Univariate linear model

Analytical chemists always face a problem in comparison of the performance between analytical instruments. There is no simple rule to justify which one is better because of the variations between the instrumental responses. In order to correct this, a standardization approach is generally adopted. However, a calibration model as developed on an instrument cannot be employed for the other instrument in the real situation. Walczak et al. [28] suggested a new standardization method for comparing the performance between two near-infrared (NIR) spectrometers in the wavelet domain. In their proposed method, the NIR spectra from two different spectrometers were transformed to the wavelet domain at resolution level (J — 1). Suppose and correspond to the NIR spectra from Instruments 1 and 2, respectively, in the wavelet domain. A univariate linear model is applied to determine the transfer parameters t between and... 

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. 

Univariate and Multivariate General Linear Models Theory and Applications Using SAS Software by Neil H.Timm andTammy A. Mieczkowski... 

Traditionally, the determination of a difference in costs between groups has been made using the Student s r-test or analysis of variance (ANOVA) (univariate analysis) and ordinary least-squares regression (multivariable analysis). The recent proposal of the generalized linear model promises to improve the predictive power of multivariable analyses. 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

When simple univariate or multivariate linear models are inadequate, higher-order models can be pursued. For example, in the case of only one instrument response (wavelength), Equation 5.8... 

It is well known that the accuracy of CE determinations using univariate calibration models, such as linear regression, relies on the selectivity of the electrophoretic data. Peaks of analytes must be baseline resolved and the occurrence of comigrations and minor impurities should be avoided. Note that peak contaminations lead to wrong integrations, and, consequently, the concentrations estimated from these data may be unreliable. 

In its simplest form, a direct, univariate cahbration method proceeds by assuming there is a mathematical model that relates analytical instmment response to concentration. Traditionally in analytical chemistry, the model is assumed to be a linear relation between response r and concentration c ... 

We now consider the case in which, again, the independent variable jc, is considered to be accurately known, but now we suppose that the variances in the dependent variable y, are not constant, but may vary (either randomly or continuously) with JC . To show the basis of the method we use the simple linear univariate model, written as Eq. (2-76). 

In this way the child spectrum is transformed into a spectrum as if measured on the parent instrument. In a more refined implementation one establishes the highest correlating wavelength channel through quadratic interpolation and, subsequently, the corresponding intensity at this non-observed channel through linear interpolation. In this way a complete spectrum measured on the child instrument can be transformed into an estimate of the spectrum as if it were measured on the parent instrument. The calibration model developed for the parent instrument may be applied without further ado to this spectram. The drawback of this approach is that it is essentially univariate. It cannot deal with complex differences between dissimilar instruments. 

Among nonlocal methods, those based on linear projection are the most widely used for data interpretation. Owing to their limited modeling ability, linear univariate and multivariate methods are used mainly to extract the most relevant features and reduce data dimensionality. Nonlinear methods often are used to directly map the numerical inputs to the symbolic outputs, but require careful attention to avoid arbitrary extrapolation because of their global nature. 

Again, however, if there is perfect linearity in the relationship of the absorbance at both wavelengths with respect to the concentrations of the components, one should equally well be able to extrapolate the model beyond the range of either or both components in the calibration set, just as in the univariate case. 

Depending on the data structure, different types of models are possible to be applied for data analysis. Thus, when data are ordered in one direction, linear univariant models can be applied (see (1)), and nonlinear models as well (see (2)). For data ordered in two directions, bilinear models can be applied (see (3)) or nonbilinear models. Finally, for data ordered in three directions, trilinear models can be applied (see (4)) or, failing that, nontrilinear models. 

The most common, straightforward multivariate calibration model is the natural extent of the univariate calibration, the linear equation for which is... 

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