A novel construction algorithm has been developed for the generalized Gaussian kernel model. Each kernel regressor in the pool of candidate regressors has an individual **diagonal** **covariance** **matrix**, which is determined by maximizing the absolute value of the correlation between the regressor and the training data using a repeated weighted search optimization. The standard orthogonal least squares

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A novel technique is proposed to construct sparse regression models based on the orthogonal least squares method with tunable kernels. The proposed technique tunes the centre vector and **diagonal** **covariance** **matrix** of individual regressor by incrementally minimising the training mean square error using a guided random search algorithm, and it offers a state-of-the- art method for constructing very sparse models that generalise well.

A novel technique is presented to construct sparse regression models based on the orthogonal least square method with boosting. This technique tunes the mean vector and **diagonal** **covariance** **matrix** of individual re- gressor by incrementally minimizing the training mean square error. An efficient weighted optimization method is developed based on boosting to append regressors one by one in an orthogonal forward selection procedure. Experimental results obtained using this construction technique demonstrate that it offers a viable alternative to the existing state-of-art kernel modeling methods for constructing parsimonious regression models.

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In the present paper we now build a theory as envisioned in Martellosio (2010) at an even more general level. In particular, we allow for general invariant tests including randomized ones, we employ weaker conditions on the underlying **covariance** model as well as on the distributions of the disturbances (e.g., we even allow for distributions that are not absolutely continuous). One aspect of our theory is to show how invariance of the tests considered can be used to convert Martellosio’s intuition about the "concentration" e¤ect into a precise mathematical argument. Furthermore, advantages of this approach over the approach in Mynbaev (2012) are that (i) standard weak con- vergence arguments can be used (avoiding the need for new tools to handle the "stretch-out" e¤ect), (ii) more general classes of tests can be treated, and (iii) much weaker distributional assumptions are required. The general theory built in this paper is then applied to tests for spatial autocorrela- tion, which, in particular, leads to correct versions of the results in Martellosio (2010) that pertain to spatial models. 1 A further contribution of the present paper is a characterization of the situation

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As the issue of robustness of face recognition based on depth image sets, we propose that multiple Kinect images is being as a set of images, and depth data captured is used to automatically estimate poses and crop face area. Firstly, divide image sets into c subsets, and divide the images in all the subsets into image blocks of 4×4. Then, simulate images in sets as a form of image blocks, dividing in accordance with posture. Each set is represented using **covariance** **matrix**. Finally, the simulation of images in subsets is on Riemannian manifold. In order to classify, separately learnt SVM models for each image subset on the Lie group of Riemannian manifold and introduce a fusion strategy to combine results from all image subsets. We have verified the effectiveness of the proposed method on the three largest public Kinect face data sets Curtin Faces, Biwi Kinect and UWA Kinect. Compared to other advanced methods, the recognition rate has improved greater, the standard deviation is kept low, with robust to the number of image sets, image sub-setting number and spatial resolution.

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precursors, time steps, Brownian motion and initial conditions. The results obtained by the numerical experiments are compared on mean with the deterministic model (DM) of the point kinetics which is calculated by the implicit Euler scheme, since the deterministic formulation does not have standard deviation values in the tables represented by do Not Apply (NA). They will also be compared with respect to the mean and standard deviation with other stochastic schemes reported in the literature, such as: SPCA (Stochastic piecewise Constant Approximation) and MC (Monte Carlo) [1], EM (Euler- Maruyama) and T 1.5 (Taylor 1.5) [2], FSNPK (Fractional stochastic point kinetics equations) [3], SSPK (Simplified Stochastic Point Kinetics Equations) [4], AEM (Analytical Exponential Model) [5], Double DDM (Double Diagonalization-Decomposition Method) [6], ESM (Efficient Stochastic Model) [7], IEM (Implicit Euler- Maruyama) [8]. The values reported in the literature have been written with four significant figures, in cases where fewer significant figures are reported it has been completed with zeros. In this work, the results in the tables are presented by the acronym IM and IMwDB denoting Implicit Milstein and Implicit Milstein with **Diagonal** Brownian, respectively.

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Abstract--Digital multimedia content protection has increasingly become an important issue. As image watermarking is identified as a major technology used in Copyright and Content Protection, a key necessity for image watermarking is the improvement of its impalpability and robustness. To meet this requirement, in this paper, an algorithm for watermarking of digital images based on singular value decomposition (SVD) is considered. At the transmitter, the host image is split into blocks and by performing SVD transformation the D components are detected. All the D components of the host image are now modified in accordance with the embedding criteria and the type of watermark bit. During extraction, the **diagonal** **matrix** of each block of the watermarked image has been observed and the reconstruction of watermark is performed.

