Polynomial Filters Research Articles

Comparative Analysis of Quasi-Linear Kalman-Type Algorithms in Estimating a Markov Sequence with Nonlinearities in the System and Measurement Equations

The so-called Kalman type algorithms (KTA) are considered, among them quasi-linear KTAs introduced as a separate class, the features of which are the Gaussian approximation of the a posteriori probability density function (p.d.f.) at each step and the procedure for processing the current measurement based on the ideology of a linear optimal algorithm. The unified structure of such algorithms and their features are discussed. Two groups are distinguished the quasi-linear KTA: the first is algorithms using Taylor series expansion of the nonlinear functions, and the second is the so-called linear regression KTA. The methods of their designing are considered, and the common features are described. Detailed attention is paid to the following KTAs: the extended Kalman filter (EKF), polynomial filters of the second and third order (PF2 and PF3), as representatives of the first group, and the Unscented and Cubature Kalman filters (UKF and CKF), as representatives the second one. Their comparative analysis is carried out using the estimation problem of a scalar Markov sequence in the presence of nonlinearities in the shaping filter and in the measurement equations. For all the studied algorithms, the formulas are given in a form convenient for comparison. Based on these formulas, possible causes of a decrease in accuracy and a violation of the consistency properties are identified. Using the previously proposed procedure based on the method of statistical tests, predictive simulation was carried out, which made it possible to confirm the conclusions obtained previously on the basis of an analysis of the formulas for the algorithms being compared. The simulation also allowed to compare the computational complexity of the compared algorithms. The results of the study may be useful to developers involved in the processing of measurement information when choosing a filtering algorithm for solving specific practical estimation problems.

Improving Real-Time Trend Estimates Using Local Parametrization of Polynomial Regression Filters

This paper examines and compares real-time estimates of the trend-cycle component using moving averages constructed with local polynomial regression. It enables the reproduction of Henderson’s symmetric and Musgrave’s asymmetric filters used in the X-13ARIMA-SEATS seasonal adjustment algorithm. This paper proposes two extensions of local polynomial filters for real-time trend-cycle estimates: first including a timeliness criterion to minimize the phase shift; second with procedure for parametrizing asymmetric filters locally while they are generally parametrized globally, which can be suboptimal around turning points. An empirical comparison, based on simulated and real data, shows that modeling polynomial trends that are too complex introduces more revisions without reducing the phase shift, and that local parametrization reduces the delay in detecting turning points and reduces revisions. The results are reproducible and all the methods can be easily applied using the R package rjd3filters.

Asymptotics of solutions of special second-order linear recurrencies with polynomial coefficients and boundary effects of polynomial filters

In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.

Polynomial filter-based adaptive fault-tolerant control for a class of nonlinear systems with multiplicative faults

In this article, the problem of fault-tolerant control is investigated for polynomial nonlinear systems (PNSs) subject to multiplicative faults and measurement noise. Firstly, a matrix Taylor expansion technique is applied to transform the PNS into a linear parameter varying system. Then the polynomial filter is constructed to get estimations of states and unmatched disturbances, which reduces the effect of measurement noise. A novel adaptive fault-tolerant controller is proposed to compensate for multiplicative faults and unmatched disturbances. Furthermore, the condition of the parameter-dependent linear matrix inequality (PDLMI) is deduced, which guarantees the boundedness of tracking errors and estimation errors, and the sum of squares (SOS) decomposition technique is utilized to solve PDLMI. Finally, the simulation of a rotary steerable drilling tool system confirms the feasibility of the proposed adaptive fault-tolerant control framework.

FAQ: How do I estimate the output gap?

Abstract I investigate the properties of output gaps in New Keynesian DSGE models and study the relationship between theory-based quantities and the estimates obtained with standard approaches. Theoretical gaps display low frequency variations, have similar frequency domain representation as potentials and are generally correlated with them. Potentials have important business cycle variability. Existing statistical approaches fail to recognize these features and generate distorted estimates. Gaps are best estimated with a Polynomial filter. Explanations for the outcomes are given. I propose a statistical procedure reducing estimation biases.

Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction

In recent years, spectral graph neural networks, characterized by polynomial filters, have garnered increasing attention and have achieved remarkable performance in tasks such as node classification. These models typically assume that eigenvalues for the normalized Laplacian matrix are distinct from each other, thus expecting a polynomial filter to have a high fitting ability. However, this paper empirically observes that normalized Laplacian matrices frequently possess repeated eigenvalues. Moreover, we theoretically establish that the number of distinguishable eigenvalues plays a pivotal role in determining the expressive power of spectral graph neural networks. In light of this observation, we propose an eigenvalue correction strategy that can free polynomial filters from the constraints of repeated eigenvalue inputs. Concretely, the proposed eigenvalue correction strategy enhances the uniform distribution of eigenvalues, thus mitigating repeated eigenvalues, and improving the fitting capacity and expressive power of polynomial filters. Extensive experimental results on both synthetic and real-world datasets demonstrate the superiority of our method.

A Graph Dynamics Prior for Relational Inference

Relational inference aims to identify interactions between parts of a dynamical system from the observed dynamics. Current state-of-the-art methods fit the dynamics with a graph neural network (GNN) on a learnable graph. They use one-step message-passing GNNs---intuitively the right choice since non-locality of multi-step or spectral GNNs may confuse direct and indirect interactions. But the effective interaction graph depends on the sampling rate and it is rarely localized to direct neighbors, leading to poor local optima for the one-step model. In this work, we propose a graph dynamics prior (GDP) for relational inference. GDP constructively uses error amplification in non-local polynomial filters to steer the solution to the ground-truth graph. To deal with non-uniqueness, GDP simultaneously fits a ``shallow'' one-step model and a polynomial multi-step model with shared graph topology. Experiments show that GDP reconstructs graphs far more accurately than earlier methods, with remarkable robustness to under-sampling. Since appropriate sampling rates for unknown dynamical systems are not known a priori, this robustness makes GDP suitable for real applications in scientific machine learning. Reproducible code is available at https://github.com/DaDaCheng/GDP.

Development of methods and algorithms for calculating ventilation and perfusion in the EIT

The work is devoted to method and algorithm for calculating lung ventilation and perfusion based on electrical impedance tomography, which consists in isolating the ventilation or perfusion component of the voltage change on the surface of the chest cavity by filtering, followed by calculating a impedance integral parameter and average value of the voltage change, the law of change of which has a high correlation with lung ventilation. Main approaches to filtering discrete signals, which are measurement data in the calculation of ventilation and lung perfusion based on EIT during mechanical ventilation, namely polynomial filters, a simple Fourier filter and a filter based on discrete wavelet transformation, have been identified and investigated. A block diagram of an algorithm for implementing a simple Fourier filter has been developed.

Learning High-Order Graph Convolutional Networks via Adaptive Layerwise Aggregation Combination.

Graph convolutional networks have attracted wide attention for their expressiveness and empirical success on graph-structured data. However, deeper graph convolutional networks with access to more information can often perform worse because their low-order Chebyshev polynomial approximation cannot learn adaptive and structure-aware representations. To solve this problem, many high-order graph convolution schemes have been proposed. In this article, we study the reason why high-order schemes have the ability to learn structure-aware representations. We first prove that these high-order schemes are generalized Weisfeiler-Lehman (WL) algorithm and conduct spectral analysis on these schemes to show that they correspond to polynomial filters in the graph spectral domain. Based on our analysis, we point out twofold limitations of existing high-order models: 1) lack mechanisms to generate individual feature combinations for each node and 2) fail to properly model the relationship between information from different distances. To enable a node-specific combination scheme and capture this interdistance relationship for each node efficiently, we propose a new adaptive feature combination method inspired by the squeeze-and-excitation module that can recalibrate features from different distances by explicitly modeling interdependencies between them. Theoretical analysis shows that models with our new approach can effectively learn structure-aware representations, and extensive experimental results show that our new approach can achieve significant performance gain compared with other high-order schemes.

Fourier-based and Rational Graph Filters for Spectral Processing.

