On the optimal designs for the prediction of complex Ornstein-Uhlenbeck processes

Physics, chemistry, biology or finance are just some examples out of the many fields where complex Ornstein-Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, the small deviation in the Earth's axis of rotation. A common problem in these applications is deciding how to choose a set of a sample locations in order to predict a random process in an optimal way. We study the optimal design problem for the prediction of a complex OU process on a compact interval with respect to integrated mean square prediction error (IMSPE) and entropy criteria. We derive the exact forms of both criteria, moreover, we show that optimal designs based on entropy criterion are equidistant, whereas the IMSPE based ones may differ from it. Finally, we present some numerical experiments to illustrate selected cases of optimal designs for small number of sampling locations.


Introduction
Physics, chemistry, biology or finance are just some examples out of the many fields where random processes have various applications in statistical modelling. In the current paper we study the problem of optimal design for the prediction of a complex Ornstein-Uhlenbeck (OU) process on a compact interval with respect to the Integrated Mean Square Prediction Error (IMSPE) and entropy criteria. A complex OU process (see e.g. Arató, 1982), defined in detail in Section 2, is used in the study of rotating waves in time-dependent reaction diffusion systems (Beyn and Lorenz, 2008;Otten, 2015), it can also describe the motion of a charged test particle in a constant magnetic field (Balescu, 1997), and has several applications in financial mathematics as well (see e.g. Barndorff-Nielsen and Shephard, 2001). An important application of the complex OU is the so-called Chandler wobble, the small deviation in the Earth's axis of rotation relative to the solid earth (Lambeck, 1980). The uniqueness of this motion is the change of about 9 meters (30 ft) in the point where the axis intersects the Earths surface and it has a period of 435 days (14 months). Recently monitoring of the polar motion is done by the International Earth Rotation Service (IERS). Kolmogorov described the Chandler Wobble using the model where Z 1 (t) and Z 2 (t) are the coordinates of the deviation of the instantaneous pole from the North Pole (Arató et al., 1962). In this model the first term is a periodical component, whereas the second one Y (t) is a complex OU process.
For the complex OU process Baran et al. (2018) derived the exact form of the Fisher information matrix (FIM) and investigated the properties of D-optimal designs for estimation of model parameters. Here we derive exact form of the IMSPE already studied e.g. in Baran et al. (2013), Baran et al. (2015) or Crary (2002), and show that in contrast to the case of classical real OU process on a compact interval investigated by Baldi Antognini and Zagoraiou (2010), the equidistant design is usually not optimal. We also investigate the properties of the optimal design with respect to the entropy criterion (Shewry and Wynn, 1987;Baldi Antognini and Zagoraiou, 2010). The paper is organized as follows. Section 2 gives the necessary definitions connected to the complex OU process, Sections 3 and 4 contain the results about IMSPE and entropy criteria, respectively, whereas in Section 5 some simulation results are presented. The paper ends with the concluding Section 6 and to maintain the continuity of the explanation, proofs are provided in the Appendix.

Complex Ornstein-Uhlenbeck process with a trend
Let Y (t) = Y 1 (t) + iY 2 (t), t ≥ 0, be a complex valued stationary OU process defined by the stochastic differential equation (SDE) , is a standard complex Brownian motion, with W 1 (t) and W 2 (t) being independent standard Brownian motions, and ξ = ξ 1 + iξ 2 , where ξ 1 and ξ 2 are centered normal random variables that are chosen according to stationarity (Arató, 1982). In the current paper we consider the shifted complex stochastic process Z(t) = Z 1 (t) + iZ 2 (t), defined as where m = m 1 + im 2 , m 1 , m 2 ∈ R, and Y (t), t ≥ 0, is a complex valued stationary OU process, observed in design points taken from the non-negative half-line R + .
In order to simplify calculations, instead of the complex process Y (t) one can use the two-dimensional real valued stationary OU process Y 1 (t), Y 2 (t) ⊤ defined by the SDE Note that in physics (2.3) this is called A-Langevin equation, see e.g. Balescu Balescu (1997). The real and imaginary parts of a complex OU process form a two-dimensional real OU process satisfying (2.3), and conversely, if will be a complex OU process which solves (2.1). Naturally, EY 1 (t) = EY 2 (t) = 0, and the covariance matrix function of the process Y 1 (t), Y 2 (t) ⊤ is given by This results in a complex covariance function of the complex OU process Y (t) defined by (2.1) of the form behaving like a damped oscillation with frequency ω.
In the next steps we will consider the damping parameter λ, frequency ω and standard deviation σ as known. Nevertheless a promising direction moving forward could be the examination of models where the above mentioned parameters should also be estimated. Note that the estimation of σ can be done on the basis of a single realization of the complex process, see e.g. Arató (1982, Chapter 4). Now, without loss of generality, one can set the variances of Y 1 (t) and Y 2 (t) to be equal to one, that is σ 2 /(2λ) = 1, which reduces R(τ ) to a correlation matrix function. In Arató et al. (1999) we can find more results on the maximum-likelihood estimation of the covariance parameters.

