Ensemble Kalman Filter Algorithm and Data Type

LICENSE

@@ -3,6 +3,11 @@ MIT License
 © Crown Copyright 2017-2022 Defence Science and Technology Laboratory UK
 © Crown Copyright 2018-2022 Defence Research and Development Canada / Recherche et développement pour la défense Canada
 © Copyright 2018-2022 University of Liverpool UK
+© Copyright 2020-2022 John Hiles
+
+Portions of this work were created while John Hiles was employed by Wright
+State University under Contract No. FA8650-18-2-1645 to the Air Force Research
+Laboratory, and the University has released its rights to this Student work.
 
 Permission is hereby granted, free of charge, to any person obtaining a copy
 of this software and associated documentation files (the "Software"), to deal

docs/source/stonesoup.predictor.rst

@@ -19,6 +19,12 @@ Particle
 .. automodule:: stonesoup.predictor.particle
     :show-inheritance:
 
+Ensemble
+--------
+
+.. automodule:: stonesoup.predictor.ensemble
+    :show-inheritance:
+
 Categorical
 -----------
 

docs/source/stonesoup.updater.rst

@@ -19,6 +19,12 @@ Particle
 .. automodule:: stonesoup.updater.particle
     :show-inheritance:
 
+Ensemble
+--------
+
+.. automodule:: stonesoup.updater.ensemble
+    :show-inheritance:
+
 Information
 -----------
 

stonesoup/predictor/ensemble.py

@@ -0,0 +1,52 @@
+from .base import Property
+from ..models.transition import TransitionModel
+from .kalman import KalmanPredictor
+from ..types.array import StateVectors
+from ..types.state import State
+from ..types.prediction import Prediction
+
+
+class EnsemblePredictor(KalmanPredictor):
+    r"""Ensemble Kalman Filter Predictor class
+
+    The EnKF predicts the state by treating each column of the ensemble matrix
+    as a state vector. The state is propagated through time by applying the
+    transition function to each member (vector) of the ensemble.
+
+    .. math::
+
+        \hat{X}_k = [f(x_1), f(x_2), ..., f(x_M)]
+
+    """
+    transition_model: TransitionModel = Property(doc="The transition model to be used.")
+
+    def predict(self, prior, timestamp=None, **kwargs):
+        """Ensemble Kalman Filter prediction step
+
+        Parameters
+        ----------
+        prior : :class:`~.EnsembleState`
+            A prior state object
+        control_input : :class:`~.State`, optional
+            The control input. It will only have an effect if
+            :attr:`control_model` is not `None` (the default is `None`)
+        timestamp: :class:`datetime.datetime`, optional
+            A timestamp signifying when the prediction is performed
+            (the default is `None`)
+
+        Returns
+        -------
+        : :class:`~.EnsembleStatePrediction`
+            The predicted state
+        """
+
+        # Compute time_interval
+        time_interval = self._predict_over_interval(prior, timestamp)
+        # This block of code propagates each column through the transition model.
+        pred_ensemble = StateVectors(
+            [self.transition_model.function(State(state_vector=ensemble_member),
+                                            noise=True, time_interval=time_interval)
+             for ensemble_member in prior.ensemble.T])
+
+        return Prediction.from_state(prior, pred_ensemble, timestamp=timestamp,
+                                     transition_model=self.transition_model)

