Estimating linear filters with errors in variables using the Hilbert transform

In this paper we present a consistent estimator for a linear filter (distributed lag) when the independent variable is subject to observational error. Unlike the standard errors-in-variables estimator which uses instrumental variables, our estimator works directly with observed data. It is based on the Hilbert transform relationship between the phase and the log gain of a minimum phase-lag linear filter. The results of using our method to estimate a known filter and to estimate the relationship between consumption and income demonstrate that the method performs quite well even when the noise-to-signal ratio for the observed independent variable is large. We also develop a criterion for determining whether an estimated phase function is minimum phase-lag.

Signal-to-noise ratio estimation in digital computer simulation of lowpass and bandpass systems with applications to analog and digital communications, volume 3

Techniques are developed to estimate power gain, delay, signal-to-noise ratio, and mean square error in digital computer simulations of lowpass and bandpass systems. The techniques are applied to analog and digital communications. The signal-to-noise ratio estimates are shown to be maximum likelihood estimates in additive white Gaussian noise. The methods are seen to be especially useful for digital communication systems where the mapping from the signal-to-noise ratio to the error probability can be obtained. Simulation results show the techniques developed to be accurate and quite versatile in evaluating the performance of many systems through digital computer simulation

Online Bearing Remaining Useful Life Prediction Based on a Novel Degradation Indicator and Convolutional Neural Networks

In industrial applications, nearly half the failures of motors are caused by the degradation of rolling element bearings (REBs). Therefore, accurately estimating the remaining useful life (RUL) for REBs are of crucial importance to ensure the reliability and safety of mechanical systems. To tackle this challenge, model-based approaches are often limited by the complexity of mathematical modeling. Conventional data-driven approaches, on the other hand, require massive efforts to extract the degradation features and construct health index. In this paper, a novel online data-driven framework is proposed to exploit the adoption of deep convolutional neural networks (CNN) in predicting the RUL of bearings. More concretely, the raw vibrations of training bearings are first processed using the Hilbert-Huang transform (HHT) and a novel nonlinear degradation indicator is constructed as the label for learning. The CNN is then employed to identify the hidden pattern between the extracted degradation indicator and the vibration of training bearings, which makes it possible to estimate the degradation of the test bearings automatically. Finally, testing bearings' RULs are predicted by using a $\epsilon$-support vector regression model. The superior performance of the proposed RUL estimation framework, compared with the state-of-the-art approaches, is demonstrated through the experimental results. The generality of the proposed CNN model is also validated by transferring to bearings undergoing different operating conditions

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed

Deformable kernels for early vision

Early vision algorithms often have a first stage of linear-filtering that `extracts' from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs. A technique is presented that allows: 1) computing the best approximation of a given family using linear combinations of a small number of `basis' functions; and 2) describing all finite-dimensional families, i.e., the families of filters for which a finite dimensional representation is possible with no error. The technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations. The relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated. Experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi-scale 2D edge-detection kernels. The implementation issues are also discussed

A Hilbert Space Theory of Generalized Graph Signal Processing

Graph signal processing (GSP) has become an important tool in many areas such as image processing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals

Learning Sets with Separating Kernels

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.Comment: final versio