A partial Granger causality based method for analysis of parameter interactions in bioreactors

Abstract

In this article, we utilize the concept of partial Granger causality to study the penicillin production process under several operating conditions. We propose a graph-theoretic template (causal network) based method for intelligent process monitoring. We validate our results with the aid of existing knowledge and available literature. The proposed method is quite general and can be extended to analyze several physical, chemical or biological systems.

Introduction

Process monitoring is of great practical importance. Intelligent process monitoring has numerous aspects and it is mainly concerned with the operation of the process at its optimal operating conditions. Also, early detection of faults and its diagnosis is quite critical in avoiding product quality deterioration, equipment damage, and personal safety. Given the fact that bioreactors (or fermentors) exhibit a fairly complex dynamics, it is important to analyze this behaviour in a more meaningful manner. Hence, fault detection and diagnosis (FDD) becomes an inherent part of such analysis tool. FDD comprises of four important sub-problems i.e. fault detection (detect abnormal situation), fault isolation (find apparent root cause of the fault, preferably in an automated manner), fault estimation (determine the size of the fault) and fault reconstruction (find normal value if there was no fault) (Venkatasubramanian, Rengaswamy, Yin, & Kavuri, 2003a).

For each category of the sub-problem outlined above, researchers have developed numerous algorithms based on statistical analysis, artificial intelligence, machine learning etc. These algorithms are broadly classified as: quantitative model based methods, qualitative model based methods and process history based methods (Venkatasubramanian et al., 2003a). Quantitative model based methods principally deals with phenomenology of the problem. These approaches are mainly based on input–output, state-space, first principle (based on physics of the system), and frequency response models. A Kalman filter is a well-known technique in this category (Venkatasubramanian et al., 2003a). Qualitative models, on the other hand, are essentially knowledge based expert systems. An expert system consists of a set of large if-then-else rules as a knowledge base and an inference engine which searches through the knowledge base to derive conclusions from the available facts. Fault trees, signed directed graphs (SDG) are the representative popular techniques (Venkatasubramanian, Rengaswamy, Yin, & Kavuri, 2003b). Finally, process history based methods require large amount of historical data. The data should contain the typical trends and previous fault information for effective monitoring methods to be built. Artificial neural networks, machine learning based methods like support vector machine (SVM) fall under this category. For further details, we refer Kulkarni, Jayaraman, and Kulkarni (2005) and Venkatasubramanian, Rengaswamy, Yin, and Kavuri (2003c)).

In this work, we present clever representation of parameter interactions that might be helpful to isolate faults in an important bioprocess, namely penicillin production process. Bioreactors exhibit complex dynamics and it is important to know these dynamical characteristics more meaningfully to understand the different operating situations. Lot of methods has been developed so far with differing complexity to tackle this problem (Cinar, Parulekar, Undey, & Birol, 2003). Also, how parameter interactions change with change in operating conditions is of utmost importance to process operator to design more efficient control systems and to operate process at optimum conditions. To this end, Patil and Kulkarni (2007) have recently developed a visualization framework for a bioreactor which depicts parameter interactions as an undirected graph for different operating conditions. This framework was found to be consistent with underlying system dynamics. However, the framework does not contain causal connections (i.e. directed graph) to pinpoint apparent root cause of the abnormal situation. We extend this work further and introduce the partial Granger causality concept for the first time in the framework of fault identification problem.

Granger causality concept is known to the econometricians from decades and it has been successfully used in many economics related problems (Granger, 1969). Recently, Krishna and Guo (2008) introduced partial Granger causality to analyze multi-parameter data derived from biological networks. We use this method to analyze the penicillin production process under different operating conditions and propose a strategy to visualize and understand the interaction patterns among multiple time series representing a set of stochastic processes. The proposed method depends on the statistical interdependence among multi-parameter time series data. This interdependence could be causal in nature and simple symmetric measures like ordinary coherence may not be suitable for measuring it. Wiener (1956) proposed an idea to measure the causal influence of one time series on another by conceiving a notion that the prediction of one time series could be improved by incorporating the knowledge of the second one. Granger (1969) formalized this concept for linear auto-regression model of causal influences and Geweke (1982) extended Granger causality to propose a measure of interdependence between two sets of time series. In this work, we use the partial Granger causality concept to calculate the interdependence between two time series by eliminating the effect of all other time series (i.e. all other variables) in the system and propose a graph-theoretic method for the analysis of parametric interactions in the bioreactor.

An operating environment like bioreactors can be perceived as a multivariate system with each component representing an individual stochastic process. The operational behaviour of the reactor is due to the interactions taking place between the individual components over a period of time. Time is an important entity in the whole operating process, as some components in the system may get activated or inhibited due to input (or lack of it) from the other components, either simultaneously or subsequently. We discuss in our manuscript that due to the temporal and multivariate nature of the system under study, Partial Granger causality is a suitable framework to model it.

The article is arranged as follows. Next section explains the details of the Geweke's causal model and partial Granger causality concept followed by the description of the dataset. Results and discussion outlines the characteristics of the proposed methodology and discuss the advantages and disadvantages of the method. Finally, conclusions summarize the results.