Univariate linear regression

On the other hand, the huge efforts made by atomic spectroscopists to resolve interferences and optimise the instrumental measurement devices to increase accuracy and precision led to a point where many of the difficulties that have to be solved nowadays cannot be described by simple univariate, linear regression methods (Chapter 1 gives an extensive review of some typical problems shown by several atomic techniques). Sometimes such problems cannot even be addressed by multivariate regression methods based on linear relationships, as is the case for the regression methods described in the previous two chapters. 

ANNs were compared with univariate linear regression in order to calibrate an XRF spectrometer to quantify Ti, V, Fe, Ni and Cu in polymetallic ores by Kierzek et al. [85]. 

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. 

The corresponding loadings for each factor, p and Qa, are simply estimated by univariate linear regression (projection) of the X and y variables on ta. Thus, PLSR consists of extracting a few linear combinations of the many X variables, ta, a = 1,2,..., A. These estimated factors ta are used as regressor for both X and y. Each X combination ta is defined so as to maximize its covariance with y. 

Univariate Linear Regression Introducing the Least Squares Concept... 

Carrying through the treatment as before yields Eqs. (2-78) as the normal equations for weighted linear univariate least-squares regression. 

More on Simple linear least squares regression (SLLSR), also known as Simple least squares regression (SLSR) or univariate least squares regression... 

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. 

Chapter three presents the basic ideas of classical univariate calibration. These constitute the standpoint from which the natural and intuitive extension of multiple linear regression (MLR) arises. Unfortunately, this generalisation is not suited to many current laboratory tasks and, therefore, the problems associated with its use are explained in some detail. Such problems justify the use of other more advanced techniques. The explanation of what the... 

Univariate and multivariate spectroscopy was applied to the analysis of spironolactone in presence of chlorthalidone [17]. Satisfactory results were obtained by partial least squares regression, with the calibration curve being linear over the range 2.92 -14.6 pg/mL. A kinetic-spectrophotometric method was described for the determination of spironolactone and canrenone in urine that also used a partial least-squares regression method [18]. After the compounds were extracted from urine, the spectra were recorded at 400 - 520 nm for 10 minutes at 30 second intervals. The relative error was less than 5%. 

Univariate calibration involves relating two single variables to each other, and is often called linear regression. It is easy to perform using most data analysis packages. 

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