Source code for stonesoup.updater.chernoff

import numpy as np

from ..base import Property
from .base import Updater
from ..types.prediction import MeasurementPrediction
from ..types.update import Update

[docs]class ChernoffUpdater(Updater): r"""A class which performs state updates using the Chernoff fusion rule. In this context, measurements come in the form of states with a mean and covariance (compared to traditional measurements which contain solely a mean). The measurements are expected to come as :class:`~.GaussianDetection` objects. The Chernoff fusion rule is written as [#]_ .. math:: p_{\omega}(x_{k}) = \frac{p_{1}(x_{k})^{\omega}p_{2}(x_{k})^{1-\omega}} {\int p_{1}(x)^{\omega}p_{2}(x)^{1-\omega} \mathrm{d} x} where :math:`\omega` is a weighting parameter in the range :math:`(0,1]`, which can be found using an optimization algorithm. In situations where :math:`p_1(x)` and :math:`p_2(x)` are multivariate Gaussian distributions, the above formula is equal to the Covariance Intersection Algorithm from Julier et al [#]_. Let :math:`(a,A)` and :math:`(b,B)` be the means and covariances of the measurement and prediction respectively. The Covariance Intersection Algorithm was reformulated for use in Bayesian state estimation by Clark and Campbell [#]_, yielding formulas for the updated covariance and mean, :math:`D` and :math:`d`, and the innovation covariance matrix, :math:`V`, as follows: .. math:: D &= \left ( \omega A^{-1} + (1-\omega)B^{-1} \right )\\ d &= D \left ( \omega A^{-1}a + (1-\omega)B^{-1}b \right )\\ V &= \frac{A}{1-\omega} + \frac{B}{\omega} In filters where gating is required, the gating region can be written using the innovation covariance matrix as: .. math:: \mathcal{V}(\gamma) = \left\{ (a,A) : (a-b)^T V^{-1} (a-b) \leq \gamma \right\} The specifics for implementing the Covariance Intersection Algorithm in several popular multi-target tracking algorithms was expanded upon by Clark et al [#]_. The work includes a discussion of Stone Soup and can be used to apply this class to a tracking algorithm of choice. Note ---- If you have tracks that you would like to use as measurements for this updater, the :class:`~.Tracks2GaussianDetectionFeeder` class can be used to convert the tracks to the appropriate format. References ---------- .. [#] Hurley, M., “An information theoretic justification for covariance intersection and its generalization,” in [Proceedings of the Fifth International Conference on Information Fusion. FUSION 2002.(IEEE Cat. No. 02EX5997) ], 1, 505–511, IEEE (2002). .. [#] Julier, S., Uhlmann, J., and Durrant-Whyte, H., “A new method for the nonlinear transformation of means and covariances in filters and estimators,” IEEE Transactions on automatic control 45(3), 477–482 (2000). .. [#] Clark, D. and Campbell, M., “Integrating covariance intersection into Bayesian multi-target tracking filters,” preprint on TechRxiv. submitted to IEEE Transactions on Aerospace and Electronic Systems. .. [#] Clark, D. and Hunter, E. and Balaji, B. and O'Rourke, S., “Centralized multi-sensor multi-target data fusion with tracks as measurements,” to be submitted to SPIE Defense and Security Symposium 2023. """ omega: float = Property( default=0.5, doc="A weighting parameter in the range :math:`(0,1]`")
[docs] def predict_measurement(self, predicted_state, measurement_model=None, **kwargs): r""" This function predicts the measurement of a state in situations where measurements consist of a covariance and state vector. Parameters ---------- predicted_state : :class:`~.GaussianState` The predicted state :math:`\mathbf{x}_{k|k-1}` measurement_model : :class:`~.MeasurementModel` The measurement model. If omitted, the updater will use the model that was specified on initialization. Returns ------- : :class:`~.MeasurementPrediction` The measurement prediction """ measurement_model = self._check_measurement_model(measurement_model) # The innovation covariance uses the noise covariance from the measurement model state_covar_m = measurement_model.noise_covar innov_covar = 1/(*state_covar_m + 1/*predicted_state.covar # The predicted measurement and measurement cross covariance can be taken from # the predicted state predicted_meas = predicted_state.state_vector meas_cross_cov = predicted_state.covar # Combine everything into a GaussianMeasurementPrediction object return MeasurementPrediction.from_state(predicted_state, predicted_meas, innov_covar, predicted_state.timestamp, cross_covar=meas_cross_cov)
[docs] def update(self, hypothesis, force_symmetric_covariance=False, **kwargs): r""" Given a hypothesis, calculate the posterior mean and covariance. Parameters ---------- hypothesis : :class:`~.Hypothesis` Hypothesis with the predicted state and the actual/associated measurement which should be used for updating. If the hypothesis does not contain a measurement prediction, one will be calculated. force_symmetric_covariance: bool A flag to force the output covariance matrix to be symmetric by way of a simple geometric combination of the matrix and transpose. Default is False. Returns ------- : :class:`~.Update` The state posterior, saved in a generic :class:`~.Update` object. """ # Get the predicted state out of the hypothesis. These are 'B' and 'b', the # covariance and mean of the predicted Gaussian predicted_covar = hypothesis.prediction.covar predicted_mean = hypothesis.prediction.state_vector # Extract the vector and covariance from the measurement. These are 'A' and 'a', the # covariance and mean of the Gaussian measurement. measurement_covar = hypothesis.measurement.covar measurement_mean = hypothesis.measurement.state_vector # Predict the measurement if it is not already done if hypothesis.measurement_prediction is None: hypothesis.measurement_prediction = self.predict_measurement( hypothesis.prediction, measurement_model=hypothesis.measurement.measurement_model, **kwargs ) # Calculate the updated mean and covariance from covariance intersection posterior_covariance = np.linalg.inv(*np.linalg.inv(measurement_covar) + (*np.linalg.inv(predicted_covar)) posterior_mean = posterior_covariance @ (*np.linalg.inv(measurement_covar) @ measurement_mean + (*np.linalg.inv(predicted_covar) @ predicted_mean) # Optionally force the posterior covariance to be a symmetric matrix if force_symmetric_covariance: posterior_covariance = \ (posterior_covariance + posterior_covariance.T)/2 # Return the updated state return Update.from_state(hypothesis.prediction, posterior_mean, posterior_covariance, hypothesis, hypothesis.measurement.timestamp)