# Source code for statsmodels.tsa.statespace.kalman_filter

"""
State Space Representation and Kalman Filter

Author: Chad Fulton
License: Simplified-BSD
"""

import contextlib
from warnings import warn

import numpy as np
from .representation import OptionWrapper, Representation, FrozenRepresentation
from .tools import 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
FILTER_CHANDRASEKHAR = 0x200   # Herbst (2015)

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_MEAN = 0x01
MEMORY_NO_FORECAST_COV = 0x02
MEMORY_NO_FORECAST = MEMORY_NO_FORECAST_MEAN | MEMORY_NO_FORECAST_COV
MEMORY_NO_PREDICTED_MEAN = 0x04
MEMORY_NO_PREDICTED_COV = 0x08
MEMORY_NO_PREDICTED = MEMORY_NO_PREDICTED_MEAN | MEMORY_NO_PREDICTED_COV
MEMORY_NO_FILTERED_MEAN = 0x10
MEMORY_NO_FILTERED_COV = 0x20
MEMORY_NO_FILTERED = MEMORY_NO_FILTERED_MEAN | MEMORY_NO_FILTERED_COV
MEMORY_NO_LIKELIHOOD = 0x40
MEMORY_NO_GAIN = 0x80
MEMORY_NO_SMOOTHING = 0x100
MEMORY_NO_STD_FORECAST = 0x200
MEMORY_CONSERVE = (
MEMORY_NO_FORECAST_COV | MEMORY_NO_PREDICTED | MEMORY_NO_FILTERED |
MEMORY_NO_LIKELIHOOD | MEMORY_NO_GAIN | MEMORY_NO_SMOOTHING
)

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, int} 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. See Also -------- FilterResults statsmodels.tsa.statespace.representation.Representation 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. """ filter_methods = [ 'filter_conventional', 'filter_exact_initial', 'filter_augmented', 'filter_square_root', 'filter_univariate', 'filter_collapsed', 'filter_extended', 'filter_unscented', 'filter_concentrated', 'filter_chandrasekhar' ] 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. """ filter_chandrasekhar = OptionWrapper('filter_method', FILTER_CHANDRASEKHAR) """ (bool) Flag for filtering with Chandrasekhar recursions. """ 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_mean', 'memory_no_forecast_cov', 'memory_no_forecast', 'memory_no_predicted_mean', 'memory_no_predicted_cov', 'memory_no_predicted', 'memory_no_filtered_mean', 'memory_no_filtered_cov', '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_mean = OptionWrapper( 'conserve_memory', MEMORY_NO_FORECAST_MEAN) """ (bool) Flag to prevent storing forecasts and forecast errors. """ memory_no_forecast_cov = OptionWrapper( 'conserve_memory', MEMORY_NO_FORECAST_COV) """ (bool) Flag to prevent storing forecast error covariance matrices. """ @property def memory_no_forecast(self): """ (bool) Flag to prevent storing all forecast-related output. """ return self.memory_no_forecast_mean or self.memory_no_forecast_cov @memory_no_forecast.setter def memory_no_forecast(self, value): if bool(value): self.memory_no_forecast_mean = True self.memory_no_forecast_cov = True else: self.memory_no_forecast_mean = False self.memory_no_forecast_cov = False memory_no_predicted_mean = OptionWrapper( 'conserve_memory', MEMORY_NO_PREDICTED_MEAN) """ (bool) Flag to prevent storing predicted states. """ memory_no_predicted_cov = OptionWrapper( 'conserve_memory', MEMORY_NO_PREDICTED_COV) """ (bool) Flag to prevent storing predicted state covariance matrices. """ @property def memory_no_predicted(self): """ (bool) Flag to prevent storing predicted state and covariance matrices. """ return self.memory_no_predicted_mean or self.memory_no_predicted_cov @memory_no_predicted.setter def memory_no_predicted(self, value): if bool(value): self.memory_no_predicted_mean = True self.memory_no_predicted_cov = True else: self.memory_no_predicted_mean = False self.memory_no_predicted_cov = False memory_no_filtered_mean = OptionWrapper( 'conserve_memory', MEMORY_NO_FILTERED_MEAN) """ (bool) Flag to prevent storing filtered states. """ memory_no_filtered_cov = OptionWrapper( 'conserve_memory', MEMORY_NO_FILTERED_COV) """ (bool) Flag to prevent storing filtered state covariance matrices. """ @property def memory_no_filtered(self): """ (bool) Flag to prevent storing filtered state and covariance matrices. """ return self.memory_no_filtered_mean or self.memory_no_filtered_cov @memory_no_filtered.setter def memory_no_filtered(self, value): if bool(value): self.memory_no_filtered_mean = True self.memory_no_filtered_cov = True else: self.memory_no_filtered_mean = False self.memory_no_filtered_cov = False 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 def _clone_kwargs(self, endog, **kwargs): # See Representation._clone_kwargs for docstring kwargs = super(KalmanFilter, self)._clone_kwargs(endog, **kwargs) # Get defaults for options kwargs.setdefault('filter_method', self.filter_method) kwargs.setdefault('inversion_method', self.inversion_method) kwargs.setdefault('stability_method', self.stability_method) kwargs.setdefault('conserve_memory', self.conserve_memory) kwargs.setdefault('filter_timing', self.filter_timing) kwargs.setdefault('tolerance', self.tolerance) kwargs.setdefault('loglikelihood_burn', self.loglikelihood_burn) return