Multivariate models

Let us now consider models that have only more than one measured variable (w l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

In many engineering applications, however, we can easily reduce the problem to the univariate tests presented in the previous section by assuming that the covariance matrix of the errors can be written as 

An independent estimate of COV(s), t, that is required for the adequacy tests can be obtained by performing N/t repeated experiments as 

---

The relationship between a criterion variable and two or more predictor variables is given by a linear multivariate model ... 

As mentioned above, there are two options for installing an APC. One option is to install an APC within the DCS framework, and the other is to install a multivariable modeling/control package in a host computer. Each has advantages and disadvantages, as indicated below. 

The multivariable modeling/control package is able to hold more tightly against constraints and recover more quickly from disturbances. This results in an incremental capacity used to justify multivariable control. An extensive test run is necessary to measure the response of unit variables. 

A multivariable model is like a black box. The constraints go in and the signals come out. Operators do not trust a system that takes the unit away from them. Successful installations require good training and continual communication. The operators must know the interconnections in the system. 

Flury B (1988) Common principal components and related multivariate models. Wiley, New York... 

Instrument standardization, v - a procedure for standardizing the response of multiple instruments such that a common multivariate model is applicable for measurements conducted across these instruments, the standardization being accomplished via adjustment of the spectrophotometer hardware or via mathematical treatment of one or a series of collected spectra. 

Multivariate model, n - a multivariate mathematical rule or formula used to calculate a physical, chemical, or quality parameter from the measured spectrum. 

Outlier detection methods, n - statistical tests which are conducted to determine if the analysis of a spectrum using a multivariate model represents an interpolation of the model. 

Physical correction, n - a type of post processing where the correction made to the numerical value produced by the multivariate model is based on a separate physical measurement of, for example, sample density, sample pathlength, or particulate... 

Much effort has been devoted to the development of reliable calculation methods for the prediction of the retention behaviour of analyses with well-known chemical structure and physicochemical parameters. Calculations can facilitate the rapid optimization of the separation process, reducing the number of preliminary experiments required for optimization. It has been earlier recognized that only one physicochemical parameter is not sufficient for the prediction of the retention of analyte in any RP-HPLC system. One of the most popular multivariate models for the calculation of the retention parameters of analyte is the linear solvation energy relationship (LSER) ... 

Nature is sometimes difficult but never insidious. Fortunately, the similarity principle—compounds with similar chemical structures often possess similar properties or activities—is valid in many cases, and thus gives multivariate models a chance. 

A simple PCA says a lot about a chemical process, but a step ahead should be to search for equations relating dependent variables (yh y2,. .., yjt. ..ym) to independent ones (xhx2, x . ..x ) these equations are called models. Multivariate models allow to mathematically describe the relationship between all the ys and all the xs. 

Cui Y. Song X. Chuang K. Venkatramani C. Lee S. Gallegos G. Venkateshwaran,T. Xie, M. Variable selection in multivariate modeling of drug product formula and manufacturing process. Journal of Pharmaceutical Sciences, 2012, in press. 

Cuba, W. and Cruciani, G. Molecular field-derived descriptors for the multivariate modelling of pharmacokinetic data, in Mdecular Modelling and Prediction of Bioactivity, Proceedings of the 12th European Symposium on Quantitative Structure-Activity Relationships (QSAR 98), Gundertofte, K. and Jorgensen, F.S. (Eds). Plenum Press, New York, 2000, 89-95. 

The combination of empirical and multivariate modeling elements makes chemometrics both very powerful and very dangerous. The power of chemometrics is that it can be used to model systems that are both unknown and complex. Furthermore, these models are not restricted by theoretical constraints - which can be a big advantage if large deviations from theoretical behavior are known to be present in your system. 

However, there are prices to pay for the advantages above. Most empirical modeling techniques need to be fed large amounts of good data. Furthermore, empirical models can only be safely applied to conditions that were represented in the data used to build the model (i.e., extrapolation of such models is very dangerous). In addition, the availability of multiple response variables for building a model results in the temptation to overfit models, in order to obtain artificially optimistic results. Finally, multivariate models are usually much more difficult to explain to others, especially those not well versed in math and statistics. 

It was mentioned earlier that empirical multivariate modeling often reqnires a very large amount of data. These data can contain a very large number of samples (IN), a very large number of variables (M) per sample, or both. In the case of PAT, where spectroscopic analytical methods are often used, the number of variables collected per process sample can range from the hundreds to the thonsands ... 

One such study details the effects of temperature variation on substrate and metabolite concentration predictions, and used an artificial neural network creating a nonlinear multivariate model to improve concentration predictions. Another study notes the effects of temperature on the mid-infrared spectral data as well, but also noted that the sensor was not affected by reactor operating conditions such as agitation, airflow and backpressure. ... 

Constructing a multivariate model is a time-consuming task that involves various operations including the preparation of a sizeable number of samples, recording their spectra and developing and validating the cali-... 

Several statistics from the models can be used to monitor the performance of the controller. Square prediction error (SPE) gives an indication of the quality of the PLS model. If the correlation of all variables remains the same, the SPE value should be low, and indicate that the model is operating within the limits for which it was developed. Hotelling s 7 provides an indication of where the process is operating relative to the conditions used to develop the PLS model, while the Q statistic is a measure of the variability of a sample s response relative to the model. Thus the use of a multivariate model (PCA or PLS) within a control system can provide information on the status of the control system. 

An example of the use of soft sensors is given by the automation of a penicillin production dependent on strict adherence to certain hmits in the fermentation process since such biological systems are sensitive to changes in operational conditions. An important issue in the use of soft sensors is what to do if one or more of the input variables are not available due, for example, to sensor failure or maintenance needs. Under such circumstances, one must rely on multivariate models to reconstruct or infer the missing sensor variable. ... 

Another direction of development of the data set is to strengthen the in vitro-in vivo correlations and develop multivariate models to predict in vivo endpoints, such as therapeutic effects and adverse events. In this respect, it will be interesting to examine which data (among in silico descriptors, in vitro primary and secondary data, in vitro functional data, etc.) are most appropriate to derive robust and predictive models. 

Multivariate models using neural networks, support vector machines and least median squares regression have been used to predict hERG activity [96-98]. These types of models function more as computational black box assays. 

Dalsey, J. M. Knelp, T. J. "Atmospheric Particulate Organic Matter Multivariate Models for Identifying Sources and Estimating Their Contributions to the Ambient Aerosol", This Symposium. 

Multivariate models have been successful in identifying source contributions in urban areas. They are not independent of Information on source composition since the chemical component associations they reveal must be verified by source emissions data. Linear regressions can produce the typical ratio of chemical components in a source but only under fairly restrictive conditions. Factor and principal components analysis require source composition vectors, though it is possible to refine these source composition estimates from the results of the analysis (6.17). 

The most important caveat to be heeded in the use of multivariate models is that though origination in the same source will cause two chemical components to correlate, the converse, that chemical components which correlate must have originated in the same source, is not true. 

---