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Polynomial Regression on Riemannian Manifolds, presentation, 2012

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D’un point de vue général, la méthode statistique de régression consiste à estimer la relation mathématique entre un ensemble de variables, appelées variables explicatives ou descriptives ou indépendantes, et une variable observée ou mesurée. On cherche donc à déterminer, parmi une certaine classe de fonctions, la fonction qui décrive de façon optimale (en un certain sens) cette relation. La régression polynomiale consiste à estimer la relation entre variables explicatives et données observées à l’aide d’une fonction polynomiale de degré fixé k. Le nombre de paramètres inconnus est alors k + 1 et ils sont le plus souvent estimés en minimisant un critère des moindres carrés, qui est le carré de la distance euclidienne entre les valeurs observées et les valeurs prédites par le modèle polynomial. L’un des problèmes à résoudre dans ce contexte est évidemment le choix du degré du polynôme.

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Polynomial Regression on Riemannian Manifolds, presentation, 2012

  1. 1. Polynomial Regression on Riemannian Manifolds Albert Thomas, Florent Renucci
  2. 2. Sommaire Introduction I - Définition d’un polynôme sur une variété riemannienne II – Intégration - Méthode d’Euler III – Résultats de l’algorithme sur la sphère IV – Régression polynomiale V – Application à l’évolution d’un crâne de rat Conclusion et discussion
  3. 3. Introduction Objectif : adapter la régression polynomiale paramétrique aux variétés riemanniennes. Régression : Etablissement d’un lien entre les variables explicatives et la variable à estimer.  déterminer la fonction, parmi une classe de fonctions, qui décrive ce lien de manière optimale. Critère : moindres carrés.
  4. 4. I - Définition d’un polynôme sur une variété riemannienne Un polynôme de degré k est entièrement caractérisé par la donnée de : Par analogie, sur une variété riemannienne munie de la dérivée covariante :
  5. 5. II - Intégration • Pas de formule explicite : calcul numérique Conditions initiales :
  6. 6. II - Intégration • Schéma d’Euler
  7. 7. III - Exemple de la sphère • Géodésique :
  8. 8. IV – Régression polynomiale • N observations y1, . . . , yN ∈ M aux temps t1,...,tN. Calcul du polynôme riemannien. γ de degré k.  Déterminer qui minimisent le critère Minimiser sous contraintes
  9. 9. IV – Régression polynomiale Minimiser sous contraintes  Multiplicateurs de Lagrange, minimisation du lagrangien à l’aide de la méthode des variations.
  10. 10. IV – Régression polynomiale Critère à minimiser : : écart par rapport à la valeur prédite par une constante : somme des carrés des écarts On peut donc définir par analogie :
  11. 11. V - Croissance d’un crâne de rat • Kendal shape space R21 = 0.79 R22 = 0.85 R23 = 0.87
  12. 12. Conclusion Point positifs : • Définition d’une régression polynomiale sur une variété • Applications Critiques : • Choix du degré du polynôme • Choix du pas d’intégration

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