Use Of Covariance Matrices Research Articles

The use of covariance matrices in laboratory processing of geo-environmental data of field survey for development projects of land use planning

The paper introduces the possibility of using covariance matrices in the factor analysis. The factors affect the effectiveness of issues’ solution, concerning the involvement of unused agricultural lands in the active economic turnover in land use planning projects. The usage of the analysis tool, proposed by the authors, helps to improve the efficiency and objectivity of project decision substantiation in land use planning.

Residual least-squares collocation: use of covariance matrices from high-resolution global geopotential models

The paper presents a modified formulation of least-squares collocation. This residual least-squares collocation (RLSC) includes a remove--compute--restore procedure with a high-resolution global geopotential model (GGM) and a topographic gravitational potential model. In contrast to previous approaches, in RLSC, the remaining input residuals are modeled with error covariance matrices instead of signal covariance matrices. Therefore, we include the full variance--covariance information of a high-resolution GGM, namely the XGM2016, to the procedure. The included covariance matrices are anisotropic and location-dependent and enable a realistic error modeling of a target area. This fact represents an advantage over covariance matrices derived from signal degree variances or empirical covariance fitting. Additionally, due to the stochastic modeling of all involved components, RLSC provides realistic accuracy estimates. In a synthetic closed-loop test case with a realistic data distribution in the Andes we demonstrate the advantages of RLSC for regional geoid modeling and quantify the benefit which results mainly from a rigorously handled high-resolution GGM. In terms of root mean square deviations from the true reference solution, RLSC delivers an improvement of about 30% compared to a standard LSC approach, where the benefit is particularly pronounced in areas with a sparse data distribution. This improved performance, together with the fact that the resulting stochastic error estimates better reflect the true errors, might be an important aspect for the application of RLSC to derive gravity potential values and their uncertainties at reference stations of the international height reference system.

Variational assimilation with covariance matrices of observation data errors for the model of the Baltic Sea dynamics

Abstract The mathematical model of the Baltic Sea dynamics developed at the Institute of Numerical Mathematics of RAS is considered. The problem of variational assimilation of average daily data for the sea surface temperature (SST) is formulated and studied with the use of covariance matrices of observation data errors. Based on variational assimilation of satellite observation data, we propose an algorithm for solving the inverse problem of the heat flux reconstruction on the sea surface. The results of numerical experiments on reconstruction of the heat flux function are presented for the problem of variational assimilation of observation SST data.

The Prime Diabetes Model: Novel Methods for Estimating Long-Term Clinical and Cost Outcomes in Type 1 Diabetes Mellitus.

Recent publications describing long-term follow-up from landmark trials and diabetes registries represent an opportunity to revisit modeling options in type 1 diabetes mellitus (T1DM). To develop a new product-independent model capable of predicting long-term clinical and cost outcomes. After a systematic literature review to identify clinical trial and registry data, a model was developed (the PRIME Diabetes Model) to simulate T1DM progression and complication onset. The model runs as a patient-level simulation, making use of covariance matrices for cohort generation and risk factor progression, and simulating myocardial infarction, stroke, angina, heart failure, nephropathy, retinopathy, macular edema, neuropathy, amputation, hypoglycemia, ketoacidosis, mortality, and risk factor evolution. Several approaches novel to T1DM modeling were used, including patient characteristics and risk factor covariance, a glycated hemoglobin progression model derived from patient-level data, and model averaging approaches to evaluate complication risk. Validation analyses comparing modeled outcomes with published studies demonstrated that the PRIME Diabetes Model projects long-term patient outcomes consistent with those reported for a number of long-term studies. Macrovascular end points were reliably reproduced across five different populations and microvascular complication risk was accurately predicted on the basis of comparisons with landmark studies and published registry data. The PRIME Diabetes Model is product-independent, available online, and has been developed in line with good practice guidelines. Validation has indicated that outcomes from long-term studies can be reliably reproduced. The model offers new approaches to long-standing challenges in diabetes modeling and may become a valuable tool for informing health care policy.

Riemannian Approaches in Brain-Computer Interfaces: A Review.

