Canonical Measure of Correlation (CMC) and Canonical Measure of Distance (CMD) between sets of data. Part 3. Variable selection in classification

In multivariate regression and classification issues variable selection is an important procedure used to select an optimal subset of variables with the aim of producing more parsimonious and eventually more predictive models. Variable selection is often necessary when dealing with methodologies that produce thousands of variables, such as Quantitative Structure-Activity Relationships (QSARs) and highly dimensional analytical procedures.In this paper a novel method for variable selection for classification purposes is introduced. This method exploits the recently proposed Canonical Measure of Correlation between two sets of variables (CMC index). The CMC index is in this case calculated for two specific sets of variables, the former being comprised of the independent variables and the latter of the unfolded class matrix. The CMC values, calculated by considering one variable at a time, can be sorted and a ranking of the variables on the basis of their class discrimination capabilities results. Alternatively, CMC index can be calculated for all the possible combinations of variables and the variable subset with the maximal CMC can be selected, but this procedure is computationally more demanding and classification performance of the selected subset is not always the best one.The effectiveness of the CMC index in selecting variables with discriminative ability was compared with that of other well-known strategies for variable selection, such as the Wilks’ Lambda, the VIP index based on the Partial Least Squares-Discriminant Analysis, and the selection provided by classification trees.A variable Forward Selection based on the CMC index was finally used in conjunction of Linear Discriminant Analysis. This approach was tested on several chemical data sets. Obtained results were encouraging.     

Chemistry, commentary, and community: discussion of “an examination of the vaporization enthalpies and vapor pressures of pyrazine, pyrimidine, pyridazine, and 1,3,5-triazine” by Lipkind and Chickos

Vaporization enthalpies for the isomeric diazines were discussed within the context of recent measurements and estimation techniques. It is suggested that pyridazine shows enigmatic behavior.

Canonical Measure of Correlation (CMC) and Canonical Measure of Distance (CMD) between sets of data: Part 2. Variable reduction

This paper proposes a new method for determining the subset of variables that reproduce as well as possible the main structural features of the complete data set. This method can be useful for pre-treatment of large data sets since it allows discarding variables that contain redundant information. Reducing the number of variables often allows one to better investigate data structure and obtain more stable results from multivariate modelling methods.The novel method is based on the recently proposed canonical measure of correlation (CMC index) between two sets of variables [R. Todeschini, V. Consonni, A. Manganaro, D. Ballabio, A. Mauri, Canonical Measure of Correlation (CMC) and Canonical Measure of Distance (CMD) between sets of data. Part 1. Theory and simple chemometric applications, Anal. Chim. Acta submitted for publication (2009)]. Following a stepwise procedure (backward elimination), each variable in turn is compared to all the other variables and the most correlated is definitively discarded. Finally, a key subset of variables being as orthogonal as possible are selected. The performance was evaluated on both simulated and real data sets. The effectiveness of the novel method is discussed by comparison with results of other well known methods for variable reduction, such as Jolliffe techniques, McCabe criteria, Krzanowski approach and its modification based on genetic algorithms, loadings of the first principal component, Key Set Factor Analysis (KSFA), Variable Inflation Factor (VIF), pairwise correlation approach, and K correlation analysis (KIF). The obtained results are consistent with those of the other considered methods; moreover, the advantage of the proposed CMC method is that calculation is very quick and can be easily implemented in any software application.     

In defence of the correlation coefficient

The relationship between calibrated range, residual standard deviation and correlation coefficient r is discussed. It is shown that in typical chemical calibration applications with appropriately distributed calibration points (particularly, with range comparable to mean and with approximately even or with ‘successive dilution’ spacing), the linear correlation coefficient has valid application as a routine criterion for acceptable fit if used with due care.

Structure-specific magnetic field inhomogeneities and its effect on the correlation time

We describe the relationship between the correlation time and microscopic spatial inhomogeneities in the static magnetic field. The theory takes into account diffusion of nuclear spins in the inhomogeneous field created by magnetized objects. A simple general expression for the correlation time is obtained. It is shown that the correlation time is dependent on a characteristic length, the diffusion coefficient of surrounding medium, the permeability of the surface and the volume fraction of the magnetized objects. For specific geometries (spheres and cylinders), exact analytical expressions for the correlation time are given. The theory can be applied to contrast agents (magnetically labeled cells), capillary network, BOLD effect and so forth.     

Multivariate time changes for Lévy asset models: Characterization and calibration

We build a theoretical framework for multivariate subordination of Brownian motions, with a common and an idiosyncratic component. This follows economic intuition and introduces generalizations of some well known multivariate Lévy processes for financial applications: the compound Poisson, NIG, Variance Gamma and CGMY. In most cases we obtain the characteristic function in closed form. The extension is first kept parsimonious, by adding one parameter only. The empirical fit of (linear) dependence is then increased, by allowing for dependent Brownian motions.     

Combining the least correlation design, wavelet packet transform and correlation coefficient test to reduce the size of calibration set for NIR quantitative analysis in multi-component systems

The paper focuses on solving a common and important problem of NIR quantitative analysis in multi-component systems: how to significantly reduce the size of the calibration set while not impairing the predictive precision. To cope with the problem orthogonal discrete wavelet packet transform (WPT), the least correlation design and correlation coefficient test (r-test) have been combined together. As three examples, a two-component carbon tetrachloride system with 21 calibration samples, a two-component aqueous system with 21 calibration samples, and a two-component aqueous system with 41 calibration samples have been treated with the proposed strategy, respectively. In comparison with some previous methods based on much more calibration samples, the results out of the strategy showed that the predictive ability was not obviously decreased for the first system while being clearly strengthened for the second one, and the predictive precision out of the third one was even satisfactory enough for most cases of quantitative analysis. In addition, all important factors and parameters related to our strategy are discussed in detail.     

Chemometrical Interpretation of Lake Waters after their Chemical Analysis by Using Aas,Flame Photometry and Titrimetric Techniques

This article reports the distribution of trace metals and major ions in lake waters of Sultansazl L < L , a reedfield near Kayseri, Turkey. The determination of elements such as Cr, Ni, Cu, Cd and Pb in lake water samples was performed by AAS after a preconcentration step using a column packed with Amberlite XAD-16 resin. Both flame- and graphite furnace-atomic absorption spectrometry was used to determine these metals in lake water samples. The concentrations of the metals bound to humic substances, and free metal ions were determined after their sorption-elution on the resin, separately. The column method optimized with sodium tetraborate reagent was used in determining the free metal ions. In the determinations of Ca²⁺ and Mg²⁺, K⁺ , and SO2-4 , Cl ^m and total hardness, flame atomic absorption spectrometry, flame photometry, and titrimetry were used, respectively. In order to evaluate the analytical data by multivariate statistical techniques which enable feature reduction and grouping of the pollutant sources in lake waters from their chemical composition, principal component analysis (PCA), cluster analysis (CA) and correlation analysis were used. As a consequence of multivariate statistical evaluation, main anthropogenic sources like traffic, industry and agricultural processes were drawn to be responsible from the pollution in the environment investigated.     

函数连接型网络应用于维生素B族四组分同时测定

报道了函数连接型网络（ＦＬＮ）用于维生素Ｂ族四组分同时测定。采用相关系数和标准偏差从原始紫外光谱数据中挑选１１个波长点供网络处理。在函数连接型网络中，非线性输入模式得到了增强，并使用了推广的δ学习规则。预测结果极行，其相关系数和标准偏差分别为０．９９９０４和０．２６８８５。