace or prefix not in self._kalman_filters if not create_filter: kalman_filter = self._kalman_filters[prefix] create_filter = ( not kalman_filter.conserve_memory == conserve_memory or not kalman_filter.loglikelihood_burn == loglikelihood_burn ) # If the dtype-specific _kalman_filter does not exist (or if we need # to re-create it), create it if create_filter: if prefix in self._kalman_filters: # Delete the old filter del self._kalman_filters[prefix] # Setup the filter cls = self.prefix_kalman_filter_map[prefix] self._kalman_filters[prefix] = cls( self._statespaces[prefix], filter_method, inversion_method, stability_method, conserve_memory, filter_timing, tolerance, loglikelihood_burn ) # Otherwise, update the filter parameters else: kalman_filter = self._kalman_filters[prefix] kalman_filter.set_filter_method(filter_method, False) kalman_filter.inversion_method = inversion_method kalman_filter.stability_method = stability_method kalman_filter.filter_timing = filter_timing kalman_filter.tolerance = tolerance # conserve_memory and loglikelihood_burn changes always lead to # re-created filters return prefix, dtype, create_filter, create_statespace
[docs] def set_filter_method(self, filter_method=None, **kwargs): r""" Set the filtering method The filtering method controls aspects of which Kalman filtering approach will be used. Parameters ---------- filter_method : integer, optional Bitmask value to set the filter method to. See notes for details. **kwargs Keyword arguments may be used to influence the filter method by setting individual boolean flags. See notes for details. Notes ----- The filtering method is defined by a collection of boolean flags, and is internally stored as a bitmask. The methods available are: FILTER_CONVENTIONAL = 0x01 Conventional Kalman filter. FILTER_UNIVARIATE = 0x10 Univariate approach to Kalman filtering. Overrides conventional method if both are specified. FILTER_COLLAPSED = 0x20 Collapsed approach to Kalman filtering. Will be used *in addition* to conventional or univariate filtering. FILTER_CONCENTRATED = 0x20 Use the concentrated log-likelihood function. Will be used *in addition* to the other options. Note that only the first method is available if using a Scipy version older than 0.16. If the bitmask is set directly via the `filter_method` argument, then the full method must be provided. If keyword arguments are used to set individual boolean flags, then the lowercase of the method must be used as an argument name, and the value is the desired value of the boolean flag (True or False). Note that the filter method may also be specified by directly modifying the class attributes which are defined similarly to the keyword arguments. The default filtering method is FILTER_CONVENTIONAL. Examples -------- >>> mod = sm.tsa.statespace.SARIMAX(range(10)) >>> mod.ssm.filter_method 1 >>> mod.ssm.filter_conventional True >>> mod.ssm.filter_univariate = True >>> mod.ssm.filter_method 17 >>> mod.ssm.set_filter_method(filter_univariate=False, ... filter_collapsed=True) >>> mod.ssm.filter_method 33 >>> mod.ssm.set_filter_method(filter_method=1) >>> mod.ssm.filter_conventional True >>> mod.ssm.filter_univariate False >>> mod.ssm.filter_collapsed False >>> mod.ssm.filter_univariate = True >>> mod.ssm.filter_method 17 """ if filter_method is not None: self.filter_method = filter_method for name in KalmanFilter.filter_methods: if name in kwargs: setattr(self, name, kwargs[name])
[docs] def set_inversion_method(self, inversion_method=None, **kwargs): r""" Set the inversion method The Kalman filter may contain one matrix inversion: that of the forecast error covariance matrix. The inversion method controls how and if that inverse is performed. Parameters ---------- inversion_method : integer, optional Bitmask value to set the inversion method to. See notes for details. **kwargs Keyword arguments may be used to influence the inversion method by setting individual boolean flags. See notes for details. Notes ----- The inversion method is defined by a collection of boolean flags, and is internally stored as a bitmask. The methods available are: INVERT_UNIVARIATE = 0x01 If the endogenous time series is univariate, then inversion can be performed by simple division. If this flag is set and the time series is univariate, then division will always be used even if other flags are also set. SOLVE_LU = 0x02 Use an LU decomposition along with a linear solver (rather than ever actually inverting the matrix). INVERT_LU = 0x04 Use an LU decomposition along with typical matrix inversion. SOLVE_CHOLESKY = 0x08 Use a Cholesky decomposition along with a linear solver. INVERT_CHOLESKY = 0x10 Use an Cholesky decomposition along with typical matrix inversion. If the bitmask is set directly via the `inversion_method` argument, then the full method must be provided. If keyword arguments are used to set individual boolean flags, then the lowercase of the method must be used as an argument name, and the value is the desired value of the boolean flag (True or False). Note that the inversion method may also be specified by directly modifying the class attributes which are defined similarly to the keyword arguments. The default inversion method is `INVERT_UNIVARIATE | SOLVE_CHOLESKY` Several things to keep in mind are: - If the filtering method is specified to be univariate, then simple division is always used regardless of the dimension of the endogenous time series. - Cholesky decomposition is about twice as fast as LU decomposition, but it requires that the matrix be positive definite. While this should generally be true, it may not be in every case. - Using a linear solver rather than true matrix inversion is generally faster and is numerically more stable. Examples -------- >>> mod = sm.tsa.statespace.SARIMAX(range(10)) >>> mod.ssm.inversion_method 1 >>> mod.ssm.solve_cholesky True >>> mod.ssm.invert_univariate True >>> mod.ssm.invert_lu False >>> mod.ssm.invert_univariate = False >>> mod.ssm.inversion_method 8 >>> mod.ssm.set_inversion_method(solve_cholesky=False, ... invert_cholesky=True) >>> mod.ssm.inversion_method 16 """ if inversion_method is not None: self.inversion_method = inversion_method for name in KalmanFilter.inversion_methods: if name in kwargs: setattr(self, name, kwargs[name])
[docs] def set_stability_method(self, stability_method=None, **kwargs): r""" Set the numerical stability method The Kalman filter is a recursive algorithm that may in some cases suffer issues with numerical stability. The stability method controls what, if any, measures are taken to promote stability. Parameters ---------- stability_method : integer, optional Bitmask value to set the stability method to. See notes for details. **kwargs Keyword arguments may be used to influence the stability method by setting individual boolean flags. See notes for details. Notes ----- The stability method is defined by a collection of boolean flags, and is internally stored as a bitmask. The methods available are: STABILITY_FORCE_SYMMETRY = 0x01 If this flag is set, symmetry of the predicted state covariance matrix is enforced at each iteration of the filter, where each element is set to the average of the corresponding elements in the upper and lower triangle. If the bitmask is set directly via the `stability_method` argument, then the full method must be provided. If keyword arguments are used to set individual boolean flags, then the lowercase of the method must be used as an argument name, and the value is the desired value of the boolean flag (True or False). Note that the stability method may also be specified by directly modifying the class attributes which are defined similarly to the keyword arguments. The default stability method is `STABILITY_FORCE_SYMMETRY` Examples -------- >>> mod = sm.tsa.statespace.SARIMAX(range(10)) >>> mod.ssm.stability_method 1 >>> mod.ssm.stability_force_symmetry True >>> mod.ssm.stability_force_symmetry = False >>> mod.ssm.stability_method 0 """ if stability_method is not None: self.stability_method = stability_method for name in KalmanFilter.stability_methods: if name in kwargs: setattr(self, name, kwargs[name])
