Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden Markov models

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the L2-risk of the nonparametric spectral estimators. When the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.Comment: 27 pages, 2 figures. arXiv admin note: text overlap with arXiv:1501.0478

Estimating Granger causality from Fourier and wavelet transforms of time series data

Experiments in many fields of science and engineering yield data in the form of time series. The Fourier and wavelet transform-based nonparametric methods are used widely to study the spectral characteristics of these time series data. Here, we extend the framework of nonparametric spectral methods to include the estimation of Granger causality spectra for assessing directional influences. We illustrate the utility of the proposed methods using synthetic data from network models consisting of interacting dynamical systems.Comment: 6 pages, 2 figure

On estimating extremal dependence structures by parametric spectral measures

Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional spaces. Asymptotic distributions are derived for the estimated parameters and the resulting estimates of the spectral measure and the extreme value copula. Finite sample properties are illustrated by a simulation study

Nonparametric Estimation of Multi-View Latent Variable Models

Spectral methods have greatly advanced the estimation of latent variable models, generating a sequence of novel and efficient algorithms with strong theoretical guarantees. However, current spectral algorithms are largely restricted to mixtures of discrete or Gaussian distributions. In this paper, we propose a kernel method for learning multi-view latent variable models, allowing each mixture component to be nonparametric. The key idea of the method is to embed the joint distribution of a multi-view latent variable into a reproducing kernel Hilbert space, and then the latent parameters are recovered using a robust tensor power method. We establish that the sample complexity for the proposed method is quadratic in the number of latent components and is a low order polynomial in the other relevant parameters. Thus, our non-parametric tensor approach to learning latent variable models enjoys good sample and computational efficiencies. Moreover, the non-parametric tensor power method compares favorably to EM algorithm and other existing spectral algorithms in our experiments

Methods for Estimating Fluxes and Absorptions of Faint X-ray Sources

X-ray sources with very few counts can be identified with low-noise X-ray detectors such as ACIS onboard the Chandra X-ray Observatory. These sources are often too faint for parametric spectral modeling using well-established methods such as fitting with XSPEC. We discuss the estimation of apparent and intrinsic broad-band X-ray fluxes and soft X-ray absorption from gas along the line of sight to these sources, using nonparametric methods. Apparent flux is estimated from the ratio of the source count rate to the instrumental effective area averaged over the chosen band. Absorption, intrinsic flux, and errors on these quantities are estimated from comparison of source photometric quantities with those of high S/N spectra that were simulated using spectral models characteristic of the class of astrophysical sources under study. The concept of this method is similar to the long-standing use of color-magnitude diagrams in optical and infrared astronomy, with X-ray median energy replacing color index and X-ray source counts replacing magnitude. Our nonparametric method is tested against the apparent spectra of 2000 faint sources in the Chandra observation of the rich young stellar cluster in the M17 HII region. We show that the intrinsic X-ray properties can be determined with little bias and reasonable accuracy using these observable photometric quantities without employing often uncertain and time-consuming methods of non-linear parametric spectral modeling. Our method is calibrated for thermal spectra characteristic of stars in young stellar clusters, but recalibration should be possible for some other classes of faint X-ray sources such as extragalactic AGN.Comment: Accepted for publication in The Astrophysical Journal. 39 pages, 15 figure