kwargs @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 : int, 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 Conventional Kalman filter. FILTER_UNIVARIATE Univariate approach to Kalman filtering. Overrides conventional method if both are specified. FILTER_COLLAPSED Collapsed approach to Kalman filtering. Will be used *in addition* to conventional or univariate filtering. FILTER_CONCENTRATED 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 : int, 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 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 Use an LU decomposition along with a linear solver (rather than ever actually inverting the matrix). INVERT_LU Use an LU decomposition along with typical matrix inversion. SOLVE_CHOLESKY Use a Cholesky decomposition along with a linear solver. INVERT_CHOLESKY 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 : int, 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 : int, 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 Store all intermediate matrices. This is the default value. MEMORY_NO_FORECAST_MEAN Do not store the forecast or forecast errors. If this option is used, the predict method from the results class is unavailable. MEMORY_NO_FORECAST_COV Do not store the forecast error covariance matrices. MEMORY_NO_FORECAST 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_MEAN Do not store the predicted state. MEMORY_NO_PREDICTED_COV Do not store the predicted state covariance matrices. MEMORY_NO_PREDICTED Do not store the predicted state or predicted state covariance matrices. MEMORY_NO_FILTERED_MEAN Do not store the filtered state. MEMORY_NO_FILTERED_COV Do not store the filtered state covariance matrices. MEMORY_NO_FILTERED Do not store the filtered state or filtered state covariance matrices. MEMORY_NO_LIKELIHOOD Do not store the vector of loglikelihood values for each observation. Only the sum of the loglikelihood values is stored. MEMORY_NO_GAIN Do not store the Kalman gain matrices. MEMORY_NO_SMOOTHING Do not store temporary variables related to Kalman smoothing. If this option is used, smoothing is unavailable. MEMORY_NO_STD_FORECAST Do not store standardized forecast errors. MEMORY_CONSERVE Do not store any intermediate matrices. 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 : int, 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] @contextlib.contextmanager def fixed_scale(self, scale): """ fixed_scale(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. """ # Handle memory conservation if conserve_memory is None: conserve_memory = self.conserve_memory | MEMORY_NO_SMOOTHING conserve_memory_cache = self.conserve_memory self.set_conserve_memory(conserve_memory) # 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) # Resent memory conservation self.set_conserve_memory(conserve_memory_cache) 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. """ kwargs.setdefault('conserve_memory', MEMORY_CONSERVE ^ MEMORY_NO_LIKELIHOOD) kfilter = self._filter(**kwargs) loglikelihood_burn = kwargs.get('loglikelihood_burn', self.loglikelihood_burn) if not (kwargs['conserve_memory'] & MEMORY_NO_LIKELIHOOD): loglike = np.sum(kfilter.loglikelihood[loglikelihood_burn:]) else: loglike = np.sum(kfilter.loglikelihood) # 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 if not (kwargs['conserve_memory'] & MEMORY_NO_LIKELIHOOD): scale = np.sum(kfilter.scale[d:]) / nobs_k_endog else: scale = kfilter.scale[0] / 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.setdefault('conserve_memory', MEMORY_CONSERVE ^ MEMORY_NO_LIKELIHOOD) else: kwargs.setdefault( 'conserve_memory', MEMORY_CONSERVE ^ (MEMORY_NO_FORECAST | 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 : ndarray An (nsimulations x k_endog) array of simulated observations. simulated_states : ndarray 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] required_shape = (nsimulations, self.k_endog) try: measurement_shocks = measurement_shocks.reshape(required_shape) except ValueError: raise ValueError('Provided measurement shocks are not of the' ' appropriate shape. Required %s, got %s.' % (str(required_shape), str(measurement_shocks.shape))) 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] required_shape = (nsimulations, self.k_posdef) try: state_shocks = state_shocks.reshape(required_shape) except ValueError: raise ValueError('Provided state shocks are not of the' ' appropriate shape. Required %s, got %s.' % (str(required_shape), str(state_shocks.shape))) 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) # Handle