Data are represented as graphs in a wide range of applications, such as Computer Vision (e.g., images) and Graphics (e.g., 3D meshes), network analysis (e.g., social networks), and bio-informatics (e.g., molecules). In this context, our overall goal is the definition of novel Fourier-based and graph filters induced by rational polynomials for graph processing, which generalise polynomial filters and the Fourier transform to non-Euclidean domains. For the efficient evaluation of discrete spectral Fourier-based and wavelet operators, we introduce a spectrum-free approach, which requires the solution of a small set of sparse, symmetric, and well-conditioned linear systems and is oblivious of the evaluation of the Laplacian or kernel spectrum. Approximating arbitrary graph filters with rational polynomials provides a more accurate and numerically stable alternative with respect to polynomials. To achieve these goals, we also study the link between spectral operators, wavelets, and filtered convolution with integral operators induced by spectral kernels.

Complex exponential graph convolutional networks

Graph convolutional networks based on spectral methods have recently achieved considerable success in processing non-Euclidean structured data such as graphs. In this paper, we propose the complex exponential graph convolutional network (CEGCN), which is a novel spectral convolutional architecture for performing deep learning on graphs. Specifically, we design a complex exponential polynomial filter with powerful expressive ability and whose zeros can be changed to prevent the over-smoothing issue. Furthermore, a CEGCN variant, called CEGCN-hp, combines the graph Fourier transform-based and high-pass filters to capture the high-frequency components in the graph spectral domain. We perform a spectral analysis to illustrate the motivation and expressive power of the proposed model. Experimental results show that our model matches or outperforms various state-of-the-art baselines on three downstream tasks: semi-supervised node classification, community detection, and graph classification. Our implementation is available online.1

Polynomial filters for camera-based structural intensity analysis on curved plates

Structural intensity can be used as a measure to detect energy sources, sinks, and transfer paths in solid structures. It provides a valuable design tool for vibro-acoustic problems next to modal analysis. However, it has been proven very challenging to determine structural intensity experimentally for the general case of curved plates. This is due to the requirement of an accurate measurement of the full 3D vibrational field and the computation of spatial gradients thereof. The purpose of this paper is to propose a mesh-free, inherently smoothing, polynomial filtering approach and investigate its application for structural intensity analysis on curved plates based on a stereo camera measurement. Numerical studies are conducted to determine reasonable algorithm parameters (discretization, fitting radius, and the number of iterations) and their performance for different levels of uncertainty and various displacement fields. The results show that, depending on the uncertainty in the measurement, optimal values of the algorithm parameters exist. Furthermore, it is essential to achieve a high signal-to-noise ratio in the camera measurement. The computation of structural intensity is validated experimentally on the case of a flat plate and a curved plate (mockup oil pan). For the flat plate, good agreement with a reference finite element simulation is obtained. While it is not possible to derive quantitative results for the mockup oil pan under realistic measurement uncertainty, qualitative conclusions can be still drawn.

Transferability of graph neural networks: An extended graphon approach

We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability: the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set. In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and, 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons.

Exploiting Unit Circle Roots for Improved Radar Moving Target Detection in Low Rank Clutter With Limited Secondary Data

Measured radar clutter often exhibits a low-rank structure that can be exploited for designing adaptive moving target detectors to reduce the amount of secondary data necessary for desirable detection performance. It has been shown recently that for symmetrically spaced pulse trains significant performance improvements can be achieved by constraining the roots of the Adaptive Matched Filter (AMF) polynomial onto the unit circle (UC) that satisfies a theoretical property of the known-covariance AMF. In this paper, the UC-roots approach is extended to develop a generalized likelihood ratio test (GLRT) for radar moving target detection in low-rank clutter with limited secondary data. The proposed unit circle GLRT (UC-GLRT) uses the properties of orthogonal projection matrices to enforce the UC roots constraint. The asymptotic performance analysis of the proposed UC-GLRT conducted in this paper is verified by simulation. Simulation studies with limited secondary data demonstrate the superior performance of the UC-GLRT approach over several existing detectors. Clutter rank and target velocity mismatch issues are also studied.

Digraph Filter Design Based on Directed Laplacian Matrix and Least Squares Method

In this paper, the digraph filter design based on directed Laplacian matrix (DLM) and least squares method is presented. First, the eigen-decomposition of DLM is used to define the digraph Fourier transform (DGFT). Then, the spectral properties of DGFT are studied and applied to specify the ideal spectral response of the digraph filter. Next, the coefficients of polynomial and rational digraph filters are determined by the least squares method which minimizes the integral absolute squared errors between ideal response and actual spectral response of filter. The matrix inversion can be used to compute the optimal solution. Finally, the proposed method is compared with conventional design methods to evaluate performance and the signal denoising application examples are demonstrated to show the effectiveness of the designed digraph filters.

