Multivariate linear regression

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

BI- AND MULTIVARIATE DATA Table 2.1. Linear Regression Equations... 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

Step 4. The RRR model coefficients are then found by a multivariate linear regression of the RRR fit, Y = ( a] + In y ) original X, which should have a... 

L. Breiman and J.H. Friedman, Predicting multivariate responses in multiple linear regression. J. Roy. Stat. Soc. B59 (1997) 3-37. 

The offset a, and the multiplication constant bj are estimated by simple linear regression of the ith individual spectrum on the reference spectrum z. For the latter one may take the average of all spectra. The deviation e, from this fit carries the unique information. This deviation, after division by the multiplication constant, is used in the subsequent multivariate calibration. For the above correction it is not mandatory to use the entire spectral region. In fact, it is better to compute the offset and the slope from those parts of the wavelength range that contain no relevant chemical information. However, this requires spectroscopic knowledge that is not always available. 

Multiple linear regression (MLR) is a classic mathematical multivariate regression analysis technique [39] that has been applied to quantitative structure-property relationship (QSPR) modeling. However, when using MLR there are some aspects, with respect to statistical issues, that the researcher must be aware of ... 

Nonlinearity is a subject the specifics of which are not prolifically or extensively discussed as a specific topic in the multivariate calibration literature, to say the least. Textbooks routinely cover the issues of multiple linear regression and nonlinearity, but do not cover the issue with full-spectrum methods such as PCR and PLS. Some discussion does exist relative to multiple linear regression, for example in Chemometrics A Textbook by D.L. Massart et al. [6], see Section 2.1, Linear Regression (pp. 167-175) and Section 2.2, Non-linear Regression, (pp. 175-181). The authors state,... 

Multivariate curve resolution, 6 54—56 Multivariate linear regression, 6 32—35 Multivariate optical elements (MOE), 6 68 Multiwalled carbon nanotubes (MWCNTs), 77 48, 49 22 720 26 737. See also Carbon nanotubes (CNTs) Multiwall nanotubes (MWNTs) synthesis of, 26 806 Multiwall fullerenes, 12 231 Multiwall nanotubes (MWNTs), 12 232 Multiwall paper bags, 78 11 Multiway analysis, 6 57-63 Multiyear profitability analysis, 9 535-537 Multiyear venture analysis, 0 537-544 sample, 9 542-S44 Mummification, 5 749 Mumps vaccine, 25 490 491 Mumps virus, 3 137 Municipal biosolids, as biomass, 3 684 Municipal distribution, potential for saline water use in, 26 55-56 Municipal effluents, disposal of, 26 54 Municipal landfill leachate, chemicals found in, 25 876t... 

Opfermann, J., Kinetic analysis using multivariate non-linear regression. 

A calc=C p Y p % component spectra via multivariate linear regression... 

While multiple linear regression aims at relating a single y-variable with several x-variables, multivariate linear regression relates several y-variables with several x-variables. Having available n observations for a number q of y-variables... 

Using multivariable linear regression, a set of equations can be derived from the parameterized data. Statistical analysis yields the "best equations to fit the en irical data. This mathematical model forms a basis to correlate the biologicsd activity to the chemical structures. 

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