Covariance Estimation Research Articles

Separable expansions for covariance estimation via the partial inner product

Summary The nonparametric estimation of covariance lies at the heart of functional data analysis, whether for curve or surface-valued data. The case of a two-dimensional domain poses both statistical and computational challenges, which are typically alleviated by assuming separability. However, separability is often questionable, sometimes even demonstrably inadequate. We propose a framework for the analysis of covariance operators of random surfaces that generalizes separability while retaining its major advantages. Our approach is based on the expansion of the covariance into a series of separable terms. The expansion is valid for any covariance over a two-dimensional domain. Leveraging the key notion of the partial inner product, we generalize the power iteration method to general Hilbert spaces, and show how the aforementioned expansion can be efficiently constructed in practice at the level of the surface observations. Truncation of the expansion and retention of the leading terms automatically induces a nonparametric estimator of the covariance, whose parsimony is dictated by the truncation level. The resulting estimator can be calculated, stored and manipulated with little computational overhead relative to separability. Consistency and rates of convergence are derived under mild regularity assumptions, illustrating the trade-off between bias and variance regulated by the truncation level. The merits and practical performance of the proposed methodology are demonstrated in a comprehensive simulation study.

A sequential MUSIC algorithm for scatterers detection in SAR tomography enhanced by a robust covariance estimator

Synthetic aperture radar (SAR) tomography (TomoSAR) is an appealing tool for the extraction of height information of urban infrastructures. Due to the widespread applications of the multiple signal classification (MUSIC) algorithm in source localization, it is a suitable solution in TomoSAR when multiple snapshots (looks) are available. While the classical MUSIC algorithm aims to estimate the whole reflectivity profile of scatterers, sequential MUSIC algorithms are suited for the detection of sparse point-like scatterers. In this class of methods, successive cancellation is performed through orthogonal complement projections on the MUSIC power spectrum. In this work, a new sequential MUSIC algorithm named recursive covariance cancelled MUSIC (RCC-MUSIC), is proposed. This method brings higher accuracy in comparison with the previous sequential methods at the cost of a negligible increase in computational cost. Furthermore, to improve the performance of RCC-MUSIC, it is combined with the recent method of covariance matrix estimation called correlation subspace. Utilizing the correlation subspace method results in a denoised covariance matrix which in turn, increases the accuracy of subspace-based methods. Several numerical examples are presented to compare the performance of the proposed method with the relevant state-of-the-art methods. As a subspace method, simulation results demonstrate the efficiency of the proposed method in terms of estimation accuracy and computational load.

Relationship between Obuchowski–Rockette–Hillis and Gallas methods for analyzing multi-reader diagnostic imaging data with empirical AUC as the reader performance measure

For analyzing multireader multicase (MRMC) diagnostic imaging data when the reader performance measure of interest is the area under the receiver-operating-characteristic curve (AUC), two popular methods of analysis that allow conclusions to generalize to both the reader and case populations are the method developed by Obuchowski, Rockette and Hillis (ORH) and the method primarily developed by Gallas (Gallas). While the ORH method is a general method that is applicable to most reader performance metrics, the Gallas method is limited to those metrics for which an unbiased variance estimate exists. Previously it was not known if the ORH method could be adapted so as to produce the same variance estimate as the Gallas method. In this paper, I show that a recently proposed version of the OR method produces the same unconstrained variance statistic as the Gallas method. However, the two methods differ in their approaches to constraining the variance estimate to be nonnegative and in their degrees-of-freedom estimates. These two differences are discussed and recommendations given. In addition, several contributions to the development of the ORH method are made, including determining sufficient conditions for unbiased variance estimates and providing justification for the ORH variance constraints and covariance estimation method.

