47,186 research outputs found
AReS and MaRS - Adversarial and MMD-Minimizing Regression for SDEs
Stochastic differential equations are an important modeling class in many
disciplines. Consequently, there exist many methods relying on various
discretization and numerical integration schemes. In this paper, we propose a
novel, probabilistic model for estimating the drift and diffusion given noisy
observations of the underlying stochastic system. Using state-of-the-art
adversarial and moment matching inference techniques, we avoid the
discretization schemes of classical approaches. This leads to significant
improvements in parameter accuracy and robustness given random initial guesses.
On four established benchmark systems, we compare the performance of our
algorithms to state-of-the-art solutions based on extended Kalman filtering and
Gaussian processes.Comment: Published at the Thirty-sixth International Conference on Machine
Learning (ICML 2019
Statistical inference in mechanistic models: time warping for improved gradient matching
Inference in mechanistic models of non-linear differential equations is a challenging problem in current computational statistics. Due to the high computational costs of numerically solving the differential equations in every step of an iterative parameter adaptation scheme, approximate methods based on gradient matching have become popular. However, these methods critically depend on the smoothing scheme for function interpolation. The present article adapts an idea from manifold learning and demonstrates that a time warping approach aiming to homogenize intrinsic length scales can lead to a significant improvement in parameter estimation accuracy. We demonstrate the effectiveness of this scheme on noisy data from two dynamical systems with periodic limit cycle, a biopathway, and an application from soft-tissue mechanics. Our study also provides a comparative evaluation on a wide range of signal-to-noise ratios
Graph matching with a dual-step EM algorithm
This paper describes a new approach to matching geometric structure in 2D point-sets. The novel feature is to unify the tasks of estimating transformation geometry and identifying point-correspondence matches. Unification is realized by constructing a mixture model over the bipartite graph representing the correspondence match and by affecting optimization using the EM algorithm. According to our EM framework, the probabilities of structural correspondence gate contributions to the expected likelihood function used to estimate maximum likelihood transformation parameters. These gating probabilities measure the consistency of the matched neighborhoods in the graphs. The recovery of transformational geometry and hard correspondence matches are interleaved and are realized by applying coupled update operations to the expected log-likelihood function. In this way, the two processes bootstrap one another. This provides a means of rejecting structural outliers. We evaluate the technique on two real-world problems. The first involves the matching of different perspective views of 3.5-inch floppy discs. The second example is furnished by the matching of a digital map against aerial images that are subject to severe barrel distortion due to a line-scan sampling process. We complement these experiments with a sensitivity study based on synthetic data
Reparameterizing the Birkhoff Polytope for Variational Permutation Inference
Many matching, tracking, sorting, and ranking problems require probabilistic
reasoning about possible permutations, a set that grows factorially with
dimension. Combinatorial optimization algorithms may enable efficient point
estimation, but fully Bayesian inference poses a severe challenge in this
high-dimensional, discrete space. To surmount this challenge, we start with the
usual step of relaxing a discrete set (here, of permutation matrices) to its
convex hull, which here is the Birkhoff polytope: the set of all
doubly-stochastic matrices. We then introduce two novel transformations: first,
an invertible and differentiable stick-breaking procedure that maps
unconstrained space to the Birkhoff polytope; second, a map that rounds points
toward the vertices of the polytope. Both transformations include a temperature
parameter that, in the limit, concentrates the densities on permutation
matrices. We then exploit these transformations and reparameterization
gradients to introduce variational inference over permutation matrices, and we
demonstrate its utility in a series of experiments
Parameter Inference in Differential Equation Models of Biopathways using Time Warped Gradient Matching
Parameter inference in mechanistic models of biopathways based on systems
of coupled differential equations is a topical yet computationally challenging problem,
due to the fact that each parameter adaptation involves a numerical integration of the
differential equations. Techniques based on gradient matching, which aim to minimize
the discrepancy between the slope of a data interpolant and the derivatives predicted
from the differential equations, offer a computationally appealing shortcut to the inference
problem. However, gradient matching critically hinges on the smoothing scheme
for function interpolation, with spurious wiggles in the interpolant having a dramatic
effect on the subsequent inference. The present article demonstrates that a time warping
approach aiming to homogenize intrinsic functional length scales can lead to a signifi-
cant improvement in parameter estimation accuracy. We demonstrate the effectiveness
of this scheme on noisy data from a dynamical system with periodic limit cycle and a
biopathway
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High-speed multi-dimensional relative navigation for uncooperative space objects
This work proposes a high-speed Light Detection and Ranging (LIDAR) based navigation architecture that is appropriate for uncooperative relative space navigation applications. In contrast to current solutions that exploit 3D LIDAR data, our architecture transforms the odometry problem from the 3D space into multiple 2.5D ones and completes the odometry problem by utilizing a recursive filtering scheme. Trials evaluate several current state-of-the-art 2D keypoint detection and local feature description methods as well as recursive filtering techniques on a number of simulated but credible scenarios that involve a satellite model developed by Thales Alenia Space (France). Most appealing performance is attained by the 2D keypoint detector Good Features to Track (GFFT) combined with the feature descriptor KAZE, that are further combined with either the H∞ or the Kalman recursive filter. Experimental results demonstrate that compared to current algorithms, the GFTT/KAZE combination is highly appealing affording one order of magnitude more accurate odometry and a very low processing burden, which depending on the competitor method, may exceed one order of magnitude faster computation
Approximate Bayesian Computation in State Space Models
A new approach to inference in state space models is proposed, based on
approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood
function by matching observed summary statistics with statistics computed from
data simulated from the true process; exact inference being feasible only if
the statistics are sufficient. With finite sample sufficiency unattainable in
the state space setting, we seek asymptotic sufficiency via the maximum
likelihood estimator (MLE) of the parameters of an auxiliary model. We prove
that this auxiliary model-based approach achieves Bayesian consistency, and
that - in a precise limiting sense - the proximity to (asymptotic) sufficiency
yielded by the MLE is replicated by the score. In multiple parameter settings a
separate treatment of scalar parameters, based on integrated likelihood
techniques, is advocated as a way of avoiding the curse of dimensionality. Some
attention is given to a structure in which the state variable is driven by a
continuous time process, with exact inference typically infeasible in this case
as a result of intractable transitions. The ABC method is demonstrated using
the unscented Kalman filter as a fast and simple way of producing an
approximation in this setting, with a stochastic volatility model for financial
returns used for illustration
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