Univariate dispersion

VALLEY is a steady-state, univariate Gaussian plmne dispersion algoridun designed for estimating either 24-hour or aimual concentrations resulting from emissions from up to 50 (total) point and area sources. 

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

The index is essentially a distance measure, Mahalanobis" squared distance (D ), which expresses the multivariate distance between the observation point and the common mean of the reference values, taldng into account the dispersion and correlation of the variables.More interpreta-tional guidance may be obtained from this distance by expressing it as a percentile analogous to the percentile presentation of univariate observed values. Also, the index of atypicality has a multivariate counterpart. ... 

Statistical indices are fundamental numerical quantities measuring some statistical property of one or more variables. They are applied in any statistical analysis of data and hence in most of Q S AR methods as well as in some algorithms for the calculation of molecular descriptors. The most important univariate statistical indices are indices of central tendency and indices of dispersion, the former measuring the center of a distribution, the latter the dispersion of data in a distribution. Among the bivariate statistical indices, the correlation measures play a fundamental role in all the sciences. Other important statistical indices are the diversity indices, which are related to the injbrmationcontentofavariahle,the —> regressiowparameters, used for regression model analysis, and the —> classification parameters, used for classification model analysis. 

The other property of a set of univariate data that must be specified to give an adequate summary description is the dispersion, the extent of the spread (scatter) of the n values around the mean reflecting the extent of random error, i.e., the precision. The simplest parameter describing the dispersion is the range, the difference ( hnax min) between maximum and minimum values. Obviously this parameter is extremely susceptible to distortion from extreme values (possible outUers). The most commonly used measure of dispersion is the standard deviation of the data set s defined as follows ... 

This standardization approach (usually referred to as the slope/bias correction ) consists of computing predicted y-values for the standardization samples with the calibration model. These transfers are most often done between instruments using the same dispersion device, in otherwords, Fourier transform to Fourier transform, or grating to grating. The procedure is as follows. Predicted y-values are computed with the standardization spectra collected in both calibration and predicted steps. The predicted y-values obtained with spectra collected in the calibration step are then plotted against those obtained with spectra collected in the prediction step, and a univariate bias or slope/bias correction is applied to these points by ordinary least squares (OLS). For new spectra collected in the prediction step, the calibration model computes y-values and the obtained predictions are corrected by the univariate linear model, yielding standardized predictions. 

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