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During the processing and planting of soybeans, it is greatly significant that a reliable, rapid, and accurate technique is used to detect soybean varieties. Traditional chemical analysis methods of soybean variety sampling (e.g., mass spectrometry and high-performance liquid chromatography) are destructive and time-consuming. In this paper, a robust and accurate method for nondestructive soybean classification is developed through hyperspectral imaging and ensemble machine learning algorithms. Image acquisition, preprocessing, and feature selection are used to obtain different types of soybean hyperspectral features. Based on these features, one of ensemble classifiers-random subspace linear discriminant (RSLD) algorithm is used to classify soybean seeds. Compared with the linear discrimination (LD) and linear support vector machine (LSVM) methods, the results show that the RSLD algorithm in this paper is more stable and reliable. In classifying soybeans in 10, 15, 20, and 25 categories, the RSLD method achieves the highest classification accuracy. When 155 features are used to classify 15 types of soybeans, the classification accuracy of the RSLD method reaches 99.2%, while the classification accuracies of the LD and LSVM methods are only 98.6% and 69.7%, respectively. Therefore, the ensemble classification algorithm RSLD can maintain high classification accuracy when different types and different classification features are used.

We provide a recursive formula for the dimension of the algebraic sum of finitely many subspaces in a finite-dimensional vector space over an arbitrary field.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Let d and k be integers with \\[1 \\le k \\le d-1\\]. Let \\[\\Lambda \\] be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in \\[\\Lambda \\cap K\\]. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional \\[n \\times \\cdots \\times n\\] grid is at least \\[\\Omega \\bigl (n^{d(d-k)/(d-1)-\\varepsilon }\\bigr )\\] and at most \\[O\\bigl (n^{d(d-k)/(d-1)}\\bigr )\\], where \\[\\varepsilon >0\\] is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover \\[\\Lambda \\cap K\\]. We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For \\[d \\ge 3\\] and \\[\\varepsilon \\in (0,1)\\], we show that there is an integer \\[r=r(d,\\varepsilon )\\] such that for all positive integers n, m the following statement is true. There is a set of n points in \\[\\mathbb {R}^d\\] and an arrangement of m hyperplanes in \\[\\mathbb {R}^d\\] with no \\[K_{r,r}\\] in their incidence graph and with at least \\[\\Omega \\bigl ((mn)^{1-(2d+3)/((d+2)(d+3)) - \\varepsilon }\\bigr )\\] incidences if d is odd and \\[\\Omega \\bigl ((mn)^{1-(2d^2+d-2)/((d+2)(d^2+2d-2)) -\\varepsilon }\\bigr )\\] incidences if d is even.

Electrocardiogram (ECG) is conducted to monitor the electrical activity of the heart by presenting small amplitude and duration signals; as a result, hidden information present in ECG data is difficult to determine. However, this concealed information can be used to detect abnormalities. In our study, a fast feature-fusion method of ECG heartbeat classification based on multi-linear subspace learning is proposed. The method consists of four stages. First, baseline and high frequencies are removed to segment heartbeat. Second, as an extension of wavelets, wavelet-packet decomposition is conducted to extract features. With wavelet-packet decomposition, good time and frequency resolutions can be provided simultaneously. Third, decomposed confidences are arranged as a two-way tensor, in which feature fusion is directly implemented with generalized N dimensional ICA (GND-ICA). In this method, co-relationship among different data information is considered, and disadvantages of dimensionality are prevented; this method can also be used to reduce computing compared with linear subspace-learning methods (PCA). Finally, support vector machine (SVM) is considered as a classifier in heartbeat classification. In this study, ECG records are obtained from the MIT-BIT arrhythmia database. Four main heartbeat classes are used to examine the proposed algorithm. Based on the results of five measurements, sensitivity, positive predictivity, accuracy, average accuracy, and t -test, our conclusion is that a GND-ICA-based strategy can be used to provide enhanced ECG heartbeat classification. Furthermore, large redundant features are eliminated, and classification time is reduced.

Consider a random walk Si =ξ₁ + … +ξi, i ∊ N, whose increments ξ₁, ξ₂, … are independent identically distributed random vectors in ℝ d such that ξ₁ has the same law as -ξ₁ and ℙ[ξ₁ ∊ H] = 0 for every affine hyperplane H ⊂ ℝd. Our main result is the distribution-free formula E ∑ 1 ≤ i 1 < … < i k ≤ n 1 0 ∉ Conv ( S i , … , S i k ) = 2 n k B ( k , d − 1 ) + B ( k , d − 3 ) + … 2 k k ! where the B(k, j)’s are defined by their generating function (t + 1)(t + 3) … (t + 2k-1) = ∑ j = 0 k B ( k , j ) t j . The expected number of k-tuples above admits the following geometric interpretation: it is the expected number of k-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type Bn that are not intersected by a generic linear subspace L ⊂ ℝ n of codimension d. The case d = 1 turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.

Let X , Y be two linear subspaces of the m -dimensional complex space ℂ m with m > 1. The dimensions of the subspaces X and Y are k and m − k respectively and let F : ℂ m → ℂ m be a non-degenerate linear operator. In this work, we study the properties of the intersection between the subspace Y and the n -iteration of the subspace X under F . In the case when the dimension of the subspace X is either one or two, we give some results about a geometrical classification when we obtain an infinite set of moments n of no transversality between the space Y and the n -iteration of X under F .

The averaged alternating modified reflections (AAMR) method is a projection algorithm for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method can be seen as an adequate modification of the Douglas–Rachford method that yields a solution to the best approximation problem. In this paper, we consider the particular case of two subspaces in a Euclidean space. We obtain the rate of linear convergence of the AAMR method in terms of the Friedrichs angle between the subspaces and the parameters defining the scheme, by studying the linear convergence rates of the powers of matrices. We further optimize the value of these parameters in order to get the minimal convergence rate, which turns out to be better than the one of other projection methods. Finally, we provide some numerical experiments that demonstrate the theoretical results.

Craniofacial reconstruction aims to estimate an individual’s facial appearance from its skull. It can be applied in many multimedia services such as forensic medicine, archaeology, face animation etc. In this paper, a statistical learning based method is proposed for 3D craniofacial reconstruction. In order to well represent the craniofacial shape variation and better utilize the relevance between different local regions, two tensor models are constructed for the skull and the face skin respectively, and multi-linear subspace analysis is used to extract the craniofacial subspace features. A partial least squares regression (PLSR) based mapping from skull subspace to skin subspace is established with the attributes such as age and BMI being considered. For an unknown skull, the 3D face skin is reconstructed using the learned mapping with the help of the skin tensor model. Compared with some other statistical learning based method in literature, the proposed method more directly and properly reflects the shape relationship between the skull and the face. In addition, the proposed method has little manual intervention. Experimental results show that the proposed method is valid.

Working in the setting of Banach spaces, we give a simpler proof of a result concerning the Ulam stability of the composition of operators. Several applications are provided. Then, we give an example of a discrete semigroup with Ulam unstable members and an example of Ulam stable operators on a Banach space, such that their sum is not Ulam stable. Another example is concerned with a C 0 -semigroup ( T t ) t ≥ 0 of operators for which each T t is Ulam stable. We present an open problem concerning the Ulam stability of the members of the Bernstein C 0 -semigroup. Two other possible problems are mentioned at the end of the paper.