Size of random Galois lattices and number of frequent itemsets

19 pagesWe compute the mean and the variance of the size of the Galois lattice built from a random matrix with i.i.d. Bernoulli(p) entries. Then, obseving that closed frequent itemsets are in bijection with winning coalitions, we compute the mean and the variance of the number of closed frequent itemsets. This can be of interest for mining association rules

Size of random Galois lattices and number of closed frequent itemsets

AbstractGiven a sample of binary random vectors with i.i.d. Bernoulli(p) components, that is equal to 1 (resp. 0) with probability p (resp. 1−p), we first establish a formula for the mean of the size of the random Galois lattice built from this sample, and a more complex one for its variance. Then, noticing that closed α-frequent itemsets are in bijection with closed α-winning coalitions, we establish similar formulas for the mean and the variance of the number of closed α-frequent itemsets. This can be interesting for the study of the complexity of some data mining problems such as association rule mining, sequential pattern mining and classification

Maximal and stochastic Galois lattices

AbstractWe present a general formula for the intent–extent mappings of a Galois lattice generated by individual descriptions which lie in any arbitrary lattice.The formulation is unique if a natural maximality condition is required. This formulation yields, as particular cases, formal concept binary Galois lattices of Wille, those defined by Brito or Blyth–Janowitz, as well as fuzzy or stochastic Galois lattices.For the case of random descriptors we show that the nodes of Galois lattices defined by distributions are limit of empirical Galois lattices nodes. Choquet capacities, t-norms and t-conorms appear as natural valuations of these lattices

Vitamin D Status and COVID-19: Some Implications

Abstract Background Vitamin D is essential for the maintenance of good health and its status is defined by the level of serum 25-hydroxyvitamin (25(OH)D). Negative correlations between mean levels of 25(OH)D per country and the number of COVID-19 confirmed cases per one million population, and COVID-19 mortality per one million population, were recently observed. The aim of this study was to identify levels of 25(OH)D below which, rates of COVID-19 confirmed cases, mortality and lethality, increase significantly.Methods A data table found in the literature, containing a list of twenty countries and their corresponding mean level of 25(OH)D was updated with COVID-19 latest numbers of confirmed cases and mortality rates. Cut points of 25(OH)D below which rates were significantly higher were found according to various statistical criteria: absolute difference of means, t-test p-values, between class variance, entropy. Thresholds of 25(OH)D below which there can be a significant rise of COVID-19 confirmed cases, mortality and lethality, were found performing a Gaussian kernel regression. Results All the criteria yielded 25(OH)D cut off points at 50 ± 10 nmol/L for Cases and Mortality variables and at 60 ± 10 nmol/L for Lethality variable. A Gaussian kernel regression showed a significant rise in the rates of COVID-19 confirmed cases, mortality and lethality, below 25(OH)D levels of 60 ± 6 nmol/L. Conclusion In this study, our method provided a reliable estimate (95 % confidence interval) of 25(OH)D levels, of 60 ± 6 nmol/L, in the range of vitamin D insufficiency, below which rates of COVID-19 confirmed cases per one million population, rates of COVID-19 mortality per one million population increased. We found that 25(OH)D levels below 50 ± 10 nmol/L, mainly in the range of vitamin D deficiency, associate with highest COVID-19 mortality rates. Therefore, we suggest that 25(OH)D concentrations should be above 60 ± 6 nmol/L to reduce morbidity and mortality during the COVID-19 pandemic.</jats:p

Estimation du seuil de vitamine D qui peut induire une stimulation de la sécrétion de la parathormone chez des femmes africaines vivant en région parisienne

L'algorithme EM a été utilisé pour estimer la distribution de la iPTH comme mélange de Gaussiennes et l'algorithme SEMc pour estimer la distribution de la 25OHD, dont les valeurs inférieures à 17,5 nmol/l étaient censurées à 17,5nmol/l. Le seuil de levée de taux de iPTH est évalué à partir de la fonction de régression de l'iPTH par rapport à la 25OHD. Pour estimer cette fonction nous avons utilisé les estimations précédentes de mélange de Gaussiennes afin écarter les observations statistiquement extrêmes et améliorer le R2. La méthode de régression choisie est une méthode non paramétrique à noyaux où la forme de la courbe s'adapte aux données. Nous avons utilisé une version récente de cette méthode implémentée dans le logiciel R version « 2.10 » package « np ». Plusieurs types de noyaux ont été testés et leur paramètre de lissage sélectionné par critère de pénalité. Finalement, le seuil de levée proposé ( 48 mol/L +/- 2) a été, lorsque l'on a fait décroître la 25OHD, la première valeur de 25OHD à partir de laquelle la fonction de régression estimée est strictement croissante. Par Bootstrap, on obtient une région de confiance de 95% de la fonction de régression

Estimation of the 25(OH) Vitamin D Threshold below which Secondary Hyperparathyroidism may Occur Among African Migrant Women in Paris

Purpose: It has previously been shown that there is a threshold of serum 25-hydroxy vitamin D below which secondary hyperparathyroidism may occur. Our purpose was to estimate this threshold in a population of migrant African women living in Paris.Methods: Between February 2008 and December 2009, measurements of 25-hydroxy vitamin D and parathormone were performed in 165 African migrant women living in Paris. The estimation of 25-hydroxy vitamin D and parathormone marginal distributions as Gaussian mixtures, combined with a nonparametric kernel estimation method of the regression function of parathormone on 25-hydroxy vitamin D, provided the desired estimate of serum 25-hydroxy vitamin D threshold.Results: In our sample of African women, our method has shown that serum 25-hydroxy vitamin D levels below 48 ± 2 nmol/L may induce an increase in parathormone concentrations.Conclusion: In this sample, our method provided a reliable estimate (95 % confidence interval) of the serum 25-hydroxy vitamin D level below which an increase in parathormone concentrations can be observed. </jats:p