# An adaptive scheduling tool to optimize measurements to reach a scientific objective: methodology and application to the measurements of stellar orbits in the Galactic Center

A. Hees, A. Dehghanfar, T. Do, A.M. Ghez, G.D. Martinez, R. Campbell, J.R. Lu

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A. Hees, A. Dehghanfar, T. Do, A.M. Ghez, G.D. Martinez, et al.. An adaptive scheduling tool to optimize measurements to reach a scientific objective: methodology and application to the measurements of stellar orbits in the Galactic Center. The Astrophysical Journal, 2019, 880, pp.87. 10.3847/1538-4357/ab2ae0. hal-02179521

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# An Adaptive Scheduling Tool to Optimize Measurements to Reach a Scientific Objective: Methodology and Application to Measurements of Stellar Orbits in the Galactic Center

A. Hees¹,², A. Dehghanfar²,³, T. Do², A. M. Ghez², G. D. Martinez², R. Campbell⁴, and J. R. Lu⁵

¹SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, LNE, 61 avenue de l'Observatoire, F-75014 Paris, France

²Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA

³Institut de Planétologie et d'Astrophysique de Grenoble (IPAG), 120 rue de la Piscine, F-38041 Grenoble, France

⁴W. M. Keck Observatory, 65-1120 Mamalahoa Highway, Kamuela, HI 96743, USA

⁵Astronomy Department, University of California, Berkeley, CA 94720, USA

Received 2019 March 6; revised 2019 June 4; accepted 2019 June 17; published 2019 July 30

## Abstract

In various fields of physics and astronomy, access to experimental facilities or to telescopes is becoming more and more competitive and limited. It therefore becomes important to optimize the type of measurements and their scheduling to reach a given scientific objective and to increase the chances of success of a scientific project. In this communication, extending the work of Ford and of Loredo et al., we present an efficient adaptive scheduling tool aimed at prioritizing measurements in order to reach a scientific goal. The algorithm, based on the Fisher matrix, can be applied to a wide class of measurements. We present this algorithm in detail and discuss some practicalities such as systematic errors or measurement losses due to contingencies (such as weather, experimental failure, ...). As an illustration, we consider measurements of the short-period star S0-2 in our Galactic Center (GC). We show that the radial velocity measurements at the two turning points of the radial velocity curve are more powerful for detecting the gravitational redshift than measurements at the maximal relativistic signal. We also explicitly present the methodology that was used to plan measurements in order to detect the relativistic redshift considering systematics and possible measurement losses. For the future, we identify the astrometric turning points to be highly sensitive to the relativistic advance of the periastron. Finally, we also identify measurements particularly sensitive to the distance to our GC: the radial velocities around periastron and the astrometric measurements just before closest approach and at the maximal R.A. astrometric turning point.

**Key words:** Galaxy: center – methods: data analysis – methods: statistical

## 1. Introduction

In all fields of physics, measurements and observations are costly and access to experimental facilities or telescopes becomes more and more competitive. It is therefore of prime importance to optimize measurements in order to make the most out of the time allocated to measurements or observations. Many examples of measurements that would benefit from an optimization can be found, such as measurements of stellar orbits in the Galactic Center (GC) in order to infer astrophysical and fundamental physics properties (see, e.g., Boehle et al. 2016; Gillessen et al. 2017; GRAVITY Collaboration et al. 2018; Do et al. 2019), measurements of exoplanets in order to infer their orbital parameters (see, e.g., Konopacky et al. 2016), measurements of asteroids in order to infer orbital parameters, detect relativistic effects, or infer their internal properties (see, e.g., Farnocchia et al. 2013; Hees et al. 2015; Greenberg et al. 2017; Verma et al. 2017), lunar laser ranging measurements to infer the properties of the Moon and to perform tests of fundamental physics (see, e.g., Nordtvedt 1998; Bourgoin et al. 2016, 2017; Viswanathan et al. 2018), radioscience tracking of spacecraft through the Deep Space Network antennas to follow the spacecraft motion or to infer planetary properties (see, e.g., Moyer 2003), specific measurements using atomic sensors in order to probe fundamental physics (see, e.g., Pihan-Le Bars et al. 2017; Safronova et al. 2018), etc. This list is not exhaustive and illustrates the wide range of projects that would benefit from an optimization of their measurements to reach a specific scientific objective.

