An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value

The study describes students’ patterns of thinking for statistical problems set in two different contexts. Fifteen students representing a wide range of experiences with high school mathematics participated in problem-solving clinical interview sessions. At one point during the interviews, each solved a problem that involved determining the typical value within a set of incomes. At another point, they solved a problem set in a signal-versus-noise context [Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289]. Several patterns of thinking emerged in the responses to each task. In responding to the two tasks, some students attempted to incorporate formal measures, while others used informal estimating strategies. The different types of thinking employed in using formal measures and informal estimates are described. The types of thinking exhibited in the signal-versus-noise context are then compared against those in the typical value context. Students displayed varying amounts of attention to both data and context in formulating responses to both problems. Suggestions for teachers in regard to helping students attend to both data and context when analyzing statistical data are given.     

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Worst Case Setting

We consider approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces with finite-order weights. This means we consider functions of d variables that can be represented as sums of functions of at most q^* variables. Here, q^* is fixed (and presumably small) and d may be arbitrarily large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that use O(ε−p) function or linear functional evaluations to achieve an error ε in the worst case setting. Based on these algorithms A_1,ε, we provide a construction of polynomial-time algorithms A_d,ε for the general d-variate problem with the number of evaluations bounded roughly by ε^−pd^q* to achieve an error ε in the worst case setting.     

Varimax extended rotation applied to multivariate spectroscopic image analysis

A new approach for spectral image analysis called the Varimax extended rotation (VER) has been developed. VER employs a four-step procedure to resolve image data. In the first step, the data are pretreated to ensure they are is in a form suitable for principal component analysis. The second step involves reducing the dimensionality of the data using principal component analysis. In the third step, the significant principal components are rotated to identify single component regions in the spectral image. The fourth step uses alternating least squares (ALS) to improve the estimates of the spectral profiles of each component. Results from simulated and real Raman imaging data of water in oil emulsions demonstrate the efficacy and efficiency of the proposed method.     

Simultaneous multielemental determination using a low-resolution inductively coupled plasma spectrometer/diode array detection system

This work presents a versatile method for simultaneous multielemental analysis using detection with photodiode arrays and multivariate calibration techniques. A multidetection system based on an array of 1024 photodiodes was built and adapted to a commercial plasma emission scanning spectrometer. Spectral data were acquired at low resolution, allowing simultaneous monitoring of a broad spectral range and resulting in very informative but considerably overlapped spectra. Partial least squares and principal component regressions were employed to minimize overlapping problems. A numerical procedure for window selection was also developed.

The new method was applied to the simultaneous determination of manganese, molybdenum, nickel, chromium and iron in steel samples, resulting in average relative prediction errors of 2.1% for Mn, 3.4% for Mo, 0.6% for Cr, 1.5% for Ni and 0.8% for Fe. These errors are comparable to those observed with conventional scanning detection systems and separate univariate calibrations, but the new method allows simultaneous determination of the five elements, with data acquisition significantly faster than in scanning instruments.     

多元时间序列模型协方差矩阵的递归算法

本文推导了多元时序模型的协方差矩阵与模型参数的关系式，并给出了计算多维时序过程自协方差矩阵的递归算法     

Asymptotic Normality of the Recursive M-estimators of the Scale Parameters

In this paper, the limit distributions of the recursive M-estimators of scatter parameters in a multivariate linear model setting are studied. Under some mild conditions, the asymptotic normality of the recursive M-esimtators is established. Some Monte Carlo simulation results are presented to illustrate the performance of the recursive M-estimators.     

A characterization of multivariate normal distribution and its application

Let X₁, X₂, …, X_n be i.i.d. d-dimensional random vectors with a continuous density. Let and . In this paper we find that the distribution of Z_k (or Y_k) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.     

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

In this paper we revisit the Bialynicki-Birula and Mycielski uncertainty principle and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Rényi entropies. We recall that in both Shannon and Rényi cases, and for a given dimension n$n$, the only case of equality occurs for Gaussian random vectors. We show that as n$n$ grows, however, the bound is also asymptotically attained in the cases of n$n$-dimensional Student-t$t$ and Student-r$r$ distributions. A complete analytical study is performed in a special case of a Student-t$t$ distribution. We also show numerically that this effect exists for the particular case of a n$n$-dimensional Cauchy variable, whatever the Rényi entropy considered, extending the results of Abe and illustrating the analytical asymptotic study of the Student-t$t$ case. In the Student-r$r$ case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a “Gaussianization” of the vector when the dimension increases.