stonesoup/predictor/tests/test_ensemble.py

@@ -0,0 +1,44 @@
+# coding: utf-8
+import datetime
+
+import numpy as np
+
+from ...models.transition.linear import ConstantVelocity
+from ...predictor.ensemble import EnsemblePredictor
+from ...types.state import GaussianState, EnsembleState
+from ...types.array import StateVector, CovarianceMatrix
+
+
+def test_ensemble():
+    # Initialise a transition model
+    transition_model = ConstantVelocity(noise_diff_coeff=0)
+
+    # Define time related variables
+    timestamp = datetime.datetime(2021, 2, 27, 17, 27, 48)
+    timediff = 1  # 1 second
+    new_timestamp = timestamp + datetime.timedelta(seconds=timediff)
+    time_interval = new_timestamp - timestamp
+
+    # Define prior state
+    mean = StateVector([[10], [10]])
+    covar = CovarianceMatrix(np.eye(2))
+    gaussian_state = GaussianState(mean, covar, timestamp)
+    num_vectors = 50
+    prior_state = EnsembleState.from_gaussian_state(gaussian_state, num_vectors)
+    prior_ensemble = prior_state.ensemble
+
+    # Create Predictor object, run prediction
+    predictor = EnsemblePredictor(transition_model)
+    prediction = predictor.predict(prior_state, timestamp=new_timestamp)
+
+    # Evaluate mean and covariance
+    eval_ensemble = transition_model.matrix(timestamp=new_timestamp,
+                                            time_interval=time_interval) @ prior_ensemble
+    eval_mean = StateVector((np.average(eval_ensemble, axis=1)))
+    eval_cov = np.cov(eval_ensemble)
+
+    # Compare evaluated mean and covariance with predictor results
+    assert np.allclose(prediction.mean, eval_mean)
+    assert np.allclose(prediction.ensemble, eval_ensemble)
+    assert np.allclose(prediction.covar, eval_cov)
+    assert prediction.timestamp == new_timestamp

stonesoup/types/prediction.py

@@ -3,8 +3,9 @@
 
 from .array import CovarianceMatrix
 from .base import Type
-from .state import (State, GaussianState, ParticleState, SqrtGaussianState, InformationState,
-                    TaggedWeightedGaussianState, WeightedGaussianState, CategoricalState)
+from .state import (State, GaussianState, ParticleState, EnsembleState,
+                    SqrtGaussianState, InformationState, TaggedWeightedGaussianState,
+                    WeightedGaussianState, CategoricalState)
 from ..base import Property
 from ..models.transition.base import TransitionModel
 from ..types.state import CreatableFromState, CompositeState
@@ -118,6 +119,20 @@ class ParticleMeasurementPrediction(MeasurementPrediction, ParticleState):
     """
 
 
+class EnsembleStatePrediction(Prediction, EnsembleState):
+    """EnsembleStatePrediction type
+
+    This is a simple Ensemble measurement prediction object.
+    """
+
+
+class EnsembleMeasurementPrediction(MeasurementPrediction, EnsembleState):
+    """EnsembleMeasurementPrediction type
+
+    This is a simple Ensemble measurement prediction object.
+    """
+
+
 class CategoricalStatePrediction(Prediction, CategoricalState):
     """Categorical state prediction type"""
 

stonesoup/types/state.py

@@ -7,7 +7,7 @@
 
 import numpy as np
 
-from .array import StateVector, CovarianceMatrix, PrecisionMatrix
+from .array import StateVector, StateVectors, CovarianceMatrix, PrecisionMatrix
 from .base import Type
 from .numeric import Probability
 from .particle import Particles
@@ -474,6 +474,126 @@ def covar(self):
 State.register(ParticleState)  # noqa: E305
 
 
+class EnsembleState(Type):
+    r"""Ensemble State type
+
+    This is an Ensemble state object which describes the system state as a
+    ensemble of state vectors for use in Ensemble based filters.
+
+    This approach is functionally identical to the Particle state type except
+    it doesn't use any weighting for any of the "particles" or ensemble members.
+    All "particles" or state vectors in the ensemble are equally weighted.
+
+    .. math::
+
+        \mathbf{X} = [x_1, x_2, ..., x_M]
+
+    """
+
+    ensemble: StateVectors = Property(doc='''An ensemble of state vectors which represent
+                                the state''')
+
+    timestamp: datetime.datetime = Property(
+        default=None, doc="Timestamp of the state. Default None.")
+
+    @classmethod
+    def from_gaussian_state(self, gaussian_state, num_vectors):
+        """
+        Returns an EnsembleState instance, from a given
+        GaussianState object.
+
+        Parameters
+        ----------
+        gaussian_state : :class:`~.GaussianState`
+            The GaussianState used to create the new EnsembleState.
+        num_vectors : int
+            The number of desired column vectors present in the ensemble.
+        Returns
+        -------
+        :class:`~.EnsembleState`
+            Instance of EnsembleState.
+        """
+        mean = gaussian_state.state_vector.reshape((gaussian_state.ndim,))
+        covar = gaussian_state.covar
+        timestamp = gaussian_state.timestamp
+
+        return EnsembleState(ensemble=self.generate_ensemble(mean, covar, num_vectors),
+                             timestamp=timestamp)
+
+    @classmethod
+    def generate_ensemble(self, mean, covar, num_vectors):
+        """
+        Returns a StateVectors wrapped ensemble of state vectors, from a given
+        mean and covariance matrix.
+
+        Parameters
+        ----------
+        mean : :class:`~.numpy.ndarray`
+            The mean value of the distribution being sampled to generate
+            ensemble.
+        covar : :class:`~.numpy.ndarray`
+            The covariance matrix of the distribution being sampled to
+            generate ensemble.
+        num_vectors : int
+            The number of desired column vectors present in the ensemble,
+            or the number of "samples".
+        Returns
+        -------
+        :class:`~.EnsembleState`
+            Instance of EnsembleState.
+        """
+        # This check is necessary, because the StateVector wrapper does
+        # funny things with dimension.
+        rng = np.random.default_rng()
+        if mean.ndim != 1:
+            mean = mean.reshape(len(mean))
+        try:
+            ensemble = StateVectors(
+                                    [StateVector((rng.multivariate_normal(mean, covar)))
+                                     for n in range(num_vectors)])
+        # If covar is univariate, then use the univariate noise generation function.
+        except ValueError:
+            ensemble = StateVectors(
+                [StateVector((rng.normal(mean, covar))) for n in range(num_vectors)])
+
+        return ensemble
+
+    @property
+    def ndim(self):
+        """Number of dimensions in state vectors"""
+        return np.shape(self.ensemble)[0]
+
+    @property
+    def num_vectors(self):
+        """Number of columns in state ensemble"""
+        return np.shape(self.ensemble)[1]
+
+    @property
+    def mean(self):
+        """The state mean, numerically equivalent to state vector"""
+        return np.average(self.ensemble, axis=1)
+
+    @property
+    def state_vector(self):
+        """State mean in StateVector wrapper."""
+        return StateVector(self.mean)
+
+    @property
+    def covar(self):
+        """Sample covariance matrix for ensemble"""
+        return np.cov(self.ensemble)
+
+    @property
+    def sqrt_covar(self):
+        """sqrt of sample covariance matrix for ensemble, useful for
+        some EnKF algorithms"""
+        return ((self.ensemble - np.tile(self.mean, self.num_vectors)) /
+                np.sqrt(self.num_vectors - 1))
+
+
+State.register(EnsembleState)  # noqa: E305
+
+
 class CategoricalState(State):
     r"""CategoricalState type.
 

stonesoup/types/tests/test_state.py

@@ -6,13 +6,15 @@
 import scipy.linalg
 
 from ..angle import Bearing
-from ..array import StateVector, CovarianceMatrix
+from ..array import StateVector, StateVectors, CovarianceMatrix
 from ..groundtruth import GroundTruthState
 from ..numeric import Probability
 from ..particle import Particle
 from ..state import CreatableFromState
-from ..state import State, GaussianState, ParticleState, StateMutableSequence, \
-    WeightedGaussianState, SqrtGaussianState, CategoricalState, CompositeState
+from ..state import State, GaussianState, ParticleState, EnsembleState, \
+    StateMutableSequence, WeightedGaussianState, SqrtGaussianState, CategoricalState, \
+    CompositeState
+
 from ...base import Property
 
