Model Order Estimation Research Articles

Order estimation for autoregressive models using criteria based on stochastic complexity

In this paper, we are interested in the order estimation of an autoregressive model using the information criterion developed by El Matouat and Hallin (1996), which is based on stochastic complexity. This criterion is a generalization of the Hannan and Quinn criterion and provides a convergence of the model order estimator, but it depends on a parameter that is sensitive to the sample size. In order to select the exact order of the candidate model, we propose a method for identifying the values of this parameter from the sample using the information contained in sub-samples of increasing size. To study the performance of the proposed method in comparison with the usual criteria, we simulated samples from autoregressive models on which we applied our procedure. Simulation results support the relevance of our procedure when compared to the Akaike criterion, the Hannan and Quinn criterion, and the Schwarz criterion.

Model order estimation in multi-component chirp signals

Real datasets exhibiting non-constant frequencies, can be analyzed with the help of multi-component chirp signals adequately. This requires estimation of the number of chirp components in practice. The problem of estimation becomes difficult when the data is composed of moderate to large number of chirp components. We propose Bayesian approach to estimate the number of components of a multi-component chirp model using an approximation to the mode of the posterior distribution. This approximation is motivated from the relation between mode of the obtained posterior and the sequential least squares estimators. Extensive numerical simulations highlight superior performance of the proposed estimators than that of the estimators based on Akaike information criterion, Bayesian information criterion and asymptotic maximum a posteriori probability, for a large range of signal-to-noise ratio levels. We further illustrate the implementation of the proposed estimators through two real data examples.

On autoregressive and neural methods for massive-MIMO channel de-noising

In modern wireless communication systems, the Multiple-Input Multiple-Output (MIMO) technology allows to greatly increase power efficiency, the serving area, and the overall cell throughput through the use of the antenna array beamforming. Nevertheless, the MIMO systems require accurate channel state knowledge to apply correct precoding. In 5G Time Division Duplex (TDD) systems, the Channel State Information (CSI) is obtained via Sounding Reference Signals (SRS) transmitted by the User Equipment (UE). UEs have limited power capabilities and thus cannot achieve high Uplink (UL) Signal-to-Noise Ratio (SNR) on gNodeB (gNB) in large bandwidth. There are multiple techniques that can be applied to improve the accuracy of Channel Estimation (CE) in noisy conditions. In this paper, we describe a classical method, namely the Vector Autoregression (VAR) with adaptive model order estimation, as well as a modern Deep Neural Network (DNN) approach for the massive-MIMO channel estimation de-noising problem. The developed methods and signal pre and postprocessing steps are described, followed by their performance evaluation in a set of realistic simulations. The designed algorithms provide results outperforming the baseline spatio-temporal windowing approaches by approximately equal to 2dB effective Downlink (DL) Signal-to-Interference-plus-Noise Ratio (SINR) metric in single and multi-user MIMO scenarios. Extensive simulation results demonstrate the robustness of the developed methods to the dynamic channel conditions.

An inverse method for characterization of dynamic response of 2D structures under stochastic conditions

The reliable estimation of the wavenumber space (k-space) of the plates remains a long-term concern for acoustic modeling and structural dynamic behavior characterization. Most current analyses of wavenumber identification methods are based on the deterministic hypothesis. To this end, an inverse method is proposed for identifying wave propagation characteristics of two-dimensional structures under stochastic conditions, such as wavenumber space, dispersion curves, and band gaps. The proposed method is developed based on an algebraic identification scheme in the polar coordinate system framework, thus named Algebraic K-Space Identification (AKSI) technique. Additionally, a model order estimation strategy and a wavenumber filter are proposed to ensure that AKSI is successfully applied. The main benefit of AKSI is that it is a reliable and fast method under four stochastic conditions: (A) High level of signal noise; (B) Small perturbation caused by uncertainties in measurement points’ coordinates; (C) Non-periodic sampling; (D) Unknown structural periodicity. To validate the proposed method, we numerically benchmark AKSI and three other inverse methods to extract dispersion curves on three plates under stochastic conditions. One experiment is then performed on an isotropic steel plate. These investigations demonstrate that AKSI is a good in-situ k-space estimator under stochastic conditions.

