2,553 research outputs found
A variational approach to path estimation and parameter inference of hidden diffusion processes
We consider a hidden Markov model, where the signal process, given by a
diffusion, is only indirectly observed through some noisy measurements. The
article develops a variational method for approximating the hidden states of
the signal process given the full set of observations. This, in particular,
leads to systematic approximations of the smoothing densities of the signal
process. The paper then demonstrates how an efficient inference scheme, based
on this variational approach to the approximation of the hidden states, can be
designed to estimate the unknown parameters of stochastic differential
equations. Two examples at the end illustrate the efficacy and the accuracy of
the presented method.Comment: 37 pages, 2 figures, revise
A selective overview of nonparametric methods in financial econometrics
This paper gives a brief overview on the nonparametric techniques that are
useful for financial econometric problems. The problems include estimation and
inferences of instantaneous returns and volatility functions of
time-homogeneous and time-dependent diffusion processes, and estimation of
transition densities and state price densities. We first briefly describe the
problems and then outline main techniques and main results. Some useful
probabilistic aspects of diffusion processes are also briefly summarized to
facilitate our presentation and applications.Comment: 32 pages include 7 figure
Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression
In this paper, we revisit batch state estimation through the lens of Gaussian
process (GP) regression. We consider continuous-discrete estimation problems
wherein a trajectory is viewed as a one-dimensional GP, with time as the
independent variable. Our continuous-time prior can be defined by any
nonlinear, time-varying stochastic differential equation driven by white noise;
this allows the possibility of smoothing our trajectory estimates using a
variety of vehicle dynamics models (e.g., `constant-velocity'). We show that
this class of prior results in an inverse kernel matrix (i.e., covariance
matrix between all pairs of measurement times) that is exactly sparse
(block-tridiagonal) and that this can be exploited to carry out GP regression
(and interpolation) very efficiently. When the prior is based on a linear,
time-varying stochastic differential equation and the measurement model is also
linear, this GP approach is equivalent to classical, discrete-time smoothing
(at the measurement times); when a nonlinearity is present, we iterate over the
whole trajectory to maximize accuracy. We test the approach experimentally on a
simultaneous trajectory estimation and mapping problem using a mobile robot
dataset.Comment: Submitted to Autonomous Robots on 20 November 2014, manuscript #
AURO-D-14-00185, 16 pages, 7 figure
Parameter estimation for macroscopic pedestrian dynamics models from microscopic data
In this paper we develop a framework for parameter estimation in macroscopic
pedestrian models using individual trajectories -- microscopic data. We
consider a unidirectional flow of pedestrians in a corridor and assume that the
velocity decreases with the average density according to the fundamental
diagram. Our model is formed from a coupling between a density dependent
stochastic differential equation and a nonlinear partial differential equation
for the density, and is hence of McKean--Vlasov type. We discuss
identifiability of the parameters appearing in the fundamental diagram from
trajectories of individuals, and we introduce optimization and Bayesian methods
to perform the identification. We analyze the performance of the developed
methodologies in various situations, such as for different in- and outflow
conditions, for varying numbers of individual trajectories and for differing
channel geometries
Double Diffusion Encoding Prevents Degeneracy in Parameter Estimation of Biophysical Models in Diffusion MRI
Purpose: Biophysical tissue models are increasingly used in the
interpretation of diffusion MRI (dMRI) data, with the potential to provide
specific biomarkers of brain microstructural changes. However, the general
Standard Model has recently shown that model parameter estimation from dMRI
data is ill-posed unless very strong magnetic gradients are used. We analyse
this issue for the Neurite Orientation Dispersion and Density Imaging with
Diffusivity Assessment (NODDIDA) model and demonstrate that its extension from
Single Diffusion Encoding (SDE) to Double Diffusion Encoding (DDE) solves the
ill-posedness and increases the accuracy of the parameter estimation. Methods:
We analyse theoretically the cumulant expansion up to fourth order in b of SDE
and DDE signals. Additionally, we perform in silico experiments to compare SDE
and DDE capabilities under similar noise conditions. Results: We prove
analytically that DDE provides invariant information non-accessible from SDE,
which makes the NODDIDA parameter estimation injective. The in silico
experiments show that DDE reduces the bias and mean square error of the
estimation along the whole feasible region of 5D model parameter space.
Conclusions: DDE adds additional information for estimating the model
parameters, unexplored by SDE, which is enough to solve the degeneracy in the
NODDIDA model parameter estimation.Comment: 22 pages, 7 figure
A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data
In financial markets, low prices are generally associated with high
volatilities and vice-versa, this well known stylized fact usually being
referred to as leverage effect. We propose a local volatility model, given by a
stochastic differential equation with piecewise constant coefficients, which
accounts of leverage and mean-reversion effects in the dynamics of the prices.
This model exhibits a regime switch in the dynamics accordingly to a certain
threshold. It can be seen as a continuous-time version of the Self-Exciting
Threshold Autoregressive (SETAR) model. We propose an estimation procedure for
the volatility and drift coefficients as well as for the threshold level.
Parameters estimated on the daily prices of 348 stocks of NYSE and S\&P 500, on
different time windows, show consistent empirical evidence for leverageeffects.
Mean-reversion effects are also detected, most markedly in crisis periods
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