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Talks recordeded on the
Ecodep Youtube Channel.
- 6/30/2021, 14:00 Denys Pommeret
(Aix-Marseille University, I2M / laboratory SAF, ISFA).
Comparing copulas, with Yves Ngounou.
Abstract .
Copulas are still extensively studied and used to model the dependence of multivariate
observations. Many applications can be found in fields such as energy, environment or ecology.
In a one-sample case, there are many tests to compare an observed copula to a target copula.
In the two-sample case, Rémillard and Scaillet (2009) proposed a test to compare two nonparametric
copulas, that is to test H_{0}: C_{1 } = C_{2 },
where C_{1 } and C_{2 } are two copulas observed on two iid samples, which
may be paired. To our knowledge, there is no extension to the K sample case. However, the
increasing amount of data requires sometimes more comprehensive analyzes. It is in this sense that
we propose an equality test of K copulas simultaneously when K populations are observed.
We propose to test the following hypothesis: H_{0}: C_{1 } = C_{2 }= ⋯ = C_{K },
from K iid samples, possibly paired. It is therefore a generalization of Rémillard and
Scaillet (2009). However, we obtain the exact
asymptotic distribution of the test statistic and the convergence of the test. The idea of the test is to
transform the observations to uniform laws, then to use the decomposition of the density of the
copula in the Legendre polynomials orthogonal basis. Returning to the copula function we obtain
what are called copula coefficients which characterize each copula. The test then amounts to
simultaneously comparing these coefficients. We provide some illustrations of this method, in
particular we suggest a clustering algorithm to classify populations with similar forms of dependence.
Reference.
YI Ngounou B, D Pommeret (2020) Nonparametric estimation of copulas and copula densities
by orthogonal projections.
Arxiv-2010.15351.
- 6/23/2021, 14:00 Jüri Lember
(University of Tartu, Estonia).
An evolution model that satisfies detailed balance.
Abstract, Slides.
Reference. J Lember, C Watkins (2020).
An Evolutionary Model that Satisfies Detailed Balance. Methodol. Comput. Appl. Probab.
doi.org/10.1007/s11009-020-09835-5.
- 6/9/2021, 14:00 Hansjörg Albrecher (UNIL, Lausanne)
Asymptotic Analysis of the Greenwood Statistic and Extensions.
Abstract.
We revisit and unify the asymptotic analysis of the classical Greenwood statistic comprising the ratio of the sum of squares and the sum squared of independent and identically distributed random variables with regularly varying tails. We discuss some of its application areas and extend the analysis to the situation of arbitrary powers. Finally, we study the robustness of the asymptotic expressions when some of the terms in the statistic are dropped. Part of the talk is based on recent joint work with Brandon Garcia-Flores.
References.
H Albrecher, S Ladoucette, J Teugels (2010). Asymptotics of the sample coefficient of variation and the sample dispersion.
JSPI, 140-2, 358-368.
Preprint.
H Albrecher, J Teugels (2007). Asymptotics analysis of a measure of variation.
Theor. Probability and Math. Statist. 74, 1–10.
- 6/2/2021, 14:00 Antonio Cuevas (Dept. Mat.
UAM, Madrid)
On the shape restrictions used in set estimation.
Abstract
Set estimation techniques deal with the problem of reconstructing a compact set S from a random sample of points.
Some shape restrictions on the target set S (often inspired on convexity-related notions) appear in a natural way in this context.
They are used at least in two ways: first, as a tool to simplify calculations when proving asymptotic results. Second, as a guide to construct natural estimators via the "hull paradigm" (the estimator is defined as the "minimal set" including the sample and fulfilling the assumed shape restriction). We will review here some recent contributions providing examples of both situations.
This talk is based on joint work with Alejandro Cholaquidis (Universidad de la República, Uruguay) and Catherine Aaron (Université de Clermont-Ferrand, France).
References.
C Aaron, A Cholaquidis, A Cuevas (2017).
Detection of low dimensionality and data denoising via set estimation techniques.
Electronic Journal of Statistics, 11, 4596-4628.
A Cholaquidis, A Cuevas (2020). Set estimation under biconvexity restrictions.
ESAIM: Probability and Statistics, 24, 770-788.
- 5/5/2021, 14:00 Kamila Kare (SAMM, Paris 1, Panthéon-Sorbonne)
Data Driven Model Selection for Same-Realization Predictions in Autoregressive Processes.
Slides.
Abstract. This paper is about the one-step ahead prediction of the future of observations
drawn from an infinite-order autoregressive AR(∞) process.It aims to design penalties
(completely data driven)
ensuring that the selected model verifies the efficiency property but in the non asymptotic framework.
We present an oracle inequality with a leading constant equal to one. Moreover, we also show that the
excess risk of the selected estimator enjoys the best bias-variance trade-off over the considered
collection.
To achieve these results, we needed to overcome the dependence difficulties by following a classical
approach which consists in restricting to a set where the empirical covariance matrix is equivalent
to the theoretical one. We show that this event happens with probability larger than
1-c_{0}/n^{3} with c_{0}>0.
