# 8 - Joint probabilistic data association tutorial

When we have multiple targets we’re going to want to arrive at a globally-consistent collection of associations for PDA, in much the same way as we did for the global nearest neighbour associator. This is the purpose of the joint probabilistic data association (JPDA) filter.

Similar to the PDA, the JPDA algorithm calculates hypothesis pairs for every measurement for every track. The probability of a track-measurement hypothesis is calculated by the sum of normalised conditional probabilities that every other track is associated to every other measurement (including missed detection). For example, with 3 tracks $$(A, B, C)$$ and 3 measurements $$(x, y, z)$$ (including missed detection $$None$$), the probability of track $$A$$ being associated with measurement $$x$$ ($$A \to x$$) is given by:

$\begin{split}p(A \to x) &= \bar{p}(A \to x \cap B \to None \cap C \to None) +\\ &+ \bar{p}(A \to x \cap B \to None \cap C \to y) +\\ &+ \bar{p}(A \to x \cap B \to None \cap C \to z) +\\ &+ \bar{p}(A \to x \cap B \to y \cap C \to None) +\\ &+ \bar{p}(A \to x \cap B \to y \cap C \to z) +\\ &+ \bar{p}(A \to x \cap B \to z \cap C \to None) +\\ &+ \bar{p}(A \to x \cap B \to z \cap C \to y)\end{split}$

where $$\bar{p}(\textit{multi-hypothesis})$$ is the normalised probability of the multi-hypothesis.

This is demonstrated for 2 tracks associating to 3 measurements in the diagrams below:

Where the probability (for example) of the orange track associating to the green measurement is $$0.25$$. The probability of every possible association set is calculated. These probabilities are then normalised.

A track-measurement hypothesis weight is then recalculated as the sum of the probabilities of every occurrence where that track associates to that measurement.

## Simulate ground truth

As with the multi-target data association tutorial, we simulate two targets moving in the positive x, y Cartesian plane (intersecting approximately half-way through their transition). We then add truth detections with clutter at each time-step.

from datetime import datetime
from datetime import timedelta
import numpy as np
from scipy.stats import uniform

from stonesoup.models.transition.linear import CombinedLinearGaussianTransitionModel, \
ConstantVelocity
from stonesoup.types.groundtruth import GroundTruthPath, GroundTruthState
from stonesoup.types.detection import TrueDetection
from stonesoup.types.detection import Clutter
from stonesoup.models.measurement.linear import LinearGaussian

np.random.seed(1991)

truths = set()

start_time = datetime.now()
transition_model = CombinedLinearGaussianTransitionModel([ConstantVelocity(0.005),
ConstantVelocity(0.005)])

truth = GroundTruthPath([GroundTruthState([0, 1, 0, 1], timestamp=start_time)])
for k in range(1, 21):
truth.append(GroundTruthState(
transition_model.function(truth[k-1], noise=True, time_interval=timedelta(seconds=1)),
timestamp=start_time+timedelta(seconds=k)))

truth = GroundTruthPath([GroundTruthState([0, 1, 20, -1], timestamp=start_time)])
for k in range(1, 21):
truth.append(GroundTruthState(
transition_model.function(truth[k-1], noise=True, time_interval=timedelta(seconds=1)),
timestamp=start_time+timedelta(seconds=k)))

# Plot ground truth.
from stonesoup.plotter import Plotter
plotter = Plotter()
plotter.ax.set_ylim(0, 25)
plotter.plot_ground_truths(truths, [0, 2])

# Generate measurements.
all_measurements = []

measurement_model = LinearGaussian(
ndim_state=4,
mapping=(0, 2),
noise_covar=np.array([[0.75, 0],
[0, 0.75]])
)

prob_detect = 0.9  # 90% chance of detection.

for k in range(20):
measurement_set = set()

for truth in truths:
# Generate actual detection from the state with a 10% chance that no detection is received.
if np.random.rand() <= prob_detect:
measurement = measurement_model.function(truth[k], noise=True)
groundtruth_path=truth,
timestamp=truth[k].timestamp,
measurement_model=measurement_model))

# Generate clutter at this time-step
truth_x = truth[k].state_vector[0]
truth_y = truth[k].state_vector[2]
for _ in range(np.random.randint(10)):
x = uniform.rvs(truth_x - 10, 20)
y = uniform.rvs(truth_y - 10, 20)
measurement_model=measurement_model))
all_measurements.append(measurement_set)

# Plot true detections and clutter.
plotter.plot_measurements(all_measurements, [0, 2], color='g')

from stonesoup.predictor.kalman import KalmanPredictor
predictor = KalmanPredictor(transition_model)

from stonesoup.updater.kalman import KalmanUpdater
updater = KalmanUpdater(measurement_model)


Initial hypotheses are calculated (per track) in the same manner as the PDA. Therefore, in Stone Soup, the JPDA filter uses the PDAHypothesiser to create these hypotheses. Unlike the PDA data associator, in Stone Soup, the JPDA associator takes this collection of hypotheses and adjusts their weights according to the method described above, before returning key-value pairs of tracks and detections to be associated with them.

from stonesoup.hypothesiser.probability import PDAHypothesiser
# This doesn't need to be created again, but for the sake of visualising the process, it has been
hypothesiser = PDAHypothesiser(predictor=predictor,
updater=updater,
clutter_spatial_density=0.125,
prob_detect=prob_detect)

from stonesoup.dataassociator.probability import JPDA
data_associator = JPDA(hypothesiser=hypothesiser)


## Running the JPDA filter

from stonesoup.types.state import GaussianState
from stonesoup.types.track import Track
from stonesoup.types.array import StateVectors
from stonesoup.functions import gm_reduce_single
from stonesoup.types.update import GaussianStateUpdate

prior1 = GaussianState([[0], [1], [0], [1]], np.diag([1.5, 0.5, 1.5, 0.5]), timestamp=start_time)
prior2 = GaussianState([[0], [1], [20], [-1]], np.diag([1.5, 0.5, 1.5, 0.5]), timestamp=start_time)

tracks = {Track([prior1]), Track([prior2])}

for n, measurements in enumerate(all_measurements):
hypotheses = data_associator.associate(tracks,
measurements,
start_time + timedelta(seconds=n))

# Loop through each track, performing the association step with weights adjusted according to
# JPDA.
for track in tracks:
track_hypotheses = hypotheses[track]

posterior_states = []
posterior_state_weights = []
for hypothesis in track_hypotheses:
if not hypothesis:
posterior_states.append(hypothesis.prediction)
else:
posterior_state = updater.update(hypothesis)
posterior_states.append(posterior_state)
posterior_state_weights.append(hypothesis.probability)

means = StateVectors([state.state_vector for state in posterior_states])
covars = np.stack([state.covar for state in posterior_states], axis=2)
weights = np.asarray(posterior_state_weights)

# Reduce mixture of states to one posterior estimate Gaussian.
post_mean, post_covar = gm_reduce_single(means, covars, weights)

# Add a Gaussian state approximation to the track.
track.append(GaussianStateUpdate(
post_mean, post_covar,
track_hypotheses,
track_hypotheses[0].measurement.timestamp))


Plot the resulting tracks.

plotter.plot_tracks(tracks, [0, 2], uncertainty=True)
plotter.fig


## References

1. Bar-Shalom Y, Daum F, Huang F 2009, The Probabilistic Data Association Filter, IEEE Control Systems Magazine

Total running time of the script: ( 0 minutes 2.400 seconds)

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