Optimal design with respect to the IMSPE criterion
Assume we observe our complex process Z(t) at design points 0 ≤ t 1 < t 2 < . . . < t n resulting in the 2n-dimensional real vector The main aim of the kriging technique consists of predicting the output of the investigated process or field on the experimental region, and for any untried location x ∈ X , which in our case lies in a closed interval X ⊂ R, the estimation procedure is focused on the best linear unbiased estimator (BLUE) Z(x) of Z(x). Considering again the two-dimensional vector form of the complex process, the real and imaginary parts Z 1 (x) and Z 2 (x), respectively, of the BLUE are given as and ( m 1 , m 2 ) ⊤ is the generalized least squares estimator of (m 1 , m 2 ) ⊤ , that is Thus, a natural criterion is to minimize suitable functionals of the mean squared prediction error (MSPE) given by where I k and O k , k ∈ N, denote the k-dimensional unit and null matrices, respectively. Since the prediction accuracy is often related to the entire prediction region X , the design criterion IMSPE is given by Without loss of generality we may assume X = [0, 1], therefore, as extrapolative prediction is not advisable in kriging, we can set t 1 = 0 and t n = 1.
Example 3.2 As an illustration consider a three-point design, that is n = 3, t 1 = 0, t 2 := d, t 3 = 1. Figure 1a shows the mean squared prediction error (MSPE) function for λ = 1, ω = 4 together with the corresponding contour plot (Figure 1b). In Figures  1c and 1d the IMSPE corresponding to the equidistant three-point design for the prediction as function of λ and ω and its contour plot, respectively, are given. Straightforward calculation shows that in this case d = 1 2 is a minimizer of IMSPE Z for all possible values of λ and ω, that is the optimal design is equidistant.
Example 3.3 Consider now the four-point design {0, d 1 , d 1 + d 2 , 1}. In this case the partial derivatives of IMSPE Z with respect to d 1 and d 2 at d 1 = d 2 = 1/3 are not necessary zero, that is in general, one cannot state that the equidistant design is optimal.

Optimal information gain for complex OU process
Another approach to optimal design is to find locations which maximize the amount of obtained information. Following the ideas of Shewry and Wynn (1987) one has to maximize the entropy Ent(Z) of the observations corresponding to the chosen design which in our Gaussian case forms a 2n-dimensional normal vector with covariance matrix σ 2 2λ C(n), that is Ent(Z) = n 1 + ln πσ 2 λ + 1 2 ln det C(n). For any sample size the equidistant design d 1 = d 2 = . . . = d n−1 is optimal with respect to the entropy criterion.

Numerical experiments
In order to compare the performances of the two examined criteria we compare the optimal values of IMSPE( Z) calculated using fmincon function of Matlab to its values corresponding to the equispaced design which is optimal for the entropy criterion. In Table 1 the values of IMSPE are given for both designs together with the relative efficiency of the equispaced monotonic design with respect to the optimal one for various sample sizes and combinations of damping parameter λ and frequency ω. Observing that for the three-point design (n = 3), the efficiency of the equispaced design is 100%, so in this case the equidistant design is optimal. In the other cases it is noticeable, that the optimal values of IMSPE( Z) are better than the optimal values of entropy criterion. The damping parameter λ and frequency ω used for the simulations are estimated based on public pole coordinates from the IERS EOP C01 IAU2000. For the estimation as a first step we give the model that fits best the real polar motion in the least squares sense. Then, according to the Kolmogorov's model (1.1) of the Chandler wobble, deviation of the real and fitted yearly polar motion results in the required complex OU process. The estimation of parameters based on the given complex OU process and the necessary maximum likelihood estimators for λ and ω are given in Arató (1968Arató ( , 1982. As an example Figures 2 shows the real and fitted yearly polar motion based on data from the past three years, together with the corresponding deviation modelled as a complex OU process.

Conclusions
We derive the exact form of the IMSPE for a shifted complex OU process on a compact interval and show that optimal design for prediction based on IMSPE may well differ from the equidistant one. This is in contrast both to the D-optimal design for estimation (Baran et al., 2018) and to the case of the classical real OU process (Baldi Antognini and Zagoraiou, 2010), but similar to the optimal design for the prediction of OU sheets on a monotonic set (Baran et al., 2013). We also investigate the properties of the optimal design with respect to entropy criterion and we show that the optimal designs are equidistant. Simulations illustrate selected cases of optimal designs for small number of sampling locations. The damping parameter λ and frequency parameter ω used for the simulations are estimated based on real data (pole coordinates from the IERS EOP C01 IAU2000), which is a well known application of the complex OU process, namely Kolmogorov's model (1.1) of the Chandler wobble. Since the above discussed designs depend on values of damping and frequency parameters, obtained optimal designs are only locally optimal. Such knowledge may be crucial for experiments to increase efficiency of design in practical setups. governmental S&T cooperation program TÉT 15-1-2016-0046. Kinga Sikolya acknowledges the support of theÚNKP-17-4 New National Excellence Program of the Ministry of Human Capacities. The project has been supported by the European Union, co-financed by the European Social Fund. Sándor Baran is grateful for the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
A.2 Proof of Theorem 4.1 Following the idea of Baldi Antognini and Zagoraiou (2010, Lemma 3.1) the covariance matrix C(n) can be written as C(n) = LDU, where L is a lower, U is an upper block triangular matrix with I 2 as blocks in the main diagonal and D is block diagonal matrix with Since ∂ 2 f (d) ∂d 2 = − 8λ 2 e 2λd (e 2λd − 2) 2 < 0 for any d ∈ (0, 1), f (x) is a concave function of x, and the result follows from Schur-concavity of the entropy criterion.