, AUC(t) sur des données simulées suivant le C-mix ; et à gauche la Figure 1.6b et les performances en sélection de variables du C-mix sur des données simulées suivant le modèle de Cox, pour différentes configurations ("confusion rate, Échantillon de résultats graphiques provenant de la Section 4.4, avec à droite la Figure 1.6a et les performances des modèles considérés en terme d

, nant l'application des modèles considérés sur un jeu de données génétiques de patients atteints d'un cancer du sein. La Figure 1.7a donne les résultats en terme d'AUC(t), la Figure 1.7b en terme d'AUC en utilisant, pour un patient i donné, la fonction de survie estimée et évaluée au temps -soit? i ( |X i = x i ) -pour prédire la quantité binaire T i > pour différents , et la Figure 1.7c donne les estimateurs de Kaplan-Meier des deux groupes identifiés par le C-mix, p.35

, Échantillon de résultats graphiques provenant de la Section 6.5, à consulter pour plus de détails

, 56 2.2 Illustration of the different steps followed in the patients selection phase. n is the number of patients and N the number of stays, p.57

, Top right : repartition of the number of hemoglobin measurement per visit ; bottom right : histogram of the hemoglobin mean

, Hemoglobin average kinetics in g/dL (bold line) with 95% Gaussian confidence interval (bands) for different subpopulations of patients, p.59

, Top right : repartition of the number of measurement per visit ; bottom right : histogram of the white blood cell count mean, White blood cell count average kinetics in 10 9 /L (bold line) with 95% Gaussian confidence interval (bands)

, Top right : repartition of the number of neutrophils measurement per visit ; bottom right : histogram of the neutrophils mean, Left : neutrophils average kinetics in 10 9 /L (bold line) with 95% Gaussian confidence interval (bands)

, Top right : repartition of the number of eosinophils measurement per visit ; bottom right : histogram of the eosinophils mean, Left : eosinophils average kinetics in 10 9 /L (bold line) with 95% Gaussian confidence interval (bands), p.61

, Top right : repartition of the number of platelets measurement per visit ; bottom right : histogram of the platelets mean

, Top right : repartition of the number of CRP measurement per visit ; bottom right : histogram of the CRP mean, 11 Left : CRP average kinetics in mg/L (bold line) with 95% Gaussian confidence interval (bands)

, Individual LDH trajectories with the color gradient corresponding to the patient age : blue means young and red means old, p.63

, Top right : repartition of the number of LDH measurement per visit ; bottom right : histogram of the LDH mean, 14 Left : LDH average kinetics in mg/L (bold line) with 95% Gaussian confidence interval (bands)

, Top right : repartition of the number of temperature measurement per visit ; bottom right : histogram of the temperature mean, 15 Left : temperature average kinetics in ? Celsius (bold line) with 95% Gaussian confidence interval (bands)

, Temperature average kinetics in ? Celsius (bold line) with 95% Gaussian confidence interval (bands) with patients grouped according the their sex

, Percentage of patients with temperature below 38 ? Celsius according to time

, Illustration of the problem of censored data that cannot be labeled when using a threshold . ? i = 1 {T i ?C i } is the censoring indicator which is equal to 1 if Y i is censored and 0 otherwise, the binary outcome setting, p.80

, We arbitrarily shows only the tests with corresponding p-values below the level ? = 5%, with the classical Bonferroni multitests correction, p.86

, Pearson correlation matrix for comparing covariates selection similarities between methods. Red means high correlations, vol.87

, Covariates boxplot comparison between the most significant C-mix groups

, Note that for RF and GB models, coefficients are, by construction, always positive, B.1Comparison of covariates importance, ordered on the C-mix estimates

. .. , Graphical model representation of the C-mix, vol.104

, 2) while columns correspond to different gap values (the problem becomes more difficult as the gap value decreases). Surprisingly, our method gives almost always the best results, even under model misspecification (see Cox PH and CURE simulation cases on the second and third rows), Average (bold lines) and standard deviation (bands) for, p.117

, 2 and obtained after 100 simulated data for each (gap, r cf ) configuration (a grid of 20x20 different configurations is considered). A Gaussian interpolation is then performed to obtain smooth figures. Note that the gap values are log-scaled. Rows correspond to the model simulated while columns correspond to the model under consideration for the variable selection evaluation procedure. Our method gives the best results in terms of variable selection, even under model misspecification, Average AUC, p.118

, We observe that C-mix model leads to the best results (higher is better) and outperforms both Cox PH and CURE in all cases. Results are similar in terms of performances for the C-mix model with geometric or Weibull distributions, p.119

, Estimated survival curves per subgroups (blue for low risk and red for high risk) with the corresponding 95 % confidence bands for the C-mix and CURE models : BRCA in column (a)

, 2 with p = 50 and K j equals to 1, 2 and 3 (for all j ? {1, . . . , p}) for the left, center and right sub-figures respectively) for varying n. The lower m 1 the best result : the binacox outperforms clearly other methods when there are more than one cut-point, and is competitive with other methods when there is only one cut-points with poorer performances when n is small because of an overestimation of K j in this case, Average (bold) m 1 scores and standard deviation (bands) obtained on 100, p.199