Polynomial chaos Kalman filter for target tracking applications

In this paper, an approximate Gaussian state estimator is developed based on generalised polynomial chaos expansion for target tracking applications. Motivated by the fact that calculating conditional moments in an approximate Gaussian filter involves computing integrals with respect to Gaussian density, the authors approximate the non-linear dynamics using polynomial chaos expansion. Second-order as well as third-order polynomial chaos expansions were used for approximate filtering, to derive the necessary recursive algorithm and also provide certain algebraic simplifications which reduce the computational burden without significantly affecting the filtering performance. Two comprehensive numerical experiments for multivariate systems, including one for a multi-model system, demonstrate the potential of the new algorithms.

Optimum Chebyshev filter with an equalised group delay response

ABSTRACT The first part of the paper discusses designing of the lowpass filter that gives both a constant group delay over the passband and a sharp cut-off slope, which is one of the most difficult challenges in continuous-time signal processing. Furthermore, equalisation of the group delay distortion of the Chebyshev filter has been performed through a cascade connection of the allpass filters. The Newton–Raphson method is applied for solving the system of nonlinear equations that calculate the allpass filter parameters for equalising the constant group delay in an equiripple sense. This procedure gives nearly symmetric impulse response of the resulting filtering system, as well as a longer duration. In the second part of the paper, two solutions for synthesising polynomial filter transfer functions are discussed: filter with monotonic amplitude characteristic (LSM) and filter using Chebyshev polynomial. For the purpose of comparison, the passband ripple factor of the Chebyshev filter is necessary to be adjusted so that both filters have the same stopband insertion loss, group delay and transient response. It is shown that Chebyshev filter is preferable since the passband magnitude distortions and sensitivity to the element deviation tolerance are significantly superior.

A Nonlinear Polynomial Filter for Sidelobe Suppression of Spaceborne and Airborne Synthetic Aperture Radar (SAR) Images With High Resolution

A nonlinear polynomial filter (NPF) was proposed for the sidelobe suppression in synthetic aperture radar (SAR) imagery with high resolution. The single-sinc and multiple-sinc cases were examined. In the single-sinc case, one target existed, and the other surrounding targets were absent or weak scatterers that can be ignored. Otherwise, the cases were considered as multiple-sinc cases. A judge function was computed using the underlying sample and two adjacent samples for every sample in the image domain. This function was the basis to determine whether the related sample fell under the single-sinc case or not. If it was the single-sinc case, the judge function was also used to reduce the sidelobes. A polynomial filter was used for the sidelobe suppression in the multiple-sinc cases. Using the polynomial filtering, the energy of sidelobes was reduced by dividing a polynomial function. The unequivocal determination of sidelobe suppression was achieved for SAR data. The performance of the proposed algorithm was evaluated using acquired spaceborne SAR and airborne SAR data with high resolution. The effectiveness of the sidelobe suppression was qualitatively and quantitatively examined, and the results were satisfactory.

Modeling PM2.5 and PM10 Using a Robust Simplified Linear Regression Machine Learning Algorithm

The machine learning algorithm based on multiple-input multiple-output linear regression models has been developed to describe PM2.5 and PM10 concentrations over time. The algorithm is fact-acting and allows for speedy forecasts without requiring demanding computational power. It is also simple enough that it can self-update by introducing a recursive step that utilizes newly measured values and forecasts to continue to improve itself. Starting from raw data, pre-processing methods have been used to verify the stationary data by employing the Dickey–Fuller test. For comparison, weekly and monthly decompositions have been achieved by using Savitzky–Golay polynomial filters. The presented algorithm is shown to have accuracies of 30% for PM2.5 and 26% for PM10 for a forecasting horizon of 24 h with a quarter-hourly data acquisition resolution, matching other results obtained using more computationally demanding approaches, such as neural networks. We show the feasibility of using multivariate linear regression (together with the small real-time computational costs for the training and testing procedures) to forecast particulate matter air pollutants and avoid environmental threats in real conditions.

QPGCN: graph convolutional network with a quadratic polynomial filter for overcoming over-smoothing

QPGCN: graph convolutional network with a quadratic polynomial filter for overcoming over-smoothing