Adaptive kernel Kalman filter (AKKF)

[jswright-dstl]

This PR adds the Adaptive kernel Kalman filter (AKKF) [1]. The implementation includes the following:

• Kernel classes
  • QuadraticKernel
  • QuarticKernel
  • GaussianKernel
• AdaptiveKernelKalmanPredictor
• AdaptiveKernelKalmanUpdater
• KernelParticleState
• AKKF Tutorial

The above implementations are described in [2].

[1] M. Sun, M. E. Davies, I. Proudler and J. R. Hopgood, "Adaptive Kernel Kalman Filter," 2021 Sensor Signal Processing for Defence Conference (SSPD), Edinburgh, United Kingdom, 2021, pp. 1-5, doi: 10.1109/SSPD51364.2021.9541455.
[2] J. S. Wright, J. R. Hopgood, M. E. Davies, I. K. Proudler and M. Sun, "Implementation of Adaptive Kernel Kalman Filter in Stone Soup," 2023 Sensor Signal Processing for Defence Conference (SSPD), Edinburgh, United Kingdom, 2023, pp. 1-5, doi: 10.1109/SSPD57945.2023.10256739.

[sdhiscocks]

Mainly minor comments and suggestions for equations.

Few queries mixed in there.

Haven't looked at tutorial yet.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 93.63%. Comparing base (ab6af2b) to head (93a0478).
Report is 116 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1014      +/-   ##
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+ Coverage   93.60%   93.63%   +0.03%     
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  Files         202      205       +3     
  Lines       12990    13123     +133     
  Branches     2651     2668      +17     
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+ Hits        12159    12288     +129     
- Misses        588      590       +2     
- Partials      243      245       +2     

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[sdhiscocks]

Minor doc suggestions, and change to which variable being used to reset caching.

Also found issue with weights of state being inconsistent, so proposed below to fix that.

diff --git a/stonesoup/predictor/kernel.py b/stonesoup/predictor/kernel.py
index e329809d..cdcd1361 100644
--- a/stonesoup/predictor/kernel.py
+++ b/stonesoup/predictor/kernel.py
@@ -76,7 +76,7 @@ class AdaptiveKernelKalmanPredictor(KalmanPredictor):
         prediction_covariance = kernel_t @ prior.kernel_covar @ kernel_t.T + v
         return Prediction.from_state(prior,
                                      state_vector=new_state_vector,
-                                     weight=np.squeeze(prediction_weights),
+                                     weight=prediction_weights,
                                      kernel_covar=prediction_covariance,
                                      timestamp=timestamp,
                                      transition_model=self.transition_model)
diff --git a/stonesoup/types/state.py b/stonesoup/types/state.py
index f1965f1b..ea116e7f 100644
--- a/stonesoup/types/state.py
+++ b/stonesoup/types/state.py
@@ -945,7 +945,7 @@ class KernelParticleState(State):
 
     @clearable_cached_property('state_vector', 'weight')
     def mean(self):
-        return self.state_vector @ self.weight
+        return self.state_vector @ self.weight[:, np.newaxis]
 
     @clearable_cached_property('state_vector', 'weight')
     def covar(self):
diff --git a/stonesoup/types/tests/test_state.py b/stonesoup/types/tests/test_state.py
index 3fd80f69..eba966ce 100644
--- a/stonesoup/types/tests/test_state.py
+++ b/stonesoup/types/tests/test_state.py
@@ -863,5 +863,5 @@ def test_kernel_particle_state():
     assert np.array_equal(prior.kernel_covar, prior_w_kernel_covar.kernel_covar)
     assert number_particles == len(prior)
     assert 4 == prior.ndim
-    assert np.array_equal(state_vector @ weights, prior.mean)
+    assert np.array_equal(state_vector @ weights[:, np.newaxis], prior.mean)
     assert np.array_equal(state_vector @ np.diag(weights) @ state_vector.T, prior.covar)
diff --git a/stonesoup/updater/kernel.py b/stonesoup/updater/kernel.py
index 9860675a..e9998377 100644
--- a/stonesoup/updater/kernel.py
+++ b/stonesoup/updater/kernel.py
@@ -86,14 +86,13 @@ class AdaptiveKernelKalmanUpdater(Updater):
             predicted_state.kernel_covar \
             @ np.linalg.pinv(G_yy @ predicted_state.kernel_covar
                              + self.lambda_updater * np.identity(len(predicted_state)))
-        updated_weights = \
-            np.atleast_2d(predicted_state.weight).T \
-            + Q_AKKF @ (g_y - np.atleast_2d(G_yy @ predicted_state.weight).T)
+        weights = predicted_state.weight[:, np.newaxis]
+        updated_weights = (weights + Q_AKKF@(g_y - G_yy@weights)).ravel()
         updated_covariance = \
             predicted_state.kernel_covar - Q_AKKF @ G_yy @ predicted_state.kernel_covar
 
         # Proposal Calculation
-        pred_mean = predicted_state.state_vector @ np.squeeze(updated_weights)
+        pred_mean = predicted_state.state_vector @ updated_weights
         pred_covar = np.diag(np.diag(
             predicted_state.state_vector @ updated_covariance @ predicted_state.state_vector.T))
 
diff --git a/stonesoup/updater/tests/test_kernel.py b/stonesoup/updater/tests/test_kernel.py
index ceea0c64..06c671f7 100644
--- a/stonesoup/updater/tests/test_kernel.py
+++ b/stonesoup/updater/tests/test_kernel.py
@@ -108,7 +108,7 @@ def test_kernel_updater(kernel, measurement_model, c, ialpha):
     assert update.hypothesis.measurement.timestamp == gt_state.timestamp
     assert np.allclose(update.state_vector, prediction.state_vector)
     assert np.allclose(update.proposal, StateVectors(new_state_vector.T), atol=1e0)
-    assert np.allclose(update.weight, updated_weights)
+    assert np.allclose(update.weight, updated_weights.ravel())
     assert np.allclose(update.kernel_covar, updated_covariance)