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, ) de période 20 points pour 1000 point représenté, et sa trajectoire dans SO(3)
, 11) et le signal de la Fig. 2.7-(c) calculée à partir de l'équation 2.31. La première ligne correspond à une interpolation avec 200 points, la seconde avec 50 points
, ce qui se traduit par le fait que les deux courbes fermées rouges et bleues ne peuvent pas se transformer l'une en l'autre par transformation continue. Imaginons que ces deux courbes soient des élastiques. Nous ne pouvons pas passer de l'élastique rouge à l'élastique bleu sans le casser
, Dans la rangée du haut sont représentés succesivement un 0-simplexe, un 1-simplexe et un 2-simplexe. Dans la rangée du bas ce sont les faces de ces simplexes qui sont représentées
, de deux triangles rouges, vol.8
, Quelques étapes de la filtration de Vietoris-Rips. Un cycle se crée à la troisième étape (également à la deuxième en tant que bord du triangle). Il disparaîtra lorsque les quatre cercles centrés aux quatre sommets correspondant s'intersecteront, p.79
, Le point isolé dans le diagramme de persistance traduit l'organisation circulaire de ce nuage de points
, En faisant varier ?, nous obtenons la séquence des nombres Betti
, La trajectoire sur le groupe SO(3) formée par les matrices de dilations de ce signal. La trajectoire est représenté à l'intérieur d'une boule unité, échantillons on été tirés, avec 20 point par périodes.(b)
, Le diagramme de persistance à l'ordre 1 du signal de la Fig. 3.7-(a)
, Une réalisation de chaque modèle utilisé dans notre classification
, 32 sequences de 10 parcors chacune ont été utilisées pour obtenir le nuage de matrice de dilation. Les diagrammes de persistance ont été obtenus avec le package Dionysus sous Python, Les diagrammes de persistance pour une réalisation de chacun des processus figurant à la Fig. 3.9
, Projection des données après une réduction de dimension à deux composantes principales, (a) Les données ne peuvent pas être différenciées avec seulement deux composantes principales. (b) Certains processus sont néanmoins bien distinguables, vol.87
, entropie persistante normalisée à l'ordre i = 0, 1 ; f i , les caractéristique polynomiales à l'ordre i, b i les caractéristique de Betti à l'odre i. Le rôle singulier de l'entropie persistante est clairement visible, Contributions des caractéristiques aux axes principaux PC1 et PC2. H i désigne l, vol.87
, Haut : un exemple du nuage de point obtenu pour chaque niveaux de bruit. Bas : les histogrammes des valeurs d'entropies persistantes correspondants
, Histogramme des valeurs de l'entropie normalisée à l'ordre 0 pour 500 réalisations par processus
, Histogramme des valeurs de l'entropie normalisée à l'ordre 1 pour 500 réalisations par processus
, Séquence de Betti moyenne obtenue après 500 réalisations pour chacun des processus
, Les points du diagramme noir sont associés aux points du diagramme rouge de sorte à ce que la somme des déplacements soit minimale
,
, (a) évolution temporelle de Bar max (k), (b) évolution temporelle de PE norm (k) pour le processus Fig.4.1. La longueur de la fenêtre glissante est N = 4. Les longueurs de la DFA sont de L = 50 et
, (b) Les dates correspondantes aux trois détections sur la période considérée pour une longueur de la DFA de L = 24, Deux critères sont minimisés : f (L), p.107
, 11-25) et estimation du nombre d'arrivée maximal (2017-01-15) obtenue avec le détecteur calibré pour l, Détection d'un nombre d'arrivées anormal, pp.2016-2017, 2016.
, Comparaison entre l'indicateur Bar max et trois autres méthode de détection de changement : une carte de contrôle, un cusum et un kurtosis local, p.109
,
, Un triangle dans un espace à deux dimensions
,
, Un tetraedre est un 3-simplexes. Ces faces sont des 2-simplexes ou des 1-simplexes, vol.117
,
,
,
, 3 un cercle est homéomorphe à un triangle
,
, 4 un disque est homéomorphe à un triangle. Le triangle est orienté dans le sens trigonométrique
,
, Une sphère, homéomorphe à un tétraèdre. Par souci de clarté, toutes les orientations non pas été représentées
,
, Un tore, qui est homéomorphe à deux triangles collés dont tous les sommets on été identifiés. Par souci de clarté, toutes les orientations non pas été représentées, p.127
, Table des figures