# Estimators of a Noisy Centered Ornstein-Uhlenbeck Process and Its Noise Variance

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration considers three estimators for a noisy centered Ornstein–Uhlenbeck process. This work is a logical sequel to [1]; they both consider a classic "AR1 plus noise" model for time series, but in [1], the noise variance was assumed to be known. Recall the setting and notations in [1]: an underlying process, assumed to be a stationary solution to the differential equation

[more]
Contributed by: Didier A. Girard (October 2018)

(CNRS-LJK and Univ. Grenoble Alpes)

Open content licensed under CC BY-NC-SA

## Details

Each realization of the underlying process (a non-noisy dataset) is generated by using the built-in function OrnsteinUhlenbeckProcess. Changing consists of adding a standard Gaussian white noise series that is amplified by the chosen to such a non-noisy dataset.

Snapshot 1: Select "diffusion coefficient parameter" 32 (the value of from which the non-noisy data is simulated), being fixed, and choose . This setting (with ) is also analyzed in Snapshot 1 of [1]. There it was observed that CGEMEV with a priori known was much more efficient than the neglecting-errors-ML; this holds as long as the noise level stays greater than, say, . The two methods—ML taking account of an unknown and CGEMEV-ML—have very similar statistical efficiencies.

The third method, CGEMEV using , is obtained by clicking "variogram extrapolation at 0+" instead of "hybrid EV-ML". It is not so bad considering that it is sometimes 10 times faster than CGEMEV-ML. First, as in ML or CGEMEV-ML, a lower bound is imposed on ; precisely is used as soon as . By changing the generation-seed, this lower bound is often active when . This is no longer the case when we come back to . More precisely, for a seed of 6499, you can see that CGEMEV❘ gives a slight underestimation of and an overestimation of ; for other seeds, these errors of estimation are in the opposite direction (see, for instance, a seed of 6517). These "small" inaccuracies of the nonparametric are very often corrected by CGEMEV-ML.

Snapshot 2: Keeping and selecting a small (here ) as the "diffusion coefficient parameter" (so that we are close to the near unit root situation), a noise with can no longer be considered as a negligible noise. This claim can be easily demonstrated by playing with [1] where "not negligible noise" means that the neglecting-errors-ML method cannot be trusted. The good news is that with such small , the noise level can now be reliably estimated, even when it is only , either by ML or CGEMEV-ML. Concerning the starting point (here randomized) required by the iterative maximization of the likelihood, it may sometimes have a non-negligible impact; by changing the randomization seed from 321 to 1492, you can see a weak impact in this setting, even though the estimation of is now slightly worse than the one by CGEMEV-ML. However, if the iteration is stopped too early (try 50 instead of 100), the impact is strong; the seed 2018 greatly deteriorates the ML estimator of .

Snapshot 3: The meaning of "small " in the previous statement depends on , as expected; here with and , the noise is still reliably estimated. This estimation is less accurate as increases; this is the case already with , even for ML where the upper bound , imposed on the SNR, is often active, yet the estimation of the diffusion coefficient by where is arbitrarily set at is still almost as efficient as ML in such a case.

References

[1] D. A. Girard. "Estimating a Centered Ornstein–Uhlenbeck Process under Measurement Errors" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/EstimatingACenteredOrnsteinUhlenbeckProcessUnderMeasurementE.

[2] D. A. Girard, "Efficiently Estimating Some Common Geostatistical Models by ‘Energy–Variance Matching’ or Its Randomized ‘Conditional–Mean’ Versions," *Spatial Statistics*, 21(Part A), 2017 pp. 1–26. doi:10.1016/j.spasta.2017.01.001.

[3] M. Katzfuss and N. Cressie, "Bayesian Hierarchical Spatio‐temporal Smoothing for Very Large Datasets,"
*Environmetrics*, 23(1), 2012 pp. 94–107. doi:10.1002/env.1147.

[4] C. Gu, *Smoothing Spline ANOVA Models*, 2nd ed., New York: Springer, 2013.

[5] D. A. Girard. "Three Alternatives to the Likelihood Maximization for Estimating a Centered Matérn (3/2) Process" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ThreeAlternativesToTheLikelihoodMaximizationForEstimatingACe.

[6] D. A. Girard. "Estimating a Centered Matérn (1) Process: Three Alternatives to Maximum Likelihood via Conjugate Gradient Linear Solvers" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/EstimatingACenteredMatern1ProcessThreeAlternativesToMaximumL.

[7] D. A. Girard. "Estimating a Centered Isotropic Matérn Field from a (Possibly Incomplete and Noisy) Lattice Observation." (Oct 1, 2018) github.com/didiergirard/CGEMEV.

[8] J. Staudenmayer and J.P. Buonaccorsi, "Measurement Error in Linear Autoregressive Models,"
*Journal of the American Statistical Association*, 100(471), 2005 pp. 841–852. doi.org/10.1198/016214504000001871.

## Snapshots

## Permanent Citation