[docs] def set_conserve_memory(self, conserve_memory=None, **kwargs): r""" Set the memory conservation method By default, the Kalman filter computes a number of intermediate matrices at each iteration. The memory conservation options control which of those matrices are stored. Parameters ---------- conserve_memory : integer, optional Bitmask value to set the memory conservation method to. See notes for details. **kwargs Keyword arguments may be used to influence the memory conservation method by setting individual boolean flags. See notes for details. Notes ----- The memory conservation method is defined by a collection of boolean flags, and is internally stored as a bitmask. The methods available are: MEMORY_STORE_ALL = 0 Store all intermediate matrices. This is the default value. MEMORY_NO_FORECAST = 0x01 Do not store the forecast, forecast error, or forecast error covariance matrices. If this option is used, the `predict` method from the results class is unavailable. MEMORY_NO_PREDICTED = 0x02 Do not store the predicted state or predicted state covariance matrices. MEMORY_NO_FILTERED = 0x04 Do not store the filtered state or filtered state covariance matrices. MEMORY_NO_LIKELIHOOD = 0x08 Do not store the vector of loglikelihood values for each observation. Only the sum of the loglikelihood values is stored. MEMORY_NO_GAIN = 0x10 Do not store the Kalman gain matrices. MEMORY_NO_SMOOTHING = 0x20 Do not store temporary variables related to Klaman smoothing. If this option is used, smoothing is unavailable. MEMORY_NO_SMOOTHING = 0x20 Do not store standardized forecast errors. MEMORY_CONSERVE Do not store any intermediate matrices. Note that if using a Scipy version less than 0.16, the options MEMORY_NO_GAIN, MEMORY_NO_SMOOTHING, and MEMORY_NO_STD_FORECAST have no effect. If the bitmask is set directly via the `conserve_memory` argument, then the full method must be provided. If keyword arguments are used to set individual boolean flags, then the lowercase of the method must be used as an argument name, and the value is the desired value of the boolean flag (True or False). Note that the memory conservation method may also be specified by directly modifying the class attributes which are defined similarly to the keyword arguments. The default memory conservation method is `MEMORY_STORE_ALL`, so that all intermediate matrices are stored. Examples -------- >>> mod = sm.tsa.statespace.SARIMAX(range(10)) >>> mod.ssm..conserve_memory 0 >>> mod.ssm.memory_no_predicted False >>> mod.ssm.memory_no_predicted = True >>> mod.ssm.conserve_memory 2 >>> mod.ssm.set_conserve_memory(memory_no_filtered=True, ... memory_no_forecast=True) >>> mod.ssm.conserve_memory 7 """ if conserve_memory is not None: self.conserve_memory = conserve_memory for name in KalmanFilter.memory_options: if name in kwargs: setattr(self, name, kwargs[name])
[docs] def set_filter_timing(self, alternate_timing=None, **kwargs): r""" Set the filter timing convention By default, the Kalman filter follows Durbin and Koopman, 2012, in initializing the filter with predicted values. Kim and Nelson, 1999, instead initialize the filter with filtered values, which is essentially just a different timing convention. Parameters ---------- alternate_timing : integer, optional Whether or not to use the alternate timing convention. Default is unspecified. **kwargs Keyword arguments may be used to influence the memory conservation method by setting individual boolean flags. See notes for details. """ if alternate_timing is not None: self.filter_timing = int(alternate_timing) if 'timing_init_predicted' in kwargs: self.filter_timing = int(not kwargs['timing_init_predicted']) if 'timing_init_filtered' in kwargs: self.filter_timing = int(kwargs['timing_init_filtered'])
[docs] @contextmanager def fixed_scale(self, scale): """ Context manager for fixing the scale when FILTER_CONCENTRATED is set Parameters ---------- scale : numeric Scale of the model. Notes ----- This a no-op if scale is None. This context manager is most useful in models which are explicitly concentrating out the scale, so that the set of parameters they are estimating does not include the scale. """ # If a scale was provided, use it and do not concentrate it out of the # loglikelihood if scale is not None and scale != 1: if not self.filter_concentrated: raise ValueError('Cannot provide scale if filter method does' ' not include FILTER_CONCENTRATED.') self.filter_concentrated = False self._scale = scale obs_cov = self['obs_cov'] state_cov = self['state_cov'] self['obs_cov'] = scale * obs_cov self['state_cov'] = scale * state_cov try: yield finally: # If a scale was provided, reset the model if scale is not None and scale != 1: self['state_cov'] = state_cov self['obs_cov'] = obs_cov self.filter_concentrated = True self._scale = None
def _filter(self, filter_method=None, inversion_method=None, stability_method=None, conserve_memory=None, filter_timing=None, tolerance=None, loglikelihood_burn=None, complex_step=False): # Initialize the filter prefix, dtype, create_filter, create_statespace = ( self._initialize_filter( filter_method, inversion_method, stability_method, conserve_memory, filter_timing, tolerance, loglikelihood_burn ) ) kfilter = self._kalman_filters[prefix] # Initialize the state self._initialize_state(prefix=prefix, complex_step=complex_step) # Run the filter kfilter() return kfilter
[docs] def filter(self, filter_method=None, inversion_method=None, stability_method=None, conserve_memory=None, filter_timing=None, tolerance=None, loglikelihood_burn=None, complex_step=False): r""" Apply the Kalman filter to the statespace model. Parameters ---------- filter_method : int, optional Determines which Kalman filter to use. Default is conventional. inversion_method : int, optional Determines which inversion technique to use. Default is by Cholesky decomposition. stability_method : int, optional Determines which numerical stability techniques to use. Default is to enforce symmetry of the predicted state covariance matrix. conserve_memory : int, optional Determines what output from the filter to store. Default is to store everything. filter_timing : int, optional Determines the timing convention of the filter. Default is that from Durbin and Koopman (2012), in which the filter is initialized with predicted values. tolerance : float, optional The tolerance at which the Kalman filter determines convergence to steady-state. Default is 1e-19. loglikelihood_burn : int, optional The number of initial periods during which the loglikelihood is not recorded. Default is 0. Notes ----- This function by default does not compute variables required for smoothing. """ if conserve_memory is None: conserve_memory = self.conserve_memory | MEMORY_NO_SMOOTHING # Run the filter kfilter = self._filter( filter_method, inversion_method, stability_method, conserve_memory, filter_timing, tolerance, loglikelihood_burn, complex_step) # Create the results object results = self.results_class(self) results.update_representation(self) results.update_filter(kfilter) return results