The proposed data driven criteria are based on the minimization of the penalized criterion akin
to the Mallows's C_{p}. Monte Carlo experiments are performed to highlight the obtained results.
Reference.
K Kare (2021). Data Driven Model Selection for Same-Realization Predictions in Autoregressive Processes.
Hal Preprint.
- 4/28/2021, 14:00 Alexander Kreiss (KU Leuven)
Non-Parametric Modelling of Interactions Among Vertices in Dynamic Networks.
Slides.
Abstract. We will consider dynamic networks in which the vertices (the actors) can interact
with each other along the edges of the network. We assume that over the observation period [0,T]
the number of vertices remains fixed while the edges between them may change randomly over time.
The occurrence of interactions between the actors is modelled by specifying a Cox-Type model
which allows for additional, time-varying covariates. Our interest lies in non-parametrically estimating
the (possibly) time-varying effect of the covariates on the interactions. To this end, we introduce a
kernel-based local likelihood estimator and study its asymptotic (as the network grows) performance.
Moreover, we introduce two test statistics which evaluate the fit of the non-parametric compared to parametric models. From a theoretical point of view a particular challenge when handling this type of data is that neighboring actors in the network influence each other and cannot be treated as independent. We introduce therefore weak dependence measures on dynamic networks based on correlation, mixing and temporal m-dependence. The results are illustrated on bike sharing data.
This is partially joint work with Enno Mammen (Heidelberg) and Wolfgang Polonik (UC Davis).
References.
A Kreiß, E Mammen, W Polonik (2019) Nonparametric inference for continuous-time event counting and link-based dynamic network models.
https://doi.org/10.1214/19-EJS1588.
A Kreiß (2019) Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-Processes.
https://arxiv.org/abs/1906.03179.
A Kreiß, E Mammen, W Polonik (2021) Testing For a Parametric Baseline-Intensity in Dynamic Interaction Networks.
https://arxiv.org/abs/2103.14668.
- 4/7/2021, 14:00 Diu Tran (University of Jyväskylä, Helsinki)
Statistical inference for Vasicek-type model driven by Hermite processes.
Slides.
Abstract Let Z denote a Hermite process of order q >= 1 and self-similarity parameter
H ∈ (1/2, 1).
This process is H-self-similar, has stationary increments and exhibits long-range dependence.
When q = 1, it corresponds
to the well-known fractional Brownian motion, whereas it is not Gaussian as soon as q >= 2.
In the talk, we deal with a Vasicek-type model driven by Z, of the form dX_{t} = a(b − X_{t})dt + dZ_{t}.
This model includes the fractional Vasicek model and Hermite-driven Ornstein-Uhlenbeck process.
Here, a > 0 and b ∈ R are considered as unknown drift parameters.
We provide estimators for a and b based on
continuous-time observations. For all possible values of H and q, we prove strong consistency and we analyze the asymptotic fluctuations.
This is a first step to estimate parameters of a stochastic equation driven by a Hermite process. Joint work with Prof.
Ivan Nourdin from University of Luxembourg.
Reference. I Nourdin, D Tran (2019): Statistical inference for Vasicek-type model driven by Hermite processes.
Stoch. Proc. Appl., 129, no. 10, pp. 3774-3791.
ArXiv.
- 3/31/2021, 15:00 Frederic Barraquand (IMB Bordeaux)
Inferring species interactions using Granger causality and convergent cross mapping. Slides.
Abstract. How to reliably infer interactions between species from time series of their population densities is a long-standing goal of statistical ecology. Usually this inference is done using multivariate (linear) autoregressive models, defining interactions through Granger causality: x causes y whenever x helps predicting future y values. However, the entangled nature of nonlinear ecological systems has suggested an alternative causal inference method based on attractor reconstruction, convergent cross mapping, which is increasingly popular in ecology. Here, we compare the two methods. They uncover interactions with surprisingly similar performance for predator-prey cycles, 2-species chaotic or stochastic competition, as well as 10- and 20-species networks. Thus, contrary to intuition, linear Granger causality remains useful to infer interactions in highly nonlinear ecological networks. We conclude on inevitable similarities between Granger-causal methods and convergent cross mapping due to interaction definitions, and provide suggestions to improve many-species interaction inference.
Reference.
F Barraquand, C Picoche, M Detto, F Hartig (2019). Inferring species interactions using Granger causality and convergent cross mapping.
https://arxiv.org/abs/1909.00731.
- 3/24/2021, 14:00 Benjamin Poignard
(Riken AIP, Osaka) An introduction to sparsity: modelling, properties and applications. Slides.
Abstract. The application domains of sparse modelling have been substantially widened by the availability of high-dimensional data. In particular, high-dimensional statistical modelling is concerned with the significantly large number of parameters to estimate. To tackle the over-fitting issue, penalised/regularized estimation methods have been gaining much attention. In this talk, I will introduce the concept of sparsity together with the standard penalisation methods for sparse modelling and the implications in terms of statistical properties. To illustrate the relevance of sparse modelling, I will present some applications to models that typically suffer from the so-called curse of dimensionality.