, Average (bold) m 2 scores and standard deviation (bands) obtained on 100 simulated datasets according to Section 6.4.2 with p = 50 for varying n. It turns out that MT-B and MT-LS tend to detect a cut-point while there is not (no matter the value of n), and that the binacox overestimates the number of cut-points for small n values but detects well S for p = 50 on the simulated data when n > 1000, p.199

. .. , The binacox detects multiple cut-points and sheds light on non-linear effects for various genes. The BH thresholds are plotted for informational purposes, but are unusable in practice, Illustration of the results obtained on the top?10 features ordered according to the binacox ? j,? TV values on the GBM dataset

, The green triangle points out the value of ? ?1 that gives the minimum score (best training score), while the ? ?1 value we automatically select (the red triangle) is the smallest value such that the score is within one standard error of the minimum, wich is a classical trick, A.1Learning curves obtained for various ?, in blue on the changing test sets of the cross-validation, and in orange on the validation set. Bold lines represent average scores on the folds and bands represent Gaussian 95% confidence intervals

, Within a block (separated with the dotted pink line), the different colors represent? j,l with corresponding µ j,l in distinct estimated I j,k . When a? j,l is "isolated, A.2Illustration of the denoising step on the cut-points detection phase, p.206

, TV obtained on univariate binacox fits for the three considered datasets. Top?P selected features appear in red, and it turns out that taking P = 50 coincides with the elbow (represented with the dotted grey lines) in each three curves

, A.4Illustration of the results obtained on the top?10 features ordered according to the binacox ? j,? TV values on the BRCA dataset, p.208

, A.5Illustration of the results obtained on the top?10 features ordered according to the binacox ? j,? TV values on the KIRC dataset, p.209

, In this scenario, the algorithm detects an extra cut-points and K j = 5 = s j while K j = 4, vol.17

A. , Organisation des données à l'HEGP

, Diagramme de classes de l'entrepôt I2B2

, 256 A.4 Diagramme des classes après le premier nettoyage avec en vert des exemples représentatifs pour chacun des attributs, p.256

, A.6 Illustration de la présence d'ADN dans chaque cellule, p.258

A. , Illustration de cellules cancéreuses entourées de cellules saines, p.259

A. ,

.. .. Le,

, Visualisation d'un sous échantillon des données relatives au cancer BRCA, en prenant ici les n = 50 premiers patients et les p = 200 premiers gènes, où les valeurs d'expression sont normalisées dans le segment

, Une valeur de 0 est représentée par un carré de couleur blanche, couleur qui tend vers le vert foncé à mesure que la valeur est proche de 1

, C-index comparison between geometric or Weibull parameterizations for the C-mix model on the three TCGA data sets considered (with d = 300)

, with corresponding C-index in parenthesis and best result in bold in each case. This times concern the learning task for each model with the best hyper parameter selected after the cross validation procedure. It turns out that our method is by far the fastest in addition to providing the best performances, Computing time comparison in second on the BRCA dataset (n = 1211), vol.122

. .. , C.1Hyper-parameters choice for simulation, p.128

, 100 simulated data for different dimension d and different screening method (including no screening), C.2Average performances and standard deviation

, 100 simulated data for different dimension d with the times simuted with a mixture of gammas. For each configuration, the best result appears in bold, C.3Average performances and standard deviation

, on 100 simulated data for different configurations (d, r c ), with geometric distributions for the C-mix model. For each configuration, the best result appears in bold, F.1Average C-index and standard deviation

G. , 20 selected genes per model for the BRCA cancer, with the corresponding effects. Dots (·) mean zeros

G. , 20 selected genes per model for the GBM cancer, with the corresponding effects. Dots (·) mean zeros

G. , 20 selected genes per model for the KIRC cancer, with the corresponding effects. Dots (·) mean zeros

. .. Table, 148 5.1 Baselines considered in our experiments. Note that Group L1 and Group TV are considered on binarized features, p.154

, Basic informations about the 9 considered datasets, p.155

. .. , 193 6.1 Estimated cut-points values for each method on the top?10 genes presented in Figure 6.1 for the GBM cancer. Dots (·) mean "no cutpoint detected". The binacox identifies much more cut-points than the univariate MT-B and MT-LS methods. But all cut-points detected by those two methods are also detected by the binacox, p.201

, On the three datasets, the binacox method gives by far the best results (in bold), C-index comparison for Cox PH model trained on continuous features vs. on its binarized version constructed using the considered methods cut-points estimates, and the CoxBoost and RSF methods

. .. , A.1Estimated cut-points values for each method on the top?10 genes presented in Figure 6.A.4 for the BRCA cancer, p.208

. .. , A.2Estimated cut-points values for each method on the top?10 genes presented in Figure 6.A.5 for the KIRC cancer, p.209

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