Spectral regions selection to improve prediction ability of PLS models by changeable size moving window partial least squares and searching combination moving window partial least squares

Changeable size moving window partial least squares (CSMWPLS) and searching combination moving window partial least squares (SCMWPLS) are proposed to search for an optimized spectral interval and an optimized combination of spectral regions from informative regions obtained by a previously proposed spectral interval selection method, moving window partial least squares (MWPLSR) [Anal. Chem. 74 (2002) 3555]. The utilization of informative regions aims to construct better PLS models than those based on the whole spectral points. The purpose of CSMWPLS and SCMWPLS is to optimize the informative regions and their combination to further improve the prediction ability of the PLS models. The results of their application to an open-path (OP)/FT-IR spectra data set show that the proposed methods, especially SCMWPLS can find out an optimized combination, with which one can improve, often significantly, the performance of the corresponding PLS model, in terms of low prediction error, root mean square error of prediction (RMSEP) with the reasonable latent variable (LVs) number, comparing with the results obtained using whole spectra or direct combination of informative regions for a compound. Regions consisting of the combinations obtained can easily be explained by the existence of IR absorption bands in those spectral regions.     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

Schur positivity and log-concavity related to longest increasing subsequences

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen’s log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen’s original conjecture.     

带反馈非对称信道的最优纠错编码

带反馈非对称信道的最优纠错编码等同于Ulam-Renyi问题的半谎搜索模型的最少提问次数qw(m).本文证明了qw(m)的取值分布具有非交叉性.借助于这一良好的分布特性,原始的确定最少提问次数精确值问题可以被简化为确定一个临界数问题并且这个临界数的存在性和唯一性得到了证明.同时本文也改进了Cicalese等人的结果,在更大的范围内明确地给出了最少提问次数qw(m)的精确值.     

Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions

In a recent paper, Backelin, West and Xin describe a map φ^* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ^*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ^* commutes with taking the inverse of a permutation. In the BWX paper, the definition of φ^* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ^* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group _n that avoid T equals the number of permutations of _n that avoid T′. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of _n that avoid T is equal to the number of involutions of _n avoiding T′, as recently conjectured by Jaggard. Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

正交饱和效应模型及其统计分析

文\cite{14}讨论了2水平正交饱和设计的一种统计分析方法---零效应搜索法, 本文将此方法推广到一般的饱和模型---正交饱和效应模型. 首先给出正交饱和效应模型的定义及其相关的参数估计和平方和分解, 然后用零效应搜索法对显著效应的检验问题进行分析, 模拟结果显示效果很好.     

Quasi-random 2- colorings of point sets

Given an arbitrary set of N points on the plane, one can two-color the points red and blue in such a way that the difference of the numbers of red and blue points in any half-plane has absolute value less than N^1/4(log N)⁴. This is essentially best possible.     

Cuckoo hashing

We present a simple dictionary with worst case constant lookup time, equaling the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. [SIAM J. Comput. 23 (4) (1994) 738–761]. The space usage is similar to that of binary search trees. Besides being conceptually much simpler than previous dynamic dictionaries with worst case constant lookup time, our data structure is interesting in that it does not use perfect hashing, but rather a variant of open addressing where keys can be moved back in their probe sequences. An implementation inspired by our algorithm, but using weaker hash functions, is found to be quite practical. It is competitive with the best known dictionaries having an average case (but no nontrivial worst case) guarantee on lookup time.     

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x∈[n], and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [n]=A₀∪?∪At−1 and asks for a such that x∈Aa. Carole responds with some b, and if a≠b, then x accumulates a lie (a,b). Carole's string of lies for x must be in the channel C. Paul wins if he determines x within q rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space [n] for which Paul can guarantee finding the distinguished element is as q→∞, where Ek(C) is the number of lie strings in C of maximum length k. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel C.