[docs] def loglike(self, **kwargs): r""" Calculate the loglikelihood associated with the statespace model. Parameters ---------- **kwargs Additional keyword arguments to pass to the Kalman filter. See `KalmanFilter.filter` for more details. Returns ------- loglike : float The joint loglikelihood. """ if self.memory_no_likelihood: raise RuntimeError('Cannot compute loglikelihood if' ' MEMORY_NO_LIKELIHOOD option is selected.') kwargs['conserve_memory'] = MEMORY_CONSERVE ^ MEMORY_NO_LIKELIHOOD kfilter = self._filter(**kwargs) loglikelihood_burn = kwargs.get('loglikelihood_burn', self.loglikelihood_burn) loglike = np.sum(kfilter.loglikelihood[loglikelihood_burn:]) # Need to modify the computed log-likelihood to incorporate the # MLE scale. if self.filter_method & FILTER_CONCENTRATED: d = max(loglikelihood_burn, kfilter.nobs_diffuse) nobs_k_endog = np.sum( self.k_endog - np.array(self._statespace.nmissing)[d:]) # In the univariate case, we need to subtract observations # associated with a singular forecast error covariance matrix nobs_k_endog -= kfilter.nobs_kendog_univariate_singular scale = np.sum(kfilter.scale[d:]) / nobs_k_endog loglike += -0.5 * nobs_k_endog # Now need to modify this for diffuse initialization, since for # diffuse periods we only need to add in the scale value part if # the diffuse forecast error covariance matrix element was singular if kfilter.nobs_diffuse > 0: nobs_k_endog -= kfilter.nobs_kendog_diffuse_nonsingular loglike += -0.5 * nobs_k_endog * np.log(scale) return loglike
[docs] def loglikeobs(self, **kwargs): r""" Calculate the loglikelihood for each observation associated with the statespace model. Parameters ---------- **kwargs Additional keyword arguments to pass to the Kalman filter. See `KalmanFilter.filter` for more details. Notes ----- If `loglikelihood_burn` is positive, then the entries in the returned loglikelihood vector are set to be zero for those initial time periods. Returns ------- loglike : array of float Array of loglikelihood values for each observation. """ if self.memory_no_likelihood: raise RuntimeError('Cannot compute loglikelihood if' ' MEMORY_NO_LIKELIHOOD option is selected.') if not self.filter_method & FILTER_CONCENTRATED: kwargs['conserve_memory'] = MEMORY_CONSERVE ^ MEMORY_NO_LIKELIHOOD kfilter = self._filter(**kwargs) llf_obs = np.array(kfilter.loglikelihood, copy=True) loglikelihood_burn = kwargs.get('loglikelihood_burn', self.loglikelihood_burn) # If the scale was concentrated out of the log-likelihood function, # then the llf_obs above is: # -0.5 * k_endog * log 2 * pi - 0.5 * log |F_t| # and we need to add in the effect of the scale: # -0.5 * k_endog * log scale - 0.5 v' F_t^{-1} v / scale # and note that v' F_t^{-1} is in the _kalman_filter.scale array # Also note that we need to adjust the nobs and k_endog in both the # denominator of the scale computation and in the llf_obs adjustment # to take into account missing values. if self.filter_method & FILTER_CONCENTRATED: d = max(loglikelihood_burn, kfilter.nobs_diffuse) nmissing = np.array(self._statespace.nmissing) nobs_k_endog = np.sum(self.k_endog - nmissing[d:]) # In the univariate case, we need to subtract observations # associated with a singular forecast error covariance matrix nobs_k_endog -= kfilter.nobs_kendog_univariate_singular scale = np.sum(kfilter.scale[d:]) / nobs_k_endog # Need to modify this for diffuse initialization, since for # diffuse periods we only need to add in the scale value if the # diffuse forecast error covariance matrix element was singular nsingular = 0 if kfilter.nobs_diffuse > 0: d = kfilter.nobs_diffuse Finf = kfilter.forecast_error_diffuse_cov singular = np.diagonal(Finf).real <= kfilter.tolerance_diffuse nsingular = np.sum(~singular, axis=1) scale_obs = np.array(kfilter.scale, copy=True) llf_obs += -0.5 * ( (self.k_endog - nmissing - nsingular) * np.log(scale) + scale_obs / scale) # Set any burned observations to have zero likelihood llf_obs[:loglikelihood_burn] = 0 return llf_obs
[docs] def simulate(self, nsimulations, measurement_shocks=None, state_shocks=None, initial_state=None): r""" Simulate a new time series following the state space model Parameters ---------- nsimulations : int The number of observations to simulate. If the model is time-invariant this can be any number. If the model is time-varying, then this number must be less than or equal to the number measurement_shocks : array_like, optional If specified, these are the shocks to the measurement equation, :math:`\varepsilon_t`. If unspecified, these are automatically generated using a pseudo-random number generator. If specified, must be shaped `nsimulations` x `k_endog`, where `k_endog` is the same as in the state space model. state_shocks : array_like, optional If specified, these are the shocks to the state equation, :math:`\eta_t`. If unspecified, these are automatically generated using a pseudo-random number generator. If specified, must be shaped `nsimulations` x `k_posdef` where `k_posdef` is the same as in the state space model. initial_state : array_like, optional If specified, this is the state vector at time zero, which should be shaped (`k_states` x 1), where `k_states` is the same as in the state space model. If unspecified, but the model has been initialized, then that initialization is used. If unspecified and the model has not been initialized, then a vector of zeros is used. Note that this is not included in the returned `simulated_states` array. Returns ------- simulated_obs : array An (nsimulations x k_endog) array of simulated observations. simulated_states : array An (nsimulations x k_states) array of simulated states. """ time_invariant = self.time_invariant # Check for valid number of simulations if not time_invariant and nsimulations > self.nobs: raise ValueError('In a time-varying model, cannot create more' ' simulations than there are observations.') # Check / generate measurement shocks if measurement_shocks is not None: measurement_shocks = np.array(measurement_shocks) if measurement_shocks.ndim == 0: measurement_shocks = measurement_shocks[np.newaxis, np.newaxis] elif measurement_shocks.ndim == 1: measurement_shocks = measurement_shocks[:, np.newaxis] if not measurement_shocks.shape == (nsimulations, self.k_endog): raise ValueError('Invalid shape of provided measurement' ' shocks. Required (%d, %d)' % (nsimulations, self.k_endog)) elif self.shapes['obs_cov'][-1] == 1: measurement_shocks = np.random.multivariate_normal( mean=np.zeros(self.k_endog), cov=self['obs_cov'], size=nsimulations) # Check / generate state shocks if state_shocks is not None: state_shocks = np.array(state_shocks) if state_shocks.ndim == 0: state_shocks = state_shocks[np.newaxis, np.newaxis] elif state_shocks.ndim == 1: state_shocks = state_shocks[:, np.newaxis] if not state_shocks.shape == (nsimulations, self.k_posdef): raise ValueError('Invalid shape of provided state shocks.' ' Required (%d, %d).' % (nsimulations, self.k_posdef)) elif self.shapes['state_cov'][-1] == 1: state_shocks = np.random.multivariate_normal( mean=np.zeros(self.k_posdef), cov=self['state_cov'], size=nsimulations) # Get the initial states if initial_state is not None: initial_state = np.array(initial_state) if initial_state.ndim == 0: initial_state = initial_state[np.newaxis] elif (initial_state.ndim > 1 and not initial_state.shape == (self.k_states, 1)): raise ValueError('Invalid shape of provided initial state' ' vector. Required (%d, 1)' % self.k_states) elif self.initialization == 'known': initial_state = np.random.multivariate_normal( self._initial_state, self._initial_state_cov) elif self.initialization == 'stationary': from scipy.linalg import solve_discrete_lyapunov # (I - T)^{-1} c = x => (I - T) x = c initial_state_mean = np.linalg.solve( np.eye(self.k_states) - self['transition', :, :, 0], self['state_intercept', :, 0]) R = self['selection', :, :, 0] Q = self['state_cov', :, :, 0] selected_state_cov = R.dot(Q).dot(R.T) initial_state_cov = solve_discrete_lyapunov( self['transition', :, :, 0], selected_state_cov) initial_state = np.random.multivariate_normal( initial_state_mean, initial_state_cov) elif self.initialization == 'approximate_diffuse': initial_state = np.zeros(self.k_states) elif self.initialization is not None: out = self.initialization(model=self) initial_state = out[0] + np.random.multivariate_normal( np.zeros_like(out[0]), out[2]) else: initial_state = np.zeros(self.k_states) return self._simulate(nsimulations, measurement_shocks, state_shocks, initial_state)