Robust inference under time‐varying volatility: A real‐time evaluation of professional forecasters

SummaryIn many forecast evaluation applications, standard tests as well as tests allowing for time‐variation in relative forecast ability build on heteroskedasticity‐and‐autocorrelation consistent (HAC) covariance estimators. Yet, the finite‐sample performance of these asymptotics is often poor. “Fixed‐ ” asymptotics, used to account for long‐run variance estimation, improve finite‐sample performance under homoskedasticity, but lose asymptotic pivotality under time‐varying volatility. Moreover, loss of pivotality due to time‐varying volatility is found in the standard HAC framework in certain cases as well. We prove a wild bootstrap implementation to restore asymptotically pivotal inference for the above and new CUSUM‐ and Cramér‐von Mises‐based tests in a fairly general setup, allowing for estimation uncertainty from either a rolling window or a recursive approach when fixed‐ asymptotics are adopted to achieve good finite‐sample performance. We then investigate the (time‐varying) performance of professional forecasters relative to naive no‐change and model‐based predictions in real‐time. We exploit the Survey of Professional Forecasters (SPF) database and analyze nowcasts and forecasts at different horizons for output and inflation. We find that not accounting for time‐varying volatility seriously affects outcomes of tests for equal forecast ability: wild bootstrap inference typically yields convincing evidence for advantages of the SPF, while tests using non‐robust critical values provide remarkably less. Moreover, we find significant evidence for time‐variation of relative forecast ability, the advantages of the SPF weakening considerably after the “Great Moderation.”

Estimation of Resolution and Covariance of Ambient Seismic Source Distributions: Full Waveform Inversion and Matched Field Processing

AbstractBoth natural and anthropogenic seismic sources generate so‐called ambient seismic waves. One in turn can use ambient seismic waves to estimate these source distributions and study source characteristics, for instance the source mechanism. A commonly used estimation method is called matched field processing (MFP), and the MFP results are inherently smeared by the array geometry. Another approach to estimate ambient seismic sources is to apply full waveform inversion (FWI) to the crosscorrelations of ambient seismic wave recordings. Both methods have pros and cons, but the model resolution and uncertainty in these two methods are important for the interpretation. Unfortunately, this topic has attracted little attention in the past. We propose to estimate both the model resolution matrix and model covariance matrix of the inversion using singular value decomposition. We demonstrate our estimates using two examples, one of which is an actual field array geometry. We quantitatively compare the model resolution of the two methods and discuss the model null space. We demonstrate that FWI has superior resolution with enough independent data and should be used when computational resources permit.

Brain metabolic connectivity reconfiguration in the semantic variant of primary progressive aphasia

Functional network-level alterations in the semantic variant of Primary Progressive Aphasia (sv-PPA) are relevant to understanding the clinical features and the neural spreading of the pathology. We assessed the effect of neurodegeneration on brain systems reorganization in early sv-PPA, using advanced brain metabolic connectivity approaches. Forty-four subjects with sv-PPA and forty-four age-matched healthy controls (HC) were included. We applied two multivariate approaches to [18F]FDG-PET data – i.e., sparse inverse covariance estimation and seed-based interregional correlation analysis – to assess the integrity of (i) the whole-brain metabolic connectivity and (ii) the connectivity of brain regions relevant for cognitive and behavioral functions. Whole-brain analysis revealed a global-scale connectivity reconfiguration in sv-PPA, with widespread changes in metabolic connections of frontal, temporal, and parietal regions. In comparison to HC, the seed-based analysis revealed a) functional isolation of the left anterior temporal lobe (ATL), b) decreases in temporo-occipital connections and contralateral homologous regions, c) connectivity increases to the dorsal parietal cortex from the spared posterior temporal cortex, d) a disruption of the large-scale limbic brain networks. In sv-PPA, the severe functional derangement of the left ATL may lead to an extensive connectivity reconfiguration, encompassing several brain regions, including those not yet affected by neurodegeneration. These findings support the hypothesis that in sv-PPA the focal vulnerability of the core region (i.e., ATL) can potentially drive the widespread cerebral connectivity changes, already present in the early phase.

Covariance Estimation for Matrix-valued Data

Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and an S&P 500 stock data analysis. Supplementary materials for this article are available online.

Simple Direction-of-Arrival Estimation under Nonuniform Noise Scenarios

The uniform white noise assumption is widely applied for most of the existing direction-of-arrival (DOA) estimation methods, which however may degrade the estimation performance. In practical scenarios, the nonuniform white noise model is more adequate. In this paper, we propose a simple DOA estimation method in the presence of spatially nonuniform white noise with unknown noise covariance matrix. The estimate of the unknown noise covariance matrix is firstly obtained by utilizing the shift property of identical subarrays. Then, the received signal covariance matrix is prewhitened by the estimated noise covariance, which can overcome the biased estimate of signal covariance. Finally, the performance improvement is verified by simulations in terms of resolution ability and estimation accuracy.