Identifying what the optimal measurements are (type of measurements and scheduling) in order to reach a given scientific objective is not an easy task considering the large parameter space over which one needs to optimize. Loredo & Chernoff (2003), Loredo (2004), and Loredo et al. (2010, 2012) developed a Bayesian adaptive scheduling algorithm aiming at seeking optimized observing times for radial velocity characterization of elliptical orbits in the context of exoplanets. Another adaptive scheduling algorithm dedicated to searches for exoplanets has been proposed by Ford (2008). Both these algorithms are based on maximizing the global increase in information entropy due to one additional measurement and iterating. Maximizing the global information entropy is also known as maximum entropy sampling (Loredo et al. 2012). In this communication, we present another adaptive scheduling algorithm that presents three differences with respect to previous works: (i) the optimization can be done on a specific parameter of interest, for a model comparison, instead of on the global information content only; (ii) the algorithm can be applied to any type of measurement; and (iii) the algorithm is extremely fast and efficient from a computational point of view. The algorithm presented in this communication is a simple but efficient one designed to prioritize measurements based on their sensitivity to a scientific objective. This algorithm relies on the computation of the Fisher matrix (Fisher 1935) and assumes Gaussian statistics on measurement errors. Practically, this adaptive scheduling tool takes as an input existing measurements, the model that is fitted to these measurements, and the criterion on which one wants to optimize, and it uses a greedy algorithm to identify a set of measurements that is optimized
with respect to the given criterion. It is important to emphasize that it is possible to take into account a possible observational window due to seeing of a target from Earth, maintenance time of the facility, etc. In this communication, we consider three criteria for which we can prioritize our measurements: (i) the global information content within a data set (similarly to the work from Ford 2008 and Loredo et al. 2012), (ii) the uncertainty related to a scientific parameter that is estimated using the measurements, and (iii) the ratio of the Bayesian evidence between two different models, a criterion used to compare two models. The set of optimized measurements will in general depend on the objective considered. The method presented in this paper is based on the computation of the Fisher matrix (Ly et al. 2017), whose inverse can be used as an estimate of the covariance matrix (see, e.g., Vallisneri 2008; Heavens 2009 or the appendix from Albrecht et al. 2006). Instead of computing the inverse of the Fisher matrix a large number of times, the algorithm uses an efficient technique to update it.

The algorithm, fully described in Section 2, can be applied to a wide class of measurements and scientific projects. In Section 3, we compare this algorithm to existing work in the literature, discuss some of its current limitations, and also discuss some practicalities related to systematic effects, measurement losses due to contingencies, and experimental windows. In Section 4, we present an application of this tool to the case of measurements of stellar orbits in the GC. It has long been anticipated that measurements of the bright short-period star S0-2 (also named S2) orbiting the supermassive black hole (SMBH) in our GC can be used to detect and probe relativistic effects (Jaroszynski 1998; Rubilar & Eckart 2001; Alexander 2005; Weinberg et al. 2005; Zucker et al. 2006; Psaltis 2008; Will 2008; Angélil & Saha 2010; Angélil et al. 2010; Iorio 2011; Johannsen 2016; Johannsen et al. 2016; Hees et al. 2017; Parsa et al. 2017). The detection of such relativistic effects relies on the measurements with a long time baseline of this star, which has been measured astrometrically since the early 1990s and spectroscopically since the early 2000s at Keck by the Galactic Center Orbit Initiative (GCOI) (Ghez et al. 1998, 2000, 2003, 2005a, 2005b, 2008; Boehle et al. 2016) and at the New Technology Telescope and the Very Large Telescope (VLT) (Eckart & Genzel 1997; Genzel et al. 1997; Eckart et al. 2002; Schödel et al. 2002; Eisenhauer et al. 2003, 2005; Gillessen et al. 2009, 2017), combined with new high-accuracy measurements. The question of interest is: given the existing measurements of the star S0-2, what are the future measurements that will increase the chances of detection of relativistic effects? First, we show how the algorithm developed in this communication has been used to plan the 2018 GCOI campaign of observations in order to successfully measure the relativistic redshift (Do et al. 2019) (for another measurement, see also GRAVITY Collaboration et al. 2018). Second, we identify clearly what the measurements are within S0-2's orbit that are relevant to detecting the relativistic advance of periastron in the near future. Third, we identified what measurements are susceptible to improving the measurement of the distance to our GC, R₀, an important point to plan future measurements but also to search for systematic effects in existing measurements in order to understand the tension between the two estimates of R₀ from the GRAVITY Collaboration (GRAVITY Collaboration et al. 2018) and the