Although promising from numerous applications, current brain-computer interfaces (BCIs) still suffer from a number of limitations. In particular, they are sensitive to noise, outliers and the non-stationarity of electroencephalographic (EEG) signals, they require long calibration times and are not reliable. Thus, new approaches and tools, notably at the EEG signal processing and classification level, are necessary to address these limitations. Riemannian approaches, spearheaded by the use of covariance matrices, are such a very promising tool slowly adopted by a growing number of researchers. This article, after a quick introduction to Riemannian geometry and a presentation of the BCI-relevant manifolds, reviews how these approaches have been used for EEG-based BCI, in particular for feature representation and learning, classifier design and calibration time reduction. Finally, relevant challenges and promising research directions for EEG signal classification in BCIs are identified, such as feature tracking on manifold or multi-task learning.

Using Data From the Microsoft Kinect 2 to Quantify Upper Limb Behavior: A Feasibility Study.

The objective of this study was to assess whether the novel application of a machine learning approach to data collected from the Microsoft Kinect 2 (MK2) could be used to classify differing levels of upper limb impairment. Twenty-four healthy subjects completed items of the Wolf Motor Function Test (WMFT), which is a clinically validated metric of upper limb function for stroke survivors. Subjects completed the WMFT three times: 1) as a healthy individual; 2) emulating mild impairment; and 3) emulating moderate impairment. A MK2 was positioned in front of participants, and collected kinematic data as they completed the WMFT. A classification framework, based on Riemannian geometry and the use of covariance matrices as feature representation of the MK2 data, was developed for these data, and its ability to successfully classify subjects as either "healthy," "mildly impaired," or "moderately impaired" was assessed. Mean accuracy for our classifier was 91.7%, with a specific accuracy breakdown of 100%, 83.3%, and 91.7% for the "healthy," "mildly impaired," and "moderately impaired" conditions, respectively. We conclude that data from the MK2 is of sufficient quality to perform objective motor behavior classification in individuals with upper limb impairment. The data collection and analysis framework that we have developed has the potential to disrupt the field of clinical assessment. Future studies will focus on validating this protocol on large populations of individuals with actual upper limb impairments in order to create a toolkit that is clinically validated and available to the clinical community.

Use of Covariance Matrices in a Consistent (Multiscale) Data Assimilation for Improvement of Basic Nuclear Parameters in Nuclear Reactor Applications: from Meters to Femtometers

A new approach is proposed, the consistent data assimilation, that allows to link the integral data experiment results to basic nuclear parameters employed by evaluators to generate ENDF/B point energy files in order to improve them. A practical example is provided where the sodium neutron propagation experiments, EURACOS and JANUS-8, are used to improve via modifications of {sup 23}Na nuclear parameters (like scattering radius, resonance parameters, Optical model parameters, Statistical Hauser-Feshbach model parameters, and Preequilibrium Exciton model parameters) the agreement of calculation versus experiments for a series of measured reaction rate detectors slopes. Future work involves comparison of results against a more traditional multigroup adjustments, and extension to other isotope of interest in the reactor community as {sup 56}Fe, actinides, and fission products.

The structure of dynamic correlations in multivariate stochastic volatility models

This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures can easily be obtained. Both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord through the use of covariance matrices. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure, and simulated data show that the estimation method works well. Various multivariate conditional volatility and MSV models are compared via simulation, including an evaluation of alternative VaR estimators. The DC MSV model is also estimated using three sets of empirical data, namely Nikkei 225 Index, Hang Seng Index and Straits Times Index returns, and significant dynamic correlations are found. The Dynamic Conditional Correlation (DCC) model is also estimated, and is found to be far less sensitive to the covariation in the shocks to the indexes. The correlation process for the DCC model also appears to have a unit root, and hence constant conditional correlations in the long run. In contrast, the estimates arising from the DC MSV model indicate that the dynamic correlation process is stationary.

Improved covariance matrices for point target detection in hyperspectral data

Our goals in hyperspectral point target detection have been to develop a methodology for algorithm comparison and to advance point target detection algorithms through the fundamental understanding of spatial and spectral statistics. In this paper, we review our methodology as well as present new metrics. We demonstrate improved performance by making better estimates of the covariance matrix. We have found that the use of covariance matrices of statistical stationary segments in the matched-filter algorithm improves the receiver operating characteristic curves; proper segment selection for each pixel should be based on its neighboring pixels. We develop a new type of local covariance matrix, which can be implemented in principal-component space and which also shows improved performance based on our metrics. Finally, methods of fusing the segmentation approach with the local covariance matrix dramatically improve performance at low false-alarm rates while maintaining performance at higher false-alarm rates.