Robust covariance estimation with noisy high-frequency financial data

We propose consistent and efficient robust different time-scales estimators to mitigate the heavy-tail effect of high-frequency financial data. Our estimators are based on minimising the Huber loss function with a suitable threshold. We show these estimators are guaranteed to be robust to measurement noise of certain types and jumps. With only finite fourth moments of the observation log-price data, we develop the sub-Gaussian concentration of our estimators around the volatility. We conduct the simulation studies to show the finite sample performance of the proposed estimation methods. The simulation studies imply that our methods are also robust to financial data in the presence of jumps. Empirical studies demonstrate the practical relevance and advantages of our estimators.

Impact of fish feed formulation training on feed use and farmers' income: Evidence from Ghana

Feed accounts for 60–80% of tilapia production costs, and high feed cost and limited feed access are major issues faced by fish farmers. A potential solution is for farmers to produce their own feeds using cheaper and locally available ingredients. This paper evaluates the feed formulation training implemented in Ghana as part of the Fisheries Commission's activities under the Ghana Aquaculture for Food and Jobs program. This paper analyzes baseline and follow-up survey data using difference-in-difference estimation and analysis of covariance. One year after the training, results show positive impact on the farmers' knowledge of feed formulation, quantity of feed formulated, feeding and management practices, productivity, and income. Feed formulation training encouraged more farmers to formulate their own feeds and maintained or increased total feeds used in their facilities from 2018/2019 to 2020/21. Among the farmers who did not receive training, feed formulation activities and feed quantity used reduced from 2018/19 to 2020/21. A common practice is the use of locally available quality commercial feed as starter feed for smaller fish and own-formulated feeds as a complement for or alternative to commercial growout feeds for bigger fish. Especially when constrained by lack of funds, many farmers reduce the use of commercial feeds and formulate their feeds using cheaper and locally available raw materials to maintain similar levels of feeding intensity. Feed formulation acts as an important coping strategy in the context of small-scale farmers with limited financial resources. This study recommends scaling out feed formulation training and expanding research on optimal fish nutrition using locally available and low-cost raw materials, in combination with commercial feeds, to maximize productivity and incomes for farmers.

Robust functional principal component analysis via a functional pairwise spatial sign operator.

Functional principal component analysis (FPCA) has been widely used to capture major modes of variation and reduce dimensions in functional data analysis. However, standard FPCA based on the sample covariance estimator does not work well if the data exhibits heavy-tailedness or outliers. To address this challenge, a new robust FPCA approach based on a functional pairwise spatial sign (PASS) operator, termed PASS FPCA, is introduced. We propose robust estimation procedures for eigenfunctions and eigenvalues. Theoretical properties of the PASS operator are established, showing that it adopts the same eigenfunctions as the standard covariance operator and also allows recovering ratios between eigenvalues. We also extend the proposed procedure to handle functional data measured with noise. Compared to existing robust FPCA approaches, the proposed PASS FPCA requires weaker distributional assumptions to conserve the eigenspace of the covariance function. Specifically, existing work are often built upon a class of functional elliptical distributions, which requires inherently symmetry. In contrast, we introduce a class of distributions called the weakly functional coordinate symmetry (weakly FCS), which allows for severe asymmetry and is much more flexible than the functional elliptical distribution family. The robustness of the PASS FPCA is demonstrated via extensive simulation studies, especially its advantages in scenarios with nonelliptical distributions. The proposed method was motivated by and applied to analysis of accelerometry data from the Objective Physical Activity and Cardiovascular Health Study, a large-scale epidemiological study to investigate the relationship between objectively measured physical activity and cardiovascular health among olderwomen.