time-varying case tvp = (self.shapes['obs_cov'][-1] > 1 or self.shapes['state_cov'][-1] > 1) if tvp and measurement_shocks is None: measurement_shocks = np.zeros((nsimulations, self.k_endog)) for i in range(nsimulations): measurement_shocks[i] = np.random.multivariate_normal( mean=np.zeros(self.k_endog), cov=self['obs_cov', ..., i]) if tvp and state_shocks is None: state_shocks = np.zeros((nsimulations, self.k_posdef)) for i in range(nsimulations): state_shocks[i] = np.random.multivariate_normal( mean=np.zeros(self.k_posdef), cov=self['state_cov', ..., i]) # 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 is not None: out = self.initialization(model=self) initial_state = out[0] + np.random.multivariate_normal( np.zeros_like(out[0]), out[2]) else: # TODO: deprecate this, since we really should not be simulating # unless we have an initialization. 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): raise NotImplementedError('Simulation only available through' ' the simulation smoother.')
[docs] def impulse_responses(self, steps=10, impulse=0, orthogonalized=False, cumulative=False, direct=False): 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 : bool, optional Whether or not to perform impulse using orthogonalized innovations. Note that this will also affect custum impulse vectors. Default is False. cumulative : bool, optional Whether or not to return cumulative impulse responses. Default is False. Returns ------- impulse_responses : ndarray 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. TODO: add note about how for time-varying systems this is - perhaps counter-intuitively - returning the impulse response within the given model (i.e. starting at period 0 defined by the model) and it is *not* doing impulse responses after the end of the model. To compute impulse responses from arbitrary time points, it is necessary to clone a new model with the appropriate system matrices. """ # We need to add an additional step, since the first simulated value # will always be zeros (note that we take this value out at the end). steps += 1 # For time-invariant models, add an additional step. This is the # default for time-invariant models based on the expected behavior for # ARIMA and VAR models: we want to record the initial impulse and also # steps values of the responses afterwards. if (self._design.shape[2] == 1 and self._transition.shape[2] == 1 and self._selection.shape[2] == 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 time-varying design, transition, or selection matrices, # then we can't produce more IRFs than we have time points time_invariant_irf = ( self._design.shape[2] == self._transition.shape[2] == self._selection.shape[2] == 1) # Note: to generate impulse responses following the end of a # time-varying model, one should clone the state space model with the # new time-varying model, and then compute the IRFs using the cloned # model if not time_invariant_irf and steps > self.nobs: raise ValueError('In a time-varying model, cannot create more' ' impulse responses than there are' ' observations') # Impulse responses only depend on the design, transition, and # selection matrices. We set the others to zeros because they must be # set in the call to clone. # Note: we don't even need selection after the first point, because # the state shocks will be zeros in every period except the first. sim_model = self.clone( endog=np.zeros((steps, self.k_endog), dtype=self.dtype), obs_intercept=np.zeros(self.k_endog), design=self['design', :, :, :steps], obs_cov=np.zeros((self.k_endog, self.k_endog)), state_intercept=np.zeros(self.k_states), transition=self['transition', :, :, :steps], selection=self['selection', :, :, :steps], state_cov=np.zeros((self.k_posdef, self.k_posdef))) # Get the impulse response function via simulation of the state # space model, but with other shocks set to zero measurement_shocks = np.zeros((steps, self.k_endog)) state_shocks = np.zeros((steps, self.k_posdef)) state_shocks[0] = impulse initial_state = np.zeros((self.k_states,)) irf, _ = sim_model.simulate( steps, measurement_shocks=measurement_shocks, state_shocks=state_shocks, initial_state=initial_state) # Get the cumulative response if requested if cumulative: irf = np.cumsum(irf, axis=0) # Here we ignore the first value, because it is always zeros (we added # an additional step at the top to account for this). return irf[1:]
