Data analysis least-squares

A eomputer program, PROG2, was developed to fit the data by least squares of a polynomial regression analysis. The data of temperature (independent variable) versus heat eapaeity (dependent variable) were inputted in the program for an equation to an nth degree... 

Figure 2. NPT study of ezFTC in PMMA (9% by weight). The crossing point for the two fit functions defines die simulated glass transition temperature of the composite. The two fit lines are linear fils to the data using least squares analysis...

Instead of calculating the solubility at each of the test points and comparing it with the experimental value we fit the coefficients of the reduced cubic model to the data by least squares multi-linear regression, and investigate the goodness of fit by analysis of variance. The resulting equation is ... 

Using the spectrophotometer, the absorbance of the samples at 617 nm was determined versus distilled water. The ahsorhance at 750 nm also was determined and subtracted from the 617 nm absorbance to yield a corrected absorbance. A first-order rate constant for the corrected absorbance versus time then was calculated using an HP4IC calculator programmed to fit the data by least squares analysis. 

The second example represents a large-scale human metabolomics study that was performed with LC/MS [54]. The aim of this study was to identify potential biomarkers from lipid profiles of some 600 human plasma samples. Lipids were extracted from plasma samples and subjected to LC/ESI-MS analysis. Several different classes of lipids, such as phosphatidylcholines, lysophosphatidyl-cholines, triglycerides, diglycerides, sphingomyelins, and cholesterol esters were the target of this study. To detect small differences in metabolic profiles, statistical methods were used to process this large set of data. Partial least-squares discriminant analysis of the data could locate potential biomarkers. 

At all but the simplest level, treatment of the results from a time-domain experiment involves some mathematical procedure such as non-linear least squares analysis. Least squares analysis is generally carried out by some modification of the Newton-Raphson method, that proposed by Marquardt currently being popular [21, 22]. There is a fundamental difficulty in that the normal equations that must be solved as part of the procedure are often ill-conditioned. This means that rather than having a single well-defined solution, there is a group of solutions all of which are equally valid. This is particularly troublesome where there are exponential components whose time constants differ by less than a factor of about three. It is easy to demonstrate that the behaviour is multi-exponential, but much more difficult to extract reliable parameters. The fitting procedure is also dependent on the model used and it is often quite difficult to determine the number of exponentials needed to adequately represent the data. Various procedures have been suggested to overcome these difficulties, but none has yet received wide acceptance in solid-state NMR [23-26]. 

Data processing by a computer may involve relatively simple mathematical operations such as calculation of concentrations, data averaging, least-squares analysis, statistical analysis, and integration to obtain peak areas. More contplex calculations may involve the... 

The viscosity of a 5,000-ppm solution of partially hydrolyzed polyacrylamide was measured at several shear rates with a cone-and-plate rheometer. Table 5.59 gives experimental data as a function of shear rate. Plot the viscosity vs. shear rate on log-log paper and determine the power-law exponent, n, and the power-law constant, K, from the experimental data using least-squares analysis and correlate the viscosity as a function of shear rate. Compare values of the viscosity from the correlation with experimental data. 

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

The ability of partial least squares to cope with data sets containing very many x values is considered by its proponents to make it particularly suited to modern-day problems, where it is very easy to compute an extremely large number of descriptors for each compound (as in CoMFA). This contrasts with the traditional situation in QSAR, where it could be time-consuming to measure the required properties or where the analysis was restricted to traditional substituent constants. 

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

Values for fQi and K 2 for acids of the form H2A are determined from a least-squares analysis of data from a potentiometric titration. 

The pyrimidine ring is virtually flat. Its corrected bond lengths, as determined by a least-squares analysis of the crystal structure data for a unit cell of four molecules, are shown in formula (2) (60AX80), and the bond angles derived from these data show good agreement with those (3) derived by other means (63JCS5893) for comparison, each bond... 

The weighted least-squares analysis is important for estimating parameter involving exponents. Examples are the eoneentration time data... 

It can be argued that the main advantage of least-squares analysis is not that it provides the best fit to the data, but rather that it provides estimates of the uncertainties of the parameters. Here we sketch the basis of the method by which variances of the parameters are obtained. This is an abbreviated treatment following Bennett and Franklin.We use the normal equations (2-73) as an example. Equation (2-73a) is solved for <2o-... 

Table 6-1. Data for Weighted Linear Least-Squares Arrhenius Analysis...

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