Performing least squares regression

Metamodels, such as (6), become useful when appropriate values are assigned to the unknown parameters. Experimental design—choosing the levels of the controllable factors at which simulation experiments will be performed—is needed to ensure good parameter estimates see Chapter 85 of the Handbook for guidance. Once the experiments have been performed, least-squares regression is a standard procedure for estimating the parameters see Chapter 87 of the Handbook. 

This technique is invasive however, the particle can be designed to be neutrally buoyant so that it well represents the flow of the phase of interest. An array of detectors is positioned around the reactor vessel. Calibration must be performed by positioning the particle in the vessel at a number of known locations and recording each of the detector counts. During actual measurements, the y-ray emissions from the particle are monitored over many hours as it moves freely in the system maintained at steady state. Least-squares regression methods can be applied to evaluate the temporal position of the particle and thus velocity field [13, 14]. This technique offers modest spatial resolutions of 2-5 mm and sampling frequencies up to 25 Hz. 

Straightedge, performing a conventional y-on-x least-squares regression, and performing a least-squares regression where each of the data points is weighted according to its analytical errors. 

Regression can be performed directly with the values of the variables (ordinary least-squares regression, OLS) but in the most powerful methods, such as principal component regression (PCR) and partial least-squares regression (PLS), it is done via a small set of intermediate linear latent variables (the components). This approach has important advantages ... 

Haaland et al. [91] developed a so-called multi-window classical least-squares method for ICP-OES measurements [charge-couple device (CCD) detector arrays]. Essentially, it consisted in performing a classical least-squares regression in each of the spectral windows which were measured and combining the concentration predictions (for a given analyte). The methodology was compared with PLS and it proved superior and capable of handling interferences from several concomitants. 

Cd2+ and the Pb2+ and all electrodes display the two peaks but to different extents. Despite the peak overlap, the electrode array can be calibrated for each metal ion using a three-way partial least squares regression (AT-PLS) [53]. The electrode array was employed to analyse three test samples of known concentration of Cu2+, Cd2+ and Pb2+ and the concentrations of each analyte predicted by the calibrated electrode array are shown in Table 10.1. As can be seen from Table 10.1 there is reasonable agreement between the actual and predicted values despite the fact that all electrodes respond to all analytes and that the electrochemical responses to lead and cadmium overlap. Further improvements would be expected if the calibrations were performed with a box experimental design, which encompassed the linear range of all the sensors. 

Figure 5 Least-square regression lines fit for the linear relationship between mean and standard deviation of PVT reaction times (msec). Data are from n = 13 subjects undergoing 88 hr (3.67 days) of total sleep deprivation. This figure illustrates that while all subjects experienced a decline in neurobehavioral performance on the PVT, as illustrated by increased reaction times when responding to the visual stimuli, there is a significant degree of interindividual variability in the magnitude of neurobehavioral impairment, evident by the differing lengths of the lines fit to the data. (From Ref. 44.)...

The elimination of outliers is possible in an additional step. For this purpose the reweighted least squares regression (RLS regression) is performed, the weighting factor Wi being [ROUSSEEUW and LEROY, 1987] ... 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

Standard statistical packages for computing models by least-squares regression typically perform an analysis of variance (ANOVA) based upon the relationship shown in Equation 5.15 and report these results in a table. An example of a table is shown in Table 5.3 for the water model computed by least squares at 1932 nm. 

Partial Least Squares regression (PLS) is usually performed on a - data matrix to search for a correlation between the thousands of CoMFA descriptors and biological response. However, usually after - variable selection, the PLS model is transformed into and presented as a multiple regression equation to allow comparison with classical QSAR models. 

Due to the large number of descriptors (commonly 15,000 - 20,000 for each field), the multivariate regression analysis is usually performed by partial least squares regression (PLS), with or without - variable selection. Moreover, a similarity matrix can be calculated from distance functions based on interaction fields between pairs of molecules. 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

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