References.
B Poignard, J-D Fermanian (2021). High-dimensional penalized arch processes.
Econometric Reviews
Volume 40, 2021 - Issue 1.
B Poignard, M Asai (2020). A Penalised OLS Framework for High-Dimensional
Multivariate Stochastic Volatility Models.
Papers In Economics
& Business, Discussion Paper 20-02.
- 3/10/2021, 16:00 Julien Randon-Furling (Paris 1, Panthéon Sorbonne)
Convex Hulls of Random Walks.
Slides.
Abstract.
This talk will cover a range of results on the convex hull of random walks in the plane and
in higher dimension:
expected perimeter length in the planar case, expected number of faces on the boundary,
expected d-dimensional volume, and other geometric properties of such random convex polytopes.
Applications in ecology include estimations of animals' home ranges and minimal habitat sizes in conservation parks.
References.
J Randon-Furling, D Zaporozhets (2020).
Convex hulls of several multidimensional Gaussian random walkls.
arXiv:2007.02768.
J Randon-Furling, F Wespi (2017).
Facets on the convex hull of d-dimensional Brownian and Lévy motion..
Physical Review E.
SN Majumdar, A Comtet, J Randon-Furling (2010).
Random convex hulls and extreme value statistics.
Journal of Statistical Physics.
- 1/27/2021, 15:00 Benjamin Bobbia (CYU & Univ. Franche-Comté)
Extreme quantile regression: A coupling approach and Wasserstein distance. Slides.
Abstract.
In this work, we develop two coupling approaches for extreme quantile regression. We
consider i.i.d copies of Y in R and X in R^d and we want an estimation of the conditional
quantile of Y given X = x of order 1- alpha for a very small alpha > 0.
We introduce the proportional tail model, strongly inspired by the heteroscedastic
extremes developed by Einmahl, de Haan and Zhou. The main assumption is that the
tail distribution of Y is asymptotically proportional to the conditional tail of Y given
X = x. We propose and study estimators of both model parameters and conditional
quantile, which are studied by coupling methods.
References.
B Bobbia, C Dombry, D Varron (2020). The coupling method in extreme value theory.
https://arxiv.org/pdf/1912.03155
B Bobbia, C Dombry, D Varron (2020).
Extreme quantile regression in a proportional tail framework.
https://arxiv.org/pdf/2002.01740
- 12/2/2020, 16:00 Rolando Rebolledo
(University of Valparaiso) Open-system approach to ecological networks.
Abstract,
Talk, and
Slides.
Reference.
R Rebolledo, SA Navarrete, S Kéfi, S Rojas, PA Marquet.
An Open-System Approach to Complex Biological Networks.
SIAM Journal on Applied Mathematics, 79(2):619–640, 2019.
- 11/25/2020, 16:00, Félix Cheysson (Agro-Paristech Paris)
Properties of Hawkes processes. Talk,
Slides.
Abstract. Hawkes processes are a family of stochastic processes for which the
occurrence of any event increases the probability of further events occurring. When count
data are only observed in discrete time, we propose a spectral approach for the estimation
of Hawkes processes, by means of Whittle's parameter estimation method. To get asymptotic
properties for the estimator, we prove alpha-mixing properties for the series of counts,
using the Galton-Watson properties of the cluster representation of Hawkes processes.
Simulated datasets and an application to the incidence of measles in France illustrate
the performances of the estimation, notably of the Hawkes excitation function, even when
the time between observations is large.
- 11/18/2020, 16:00, Marc Lavielle
(INRIA & CMAP, Polytechnique) Modelling the COVID 19 pandemic requires a model...
but also data! Talk,
Slides.
Abstract.
I will present in this talk some models for different Covid-19 data. I will first propose a
SIR-type model for the data provided by the Johns-Hopkins University for several countries:
these are the daily numbers of confirmed cases and deaths. The same model is used for all
countries but the parameters of the model change from one country to another to reflect
differences in dynamics. In particular, the model incorporates a time-dependent transmission
rate, whose variations are thought to be related to the public health measures taken by the
country in question.
I will then present a model for French hospital data provided by Santé Publique France: daily
numbers of hospitalization, admissions in intensive care units, deaths and hospital
discharges.
The proposed models may seem relatively simple, but it must be understood that they
do not pretend to describe the spread of the pandemic in a precise and detailed way.
Their role is to adjust the available data and provide reliable forecasts: their
complexity is therefore adjusted to the amount of information available in the data.
Indeed, very few parameters are needed to properly describe the outcome of interest
and the prediction proves to be stable over time.
Two interactive web applications are available to visualize the data and the adjusted
models:
http://shiny.webpopix.org/covidix/app1/
for JHU data,
http://shiny.webpopix.org/covidix/app3/
for SPF data.