Model calibration

This calibration model for the best-fit fit line requires that the line pass through the centroid of the points (X, Y). It can be shown that ... 

A solvent free, fast and environmentally friendly near infrared-based methodology was developed for the determination and quality control of 11 pesticides in commercially available formulations. This methodology was based on the direct measurement of the diffuse reflectance spectra of solid samples inside glass vials and a multivariate calibration model to determine the active principle concentration in agrochemicals. The proposed PLS model was made using 11 known commercial and 22 doped samples (11 under and 11 over dosed) for calibration and 22 different formulations as the validation set. For Buprofezin, Chlorsulfuron, Cyromazine, Daminozide, Diuron and Iprodione determination, the information in the spectral range between 1618 and 2630 nm of the reflectance spectra was employed. On the other hand, for Bensulfuron, Fenoxycarb, Metalaxyl, Procymidone and Tricyclazole determination, the first order derivative spectra in the range between 1618 and 2630 nm was used. In both cases, a linear remove correction was applied. Mean accuracy errors between 0.5 and 3.1% were obtained for the validation set. 

Different calibration models, such as classical least squares and multivariate calibration approaches have been considered. 

The depth profiles in Fig. 3.26 show that the typical flat channeling implantation profile is generated with low doses only. Increasing the dose superimposes the normal implantation profile shape. Undertaking such experiments with homogeneous wafers enables the production of calibrating models for semiconductor production. 

Purpose Generate a data set that superimposes normally distributed noise on a linear calibration model to study the effects of the adjustable parameters. A whole calibration—measurement—evaluation sequence can be optimized for quality of the results and total costs. 

In the model simulations, the settling and decanting phase were characterized by a reactive point-settler model. The simulations were carried out using matlab 6.5 simulation platform. A systematic model calibration methodology as described in Fig. 2 was applied to the SBR. Fig. 3. shows the simulation results from the calibrated model. The model predicted the dynamics of the SBR with good accuracy. 

The set of possible dependent properties and independent predictor variables, i.e. the number of possible applications of predictive modelling, is virtually boundless. A major application is in analytical chemistry, specifically the development and application of quantitative predictive calibration models, e.g. for the simultaneous determination of the concentrations of various analytes in a multi-component mixture where one may choose from a large arsenal of spectroscopic methods (e.g. UV, IR, NIR, XRF, NMR). The emerging field of process analysis,... 

The ultimate goal of multivariate calibration is the indirect determination of a property of interest (y) by measuring predictor variables (X) only. Therefore, an adequate description of the calibration data is not sufficient the model should be generalizable to future observations. The optimum extent to which this is possible has to be assessed carefully when the calibration model chosen is too simple (underfitting) systematic errors are introduced, when it is too complex (oveifitting) large random errors may result (c/. Section 10.3.4). 

In many applications the goal of predictive modelling is not a detailed understanding of the relation between dependent and independent variables. Ability to interpret the model, therefore, is not a requirement perse. This should not preclude the exploitation of available background knowledge on the problem at hand during calibration modelling. A model that can be sensibly interpreted certainly adds value and confidence to the calibration result. 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

We chose the number of PCs in the PCR calibration model rather casually. It is, however, one of the most consequential decisions to be made during modelling. One should take great care not to overfit, i.e. using too many PCs. When all PCs are used one can fit exactly all measured X-contents in the calibration set. Perfect as it may look, it is disastrous for future prediction. All random errors in the calibration set and all interfering phenomena have been described exactly for the calibration set and have become part of the predictive model. However, all one needs is a description of the systematic variation in the calibration data, not the... 

There are two points of view to take into account when setting up a trmning set for developing a predictive multivariate calibration model. One viewpoint is that the calibration set should be representative for the population for which future predictions are to be made. This will generally lead to a distribution of objects in experimental space that has a higher density towards the center, tailing out to the boundaries. Another consideration is that it is better to spread the samples more or... 