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Abstract—In this paper, an array aperture extension algorithm is developed for two-dimensional (2-D) direction-of-arrival (DOA) estimation with L-shaped array. We enlarge the dimension of the **covariance** **matrix** by using the rotational invariance in conjunction with the property that the signal **covariance** **matrix** is real **diagonal** **matrix**. Estimation of DOAs is performed by processing this larger dimensional **matrix**. The simulation results indicate that our method can improve the DOA estimation accuracy.

R(z) = U(z)Λ(z)U P (z) , (2) where U(z) is a paraunitary **matrix** of eigenvectors and Λ(z) is a **diagonal** parahermitian **matrix** of eigenvalues. In most standard cases, these factors can be selected to be analytic [2]. This is closely related to the McWhirter decomposition [3], where the factors U (z) and a spectrally majorised Λ(z) are approximated by polynomials, i.e. are of finite order, while the terms on the r.h.s. of (2) are generally algebraic or tran- scendental.

Over the past several years, many types of statistics have been proposed to test various equalities of **covariance** matrices. The first type is a class of statistics based on the likelihood ratio (LR). Mauchly [3] was one of the earlier attempts whose approach was based on the likelihood ratio. The statistic of Mauchly de- pends on the determinant and the trace of sample **covariance** **matrix**. It requires that the sample **covariance** **matrix** is non-singular, which is the case with proba- bility one when the sample size is larger than the dimension. Gupta and Xu [4] generalized the likelihood ratio test to non-normal distributions by deriving the asymptotic expansion of the test statistic under the null hypothesis when the sample size is moderate. Latterly, Jiang et al. [5] proved that the likelihood ratio test statistic has an asymptotic normal distribution under two different assump- tions by the aid of Selberg integrals. Also, the first type of statistics can be ex- tended to analyze high-dimensional data. For instance, Bai et al. [6] used central limit theorems for linear spectral statistics of sample **covariance** matrices and of random F-matrices, and proposed a modification of the likelihood ratio test to cope with high-dimensional effects. In the following, Niu et al. [7] considered testing mean vector and **covariance** **matrix** simultaneously with high-dimensional non-Gaussian data. Niu et al. applied the central limit theorem for linear spectral statistics of sample **covariance** matrices and established new modification for the likelihood ratio test. The second type is a class of statistics based on empirical distance. Let Z Z 1 , , , 2 Z N be a p -dimensional random sample drawn from a

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In the previous sections, the asymptotic **covariance** **matrix** of parameter estimation obtained by different estimation methods based on a transformation of the empirical quantiles of the two-parameters Weibull distribution, was derived. The optimal general case for the quantiles with p 1 = 0.23875930, p 2 = 0.92656148 is deduced to

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Bilinear **diagonal** models for knowledge graph embedding (KGE), such as DistMult and Com- plEx, balance expressiveness and computa- tional efficiency by representing relations as di- agonal matrices. Although they perform well in predicting atomic relations, composite rela- tions (relation paths) cannot be modeled nat- urally by the product of relation matrices, as the product of **diagonal** matrices is commuta- tive and hence invariant with the order of rela- tions. In this paper, we propose a new bilin- ear KGE model, called BlockHolE, based on block circulant matrices. In BlockHolE, rela- tion matrices can be non-commutative, allow- ing composite relations to be modeled by ma- trix product. The model is parameterized in a way that covers a spectrum ranging from diag- onal to full relation matrices. A fast compu- tation technique is developed on the basis of the duality of the Fourier transform of circu- lant matrices.

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The **covariance** matrices of these multivariate dis- tributions are estimated, which are then used in a LDA-based classification. The estimation of covari- ance matrices is notoriously susceptible to outlier interference [4]. To solve this problem, one can turn to robust statistics. This branch deals with outliers and corrupted data. Robust estimation has seen a good amount of mathematical development start- ing from Hampel [5], who defined the concept of robustness from the influence function, called the influence curve back then, and some measures that can be derived from it. A public start on the ro- bust estimation of **covariance** matrices was made by Gnanadesiken and Kettenring [6]. Maronna [7] was the first to formulate robust M-estimators in this con- text. Subsequently many approaches to this problem were described, including the minimum determinant **covariance** (MCD) [8], projection methods [9] and computing the smallest volume ellipsoid over the set of data [10].

