# Source code for stonesoup.hypothesiser.probability

```
from scipy.stats import multivariate_normal as mn
from .base import Hypothesiser
from ..base import Property
from ..types.detection import MissedDetection
from ..types.hypothesis import SingleProbabilityHypothesis
from ..types.multihypothesis import MultipleHypothesis
from ..types.numeric import Probability
from ..predictor import Predictor
from ..updater import Updater
[docs]class PDAHypothesiser(Hypothesiser):
"""Hypothesiser based on Probabilistic Data Association (PDA)
Generate track predictions at detection times and calculate probabilities
for all prediction-detection pairs for single prediction and multiple
detections.
"""
predictor: Predictor = Property(doc="Predict tracks to detection times")
updater: Updater = Property(doc="Updater used to get measurement prediction")
clutter_spatial_density: float = Property(
doc="Spatial density of clutter - tied to probability of false detection")
prob_detect: Probability = Property(
default=Probability(0.85),
doc="Target Detection Probability")
prob_gate: Probability = Property(
default=Probability(0.95),
doc="Gate Probability - prob. gate contains true measurement "
"if detected")
[docs] def hypothesise(self, track, detections, timestamp, **kwargs):
r"""Evaluate and return all track association hypotheses.
For a given track and a set of N detections, return a
MultipleHypothesis with N+1 detections (first detection is
a 'MissedDetection'), each with an associated probability.
Probabilities are assumed to be exhaustive (sum to 1) and mutually
exclusive (two detections cannot be the correct association at the
same time).
Detection 0: missed detection, none of the detections are associated
with the track.
Detection :math:`i, i \in {1...N}`: detection i is associated
with the track.
The probabilities for these detections are calculated as follow:
.. math::
\beta_i(k) = \begin{cases}
\frac{\mathcal{L}_{i}(k)}{1-P_{D}P_{G}+\sum_{j=1}^{m(k)}
\mathcal{L}_{j}(k)}, \quad i=1,...,m(k) \\
\frac{1-P_{D}P_{G}}{1-P_{D}P_{G}+\sum_{j=1}^{m(k)}
\mathcal{L}_{j}(k)}, \quad i=0
\end{cases}
where
.. math::
\mathcal{L}_{i}(k) = \frac{\mathcal{N}[z_{i}(k);\hat{z}(k|k-1),
S(k)]P_{D}}{\lambda}
:math:`\lambda` is the clutter density
:math:`P_{D}` is the detection probability
:math:`P_{G}` is the gate probability
:math:`\mathcal{N}[z_{i}(k);\hat{z}(k|k-1),S(k)]` is the likelihood
ratio of the measurement :math:`z_{i}(k)` originating from the track
target rather than the clutter.
NOTE: Since all probabilities have the same denominator and are
normalized later, the denominator can be discarded.
References:
[1] "The Probabilistic Data Association Filter: Estimation in the
Presence of Measurement Origin Uncertainty" -
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5338565
[2] "Robotics 2 Data Association" (Lecture notes) -
http://ais.informatik.uni-freiburg.de/teaching/ws10/robotics2/pdfs/rob2-15-dataassociation.pdf
Parameters
----------
track : Track
The track object to hypothesise on
detections : set of :class:`~.Detection`
The available detections
timestamp : datetime.datetime
A timestamp used when evaluating the state and measurement
predictions. Note that if a given detection has a non empty
timestamp, then prediction will be performed according to
the timestamp of the detection.
Returns
-------
: :class:`~.MultipleHypothesis`
A container of :class:`~.SingleProbabilityHypothesis` objects
"""
hypotheses = list()
# Common state & measurement prediction
prediction = self.predictor.predict(track, timestamp=timestamp, **kwargs)
# Missed detection hypothesis
probability = Probability(1 - self.prob_detect*self.prob_gate)
hypotheses.append(
SingleProbabilityHypothesis(
prediction,
MissedDetection(timestamp=timestamp),
probability
))
# True detection hypotheses
for detection in detections:
# Re-evaluate prediction
prediction = self.predictor.predict(
track, timestamp=detection.timestamp, **kwargs)
# Compute measurement prediction and probability measure
measurement_prediction = self.updater.predict_measurement(
prediction, detection.measurement_model, **kwargs)
# Calculate difference before to handle custom types (mean defaults to zero)
# This is required as log pdf coverts arrays to floats
log_pdf = mn.logpdf(
(detection.state_vector - measurement_prediction.state_vector).ravel(),
cov=measurement_prediction.covar)
pdf = Probability(log_pdf, log_value=True)
probability = (pdf * self.prob_detect)/self.clutter_spatial_density
# True detection hypothesis
hypotheses.append(
SingleProbabilityHypothesis(
prediction,
detection,
probability,
measurement_prediction))
return MultipleHypothesis(hypotheses, normalise=True, total_weight=1)
```