An efficient algorithm for calculating the likelihood andlikelihood gradient of ARMA models

Exact analytical expressions are obtained for the likelihood and likelihood gradient stationary autoregressive moving average (ARMA) models. Denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p+2q+1)N +o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p+6q+2)N+o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2 p+6q+2)N+o(N) multiply-add operations per iteration     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

Computing the width of a set

For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n +I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O( n²). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices     

A formal analysis of the fault-detecting ability of testing methods

Several relationships between software testing criteria, each induced by a relation between the corresponding multisets of subdomains, are examined. The authors discuss whether for each relation R and each pair of criteria, C₁ and C₂ , R(C₁, C₂) guarantees that C₁ is better at detecting faults than C₂ according to various probabilistic measures of fault-detecting ability. It is shown that the fact that C ₁ subsumes C₂ does not guarantee that C₁ is better at detecting faults. Relations that strengthen the subsumption relation and that have more bearing on fault-detecting ability are introduced     

Serial and parallel algorithms for the medial axis transform

An O(n²) time serial algorithm is developed for obtaining the medial axis transform (MAT) of an n×n image. An O(log n) time CREW PRAM algorithm and an O(log² n) time SIMD hypercube parallel algorithm for the MAT are also developed. Both of these use O(n²) processors. Two problems associated with the MAT, the area and perimeter reporting problem, are studied. An O(log n) time hypercube algorithm is developed for both of them, where n is the number of squares in the MAT, and the algorithms use O(n²) processors     

基于机器视觉的枸杞分级方法

针对目前传统的枸杞分级主要采用人工方法,费时费力且效率不高的缺点,提出了一种基于机器视觉技术对枸杞进行自动分类的方法。采用数字图像处理技术对枸杞图像进行了预处理、分割,从而提取枸杞的色泽、大小及形状等特征参数;用K-means算法对特征进行聚类,得到枸杞相应等级的基准;根据聚类分析得到的基准采用最小距离分类器对枸杞进行分级。实验结果表明,该方法能够准确快速地对不同色泽和大小的枸杞进行分类。     

The minimal dimension of stable faces required to guaranteestability of a matrix polytope: D-stability

The problem of determining whether a polytope P of n ×n matrices is D-stable-i.e. whether each point in P has all its eigenvalues in a given nonempty, open, convex, conjugate-symmetric subset D of the complex plane-is discussed. An approach which checks the D-stability of certain faces of P is used. In particular, for each D and n the smallest integer m such that D-stability of every m-dimensional face guarantees D-stability of P is determined. It is shown that, without further information describing the particular structure of a polytope, either (2n-4)-dimensional or (2n-2)-dimensional faces need to be checked for D-stability, depending on the structure of D. Thus more work needs to be done before a computationally tractable algorithm for checking D-stability can be devised     

M-dimensional Cayley-Hamilton theorem

The theorem states that every block square matrix satisfies its own m-D (m-dimensional, m⩾1) matrix characteristic polynomial. The exact statement and a simple proof of this theorem are given. The theorem refers to a matrix A subdivided into m blocks, and hence having dimension at least m. The conclusion is that every square matrix A with dimension M satisfies several m-D characteristic matrix polynomials with degrees N₁ . . ., N _m, such that N₁+ . . . +N_m ⩽M     

Mapping nested loop algorithms into multidimensional systolicarrays

Consideration is given to transforming depth p-nested for loop algorithms into q-dimensional systolic VLSI arrays where 1⩽q⩽p-1. Previously, there existed complete characterizations of correct transformation only for the cases where q=p-1 or q=1. This gap is filled by giving formal necessary and sufficient conditions for correct transformation of a p-nested loop algorithm into a q-dimensional systolic array for any q, 1⩽q⩽p-1. Practical methods are presented. The techniques developed are applied to the automatic design of special purpose and programmable systolic arrays. The results also contribute toward automatic compilation onto more general purpose programmable arrays. Synthesis of linear and planar systolic array implementations for a three-dimensional cube-graph algorithm and a reindexed Warshall-Floyd path-finding algorithm are used to illustrate the method     

Constant time algorithms for the transitive closure and somerelated graph problems on processor arrays with reconfigurable bussystems

The transitive closure problem in O(1) time is solved by a new method that is far different from the conventional solution method. On processor arrays with reconfigurable bus systems, two O (1) time algorithms are proposed for computing the transitive closure of an undirected graph. One is designed on a three-dimensional n×n×n processor array with a reconfigurable bus system, and the other is designed on a two-dimensional n²×n² processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Using the O(1) time transitive closure algorithms, many other graph problems are solved in O(1) time. These problems include recognizing bipartite graphs and finding connected components, articulation points, biconnected components, bridges, and minimum spanning trees in undirected graphs     

Parallel binary search

Two arrays of numbers sorted in nondecreasing order are given: an array A of size n and an array B of size m, where n<m. It is required to determine, for every element of A, the smallest element of B (if one exists) that is larger than or equal to it. It is shown how to solve this problem on the EREW PRAM (exclusive-read exclusive-write parallel random-access machine) in O(logm logn/log log m) time using n processors. The solution is then extended to the case in which fewer than n processors are available. This yields an EREW PRAM algorithm for the problem whose cost is O(n log m, which is O(m)) for n⩽m/log m. It is shown how the solution obtained leads to an improved parallel merging algorithm     

