Forecasting polynomial dynamics
Introduction
Polynomial dynamical systems have been used in reverse engineering of gene regulatory networks [1], [2]. Using the algorithm of Laubenbacher and Stigler [1], one can generate a multivariate time series that simulates the network evolution. For a fixed time, the different components of the series correspond to the nodes (genes) in a finite dynamical system.
Time series forecasting, which has largely been motivated by market analysis, has gradually evolved to include nonlinear techniques. Two notable methodologies in this area are neural networks and fuzzy control. Generally, neural control is suited for numerical input–output pairs, whereas fuzzy control is an effective approach to utilizing linguistic rules. When fuzzy rules are generated from numerical data pairs, the two kinds of information are combined into a common framework [3].
The goal of this paper is to compare the Laubenbacher–Stigler algorithm [1] against the rule-based formalism for time series prediction. The time series generated form polynomial dynamics will be regarded as the observed, or actual, time series. Although a multivariate time series can, in principle, be processed by a fuzzy controller, we first convert the multivariate series to a time series with a single component. The values at successive time steps are then used to learn the fuzzy rules needed by the fuzzy controller. A library of rules allows one to use values at consecutive time steps to predict the future values.
The algebraic approach of Laubenbacher and Stigler [1] interpolates the first few time steps to create a solution that fits the data. Then, the dynamics of the network is assumed to be contained in the solution. One could extend this procedure to a larger number of time steps, that is, for longer initial epochs. Such a procedure, however, would be neither practical nor effective. In a rule-based system, a library of rules is created that encapsulates the known behavior of a dynamical system. As the system evolves, the library can be supplemented by new rules as the additional information becomes available. This results in an adaptive procedure, based on the updated fuzzy rule base, to predict future values [3]. As a supplementary benefit, fuzzy controllers can operate in real time; their learning does not require many iterations to converge.
The remainder of this paper is organized as follows: in Section 2, we summarize the algebraic approach of Laubenbacher and Stigler and apply it to a simulated network on six genes. The rule-based forecasting is the subject of Section 3, in which we describe the principle of a fuzzy controller and use it, subsequently, to predict the evolution of the simulated network of Section 2. In Section 4, we use the binary model of Albert and Othmer [6], supplemented by bit changes that simulate mutations, to show how anomalies can be detected through a rule-based system. Simply put, anomaly detection employs the difference signal, that is, the difference between the forecasted and observed time series. Since the original algebraic algorithm was developed for biological applications, the example considered in Section 4 is taken from biology.
Section snippets
Algebraic approach
As in the method of Laubenbacher and Stigler [1], we generate a discretized time series for the genes . The genes are also called nodes of a state. For p levels of discretization, the network evolves starting with a time series over the finite field of integers modulus p with m time steps (m states). This time series is encoded in terms of a matrix S of size .
The first stage of the algorithm is to compute a particular interpolating polynomial .
Rule-based forecasting
In a fuzzy controller [7], a numeric input is transformed to a numeric output through a combined process of fuzzification–defuzzification. At the heart of the controller is a library of fuzzy rules that are constructed from a training pattern. In Fig. 1, a triple , which forms a numeric input, is converted to a numeric output y. For three antecedents and one consequent, there are r fuzzy rules of the form “If is positive large and is negative medium and is positive small,
Anomaly detection
Albert and Othmer [6] analyzed a Boolean network of regulatory interactions, in which the interactions are formulated as logical functions. In this model, a line of 12 cells corresponds to three spatial regions that become parasegments, resulting in periodic gene repetition every fourth cell. With 15 genes per cell, there are 60 different nodes. Actually, to illustrate spatial patterns, Albert and Othmer employ 180 nodes. We followed the evolution equations using the Boolean updating functions
Conclusions
The rule-based forecasting has been applied to predict the evolution of gene networks, using both simulated and realistic models. This approach is especially well-suited to study anomalous behavior of a network as the anomaly results in large forecast errors. There are two restrictions of this methodology, manifest for large numbers of nodes, that should be mentioned. First, when we use Eq. (6), we end up with numbers exceeding the values of long integers in most programming languages. In C
Acknowledgement
The authors thank Hiren Maharaj for contributing ideas and helpful discussions.
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