# Introduction to multivariate data analysis in chemical engineering

## Multivariate data analysis methods are being used more and more beyond chemical engineering and have useful, practical uses for process control, though there are challenges to using multivariate data.

Multivariate data analysis (MVA) is the investigation of many variables, simultaneously, in order to understand the relationships that may exist between them. Multivariate data analysis methods have been around for decades, but until recently, have primarily been used in laboratories and specialist technical groups, rarely being applied to production processes.

Most chemical manufacturing processes are highly multivariate in nature due to the complex reactions involved i.e. there are a large number of variables which are typically very interactive. Complex systems require multiple measurements to fully understand them.

However, the Statistical Process Control (SPC) tools used in chemical engineering still rely largely on univariate (i.e. one variable at a time) methods, which do not show the full picture of complex chemical processes despite collecting masses of data through instruments and control systems. These SPC tools use traditional statistical approaches such as mean, standard deviation and Student’s-t, which only look at single variables individually.

While univariate statistics can be useful for investigating simple systems, they tend to fail when more complex systems are analyzed. This is because they cannot detect relationships that may exist between the variables being studied, as they treat all such variables as being independent of each other.

This relationship among variables is known as covariance or correlation, and is a central theme in MVA. Covariance describes the influence that one variable has on others, and process upsets will typically be caused by several variables acting together.

A common example is the relationship between Temperature and pH, as shown below. Suppose I am the Plant Manager at a chemical plant. My process has been running smoothly until suddenly, the product quality starts to deteriorate and I have to make a decision what to do.

I have two control charts at my disposal for the measurements performed on the system. These are supposed to be related to product quality and are meant to serve as an indicator that the process is in control.

The ‘pH’ and ‘Temperature’ control charts indicate that nothing is wrong, but in fact there is. The point marked with a red dot in both charts is an abnormal situation, but how can I detect this?

Obviously univariate statistics have failed me! What if I plot the points of both control charts against each other to form a simple multivariate control chart? The plot labeled ‘Multivariate view’ shows this.

The square region formed by dotted red lines in the ‘Multivariate view’ plot shows the univariate limits my process is allowed to operate in, however, on this simple multivariate graph it can be seen that variable 1 and variable 2 are related to each other, i.e. higher temperature corresponds to higher pH. Accordingly, the multivariate limits, indicated by the ellipse in the figure, are very different from the two univariate limits, indicated by the dashed lines in the figure. The red highlighted point in the figure is well inside the two univariate limits but is outside the multivariate control limits. In this case, if multivariate control charts were applied the process operator would be able to detect the process deviation.

This is a simple example of how multivariate methods enable superior Early Event Detection capabilities compared to univariate control charts, especially when systems are complex and the number of input variables becomes large i.e. greater than 10.

**Typical multivariate techniques**

The main multivariate techniques are Exploratory Data Analysis, Regression/Prediction methods, and Classification methods.

Exploratory data analysis (EDA) attempts to find the hidden structure or underlying patterns in large, complex data sets. This gives a better understanding of the process and can lead to insights that would not have been observed otherwise. EDA methods include Cluster analysis and Principal Component Analysis (PCA). An example application of exploratory data analysis is checking for contaminants in a process or feedstock, or identifying by-products caused by incorrect process settings.

Regression analysis involves developing a model from available data to predict a desired response or responses for future measurements. Multivariate regression is an extension of the simple straight line model case, where there are many independent variables and at least one dependent variable. Regression methods include Multiple Linear Regression (MLR), Principal Component Regression (PCR) and Partial Least Squares Regression (PLSR). Common applications include predicting purity, yield or end product quality from input raw material quality.

Classification is the separation (or sorting) of a group of objects into one or more classes based on distinctive features in the objects. Classification methods include Linear Discriminant Analysis (LDA), SIMCA, and Support Vector Machine Classification (SVM-C). Example applications include grouping products according to similar characteristics or quality grades.

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