 
@@ -218,6 +220,74 @@ def test_particlestate_angle():
     assert np.allclose(state.covar, CovarianceMatrix([[0.01, -1.5], [-1.5, 225]]))
 
 
+def test_ensemblestate():
+
+    # 1D
+    state_vector1 = StateVector(np.array([1.5]))
+    state_vector2 = StateVector(np.array([0.5]))
+    list_of_state_vectors = [state_vector1, state_vector2]
+    ensemble = StateVectors(list_of_state_vectors)
+
+    # Test state without timestamp
+    state = EnsembleState(ensemble)
+    assert np.allclose(state.state_vector, StateVector([[1]]))
+    assert np.allclose(state.covar, CovarianceMatrix([[0.5]]))
+
+    # Test state with timestamp
+    timestamp = datetime.datetime(2021, 2, 25, 22, 29, 2)
+    state = EnsembleState(ensemble, timestamp=timestamp)
+    assert np.allclose(state.state_vector, StateVector([[1]]))
+    assert np.allclose(state.covar, CovarianceMatrix([[0.5]]))
+    assert state.timestamp == timestamp
+
+    # 2D
+    state_vector1 = StateVector(np.array([1.5, 0.75]))
+    state_vector2 = StateVector(np.array([0.5, 1.25]))
+    ensemble = StateVectors([state_vector1, state_vector2])
+
+    state = EnsembleState(ensemble)
+    assert np.allclose(state.state_vector, StateVector([[1], [1]]))
+    assert np.allclose(state.covar, CovarianceMatrix([[0.5, -0.25], [-0.25, 0.125]]))
+    assert np.allclose(state.sqrt_covar @ state.sqrt_covar.T, state.covar)
+
+    # Test generate_ensemble class method.
+    # 1 Dimensional
+    test_mean_1d = np.array([0])
+    test_covar_1d = np.array([1])
+    ensemble1d = state.generate_ensemble(mean=test_mean_1d,
+                                         covar=test_covar_1d, num_vectors=5)
+    assert np.shape(ensemble1d) == (1, 5)
+    assert isinstance(ensemble1d, StateVectors)
+
+    # 2 Dimensional
+    # Lets pass in a state vector mean(as opposed to an array) while we're at it
+    test_mean_2d = StateVector([1, 1])
+    test_covar_2d = CovarianceMatrix(np.eye(2))
+    ensemble2d = state.generate_ensemble(mean=test_mean_2d,
+                                         covar=test_covar_2d, num_vectors=5)
+    assert np.shape(ensemble2d) == (2, 5)
+    assert isinstance(ensemble2d, StateVectors)
+
+
+def test_ensemblestate_gaussian_init():
+    """Test initialising with an existing gaussian state object"""
+
+    # Initialize GaussianState
+    mean = StateVector([[25], [25], [25], [25]])
+    covar = CovarianceMatrix(np.eye(4))
+    timestamp = datetime.datetime(2021, 2, 26, 16, 35, 42)
+    gaussian_state = GaussianState(mean, covar, timestamp)
+    # Generate EnsembleState
+    num_vectors = 50
+    ensemble_state = EnsembleState.from_gaussian_state(gaussian_state, num_vectors)
+
+    assert isinstance(ensemble_state.state_vector, StateVector)
+    assert isinstance(ensemble_state.ensemble, StateVectors)
+    assert isinstance(ensemble_state.covar, CovarianceMatrix)
+    assert isinstance(ensemble_state.timestamp, datetime.datetime)
+    assert ensemble_state.timestamp == timestamp
+
+
 def test_state_mutable_sequence_state():
     state_vector = StateVector([[0]])
     timestamp = datetime.datetime(2018, 1, 1, 14)