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The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined.

Animals must adapt their dietary choices to meet their nutritional needs. How these needs are detected and translated into nutrient-specific appetites that drive food-choice behaviors is poorly defined. Here, we show that the enteroendocrine cells (EECs) of the adult female Drosophila midgut sense nutrients and in response release neuropeptide F (NPF), an ortholog of mammalian NPY-family gut-brain hormones. Gut-derived NPF acts via effects on glucagon-like adipokinetic hormone (AKH) signaling to induce sugar satiety and to drive hunger for protein-rich food, and on adipose tissue to promote storage of ingested nutrients. Suppression of gut NPF leads to overconsumption of dietary sugar while decreasing intake of protein-rich yeast. Furthermore, we show a female-specific function of gut-derived NPF in the suppression of AKH signaling after mating. This induces a dietary switch that promotes preference for protein-containing food to support reproduction. Together, our findings suggest that the gut NPF-AKH axis regulates appetite that drives specific food choices to ensure homeostatic consumption of nutrients, providing insight into the hormonal mechanisms that underlie nutrient-specific hungers.

Animals must adapt their dietary choices to meet their nutritional needs. How these needs are detected and translated into nutrient-specific appetites that drive food-choice behaviours is poorly understood. Here we show that enteroendocrine cells of the adult female Drosophila midgut sense nutrients and in response release neuropeptide F (NPF), which is an ortholog of mammalian neuropeptide Y-family gut-brain hormones. Gut-derived NPF acts on glucagon-like adipokinetic hormone (AKH) signalling to induce sugar satiety and increase consumption of protein-rich food, and on adipose tissue to promote storage of ingested nutrients. Suppression of NPF-mediated gut signalling leads to overconsumption of dietary sugar while simultaneously decreasing intake of protein-rich yeast. Furthermore, gut-derived NPF has a female-specific function in promoting consumption of protein-containing food in mated females. Together, our findings suggest that gut NPF-to-AKH signalling modulates specific appetites and regulates food choice to ensure homeostatic consumption of nutrients, providing insight into the hormonal mechanisms that underlie nutrient-specific hungers.

Animals must adapt their dietary choices to meet their nutritional needs. How these needs are detected and translated into nutrient-specific appetites that drive food-choice behaviors is poorly defined. Here, we show that the enteroendocrine cells (EECs) of the adult female Drosophila midgut sense nutrients and in response release neuropeptide F (NPF), an ortholog of mammalian NPY-family gut-brain hormones. Gut-derived NPF acts via effects on glucagon-like adipokinetic hormone (AKH) signaling to induce sugar satiety and to drive hunger for protein-rich food, and on adipose tissue to promote storage of ingested nutrients. Suppression of gut NPF leads to overconsumption of dietary sugar while decreasing intake of protein-rich yeast. Furthermore, we show a female-specific function of gut-derived NPF in the suppression of AKH signaling after mating. This induces a dietary switch that promotes preference for protein-containing food to support reproduction. Together, our findings suggest that the gut NPF-AKH axis regulates appetite that drives specific food choices to ensure homeostatic consumption of nutrients, providing insight into the hormonal mechanisms that underlie nutrient-specific hungers.

Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth enables simultaneous robust estimation of location and scatter, and nonparametric inference on these. A handful of remarks on the definition and the properties of the scatter halfspace depth are provided. It is argued that the currently used notion of this depth is well suited especially for symmetric random vectors. The scatter halfspace depth closely relates to an appropriate distance of matrix-generated ellipsoids from an upper level set of the (location) halfspace depth function. Several modifications and extensions to the scatter halfspace depth are considered, with their theoretical properties outlined.

The intestine is a central regulator of metabolic homeostasis. Dietary inputs are absorbed through the gut, which senses their nutritional value and relays hormonal information to other organs to coordinate systemic energy balance. However, the specific gut hormones that communicate energy availability to target organs to induce appropriate metabolic and behavioral responses are poorly defined. Here we show that the enteroendocrine cells (EECs) of the Drosophila gut sense nutrient stress via the intracellular TOR pathway, and in response secrete the peptide hormone allatostatin C (AstC), a Drosophila Somatostatin homolog. Gut-derived AstC induces secretion of glucagon-like adipokinetic hormone (AKH) via its receptor AstC-R2, a homolog of mammalian somatostatin receptors, to coordinate food intake and energy mobilization. Loss of gut AstC or its receptor in the AKH-producing cells impairs lipid and sugar mobilization during fasting, leading to hypoglycemia. Our findings illustrate a nutrient-responsive endocrine mechanism that maintains energy homeostasis under nutrient-stress conditions, a function that is essential to health and whose failure can lead to metabolic disorders.

We give examples of different multivariate probability distributions whose halfspace depths coincide at all points of the sample space.

