8 research outputs found
Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs
Approximate Bayesian Computation (ABC) has become one of the major tools of
likelihood-free statistical inference in complex mathematical models.
Simultaneously, stochastic differential equations (SDEs) have developed to an
established tool for modelling time dependent, real world phenomena with
underlying random effects. When applying ABC to stochastic models, two major
difficulties arise. First, the derivation of effective summary statistics and
proper distances is particularly challenging, since simulations from the
stochastic process under the same parameter configuration result in different
trajectories. Second, exact simulation schemes to generate trajectories from
the stochastic model are rarely available, requiring the derivation of suitable
numerical methods for the synthetic data generation. To obtain summaries that
are less sensitive to the intrinsic stochasticity of the model, we propose to
build up the statistical method (e.g., the choice of the summary statistics) on
the underlying structural properties of the model. Here, we focus on the
existence of an invariant measure and we map the data to their estimated
invariant density and invariant spectral density. Then, to ensure that these
model properties are kept in the synthetic data generation, we adopt
measure-preserving numerical splitting schemes. The derived property-based and
measure-preserving ABC method is illustrated on the broad class of partially
observed Hamiltonian type SDEs, both with simulated data and with real
electroencephalography (EEG) data. The proposed ingredients can be incorporated
into any type of ABC algorithm and directly applied to all SDEs that are
characterised by an invariant distribution and for which a measure-preserving
numerical method can be derived.Comment: 35 pages, 21 figure
Network inference in a stochastic multi-population neural mass model via approximate Bayesian computation
In this article, we propose a 6N-dimensional stochastic differential equation
(SDE), modelling the activity of N coupled populations of neurons in the brain.
This equation extends the Jansen and Rit neural mass model, which has been
introduced to describe human electroencephalography (EEG) rhythms, in
particular signals with epileptic activity. Our contributions are threefold:
First, we introduce this stochastic N-population model and construct a reliable
and efficient numerical method for its simulation, extending a splitting
procedure for one neural population. Second, we present a modified Sequential
Monte Carlo Approximate Bayesian Computation (SMC-ABC) algorithm to infer both
the continuous and the discrete model parameters, the latter describing the
coupling directions within the network. The proposed algorithm further develops
a previous reference-table acceptance rejection ABC method, initially proposed
for the inference of one neural population. On the one hand, the considered
SMC-ABC approach reduces the computational cost due to the basic
acceptance-rejection scheme. On the other hand, it is designed to account for
both marginal and coupled interacting dynamics, allowing to identify the
directed connectivity structure. Third, we illustrate the derived algorithm on
both simulated data and real multi-channel EEG data, aiming to infer the
brain's connectivity structure during epileptic seizure. The proposed algorithm
may be used for parameter and network estimation in other multi-dimensional
coupled SDEs for which a suitable numerical simulation method can be derived.Comment: 28 pages, 11 figure
Granger Causal Inference in Multivariate Hawkes Processes by Minimum Message Length
Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used
to model various real-life phenomena: earthquakes, operations on stock markets,
neuronal activity, virus propagation and many others. In this paper, we focus
on MHPs with exponential decay kernels and estimate connectivity graphs, which
represent the Granger causal relations between their components. We approach
this inference problem by proposing an optimization criterion and model
selection algorithm based on the minimum message length (MML) principle. MML
compares Granger causal models using the Occam's razor principle in the
following way: even when models have a comparable goodness-of-fit to the
observed data, the one generating the most concise explanation of the data is
preferred. While most of the state-of-art methods using lasso-type penalization
tend to overfitting in scenarios with short time horizons, the proposed
MML-based method achieves high F1 scores in these settings. We conduct a
numerical study comparing the proposed algorithm to other related classical and
state-of-art methods, where we achieve the highest F1 scores in specific sparse
graph settings. We illustrate the proposed method also on G7 sovereign bond
data and obtain causal connections, which are in agreement with the expert
knowledge available in the literature.Comment: 23 pages, 5 figure
A splitting method for SDEs with locally Lipschitz drift : illustration on the FitzHugh-Nagumo model
In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms
Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motion
We provide a comparative analysis of qualitative features of different numerical methods for the inhomogeneous geometric Brownian motion (IGBM). The limit distribution of the IGBM exists, its conditional and asymptotic mean and variance are known and the process can be characterised according to Feller’s boundary classification. We compare the frequently used Euler–Maruyama and Milstein methods, two Lie–Trotter and two Strang splitting schemes and two methods based on the ordinary differential equation (ODE) approach, namely the classical Wong–Zakai approximation and the recently proposed log-ODE scheme. First, we prove that, in contrast to the Euler–Maruyama and Milstein schemes, the splitting and ODE schemes preserve the boundary properties of the process, independently of the choice of the time discretisation step. Second, we prove that the limit distribution of the splitting and ODE methods exists for all stepsize values and parameters. Third, we derive closed-form expressions for the conditional and asymptotic means and variances of all considered schemes and analyse the resulting biases. While the Euler–Maruyama and Milstein schemes are the only methods which may have an asymptotically unbiased mean, the splitting and ODE schemes perform better in terms of variance preservation. The Strang schemes outperform the Lie–Trotter splittings, and the log-ODE scheme the classical ODE method. The mean and variance biases of the log-ODE scheme are very small for many relevant parameter settings. However, in some situations the two derived Strang splittings may be a better alternative, one of them requiring considerably less computational effort than the log-ODE method. The proposed analysis may be carried out in a similar fashion on other numerical methods and stochastic differential equations with comparable features
Stochastic Oscillations Motivated by the Noisy and Rhythmic Firing Activity of Neurons
A human brain contains billions of neurons. These are extremely complex dynamical systems that are affected by intrinsic channel noise and extrinsic synaptic noise. Oscillatory behaviour is a phenomena that arises in single neurons as well as in neuronal networks. Based on the noisy and rhythmic firing activity of nerve cells, the aim of this thesis is to provide an overview on the existing theory of stochastic oscillations. The main question is how oscillations can be defined mathematically in a stochastic setting. This work suggests two very different definitions of a stochastic oscillator according to certain well-known deterministic tools. The first one states that the solution of a specific two-dimensional stochastic differential equation is a stochastic oscillator if it has infinitely many simple zeros almost surely. The second definition of a stochastic oscillator corresponds to the concept of a random periodic solution that is based on the cocycle property. Two particular stochastic equations are introduced each satisfying one of these definitions. For this purpose, two detailed proofs on the validity of the first and the second definition, respectively, are presented. A second topic addressed by this work is the stability of stochastic equations. The system energy carries information on how the trajectory propagates in the phase plane. Besides, Lyapunov exponents describe the asymptotic exponential growth of random dynamical systems. Finally, this work provides an application of the developed theory to specific standard oscillatory models as well as to the Van der Pol oscillator on which the FitzHugh- Nagumo neuron model is based. Sample path simulations of the stochastic equations are provided by the implementation of the Euler-Maruyama method in MATLAB. Furthermore, an exact simulation method applied to the stochastic harmonic oscillator equation is introduced.submitted by Irene TubikanecUniversität Linz, Masterarbeit, 2017(VLID)170673
Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis
International audienceOscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure