Calibration data set

It is important to emphasize that MSEC is only an indication of how well the regression was able to fit the calibrations data set. It is a major blunder to... 

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.

RIA PREC.dat Two hundred thirty-eight calibration data sets were collected and analyzed for repeatability (within group CV) and plotted against the mean concentration. In a double-logarithmic plot the pattern seen in Fig. 4.6 appears. 

Factors affecting the integrity of spectroscopic data include the variations in sample chemistry, the variations in the physical condition of samples, and the variation in measurement conditions. Calibration data sets must represent several sample spaces to include compositional space, instrument space, and measurement or experimental condition space (e.g., sample handling and presentation spaces). Interpretive spectroscopy where spectra-structure correlations are understood is a key intellectual process in approaching spectroscopic measurements if one is to achieve an understanding in the X and Y relationships of these measurements. 

Treatment of a real, imperfect calibration data set revealed the full complexity and breadth of the calibration curve -detection limit problem, ranging from varying statistical weights to an uncertain model and data containing possible blunders to an artificially imposed response threshold. 

In the text which follows we shall examine in numerical detail the decision levels and detection limits for the Fenval-erate calibration data set ( set-B ) provided by D. Kurtz (17). In order to calculate said detection limits it was necessary to assign and fit models both to the variance as a function of concentration and the response (i.e., calibration curve) as a function of concentration. No simple model (2, 3 parameter) was found that was consistent with the empirical calibration curve and the replication error, so several alternative simple functions were used to illustrate the approach for calibration curve detection limits. A more appropriate treatment would require a new design including real blanks and Fenvalerate standards spanning the region from zero to a few times the detection limit. Detailed calculations are given in the Appendix and summarized in Table V. 

Raman spectroscopy s sensitivity to the local molecular enviromnent means that it can be correlated to other material properties besides concentration, such as polymorph form, particle size, or polymer crystallinity. This is a powerful advantage, but it can complicate the development and interpretation of calibration models. For example, if a model is built to predict composition, it can appear to fail if the sample particle size distribution does not match what was used in the calibration set. Some models that appear to fail in the field may actually reflect a change in some aspect of the sample that was not sufficiently varied or represented in the calibration set. It is important to identify any differences between laboratory and plant conditions and perform a series of experiments to test the impact of those factors on the spectra and thus the field robustness of any models. This applies not only to physical parameters like flow rate, turbulence, particulates, temperature, crystal size and shape, and pressure, but also to the presence and concentration of minor constituents and expected contaminants. The significance of some of these parameters may be related to the volume of material probed, so factors that are significant in a microspectroscopy mode may not be when using a WAl probe or transmission mode. Regardless, the large calibration data sets required to address these variables can be burdensome. 

The factors that influence the optimal cross-validation method, as well as the parameters for that method, are the number of calibration samples (AO, the arrangement order of the samples in the calibration data set, whether the samples arise from a design of experiments (DOE, Section 12.2.6), the presence or absence of replicate samples, and the specific objective of the cross-validation experiment. In addition, there are two traps that one needs to be aware of when setting up a cross-validation experiment. 

Figure 12.27 (A) Scatter plot of the Hotelling P and Q residual statistics associated with the samples in the process spectroscopy calibration data set, obtained from a PCA model built on the data after obvious outliers were removed. The dashed lines represent the 95% confidence limit of the respective statistic. (B) The spectra used to generate the plot in (A), denoting one of the outlier samples.

Figure 12.29 Time-series plot of the y-residuals obtained from a PLS model developed using the process spectroscopy calibration data set (solid line), after removal of sample and variable outliers as discussed earlier. The measured y-values (dashed line) are also provided for reference.

Figure 12.31 Illustration of a hybrid calibration strategy. (A) Scatter plot of first two PCA scores obtained from a process analytical calibration data set containing both synthesized standards (circles) and actual process samples (triangles). (B) Results of a PLS regression fit to the property of Interest, using all of the calibration samples represented In (A).

Step 1 Include fouled spectra in the calibration data set. 

We must not accept outliers in a calibration data set. We also can test this with an statistical outlier test. 

Tlie operating ranges of the prediction model based on the calibration data set follow. Predicting future samples from outside this operating i nge is extrapolating and ma produce unreliable results. 

Calibration The process of constructing a model that is used to predict characteristics or properties of unknown samples. The model is constructed from a calibration data set with measured multivariate responses (R) and corresponding known sample concentrations or physical characteristics of interest (O-... 

A simple and classical method is Wold s criterion [39], which resembles the well-known F-test, defined as the ratio between two successive values of PRESS (obtained by cross-validation). The optimum dimensionality is set as the number of factors for which the ratio does not exceeds unity (at that moment the residual error for a model containing A components becomes larger than that for a model with only A - 1 components). The adjusted Wold s criterion limits the upper ratio to 0.90 or 0.95 [35]. Figure 4.17 depicts how this criterion behaves when applied to the calibration data set of the working example developed to determine Sb in natural waters. This plot shows that the third pair (formed by the third and fourth factors) yields a PRESS ratio that is slightly lower than one, so probably the best number of factors to be included in the model would be three or four. 

To develop our calibration data set using an experimental design (see Chapter 2) in order to be, hopefully, reasonably sure that all the experimental domain is represented by the standard solutions to be prepared. 

Both CCA and WA require a calibration data set, as already described. For best results, these data sets should be developed carefully and with considerable forethought. The lakes chosen should include the range of limnological conditions that investigators anticipate inferring from the stratigraphic data from cored lakes. Several chemical and other environmental characteristics likely to influence the distribution of taxa should also be measured accurately on as many samples as possible to characterize temporal variability. Insufficient or inadequate chemistry data is an important source of error associated with inferred values. 

Once WA values for an environmental characteristic (e.g., water chemistry) have been calculated for taxa in a calibration data set, the information can be used to infer that characteristic from sediment core samples and consequently to reconstruct past conditions. The first step is to determine the percent abundance of each taxon in the sediment core assemblages. The taxon abundance is then multiplied by the WA value for that taxon (determined from the calibration data set). These products are summed for all taxa and are standardized by the sum of the relative abundances of the taxa in that sample to obtain an inferred value, namely... 

Another statistical issue is the relationship between the composition in the calibration set used to derive the transfer functions and the lakes for which the transfer functions will be applied. Calibration data sets should be modified if used to reconstruct chemistry of different types of lakes. A subset of calibration lakes can be selected that does not contain lakes so different that they might unduly influence optimum environmental values for a taxon (for example, saline lakes can be removed from a calibration data set to be used for generating data to infer trophic-state change in low-conductivity lakes). 

Tolerance weighting does not always work as well as expected in diatom reconstructions (17), perhaps because calibration data sets have not been large enough to provide accurate estimates of tolerance. Comparisons of tolerance-weighted versus unweighted approaches have shown that the simplest approach works best with diatoms (17, 22), but chrysophyte inference models perform best when tolerances are included (50, 51). 

In some studies the r2 and SE are calculated by using with data only from the original calibration data set. Thus, the inferred values used in the correlation analysis are for the same lakes used to develop the equations. 

Although all of these cross-validation methods can be used effectively, there could be an optimal method for a given application. The factors that most often influence the optimal cross-validation method are the design of the calibration experiment, the order of the samples in the calibration data set, and the total number of calibration samples (N). 

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