GCOI team (Do et al. 2019). Finally, our conclusions are presented in Section 5.

## 2. Methodology

### 2.1. Formulation of the Problem

The goal of most measurements performed is to extract some scientific information by fitting a model to the observations. We will assume the measurements errors to be normally distributed and we will denote one measurement by $(m_i, t_i, \sigma_i)$ where $m_i$ is the type of measurement, $t_i$ is the time of the measurement, and $\sigma_i$ is the measurement uncertainty. In addition, we will denote the model used to analyze the measurements as $M(\{m_i, t_i\}; \mathbf{p})$ where the vector $\mathbf{p}$ denotes all the $N_{\text{param}}$ parameters that are included in the fit. These include the parameters that are scientifically relevant but can also include other parameters such as nuisance parameters or noise parameters. In some cases, one is interested in comparing two models $M_1(\{m_i, t_i\}; \mathbf{p}_1)$ and $M_2(\{m_i, t_i\}; \mathbf{p}_2)$ (which do not necessarily depend on the same number of parameters) in order to identify the one favored by the data.

In addition, we will denote the set of existing measurements by $\mathcal{O}_{\text{existing}} = \{(m_i, t_i, \sigma_i)\}$ and the set of possible (future) measurements by $\mathcal{O}_{\text{possible}} = \{(m_j, t_j, \sigma_j)\}$. Note that the set of possible future measurements is discrete and takes into account experimental/observational constraints (observational windows, maintenance time for experimental facilities, etc.).

The question we are addressing in this communication can be formulated as follows: given a set of existing measurements, how can the measurements from a set of possible future measurements be prioritized in order to increase the sensitivity with respect to a given scientific objective?

Three types of scientific objective will be considered: (i) a case where one is interested by maximizing the global quantity of information (the information entropy) related to the full posterior distribution, similarly to what has been developed by Ford (2008) and Loredo et al. (2012), (ii) a case where one focuses on one scientific parameter $p_k$ and where one wants to minimize the uncertainty $\delta_{p_k}$ related to the estimation of that specific parameter, and (iii) a case where one wants to maximize the ratio of the Bayesian evidence between two models in order to maximize the chances of discriminating one model from the other.

### 2.2. The Fisher Matrix as a Tool to Evaluate Quickly the Uncertainties and the Information Entropy

The implementation of the adaptive scheduling tool presented in this paper is based on the Fisher matrix. This implicitly assumes that the posterior probability distribution function can be approximated by a multivariate normal distribution. In particular, this is the case when the errors follow a Gaussian distribution and the model can be approximated by its linearization around the maximum likelihood. Under this assumption, the Fisher matrix is a good estimate of the inverse of the covariance matrix for unbiased estimators. The Fisher matrix, a square matrix of dimension $N_{\text{param}}$, can be computed by (see, e.g., Vallisneri 2008; Heavens 2009 or the appendix from Albrecht et al. 2006)

$$F = P^T \cdot P, \qquad (1)$$
where **P** is the matrix of the partial derivatives of the model with respect to the parameters

$$[\mathbf{P}]_{ik} = \frac{1}{\sigma_i} \frac{\partial M(i; \mathbf{p})}{\partial p_k}, \qquad (2)$$

where $\sigma_i$ corresponds to the uncertainty of the $i$th measurement. The matrix of the partial derivatives **P** has dimension $N_{\text{measurements}} \times N_{\text{param}}$.

