X2 X3 X4






2.665e-01 2.188e-01 1.507e-01 7.433e-02 4.312e-01 3.819e-01 3.240e-01 2.612e-01 2.614e-01 2.013e-01 4.533e-01 4.568e-01 4.098e-01 2.905e-01 4.802e-01

3.664e-01 2.679e-01 2.190e-01 1.508e-01 5.043e-01 5.770e-01 5.125e-01 4.315e-01 3.829e-01 2.616e-01 5.979e-01 5.800e-01 6.041e-01 4.535e-01 6.268e-01

2.493e-01 3.651e-01 2.675e-01 2.186e-01 4.183e-01 4.283e-01 4.398e-01 5.029e-01 5.751e-01 4.302e-01 5.318e-01 6.052e-01 5.422e-01 5.959e-01 6.257e-01

1.250e-01 2.476e-01 3.627e-01 2.665e-01 3.214e-01 4.098e-01 3.486e-01 4.152e-01 4.260e-01 4.999e-01 4.431e-01 4.752e-01 5.347e-01 5.287e-01 5.562e-01

finding the simplest functional relationships (fewest variables in an equation) that describe the system with the desired accuracy. For example, if measurement noise is about 10% of a signal, we would be content by about 90% of the information we could gather about the system by considering all the coordinates. This leads to

The conventional usage of the methods introduced in this chapter in system modeling is to reconstruct a phase space using a scalar measurement. In the following example we will demonstrate how these concepts can be used to narrow down the phase space using multivariate measurements from a fermentation process [59].

Example 10 Phase space reduction

Consider the starch fermentation by recombinant Saccharomyces cerevisiae in a batch reactor (Figure 5.18). A series of experiments were conducted in the absence of oxygen supply by changing initial starch concentrations, and time courses of

• V, extracellular protein (mg/L) concentrations, and

activities were measured using appropriate analytical techniques [57, 60]. A single run of the experiment generates snapshots of the measured quantities like the one shown in Figure 5.19. Note that measurements were made at varying time intervals, and different measurements are not necessarily synchronous. The focus is different from the previous examples in that we concentrate on the underlying non-chaotic dynamics. Although the techniques described in this chapter are applicable to chaotic and non-chaotic systems, most of the nonlinear time series analysis tools require a dense phase space. As we demonstrated with numerous examples, chaotic systems fulfill this requirement by populating the phase space, e.g., by strange attractors. In the present example, this was achieved by repeated experiments. Figure 5.19 shows only a single realization of the experiment. When we aggregate data from many experiments involving the same system, due to variations in the controlled conditions (e.g., initial conditions) and uncontrolled conditions (e.g., environmental conditions), the phase space will be dense, and we will be able to use the tools introduced earlier.

Next, we classify pairs of measurements as Independent, Coupled or Redundant using the heuristic scheme described in Figure 5.20. When a measurement pair is found to be Independent, we will conclude that the



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