We provide exact expressions for the scatter halfspace depth of a rich class of symmetric multivariate distributions, and discuss their properties. As special cases, we recover some results of Chen et al. (2018) and Paindaveine and Van Bever (2018), using simpler proof techniques.

This paper develops a new class of functional depths. A generic member of this class is coined Jth order kth moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.

Elliptical models are the most important family of multivariate probability distributions. We explore the properties of these distributions with respect to their halfspace depth and their illumination. The densities of elliptically symmetric distributions are expressed only in terms of the depth, the illumination, and a univariate function that can be estimated from the data. These observations set the ground for robust and nonparametric inference for (nearly) elliptical models based on the use of depth and illumination.

The intestine is a central regulator of metabolic homeostasis. Dietary inputs are absorbed through the gut, which senses their nutritional value and relays hormonal information to other organs to coordinate systemic energy balance. However, the specific gut hormones that communicate energy availability to target organs to induce appropriate metabolic and behavioral responses are poorly defined. Here we show that the enteroendocrine cells (EECs) of theDrosophilagut sense nutrient stress via the intracellular TOR pathway, and in response secrete the peptide hormone allatostatin C (AstC). Gut-derived AstC induces secretion of glucagon-like adipokinetic hormone (AKH) via its receptor AstC-R2, a homolog of mammalian somatostatin receptors, to coordinate food intake and energy mobilization. Loss of gutAstCor its receptor in the AKH-producing cells impairs lipid and sugar mobilization during fasting, leading to hypoglycemia. Our findings illustrate a nutrient-responsive endocrine mechanism that maintains energy homeostasis under nutrient-stress conditions, a function that is essential to health and whose failure can lead to metabolic disorders.

The regulation of systemic energy balance involves the coordinated activity of specialized organs, which control nutrient uptake, utilization and storage to promote metabolic homeostasis during environmental challenges. The humoral signals that drive such homeostatic programs are largely unidentified. Here we show that three pairs of central neurons in adultDrosophilarespond to internal water and nutrient availability by releasing Capa-1 and -2 hormones that signal through the Capa receptor (CapaR) to exert systemic metabolic control. Loss of Capa/CapaR signaling leads to intestinal hypomotility and impaired nutrient absorption, which gradually deplete internal nutrient stores and reduce organismal lifespan. Conversely, hyperactivation of the Capa circuitry stimulates fluid and waste excretion. Furthermore, we demonstrate that Capa/CapaR regulates energy metabolism by modulating the release of the glucagon-like adipokinetic hormone, which governs lipolysis in adipose tissue to stabilize circulating energy levels. Altogether, our results uncover a novel inter-tissue program that plays a central role in coordinating post-prandial responses that are essential to maintain adult viability.

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in R d . If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.

The activation of a neuroendocrine system that induces a surge in steroid production is a conserved initiator of the juvenile-to-adult transition in many animals. The trigger for maturation is the secretion of brain-derived neuropeptides, yet the mechanisms controlling the timely onset of this event remain ill-defined. Here, we show that a regulatory feedback circuit controlling the Drosophila prothoracicotropic hormone (PTTH) neuropeptide triggers maturation onset. We identify the ecdysone receptor (EcR) in the PTTH-expressing neurons (PTTHn) as a regulator of developmental maturation onset. Loss of EcR in these PTTHn impairs PTTH signaling, which delays maturation. We find that the steroid ecdysone dose-dependently affects Ptth transcription, promoting its expression at lower concentrations while inhibiting it at higher concentrations. Our findings indicate the existence of a feedback circuit in which rising ecdysone levels trigger, via EcR activity in the PTTHn, the PTTH surge that generates the maturation-inducing ecdysone peak toward the end of larval development. Because steroid feedback is also known to control the vertebrate maturation-inducing hypothalamic-pituitary-gonadal axis, our findings suggest an overall conservation of the feedback-regulatory neuroendocrine circuitry that times maturation initiation.

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

The halfspace depth characterization conjecture states that for any two distinct (probability) measures P and Q in the d-dimensional Euclidean space, there exists a point at which the halfspace depths of P and Q differ. Until recently, it was widely believed that this conjecture holds true for all integers \(d \ge 1\). In several research papers dealing with this problem, partial positive results towards the complete characterization of measures by their depths can be found. We provide a comprehensive review of this literature, point out to certain difficulties with some of these earlier results and construct examples of distinct (probability or finite) measures whose halfspace depths coincide at all points of the sample space, for all integers \(d > 1\).