An estimation of the uncertainties on the estimated parameters resulting from a fit of the model M using a set of measurements $\{(m_i, t_i, \sigma_i)\}$ can be determined from the covariance matrix $\Sigma$, which is the inverse of the Fisher matrix (see, e.g., Vallisneri 2008; Heavens 2009 or the appendix from Albrecht et al. 2006):

$$\Sigma = F^{-1}. \qquad (3)$$

In particular, the marginalized uncertainties of the estimated parameters are given by the diagonal of this matrix (Heavens 2009):

$$\delta_{p_i} = \sqrt{\Sigma_{ii}}. \qquad (4)$$

It is important to note that this covariance matrix depends on (i) the measurement data set (the timing and the type of measurements and their related uncertainties $\{(m_i, t_i, \sigma_i)\}$) and (ii) the set of fitted parameters (i.e., **p**).

In addition, Ford (2008) has shown that the expected difference in information entropy $\mathcal{I}$ that is induced by an additional measurement $k$ is given by (see Equation (25) from Ford 2008)

$$\Delta_k \mathcal{I} = \ln \left( \frac{\sigma_{\text{pred};k}}{\sigma_k} \right), \qquad (5)$$

where $\sigma_{\text{pred};k}$ is the model uncertainty for the measurement $k$ obtained while not including this measurement in the fit and $\sigma_k$ is the expected measurement uncertainty. The notation $\Delta_k x$ denotes the difference in the quantity $x$ induced by adding the measurement $k$ to the existing data set. This relationship means that the optimal measurements in order to increase the global information entropy are those with large model uncertainties (or, quoting Loredo et al. 2012, “we learn the most by sampling where we know the least”), which is called maximum entropy sampling (Sebastiani & Wynn 2000).

Finally, under the assumption that the posterior is a multivariate normal distribution, the model uncertainty can directly be related to the inverse of the Fisher matrix through

$$\frac{\sigma_{\text{pred};k}^2}{\sigma_k^2} = \tilde{\mathbf{P}}_k^T \cdot \Sigma \cdot \tilde{\mathbf{P}}_k, \qquad (6)$$

where $\Sigma$ is the covariance matrix obtained when the measurement $(m_k, t_k, \sigma_k)$ is not included in the fit and $\tilde{\mathbf{P}}_k$ is a column vector containing the partial derivatives of the measurement $k$ with respect to the fitted parameters **p**:

$$[\tilde{\mathbf{P}}_k]_i = [\mathbf{P}]_{ki} = \frac{1}{\sigma_k} \frac{\partial M(m_k, t_k; \mathbf{p})}{\partial p_i}. \qquad (7)$$

One statistical tool to compare two models using a set of measurements is to use the Bayesian evidence, defined as the probability to obtain the data given a certain model (or in other words, the evidence is the normalization of the posterior) (see,

e.g., Kass & Raftery 1995; Jaynes 2003; Liddle 2007; Gregory 2010; Gelman et al. 2013; Knuth et al. 2014). Under the assumption that the prior knowledge about the two models is uniform, the Bayesian evidence is also directly proportional to the probability of a certain model given the data set. Comparing the Bayesian evidences from fits using two different models is one way to compare these models and to discriminate them. Using the Laplace quadratic approximation, the Bayesian evidence for a model 1 is given by (Liddle 2007)

$$\mathcal{E}_1 = \mathcal{P}_{1,\text{max}} \sqrt{\frac{(2\pi)^{N_1}}{|\Sigma_1|}}, \qquad (8)$$

where $\mathcal{P}_{1,\text{max}}$ is the maximum of the posterior, $N_1$ is the number of parameters fitted in the model 1, and $|\Sigma_1|$ is the determinant of the matrix $\Sigma_1$. The difference of the log-evidence between two models is therefore given by

