WOLFRAM|DEMONSTRATIONS PROJECT

Estimating a Centered Ornstein-Uhlenbeck Process under Measurement Errors

​
diffusion coefficient
parameter
1
10
1
3
1
2
3
8
32
100
400
noise level σ
0
-4
10
-3
10
-2
10
0.05
seed for data
generation
321
​
dataset size
512
1024
2048
4096
8192
The problem of estimating the two parameters of a stationary process satisfying the differential equation
dx(t)-θx(t)dt+
c
dw(t)
, where
w
follows a standard Wiener process, from
n
observations at equidistant points of the interval
[0,1]
, has been well studied. This is also the classical problem of fitting an autoregressive time series of order 1 (AR1), the case "
n
large" yielding the "near unit root" situation. This Demonstration considers the important case where the observations may have additive measurement errors: we assume that these errors are independent, normal random variables with known variance
2
σ
.
Recall that
θ
, assumed positive, is often referred to as the mean reversion speed (here assume the constant mean of the process is zero). In geostatistics
θ
is called the inverse-range parameter. It is well known that the autoregression coefficient
ρ
in the equivalent AR1 formulation is given by
exp(-θδ)
, where
δ=1/(n-1)
.
Here we use the two parameters
c
(the diffusion coefficient) and
τ=
c/2θ
(recall that
2
τ
is then the marginal variance of the process; see the Details section in the help page for the OrnsteinUhlenbeckProcess function). We restrict ourselves to the case
2
τ
=1
(so that
σ
is also the noise-to-signal ratio).
A simple "solution" to this fitting problem is to neglect the noise, that is, to use the most appealing estimator among those available for the non-noisy case and to substitute the noisy observations, as was studied in [2]. Here as "most appealing" we choose the celebrated maximum likelihood (ML) estimator. Indeed, it is known that this estimator can be exactly and reliably calculated by first solving a simple cubic equation in
ρ
(see [3] and the references therein), the ML estimate of
2
τ
being then an explicit "Gibbs energy" (a quadratic form whose computation cost is of order
n
).
On the other hand, as soon as
σ>0
, the exact maximization of the correctly specified likelihood criterion (the one that takes into account the noise) is not so easy.
This Demonstration considers the recently proposed "CGEM-EV" approach [1]. In short, firstly
2
τ
is simply estimated by the bias-corrected empirical variance, say
2
τ
EV
; secondly an estimating equation is invoked to estimate
c
. Precisely,
c
is searched so that the conditional mean of the "candidate Gibbs energy" (where we substitute
2
τ
EV
in place of the true
2
τ
so that this conditional mean is a function of only
c
) is equal to
2
τ
EV
. It is easy to show that these two equations are unbiased, that is, they are true on average when
c
and
2
τ
are set to their true values (the averaging is ensemble-averaging, i.e. from infinitely repeated simulations of the process and of the noise under the true model). Stronger properties are studied in [1].
Implementation of CGEM-EV is much simpler than exact ML, since it reduces to one-dimensional numerical root finding. A simple fixed-point algorithm is used here. It proves to be reliable (with fast convergence) for all the settings in this Demonstration.