Moment Matching Research Articles

Two-sided Interconnection-Based Model Reduction for Quadratic-Bilinear Systems

We propose a two-sided interconnection-based method for model order reduction of quadratic-bilinear systems. Exploiting the solution of an infinite-dimensional system of Sylvester-like equations we define the “swapped” nonlinear moment by a power series representation. We then consider together the “direct” moment and the “swapped” moment and propose two families of reduced-order models that achieve two-sided approximate moment matching. We illustrate the results of the paper on the problem of reducing the 1-D Burgers’ equation.

Lattice Boltzmann method for warm fluid simulations of plasma wakefield acceleration

A comprehensive characterization of lattice Boltzmann (LB) schemes to perform warm fluid numerical simulations of particle wakefield acceleration (PWFA) processes is discussed in this paper. The LB schemes we develop hinge on the moment matching procedure, allowing the fluid description of a warm relativistic plasma wake generated by a driver pulse propagating in a neutral plasma. We focus on fluid models equations resulting from two popular closure assumptions of the relativistic kinetic equations, i.e., the local equilibrium and the warm plasma closure assumptions. The developed LB schemes can, thus, be used to disclose insights on the quantitative differences between the two closure approaches in the dynamics of PWFA processes. Comparisons between the proposed schemes and available analytical results are extensively addressed.

Closed-Loop Interpolation by Moment Matching for Linear and Nonlinear Systems

Closed-Loop Interpolation by Moment Matching for Linear and Nonlinear Systems

Probabilistic Optimal Power Flow Computation Using Stratified Quadrature Rule

In this paper, the probabilistic optimal power flow (POPF) problem is modeled as a black box uncertainty quantification problem. If distribution functions of POPF inputs are unknown, only samples are available, the multivariate Gaussian mixture model is suggested to recover the joint cumulative distribution function of POPF inputs, and a probability selection method is introduced to relate POPF inputs to an independent standard normal vector. Then, moment matching equations are proposed to characterize the uncertainty effects of POPF inputs on outputs, whereby some properties of the POPF problem are revealed, and a stratified quadrature rule (SQR) is proposed for POPF computation by combining a Hadamard matrix based quadrature rule and a discrete sine transformation matrix based quadrature rule. This newly developed SQR has a computational burden linear with the number of POPF inputs, and performs well in estimating means and standard deviations POPF outputs. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the accuracy and efficiency of the proposed method.

Unsupervised domain adaptation without source data for estimating occupancy and recognizing activities in smart buildings

Activities Recognition (AR) and Occupancy Estimation (OE) are topics of current interest. AR and OE help many smart building applications such as energy systems and provide good services for occupants. Prior research on AR and OE has typically focused on supervised machine learning methods. For a specific smart building domain, a model is trained using data collected from the current environment (domain). The created model will not generalize well when evaluated in a new related domain due to data distribution differences. Creating a model for each smart building environment is infeasible due to the lack of labeled data. Indeed, data collection is a tedious and time-consuming task. Unsupervised Domain Adaptation (UDA) is a good solution for the considered case. UDA solves the problem of the lack of labeled data in the target domain by allowing knowledge transfer across domains. Most of the previous research in UDA requires having access to source data while creating target models which leads to privacy problems. This work considers techniques that use only a trained source model instead of a huge amount of source data to make domain adaption. This research adapted and tested UDA methods called Source HypOthesis Transfer (SHOT), Higher-Order Moment Matching (HoMM), and Source data Free Domain Adaptation (SFDA) on smart building data. SHOT is a deep learning method that learns a feature encoding module for the target model to align the data representation of the target environment with the data representation of the source environment, and it freezes the hypothesis (classifier) of the source model. Data alignment is done using information maximization and self-supervised pseudo-labeling. HoMM is also a deep learning method, however, it freezes the feature encoding module, and it learns a classifier to perform data alignment. HoMM also performs pseudo-labeling to target samples to enhance data alignment. SFDA is a deep domain adaptation method that optimizes two losses to train the target model without using any source-labeled data. The target model updates its initialized weights from the source model by minimizing a first loss that uses pseudo labels of target samples using the pre-trained source model, and by minimizing a second loss that uses pseudo labels of target samples generated by the trainable target model. To prove the efficiency of SHOT, HoMM, and SFDA, this research tests them on AR and OE datasets for a different number of activities and levels of occupancy. The impressive obtained results, with scores up to 90% for OE and up to 97% for AR, show that the considered approaches can be used to transfer knowledge across different related domains.