def _simulate(self, nsimulations, measurement_shocks, state_shocks, initial_state): time_invariant = self.time_invariant # Holding variables for the simulations simulated_obs = np.zeros((nsimulations, self.k_endog), dtype=self.dtype) simulated_states = np.zeros((nsimulations+1, self.k_states), dtype=self.dtype) simulated_states[0] = initial_state # Perform iterations to create the new time series obs_intercept_t = 0 design_t = 0 state_intercept_t = 0 transition_t = 0 selection_t = 0 for t in range(nsimulations): # Get the current shocks (this accomodates time-varying matrices) if measurement_shocks is None: measurement_shock = np.random.multivariate_normal( mean=np.zeros(self.k_endog), cov=self['obs_cov', :, :, t]) else: measurement_shock = measurement_shocks[t] if state_shocks is None: state_shock = np.random.multivariate_normal( mean=np.zeros(self.k_posdef), cov=self['state_cov', :, :, t]) else: state_shock = state_shocks[t] # Get current-iteration matrices if not time_invariant: obs_intercept_t = 0 if self.obs_intercept.shape[-1] == 1 else t design_t = 0 if self.design.shape[-1] == 1 else t state_intercept_t = ( 0 if self.state_intercept.shape[-1] == 1 else t) transition_t = 0 if self.transition.shape[-1] == 1 else t selection_t = 0 if self.selection.shape[-1] == 1 else t obs_intercept = self['obs_intercept', :, obs_intercept_t] design = self['design', :, :, design_t] state_intercept = self['state_intercept', :, state_intercept_t] transition = self['transition', :, :, transition_t] selection = self['selection', :, :, selection_t] # Iterate the measurement equation simulated_obs[t] = ( obs_intercept + np.dot(design, simulated_states[t]) + measurement_shock) # Iterate the state equation simulated_states[t+1] = ( state_intercept + np.dot(transition, simulated_states[t]) + np.dot(selection, state_shock)) return simulated_obs, simulated_states[:-1]
[docs] def impulse_responses(self, steps=10, impulse=0, orthogonalized=False, cumulative=False, **kwargs): r""" Impulse response function Parameters ---------- steps : int, optional The number of steps for which impulse responses are calculated. Default is 10. Note that the initial impulse is not counted as a step, so if `steps=1`, the output will have 2 entries. impulse : int or array_like If an integer, the state innovation to pulse; must be between 0 and `k_posdef-1` where `k_posdef` is the same as in the state space model. Alternatively, a custom impulse vector may be provided; must be a column vector with shape `(k_posdef, 1)`. orthogonalized : boolean, optional Whether or not to perform impulse using orthogonalized innovations. Note that this will also affect custum `impulse` vectors. Default is False. cumulative : boolean, optional Whether or not to return cumulative impulse responses. Default is False. **kwargs If the model is time-varying and `steps` is greater than the number of observations, any of the state space representation matrices that are time-varying must have updated values provided for the out-of-sample steps. For example, if `design` is a time-varying component, `nobs` is 10, and `steps` is 15, a (`k_endog` x `k_states` x 5) matrix must be provided with the new design matrix values. Returns ------- impulse_responses : array Responses for each endogenous variable due to the impulse given by the `impulse` argument. A (steps + 1 x k_endog) array. Notes ----- Intercepts in the measurement and state equation are ignored when calculating impulse responses. """ # Since the first step is the impulse itself, we actually want steps+1 steps += 1 # Check for what kind of impulse we want if type(impulse) == int: if impulse >= self.k_posdef or impulse < 0: raise ValueError('Invalid value for `impulse`. Must be the' ' index of one of the state innovations.') # Create the (non-orthogonalized) impulse vector idx = impulse impulse = np.zeros(self.k_posdef) impulse[idx] = 1 else: impulse = np.array(impulse) if impulse.ndim > 1: impulse = np.squeeze(impulse) if not impulse.shape == (self.k_posdef,): raise ValueError('Invalid impulse vector. Must be shaped' ' (%d,)' % self.k_posdef) # Orthogonalize the impulses, if requested, using Cholesky on the # first state covariance matrix if orthogonalized: state_chol = np.linalg.cholesky(self.state_cov[:, :, 0]) impulse = np.dot(state_chol, impulse) # If we have a time-invariant system, we can solve for the IRF directly # Note that it doesn't matter if we have time-invariant intercepts, # since those don't affect the IRF anyway time_invariant = ( self._design.shape[2] == self._obs_cov.shape[2] == self._transition.shape[2] == self._selection.shape[2] == self._state_cov.shape[2]) if time_invariant: # Get the state space matrices design = self.design[:, :, 0] transition = self.transition[:, :, 0] selection = self.selection[:, :, 0] # Holding arrays irf = np.zeros((steps, self.k_endog), dtype=self.dtype) states = np.zeros((steps, self.k_states), dtype=self.dtype) # First iteration states[0] = np.dot(selection, impulse) irf[0] = np.dot(design, states[0]) # Iterations for t in range(1, steps): states[t] = np.dot(transition, states[t-1]) irf[t] = np.dot(design, states[t]) # Otherwise, create a new model else: # Get the basic model components representation = {} for name, shape in self.shapes.items(): if name in ['obs', 'obs_intercept', 'state_intercept']: continue representation[name] = getattr(self, name) # Allow additional specification warning = ('Model has time-invariant %s matrix, so the %s' ' argument to `irf` has been ignored.') exception = ('Impulse response functions for models with' ' time-varying %s matrix requires an updated' ' time-varying matrix for any periods beyond those in' ' the original model.') for name, shape in self.shapes.items(): if name in ['obs', 'obs_intercept', 'state_intercept']: continue if representation[name].shape[-1] == 1: if name in kwargs: warn(warning % (name, name), ValueWarning) elif name not in kwargs: raise ValueError(exception % name) else: mat = np.asarray(kwargs[name]) validate_matrix_shape(name, mat.shape, shape[0], shape[1], steps) if mat.ndim < 3 or not mat.shape[2] == steps: raise ValueError(exception % name) representation[name] = np.c_[representation[name], mat] # Setup the new statespace representation model_kwargs = { 'filter_method': self.filter_method, 'inversion_method': self.inversion_method, 'stability_method': self.stability_method, 'conserve_memory': self.conserve_memory, 'tolerance': self.tolerance, 'loglikelihood_burn': self.loglikelihood_burn } model_kwargs.update(representation) model = self.__class__(np.zeros(self.endog.T.shape), self.k_states, self.k_posdef, **model_kwargs) model.initialize_approximate_diffuse() model._initialize_filter() model._initialize_state() # Get the impulse response function via simulation of the state # space model, but with other shocks set to zero # Since simulate returns the zero-th period, we need to simulate # steps + 1 periods and exclude the zero-th observation. steps += 1 measurement_shocks = np.zeros((steps, self.k_endog)) state_shocks = np.zeros((steps, self.k_posdef)) state_shocks[0] = impulse irf, _ = model.simulate( steps, measurement_shocks=measurement_shocks, state_shocks=state_shocks) irf = irf[1:] # Get the cumulative response if requested if cumulative: irf = np.cumsum(irf, axis=0) return irf
