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Multi angle  Large scale reduction simple
Clausel, Marianne (Auteur de la Conférence) | CIRM (Editeur )

Consider a non-linear function $G(X_t)$ where $X_t$ is a stationary Gaussian sequence with long-range dependence. The usual reduction principle states that the partial sums of $G(X_t)$ behave asymptotically like the partial sums of the first term in the expansion of $G$ in Hermite polynomials. In the context of the wavelet estimation of the long-range dependence parameter, one replaces the partial sums of $G(X_t)$ by the wavelet scalogram, namely the partial sum of squares of the wavelet coefficients. Is there a reduction principle in the wavelet setting, namely is the asymptotic behavior of the scalogram for $G(X_t)$ the same as that for the first term in the expansion of $G$ in Hermite polynomial? The answer is negative in general. This paper provides a minimal growth condition on the scales of the wavelet coefficients which ensures that the reduction principle also holds for the scalogram. The results are applied to testing the hypothesis that the long-range dependence parameter takes a specific value. Joint work with François Roueff and Murad S. Taqqu

Keywords: long-range dependence; long memory; self-similarity; wavelet transform; estimation; hypothesis
testing
Consider a non-linear function $G(X_t)$ where $X_t$ is a stationary Gaussian sequence with long-range dependence. The usual reduction principle states that the partial sums of $G(X_t)$ behave asymptotically like the partial sums of the first term in the expansion of $G$ in Hermite polynomials. In the context of the wavelet estimation of the long-range dependence parameter, one replaces the partial sums of $G(X_t)$ by the wavelet scalogram, ...

42C40 ; 60G18 ; 62M15 ; 60G20 ; 60G22

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This talk develops a new test for local white noise which also doubles as a test for the lack of aliasing in a locally stationary wavelet process. We compare and contrast our new test with the aliasing test for stationary time series due to Hinich and co-authors. We show that the test is robust to mismatch of analysis and synthesis wavelet. We demonstrate the effectiveness of the test on some simulated examples and on an example from wind energy.

42C40 ; 60G10 ; 62M10 ; 62M15

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In this talk we address generalisation of stationary Hawkes processes in order to allow for a time-evolutive second-order analysis. A formal derivation of a time-frequency analysis via a time-varying Bartlett spectrum is given by introduction of the new class of locally stationary Hawkes process. This model is most appropriate for the analysis of (potentially very) long stretches of observed self-exciting point processes, as introduced in the stationary case by A. Hawkes (1971), in one dimension (temporal) or in a higher dimensional (i.e. spatial) context. Motivated by the concept of locally stationary autoregressive processes, we apply however inherently different techniques to describe and capture the time-varying dynamics of self-exciting point processes in the frequency domain. In particular we derive a stationary approximation of the Laplace transform of a locally stationary Hawkes process. This allows us to define a local intensity function and a local Bartlett spectrum which can be used to compute approximations of first and second order moments of the process. We will also present some insightful simulation studies and propose and discuss preliminary asymptotic results on how to estimate the first and second order structure of the process. Joint work with François Roueff and Laure Sansonnet

Keywords: locally stationary processes; Hawkes processes; Bartlett spectrum; time frequency analysis; point processes
In this talk we address generalisation of stationary Hawkes processes in order to allow for a time-evolutive second-order analysis. A formal derivation of a time-frequency analysis via a time-varying Bartlett spectrum is given by introduction of the new class of locally stationary Hawkes process. This model is most appropriate for the analysis of (potentially very) long stretches of observed self-exciting point processes, as introduced in the ...

46N30 ; 60G55 ; 62M15

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Arctic sea-ice extent has been of considerable interest to scientists in recent years, mainly due to its decreasing trend over the past 20 years. In this talk, I propose a hierarchical spatio-temporal generalized linear model (GLM) for binary Arctic-sea-ice data, where data dependencies are introduced through a latent, dynamic, spatio-temporal mixed-effects model. By using a fixed number of spatial basis functions, the resulting model achieves both dimension reduction and non-stationarity for spatial fields at different time points. An EM algorithm is used to estimate model parameters, and an MCMC algorithm is developed to obtain the predictive distribution of the latent spatio-temporal process. The methodology is applied to spatial, binary, Arctic-sea-ice data for each September over the past 20 years, and several posterior summaries are computed to detect changes of Arctic sea-ice cover. The fully Bayesian version is under development awill be discussed.
Arctic sea-ice extent has been of considerable interest to scientists in recent years, mainly due to its decreasing trend over the past 20 years. In this talk, I propose a hierarchical spatio-temporal generalized linear model (GLM) for binary Arctic-sea-ice data, where data dependencies are introduced through a latent, dynamic, spatio-temporal mixed-effects model. By using a fixed number of spatial basis functions, the resulting model achieves ...

62M30 ; 62M10 ; 62M15

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