Model Reduction by Least Squares Moment Matching for Linear and Nonlinear Systems

Model Reduction by Least Squares Moment Matching for Linear and Nonlinear Systems

Interconnection-based model order reduction - a survey

In this survey we present in an organic, complete, and accessible style the interconnection framework for model order reduction. While this framework originally started as a revisitation of the moment matching method, in the last 15 years it has expanded to provide solutions to the model order reduction problem for general classes of systems, such as nonlinear, time-delay, stochastic, and hybrid, and to give insight on topics as diverse as circuit analysis and optimal control. The main theme of the paper is the characterization of “interconnection behaviors” as primary properties to be maintained by the reduction process. The survey is enriched by historical notes on the development of the interconnection framework.

Non-parametric style transfer: Correlation-aware exact distribution matching

Early feed-forward neural methods of arbitrary image style transfer utilized the encoded feature map up to its second-order statistics, i.e., mean and variance (or covariance) of the encoded feature map. Recent methods have begun to utilize feature statistics beyond the 2nd-order statistics, e.g., a higher-order moment, slicing-based method, and histogram matching for better style representation. However, even in the distribution matching of the latest method, considering the channel-correlation and the order of duplicated values in the feature map is missing. This resulted in insufficient style transfer or artifacts on style-transferred images e.g., unexpected color or pattern co-occurrence, skewed color or style pattern into one side, noise between background and foreground, etc. In this work, we show how to make a feature transform layer to consider both channel-correlation and duplicated values in the feature map for exact feature distribution matching in style transfer. Our experimental results prove that the stylized images obtained with our method are more similar to the target style images qualitatively without losing content clearness and superior to the existing methods quantitatively in all possible style measures, e.g., 0.0311 lower sliced Wasserstein distance than the previous method and the most favored choice with 25.54% preference.

Performance Analysis of RIS-Assisted Large-Scale Wireless Networks Using Stochastic Geometry

In this paper, we investigate the performance of a reconfigurable intelligent surface (RIS) assisted large-scale network by characterizing the coverage probability and the average achievable rate using stochastic geometry. Considering the spatial correlation between transmitters (TXs) and RISs, their locations are jointly modelled by a Gauss-Poisson process (GPP). Two association strategies, i.e., nearest association and fixed association, are both discussed. For the RIS-aided transmission, the signal power distribution with a direct link is approximated by a gamma random variable using a moment matching method, and the Laplace transform of the aggregate interference power is derived in closed form. Based on these expressions, we analyze the channel hardening effect in the RIS-assisted transmission, the coverage probability, and the average achievable rate of the typical user. We derive the coverage probability expressions for the fixed association strategy and the nearest association strategy in an interference-limited scenario in closed form. Numerical results are provided to validate the analysis and illustrate the effectiveness of RIS-assisted transmission with passive beamforming in improving the system performance. Furthermore, it is also unveiled that the system performance is independent of the density of TXs with the nearest association strategy in the interference-limited scenario.

Performance evaluation of MOMA (MOment MAtching) – a remote network calibration technique for PM2.5 and PM10 sensors

Abstract. We evaluate the potential of using a previously developed remote calibration framework we name MOMA (MOment MAtching) to improve the data quality in particulate matter (PM) sensors deployed in hierarchical networks. MOMA assumes that a network of reference instruments can be used as “proxies” to calibrate the sensors given that the probability distribution over time of the data at the proxy site is similar to that at a sensor site. We use the reference network to test the suitability of proxies selected based on distance versus proxies selected based on land use similarity. The performance of MOMA for PM sensors is tested with sensors co-located with reference instruments across three Southern Californian regions, representing a range of land uses, topography and meteorology, and calibrated against a distant proxy reference. We compare two calibration approaches: one where calibration parameters get calculated and applied at monthly intervals and one which uses a drift detection framework for calibration. We demonstrate that MOMA improves the accuracy of the data when compared against the co-located reference data. The improvement was more visible for PM10 and when using the drift detection approach. We also highlight that sensor drift was associated with variations in particle composition rather than instrumental factors, explaining the better performance of the drift detection approach if wind conditions and associated PM sources varied within a month.

