CORDIC算法的优化及其硬件实现

针对采用流水结构实现CORDIC算法时存在的不足,从旋转角度范围、旋转角度精度的调整、模校正因子的分解3个方面进行了详细的分析和讨论,并给出了相应的优化设计和改进措施.实现了基于FPCA的CORDIC算法全流水结构,最后用CORDIC算法实现信号发生器加以验证.     

Functional principal component analysis for the explorative analysis of multisite–multivariate air pollution time series with long gaps

The knowledge of the urban air quality represents the first step to face air pollution issues. For the last decades many cities can rely on a network of monitoring stations recording concentration values for the main pollutants. This paper focuses on functional principal component analysis (FPCA) to investigate multiple pollutant datasets measured over time at multiple sites within a given urban area. Our purpose is to extend what has been proposed in the literature to data that are multisite and multivariate at the same time. The approach results to be effective to highlight some relevant statistical features of the time series, giving the opportunity to identify significant pollutants and to know the evolution of their variability along time. The paper also deals with missing value issue. As it is known, very long gap sequences can often occur in air quality datasets, due to long time failures not easily solvable or to data coming from a mobile monitoring station. In the considered dataset, large and continuous gaps are imputed by empirical orthogonal function procedure, after denoising raw data by functional data analysis and before performing FPCA, in order to further improve the reconstruction.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

Interpreting the Principal Component Analysis of Multivariate Density Functions

Functional principal component analysis (FPCA) as a reduction data technique of a finite number T of functions can be used to identify the dominant modes of variation of numeric three-way data.

We carry out the FPCA on multidimensional probability density functions, relate this method to other standard methods and define its centered or standardized versions. Grounded on the relationship between FPCA of densities, FPCA of their corresponding characteristic functions, PCA of the MacLaurin expansions of these characteristic functions and dual STATIS method applied to their variance matrices, we propose a method for interpreting the results of the FPCA of densities. This method is based on the investigations of the relationships between the scores of the FPCA and the moments associated to the densities.

The method is illustrated using known Gaussian densities. In practice, FPCA of densities deals with observations of multidimensional variables on T occasions. These observations can be used to estimate the T associated densities (i) by estimating the parameters of these densities, assuming that they are Gaussian, or (ii) by using the Gaussian kernel method and choosing the matrix bandwidth by the normal reference rule. Thereafter, FPCA estimate is derived from these estimates and the interpretation method is carried out to explore the dominant modes of variation of the types of three-way data encountered in sensory analysis and archaeology.