[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 : ndarray The observation vector. design : ndarray The design matrix, :math:Z. obs_intercept : ndarray The intercept for the observation equation, :math:d. obs_cov : ndarray The covariance matrix for the observation equation :math:H. transition : ndarray The transition matrix, :math:T. state_intercept : ndarray The intercept for the transition equation, :math:c. selection : ndarray The selection matrix, :math:R. state_cov : ndarray 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 : ndarray The filtered state vector at each time period. filtered_state_cov : ndarray The filtered state covariance matrix at each time period. predicted_state : ndarray The predicted state vector at each time period. predicted_state_cov : ndarray The predicted state covariance matrix at each time period. forecast_error_diffuse_cov : ndarray Diffuse forecast error covariance matrix at each time period. predicted_diffuse_state_cov : ndarray The predicted diffuse state covariance matrix at each time period. kalman_gain : ndarray The Kalman gain at each time period. forecasts : ndarray The one-step-ahead forecasts of observations at each time period. forecasts_error : ndarray The forecast errors at each time period. forecasts_error_cov : ndarray The forecast error covariance matrices at each time period. llf_obs : ndarray 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', '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 : bool, 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: if not self.memory_no_gain: 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 ) # Note: below we will set self.llf, and in the memory_no_likelihood # case we will replace self.llf_obs = None at that time. 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 dtype = self.collapsed_forecasts.dtype self.forecasts = np.zeros((self.k_endog, self.nobs), dtype=dtype) self.forecasts_error = np.zeros((self.k_endog, self.nobs), dtype=dtype) self.forecasts_error_cov = ( np.zeros((self.k_endog, self.k_endog, self.nobs), dtype=dtype) ) # 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 can_compute_mean = not (self.memory_no_forecast_mean or self.memory_no_predicted_mean) can_compute_cov = not (self.memory_no_forecast_cov or self.memory_no_predicted_cov) if can_compute_mean or can_compute_cov: 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. if can_compute_mean: 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 if can_compute_cov: 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: if can_compute_mean: 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] ) if can_compute_cov: 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) if not self.memory_no_likelihood: self.scale = np.sum(scale_obs[d:]) / nobs_k_endog else: self.scale = scale_obs[0] / 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: 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) if not self.memory_no_likelihood: self.llf_obs += -0.5 * ( (self.k_endog - nmissing - nsingular) * np.log(self.scale) + scale_obs / self.scale) else: self.llf_obs[0] += -0.5 * (np.sum( (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 # Now, save self.llf, and handle the memory_no_likelihood case if not self.memory_no_likelihood: self.llf = np.sum(self.llf_obs[self.loglikelihood_burn:]) else: self.llf = self.llf_obs[0] self.llf_obs = None
@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): r""" 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 and not self.memory_no_forecast): 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])] try: upper, _ = linalg.cho_factor(F[:, :, 0]) self._standardized_forecasts_error[mask, t] = ( linalg.solve_triangular( upper, self.forecasts_error[mask, t], trans=1)) except linalg.LinAlgError: self._standardized_forecasts_error[mask, t] = ( np.nan) 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 prediction, i.e., the first prediction will be at start. end : int, optional Zero-indexed observation number at which to end prediction, i.e., the last prediction 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, predicted 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. """ # 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 # Check validity of dynamic and warn or error if issues dynamic, ndynamic = _check_dynamic(dynamic, start, end, self.nobs) # Get the number of in-sample static predictions if dynamic is None: nstatic = min(end, self.nobs) - min(start, self.nobs) else: # (use max(., 0), since dynamic can be prior to start) nstatic = max(dynamic - start, 0) # Cannot do in-sample prediction if we do not have appropriate # arrays (we can do out-of-sample forecasting, however) if nstatic > 0 and self.memory_no_forecast_mean: raise ValueError('In-sample prediction is not available if memory' ' conservation has been used to avoid storing' ' forecast means.') # Cannot do dynamic in-sample prediction if we do not have appropriate # arrays (we can do out-of-sample