Linear regression calibration

C.I Linear Regression of Straight-Line Calibration Curves... 

A linear regression analysis should not be accepted without evaluating the validity of the model on which the calculations were based. Perhaps the simplest way to evaluate a regression analysis is to calculate and plot the residual error for each value of x. The residual error for a single calibration standard, r , is given as... 

Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regression analysis can provide an equation for the standardization. 

Construct an appropriate standard additions calibration curve, and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval. 

Linear regression of the calibration standards gives the relationship between absorbance and concentration as... 

A standard addition calibration curve of emission versus the concentration of added sodium gives, by linear regression, an equation of... 

Linear regression gives the equation for the calibration curve as Ecell = 0.027 -h 0.0303 log [Ca +]... 

Once a linear relationship has been shown to have a high probability by the value of the correlation coefficient (r), then the best straight line through the data points has to be estimated. This can often be done by visual inspection of the calibration graph but in many cases it is far better practice to evaluate the best straight line by linear regression (the method of least squares). 

We will explore the two major families of chemometric quantitative calibration techniques that are most commonly employed the Multiple Linear Regression (MLR) techniques, and the Factor-Based Techniques. Within each family, we will review the various methods commonly employed, learn how to develop and test calibrations, and how to use the calibrations to estimate, or predict, the properties of unknown samples. We will consider the advantages and limitations of each method as well as some of the tricks and pitfalls associated with their use. While our emphasis will be on quantitative analysis, we will also touch on how these techniques are used for qualitative analysis, classification, and discriminative analysis. 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

We calculate the calibration (regression) coefficients on a rank-by-rank basis using linear regression between the projections of the spectra on each individual spectral factor with the projections of the concentrations on each corresponding concentration factor of the same rank. 

Linear regression with emphasis on the use as a calibration/interpolation tool. 

One-dimensional data are plotted versus an experimental variable a prime example is the Lambert-Beer plot of absorbance vs. concentration, as in a calibration run. The graph is expected to be a straight line over an appreciable range of the experimental variable. This is the classical domain of linear regression analysis. 

If the calibration is repeated and a number of linear regression slopes b are available, these can be compared as are means. (See Section 1.5.1, but also Section 2.2.4.)... 

Figure 2.9. The confidence interval for an individual result CI( 3 ) and that of the regression line s CLj A are compared (schematic, left). The information can be combined as per Eq. (2.25), which yields curves B (and S, not shown). In the right panel curves A and B are depicted relative to the linear regression line. If e > 0 or d > 0, the probability of the point belonging to the population of the calibration measurements is smaller than alpha cf. Section 1.5.5. The distance e is the difference between a measurement y (error bars indicate 95% CL) and the appropriate tolerance limit B this is easy to calculate because the error is calculated using the calibration data set. The distance d is used for the same purpose, but the calculation is more difficult because both a CL(regression line) A and an estimate for the CL( y) have to be provided.

The algebraic solution is the classical fitting technique, as exemplified by the linear regression (Chapter 2). The advantage lies in the clear formulation of the numerical algorithm to be used and in the uniqueness of the solution. If one is free to choose the calibration concentrations and the number of... 

Calibration Each of the solutions is injected once and a linear regression is calculated for the five equidistant points, yielding, for example, Y = -0.00064 + 1.004 X, = 0.9999. Under the assumption that the software did not truncate the result, an r of this size implies a residual standard deviation of better than 0.0001 (-0.5% CV in the middle of the LO range use program SIMCAL to confirm this statement ) the calibration results are not shown in Fig. 4.39. 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

Often, it is not quite feasible to control the calibration variables at will. When the process under study is complex, e.g. a sewage system, it is impossible to produce realistic samples that are representative of the process and at the same time optimally designed for calibration. Often, one may at best collect representative samples from the population of interest and measure both the dependent properties Y and the predictor variables X. In that case, both Y and X are random, and one may just as well model the concentrations X, given the observed Y. This case of natural calibration (also known as random calibration) is compatible with the linear regression model... 

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