[docs]class FilterResults(FrozenRepresentation): """ Results from applying the Kalman filter to a state space model. Parameters ---------- model : Representation A Statespace representation Attributes ---------- nobs : int Number of observations. nobs_diffuse : int Number of observations under the diffuse Kalman filter. k_endog : int The dimension of the observation series. k_states : int The dimension of the unobserved state process. k_posdef : int The dimension of a guaranteed positive definite covariance matrix describing the shocks in the measurement equation. dtype : dtype Datatype of representation matrices prefix : str BLAS prefix of representation matrices shapes : dictionary of name,tuple A dictionary recording the shapes of each of the representation matrices as tuples. endog : array The observation vector. design : array The design matrix, :math:`Z`. obs_intercept : array The intercept for the observation equation, :math:`d`. obs_cov : array The covariance matrix for the observation equation :math:`H`. transition : array The transition matrix, :math:`T`. state_intercept : array The intercept for the transition equation, :math:`c`. selection : array The selection matrix, :math:`R`. state_cov : array The covariance matrix for the state equation :math:`Q`. missing : array of bool An array of the same size as `endog`, filled with boolean values that are True if the corresponding entry in `endog` is NaN and False otherwise. nmissing : array of int An array of size `nobs`, where the ith entry is the number (between 0 and `k_endog`) of NaNs in the ith row of the `endog` array. time_invariant : bool Whether or not the representation matrices are time-invariant initialization : str Kalman filter initialization method. initial_state : array_like The state vector used to initialize the Kalamn filter. initial_state_cov : array_like The state covariance matrix used to initialize the Kalamn filter. initial_diffuse_state_cov : array_like Diffuse state covariance matrix used to initialize the Kalamn filter. filter_method : int Bitmask representing the Kalman filtering method inversion_method : int Bitmask representing the method used to invert the forecast error covariance matrix. stability_method : int Bitmask representing the methods used to promote numerical stability in the Kalman filter recursions. conserve_memory : int Bitmask representing the selected memory conservation method. filter_timing : int Whether or not to use the alternate timing convention. tolerance : float The tolerance at which the Kalman filter determines convergence to steady-state. loglikelihood_burn : int The number of initial periods during which the loglikelihood is not recorded. converged : bool Whether or not the Kalman filter converged. period_converged : int The time period in which the Kalman filter converged. filtered_state : array The filtered state vector at each time period. filtered_state_cov : array The filtered state covariance matrix at each time period. predicted_state : array The predicted state vector at each time period. predicted_state_cov : array The predicted state covariance matrix at each time period. forecast_error_diffuse_cov : array Diffuse forecast error covariance matrix at each time period. predicted_diffuse_state_cov : array The predicted diffuse state covariance matrix at each time period. kalman_gain : array The Kalman gain at each time period. forecasts : array The one-step-ahead forecasts of observations at each time period. forecasts_error : array The forecast errors at each time period. forecasts_error_cov : array The forecast error covariance matrices at each time period. llf_obs : array The loglikelihood values at each time period. """ _filter_attributes = [ 'filter_method', 'inversion_method', 'stability_method', 'conserve_memory', 'filter_timing', 'tolerance', 'loglikelihood_burn', 'converged', 'period_converged', 'filtered_state', 'filtered_state_cov', 'predicted_state', 'predicted_state_cov', 'forecasts_error_diffuse_cov', 'predicted_diffuse_state_cov', 'tmp1', 'tmp2', 'tmp3', 'tmp4', 'forecasts', 'forecasts_error', 'forecasts_error_cov', 'llf_obs', 'collapsed_forecasts', 'collapsed_forecasts_error', 'collapsed_forecasts_error_cov', 'scale' ] _filter_options = ( KalmanFilter.filter_methods + KalmanFilter.stability_methods + KalmanFilter.inversion_methods + KalmanFilter.memory_options ) _attributes = FrozenRepresentation._model_attributes + _filter_attributes def __init__(self, model): super(FilterResults, self).__init__(model) # Setup caches for uninitialized objects self._kalman_gain = None self._standardized_forecasts_error = None
[docs] def update_representation(self, model, only_options=False): """ Update the results to match a given model Parameters ---------- model : Representation The model object from which to take the updated values. only_options : boolean, optional If set to true, only the filter options are updated, and the state space representation is not updated. Default is False. Notes ----- This method is rarely required except for internal usage. """ if not only_options: super(FilterResults, self).update_representation(model) # Save the options as boolean variables for name in self._filter_options: setattr(self, name, getattr(model, name, None))
[docs] def update_filter(self, kalman_filter): """ Update the filter results Parameters ---------- kalman_filter : statespace.kalman_filter.KalmanFilter The model object from which to take the updated values. Notes ----- This method is rarely required except for internal usage. """ # State initialization self.initial_state = np.array( kalman_filter.model.initial_state, copy=True ) self.initial_state_cov = np.array( kalman_filter.model.initial_state_cov, copy=True ) # Save Kalman filter parameters self.filter_method = kalman_filter.filter_method self.inversion_method = kalman_filter.inversion_method self.stability_method = kalman_filter.stability_method self.conserve_memory = kalman_filter.conserve_memory self.filter_timing = kalman_filter.filter_timing self.tolerance = kalman_filter.tolerance self.loglikelihood_burn = kalman_filter.loglikelihood_burn # Save Kalman filter output self.converged = bool(kalman_filter.converged) self.period_converged = kalman_filter.period_converged self.filtered_state = np.array(kalman_filter.filtered_state, copy=True) self.filtered_state_cov = np.array( kalman_filter.filtered_state_cov, copy=True ) self.predicted_state = np.array( kalman_filter.predicted_state, copy=True ) self.predicted_state_cov = np.array( kalman_filter.predicted_state_cov, copy=True ) # Reset caches has_missing = np.sum(self.nmissing) > 0 if not (self.memory_no_std_forecast or self.invert_lu or self.solve_lu or self.filter_collapsed): if has_missing: self._standardized_forecasts_error = np.array( reorder_missing_vector( kalman_filter.standardized_forecast_error, self.missing, prefix=self.prefix)) else: self._standardized_forecasts_error = np.array( kalman_filter.standardized_forecast_error, copy=True) else: self._standardized_forecasts_error = None # In the partially missing data case, all entries will # be in the upper left submatrix rather than the correct placement # Re-ordering does not make sense in the collapsed case. if has_missing and (not self.memory_no_gain and not self.filter_collapsed): self._kalman_gain = np.array(reorder_missing_matrix( kalman_filter.kalman_gain, self.missing, reorder_cols=True, prefix=self.prefix)) self.tmp1 = np.array(reorder_missing_matrix( kalman_filter.tmp1, self.missing, reorder_cols=True, prefix=self.prefix)) self.tmp2 = np.array(reorder_missing_vector( kalman_filter.tmp2, self.missing, prefix=self.prefix)) self.tmp3 = np.array(reorder_missing_matrix( kalman_filter.tmp3, self.missing, reorder_rows=True, prefix=self.prefix)) self.tmp4 = np.array(reorder_missing_matrix( kalman_filter.tmp4, self.missing, reorder_cols=True, reorder_rows=True, prefix=self.prefix)) else: self._kalman_gain = np.array( kalman_filter.kalman_gain, copy=True) self.tmp1 = np.array(kalman_filter.tmp1, copy=True) self.tmp2 = np.array(kalman_filter.tmp2, copy=True) self.tmp3 = np.array(kalman_filter.tmp3, copy=True) self.tmp4 = np.array(kalman_filter.tmp4, copy=True) self.M = np.array(kalman_filter.M, copy=True) self.M_diffuse = np.array(kalman_filter.M_inf, copy=True) # Note: use forecasts rather than forecast, so as not to interfer # with the `forecast` methods in subclasses self.forecasts = np.array(kalman_filter.forecast, copy=True) self.forecasts_error = np.array( kalman_filter.forecast_error, copy=True ) self.forecasts_error_cov = np.array( kalman_filter.forecast_error_cov, copy=True ) self.llf_obs = np.array(kalman_filter.loglikelihood, copy=True) # Diffuse