Model reduction by moment matching with preservation of global stability for a class of nonlinear models

Model reduction by time-domain moment matching naturally extends to nonlinear models, where the notion of moments has a local nature stemming from the center manifold theorem. In this paper, the notion of moments of nonlinear models is extended to the global case and is, subsequently, utilized for model order reduction of convergent Lur’e-type nonlinear models. This model order reduction approach preserves the Lur’e-type model structure, inherits the frequency-response function interpretation of moment matching, preserves the convergence property, and allows formulating a posteriori error bound. By the grace of the preservation of the convergence property, the reduced-order Lur’e-type model can be reliably used for generalized excitation signals without exhibiting instability issues. In a case study, the reduced-order model accurately matches the moment of the full-order Lur’e-type model and accurately describes the steady-state model response under input variations.

On-line tool wear monitoring under variable milling conditions based on a condition-adaptive hidden semi-Markov model (CAHSMM)

Since tool wear during machining process imposes a significant limitation on production quality as well as efficiency, on-line tool wear monitoring (TWM) is one of considerably crucial issues for intelligent manufacturing system. However, there is still a lack of universal on-line TWM techniques appropriate for the practical production process with variable machining parameters. Therefore, this study proposes a highly condition-adaptive method for tool wear state monitoring and remaining useful life (RUL) estimation under time-varying operation conditions. Multi-domain features are first extracted from vibration, acoustic emission and sound signals, and subsequently fused by the kernel principal component analysis (KPCA) to highlight discriminative information correlated to tool wear. A condition-adaptive hidden semi-Markov model (CAHSMM) is established to describe the evolution of tool wear, in which the accelerated failure time model (AFTM) embedded with Taylor tool life equation is utilized to determine the state transition probabilities that are strongly dependent on cutting parameters and duration time. Moreover, to extrapolate the Gaussian mixture models (GMMs) used to represent the observation probability beyond the range where they were trained, a Moment matching based on-line unsupervised transfer learning (OUTL) method is developed. After obtaining the state transition and observation probabilities, a modified forward algorithm is eventually implemented so as to assess tool wear online. Both the dataset from milling experiments of Ti6Al4V and PHM 2010 are employed for verifying the effectiveness of the proposed method. The results show that the presented method enables to reliably evaluate wear state and RUL of the cutter even under time-switching operation parameters, which demonstrates its promising potential for industrial applications.

Testing for the Pareto type I distribution: a comparative study

AbstractPareto distributions are widely used models in economics, finance and actuarial sciences. As a result, a number of goodness-of-fit tests have been proposed for these distributions in the literature. We provide an overview of the existing tests for the Pareto distribution, focussing specifically on the Pareto type I distribution. To date, only a single overview paper on goodness-of-fit testing for Pareto distributions has been published. However, the mentioned paper has a much wider scope than is the case for the current paper as it covers multiple types of Pareto distributions. The current paper differs in a number of respects. First, the narrower focus on the Pareto type I distribution allows a larger number of tests to be included. Second, the current paper is concerned with composite hypotheses compared to the simple hypotheses (specifying the parameters of the Pareto distribution in question) considered in the mentioned overview. Third, the sample sizes considered in the two papers differ substantially. In addition, we consider two different methods of fitting the Pareto Type I distribution; the method of maximum likelihood and a method closely related to moment matching. It is demonstrated that the method of estimation has a profound effect, not only on the powers achieved by the various tests, but also on the way in which numerical critical values are calculated. We show that, when using maximum likelihood, the resulting critical values are shape invariant and can be obtained using a Monte Carlo procedure. This is not the case when moment matching is employed. The paper includes an extensive Monte Carlo power study. Based on the results obtained, we recommend the use of a test based on the phi divergence together with maximum likelihood estimation.

Parameterizing Lognormal state space models using moment matching

In ecology, it is common for processes to be bounded based on physical constraints of the system. One common example is the positivity constraint, which applies to phenomena such as duration times, population sizes, and total stock of a system’s commodity. In this paper, we propose a novel method for parameterizing Lognormal state space models using an approach based on moment matching. Our method enforces the positivity constraint, allows for arbitrary mean evolution and variance structure, and has a closed-form Markov transition density which allows for more flexibility in fitting techniques. We discuss two existing Lognormal state space models and examine how they differ from the method presented here. We use 180 synthetic datasets to compare the forecasting performance under model misspecification and assess the estimation of precision parameters between our method and existing methods. We find that our models perform well under misspecification, and that fixing the observation variance both helps to improve estimation of the process variance and improves forecast performance. To test our method on a difficult problem, we compare the predictive performance of two Lognormal state space models in predicting the Leaf Area Index over a 151 day horizon by using a process-based ecosystem model to describe the temporal dynamics. We find that our moment matching model performs better than its competitor, and is better suited for intermediate predictive horizons. Overall, our study helps to inform practitioners about the importance of incorporating sensible dynamics when using models of complex systems to predict out-of-sample.

