Projections of determinantal point processes

Let $\mathbf x=\{x^{(1)},\dots,x^{(n)}\}$ be a space filling-design of $n$ points defined in $[0{,}1]^d$. In computer experiments, an important property seeked for $\mathbf x$ is a nice coverage of $[0{,}1]^d$. This property could be desirable as well as for any projection of $\mathbf x$ onto $[0{,}1]^\iota$ for $\iota<d$ . Thus we expect that $\mathbf x_I=\{x_I^{(1)},\dots,x_I^{(n)}\}$, which represents the design $\mathbf x$ with coordinates associated to any index set $I\subseteq\{1,\dots,d\}$, remains regular in $[0{,}1]^\iota$ where $\iota$ is the cardinality of $I$. This paper examines the conservation of nice coverage by projection using spatial point processes, and more specifically using the class of determinantal point processes. We provide necessary conditions on the kernel defining these processes, ensuring that the projected point process $\mathbf{X}_I$ is repulsive, in the sense that its pair correlation function is uniformly bounded by 1, for all $I\subseteq\{1,\dots,d\}$. We present a few examples, compare them using a new normalized version of Ripley's function. Finally, we illustrate the interest of this research for Monte-Carlo integration

Robust fluctuation analyses dealing with the plating efficiency and fluctuating final count of cells

Two specific extensions for fluctuation analysis are considered here: mutants are sampled from only a fraction of the final cultures, also called plating efficiency, and the final number of cells is no longer assumed to be constant from one culture to another. We are focusing in this paper to the extension of three specific robust methods: the classic P0 method of Luria and Delbrück, the Maximum Likelihood and a method based on the generating function of the mutant count. Unbiased estimators are thus proposed. Their asymptotic variances are computed. These statistical properties are illustrated with simulation experiments. The methods are also applied to real data sets. In particular, the results are compared with those obtained using previous methods

Time inhomogeneous mutation models with birth-date dependence

International audienceThe classic Luria-Delbrück model for fluctuation analysis is extended to the case where the split instant distributions of cells are not i.i.d.: the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non exponentially distributed lifetimes. In particular, it is possible to consider subprobability distributions, and model non exponential growth. It leads to a family of probability distributions which depend on the expected number of mutations, the death probability of mutant cells, and the split instant distributions of normal and mutant cells. This is deduced from the Bellman-Harris integral equation, written for the birthdate inhomoge-neous case. A new theorem of convergence for the final mutant counts is proved, using an analytic method. Particular examples like the Haldane model, or the case where hazard functions of the split instant distributions are proportional are studied. The Luria-Delbrück distribution with cell deaths is recovered. A computation algorithm for the probabilities is provided

Projections of determinantal point processes

In computer experiments setting, space-filling designs are used to produce inputs, viewed as point patterns. A first important property of the design is that the point pattern covers regularly the input space. A second property is the conservation of this regular covering if the point pattern is projected onto a lower dimensional space. According to the first requirement, it seems then natural to consider classes of spatial point process which generate repulsive patterns. The class of determinantal point processes (DPPs) is considered in this paper. In particular, we address the question: Can we construct a DPP such that any projection on a lower-dimensional space remains a DPP, or at least remains repulsive? By assuming a particular form for the kernel defining the DPP, we prove rigorously that the answer is positive. We propose several examples of models, and in particular stationary models, achieving this property. These models defined on a compact set of $\mathbb{R}^d$ are shown to be efficient for Monte-Carlo integration problems; we show that the same initial spatial design, defined in $\mathbb{R}^d$, can be used to efficiently estimate integrals of $\mathbb{R}^\omega$-valued for any $\omega=1,\dots,d$

Integrated photonic-based coronagraphic systems for future space telescopes

The detection and characterization of Earth-like exoplanets around Sun-like stars is a primary science motivation for the Habitable Worlds Observatory. However, the current best technology is not yet advanced enough to reach the 10^-10 contrasts at close angular separations and at the same time remain insensitive to low-order aberrations, as would be required to achieve high-contrast imaging of exo-Earths. Photonic technologies could fill this gap, potentially doubling exo-Earth yield. We review current work on photonic coronagraphs and investigate the potential of hybridized designs which combine both classical coronagraph designs and photonic technologies into a single optical system. We present two possible systems. First, a hybrid solution which splits the field of view spatially such that the photonics handle light within the inner working angle and a conventional coronagraph that suppresses starlight outside it. Second, a hybrid solution where the conventional coronagraph and photonics operate in series, complementing each other and thereby loosening requirements on each subsystem. As photonic technologies continue to advance, a hybrid or fully photonic coronagraph holds great potential for future exoplanet imaging from space.Comment: Conference Proceedings of SPIE: Techniques and Instrumentation for Detection of Exoplanets XI, vol. 12680 (2023

