Identifiability for Blind Source Separation of Multiple Finite Alphabet Linear Mixtures

We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. Our work extends and clarifies the scope of scenarios for which blind source separation becomes meaningful

Autocovariance estimation in regression with a discontinuous signal and mm-dependent errors: A difference-based approach

We discuss a class of difference-based estimators for the autocovariance in nonparametric regression when the signal is discontinuous (change-point regression), possibly highly fluctuating, and the errors form a stationary $m$-dependent process. These estimators circumvent the explicit pre-estimation of the unknown regression function, a task which is particularly challenging for such signals. We provide explicit expressions for their mean squared errors when the signal function is piecewise constant (segment regression) and the errors are Gaussian. Based on this we derive biased-optimized estimates which do not depend on the particular (unknown) autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators which depends on the signal is minimal as well. Further, we provide sufficient conditions for $\sqrt{n}$-consistency; this result is extended to piecewise Holder regression with non-Gaussian errors. We combine our biased-optimized autocovariance estimates with a projection-based approach and derive covariance matrix estimates, a method which is of independent interest. Several simulation studies as well as an application to biophysical measurements complement this paper.Comment: 41 pages, 3 figures, 3 table

Lower bounds for volatility estimation in microstructure noise models

In this paper we derive lower bounds in minimax sense for estimation of the instantaneous volatility if the diffusion type part cannot be observed directly but under some additional Gaussian noise. Three different models are considered. Our technique is based on a general inequality for Kullback-Leibler divergence of multivariate normal random variables and spectral analysis of the processes. The derived lower bounds are indeed optimal. Upper bounds can be found in Munk and Schmidt-Hieber [18]. Our major finding is that the Gaussian microstructure noise introduces an additional degree of ill-posedness for each model, respectively.Comment: 16 page

Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

We consider the models Y_{i,n}=\int_0^{i/n} \sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t denotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d. random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment. Furthermore, \sigma and \tau are unknown deterministic functions and W_t and (\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for \sigma^2 and \tau^2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. Our major finding is that the microstructure noise \epsilon_{i,n} introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of \epsilon_{i,n}. The method is illustrated by a small numerical study.Comment: 5 figures, corrected references, minor change

Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis

This paper is concerned with a novel regularisation technique for solving linear ill-posed operator equations in Hilbert spaces from data that is corrupted by white noise. We combine convex penalty functionals with extreme-value statistics of projections of the residuals on a given set of sub-spaces in the image-space of the operator. We prove general consistency and convergence rate results in the framework of Bregman-divergences which allows for a vast range of penalty functionals. Various examples that indicate the applicability of our approach will be discussed. We will illustrate in the context of signal and image processing that the presented method constitutes a locally adaptive reconstruction method

Heterogeneous Change Point Inference

We propose HSMUCE (heterogeneous simultaneous multiscale change-point estimator) for the detection of multiple change-points of the signal in a heterogeneous gaussian regression model. A piecewise constant function is estimated by minimizing the number of change-points over the acceptance region of a multiscale test which locally adapts to changes in the variance. The multiscale test is a combination of local likelihood ratio tests which are properly calibrated by scale dependent critical values in order to keep a global nominal level alpha, even for finite samples. We show that HSMUCE controls the error of over- and underestimation of the number of change-points. To this end, new deviation bounds for F-type statistics are derived. Moreover, we obtain confidence sets for the whole signal. All results are non-asymptotic and uniform over a large class of heterogeneous change-point models. HSMUCE is fast to compute, achieves the optimal detection rate and estimates the number of change-points at almost optimal accuracy for vanishing signals, while still being robust. We compare HSMUCE with several state of the art methods in simulations and analyse current recordings of a transmembrane protein in the bacterial outer membrane with pronounced heterogeneity for its states. An R-package is available online