Dynamic system classifier

Stochastic differential equations describe well many physical, biological, and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and classify complex dynamical systems is proposed within a Bayesian framework. To this end, we develop a dynamic system classifier (DSC). The DSC first abstracts training data of a system in terms of time-dependent coefficients of the descriptive stochastic differential equation. Thereby the DSC identifies unique correlation structures within the training data. For definiteness we restrict the presentation of the DSC to oscillation processes with a time-dependent frequency $\omega \left(t\right)$ and damping factor $\gamma \left(t\right)$. Although real systems might be more complex, this simple oscillator captures many characteristic features. The $\omega$ and $\gamma$ time lines represent the abstract system characterization and permit the construction of efficient signal classifiers. Numerical experiments show that such classifiers perform well even in the low signal-to-noise regime.

Figure 1

Hierarchical Bayes model for the key quantities. The two fields ${\beta }_t$ and ${\gamma }_t$ form together a signal $s$, defining the response ${\mathit{R}}^{\left(s\right)}$, with which the system reacts to the driving white noise $\xi$. The application of ${\mathit{R}}^{\left(s\right)}$ on white Gaussian noise solves Eq. (1) and consequently yields to its solution $x={\mathit{R}}^{\left(s\right)}\xi$. From sufficient training data vectors $x$ a plausible signal should be inferred in the learning phase.

Figure 2

The spectral parameters ${\alpha }_{\beta }$ and $q_{\beta }$ define together with the smoothness enforcing parameter ${\sigma }_{\beta }$ the prior for the shape of the spectral parameters ${\tau }_{\beta }\left(k\right)$. Hence, the correlation structure of ${\beta }_t$ is described by ${\tau }_{\beta }\left(k\right)$.

Figure 3

Complete Bayesian network and algorithm. The solid arrows show graphical steps by the probabilistic dependencies. The top four boxes indicate the hyperparameters ${\alpha }_{\beta /\gamma }, q_{\beta /\gamma }$, and ${\sigma }_{\beta /\omega }$. Below them follow the spectral parameters ${\tau }_{\beta /\gamma }$ and the tuple of the frequency ${\beta }_t$ and the damping factor ${\gamma }_t$, which form together the signal $s$. Each system class $i$ is thereby characterized by its $s_i$. Beneath that comes the response operator ${\mathit{R}}^{\left(s\right)}$, the training data $x_t$, and finally the observed data, according to Eq. (28). The dashed lines on the left-hand side of the figure display the work flow of the algorithm to learn the time evolution of ${\beta }_t$ and ${\gamma }_t$, including their power spectra from the training data set. On the right-hand side the dotted lines display the work flow of the algorithm to state which of the previously learned training data set explains the observed data $d$ best.

Figure 4

Six wave realizations according to Eq. (9) for the same model class, i.e., all $x_t$ share the same ${\gamma }_t$ and ${\beta }_t$. The amplifying force $\xi$ was drawn from $P\left(\xi |\mathbf{\Xi }\right)=\mathcal{G}\left(\xi ,\mathbf{\Xi }\right)$, with a given constant covariance $\tilde{\mathrm{\Xi }}=5$.

Figure 5

(a) shows the original ${\beta }_t$ as well as its reconstruction for different numbers of data realizations $x_t$ ranging from two $\left(x_{1,2}\right)$ up to six $\left(x_{1,2,3,4,5,6}\right)$. The residuals between the original ${\beta }_t$ and their reconstructions are also shown. (b) shows the same for ${\gamma }_t$ and its reconstructions. One needs at least three $x_t$ to get a proper reconstruction of the fields. Otherwise, the reconstruction shows too many features imprinted by the driving white noise.

Figure 6

The original power spectrum of ${\beta }_t$ as well as its reconstruction for different numbers of data realizations $x_t$ used in its reconstruction.

Figure 7

In (a) the training data set of $s_2$ with its reconstructed parameters, ${\omega }_{\mathrm{rec}}$ and ${\gamma }_{\mathrm{rec}}$ is shown. (b) shows the same for $s_3$.

Figure 8

(a), (b), and (c) show data realizations, Eq. (28) of $s_1$. The SNR was tuned to 10, $0.1$, and $0.001$, respectively.