Variance chemometrics

Davidian, M. and Haaland, P.D. Regression and calibration with nonconstant error variance. Chemometrics and Intelligent Laboratory Systems 1990 9 231-248. 

PLS has been introduced in the chemometrics literature as an algorithm with the claim that it finds simultaneously important and related components of X and of Y. Hence the alternative explanation of the acronym PLS Projection to Latent Structure. The PLS factors can loosely be seen as modified principal components. The deviation from the PCA factors is needed to improve the correlation at the cost of some decrease in the variance of the factors. The PLS algorithm effectively mixes two PCA computations, one for X and one for Y, using the NIPALS algorithm. It is assumed that X and Y have been column-centred as usual. The basic NIPALS algorithm can best be demonstrated as an easy way to calculate the singular vectors of a matrix, viz. via the simple iterative sequence (see Section 31.4.1) ... 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

In exploratory chemometric analyses, one must also be careful when interpreting modeling results from autoscaled data. For PAT problems, one can do exploratory analyses using PCA and other chemometric methods to assess the relative sensitivities of different analyzer responses to a property of interest. If antoscaling is done, however, such an assessment cannot be done, as relative sensitivity (i.e., variance) information for the variables has been removed. 

It was mentioned earlier that PCA is a useful method for compressing the information contained in a large number of x variables into a smaller number of orthogonal principal components that explain most of the variance in the x data. This particular compression method was considered to be one of the foundations of chemometrics, because many commonly used chemometric tools are also focused on explaining variance and dealing with colinearity. However, there are other compression methods that operate quite differently than PCA, and these can be useful as both compression methods and preprocessing methods. 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

Autoscaling can also be used when all of the variables have the same units and come from the same instrument. However, it can be detrimental if the total variance information is relevant to the problem being solved. For example, if one wants to do an exploratory chemometric analysis of a series of FTIR (Fourier transform infrared) spectra in order to determine the relative sensitivities of different wavenumbers (X-variables) to a property of interest, then it would be wise to avoid autoscaling and retain the total variance information because this information is relevant for assessing the sensitivities of different X-variables. 

Often, relationships between measured process parameters and desired product attributes are not directly measurable, but must rather be inferred from measurements that are made. This is the case with several spectroscopic measurements including that of octane number or polymer viscosity by NIR. When this is the case, these latent properties can be related to the spectroscopic measurement by using chemometric tools such as PLS and PCA. The property of interest can be inferred through a defined mathematical relation.39 Latent variables allow a multidimensional data set to be reduced to a data set of fewer variables which describe the majority of the variance related to the property of interest. This data compression using the most relevant data also removes the irrelevant or noisy data from the model used to measure properties. Latent variables are used to extract features from data, and can result in better accuracy of measurement and a reduced measurement time.4... 

By executing the steps of the analytical process, we can take advantage of most of the basic methods of chemometrics, e.g., statistics including analysis of variance, experimental design and optimization, regression modeling, and methods of time series analysis. 

Chemometric Investigation of Measurement Results 9.1.3.1 Univariate Analysis of Variance... 

CONTENTS 1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 

The most common way to reexpress a data set in chemometrics makes use of the principal components of the data. Here, the data are expressed in terms of the components of the variance-covariance matrix of the data. To get a variance-covariance matrix, we need not one spectrum, as shown in Figure 10.1, but a set of similar spectra for which the same underlying effects are operative. That is, we must have the same true signal and the same noise effects. Neither the exact contributions of signal nor the noise need be identical from spectrum to spectrum, but the same basic effects should be present over the set of data if we are to use variance-covariance matrices to discern how to retain signal and attenuate noise. 

Note diat if the scores are mean centred, the denominator equals the variance. Some authors us the expression in the denominator of this equation to denote an eigenvalue, so in certain articles it is stated that the scores of each PC are divided by their eigenvalue. As is usual in chemometrics, it is important to recognise diat there are many different schools of thought and incompatible definitions. 

In some areas of chemometrics we used a variance-covariance matrix. This is a square matrix, whose dimensions usually equal the number of variables in a dataset, for example, if there are 20 variables the matrix has dimensions 20 x 20. The diagonal elements equal the variance of each variable and the off-diagonal elements the covariances. This matrix is symmetric about the diagonal. It is usual to employ population rather than sample statistics for this calculation. 

Chemometrics in Spectroscopy Variances for uniformly distributed noise... 

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