© 2017 The Korean Statistical Society A strong law of large numbers for continuous random functions, and associated tensor product surfaces is established in the setup of discretely observed functional data. The result is shown in the framework of uniform convergence of functions, and stated without imposing any distributional assumptions. It is demonstrated that, under mild conditions, laws of large numbers for continuously observed functional data imply the corresponding laws under the discrete observational design of functions. Applications to the problem of estimation of expectation functions and covariance surfaces for discretely observed functional data are discussed. ispartof: Journal of the Korean Statistical Society vol:46 issue:4 pages:562-572 status: published

© 2017 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. A major drawback of many established depth functionals is their ineffectiveness in identifying functions outlying merely in shape. Herein, a simple modification of functional depth is proposed to provide a remedy for this difficulty. The modification is versatile, widely applicable, and introduced without imposing any assumptions on the data, such as differentiability. It is shown that many favorable attributes of the original depths for functions, including consistency properties, remain preserved for the modified depths. The powerfulness of the new approach is demonstrated on a number of examples for which the known depths fail to identify the outlying functions. Supplementary material for this article is available online. ispartof: Journal of Computational and Graphical Statistics vol:26 issue:4 pages:883-893 status: published

We investigate the monotonicity properties of the spatial depth for multivariate data. We show that the spatial depth does not decrease monotonically with respect to the deepest point. Moreover, an answer to the conjecture of Gao (2003) is provided.

Measurability, and uniform strong consistency of the integrated depths for functional data are established for the case when the random functions are Borel measurable. First consistent depths applicable to set-valued, and fuzzy data are obtained.

The human 22q11.2 chromosomal deletion is one of the strongest identified genetic risk factors for schizophrenia. Although the deletion spans a number of known genes, the contribution of each of these to the 22q11.2 deletion syndrome (DS) is not known. To investigate the effect of individual genes within this interval on the pathophysiology associated with the deletion, we analyzed their role in sleep, a behavior affected in virtually all psychiatric disorders, including the 22q11.2 DS. We identified the gene LZTR1 (night owl, nowl) as a regulator of night-time sleep in Drosophila. In humans, LZTR1 has been associated with Ras-dependent neurological diseases also caused by Neurofibromin-1 (Nf1) deficiency. We show that Nf1 loss leads to a night-time sleep phenotype nearly identical to that of nowl loss and that nowl negatively regulates Ras and interacts with Nf1 in sleep regulation. Furthermore, nowl is required for metabolic homeostasis, suggesting that LZTR1 may contribute to the genetic susceptibility to obesity associated with the 22q11.2 DS. Knockdown of nowl or Nf1 in GABA-responsive sleep-promoting neurons elicits the sleep phenotype, and this defect can be rescued by increased GABAA receptor signaling, indicating that Nowl regulates sleep through modulation of GABA signaling. Our results suggest that nowl/LZTR1 may be a conserved regulator of GABA signaling important for normal sleep that contributes to the 22q11.2 DS., Author summary Schizophrenia is a devastating mental disorder with a large genetic component to disease predisposition. One of the strongest genetic risk factors for this disorder is a relatively small genetic deletion of 43 genes on the 22nd chromosome, called 22q11.2, which confers about a 25% risk of schizophrenia development. However, it is likely that only some of these deleted genes affect disease risk, so we tested most of them individually. One of the main symptoms of schizophrenia is disturbed sleep. Sleep is an evolutionarily conserved behavior that can be easily studied in the fruit fly Drosophila melanogaster, so we investigated the effect on sleep of blocking expression of the fly homologs of most of the 22q11.2 genes and identified the gene LZTR1 (night owl, nowl) as an important sleep regulator. We found that Nowl/LZTR1 is required for inhibition of the Ras pathway and interacts genetically with the Ras inhibitor NF1. Nowl/LZTR1 appears to function in sleep by modulating inhibitory GABA signaling, which is affected in schizophrenia. Thus, this gene may underlie some of the phenotypes of the human schizophrenia-risk deletion.

Summary It is common to want to regress a scalar response on a random function. This paper presents results that advocate local linear regression based on a projection as a nonparametric approach to this problem. Our asymptotic results demonstrate that functional local linear regression outperforms its functional local constant counterpart. Beyond the estimation of the regression operator itself, local linear regression is also a useful tool for predicting the functional derivative of the regression operator, a promising mathematical object in its own right. The local linear estimator of the functional derivative is shown to be consistent. For both the estimator of the regression functional and the estimator of its derivative, theoretical properties are detailed. On simulated datasets we illustrate good finite-sample properties of the proposed methods. On a real data example of a single-functional index model, we indicate how the functional derivative of the regression operator provides an original, fast and widely applicable estimation method.

The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the associated depth-based ordering, and introduces a depth-like function for points outside the convex hull of the support of the probability measure. The illumination is, in a certain sense, dual to the halfspace depth mapping, and shares the majority of its beneficial properties. It is affine invariant, robust, uniformly consistent, and aligns well with common probability distributions.