objects self.nobs_diffuse = kalman_filter.nobs_diffuse self.initial_diffuse_state_cov = None self.forecasts_error_diffuse_cov = None self.predicted_diffuse_state_cov = None if self.nobs_diffuse > 0: self.initial_diffuse_state_cov = np.array( kalman_filter.model.initial_diffuse_state_cov, copy=True) self.predicted_diffuse_state_cov = np.array( kalman_filter.predicted_diffuse_state_cov, copy=True) if has_missing and not self.filter_collapsed: self.forecasts_error_diffuse_cov = np.array( reorder_missing_matrix( kalman_filter.forecast_error_diffuse_cov, self.missing, reorder_cols=True, reorder_rows=True, prefix=self.prefix)) else: self.forecasts_error_diffuse_cov = np.array( kalman_filter.forecast_error_diffuse_cov, copy=True) # If there was missing data, save the original values from the Kalman # filter output, since below will set the values corresponding to # the missing observations to nans. self.missing_forecasts = None self.missing_forecasts_error = None self.missing_forecasts_error_cov = None if np.sum(self.nmissing) > 0: # Copy the provided arrays (which are as the Kalman filter dataset) # into new variables self.missing_forecasts = np.copy(self.forecasts) self.missing_forecasts_error = np.copy(self.forecasts_error) self.missing_forecasts_error_cov = ( np.copy(self.forecasts_error_cov) ) # Save the collapsed values self.collapsed_forecasts = None self.collapsed_forecasts_error = None self.collapsed_forecasts_error_cov = None if self.filter_collapsed: # Copy the provided arrays (which are from the collapsed dataset) # into new variables self.collapsed_forecasts = self.forecasts[:self.k_states, :] self.collapsed_forecasts_error = ( self.forecasts_error[:self.k_states, :] ) self.collapsed_forecasts_error_cov = ( self.forecasts_error_cov[:self.k_states, :self.k_states, :] ) # Recreate the original arrays (which should be from the original # dataset) in the appropriate dimension self.forecasts = np.zeros((self.k_endog, self.nobs)) self.forecasts_error = np.zeros((self.k_endog, self.nobs)) self.forecasts_error_cov = ( np.zeros((self.k_endog, self.k_endog, self.nobs)) ) # Fill in missing values in the forecast, forecast error, and # forecast error covariance matrix (this is required due to how the # Kalman filter implements observations that are either partly or # completely missing) # Construct the predictions, forecasts if not (self.memory_no_forecast or self.memory_no_predicted): for t in range(self.nobs): design_t = 0 if self.design.shape[2] == 1 else t obs_cov_t = 0 if self.obs_cov.shape[2] == 1 else t obs_intercept_t = 0 if self.obs_intercept.shape[1] == 1 else t # For completely missing observations, the Kalman filter will # produce forecasts, but forecast errors and the forecast # error covariance matrix will be zeros - make them nan to # improve clarity of results. if self.nmissing[t] > 0: mask = ~self.missing[:, t].astype(bool) # We can recover forecasts # For partially missing observations, the Kalman filter # will produce all elements (forecasts, forecast errors, # forecast error covariance matrices) as usual, but their # dimension will only be equal to the number of non-missing # elements, and their location in memory will be in the # first blocks (e.g. for the forecasts_error, the first # k_endog - nmissing[t] columns will be filled in), # regardless of which endogenous variables they refer to # (i.e. the non- missing endogenous variables for that # observation). Furthermore, the forecast error covariance # matrix is only valid for those elements. What is done is # to set all elements to nan for these observations so that # they are flagged as missing. The variables # missing_forecasts, etc. then provide the forecasts, etc. # provided by the Kalman filter, from which the data can be # retrieved if desired. self.forecasts[:, t] = np.dot( self.design[:, :, design_t], self.predicted_state[:, t] ) + self.obs_intercept[:, obs_intercept_t] self.forecasts_error[:, t] = np.nan self.forecasts_error[mask, t] = ( self.endog[mask, t] - self.forecasts[mask, t]) # TODO: We should only fill in the non-masked elements of # this array. Also, this will give the multivariate version # even if univariate filtering was selected. Instead, we # should use the reordering methods and then replace the # masked values with NaNs self.forecasts_error_cov[:, :, t] = np.dot( np.dot(self.design[:, :, design_t], self.predicted_state_cov[:, :, t]), self.design[:, :, design_t].T ) + self.obs_cov[:, :, obs_cov_t] # In the collapsed case, everything just needs to be rebuilt # for the original observed data, since the Kalman filter # produced these values for the collapsed data. elif self.filter_collapsed: self.forecasts[:, t] = np.dot( self.design[:, :, design_t], self.predicted_state[:, t] ) + self.obs_intercept[:, obs_intercept_t] self.forecasts_error[:, t] = ( self.endog[:, t] - self.forecasts[:, t] ) self.forecasts_error_cov[:, :, t] = np.dot( np.dot(self.design[:, :, design_t], self.predicted_state_cov[:, :, t]), self.design[:, :, design_t].T ) + self.obs_cov[:, :, obs_cov_t] # Note: if we concentrated out the scale, need to adjust the # loglikelihood values and all of the covariance matrices and the # values that depend on the covariance matrices # Note: concentrated computation is not permitted with collapsed # version, so we do not need to modify collapsed arrays. self.scale = 1. if self.filter_concentrated and self.model._scale is None: d = max(self.loglikelihood_burn, self.nobs_diffuse) # Compute the scale nmissing = np.array(kalman_filter.model.nmissing) nobs_k_endog = np.sum(self.k_endog - nmissing[d:]) # In the univariate case, we need to subtract observations # associated with a singular forecast error covariance matrix nobs_k_endog -= kalman_filter.nobs_kendog_univariate_singular scale_obs = np.array(kalman_filter.scale, copy=True) self.scale = np.sum(scale_obs[d:]) / nobs_k_endog # Need to modify this for diffuse initialization, since for # diffuse periods we only need to add in the scale value if the # diffuse forecast error covariance matrix element was singular nsingular = 0 if kalman_filter.nobs_diffuse > 0: d = kalman_filter.nobs_diffuse Finf = kalman_filter.forecast_error_diffuse_cov singular = (np.diagonal(Finf).real <= kalman_filter.tolerance_diffuse) nsingular = np.sum(~singular, axis=1) # Adjust the loglikelihood obs (see `KalmanFilter.loglikeobs` for # defaults on the adjustment) self.llf_obs += -0.5 * ( (self.k_endog - nmissing - nsingular) * np.log(self.scale) + scale_obs / self.scale) # Scale the filter output self.obs_cov = self.obs_cov * self.scale self.state_cov = self.state_cov * self.scale self.initial_state_cov = self.initial_state_cov * self.scale self.predicted_state_cov = self.predicted_state_cov * self.scale self.filtered_state_cov = self.filtered_state_cov * self.scale self.forecasts_error_cov = self.forecasts_error_cov * self.scale if self.missing_forecasts_error_cov is not None: self.missing_forecasts_error_cov = ( self.missing_forecasts_error_cov * self.scale) # Note: do not have to adjust the Kalman gain or tmp4 self.tmp1 = self.tmp1 * self.scale self.tmp2 = self.tmp2 / self.scale self.tmp3 = self.tmp3 / self.scale if not (self.memory_no_std_forecast or self.invert_lu or self.solve_lu or self.filter_collapsed): self._standardized_forecasts_error = ( self._standardized_forecasts_error / self.scale**0.5) # The self.model._scale value is only not None within a fixed_scale # context, in which case it is set and indicates that we should # generally view this results object as using a concentrated scale # (e.g. for d.o.f. computations), but because the fixed scale was # actually applied to the model prior to filtering, we do not need to # make any adjustments to the filter output, etc. elif self.model._scale is not None: self.filter_concentrated = True self.scale = self.model._scale
