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Rare Events for Stationary Processes

Rare Events for Stationary Processes

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY Unité de recherche INRIA Rennes, Irisa, [r]

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Efficient Policies for Stationary Possibilistic Markov  Decision Processes

Efficient Policies for Stationary Possibilistic Markov Decision Processes

Keywords: Markov Decision process, Possibility theory, lexicographic compar- isons, possibilistic qualitative utilities 1 Introduction The classical paradigm for sequential decision making under uncertainty is the one of expected utility-based Markov Decision Processes (MDP) [11, 2], which assumes that the uncertain effects of actions can be represented by probability distributions and that utilities are additive. But the EU model is not tailored to problems where uncertainty and preferences are ordinal in essence. Alternatives to the EU-based model have been proposed to handle ordinal preferences/uncertainty. Remaining within the probabilis- tic, quantitative, framework while considering ordinal preferences has lead to quantile- based approaches [17, 15, 8, 18, 9]) Purely ordinal approaches to sequential decision under uncertainty have also been considered. In particular, possibilistic MDPs [13, 12, 4, 1] form a purely qualitative decision model with an ordinal evaluation of plausibility and preference. In this model, uncertainty about the consequences of actions is repre- sented by possibility distributions and utilities are also ordinal. The decision criteria are either the optimistic qualitative utility or its pessimistic counterpart [5]. However, it is now well known that possibilistic decision criteria suffer from the drowning effect [6]. Plausible enough bad or good consequences may completely blur the comparison between policies, that would otherwise be clearly differentiable. [6] have proposed lex- icographic refinements of possibilistic criteria for the one-step decision case, in order to remediate the drowning effect. In this paper, we propose an extension of the lexico- graphic preference relations to stationary possibilistic MDPs.
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Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes

the assumptions of the classical linear model on which standard inference techniques can be ap- plied. Further they provide interesting insights on the structure of Markovian processes, and thus have interest by themselves. The intercalary independence property was apparently first given with- out proof by Ogawara (1951) for univariate Markov processes, while the truncation property was used implicitly by him (again without proof) in the context of univariate autoregressive stationary Gaussian processes. Ogawara (1951) notes that these results have been stated without proof in Lin- nik (1949). However no proof is given by Ogawara (1951) nor (apparently) by any other author. In this section, we demonstrate and extend these results to multivariate Markov processes of order p, allowing for non-stationarity and non-normality. In order to keep things as simple as possible, we shall assume that the time index set T contains the positive integers N : T ⊇ N = {1, 2, . . . }.
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Adaptive estimating function inference for non-stationary determinantal point processes

Adaptive estimating function inference for non-stationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this paper we es- tablish asymptotic normality of estimating function estimators in a very general setting of non-stationary point processes. We then adapt this result to the case of non-stationary determi- nantal point processes which are an important class of models for repulsive point patterns. In practice often first and second order estimating functions are used. For the latter it is common practice to omit contributions for pairs of points separated by a distance larger than some trun- cation distance which is usually specified in an ad hoc manner. We suggest instead a data-driven approach where the truncation distance is adapted automatically to the point process being fit- ted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non-stationary determinantal point processes.
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Contrast estimation for parametric stationary determinantal point processes

Contrast estimation for parametric stationary determinantal point processes

From a theoretical point of view, neither the likelihood method nor the minimum contrast methods for DPPs have been studied thoroughly, even in assuming that a spectral method for C is known. In this work, we focus on parametric stationary DPPs and we prove the strong consistency and the asymptotic normality of the min- imum contrast estimators based on K and g. These questions are in connection with the general investigation of Y. Guan and M. Sherman [10], who study the asymp- totic properties of the latter estimators for stationary point processes. However the setting in [10] has a clear view to Cox processes and the assumptions involve both
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Residuals and goodness-of-fit tests for stationary marked Gibbs point processes