Two‐sided Krylov enhanced proper orthogonal decomposition methods for partial differential equations with variable coefficients

AbstractIn this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

Outage Performance of RIS-Assisted V2I Communications With Inter-Cell Interference

With the ability of enhancing desired signals, Reconfigurable Intelligent Surface (RIS) is expected to be applied to vehicular networks which require ultra-reliable communications. In this paper we propose a promising RIS-assisted vehicle-to-infrastructure (V2I) network model in highway environment. Then under Rayleigh fading channels, our goal is to derive the downlink outage probability (OP) of the proposed V2I system with considering the inter-cell interference, i.e., interference from neighbor road side units (RSUs) and RISs. To achieve this intractable derivation, the probability density functions (PDFs) of desired signal power and interference-plus noise power are calculated by leveraging Gauss-Chebyshev quadrature and moment matching. Numerical results verify the accuracy of our derivation and it is shown that RISs can enhance the desired signal with causing little interference to neighbor cells, so that the OP of V2I communications can be greatly reduced.

Performance Analysis of IRS-Aided Short-Packet NOMA Systems Over Nakagami-$m$ Fading Channels

This paper analyzes the performance of an intelligent reflecting surface (IRS)-aided short-packet communication system over Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> fading channels. In this system, the base station adopts non-orthogonal multiple access (NOMA) to transmit to a near user and a far user, and its transmission to the far user is aided by the IRS. Both continuous phase shifts (CPSs) and discrete phase shifts (DPSs) are considered for the IRS. By using order statistic and moment matching, we approximate the end-to-end channel gain to follow the Gamma distribution and then derive the approximate closed-form expressions for the average block error rate (BLER) in the high signal-to-noise ratio (SNR) regime and then conduct a refined analysis of the interplay among the minimum common blocklength, optimal power allocation, and the related parameters of IRS (e.g., the number of elements of the IRS) for achieving target BLERs at two users. Finally, numerical and simulation results demonstrate the accuracy of theoretical analysis. They also show that the IRS-aided short-packet NOMA system outperforms the system without IRS, and that DPSs with 3-bit quantization achieve a similar performance as the optimal CPSs.

Parameter estimation for a bivariate beta distribution with arbitrary beta marginals and positive correlation

We discuss a bivariate beta distribution that can model arbitrary beta-distributed marginals with a positive correlation. The distribution is constructed from six independent gamma-distributed random variates. While previous work used an approximate and sometimes inaccurate method to compute the distribution’s covariance and estimate its parameters, here, we derive all product moments and the exact covariance, which can be computed numerically. Based on this analysis we present an algorithm for estimating the parameters of the distribution using moment matching. We evaluate this inference method in a simulation study and demonstrate its practical use on a data set consisting of predictions from two correlated forecasters. Furthermore, we generalize the bivariate beta distribution to a correlated Dirichlet distribution, for which the proposed parameter estimation method can be used analogously.

On Optimality of Gamma Approximation for Lognormal Shadowing Models

In this letter, we study the information-theoretic optimality of the gamma approximation for lognormal shadowing models in order to provide a rigorous mathematical ground for this useful technique. Specifically, we adopt the Kullback-Leibler (KL) divergence as the metric quantifying the distance between the original lognormal and the approximate gamma distributions. The KL divergence resulted from the moment matching criterion, the minimum achievable KL divergence of the gamma approximation, and the statistical parameter mapping relations are all derived in closed form. By these closed-form analytical expressions, we are able to rigorously examine the utility and optimality of the gamma approximation with a benchmark. Comparing the closed-form expressions of the KL divergence by moment matching and the minimum achievable benchmark, we find that the moment matching criterion, as a heuristic method, cannot guarantee the information-theoretic optimality. We also present and discuss the relevant results to substantiate the information-theoretic optimality achieved by our proposed statistical parameter mapping relations and the corresponding analytical insights.