Fenêtrage fondé sur une distance de Mahalanobis et un seuil empirique

S_k⁻¹ " rk− µ_r d_swg(θ_k− µ_θ) # . (F.14)

En considérant que z_k suit une loi gaussienne de moyenne µ_swg et de matrice de covariance ˜S_k, la distance au carré d²_{M aha,P F} (F.14) est comparée au seuil calculé à partir de la fonction de répartition inverse de la loi du χ² et d’une probabilité de gating p_g.

Dans certains cas cependant, approcher la distribution de z_k par une loi semi-wrapped Gaussian est une trop forte simplification à cause d’une non-linéarité trop importante introduite par le modèle de mesure. La distance de Mahalanobis peut continuer à être calculée sans approximation, mais elle ne suit plus une loi du χ² de sorte que le seuil considéré n’est plus valide. Dans la sous-partie suivante, nous proposons de définir différemment le seuil de validation.

F.2.2 Fenêtrage fondé sur une distance de Mahalanobis et un seuil empirique Dans cette partie, nous proposons de calculer un seuil à partir des pseudo mesures ˜z⁽ⁱ⁾_k|k−1. En effet, {ω_k|k−1⁽ⁱ⁾ , ˜z⁽ⁱ⁾_k|k−1}_i=1,...,N représente la loi de prédiction p(z_k|z_1:k−1). Pour chaque pseudo-mesure, une distance de Mahalanobis au carré peut être calculée de la manière suivante :

 d⁽ⁱ⁾_{M aha}^2=   ˜ r_k|k−1⁽ⁱ⁾ − µ_r d_swg^θ^˜(i)_k|k−1− µ_θ^   T ˜ S_k⁻¹   ˜ r⁽ⁱ⁾_k|k−1− µ_r d_swg^θ^˜(i)_k|k−1− µ_θ^  . (F.15)

Il est possible de déterminer l’effectif empirique cumulé des distances au carré. Le seuil empirique est alors donné par la distance au carré pour laquelle p_g× 100 % des distances sont inférieures. Le principe de cette méthode est illustrée par la figure F.6. Comme p_g est généralement plus grand que 0.9, il peut être utile d’utiliser une méthode d’estimation à noyau de la fonction de répartition [Ser09] pour compenser la rareté des particules en queue de distribution.

Figure F.6 – Fenêtre de validation empirique.

Dans certains cas, cette approche montre également ses limites. Par exemple, lorsque les trajectoires de deux cibles sont proches pendant plusieurs instants. À ce moment-là, deux mesures apparaissent relativement proches à plusieurs reprises et participent potentiellement toutes les deux à l’estimation des deux pistes. La distribution sur l’état devient bi-modale. Lorsque les deux cibles s’éloignent, les deux modes deviennent nettement séparés. La fenêtre de validation décrite ci-dessus a alors tendance à grossir sans pour autant décrire fidèlement l’incertitude dans l’espace des mesures. Des mesures pertinentes pour l’estimation peuvent alors être rejetées.

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