Some identities for products and degree raising of splines

In this paper it is shown how the algebraic product of two spline functions, each represented in terms of B-splines, can again be represented as a linear combination of suitable B-splines. As a corollary to this result we obtain an explicit representation of a given B-spline function in terms of B-splines of some arbitrary higher degree. This generalizes some known results for raising the degree by one. Recurrence relations for both products and degree raising are established that may be useful for computation.Communicated by Larry L. Schumaker.     

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Phenomenology of nonlocal cellular automata

Dynamical systems with nonlocal connections have potential applications to economic and biological systems. This paper studies the dynamics of nonlocal cellular automata. In particular, all two-state, three-input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings, although it is observed that sometimes a rule can have different dynamical behaviors with different wirings. The nonlocal cellular automata rule space is studied using a mean-field parametrization which is ideal for the situation of random wiring. Nonlocal cellular automata can be considered as computers carrying out computation at the level of each component. Their computational abilities are studied from the point of view of whether they contain many basic logical gates. In particular, I ask the question of whether a three-input cellular automaton rule contains the three fundamental logical gates: two-input rules AND and OR, and one-input rule NOT. A particularly interesting edge-of-chaos nonlocal cellular automaton, the rule 184, is studied in detail. It is a system of coupled selectors or multiplexers. It is also part of the Fredkin's gate—a proposed fundamental gate for conservative computations. This rule exhibits irregular fluctuations of density, large coherent structures, and long transient times.     

Analysis on Optimal Error Exponents of Binary Classification for Source with Multiple Subclasses

We consider a binary classification problem for a test sequence to determine from which source the sequence is generated. The system classifies the test sequence based on empirically observed (training) sequences obtained from unknown sources P1 and P2. We analyze the asymptotic fundamental limits of statistical classification for sources with multiple subclasses. We investigate the first- and second-order maximum error exponents under the constraint that the type-I error probability for all pairs of distributions decays exponentially fast and the type-II error probability is upper bounded by a small constant. In this paper, we first give a classifier which achieves the asymptotically maximum error exponent in the class of deterministic classifiers for sources with multiple subclasses, and then provide a characterization of the first-order error exponent. We next provide a characterization of the second-order error exponent in the case where only P2 has multiple subclasses but P1 does not. We generalize our results to classification in the case that P1 and P2 are a stationary and memoryless source and a mixed memoryless source with general mixture, respectively.     

Differential Expression Analysis of Single-Cell RNA-Seq Data: Current Statistical Approaches and Outstanding Challenges

With the advent of single-cell RNA-sequencing (scRNA-seq), it is possible to measure the expression dynamics of genes at the single-cell level. Through scRNA-seq, a huge amount of expression data for several thousand(s) of genes over million(s) of cells are generated in a single experiment. Differential expression analysis is the primary downstream analysis of such data to identify gene markers for cell type detection and also provide inputs to other secondary analyses. Many statistical approaches for differential expression analysis have been reported in the literature. Therefore, we critically discuss the underlying statistical principles of the approaches and distinctly divide them into six major classes, i.e., generalized linear, generalized additive, Hurdle, mixture models, two-class parametric, and non-parametric approaches. We also succinctly discuss the limitations that are specific to each class of approaches, and how they are addressed by other subsequent classes of approach. A number of challenges are identified in this study that must be addressed to develop the next class of innovative approaches. Furthermore, we also emphasize the methodological challenges involved in differential expression analysis of scRNA-seq data that researchers must address to draw maximum benefit from this recent single-cell technology. This study will serve as a guide to genome researchers and experimental biologists to objectively select options for their analysis.     

基于偏序关系的Rough集模型及其应用

在标准Rough集理论的指导下,利用偏序关系性质构造了不同分类,并以此为基础探讨了上、下近似集,从而构建了基于偏序关系的Rough集模型。新模型将Rough集理论的应用范围由等价关系扩展到偏序关系。为了更好地增强模型的实用性和灵括性,一方面从程度、精度、概率等角度出发分别对其进行了扩展,另一方面引入依赖度使其适用于研究各种非严格的偏序关系。给出了实例分析,并结合现实生活中的现象阐述了模型的应用价值。     

高光谱成像的垃圾分类识别研究

高光谱成像因光谱分辨率高、图谱合一、可实现快速无损检测等特点现已广泛应用于农业、医学、遥感等领域。现有的对可回收生活垃圾检测与分类的方法,都存在检测时间长,分类效率低,而大量多种垃圾无法同时快速分拣等问题。考虑到不同类别的生活垃圾由于其主要组成分子结构的差异,对不同波长的光有不同的吸收特性。高光谱图像在记录待分类垃圾的空间信息的同时,可以获得垃圾对不同波长的光的反射率光谱信息,通过建立识别分类模型对反射率光谱信息进行分析可以实现对高光谱图像中待分类垃圾的识别与分类。收集常见纸质、塑料、木质三种材料的可回收的垃圾样本,包括塑料瓶、食品包装袋、塑料玩具(饰品)碎片、一次性筷子、雪糕棒、木制家具碎片、木制包装盒、废旧课本、广告纸、办公用纸等多种物品共30个样本,进行清洗和裁剪处理,避免样本表面污渍对样本反射率产生影响。利用高光谱成像系统采集样本在近红外(780～1 000 nm)范围内的高光谱图像,其中18个样本做训练样本集,12个样本做测试样本集。对采集的样本图像数据做预处理,包括去噪声以及黑白校正反演反射率信息等处理;通过主成分分析(PCA)方法对训练样本集感兴趣区域(ROI)进行分析,提取到的特征波段为795.815,836.869,885.619,916.409,929.239,934.37,957.463,972.858和988.253 nm;在特征波段下分别提取这三种类别垃圾的参考光谱,通过光谱角度填图法(SAM)对测试样本ROI区域内提取的测试样本点集在特征波段下与参考光谱进行匹配,由匹配程度进行样本点归类,分析结果表明,测试样本集中纸制样本(A类别)、塑料样本(B类别)、木制样本(C类别)的分类准确度分别为100%,98%和100%,测试样本点集整体的分类准确度为99.33%;通过Fisher判别方法分析训练样本集得出判别函数式和判别准则,对测试样本点集分类,评价结果为A,B和C类样本分类准确度分别为100%,100%和97%,测试样本点集整体分类准确度为99%。通过SAM和Fisher两种判别方法对测试样本集的光谱图像进行目标物的检测与分类,结果表明,利用SAM判别方法在可回收垃圾的高光谱图像中实现检测与分类有更高的分类准确度,可达到99.33%。同时,也验证了使用高光谱成像进行可回收垃圾快速分类的科学性以及可行性,对未来系统化、机械化、智能化地解决生活中可回收垃圾的分类具有一定的实用意义。     

基于图像处理的闸机口跳闸事件检测算法

文中提出了一种基于图像处理的闸机口跳闸事件检测方法。首先,利用模板匹配的方法进行预处理,得到闸门图像匹配率;其次,根据跳闸事件的连贯性特点,提取一段连续时间序列的匹配率作为分类特征;最后,采取基于最小错误率的贝叶斯分类方法对这段时间序列特征向量进行分类。实验表明,该算法可以有效的检测出闸机口的跳闸事件,具有实时性好、不需要额外的传感器、成本低的优点,具有较好的工程应用前景。