Visible extreme adaptive optics on extremely large telescopes: Towards detecting oxygen in Proxima Centauri b and analogs

Looking to the future of exo-Earth imaging from the ground, core technology developments are required in visible extreme adaptive optics (ExAO) to enable the observation of atmospheric features such as oxygen on rocky planets in visible light. UNDERGROUND (Ultra-fast AO techNology Determination for Exoplanet imageRs from the GROUND), a collaboration built in Feb. 2023 at the Optimal Exoplanet Imagers Lorentz Workshop, aims to (1) motivate oxygen detection in Proxima Centauri b and analogs as an informative science case for high-contrast imaging and direct spectroscopy, (2) overview the state of the field with respect to visible exoplanet imagers, and (3) set the instrumental requirements to achieve this goal and identify what key technologies require further development.Comment: SPIE Proceeding: 2023 / 12680-6

Fluctuation analysis on mutation models with birth-date dependence

The classic Luria-Delbrück model can be interpreted as a Poisson compound (number of mutations) of exponential mixtures (developing time of mutant clones) of geometric distributions (size of a clone in a given time). This " three-ingredients " approach is generalized in this paper to the case where the split instant distributions of cells are not i.i.d. : the lifetime of each cell is assumed to depend on its birth date. This model takes also into account cell deaths and non-exponentially distributed lifetimes. Previous results on the convergence of the distribution of mutant counts are recovered. The particular case where the instantaneous division rates of normal and mutant cells are proportional is studied. The classic Luria-Delbrück and Haldane models are recovered. Probability computations and simulation algorithms are provided. Robust estimation methods developed for the classic mutation models are extended to the new model: their properties of consistency and asymptotic normality remain true; their asymptotic variance are computed. Finally, the estimation biases induced by considering classic mutation models instead of inhomogeneous model are studied with simulation experiments