Model order estimation method for user mobility classification in heterogeneous RAN systems

The performance of modern heterogeneous Radio Access Networks (RANs) can be improved by introducing new methods and algorithms for network resource management. In particular, user mobility information can be used to control RAN resources. To determine user mobility, information about the variation of the Signal to Interference + Noise Ratio (SINR) between base stations and users can be collected and analyzed. These data are multi-linear and can be represented as tensors to extract information from them. However, for the tensors decomposition, it is necessary to know the order of the multi-linear model or the rank of the tensors. Inspired by the LaRGE (LineAr Regression of Global Eigenvalues) method, we propose a novel method for the model order estimation of the multi-linear data based on the analysis of the Complimented Global Eigenvalues (CGE) profile. We have denoted our method as Enhanced LaRGE (EnLaRGE). We have compared the EnLaRGE method with the classical methods AIC, MDL and their multi-linear versions, as well as with the M-EFT and N-D EFT methods. EnLaRGE shows good performance for the tensors with small one of the tensors dimensions. EnLaRGE method is implemented for the automatic model order estimation of multi-linear data received from the Network Simulator 3 with RAN Intelligent Controller (RIC). The results show the high precision of the multi-linear model order estimation in comparison with state-of-the-art methods. The estimated model order can be used for user mobility classification and improving heterogeneous RANs performance. This research was funded by Russian Science Foundation, grant number 23-69-10084, 318, https://rscf.ru/project/23-69-10084/

Enhanced IpD2FT-Based Synchrophasor Estimation for M Class PMUs Through Adaptive Narrowband Interferers Detection and Compensation

One of the most challenging problems in the design of next-generation Phasor Measurement Units (PMUs) is related to the need to minimize the impact of multiple harmonics and out-of-band interharmonics (OOBI) on the measurement of synchrophasors over short observation intervals. Most of the existing solutions aim at removing such narrowband disturbances before synchrophasor estimation. However, when data records are short, accurate OOBI filtering may be infeasible. The algorithm described in this paper tackles this problem through a prior estimation of the number and the frequency of the significant narrowband interferers emerging from the noise floor. This result is achieved using a non-parametric signal detector based on random matrix theory and hypothesis testing, followed by the application of the estimation of signal parameters via the rotational invariance technique (ESPRIT). Once the number and the frequency of the narrowband components are known, these parameters are used to augment the dimension of the frequency-domain linear system of equations employed by the Interpolated Dynamic Discrete Fourier Transform (IpD2FT) for synchrophasor estimation. The results of multiple simulations confirm not only that the proposed approach for narrowband components detection is robust and more accurate than other model-order estimation algorithms described in the scientific literature, but also that the synchrophasor estimation accuracy over two-cycle observation intervals is generally higher than using other for M Class PMUs algorithms when OOBIs or a mixture of OOBIs and harmonics are considered.

Model order estimation based on the correntropy of observation eigenvalues

This paper presents a novel approach for estimating the model order in the presence of observation errors. The proposed method is based on the correntropy estimation of eigenvalues in the observation space. By utilizing the bootstrap method to resample the observations, more precise estimations can be obtained. The next step in the algorithm involves partitioning the observation space, generated by the covariance matrix of mixtures, into two subspaces: the signal subspace and the noise subspace. The order of the model, corresponding to the dimension of the signal subspace, is determined using a correntropy estimator based on kernel functions. Theoretical analysis demonstrates the consistency of the proposed algorithm. Comparative evaluations against existing methods in the literature highlight the superior performance of this information-theoretic approach.