Residuals and goodness-of-fit tests for stationary marked Gibbs point processes

bution) parametrically estimated and integrated from 0 to t. The extension in higher dimension requires further developments due to the lack of natural ordering. It may be done for point pro- cesses admitting a conditional density with respect to the Poisson process. These point processes correspond to the Gibbs measures. The equilibrium in one dimension between the number of events and the integrated hazard rate may be replaced in higher dimension by the Campbell equilibrium equation or Georgii-Nguyen-Zessin formula (see Georgii ( 1976 ), Nguyen and Zessin ( 1979a ) and Section 2.3 ), which is the basis for defining the class of h −residuals where h represents a test func- tion. In particular, Baddeley et al. Baddeley et al. ( 2005 ) consider different choices of h leading to the so-called raw residuals, inverse residuals and Pearson residuals, and show that they share similarities with the residuals obtained for generalized linear models.
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Efficient Policies for Stationary Possibilistic Markov  Decision Processes

Efficient Policies for Stationary Possibilistic Markov Decision Processes

INRA-MIAT, Toulouse, France, email: regis.sabbadin@inra.fr Abstract. Possibilistic Markov Decision Processes offer a compact and tractable way to represent and solve problems of sequential decision under qualitative un- certainty. Even though appealing for its ability to handle qualitative problems, this model suffers from the drowning effect that is inherent to possibilistic decision theory. The present paper proposes to escape the drowning effect by extending to stationary possibilistic MDPs the lexicographic preference relations defined in [6] for non-sequential decision problems and provides a value iteration algorithm to compute policies that are optimal for these new criteria.
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Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domain

Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domain

distribution of a corresponding Markov chain (we refer the reader to the sur- veys [ 15 , 19 ] and to the book [ 11 ] for general references on quasi-stationary distributions; basic facts and useful results on quasi-stationary distributions are also reminded in Section 3 ). Under a similar setting but using stochastic approx- imation techniques, Benaïm and Cloez [ 3 ] and Blanchet, Glynn and Zheng[ 6 ] independently proved the almost sure convergence of the occupation measure µt toward the quasi-stationary distribution of X. These works have since been generalized to the compact state space case by Benaïm, Cloez and Panloup [ 4 ] under general criteria for the existence of a quasi-stationary distribution for X. Continuous time diffusion processes with smooth bounded killing rate on com- pact Riemanian manifolds have been recently considered by Wang, Roberts and Steinsaltz [ 21 ], who show that a similar algorithm with weights also converges toward the quasi-stationary distribution of the underlying diffusion process. Re- cently, Mailler and Villemonais [ 14 ] have proved such a convergence result for processes with smooth and bounded killing rate evolving in non-compact (more precisely unbounded) spaces using a measure-valued Pólya process representa- tion of this reinforced algorithm.
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Smaller population size at the MRCA time for stationary branching processes

Smaller population size at the MRCA time for stationary branching processes

STATIONARY BRANCHING PROCESSES YU-TING CHEN AND JEAN-FRANC ¸ OIS DELMAS Abstract. We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distri- bution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individuals involved in the last coalescent event of the genealogical tree. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mech- anism. In this case, the size of the population at the MRCA is, in mean, less by 1/3 than size of the current population size. We also provide in this case the fluctuations for the renormalized number of ancestors.
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Aggregation of predictors for non stationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes

Aggregation of predictors for non stationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes

varying sub-linear coefficients, 2) a Lipschitz assumption on the predictors and 3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach for deriving an adaptive predictor for locally stationary time varying autoregres- sive (TVAR) processes. It is obtained by aggregating a finite number of well chosen predictors, each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the TVAR coefficients. We show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness. To prove this result, a lower bound is established for the minimax rate of the prediction risk for the TVAR process. Numerical experiments complete this study. An important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online predic- tion context.
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Standard and robust intensity parameter estimation for stationary determinantal point processes

Standard and robust intensity parameter estimation for stationary determinantal point processes