Adaptive confidence regions for indirect tracking of moving tumors in radiotherapy

Target motion in the course of radiotherapy is one of the largest factors affecting the treatment quality of highly dynamic sites such as lung. A critical component of real-time motion management is not only the prediction of tumor location at a future point in time but assessment of positional uncertainty for the purposes of margin adjustment and optimization of validationschemes. In this study, we propose to investigate the ability of a confidence estimator to accurately reflect the reliability of individual target position predictions and prospectively detect large prediction errors by relying exclusively on a surrogatesignal. This work uses a Bayesian framework for indirect tracking. While constant covariance estimates are commonly used to express the uncertainty of the models involved, in this study new adaptive estimates are derived from the surrogate behavior to reflect increasing uncertainty when the breathing conditions differ from the reference conditions observed during the training step. The accuracy of the resulting 95% predicted confidence regions (CRs) is evaluated on nine breathing sequences involving changes of respiratory types (free, thoracic, abdominal, deep). The breathing motions are collected simultaneously from a lung target and two different surrogate signals (an external marker and an anatomical location within the liver). Receiver operating characteristic (ROC) analysis is performed to evaluate the ability of the predicted uncertainty to prospectively detect large predictionerrors. Higher CR accuracy is obtained when using the proposed adaptive estimates over using constant estimations: on average over the cohort, the proportion of actual target positions lying within the 95% CR is increased by 40 and 35 p.p. with the internal and external surrogates. The time-dependent inflation of the CR width tends to match the magnitude variation of the prediction errors: the adaptive CR effectively enlarges when the target position cannot be predicted reliably, which corresponds to potentially high prediction errors. More precisely, the ROC analysis indicates that the proposed uncertainty estimate can detect if prediction errors are greater than 5 mm with on average high sensitivity (90%) and modest specificity (54% and 47% from internal and external surrogates, respectively). While relying exclusively on the surrogate motion characteristics being continuously monitored, the Bayesian framework coupled to adaptive uncertainty estimations can provide reliable CR able to detect large prediction errors. The findings of this study could be further used to automatically trigger risk management mechanisms prospectively.

Galaxy blending effects in deep imaging cosmic shear probes of cosmology

ABSTRACT Upcoming deep imaging surveys such as the Vera C. Rubin Observatory Legacy Survey of Space and Time will be confronted with challenges that come with increased depth. One of the leading systematic errors in deep surveys is the blending of objects due to higher surface density in the more crowded images; a considerable fraction of the galaxies which we hope to use for cosmology analyses will overlap each other on the observed sky. In order to investigate these challenges, we emulate blending in a mock catalogue consisting of galaxies at a depth equivalent to 1.3 yr of the full 10-yr Rubin Observatory that includes effects due to weak lensing, ground-based seeing, and the uncertainties due to extraction of catalogues from imaging data. The emulated catalogue indicates that approximately 12 per cent of the observed galaxies are ‘unrecognized’ blends that contain two or more objects but are detected as one. Using the positions and shears of half a billion distant galaxies, we compute shear–shear correlation functions after selecting tomographic samples in terms of both spectroscopic and photometric redshift bins. We examine the sensitivity of the cosmological parameter estimation to unrecognized blending employing both jackknife and analytical Gaussian covariance estimators. An ∼0.025 decrease in the derived structure growth parameter S8 = σ8(Ωm/0.3)0.5 is seen due to unrecognized blending in both tomographies with a slight additional bias for the photo-z-based tomography. This bias is greater than the 2σ statistical error in measuring S8.

Investigating Confidence Intervals of Item Parameters When Some Item Parameters Take Priors in the 2PL and 3PL Models.

To reduce the chance of Heywood cases or nonconvergence in estimating the 2PL or the 3PL model in the marginal maximum likelihood with the expectation-maximization (MML-EM) estimation method, priors for the item slope parameter in the 2PL model or for the pseudo-guessing parameter in the 3PL model can be used and the marginal maximum a posteriori (MMAP) and posterior standard error (PSE) are estimated. Confidence intervals (CIs) for these parameters and other parameters which did not take any priors were investigated with popular prior distributions, different error covariance estimation methods, test lengths, and sample sizes. A seemingly paradoxical result was that, when priors were taken, the conditions of the error covariance estimation methods known to be better in the literature (Louis or Oakes method in this study) did not yield the best results for the CI performance, while the conditions of the cross-product method for the error covariance estimation which has the tendency of upward bias in estimating the standard errors exhibited better CI performance. Other important findings for the CI performance are also discussed.