Model order estimation for low-frequency oscillations in power systems by an advanced K-mean clustering approach

The estimation of low frequency modes present in the complex inter connected power system is very much crucial for taking remedial action to maintain power system stability and security. Modified total least square estimation of signal parameters via rotational invariance techniques (TLS-ESPRIT) is a popular mode estimation technique used for signal analysis, even in noisy conditions. But this technique requires a preliminary idea about the number of modes (Model order) present in the signal and its accuracy of estimation degrade when noise level increases. So the key contribution of this work is to develop a robust model order estimation technique, which will work efficiently in all range of signal and noise level. So, in this proposed work, an unsupervised sequential K-Mean++ clustering technique is implemented for the estimation of model order of low-frequency Phasor Measurement Unit (PMU) data by separating the significant eigenvalues of autocorrelation matrix (ACM) iteratively into signal and noise subspace. This two-layer K-Means++ algorithm promises a valid estimation of model order so that the insignificant eigenvalues are not clustered into the signal subspace. To validate the efficiency and robustness of the proposed method, a comparative study is carried out with the other techniques. Finally, the method is validated on two area data, real probing data obtained from Western Electricity Coordinating Council (WECC) network and oscillatory power data obtained for WSCC 9 and IEEE 39 bus system simulated on Real Time Digital Simulator (RTDS).

A constraint optimization problem for model order estimation

In this paper, we present a new method for estimating the number of terms in a sum of exponentially damped sinusoids embedded in noise. In particular, we propose to combine the shift-invariance property of the Hankel matrix associated with the signal with a constraint over its singular values to penalize small-order estimations. With this new methodology, the algebraic structure of the Hankel matrix and the statistical properties of the noise are considered. The new order estimation technique shows significant improvements over subspace-based methods. In particular, when a good separation between the noise and the signal subspaces is not possible, the new methodology outperforms known techniques. We evaluate the performance of our method using numerical experiments and compare its performance with previous results found in the literature.

Estimation of ARMA Model Order Using Artificial Neural Networks

Estimation of ARMA Model Order Using Artificial Neural Networks

Model Order Estimation in the Presence of Multipath Interference using Residual Convolutional Neural Networks

Model Order Estimation in the Presence of Multipath Interference using Residual Convolutional Neural Networks

Wave-shape function model order estimation by trigonometric regression

The adaptive non-harmonic (ANH) model is a powerful tool to compactly represent oscillating signals with time-varying amplitude and phase, and non-sinusoidal oscillating morphology. Given good estimators of instantaneous amplitude and phase we can construct an adaptive model, where the morphology of the oscillation is described by the wave-shape function (WSF), a 2π-periodic more general periodic function. In this paper, we address the problem of estimating the number of harmonic components of the WSF, a problem that remains underresearched, by adapting trigonometric regression model selection criteria into this context. We study the application of these criteria, originally developed in the context of stationary signals, to the case of signals with time-varying amplitudes and phases. We then incorporate the order estimation to the ANH model reconstruction procedure and analyze its performance for noisy AM-FM signals. Experimental results on synthetic signals indicate that these criteria enable the adaptive estimation of the waveform of non-stationary signals with non-sinusoidal oscillatory patterns, even in the presence of considerable amount of noise. We also apply our reconstruction procedure to the task of denoising simulated pulse wave signals and determine that the proposed technique performs competitively to other denoising schemes. We conclude this work by showing that our adaptive order estimation algorithm takes into account the interpatient waveform variability of the electrocardiogram (ECG) and respiratory signals by analyzing recordings from the Fantasia Database.

Scale-Invariant and consistent Bayesian information criterion for order selection in linear regression models

The Bayesian information criterion (BIC) is one of the most well-known criterion used for model order estimation in linear regression models. However, in its popular form, BIC is inconsistent as the noise variance tends to zero given that the sample size is small and fixed. Several modifications of the original BIC have been proposed that takes into account the high-SNR consistency, but it has been recently observed that the performance of the high-SNR forms of BIC highly depends on the scaling of the data. This scaling problem is a byproduct of the data dependent penalty design, which generates irregular penalties when the data is scaled and often leads to greater underfitting or overfitting losses in some scenarios when the noise variance is too small or large. In this paper, we present a new form of the BIC for order selection in linear regression models where the parameter vector dimension is small compared to the sample size. The proposed criterion eliminates the scaling problem and at the same time is consistent for both large sample sizes and high-SNR scenarios.