$$\ln \frac{\mathcal{E}_1}{\mathcal{E}_2} = \ln \frac{\Pi_{1,\text{max}}}{\Pi_{2,\text{max}}} + \frac{N_1 - N_2}{2} \ln 2\pi + \ln \frac{\mathcal{L}_{1,\text{max}}}{\mathcal{L}_{2,\text{max}}} + \frac{1}{2} \ln |\mathbf{F}_1| - \frac{1}{2} \ln |\mathbf{F}_2|, \qquad (9)$$

where $\mathcal{L}_{i,\text{max}}$ and $\pi_{i,\text{max}}$ are the likelihood and prior of the model $i$ for the parameters that maximize the posterior. The first two terms (the ratio of the priors and the number of fitted parameters) do not depend directly on the scheduling of the measurements. On the other hand, the third part of this equation depends on the actual measurements themselves. This dependence makes it hard to find the sequence of measurements optimal to discriminate between two models (since the actual measurements are not known). One solution is to consider a Gaussian likelihood and to simulate data using one of the two models. Let us assume that model 1 is the model we are trying to confirm from the data and let us use this model to simulate the measurements so that the last equation becomes

$$\ln \frac{\mathcal{E}_1}{\mathcal{E}_2} = \ln \frac{\Pi_{1,\text{max}}}{\Pi_{2,\text{max}}} + \frac{N_1 - N_2}{2} \ln 2\pi + \frac{1}{2} \ln |\mathbf{F}_1| - \frac{1}{2} \ln |\mathbf{F}_2| + \sum_i \frac{(M_1(m_i, t_i, \mathbf{p}_1) - M_2(m_i, t_i, \mathbf{p}_2))^2}{2\sigma_i^2}. \qquad (10)$$

In conclusion, in this section we have shown how the Fisher matrix can be used to characterize the different objectives that one may want to optimize on:

1. The global information entropy: it requires one to optimize over Equation (6), which depends on the inverse of the Fisher matrix.

2. The uncertainty of one specific parameter of interest ($\delta_{p_i}$): it requires one to optimize over one component of the diagonal of the inverse of the Fisher matrix.

3. A criterion to optimize the chances of discriminating a model $M_1$ from a model $M_2$: one suggestion is to optimize over an estimate of the difference in the log of the Bayesian evidence, given by Equation (10). This quantity depends on the evaluation of the models themselves and on the determinant of their Fisher matrices.

In the next two subsections, we will discuss how to estimate efficiently the Fisher matrix, its determinant, and its inverse.
## 2.3. Computation of the Partial Derivatives of the Model

As can be noticed from Equation (2), one needs to compute the partial derivatives of the model with respect to the parameters in order to compute the Fisher matrix. Three methods can be used to compute these derivatives:

1. Analytically compute the derivatives of the model. This method has the advantage of being quick to evaluate, but it requires long calculations and code. When the model relies on the integration of a set of ordinary differential equations of motion, this method also requires integration of the equations of variations (as an example, see Section 4 from Lainey et al. 2004).

2. Use automatic differentiation. The main advantage of this method is that one does not need to derive explicitly the expression for the partial derivatives.

3. Use finite differences. This method has the advantage of easy implementation (especially when the model requires the integration of differential equations) but has the disadvantage that it can be numerically unstable.

Each of these methods has pros and cons and they need to be chosen on a case-by-case basis considering each specific situation.

## 2.4. Update of the Fisher Matrix, of Its Determinant, and of Its Inverse

One of the main steps in the algorithm developed in this paper is to compute the Fisher matrix, its determinant, and its inverse. Indeed, the goal of the adaptive scheduling tool is to optimize either on one diagonal component of $\Sigma$ or on its determinant $|\Sigma|$. Therefore, we will need to evaluate these quantities for a huge number of different combinations of measurements, and it is important to compute them using a method that is computationally efficient. Recomputing the full Fisher matrix and inverting it for all the sets of measurements is computationally intensive. An alternative approach is to update the Fisher matrix as well as its determinant and its inverse.