Random Surface Covariance Estimation by Shifted Partial Tracing

The problem of covariance estimation for replicated surface-valued processes is examined from the functional data analysis perspective. Considerations of statistical and computational efficiency often compel the use of separability of the covariance, even though the assumption may fail in practice. We consider a setting where the covariance structure may fail to be separable locally—either due to noise contamination or due to the presence of a nonseparable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a nonseparable but banded component. We introduce nonparametric estimators hinging on the novel concept of shifted partial tracing, enabling computationally efficient estimation of the model under dense observation. Due to the denoising properties of shifted partial tracing, our methods are shown to yield consistent estimators even under noisy discrete observation, without the need for smoothing. Further to deriving the convergence rates and limit theorems, we also show that the implementation of our estimators, including prediction, comes at no computational overhead relative to a separable model. Finally, we demonstrate empirical performance and computational feasibility of our methods in an extensive simulation study and on a real dataset. Supplementary materials for this article are available online.

Sparse Inverse Covariance Estimation for Causal Inference in Process Data Analytics

Causal analysis plays a vital role in determining the underlying relationship among the variables in a system from the data. In this article, the sparse inverse covariance (SIC) estimation is coupled with likelihood score, and a two-step approach is proposed to address the problem of causal analysis. The estimation of SIC matrix for undirected sparse network reconstruction is performed with the <inline-formula> <tex-math notation="LaTeX">$L_{0}$ </tex-math></inline-formula>-norm constraint in the framework of greedy sparse simplex (GSS) algorithm. Furthermore, the GSS algorithm is suitably modified to incorporate the additional constraint of positive semidefiniteness of the inverse covariance matrix. To determine the causal direction among the variables, the likelihood score is computed for the associated variables in the reconstructed network in the second step. The efficacy of the proposed approach for causal analysis is illustrated using numerical examples and an industrial application on prediction of flooding and weeping in a deethanizer column associated with a fluid catalytic cracking unit. From these studies, it is observed that the proposed approach is able to recover causal connections accurately in both cases. Furthermore, the probable reasons for the occurrence of flooding and weeping phenomena in an industrial deethanizer unit are also inferred from the identified causal network.

Optimal experiment design for a bottom friction parameter estimation problem

Calibration with respect to a bottom friction parameter is standard practice within numerical coastal ocean modelling. However, when this parameter is assumed to vary spatially, any calibration approach must address the issue of overfitting. In this work, we derive calibration problems in which the control parameters can be directly constrained by available observations, without overfitting. This is achieved by carefully selecting the ‘experiment design’, which in general encompasses both the observation strategy, and the choice of control parameters (i.e. the spatial variation of the friction field). In this work we focus on the latter, utilising existing observations available within our case study regions. We adapt a technique from the optimal experiment design (OED) literature, utilising model sensitivities computed via an adjoint-capable numerical shallow water model, Thetis. The OED method uses the model sensitivity to estimate the covariance of the estimated parameters corresponding to a given experiment design, without solving the corresponding parameter estimation problem. This facilitates the exploration of a large number of such experiment designs, to find the design producing the tightest parameter constraints. We take the Bristol Channel as a primary case study, using tide gauge data to estimate friction parameters corresponding to a piecewise-constant field. We first demonstrate that the OED framework produces reliable estimates of the parameter covariance, by comparison with results from a Bayesian inference algorithm. We subsequently demonstrate that solving an ‘optimal’ calibration problem leads to good model performance against both calibration and validation data, thus avoiding overfitting.

Reconfigured metabolism brain network in asymptomatic microtubule-associated protein tau mutation carriers: a graph theoretical analysis

