Deterministic inference for stochastic systems using multiple shooting and a linear noise approximation for the transition probabilities

Abstract

Estimating model parameters from experimental data is a crucial technique for working with computational models in systems biology. Since stochastic models are increasingly important, parameter estimation methods for stochastic modelling are also of increasing interest. This study presents an extension to the 'multiple shooting for stochastic systems (MSS)' method for parameter estimation. The transition probabilities of the likelihood function are approximated with normal distributions. Means and variances are calculated with a linear noise approximation on the interval between succeeding measurements. The fact that the system is only approximated on intervals which are short in comparison with the total observation horizon allows to deal with effects of the intrinsic stochasticity. The study presents scenarios in which the extension is essential for successfully estimating the parameters and scenarios in which the extension is of modest benefit. Furthermore, it compares the estimation results with reversible jump techniques showing that the approximation does not lead to a loss of accuracy. Since the method is not based on stochastic simulations or approximative sampling of distributions, its computational speed is comparable with conventional least-squares parameter estimation methods.

Fig. 1

Outline of the simulation study Time courses are simulated (5 small graphics) and used as pseudo data for a parameter estimation (rep resented by the arrows). Each time series results in one estimator plotted in the coordinate system in the middle. As each time series is intrinsically stochastic, the estimators are random variables clustering around the true parameter value. To judge “how close” they are, two statistical quantities are displayed: their average av,. to see if the method is unbiased, and their average relative error ARE, to see how far they spread around the true value. Note that for a parameter estimation one time series is necessary. Only for the simulation study more than one time series is needed.

Fig. 2

Estimates for an immigration‐death scenario For the scenario of 21 measurements with Δt = 2 (table 1, first row, first column) and θ ⁽⁰⁾ = (0.6,0.03) the graphic shows the 100 estimates (black dots) and the true parameter value θ ⁽⁰⁾ (big grey dot). Using a deterministic model, only the ratio but not the absolute value of the parameters is identifiable. Using the MSS method, one can see that not only the ratio but also the absolute value is identifiable, however, with a slightly worse accuracy.

Fig. 3

Checking the approximation for one time course This figure checks how well the approximation works for a time course of the Immigration‐Death model. As the data sets contain 21 measurements, 21‐1 N(0, 1) random variables are drawn and a density estimation is performed with the SmoothKernelDistribution function in Mathematica 9 [39]. This action is repeated 1000 times. The solid line shows the mean of the 1000 densities and the yellow area fills the area from the 10%‐quantile to the 90%‐quantile, the grey area from the 1%‐quantile to the 99%‐quantile. The graphic shows an example of the ID11 scenario. The dashed line shows the density estimate of the (r ₁,…,r ₂₀) of equation (5). One can see that these numbers are compatible with the assumption of a N(0, 1) distribution which is a condition for the validity of the approximation.

Fig. 4

Different scenarios for Lotka–Volterra The stochastic Lotka–Volterra model can show a behaviour that is qualitatively different from the deterministic Lotka–Volterra model. Panel A shows oscillations in which the intrinsic stochasticity influences amplitude and frequency. Panel B shows a case in which the prey Y ⁽¹⁾ dies out and then after that the predator Y ⁽²⁾. Panel C shows a scenario in which the predator dies out and then the prey population explodes