Reconstruction of stochastic nonlinear dynamical models from trajectory measurements

An algorithm is presented for reconstructing stochastic nonlinear dynamical models from noisy time-series data. The approach is analytical; consequently, the resulting algorithm does not require an extensive global search for the model parameters, provides optimal compensation for the effects of dynamical noise, and is robust for a broad range of dynamical models. The strengths of the algorithm are illustrated by inferring the parameters of the stochastic Lorenz system and comparing the results with those of earlier research. The efficiency and accuracy of the algorithm are further demonstrated by inferring a model for a system of five globally and locally coupled noisy oscillators.

Figure 1

The phase portrait of the chaotic nonlinear Lorenz system (23) with the standard parameters (see text): (a) deterministic system; (b) stochastic system with strong dynamical noise, simulated with a diagonal diffusion matrix having elements $D_{11}=1500$, $D_{22}=1600$, and $D_{33}=1700$. (All quantities in the equations and figures are dimensionless in this paper.)

Figure 2

Results for (a) the posterior mean (i.e., inferred value) and (b) the posterior variance (i.e., associated uncertainty) of the model parameter $a_{21}$ corresponding to parameter $r$ of the Lorenz system (24) as a function of increasing observation duration. Curve 1, $\left\{D_{ll}\right\}=\{0.01,0.012,0.014\}$, $h=0.002$; curve 2, $\left\{D_{ll}\right\}=\{100,120,140\}$, $h=0.00002$. The time instant of steplike decrease in the variance is indicated by the vertical dashed line.

Figure 3

Demonstration of improved inference accuracy due to the prefactor (9) in the likelihood function. The true value of the parameter being inferred is indicated by the vertical dashed line. The solid and dashed curves, respectively, show the histograms of parameter values inferred by our algorithm and by the generalized least-squares method, which lacks the Jacobian prefactor. The histograms were built from an ensemble of 1000 numerical experiments with 90 000 data points each.

Figure 4

The phase portrait of the system (25) as projected onto the $(u_1,u_2,u_3)$ subspace: (a) deterministic system; (b) stochastic system with a diagonal diffusion matrix of the form $\widehat{\mathbf{D}}=100\widehat{\mathbf{I}}$. (Note the scale change between the axes of the two figures.) See Table and Fig. 5 for the values of some of the model parameters used in the simulation.

Figure 5

Accurate inference of (a) the parameters of the fifth oscillator in the system (25) and (b) the elements of the last row of the diffusion matrix $\widehat{\mathbf{Q}}$. The horizontal axes show the number of blocks of data used, with 800 points in each block, sampled at $h=0.02$.

Figure 6

Further demonstration of improved inference accuracy due to the prefactor (9) in the likelihood function. The true value of the parameter being inferred is indicated by the vertical dashed line. Histograms 1 and 2 show results obtained with our algorithm, while histograms $1^{\prime }$ and $2^{\prime }$ are due to the GLS method, showing the detrimental effect on inference accuracy of the missing prefactor. The diffusion matrix was $\widehat{\mathbf{Q}}$ for curves 1 and $1^{\prime }$, and $2\widehat{\mathbf{Q}}$ for curves 2 and $2^{\prime }$ (see text).