Multivariate statistics definition

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

Over the last 30 years, chemometrics has evolved into an interdisciplinary subdiscipline of chemistry that combines mathematical modeling, multivariate statistics, and chemical measurements. There have been numerous definitions of the... 

Chemometrics is often defined as the application of statistics and mathematics to the analysis of chemical data. Without arguing the sufficiency of this definition, it is safe to say that the application of multivariate statistical and mathematical spectral analysis methods to near-infrared (NIR) data provides an intriguing set of advantages absent in univariate analysis of NIR data. Foremost of these advantages are the abilities to preprocess NIR spectra for removal of complex background signals, perform multianalyte calibration and calibration in the presence of multiple changing chemical... 

Analysis precision, n - a statistical measure of the expected repeatability of results for an unchanging sample, produced by an analytical method or instrument for samples whose spectra represent an interpolation of a multivariate calibration. The reader is cautioned to refer to specific definitions for precision and repeatability based on the context of use. 

Some basic concepts and definitions of statistics, chemometrics, algebra, graph theory, similarity/diversity, which are fundamental tools in the development and application of molecular descriptors, are also presented in the Handbook in some detail. More attention has been paid to information content, multivariate correlation, model complexity, variable selection, and parameters for model quality estimation, as these are the characteristic components of modern QSAR/QSPR modelling. 

However, by definition, these univariate methods of hypothesis testing are inappropriate for multispecies toxicity tests. As such, these methods are an attempt to understand a multivariate system by looking at one univariate projection after another, attempting to find statistically significant differences. Often the power of the statistical tests is quite low due to the few replicates and the high inherent variance of many of the biotic variables. 

In this small-sample set, the f-test does as well as the best multivariate methods. This shows that modeling the correlation structure is not necessarily an advantage if the number of samples is low, or, alternatively, that the true correlation structure has not been captured well enough from the few samples that are available to allow meaningful inference. A definite advantage of the f-test is that it has no tunable parameters and can be applied without further optimization. It should be noted that we do not need to apply multiple-testing corrections in this context since we only use the order of the absolute size of the f-statistics to construct the ROC curves, and not a specific cut-off level a. In other applications, however, this aspect should be taken into accoimt. 

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