Modèles de mutation : étude probabiliste et estimation paramétrique

Mutation models are probabilistic descriptions of the growth of a population of cells, where mutationsoccur randomly during the process. Data are samples of integers, interpreted as final numbers ofmutant cells. These numbers may be coupled with final numbers of cells (mutant and non mutant) or a mean final number of cells.The frequent appearance in the data of very large mutant counts, usually called “jackpots”, evidencesheavy-tailed probability distributions.Any mutation model can be interpreted as the result of three ingredients. The first ingredient is about the number of mutations occuring with small probabilityamong a large number of cell divisions. Due to the law of small numbers, the number of mutations approximately follows aPoisson distribution. The second ingredient models the developing duration of the clone stemming from each mutation. Due to exponentialgrowth, most mutations occur close to the end of the experiment. Thus the developing time of arandom clone has exponential distribution. The last ingredients represents the number of mutant cells that any clone developing for a given time will produce. Thedistribution of this number depends mainly on the distribution of division times of mutants.One of the most used mutation model is the Luria-Delbrück model.In these model, division times of mutant cells were supposed to be exponentially distributed.Thus a clone develops according to a Yule process and its size at any given time follows a geometric distribution.This approach leads to a family of probability distributions which depend on the expected number of mutations and the relative fitness, which is the ratio between the growth rate of normal cells to that of mutants.The statistic purpose of these models is the estimation of these parameters. The probability for amutant cell to appear upon any given cell division is estimated dividing the mean number of mutations by the mean final number of cells.Given samples of final mutant counts, it is possible to build estimators maximizing the likelihood, or usingprobability generating function.Computing robust estimates is of crucial importance in medical applications, like cancer tumor relapse or multidrug resistance of Mycobacterium Tuberculosis for instance.The problem with classical mutation models, is that they are based on quite unrealistic assumptions: constant final number of cells,no cell deaths, exponential distribution of lifetimes, or time homogeneity. Using a model for estimation, when thedata have been generated by another one, necessarily induces a bias on estimates.Several sources of bias has been partially dealed until now: non-exponential lifetimes, cell deaths, fluctuations of the final count of cells,dependence of the lifetimes, plating efficiency. The time homogeneity remains untreated.This thesis contains probabilistic and statistic study of mutation models taking into account the following bias sources:non-exponential and non-identical lifetimes, cell deaths, fluctuations of the final count of cells, plating efficiency.Simulation studies has been performed in order to propose robust estimation methods, whatever the modeling assumptions.The methods have also been applied to real data sets, to compare the results with the estimates obtained under classical models.An R package based on the different results obtained in this work has been implemented (joint work with Rémy Drouilhetand Stéphane Despréaux) and is available on the CRAN. It includes functions dedicated to the mutation models and parameter estimation.The applications have been developed for the Labex TOUCAN (Toulouse Cancer).Les modèles de mutations décrivent le processus d’apparitions rares et aléatoires de mutations au cours de lacroissance d’une population de cellules. Les échantillons obtenus sont constitués de nombres finaux de cellules mutantes,qui peuvent être couplés avec des nombres totaux de cellules ou un nombre moyen de cellules en fin d’expérience.La loi du nombre final de mutantes est une loi à queue lourde : de grands décomptes, appelés “jackpots”,apparaissent fréquemment dans les données.Une construction générale des modèles se décompose en troisniveaux. Le premier niveau est l’apparition de mutations aléatoires au cours d’un processus de croissance de population.En pratique, les divisions cellulaires sont très nombreuses, et la probabilité qu’une de ces divisions conduise à une mutation est faible,ce qui justifie une approximation poissonnienne pour le nombre de mutations survenant pendant un temps d’observation donné.Le second niveau est celui des durées de développement des clones issus de cellules mutantes. Du fait de la croissance exponentielle,la majeure partie des mutations ont lieu à la fin du processus, et les durées de développement sont alors indépendanteset exponentiellement distribuées. Le troisième niveau concerne le nombre decellules qu’un clone issu d’une cellule mutante atteint pendant une durée de développement donnée.La loi de ce nombre dépend principalement de la loi des instants de division des mutantes.Le modèle classique, dit de Luria-Delbrück, suppose que les développements cellulaires des cellules normales aussi bien que mutantess’effectue selon un processus de Yule. On peut dans ce cas calculer expliciter la loi du nombre final de mutantes.Elle dépend de deux paramètres, qui sont le nombre moyen de mutations et le paramètre de fitness (ratio des taux de croissance des deux types de cellules).Le problème statistique consiste à estimer ces deux paramètres au vu d’un échantillon denombres finaux de mutantes. Il peut être résolu par maximisation de la vraisemblance,ou bien par une méthode basée sur la fonction génératrice. Diviser l'estimation du nombre moyen de mutations par le nombre total de cellulespermet alors d'estimer la probabilité d’apparition d’une mutation au cours d’une division cellulaire.L’estimation de cette probabilité est d’une importancecruciale dans plusieurs domaines de la médecine et debiologie: rechute de cancer, résistance aux antibiotiques de Mycobacterium Tuberculosis, etc.La difficulté provient de ce que les hypothèses de modélisation sous lesquelles la distribution du nombre final de mutants est explicitesont irréalistes.Or estimer les paramètres d’un modèle quand la réalité en suit un autre conduit nécessairement à un biais d’estimation.Il est donc nécessaire de disposer de méthodes d’estimation robustes pour lesquelles le biais, en particulier sur la probabilité de mutation,reste le moins sensible possible aux hypothèses de modélisation.Cette thèse contient une étude probabiliste et statistique de modèles de mutations prenant en compte les sources de biais suivantes : durées de vie non exponentielles, morts cellulaires,variabilité du nombre final de cellules, durées de vie non-exponentielles et non-identiquement distribuées, dilution de la population initiale.Des études par simulation des méthodes considérées sont effectuées afin de proposer, selon les caractéristiques du modèle,l’estimation la plus fiable possible. Ces méthodes ont également été appliquées à desjeux de données réelles, afin de comparer les résultats avec les estimations obtenues avec les modèles classiques.Un package R a été implémenté en collaboration avec Rémy Drouilhet et Stéphane Despréaux et est disponible sur le CRAN.Ce package est constitué des différents résultats obtenus au cours de ce travail. Il contient des fonctions dédiées aux modèles de mutations,ainsi qu'à l'estimation des paramètres. Les applications ont été développées pour le Labex TOUCAN (Toulouse Cancer)