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# Accumulated States Densities - Out-of-Sequence measurements

Smoothing a filtered trajectory is an important task in live systems. Using Rauch–Tung–Striebel retrodiction after the normal filtering has a great effect on the filtered trajectories but it is not optimal because one has to calculate the retrodiction in an own step. In this point the Accumulated-State-Densities (ASDs) can help. In the ASDs the retrodiction is calculated in the prediction and update step. We use a multistate over time which can be pruned for better performance. Another advantage is the possibility to calculate Out-of-Sequence measurements in an optimal way. A more detailed introduction and the derivation of the formulas can be found in [1].

First of all we plot the ground truth of one target moving on the Cartesian 2D plane. The target moves in a cubic function.

```
from datetime import timedelta
from datetime import datetime
import numpy as np
from stonesoup.types.groundtruth import GroundTruthPath, GroundTruthState
from stonesoup.plotter import Plotterly
plotter = Plotterly()
truth = GroundTruthPath()
start_time = datetime.now()
for n in range(1, 202, 2):
x = n -100
y = 1e-4 * (n-100)**3
varxy = np.array([[0.1, 0.], [0., 0.1]])
xy = np.random.multivariate_normal(np.array([x, y]), varxy)
truth.append(GroundTruthState(np.array([[xy[0]], [xy[1]]]),
timestamp=start_time + timedelta(seconds=n)))
# Plot the result
plotter.plot_ground_truths({truth}, [0, 1])
plotter.fig
```

Following we plot the measurements made of the ground truth. The measurements have an error matrix of variance 5 in both dimensions.

```
from scipy.stats import multivariate_normal
from stonesoup.types.detection import Detection
from stonesoup.models.measurement.linear import LinearGaussian
measurements = []
for state in truth:
x, y = multivariate_normal.rvs(
state.state_vector.ravel(), cov=np.diag([5., 5.]))
measurements.append(Detection(
[x, y], timestamp=state.timestamp))
# Plot the result
plotter.plot_measurements(measurements, [0, 1], LinearGaussian(2, (0, 1), None))
plotter.fig
```

Now we have to setup a transition model for the prediction and the `ASDKalmanPredictor`

.

```
from stonesoup.models.transition.linear import \
CombinedLinearGaussianTransitionModel, ConstantVelocity
from stonesoup.predictor.asd import ASDKalmanPredictor
transition_model = CombinedLinearGaussianTransitionModel(
(ConstantVelocity(0.2), ConstantVelocity(0.2)))
predictor = ASDKalmanPredictor(transition_model)
```

We have to do the same for the measurement model and the `ASDKalmanUpdater`

.

```
from stonesoup.updater.asd import ASDKalmanUpdater
measurement_model = LinearGaussian(
4, # Number of state dimensions (position and velocity in 2D)
(0, 2), # Mapping measurement vector index to state index
np.array([[5., 0.], # Covariance matrix for Gaussian PDF
[0., 5.]])
)
updater = ASDKalmanUpdater(measurement_model)
```

We set up the state at position (-100, -100) with velocity 0. We set max_nstep to 30.

```
from stonesoup.types.state import ASDGaussianState
prior = ASDGaussianState(multi_state_vector=[[-100.], [0.], [-100.], [0.]],
timestamps=start_time,
multi_covar=np.diag([1., 1., 1., 1.]),
max_nstep=30)
```

Last but not least we set up a track and execute the filtering. The first and last 10 steps are processed in sequence. All other measurements are divided in groups of 10 following in time. The latest one is processed first and the other 9 are used for filtering. In the end we plot the filtered trajectory. The animated plot will show the changing state estimate across max_nstep set above.

```
import matplotlib
from matplotlib import animation
matplotlib.rcParams['animation.html'] = 'jshtml'
from stonesoup.plotter import Plotter
from stonesoup.types.hypothesis import SingleHypothesis
from stonesoup.types.track import Track
ani_plotter = Plotter()
frames = []
artists = []
track = Track() # For ASD track
track2 = Track() # For Gaussian state equivalent without ASD
processed_measurements = set()
for i in range(0, len(measurements)):
if i > 10:
if i % 10 != 0: # or i%10==3:
m = measurements[i]
prediction = predictor.predict(prior, timestamp=m.timestamp)
track2.append(prediction.state) # This track will ignore OoS measurements
else:
# prediction and update of the newest measurement
m = measurements[i]
processed_measurements.add(m)
prediction = predictor.predict(prior, timestamp=m.timestamp)
hypothesis = SingleHypothesis(prediction, m)
# Used to group a prediction and measurement together
post = updater.update(hypothesis)
track.append(post)
track2.append(post.state)
prior = track[-1]
artists.extend(ani_plotter.plot_tracks(Track(track[-1].states), [0, 2], color='r'))
artists.extend(
ani_plotter.plot_measurements(processed_measurements, [0, 2], measurement_model))
frames.append(artists); artists =[]
for j in range(9, 0, -1):
# prediction and update for all OOS measurement. Beginning with the latest one.
m = measurements[i - j]
processed_measurements.add(m)
prediction = predictor.predict(prior, timestamp=m.timestamp)
hypothesis = SingleHypothesis(prediction, m)
# Used to group a prediction and measurement together
post = updater.update(hypothesis)
track.append(post)
prior = track[-1]
artists.extend(ani_plotter.plot_tracks(Track(track[-1].states), [0, 2], color='r'))
artists.extend(ani_plotter.plot_measurements(
processed_measurements, [0, 2], measurement_model))
frames.append(artists); artists = []
else:
# the first 10 steps are for beginning of the ASD so that it is numerically stable
m = measurements[i]
processed_measurements.add(m)
prediction = predictor.predict(prior, timestamp=m.timestamp)
hypothesis = SingleHypothesis(prediction, m)
# Used to group a prediction and measurement together
post = updater.update(hypothesis)
track.append(post)
track2.append(post.state)
prior = track[-1]
artists.extend(ani_plotter.plot_tracks(Track(track[-1].states), [0, 2], color='r'))
artists.extend(
ani_plotter.plot_measurements(processed_measurements, [0, 2], measurement_model))
frames.append(artists); artists = []
animation.ArtistAnimation(ani_plotter.fig, frames)
```