Robust Multi-Dimensional Model Order Estimation Using LineAr Regression of Global Eigenvalues (LaRGE)

The efficient estimation of an approximate model order is very important for real applications with multi-dimensional data if the observed low-rank data is corrupted by additive noise. In this paper, we present a novel robust method for model order estimation of noise-corrupted multi-dimensional low-rank data based on the LineAr Regression of Global Eigenvalues (LaRGE). The LaRGE method uses the multi-linear singular values obtained from the HOSVD of the measurement tensor to construct global eigenvalues. In contrast to the Modified Exponential Test (EFT) that also exploits the approximate exponential profile of the noise eigenvalues, LaRGE does not require the calculation of the probability of false alarm. Moreover, LaRGE achieves a significantly improved performance in comparison with popular state-of-the-art methods. It is well suited for the analysis of biomedical data. The excellent performance of the LaRGE method is illustrated via simulations and results obtained from EEG recordings.

Instantaneous Fundamental Frequency and Model Order Estimation of Non-Stationary Harmonic Signals

Instantaneous Fundamental Frequency and Model Order Estimation of Non-Stationary Harmonic Signals

Harmonic/interharmonic estimation using standard deviation assisted ESPRIT method

PurposeThis paper aims to accurately estimate harmonics/interharmonics in modern power system. There are several high spectral resolution techniques that have been in use for several years like Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT), Prony methods, etc. but these techniques require prior knowledge of number of modes present in the signal. Model Order (MO) estimation techniques have to make a trade-off between accuracy and their speed i.e., computational burden. Therefore, there is always a requirement of a technique that is fast as well as accurate.Design/methodology/approachThe proposed standard deviation (SD) method eliminates the requirement of energy validation test and analyses the distribution pattern, i.e. standard deviation of eigenvalues to identify the number of modes present in the signal. Signal is reconstructed using estimated modes and reconstruction error is obtained to show accuracy of the proposed estimation.FindingsSix test synthetic signals as well as one practical signal have been taken for validating the proposed method. The paper shows that proposed methodology has a better accuracy compared to modified exact model order (MEMO) method in high noise environment and takes very less computation time compared to the exact model order (EMO) method.Practical implicationsThe proposed method has been practically implemented for harmonic/interharmonic analysis at a sewage treatment plant at GIFT City, Gujarat, India. Apart from this the proposed method is modeled in python-based tool and is run into low-cost Raspberry Pi like hardware to create an onsite as well as remote monitoring device.Originality/valueSD-based approach for model order estimation is novel to this area. Further, the proposed method is compared with EMO and MEMO under varying noise conditions to check for accuracy and estimation time.

Simultaneous Model and Parameter Estimation for Joint Communication and Positioning

Joint communication and positioning based on a unified signal structure yields synergy and can complement and assist system designs enabling higher coverage and quality of service for both communication and positioning. For the time of arrival (TOA) based joint communication and positioning the TOA estimation accuracy is crucial. It is known to translate directly into position estimation accuracy. In the presence of multipath propagation, the estimation accuracy of signal arrival times in return strongly depends on the actual as well as the estimated number of the physical path parameters, the model order. In this work, we assess the performance and the mutual impact of simultaneous model order and parameter estimation for channel-estimation-based joint communication and positioning. Besides introducing a terrestrial channel-estimation-based unified joint communication and positioning system framework, we discuss and numerically compare different methods to sequentially or jointly estimate the parameters and the model order. We show that a TOA error-minimizing model order estimation is preferable over estimating the correct model order. Furthermore, we compare the performance with a proposed focused order-related lower bound. This bound determines the optimal model order for a chosen estimator. It depends on the actual and hypothetical model order and it replaces the here unsuitable Cramer-Rao lower bound. Secondly, the comparison shows that employing the parameter and model order-dependent inverse Fisher information matrix yields a close-to-optimal approach. We numerically show for a realistic channel scenario with many multipath parameters that the method is still accurate.

