Multivariate normal regression

In the multivariate linear regression module M42 first we normalize the matrix X WX to a correlation-type matrix by a transformation similar to (3.31) in order to somewhat decrease the numerical errors. This transformation... 

Lifshitz et al. (166) proposed a chi-square (x2) test for detecting adulteration in lemon juice which, for their data, is more sensitive to dilution than the multiple regression approach. The same group (133) used a multivariate normal test on five parameters (Brix, formol number, chloramine-T number, total sugars and chlorides) of Israeli orange and grapefruit juices. 

Two datasets are fist simulated. The first contains only normal samples, whereas there are 3 outliers in the second dataset, which are shown in Plot A and B of Figure 2, respectively. For each dataset, a percentage (70%) of samples are randomly selected to build a linear regression model of which the slope and intercept is recorded. Repeating this procedure 1000 times, we obtain 1000 values for both the slope and intercept. For both datasets, the intercept is plotted against the slope as displayed in Plot C and D, respectively. It can be observed that the joint distribution of the intercept and slope for the normal dataset appears to be multivariate normally distributed. In contrast, this distribution for the dataset with outliers looks quite different, far from a normal distribution. Specifically, the distributions of slopes for both datasets are shown in Plot E and F. These results show that the existence of outliers can greatly influence a regression model, which is reflected by the odd distributions of both slopes and intercepts. In return, a distribution of a model parameter that is far from a normal one would, most likely, indicate some abnormality in the data. 

Parametru/non-parametric techniques This first distinction can be made between techniques that take account of the information on the population distribution. Non parametric techniques such as KNN, ANN, CAIMAN and SVM make no assumption on the population distribution while parametric methods (LDA, SIMCA, UNEQ, PLS-DA) are based on the information of the distribution functions. LDA and UNEQ are based on the assumption that the population distributions are multivariate normally distributed. SIMCA is a parametric method that constructs a PCA model for each class separately and it assumes that the residuals are normally distributed. PLS-DA is also a parametric technique because the prediction of class memberships is performed by means of model that can be formulated as a regression equation of Y matrix (class membership codes) against X matrix (Gonzalez-Arjona et al., 1999). 

We have noted in Section 15.2 that data for Mahalanobis distance calculations should have a multivariate normal distribution. A coherent set of data for regression analysis will also meet this... 

The updatfng formulas. Under the normal linear regression assumptions, the least squares estimates maximize the likelihood function. This makes them the maximum likelihood estimates and their covariance matrix the eovariance matrix of the maximum likelihood estimates. Thus the posterior has the multivariate normal where the constants are found by the updating formulas "the posterior precision matrix equals the sum of the prior precision matrix plus the precision matrix of the "maximum likelihood estimates"... 

When we have n independent observations from the normal linear regression model where the observations all have the same known variance, the conjugate prior distribution for the regression coefficient vector /3 is multivariate normal(bo, Vq). The posterior distribution of /3 will be multivariate nor-mal y>i, Vi), where... 

Data have been analyzed from a multivariate point of view. In this way the cooperative effects of the different materials is studied and the characteristics of each sensor are easily compared with those of the other sensors. PLS was used as a regression method for calculating the capability of the set of sensors to discriminate between the volatile compounds. Volatile compounds were checked at different concentrations in order to evaluate the response of sensors in a wide concentration range. Nevertheless, the concentration variation tends to shadow the reaction of sensors with analytes, since the sensor response contains both qualitative (sensor analyte interaction) and quantitative (analyte concentration) information. In order to remove the quantitative information, data have been normalized using the linear normalization discussed in section 3. 

Shapiro Wilks W-test for normal data Shapiro Wilks W-test for exponential data Maximum studentlzed residual Median of deviations from sample median Andrew s rho for robust regression Classical methods of multiple comparisons Multivariate methods... 

The multivariate methods of data analysis, like discriminant analysis, factor analysis and principal component analysis, are often employed in chemometrics if the multiple regression method fails. Most popular in QSRR studies is the technique of principal component analysis (PCA). By PCA one reduces the number of variables in a data set by finding linear combinations of these variables which explain most of the variability [28]. Normally, 2-3 calculated abstract variables (principal components) condense most (but not all) of the information dispersed within the original multivariable data set. 

Multivariate techniques are inverse calibration methods. In normal least-squares methods, often called classical least-squares methods, the system response is modeled as a function of analyte concentration. In inverse methods, the concentrations are treated as functions of the responses. The latter has some advantages in that concentrations can be accurately predicted even in the presence of chemical and physical sources of interference. In classical methods, all components in the system need to be considered in the mathematical model produced (regression equation). 

To establish a correlation between the concentrations of different kinds of nucleosides in a complex metabolic system and normal or abnormal states of human bodies, computer-aided pattern recognition methods are required (15, 16). Different kinds of pattern recognition methods based on multivariate data analysis such as principal component analysis (PCA) (8), partial least squares (16), stepwise discriminant analysis, and canonical discriminant analysis (10, 11) have been reported. Linear discriminant analysis (17, 18) and cluster analysis were also investigated (19,20). Artificial neural network (ANN) is a branch of chemometrics that resolves regression or classification problems. The applications of ANN in separation science and chemistry have been reported widely (21-23). For pattern recognition analysis in clinical study, ANN was also proven to be a promising method (8). 

NIR.spectral bands are normally broad and often overlapping. There arc rarely clean spectral bands that allow simple correlation with analyte concentration. Instead. multivariate calibration techniques are used." Most commonly, partial least squares, principal compti-nents regression, and arlificial neural networks are eni-... 

Uq, ai,U2,... are determined by multivariate regression. Normally, a polynome of the 2nd degree is sufficient to describe the calibration function over a large concentration range. A segmented calibration curve can also be used when the calibration function is not linear. 

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