Multivariate calibration models prediction

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... 

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 ... 

Prediction limits for the estimation of an unknown concentration x, can be calculated. The calculation depends on the specific multivariate calibration model... 

Figure 9.1 shows schematically a general acoustic chemomehic data path, from acoustic emission to the final multivariate calibration model, including future prediction capabilities. 

This method can be considered a calibration transfer method that involves a simple instrument-specific postprocessing of the calibration model outputs [108,113]. It requires the analysis of a subset of the calibration standards on the master and all of the slave instmments. A multivariate calibration model built using the data from the complete calibration set obtained from the master instrument is then applied to the data of the subset of samples obtained on the slave instruments. Optimal multiplicative and offset adjustments for each instrument are then calculated using linear regression of the predicted y values obtained from the slave instrument spectra versus the known y values. 

R.D. Guenard, C.M. Wehlberg, R.J. Pell and D.M. Haaland, Importance of prediction outlier diagnostics in determining a successful inter-vendor multivariate calibration model transfer, Appl. Spectrosc., 61, 747 (2007). 

In order to construct a calibration model, the values of the parameters to be determined must be obtained by using a reference method. The optimum choice of reference method will be that providing the highest possible accuracy and precision. The quality of the results obtained with a multivariate calibration model can never exceed that of the method used to obtain the reference values, so the choice should be carefully made as the quality of the model will affect every subsequent prediction. The averaging of random errors inherent in regression methods can help construct models with a higher precision than the reference method. 

This method is probably the simplest of the software-based standardization approaches.73,74 It is applied to each X-variable separately, and requires the analysis of a calibration set of samples on both master and slave instruments. A multivariate calibration model is built using the spectra obtained from the master instrument, and then this model is applied to the spectra of the same samples obtained from the slave instrument. Then, a linear regression of the predicted Y-values obtained from the slave instrument spectra and the known Y-values is performed, and the parameters obtained from this linear regression fit are used to calculate slope and intercept correction factors. In this... 

It is important that calibration models are rigorously validated and in the first instance that all variations are accounted for in the model using diverse samples that are expected to be observed in future bioprocess runs. Some investigators attempt to keep process conditions very reproducible but such conditions are uncommon in an industrial environment. In addition, multivariate calibration models will work well if identical media (composition) and process conditions are used on each successive run. Simple modifications such as use of a different media supplier can affect the spectral background. The predictive ability of the models will then be affected as they will be challenged with samples which they have not been trained to recognise [74]. 

Many scientists hesitate to rely on multivariate calibration models due to the complexity of the statistics involved. The quality of a multivariate model is highly dependent upon the factors chosen. Incorporation of too many factors leads to an overfit model which provides a low calibration error value, but predicts unknown samples poorly. Too few factors lead to a model that neither adequately fits the calibration data nor accurately predicts new samples. Typically, factors are included or excluded from the calibration model based upon their statistical significance, but this may not provide the opti-... 

There are three statistics often employed for comparing the performances of multivariate calibration models root mean squared error of calibration (RMSEC), root mean squared error of cross validation (RMSECV), and root mean squared error of prediction (RMSEP). All three methods are based on the calculated root mean squared error (RMSE)... 

The root mean squared error of calibration (RMSEC) has been defined above. The leverage, ha, quantifies the distance of the predicted sample (at zero concentration level) to the mean of the calibration set in the -dimensional space Hq can be estimated as an average value of the leverages of a set of validation samples having zero concentration of the analyte. For a model calculated from mean-centred spectra its calculation was presented in Section 5.3 in matrix notation /zo=l//+to (T T) 4o, where to is the (.4x1) score vector of the predicted sample and T is the (7x4) matrix of scores for the calibration set. Finally, A(a,p,v) is a statistical parameter that takes into account the a and (3 probabilities of falsely stating the presence/absence of analyte, respectively, as recommended elsewhere. When the number of degrees of freedom i.e. the number of calibration samples) is high (v>25), as is usually the case in multivariate calibration models, and a =) , then A(a,(S,v) can be safely approximated to 2 ... 

By moving the measurement from the well-controlled laboratory to the process environment, the influence of external process variables such as p, T, and flow turbulence will affect the measurements. When vibrational spectra are measured on- or in-line for process analytical and control purposes, the performance variations influence the shape of the spectra in a non-linear manner. Smilde et al. [81] have assessed the influence of these temperature-induced spectral variations on the predictive ability of multivariate calibration models. 

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... 

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... 

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