The development of a calibration model is a time consuming process. Not only have the samples to be prepared and measured, but the modelling itself, including data pre-processing, outlier detection, estimation and validation, is not an automated procedure. Once the model is there, changes may occur in the instrumentation or other conditions (temperature, humidity) that require recalibration. Another situation is where a model has been set up for one instrument in a central location and one would like to distribute this model to other instruments within the organization without having to repeat the entire calibration process for all these individual instruments. One wonders whether it is possible to translate the model from one instrument (old or parent or master. A) to the others (new or children or slaves, B). 

Several approaches have been investigated recently to achieve this multivariate calibration transfer. All of these require that a small set of transfer samples is measured on all instruments involved. Usually, this is a small subset of the larger calibration set that has been measured on the parent instrument A. Let Z indicate the set of spectra for the transfer set, X the full set of spectra measured on the parent instrument and a suffix Aor B the instrument on which the spectra were obtained. The oldest approach to the calibration transfer problem is to apply the calibration model, b, developed for the parent instrument A using a large calibration set (X ), to the spectra of the transfer set obtained on each instrument, i.e. and Zg. One then regresses the predictions (=Z b ) obtained for the parent instrument on those for the child instrument yg (=Z b ), giving... 

This yields an estimate for the bias (intercept) a and slope b needed to correct predictions yg from the new (child) instrument that are based on the old (parent) calibration model, b. The virtue of this approach is its simplicity one does not need to investigate in any detail how the two sets of spectra compare, only the two sets of predictions obtained from them are related. The assumption is that the same type of correction applies to all future prediction samples. Variations in conditions that may have a different effect on different samples cannot be corrected for in this manner. 

In this way the child spectrum is transformed into a spectrum as if measured on the parent instrument. In a more refined implementation one establishes the highest correlating wavelength channel through quadratic interpolation and, subsequently, the corresponding intensity at this non-observed channel through linear interpolation. In this way a complete spectrum measured on the child instrument can be transformed into an estimate of the spectrum as if it were measured on the parent instrument. The calibration model developed for the parent instrument may be applied without further ado to this spectram. The drawback of this approach is that it is essentially univariate. It cannot deal with complex differences between dissimilar instruments. 

In the direct standardization introduced by Wang et al. [42] one finds the transformation needed to transfer spectra from the child instrument to the parent instrument using a multivariate calibration model for the transformation matrix = ZgF. The transformation matrix F (qxq) translates spectra Zg that are actually measured on the child instrument B into spectra Z that appear as if they were measured on instrument A. Predictions are then obtained by applying the old calibration model to these simulated spectra Z ... 

E. de Noord, The influence of data preprocessing on the robustness nd parsimony of multivariate calibration models. Chemom. Intell. Lab. Systems, 23 (1994) 65-70,... 

For the time invariant calibration model discussed in Section 41.2, eq. (41.14) reduces to ... 

The Mahalanobis distance measures the degree to which data fit the calibration model. It is defined as... 

Verification is the complement of calibration model predictions are compared to field observations that were not used in calibration or fidelity testing. This is usually the second half of split-sample testing procedures, where the universe of data is divided (either in space or time), with a portion of the data used for calibration/fidelity check and the remainder used for verification. In essence, verification is an independent test of how well the model (with its calibrated parameters) is representing the important processes occurring in the natural system. Although field and environmental conditions are often different during the verification step, parameters determined during calibration are not adjusted for verification. 

Validation of the model. Output from the Leggett Model has been compared with data in children and adult subjects exposed to lead in order to calibrate model parameters. The model appears to predict blood lead concentrations in adults exposed to relatively low levels of lead however, no information could be found describing efforts to compare predicted blood lead concentrations with observations in children. 

Residual standard deviation (of the calibration) estimate of the error of the calibration model... 

In many practical problems, interactions between the variables appear so that the absolute global optimum can be found heavily. As an example, wavelength selection in NIR determination of blood glucose (see Sect. 6.2.6) is considered. The aim of the selection is to find such combinations of wavelengths with which calibration models are obtained their prediction quality is as near at the global optimum as possible (Danzer et al. [2001], p 174). The number of combinations C for the selection of k wavelengths from n channels of the spectrometer is given by... 

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