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The space-time **covariance** **matrix** of a vector of M sensor measurements x[n] ∈ C M , R[τ] = E x[n]x H [n − τ] , where E{·} is the expectation operator, represents the data’s second order statistics, and is therefore central in the formu- lation of many broadband array processing problems. This includes for example broadband MIMO systems [1], cod- ing [2], beamforming [3], [4], source separation [5], angle of arrival estimation [6], scene discovery [7], and many others applications. Based on factorisations of its z-transform R(z) = P

The estimation results of the **Diagonal** BEKK model under the multivariate normal and multivariate Student's t error distributions are reported in Tables 2 and 3, respectively. It can be noticed that in comparison with the results obtained under the multivariate normal distribution, the log-likelihood value is increased and the values of all the three information criteria used in this study (Akaike, Schwarz, and Hannan–Quinn) are decreased under the multivariate Student's t error distribution. The estimated model under the multivariate Student's t error distribution is thus preferred. We notice that the estimated value of the GARCH coefficient, in particular, is equal to 0.8359 and 0.7583 for Bitcoin and Ether, respectively, indicating a relatively high degree of volatility persistence for both cryptocurrencies, with higher volatility persistence displayed in the Bitcoin market, though. Moreover, the ARCH and GARCH coefficients are highly significant for both cryptocurrencies. The significance of the estimated ARCH coefficients suggests that news/shocks in Bitcoin (Ether) are of great importance for Bitcoin's (Ether's) future volatility,

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The additive genetic covariance function 59 plays the same role in the evolution of growth trajectories that the additive genetic covariance matrix does in the stan[r]

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Determining the dimensionality of G provides an important perspective on the genetic basis of a multivariate suite of traits. Since the introduction of Fisher’s geometric model, the number of genetically independent traits underlying a set of functionally related phenotypic traits has been recognized as an important factor influencing the response to selection. Here, we show how the effective dimensionality of G can be established, using a method for the determination of the dimensionality of the effect space from a multivariate general linear model introduced by A memiya (1985). We compare this approach with two other available methods, factor-analytic modeling and bootstrapping, using a half-sib experiment that estimated G for eight cuticular hydrocarbons of Drosophila serrata. In our example, eight pheromone traits were shown to be adequately represented by only two underlying genetic dimensions by Amemiya’s approach and factor-analytic modeling of the **covariance** structure at the sire level. In contrast, boot- strapping identified four dimensions with significant genetic variance. A simulation study indicated that while the performance of Amemiya’s method was more sensitive to power constraints, it performed as well or better than factor-analytic modeling in correctly identifying the original genetic dimensions at mod- erate to high levels of heritability. The bootstrap approach consistently overestimated the number of dimensions in all cases and performed less well than Amemiya’s method at subspace recovery.

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is important in the digital calculation of a network bus impedance matrix, as z 5 is a diagonal matrix and simple rules can be derived for augmenting the bus impedance matrix of a partia[r]

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To solve many difficult mathematical problems a simple linear algebra technique is used which is called Singular value decomposition .This technique is also used in digital image watermarking for embedding and extraction process [8]. Images are considered to be square **matrix** using SVD, without the loss of image quality. So SVD technique can be easily implemented to any kind of digital images either the images may be grayscale or RGB. The orthogonal transform belongs to SVD which can decompose the given **matrix** into three equal size of matrixes of same size from which one matric is called **diagonal** and others two are orthogonal [3]. **Diagonal** **matrix** are used in digital image watermarking technique to embed the watermark into the original digital content. Square **matrix** are not require to decompose the **matrix** using SVD technique.

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In general, the space-time adaptive processing (STAP) can achieve excellent clutter suppression and moving target detection performance in the airborne multiple-input multiple-output (MIMO) radar for the increasing system degrees of freedom (DoFs). However, the performance improvement is accompanied by a dramatic increase in computational cost and training sample requirement. As one of the most efficient dimension-reduced STAP methods, the extended factored approach (EFA) transforms the full-dimension STAP problem into several small-scale adaptive processing problems, and therefore alleviates the computational cost and training sample requirement. However, it cannot effectively work in the airborne MIMO radar since sufficient training samples are unavailable. Aiming at the problem, a fast iterative method using persymmetry **covariance** **matrix** estimation in the airborne MIMO radar is proposed. In this method, the clutter **covariance** **matrix** is estimated by the original data and the constructed data. Then, the spatial weight vector in EFA is decomposed into the Kronecker product of two short-weight vectors. The bi-iterative algorithm is exploited to obtain the desired weight vectors. Simulation results demonstrate the effectiveness of our proposed method.

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