Calibration univariate

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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The IR spectra are finally analyzed to determine the effluent concentration from each reactor channel. The quantification of species concentration is performed using either univariate or multivariate calibration methods. For non-overlapping peaks, like CO, C02, and N20, we can use univariate calibration. This is simply performed by baseline correction, the peak areas and/or peak heights and then converting these values... 

As for testing other characteristics of a univariate calibration, there are also ways to test for statistical significance of the slope, to see whether unity slope adequately describes the relationship between test results and analyte concentration. These are described in the book Principles and Practice of Spectroscopic Calibration [10]. The Statistics are described there, and are called the Data Significance t test and the Slope Significance t test (or DST and SST tests ). Unless the DST is statistically significant, the SST is meaningless, though. 

To evaluate the quantitation capabilities of the experimental design, univariate calibration curves were constructed at 770 nm for four tested vapors as shown in Fig. 4.13. Upon exposure to the highest tested concentration of DCM and toluene... 

Fig. 4.13 Univariate calibration curves constructed at 770 nm for four tested vapors over 0 0.1 PI P0 concentration range. Reprinted from Ref. 15 with permission. 2008 Institute of Electrical and Electronics Engineers...

An alternative to univariate calibration is to use multivariate techniques to sense when a steady state has been reached in a chemical reaction. This approach has been successfully apphed to the detection of reaction end points [82]. A very similar technique can be used to establish deviation from steady state in a continuous process reactor. 

In many applications, one response from an instrument is related to the concentration of a single chemical component. This is referred to as univariate calibration because only one instrument response is used per sample. Multivariate calibration is the process of relating multiple responses from an instrument to a property or properties of a sample. The samples could be, for example, a mixture of chemical components in a process stream, and the goal is to predict the concentration levels of the different chemical components in the stream from infrared measurements. The methods are quite powerful, but as Dr. Einstein noted, the application of mathematics to reality is not without its limitations. It is, therefore, the obligation of the analyst to use them in a responsible manner. 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

Quantitative analysis in ICP-MS is typically achieved by several univariate calibration strategies external calibration, standard addition calibration or internal standardisation. Nevertheless multivariate calibration has also been applied, as will be presented in Chapters 3 and 4. 

Figure 3.1 Univariate calibration. Direct model (left) and inverse model (right). The arrow indicates the direction in which the prediction is made.

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

For all the mentioned reasons, there is an ongoing tendency in spectroscopic studies to manipulate samples less and perform fewer experiments but to obtain more data in each of them and use more sophisticated mathematical techniques than simple univariate calibration. Hence multivariate calibration methods are being increasingly used in laboratories where instruments providing multivariate responses are of general use. Sometimes, these models may give less precise or less accurate results than those given by the traditional method of (univariate) analysis, but they are much quicker and cheaper than classical approaches. 

The main advantage of multivariate calibration based on CLS with respect to univariate calibration is that CLS does not require selective measurements. Selectivity is obtained mathematically by solving a system of equations, without the requirement for chemical or instrumental separations that are so often needed in univariate calibration. In addition, the model can use a large number of sensors to obtain a signal-averaging effect [4], which is beneficial for the precision of the predicted concentration, making it less susceptible to the noise in the data. Finally, for the case of spectroscopic data, the Lambert Bouguer Beer s law provides a sound foundation for the predictive model. 

The quality of a model depends on the quality of the samples used to calculate it (or, to say it using the univariate approach, the quality of any traditional univariate calibration cannot be better than the quality of the standards employed to measure the analyte). Although this statement is trivial, the discussion on how many samples and which samples are required to develop a good predictive model is still open, so only general comments will be given. Below, we consider that the quality of the measurement device fits the purpose of the analytical problem. 

As in traditional methods that use univariate calibrations, the description of a method of analysis that uses multivariate calibration must also include the corresponding estimated figures of merit, including accuracy (trueness and precision), selectivity, sensitivity, linearity, limit of detection (LOD), limit of quantification (LOQ) and robustness. In this chapter, only the most common figures of merit are described. For a more extensive review, see [55]. Also, for a practical calculation of figures of merit in an atomic spectroscopic application, see [12]. 

As in univariate calibration, prediction intervals (Pis) can be constructed from the above estimated standard error of prediction, by means of a Student s /-statistic, as ... 

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... 

Calibration is the process by which a mathematical model relating the response of the analytical instrument (a spectrophotometer in this case) to specific quantities of the samples is constructed. This can be done by using algorithms (usually based on least squares regression) capable of establishing an appropriate mathematical relation such as single absorbance vs. concentration (univariate calibration) or spectra vs. concentration (multivariate calibration). 

Multivariate calibrations have become a commonly applied tool in the field of modern analytical chemistry and, specifically, in quantitative IR analysis [13,14]. PLS regression is one of several methods that utilize an entire spectral information band present in IR data, often referred to as full-spectrum calibrations. The advantages of full-spectrum calibrations, such as PLS and CLS, are improvements in precision and robustness over univariate calibrations owing to increased signal averaging from including more spectral intensities. The distinction between PLS and CLS manifests in the fact that PLS is a factor-based regression, which means the full spectra for the acquired... 

Classical calibration. There is a huge literature on univariate calibration.19 23 One of the simplest problems is to determine the concentration of a single compound using the... 

However, as in univariate calibration, the coefficients obtained using both approaches may not be exactly equal, both methods making different assumptions about error distributions. 

Univariate calibration is specific to situations where the instrument response depends only on the target analyte concentration. With multivariate calibration, model parameters can be estimated where responses depend on the target analyte in addition to other chemical or physical variables and, hence, multivariate calibration corrects for these interfering effects. For the ith calibration sample, the model with a nonzero intercept can be written as... 

Water Results from Univariate Calibration of the Water-Methanol Mixture... 

For univariate calibration, the International Union of Pure and Applied Chemistry (IUPAC) defines sensitivity as the slope of the calibration curve when the instrument response is the dependent variable, i.e., y in Equation 5.4, and the independent variable is concentration. This is also known as the calibration sensitivity, contrasted with the analytical sensitivity, which is the calibration sensitivity divided by the standard deviation of an instrumental response at a specified concentration [18], Changing concentration to act as the dependent variable, as in Equation 5.4, shows that the slope of this calibration curve, f, is related to the inverse of the calibration sensitivity. In either case, confidence intervals for concentration estimates are linked to sensitivity [1, 19-22],... 

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