Covariance correlations in collision avoidance probability calculations

Collision-avoidance calculations between orbiting objects make use of covariance matrices to characterize the uncertainty of the orbital position in space and time. Both the potential collider and target each have such a covariance matrix that integrates uncertainties in the original state vector propagated forward in time as well as uncertainties in the atmospheric model for instance due to unpredictable variations in solar activity. For objects in low-Earth orbits, the longer the propagation, the more important the atmospheric uncertainty becomes. Currently, the collision probability is computed by assuming that the covariances characterizing the position uncertainties of the two objects are independent. Because of the coherent nature of atmospheric drag, this assumption is not necessarily true, especially for the down-range uncertainty. If the solar activity goes up, both objects will have increased drag. Likewise, if the solar activity goes down, both objects will have decreased drag. In this parametric study, we use two-line element set data to demonstrate this behavior and to make some simple estimates on its magnitude. We then estimate how much this may affect the accuracy of the calculated collision probability. We conclude with recommendations for future study.

Comment concerning ‘The rotation group in plate tectonics and the representation of uncertainties of plate reconstructions’ by T. Chang, J. Stock and P. Molnar

We wish to make a few comments on Chang et al.’s paper. The use of covariance matrices or their geometric counterpart error ellipses to describe rotation errors is not a new idea. For present-day plate motion models Chase (1978) and Minster & Jordan (1978) used this description with angular variables, but these are easily converted to lengths. We believe that we (Jurdy & Stefanick 1987, referred to as J&S) were the first to treat the full three-dimensional problem of representing and combining rotation errors as tensors including optimal estimates. Chang et al.’s paper does not discuss any of these contributions. Moreover, the authors factor rotations into a mean rotation followed by a set of perturbing rotations and the latter are treated statistically. This separation is artificial and obscures the geometric content of a description of errors and does not allow a comparison of alternative estimates or the formation of an optimal estimate if they are independent. We wish to go into detail on these points. We defined our fit of rotations so that there would be an exact fit for one rotation (Stock & Molnar 1983) and then found those rotations which would correspond to an rms misfit of 10 km at the endpoints of the common boundary or at a sequence of points along the common boundary. (The results would be slightly different in the two cases.) This was done so that we could compare our method for combining errors with Stock & Molnar’s method of partial uncertainty rotations which, for combining errors, was combinatorial and required 74 rotation combinations. The agreement was very good. That was our purpose. We made no attempt to make real picks of magnetic anomalies and to estimate the individual errors and make a x 2 fit. If this were done there would be no exact fit and we would have found the rotation which gave the best fit as well as the set of rotations which gave acceptable values for x2. A rotation or a set of rotations is most intuitively represented by an axial vector or set of axial vectors ending in some error ellipsoid. If we have several estimates of a rotation due to alternative chains or different authors then we can imagine several ellipsoids (hopefully intersecting) and if the estimates are independent we can combine the results and obtain an optimal estimate and corresponding error ellipsoid which will roughly coincide with the intersection of the original ellipsoids. Geometrically this is straightforward. Representing the reconstruction of Stock & Molnar (1983) and Molnar & Stock (1985) in our description (J&S), we illustrated the combination of errors for two different chains of reconstructions for the Pacific plate relative to the North American for chron 6, 20 Ma:

Statistical measures of intrasample variability

We have implemented a multivariate statistical methodology to assess the degree and pattern of variability in skeletal samples. Specifically, it is designed to test whether variability in a skeletal sample exceeds that of a local breeding population, defined in the present instance as variability present among crania from an endogamous village. It involves the use of covariance matrices derived from hypothesis and reference samples. The methodology has been applied to subsamples of a Plains Indian craniometric database consisting of 80 variables taken on some 1000 individuals. The subsamples are assumed to reflect varying patterns and degrees of intrasample variability. The assumed group structures are assembled posthoc and subject to various analyses. The excellent results attest to the great potential of the methodology.