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Process Analytical Technology for the Food Industry pp 7–59Cite as

Multivariate Data Analysis (Chemometrics)

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Part of the book series: Food Engineering Series ((FSES))

Abstract

Chemometrics plays a key role in PAT strategies. It is essential in understanding and diagnosing real-time processes, and keeping them under multivariate statistical control. This chapter will cover design of experiments, exploratory analysis, quantitative predictive modelling, classification, multivariate process monitoring and multi-block and multi-way analyses. The objective of the chapter is to describe chemometrics methods with a main focus on understanding, interpretation and evaluating the usefulness of the results.

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Acknowledgments

The authors want to thank Dr. Mazerolles from INRA for his multi-block section review, Dr. Williams from the Canadian Grain Commission, CAMO (Oslo, Norway) and Dr. Guillaume from Cemagref for authorising the usage of their data.

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Authors and Affiliations

  1. Ondalys, ZA La Plaine, 4 rue Georges Besse, 34830, Clapiers, France

    Dr. Sylvie Roussel, Dr. Sébastien Preys & Eng. Jordane Lallemand

  2. Indatech, ZA La Plaine, 4 rue Georges Besse, 34830, Clapiers, France

    Dr. Fabien Chauchard

Authors
  1. Dr. Sylvie Roussel
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  2. Dr. Sébastien Preys
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  3. Dr. Fabien Chauchard
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  4. Eng. Jordane Lallemand
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Corresponding author

Correspondence to Sylvie Roussel .

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Editors and Affiliations

  1. University College Dublin, Belfield, Ireland

    Colm P. O'Donnell

  2. University of Reading, Reading, United Kingdom

    Colette Fagan

  3. Dublin Institute of Technology, Dublin, Ireland

    P.J. Cullen

Annex: Figures of Merit

Annex: Figures of Merit

Some statistical criteria (Figures of Merit) encountered during this chapter are mathematically defined here.

Root-mean-squared error (RMSE) is defined as the root square of the ratio of the prediction error sum of squares (PRESS) and the estimated degrees of freedom of the set. This square root value is generally preferred because the unit is the same as the original data. RMSE is calculated for a calibration set (RMSEC) and a prediction (or validation) set (RMSEP):

$$ \text{PRESS}=\sum\limits_{i=1}^{n}{( {{y}_{i}}-{{{\hat{y}}}_{i}} ){}^\text{2}} $$
(2.31)
$$ \text{RMSEC}=\sqrt{\frac{\text{PRESS}}{n-p-1}} $$
(2.32)
$$ \text{RMSEP}=\sqrt{\frac{\text{PRESS}}{n}}. $$
(2.33)

The bias is the mean error, i.e. the systematic part of the error:

$$ bs=\frac{1}{n}\sum\limits_{i=1}^{n}{( {{y}_{i}}-{{{\hat{y}}}_{i}} )}. $$
(2.34)

The coefficient of determination (R 2 ) represents the spread of the predictions. It is important not to consider it alone. For example, the R 2 value could be almost 1 whereas the bias and/or the PRESS could be high:

$$ R{}^\text{2}=1-\frac{\text{PRESS}}{\sum\limits_{i=1}^{n}{( {{y}_{i}}-\bar{y} ){}^\text{2}}}. $$
(2.35)

In classification, standard errors are defined as the proportion of misclassified samples. A confusion matrix is generally built to summarise the results. The number of correctly classified objects corresponds to n 11 and n 22, and the misclassified ones to n 21 and n 12. It can also be seen in terms of first-order errors, which are similar to the lack of sensitivity (e.g. the proportion of samples A not classified in A), or second order, which represents the lack of specificity (e.g. the proportion of B classified in A) (Table 2.4).

Table 2.4 Confusion matrix
Full size table

It is important to note that some other figures of merit are widely used in certain applications and can be found in Olivieri et al. (2006).

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Roussel, S., Preys, S., Chauchard, F., Lallemand, J. (2014). Multivariate Data Analysis (Chemometrics). In: O'Donnell, C., Fagan, C., Cullen, P. (eds) Process Analytical Technology for the Food Industry. Food Engineering Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0311-5_2

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