BackgroundStudies exploring topological properties of the metabolic network during the presymptomatic stage of genetic frontotemporal dementia (FTD) are scarce. However, such knowledge is important for understanding brain function and disease pathogenesis. Therefore, we aimed to explore FTD-specific patterns of metabolism topology reconfiguration in microtubule-associated protein tau (MAPT) mutation carriers before the onset of symptoms.MethodsSix asymptomatic carriers of the MAPT P301L mutation were compared with 12 non-carriers who all belonged to the same family of FTD. For comparison, we included 32 behavioral variant FTD (bvFTD) patients and 33 unrelated healthy controls. Each participant underwent neuropsychological assessments, genetic testing, and a hybrid positron emission tomography (PET)/magnetic resonance imaging (MRI) scan. Voxel-wise gray matter volumes and standardized uptake value ratios were calculated and compared for structural MRI and fluorodeoxyglucose (FDG)-PET, separately. The sparse inverse covariance estimation method (SICE) was applied to topological properties and metabolic connectomes of brain functional networks derived from 18F-FDG PET/MRI data. Independent component analysis was used to explore the metabolic connectivity of the salience (SN) and default mode networks (DMN).ResultsThe asymptomatic MAPT carriers performed normal global parameters of the metabolism network, whereas bvFTD patients did not. However, we revealed lost hubs in the ventromedial prefrontal, orbitofrontal, and anterior cingulate cortices and reconfigured hubs in the anterior insula, precuneus, and posterior cingulate cortex in asymptomatic carriers compared with non-carriers, which overlapped with the comparisons between bvFTD patients and controls. Similarly, significant differences in local parameters of these nodes were present between asymptomatic carriers and non-carriers. The reduction in the connectivity of lost hub regions and the enhancement of connectivity between reconfigured hubs and components of the frontal cortex were marked during the asymptomatic stage. Metabolic connectivity within the SN and DMN was enhanced in asymptomatic carriers compared with non-mutation carriers but reduced in bvFTD patients relative to controls.ConclusionsOur findings showed that metabolism topology reconfiguration, characterized by the earliest involvement of medial prefrontal areas and active compensation in task-related regions, was present in the presymptomatic phase of genetic FTD with MAPT mutation, which may be used as an imaging biomarker of increased risk of FTD.

MATLAB toolbox for ROC analysis of multi-reader multi-case diagnostic imaging studies.

A common study design for comparing the performances of diagnostic imaging tests is to obtain ratings from multiple readers of multiple cases whose true statuses are known. Typically, there is overlap between the tests, readers, and/or cases for which special analytical methods are needed to perform statistical comparisons. We present our new MATLAB MRMCaov toolbox, which is designed for multi-reader multi-case comparisons of two or more diagnostic tests. The toolbox allows for statistical comparison of reader performance metrics, such as area under the receiver operating characteristic curve (ROC AUC), with analysis of variance methods originally proposed by Obuchowski and Rockette (1995) and later unified and improved by Hillis and colleagues (2005, 2007, 2008, 2018). MRMCaov is open-source software with an integrated command-line interface for performing multi-reader multi-case statistical analysis, plotting, and presenting results. Its features (1) ROC AUC, likelihood ratios of positive or negative ratings, sensitivity, specificity, and expected utility reader performance metrics; (2) reader-specific ROC curves; (3) user-definable performance metrics; (4) test-specific estimates of mean performance along with confidence intervals and p-values for statistical comparisons; (5) support for factorial, nested, or partially paired study designs; (6) inference for random or fixed readers and cases; (7) DeLong, jackknife, or unbiased covariance estimation; and (8) compatibility with Microsoft Windows, Mac OS, and Linux.

The noise covariances of linear Gaussian systems with unknown inputs are not uniquely identifiable using autocovariance least-squares

Existing works in optimal filtering for linear Gaussian systems with arbitrary unknown inputs assume perfect knowledge of the noise covariances in the filter design. This is impractical and raises the question of whether and under what conditions one can identify the noise covariances of linear Gaussian systems with arbitrary unknown inputs. This paper considers the above identifiability question using the correlation-based autocovariance least-squares (ALS) approach. In particular, for the ALS framework, we prove that (i) the process noise covariance Q and the measurement noise covariance R cannot be uniquely jointly identified; (ii) neither Q nor R is uniquely identifiable, when the other is known. This not only helps us to have a better understanding of the applicability of existing filtering frameworks under unknown inputs (since almost all of them require perfect knowledge of the noise covariances) but also calls for further investigation of alternative and more viable noise covariance methods under unknown inputs. Especially, it remains to be explored whether the noise covariances are uniquely identifiable using other correlation-based methods. We are also interested to use regularization for noise covariance estimation under unknown inputs, and investigate the relevant property guarantees for the covariance estimates. The above topics are the main subjects of our current and future work.