Second order blind identification algorithm with exact model order estimation for harmonic and interharmonic decomposition with reduced complexity

The power distribution network is susceptible to several Power Quality (PQ) disturbances. Among those, the harmonic and interharmonic distortions should be highlighted due to their high proliferation. This work proposes the utilization of signal processing techniques to decompose the electrical voltage and/or current signals into its harmonic and interhamonic component waveforms through a Blind Source Separation (BSS) algorithm named Second Order Blind Identification (SOBI). This algorithm is normally applied to a multivariate data set, what implies in a necessity of multiple measurements in different points of the system that will be analyzed. However, Single-Channel Blind Source Separation (SCBSS) method will be proposed in this work to estimate the components via SOBI using only one measured signal point. The method works as a set of adaptive filters whose coefficients are blindly obtained via SOBI and is responsible for the components separation. An Exact Model Order (EMO) algorithm will be used to improve the performance of the SOBI algorithm in order to estimate the correct number of components to be separated. Also, the EMO will be helpful to reduce the computational complexity of the SOBI. The performance of the proposed SCBSS method will be compared to that of the SCICA (Single-Channel Independent Component Analysis) based on the well-known FastICA algorithm, which employs Higher Order Statistics (HOS). It will be shown that the proposed SCBSS overtakes the SCICA for harmonic and interharmonic decomposition in performance and in reduced computational complexity. Also, the proposed SCBSS method will be compared to EMO-ESPRIT algorithm, where will be shown that the SCBSS achieved better results in noisy and time-varying scenarios. Finally, the proposed SCBSS will be applied for the analysis of a voltage signal acquired from the simulation of a power system containing wind generation.

A low complexity model order and frequency estimation of multiple 2-D complex sinusoids

Model order and frequency estimation of multiple 2-D complex sinusoids in additive white Gaussian noise are addressed. Frequency estimation follows the coarse-fine search strategy. Coarse estimates, obtained by locating maxima of the 2-D discrete Fourier transform, are refined in a two-stage procedure. In both stages, frequency refinement is based on three-point periodogram maximization. In order to provide accurate model order estimation (MOE) for a wide signal-to-noise ratio (SNR) range, our approach combines two metrics for sinusoid detection in the 2-D frequency domain, one for low and the other for high SNR values. The proposed frequency estimation method attains the Cramér-Rao lower bound and it outperforms parametric methods in terms of the estimation accuracy and numerical efficiency. Compared with information criterion-based methods, the proposed MOE approach is numerically more efficient, it does not require estimation of noise variance and hence does not suffer from overestimation at high SNR.

Estimation of ARMA Model Order via Artificial Neural Network for Modeling Physiological Systems

Model order estimation is the most important but challenging step for system identification using an autoregressive moving average (ARMA) model. In this paper, we propose an artificial neural network (ANN) structure to estimate the best model order for ARMA modeling of linear, time-invariant systems using the system’s input and output data. The proposed algorithm creates an equivalent ANN structure corresponding to an ARMA model and chooses the best model order using the neural network’s mean squared error (MSE) loss function. The proposed method is validated on simulated ARMA model data and the performance is compared with the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). We considered three hypothetical linear systems and performed 100 Monte Carlo simulations for each model, with different data lengths, and with additive noise. For each of the three simulation models, the proposed method significantly outperformed the AIC and BIC in terms of the correct model order selection. Finally, the proposed ANN-based model order estimator was successfully applied to determine the dynamic relationship between heart rate (HR) and instantaneous lung volume (ILV) using an ARMA model. The results indicate that physiological and biological systems can be modeled with appropriate ARMA models obtained by the proposed algorithm to better understand the system dynamics.