forecasting, however) if ndynamic > 0 and self.memory_no_predicted: raise ValueError('In-sample dynamic prediction is not available if' ' memory conservation has been used to avoid' ' storing forecasted or predicted state means' ' or covariances.') # 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 # If we have dynamic prediction or forecasting, then we need to # re-apply the Kalman filter else: # Figure out the period for which we need to run the Kalman filter if dynamic is not None: kf_start = min(start, dynamic, self.nobs) else: kf_start = min(start, self.nobs) kf_end = end # Make start, end consistent with the results that we're generating start = max(start - kf_start, 0) end = kf_end - kf_start # We must at least store forecasts and predictions kwargs['conserve_memory'] = ( self.conserve_memory & ~MEMORY_NO_FORECAST & ~MEMORY_NO_PREDICTED) # Can't use Chandrasekhar recursions for prediction kwargs['filter_method'] = ( self.model.filter_method & ~FILTER_CHANDRASEKHAR) # Even if we have not stored all predicted values (means and covs), # we can still do pure out-of-sample forecasting because we will # always have stored the last predicted values. In this case, we # will initialize the forecasting filter with these values if self.memory_no_predicted: constant = self.predicted_state[..., -1] stationary_cov = self.predicted_state_cov[..., -1] # Otherwise initialize with the predicted state / cov from the # existing results, at index kf_start (note that the time # dimension of predicted_state and predicted_state_cov is # self.nobs + 1; so e.g. in the case of pure forecasting we should # be using the very last predicted state and predicted state cov # elements, and kf_start will equal self.nobs which is correct) else: constant = self.predicted_state[..., kf_start] stationary_cov = self.predicted_state_cov[..., kf_start] kwargs.update({'initialization': 'known', 'constant': constant, 'stationary_cov': stationary_cov}) # Construct the new endogenous array. endog = np.zeros((nforecast, self.k_endog)) * np.nan model = self.model.extend( endog, start=kf_start, end=kf_end - nforecast, **kwargs) # Have to retroactively modify the model's endog if ndynamic > 0: model.endog[:, -(ndynamic + nforecast):] = np.nan with model.fixed_scale(self.scale): results = model.filter() return PredictionResults(results, start, end, nstatic, ndynamic, nforecast)
[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 : ndarray The observation vector. design : ndarray The design matrix, :math:Z. obs_intercept : ndarray The intercept for the observation equation, :math:d. obs_cov : ndarray The covariance matrix for the observation equation :math:H. transition : ndarray The transition matrix, :math:T. state_intercept : ndarray The intercept for the transition equation, :math:c. selection : ndarray The selection matrix, :math:R. state_cov : ndarray The covariance matrix for the state equation :math:Q. filtered_state : ndarray The filtered state vector at each time period. filtered_state_cov : ndarray The filtered state covariance matrix at each time period. predicted_state : ndarray The predicted state vector at each time period. predicted_state_cov : ndarray The predicted state covariance matrix at each time period. forecasts : ndarray The one-step-ahead forecasts of observations at each time period. forecasts_error : ndarray The forecast errors at each time period. forecasts_error_cov : ndarray 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
[docs] def clear(self): attributes = (['endog'] + self.representation_attributes + self.filter_attributes) for attr in attributes: _attr = '_' + attr if hasattr(self, _attr): delattr(self, _attr)
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)
def _check_dynamic(dynamic, start, end, nobs): """ Verify dynamic and warn or error if issues Parameters ---------- dynamic : {int, None} The offset relative to start of the dynamic forecasts. None if no dynamic forecasts are required. start : int The location of the first forecast. end : int The location of the final forecast (inclusive). nobs : int The number of observations in the time series. Returns ------- dynamic : {int, None} The start location of the first dynamic forecast. None if there are no in-sample dynamic forecasts. ndynamic : int The number of dynamic forecasts """ if dynamic is None: return dynamic, 0 # 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) return None, 0 elif dynamic > nobs: warn('Dynamic prediction specified to begin during' ' out-of-sample forecasting period, and so has no' ' effect.', ValueWarning) return None, 0 # 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. ndynamic = max(0, min(end, nobs) - dynamic) return dynamic, ndynamic