@property def kalman_gain(self): """ Kalman gain matrices """ if self._kalman_gain is None: # k x n self._kalman_gain = np.zeros( (self.k_states, self.k_endog, self.nobs), dtype=self.dtype) for t in range(self.nobs): # In the case of entirely missing observations, let the Kalman # gain be zeros. if self.nmissing[t] == self.k_endog: continue design_t = 0 if self.design.shape[2] == 1 else t transition_t = 0 if self.transition.shape[2] == 1 else t if self.nmissing[t] == 0: self._kalman_gain[:, :, t] = np.dot( np.dot( self.transition[:, :, transition_t], self.predicted_state_cov[:, :, t] ), np.dot( np.transpose(self.design[:, :, design_t]), np.linalg.inv(self.forecasts_error_cov[:, :, t]) ) ) else: mask = ~self.missing[:, t].astype(bool) F = self.forecasts_error_cov[np.ix_(mask, mask, [t])] self._kalman_gain[:, mask, t] = np.dot( np.dot( self.transition[:, :, transition_t], self.predicted_state_cov[:, :, t] ), np.dot( np.transpose(self.design[mask, :, design_t]), np.linalg.inv(F[:, :, 0]) ) ) return self._kalman_gain @property def standardized_forecasts_error(self): """ Standardized forecast errors Notes ----- The forecast errors produced by the Kalman filter are .. math:: v_t \sim N(0, F_t) Hypothesis tests are usually applied to the standardized residuals .. math:: v_t^s = B_t v_t \sim N(0, I) where :math:`B_t = L_t^{-1}` and :math:`F_t = L_t L_t'`; then :math:`F_t^{-1} = (L_t')^{-1} L_t^{-1} = B_t' B_t`; :math:`B_t` and :math:`L_t` are lower triangular. Finally, :math:`B_t v_t \sim N(0, B_t F_t B_t')` and :math:`B_t F_t B_t' = L_t^{-1} L_t L_t' (L_t')^{-1} = I`. Thus we can rewrite :math:`v_t^s = L_t^{-1} v_t` or :math:`L_t v_t^s = v_t`; the latter equation is the form required to use a linear solver to recover :math:`v_t^s`. Since :math:`L_t` is lower triangular, we can use a triangular solver (?TRTRS). """ if self._standardized_forecasts_error is None: if self.k_endog == 1: self._standardized_forecasts_error = ( self.forecasts_error / self.forecasts_error_cov[0, 0, :]**0.5) else: from scipy import linalg self._standardized_forecasts_error = np.zeros( self.forecasts_error.shape, dtype=self.dtype) for t in range(self.forecasts_error_cov.shape[2]): if self.nmissing[t] > 0: self._standardized_forecasts_error[:, t] = np.nan if self.nmissing[t] < self.k_endog: mask = ~self.missing[:, t].astype(bool) F = self.forecasts_error_cov[np.ix_(mask, mask, [t])] upper, _ = linalg.cho_factor(F[:, :, 0]) self._standardized_forecasts_error[mask, t] = ( linalg.solve_triangular( upper, self.forecasts_error[mask, t], trans=1)) return self._standardized_forecasts_error
[docs] def predict(self, start=None, end=None, dynamic=None, **kwargs): r""" In-sample and out-of-sample prediction for state space models generally Parameters ---------- start : int, optional Zero-indexed observation number at which to start forecasting, i.e., the first forecast will be at start. end : int, optional Zero-indexed observation number at which to end forecasting, i.e., the last forecast will be at end. dynamic : int, optional Offset relative to `start` at which to begin dynamic prediction. Prior to this observation, true endogenous values will be used for prediction; starting with this observation and continuing through the end of prediction, forecasted endogenous values will be used instead. **kwargs If the prediction range is outside of the sample range, any of the state space representation matrices that are time-varying must have updated values provided for the out-of-sample range. For example, of `obs_intercept` is a time-varying component and the prediction range extends 10 periods beyond the end of the sample, a (`k_endog` x 10) matrix must be provided with the new intercept values. Returns ------- results : kalman_filter.PredictionResults A PredictionResults object. Notes ----- All prediction is performed by applying the deterministic part of the measurement equation using the predicted state variables. Out-of-sample prediction first applies the Kalman filter to missing data for the number of periods desired to obtain the predicted states. """ # Cannot predict if we do not have appropriate arrays if self.memory_no_forecast or self.memory_no_predicted: raise ValueError('Predict is not possible if memory conservation' ' has been used to avoid storing forecasts or' ' predicted values.') # Get the start and the end of the entire prediction range if start is None: start = 0 elif start < 0: raise ValueError('Cannot predict values previous to the sample.') if end is None: end = self.nobs # Prediction and forecasting is performed by iterating the Kalman # Kalman filter through the entire range [0, end] # Then, everything is returned corresponding to the range [start, end]. # In order to perform the calculations, the range is separately split # up into the following categories: # - static: (in-sample) the Kalman filter is run as usual # - dynamic: (in-sample) the Kalman filter is run, but on missing data # - forecast: (out-of-sample) the Kalman filter is run, but on missing # data # Short-circuit if end is before start if end <= start: raise ValueError('End of prediction must be after start.') # Get the number of forecasts to make after the end of the sample nforecast = max(0, end - self.nobs) # Get the number of dynamic prediction periods # If `dynamic=True`, then assume that we want to begin dynamic # prediction at the start of the sample prediction. if dynamic is True: dynamic = 0 # If `dynamic=False`, then assume we want no dynamic prediction if dynamic is False: dynamic = None ndynamic = 0 if dynamic is not None: # Replace the relative dynamic offset with an absolute offset dynamic = start + dynamic # Validate the `dynamic` parameter if dynamic < 0: raise ValueError('Dynamic prediction cannot begin prior to the' ' first observation in the sample.') elif dynamic > end: warn('Dynamic prediction specified to begin after the end of' ' prediction, and so has no effect.', ValueWarning) dynamic = None elif dynamic > self.nobs: warn('Dynamic prediction specified to begin during' ' out-of-sample forecasting period, and so has no' ' effect.', ValueWarning) dynamic = None # Get the total size of the desired dynamic forecasting component # Note: the first `dynamic` periods of prediction are actually # *not* dynamic, because dynamic prediction begins at observation # `dynamic`. if dynamic is not None: ndynamic = max(0, min(end, self.nobs) - dynamic) # Get the number of in-sample static predictions nstatic = min(end, self.nobs) if dynamic is None else dynamic # Construct the design and observation intercept and covariance # matrices for start-npadded:end. If not time-varying in the original # model, then they will be copied over if none are provided in # `kwargs`. Otherwise additional matrices must be provided in `kwargs`. representation = {} for name, shape in self.shapes.items(): if name == 'obs': continue representation[name] = getattr(self, name) # Update the matrices from kwargs for forecasts warning = ('Model has time-invariant %s matrix, so the %s' ' argument to `predict` has been ignored.') exception = ('Forecasting for models with time-varying %s matrix' ' requires an updated time-varying matrix for the' ' period to be forecasted.') if nforecast > 0: for name, shape in self.shapes.items(): if name == 'obs': continue if representation[name].shape[-1] == 1: if name in kwargs: warn(warning % (name, name), ValueWarning) elif name not in kwargs: raise ValueError(exception % name) else: mat = np.asarray(kwargs[name]) if len(shape) == 2: validate_vector_shape(name, mat.shape, shape[0], nforecast) if mat.ndim < 2 or not mat.shape[1] == nforecast: raise ValueError(exception % name) representation[name] = np.c_[representation[name], mat] else: validate_matrix_shape(name, mat.shape, shape[0], shape[1], nforecast) if mat.ndim < 3 or not mat.shape[2] == nforecast: raise ValueError(exception % name) representation[name] = np.c_[representation[name], mat] # Update the matrices from kwargs for dynamic prediction in the case # that `end` is less than `nobs` and `dynamic` is less than `end`. In # this case, any time-varying matrices in the default `representation` # will be too long, causing an error to be thrown below in the # KalmanFilter(...) construction call, because the endog has length # nstatic + ndynamic + nforecast, whereas the time-varying matrices # from `representation` have length nobs. if ndynamic > 0 and end < self.nobs: for name, shape in self.shapes.items(): if not name == 'obs' and representation[name].shape[-1] > 1: representation[name] = representation[name][..., :end] # Construct the predicted state and covariance matrix for each time # period depending on whether that time period corresponds to # one-step-ahead prediction, dynamic prediction, or out-of-sample # forecasting. # If we only have simple prediction, then we can use the already saved # Kalman filter output if ndynamic == 0 and nforecast == 0: results = self else: # Construct the new endogenous array. endog = np.empty((self.k_endog, ndynamic + nforecast)) endog.fill(np.nan) endog = np.asfortranarray(np.c_[self.endog[:, :nstatic], endog]) # Do not propagate through FILTER_CONCENTRATED, because we want # to perform prediction based on the estimated values, and one of # the estimated values is the scale (and in any case, the # obs_cov and state_cov have been updated to reflect the scale # estimate already) filter_method = self.filter_method & ~FILTER_CONCENTRATED # Setup the new statespace representation model_kwargs = { 'filter_method': filter_method, 'inversion_method': self.inversion_method, 'stability_method': self.stability_method, 'conserve_memory': self.conserve_memory, 'filter_timing': self.filter_timing, 'tolerance': self.tolerance, 'loglikelihood_burn': self.loglikelihood_burn } model_kwargs.update(representation) model = self.model.__class__( endog, self.k_states, self.k_posdef, **model_kwargs ) model.initialization = self.initialization model._initialize_filter() model._initialize_state() results = self._predict(nstatic, ndynamic, nforecast, model) return PredictionResults(results, start, end, nstatic, ndynamic, nforecast)
def _predict(self, nstatic, ndynamic, nforecast, model): # Note: this doesn't use self, and can either be a static method or # moved outside the class altogether. # Get the underlying filter kfilter = model._kalman_filter # Save this (which shares memory with the memoryview on which the # Kalman filter will be operating) so that we can replace actual data # with predicted data during dynamic forecasting endog = model._representations[model.prefix]['obs'] for t in range(kfilter.model.nobs): # Run the Kalman filter for the first `nstatic` periods (for # which dynamic computation will not be performed) if t < nstatic: next(kfilter) # Perform dynamic prediction elif t < nstatic + ndynamic: design_t = 0 if model.design.shape[2] == 1 else t obs_intercept_t = 0 if model.obs_intercept.shape[1] == 1 else t # Unconditional value is the intercept (often zeros) endog[:, t] = model.obs_intercept[:, obs_intercept_t] # If t > 0, then we can condition the forecast on the state if t > 0: # Predict endog[:, t] given `predicted_state` calculated in # previous iteration (i.e. t-1) endog[:, t] += np.dot( model.design[:, :, design_t], kfilter.predicted_state[:, t] ) # Advance Kalman filter next(kfilter) # Perform any (one-step-ahead) forecasting else: next(kfilter) # Return the predicted state and predicted state covariance matrices results = FilterResults(model) results.update_representation(model) results.update_filter(kfilter) return results
[docs]class PredictionResults(FilterResults): r""" Results of in-sample and out-of-sample prediction for state space models generally Parameters ---------- results : FilterResults Output from filtering, corresponding to the prediction desired start : int Zero-indexed observation number at which to start forecasting, i.e., the first forecast will be at start. end : int Zero-indexed observation number at which to end forecasting, i.e., the last forecast will be at end. nstatic : int Number of in-sample static predictions (these are always the first elements of the prediction output). ndynamic : int Number of in-sample dynamic predictions (these always follow the static predictions directly, and are directly followed by the forecasts). nforecast : int Number of in-sample forecasts (these always follow the dynamic predictions directly). Attributes ---------- npredictions : int Number of observations in the predicted series; this is not necessarily the same as the number of observations in the original model from which prediction was performed. start : int Zero-indexed observation number at which to start prediction, i.e., the first predict will be at `start`; this is relative to the original model from which prediction was performed. end : int Zero-indexed observation number at which to end prediction, i.e., the last predict will be at `end`; this is relative to the original model from which prediction was performed. nstatic : int Number of in-sample static predictions. ndynamic : int Number of in-sample dynamic predictions. nforecast : int Number of in-sample forecasts. endog : array The observation vector. design : array The design matrix, :math:`Z`. obs_intercept : array The intercept for the observation equation, :math:`d`. obs_cov : array The covariance matrix for the observation equation :math:`H`. transition : array The transition matrix, :math:`T`. state_intercept : array The intercept for the transition equation, :math:`c`. selection : array The selection matrix, :math:`R`. state_cov : array The covariance matrix for the state equation :math:`Q`. filtered_state : array The filtered state vector at each time period. filtered_state_cov : array The filtered state covariance matrix at each time period. predicted_state : array The predicted state vector at each time period. predicted_state_cov : array The predicted state covariance matrix at each time period. forecasts : array The one-step-ahead forecasts of observations at each time period. forecasts_error : array The forecast errors at each time period. forecasts_error_cov : array The forecast error covariance matrices at each time period. Notes ----- The provided ranges must be conformable, meaning that it must be that `end - start == nstatic + ndynamic + nforecast`. This class is essentially a view to the FilterResults object, but returning the appropriate ranges for everything. """ representation_attributes = [ 'endog', 'design', 'design', 'obs_intercept', 'obs_cov', 'transition', 'state_intercept', 'selection', 'state_cov' ] filter_attributes = [ 'filtered_state', 'filtered_state_cov', 'predicted_state', 'predicted_state_cov', 'forecasts', 'forecasts_error', 'forecasts_error_cov' ] def __init__(self, results, start, end, nstatic, ndynamic, nforecast): # Save the filter results object self.results = results # Save prediction ranges self.npredictions = start - end self.start = start self.end = end self.nstatic = nstatic self.ndynamic = ndynamic self.nforecast = nforecast def __getattr__(self, attr): """ Provide access to the representation and filtered output in the appropriate range (`start` - `end`). """ # Prevent infinite recursive lookups if attr[0] == '_': raise AttributeError("'%s' object has no attribute '%s'" % (self.__class__.__name__, attr)) _attr = '_' + attr # Cache the attribute if not hasattr(self, _attr): if attr == 'endog' or attr in self.filter_attributes: # Get a copy value = getattr(self.results, attr).copy() # Subset to the correct time frame value = value[..., self.start:self.end] elif attr in self.representation_attributes: value = getattr(self.results, attr).copy() # If a time-invariant matrix, return it. Otherwise, subset to # the correct period. if value.shape[-1] == 1: value = value[..., 0] else: value = value[..., self.start:self.end] else: raise AttributeError("'%s' object has no attribute '%s'" % (self.__class__.__name__, attr)) setattr(self, _attr, value) return getattr(self, _attr)