Statistical regression techniques

The lack of a correlation was also appreciated in a follow-up study in which the data presented in the Dutch report20 (particulate concentration, water content, carbon monoxide concentration, furnace temperature, etc.) were compared with the PCDD/F emissions using statistical regression techniques.21 The aim was to determine which operating and emission parameters were strongly related to PCDD/F emissions. The data set was divided into two groups on the basis of the type of air pollution control device installed ... 

Hence, a plot of ln(C) versus t will give a straight line with a slope of — k and an intercept of ln(Co). Because each number will have some measurement error, you will need to use statistical regression techniques to get the best values of Co and k. The technique is simple Just... 

In recent years the merging between the statistical and AI points of view on the same problem has benefited both approaches. Statistical regression techniques have been enriched by the addition of new methods... 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

A first evaluation of the data can be done by running nonparametric statistical estimation techniques like, for example, the Nadaraya-Watson kernel regression estimate [2]. These techniques have the advantage of being relatively cost-free in terms of assumptions, but they do not provide any possibility of interpreting the outcome and are not at all reliable when extrapolating. The fact that these techniques do not require a lot of assumptions makes them... 

In the calibration problem two related quantities, X and Y, are investigated where Y, the response variable, is relatively easy to measure while X, the amount or concentration variable, is relatively difficult to measure in terms of cost or effort Furthermore, the measurement error for X is small compared with that of Y The experimenter observes a calibration set of N pairs of values (x, y ), i l,...,N, of the quantities X and Y, x being the known standard amount or concentration values and y the chromatographic response from the known standard The calibration graph is determined from this set of calibration samples using regression techniques Additional values of the dependent variable Y, say y., j l,, M, where M is arbitrary, are also observed whose corresponding X values, x. are the unknown quantities of interest The statistical literature on the calibration problem considers the estimation of these unknown values, x, from the observed and the... 

Most laboratories now have access to powerful computers and an extensive array of commercially available data analysis software (e.g., Prism (GraphPad, San Diego, CA), Sigma Plot (San Rafael, CA)). This provides ready access to the use of nonlinear regression techniques for the direct analysis of binding data, together with appropriate statistical analyses. However, there remains a valuable place for the manual methods, which involve linearisation, particularly in the undergraduate arena and these have been rehearsed in the above text. 

A short-term stability study is conducted under stressed conditions to increase the rate of chemical and physical degradation of the drug product. The classical statistical procedure to establish a preliminary shelf life is based on regression techniques, which are used to estimate the parameters of the Arrhenius equation. 

Use of multiple regression techniques in the study of functional properties of food proteins is not new I76) Most food scientists have some familiarity with basic statistical concepts and some access to competent statistical advice. At least one good basic text on statistical modelling for biological scientists exists (7 ). A number of more advanced texts covering use of regression in modelling are available (, ). ... 

The objectives of this paper are to present some potential uses of regression techniques in food protein research, to discuss some desirable steps in the modelling process, to present an example of the rationale underlying development of a model, and to discuss some potential statistical problems which might arise. 

In organic chemistry, decomposition of molecules into substituents and molecular frameworks is a natural way to characterize molecular structures. In QSAR, both the Hansch-Fujita " and the Free-Wilson classical approaches are based on this decomposition, but only the second one explicitly accounts for the presence or the absence of substituent(s) attached to molecular framework at a certain position. While the multiple linear regression technique was associated with the Free-Wilson method, recent modifications of this approach involve more sophisticated statistical and machine-learning approaches, such as the principal component analysis and neural networks. ... 

In most cases, the MFTA models are built using the Partial Least Squares Regression (PLSR) technique that is suitable for the stable modeling based on the excessive and/or correlated descriptors (under-defined data sets). However, the MFTA approach is not limited to the PLSR models and can successfully employ other statistical learning techniques such as the Artificial Neural Networks (ANN) supporting the detection of the nonlinear structure-activity relationships. ... 

Eq. (6.12) describes a non-linear model. The term non-linear is used here, and only in this section, in a statistical sense. In that sense an equation such as Eq. (6.2) is a linear regression model although it describes a quadratic, and therefore curved, relationship. It is however linear in the h parameters and standard linear regression techniques can be applied to obtain them. In Eq. (6.12) the parameters are part of the exponent and transformations such as the log transform cannot help that. The regression model is then called non-linear. Non-linear regression is less evident than linear regression. Software is available but it turns out that non-linear regression can lead to unstable numerical results. How to avoid this is described in Ref. [69]. 

Here Kj is the adsorption equilibrium constant of species j. which can be a function of potential. In this case estimation of the best fit of kinetic parameters Qj, m, a, k°, Kj, Ef, requires the use of nonlinear regression techniques (84a). Although experimental data can fit an equation similar to Eq. (16), mechanistic deductions from such information alone should be restrained. It can be shown that more than one mechanism can be devised, the rate expressions of which cannot be statistically discriminated (84a). 

Corrosion rate was evaluated with respect to, 1) flux of pollutants (sulfur oxides, nitrogen oxides, oxidants, and particles) to the steel during both wet and dry periods, 2) temperature, and 3) exposure history. Different definitions of when the steel was wet were evaluated to determine the most likely "critical relative humidity." Non-linear multiple regression techniques were used to determine the statistical significance of each factor and develop a theoretically consistent environmental damage function. 

The ultimate development in the field of sample preparation is to eliminate it completely, that is, to make a chemical measurement directly without any sample pretreatment. This has been achieved with the application of chemometric near-infrared methods to direct analysis of pharmaceutical tablets and other pharmaceutical solids (74-77). Chemometrics is the use of mathematical and statistical correlation techniques to process instrumental data. Using these techniques, relatively raw analytical data can be converted to specific quantitative information. These methods have been most often used to treat near-infrared (NIR) data, but they can be applied to any instrumental measurement. Multiple linear regression or principal-component analysis is applied to direct absorbance spectra or to the mathematical derivatives of the spectra to define a calibration curve. These methods are considered secondary methods and must be calibrated using data from a primary method such as HPLC, and the calibration material must be manufactured using an equivalent process to the subject test material. However, once the calibration is done, it does not need to be repeated before each analysis. 

Quantitative structure-activity relationships QSAR. The QSAR approach pioneered by Hansch and co-workers relates biological data of congeneric structures to physical properties such as hydrophobicity, electronic, and steric effects using linear regression techniques to estimate the relative importance of each of those effects contributing to the biological effect. The molecular descriptors used can be 1-D or 3-D (3D-QSAR). A statistically sound QSAR regression equation can be used for lead optimization. 

The availability of computer packages of classification techniques has led to the waste of more valuable scientific time than any other statistical innovation (with the possible exception of multiple-regression techniques) . 

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