Note

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# 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:

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)))
truths.add(truth)
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)))
truths.add(truth)
# Plot ground truth.
from stonesoup.plotter import Plotterly
plotter = Plotterly()
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)
measurement_set.add(TrueDetection(state_vector=measurement,
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_set.add(Clutter(np.array([[x], [y]]), timestamp=truth[k].timestamp,
measurement_model=measurement_model))
all_measurements.append(measurement_set)
# Plot true detections and clutter.
plotter.plot_measurements(all_measurements, [0, 2])
plotter.fig
```

```
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
# added.
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 1.567 seconds)