5 Conclusion In this paper, we focus on the class of stationary determinantal point processes and present two estimators of the intensity parameter for which we prove asymptotic properties. Among the two estimators, one of them, namely the median-based estimator is tailored to be robust to outliers. The median-based estimator depends on a tuning estimator, the number of blocks into which the original window is divided. The empirical findings show that the results are quite sensitive to this parameter. To correct that sensitivity we propose a combined approach and define the estimator e
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Takacs-Fiksel method for stationary marked Gibbs point processes

Takacs-Fiksel method for stationary marked Gibbs point processes

In contrast, our asymptotic results are proved in a very general setting, i.e. for a large class of stationary marked Gibbs models and test functions. The method employed to prove asymptotic normality is based on a conditional centering assumption, first appeared in [24] for the Ising model and generalized to certain spatial point processes in [28]. The main restriction that this method induces is only the finite range of the Hamiltonian. There are no limitations on the space of parameters and, in particular, the possible presence of phase transition does not affect the asymptotic behavior of the estimator. Moreover, the test functions may depend on the parameters. This extension seems important to us because, as emphasized in Section 3.2.2, such test functions can lead to quick and/or explicit estimators. All the general hypotheses assumed for the asymptotic results are discussed. For this, we focus on exponential family models, that is, on models whose interaction function is linear in the parameters. We show that our integrability and regularity assumptions are not restrictive since they are valid for a large class of models such as the Multi-Strauss marked point process, the Strauss-disc type point process, the Geyer’s triplet point process, the quermass model and for all test functions used as a motivation for this work. In the setting of the exponential family models, we also discuss the classical identifiability condition which is required for the Takacs-Fiksel procedure. To the best of our knowledge, this is the first attempt to discuss it. We will specially dwell on questions like: what choices of test functions (and how many test functions) lead to a unique minimum of the contrast function? We propose general criteria and provide examples. It seems commonly admitted that to achieve the identification of the Takacs-Fiksel procedure, one should at least choose as many test functions as the number of parameters. As a consequence of our study, it appears that one should generally strictly choose more test functions than the number of parameters to achieve identification.
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ALE Methods for Determining Stationary Solutions Metal Forming Processes

ALE Methods for Determining Stationary Solutions Metal Forming Processes

[2] R. Boman and J.-P. Ponthot. ALE methods for stationary solutions of metal forming processes. In J.A. Cavas, editor, Second ESAFORM conference on Material Forming, pages 585–589, Guimaraes, Portugal, 1998. [3] A. N. Brooks and T. J. R. Hughes. Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier- stokes equations. Computer Methods in Applied Mechanics and Engineering, 32:199– 259, 1982.

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Wavelet Method for Locally Stationary Seasonal Long Memory Processes

Wavelet Method for Locally Stationary Seasonal Long Memory Processes

For the estimation procedure, we use the MB(16) wavelet filter (L = 16), and we choose the adaptive orthonormal basis using portmanteau test with p = 0.01. We partition the sampling interval [0, 1) into 2 6 = 64 subintervals (l = 6), and we get 64 local estimates for d(t). Finally, we smooth the esti- mates using two local polynomial methods, spline method and loess method. We replicate the simulations 100 times for each locally stationary 1−factor Gegenbauer process (12) using the two previous functions d(t). We carry out the code on the computer Mac OS X 10.5.1 Léopard, written in language R with the help of the package "waveslim".
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Adaptive wavelet based estimator of the memory parameter for stationary Gaussian processes

Adaptive wavelet based estimator of the memory parameter for stationary Gaussian processes

ψ(t) dt = 0 and ψ(0) = ψ(1) = 0. Remark 3 The choice of a wavelet satisfying Assumption W ( ∞) is quite restricted because of the required smoothness of ψ. For instance, the function ψ(t) = (t 2 − t + a) exp(−1/t(1 − t)) and a ≃ 0.23087577 satisfies Assumption W ( ∞). The class of ”wavelet” checking Assumption W (5/2) is larger. For instance, ψ can be a dilated Daubechies ”mother” wavelet of order d with d ≥ 6 to ensure the smoothness of the function ψ.It is also possible to apply the following theory to ”essentially” compactly supported ”mother” wavelet like the Lemari´e-Meyer wavelet. Note that it is not necessary to choose ψ being a ”mother” wavelet associated to a multi-resolution analysis of L 2 (R) as in the recent paper of Moulines et al. (2007). The whole theory can be developed without this assumption, in which case the choice of ψ is larger.
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Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes

Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes

particular, we recover the results of Knobloch and Partzsch [14], who proved that (1.1) holds for a class of diffusion processes evolving in R d (d ≥ 3), us- ing two sided estimates combined with non-trivial spectral properties of the infinitesimal generator of X. We actually prove that the two sided esti- mates are sufficient for diffusion processes in R d but also for general Markov processes, while some spectral properties can be recovered from our results. Our second result (Theorem 3.1) is based on gradient estimates of the Dirichlet semi-group obtained by Wang [22] and Priola and Wang [21]. The gradient estimates of [22] hold for Brownian motions with C 1 drift evolving in bounded manifolds with C 2 boundary ∂M and killed when they hit ∂M .
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Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes

The intercalary independence property was apparently first given without proof by Ogawara 1951 for univariate Markov processes, while the truncation property was used implicitly by him a[r]

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On the Use of Non-Stationary Policies for Infinite-Horizon Discounted Markov Decision Processes

On the Use of Non-Stationary Policies for Infinite-Horizon Discounted Markov Decision Processes

that tends to 1−γ γ ǫ when k tends to ∞. In other words, we can see here that the problem of “computing a (non stationary) approximately-optimal policy” is not harder than that of “computing approximately the value of some fixed policy”. Since the respective asymptotic errors are 1−γ γ ǫ and 1−γ 1 ǫ, it seems even simpler ! Proof of Theorem 2. The value of π k,m satisfies:

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Analysis of stationary and non-stationary long memory processes : estimation, applications and forecast

Analysis of stationary and non-stationary long memory processes : estimation, applications and forecast

3.3.2 Daubechies Wavelets The Daubechies wavelet filters represent a collection of wavelets that improve the frequency- domain characteristics of the Haar wavelet and may be still interpreted as generalized differences of adjacent averages (Daubechies, 1992). Daubechies derived these wavelets from the criteria of a compactly supported function with the maximum number of van- ishing moments. In general, there are no explicit time-domain formulae for this class for wavelet filters (when possible, filter coefficients will be provided). Daubechies first chose an extremal phase factorization, whose resulting wavelet we denote by D(L) where L is the length of the filter. An alternative factorization leads to the least asymmetric class of wavelets, which we denote by LA(L). Shann and Yen (1999) provided exact values for both the extremal phase and least asymmetric wavelets of length L ∈ {8, 10}. Longer extremal phase and least asymmetric wavelet filters do not have a closed form and have been tabulated by, for example, Daubechies (1992) and Percival and Walden (2000). For Daubechies wavelets, the number of vanishing moments is half the filter length, thus the Haar wavelet has a single vanishing moment, the D(4) wavelet has two vanishing moments, and the D(8) and LA(8) wavelets both have four vanishing moments. One implication of this property is that longer wavelet filters may produce stationary wavelet coefficients vectors from "higher degree" non-stationary stochastic processes.
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Prediction of weakly locally stationary processes by auto-regression

Prediction of weakly locally stationary processes by auto-regression

et al. ( 2015 ) . More precisely, using aggregation techniques introduced in the context of individual sequences prediction (see e.g. Cesa-Bianchi and Lugosi ( 2006 )) and statistical learning (see e.g. Barron ( 1987 )), one can aggregate su fficiently many predictors in order to build a minimax predictor which adapts to the unknown smoothness β of the time vary- ing parameter of a TVAR process. However, a crucial requirement in Giraud et al. ( 2015 ) is to rely on β-minimax-rate sequences of predictors for any β > 0. Our main contribution here is to fill this gap, hence achieving to solve the problem of the adaptive minimax-rate linear forecasting of locally stationary TVAR processes with coe fficients of any (unknown, arbitrarily large) H¨older smoothness index.
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