Decoupling of square singular systems via proportional statefeedback

Necessary and sufficient conditions for the decoupling of a solvable square singular system Ex˙(t)=Ax(t)+Bu(t ) with output y(t)=Dx(t), through an admissible control law of the form u(t)=Kx(t)+Hr(t) where H is a square nonsingular matrix. It has been shown that for a given singular system that satisfies these conditions, a propagational state feedback exists for which the system's transfer function is a diagonal, nonsingular, and proper rational matrix. The proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback     

Distributed detection with consulting sensors and communicationcost

The problem of distributed detection with consulting sensors in the presence of communication cost associated with any exchange of information (consultation) between sensors is considered. The system considered has two sensors, S1 and S2; S1 is the primary sensor responsible for the final decision u₀ , and S2 is a consulting sensor capable of relaying its decision u₂ to S1 when requested by S 1. The final decision u₀ is either based on the raw data available to S1 only, or, under certain request conditions, also takes into account the decision u₂ of sensor S2. Random and nonrandom request schemes are analyzed and numerical results are presented and compared for Gaussian and slow-fading Rayleigh channels. For each decision-making scheme, an associated optimization problem is formulated whose solution is shown to satisfy certain set design criteria that the authors consider essential for sensor fusion     

Bicoprime factorizations of the plant and their relation to right and left-coprime factorizations

In a general algebraic framework, starting with a bicoprime factorization P=N_prD^-1 N_pl, a right-coprime factorization N_p D_p^-1, a left-coprime factorization D^-1_pN_p, and the generalized Bezout identities associated with the pairs (N_p, D_p) and (D˜ _p, N˜_p) are obtained. The set of all H-stabilizing compensators for P in the unity-feedback configuration S(P, C) are expressed in terms of (N_pr, D, N _pt) and the elements of the Bezout identity. The state-space representation P=C(sI-A)^-1B is included as an example     

Vibrational feedback control in the problem of acoustic stability

The problem of absolute stability in a vibrational feedback controller is introduced and discussed. It is shown that for any rational G(s)=n(s)/d(s ) with d(s) Hurwitz and deg d(s) -deg n(s)=1 there exists a linear dynamic periodic controller that ensures, in a certain sense, the infinite sector of absolute stability. This implies that an additional dynamical element, inserted in the feedback loop, may lead to improvements in the robustness of nonlinear systems     

Functional and topological relations among banyan multistagenetworks of differing switch sizes

Relations among banyan multistage interconnection networks (MINs) of differing switch sizes are studied. If two N×N networks W and W' have switch sizes r and s, respectively, and if r>s, then W realizes a larger number of permutations than W'. Consequently, the two networks can never be equivalent. However, W may realize all the permutations of W', in which case W is said to functionally cover W' in the strict sense. More generally, W is said to functionally cover W' in the wide sense if the terminals of W can be relabeled so that W realizes all the permutations of W'. Functional covering is topologically characterized, and an optimal algorithm to decide strict functional covering is developed     

Matrix inequalities applicable to estimating solution sizes ofRiccati and Lyapunov equations

Simultaneous summation upper bounds for the eigenvalues of the matrix product XY are presented, where X, Y ϵ R^nxn, with Y symmetric and X arbitrary. These bounds are a generalization of tr (XY) bounds; the requirements on Y are relaxed, and the bound for tr (XY) is stronger than those shown in the literature     

Efficient algorithms for list ranking and for solving graphproblems on the hypercube

A hypercube algorithm to solve the list ranking problem is presented. Let n be the length of the list, and let p be the number of processors of the hypercube. The algorithm described runs in time O(n/p) when n=Ω(p ^1+ε) for any constant ε>0, and in time O(n log n/p+log³ p) otherwise. This clearly attains a linear speedup when n=Ω(p ^1+ε). Efficient balancing and routing schemes had to be used to achieve the linear speedup. The authors use these techniques to obtain efficient hypercube algorithms for many basic graph problems such as tree expression evaluation, connected and biconnected components, ear decomposition, and st-numbering. These problems are also addressed in the restricted model of one-port communication     

A VLSI constant geometry architecture for the fast Hartley andFourier transforms

An application-specific architecture for the parallel calculation of the decimation in time and radix 2 fast Hartley (FHT) and Fourier (FFT) transforms is presented. A real sequence with N=2ⁿ data items is considered as input. The system calculates the FHT and the FFT in n and n+1 stages. respectively. The modular and regular parallel architecture is based on a constant geometry algorithm using butterflies of four data items and the perfect unshuffle permutation. With this permutation, the mapping of the algorithm in VLSI technology is simplified and the communications among processors are minimized. Organization of the processor memory based on first-in, first-out (FIFO) queues facilitates a systolic data flow and permits the implementation in a direct way of the complex data movements and address sequences of the transforms. This is accomplished by means of simple multiplexing operations, using hardwired control. The total calculation time is (Nlog₂N)/4Q cycles for the FHT and N(1+log₂N)/4Q cycles for the FFT, where Q is the number of processors ( Q= 2^q, Q⩽N/4)     

Out-of-roundness problem revisited

The properties and computation of the minimum radial separation (MRS) standard for out-of-roundness are discussed. Another standard out-of-roundness measurement called the minimum area difference (MAD) center is introduced. Although the two centers have different characteristics, the approach to finding both centers shares many commonalities. An O(n log n+k) time algorithm which is used to compute the MRS center is presented. It also computes the MAD center of a simple polygon G, where n is the number of vertices of G, and k is the number of intersection points of the medial axis and the farthest-neighbor Voronoi diagram of G. The relationship between MRS and MAD is discussed