In other words, if we consider the Fisher matrix $F^{(n)}$ obtained for a set of measurements $O_n$, we can efficiently compute the Fisher matrix $F^{(n+1)}$ which is obtained from the same set of measurements with an additional measurement $(m_k, t_k, \sigma_k)$. The procedure is straightforward and is given by

$$F^{(n+1)} = F^{(n)} + \tilde{\mathbf{P}}_k^T \cdot \tilde{\mathbf{P}}_k. \quad (11)$$

Following the same principle, the determinant of the Fisher matrix can easily be updated:

$$|\mathbf{F}^{(n+1)}| = |\mathbf{F}^{(n)}|(1 + \tilde{\mathbf{P}}_k^T \cdot \mathbf{\Sigma}^{(n)} \cdot \tilde{\mathbf{P}}_k), \quad (12)$$

where $\mathbf{\Sigma}^{(n)} = [\mathbf{F}^{(n)}]^{-1}$.

Finally, the procedure to update the covariance matrix (the inverse of the Fisher matrix) is given by

$$\mathbf{\Sigma}^{(n+1)} = \mathbf{\Sigma}^{(n)} - \mathbf{U}_k^{(n)}, \quad (13)$$

where $\mathbf{U}_k^{(n)}$ is the update matrix, which depends on the measurement $(m_k, t_k, \sigma_k)$ and is given by

$$\mathbf{U}_k^{(n)} = \frac{(\mathbf{\Sigma}^{(n)} \cdot \tilde{\mathbf{P}}_k) \cdot (\mathbf{\Sigma}^{(n)} \cdot \tilde{\mathbf{P}}_k)^T}{1 + \tilde{\mathbf{P}}_k^T \cdot \mathbf{\Sigma}^{(n)} \cdot \tilde{\mathbf{P}}_k}, \quad (14)$$

where the vector $\tilde{\mathbf{P}}_k$ is given by Equation (7).

To summarize, adding one measurement $(m_k, t_k, \sigma_k)$ to a given data set $O_n$ will lead to a change in the criterion that we are optimizing on, given by the following expressions.

1. Optimization over the global information entropy (maximization):

$$\varepsilon_k = \Delta_k \mathcal{I} = \ln(\tilde{\mathbf{P}}_k^T \cdot \mathbf{\Sigma}^{(n)} \cdot \tilde{\mathbf{P}}_k), \quad (15a)$$

where the column vector $\tilde{\mathbf{P}}_k$ is given by Equation (7).

2. Optimization over one specific parameter of interest (minimization of the related uncertainty):

$$\varepsilon_k = -\Delta_k \delta_{p_i}^2 = [\mathbf{U}_k^{(n)}]_{il}, \quad (15b)$$

using Equation (14).

3. Optimization for model comparison (maximization of the ratio of the Bayesian evidence):

$$\varepsilon_k = \Delta_k \ln \frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{(M_1(m_k, t_k, \mathbf{p}_1) - M_2(m_k, t_k, \mathbf{p}_2))^2}{2\sigma_k^2} + \frac{1}{2} \ln \frac{1 + \tilde{\mathbf{P}}_{1,k}^T \cdot \mathbf{\Sigma}_1^{(n)} \cdot \tilde{\mathbf{P}}_{1,k}}{1 + \tilde{\mathbf{P}}_{2,k}^T \cdot \mathbf{\Sigma}_2^{(n)} \cdot \tilde{\mathbf{P}}_{2,k}}, \quad (15c)$$

where subscripts 1 and 2 refer to the two models to be compared.

## 2.5. Description of the Algorithm

To start the algorithm, we need a set of existing measurements $O_{existing} = \{(m_i, t_i, \sigma_i)\}$ and a set of possible future measurements $O_{possible} = \{(m_j, t_j, \sigma_j)\}$. We also need to know the scientific objective we are optimizing for: (i) maximize the global information entropy or (ii) minimize the uncertainty related to a given parameter $p_j$ or (iii) maximize the odd ratio between two models.

Our implementation of the adaptive scheduling tool is iterative. There are two different ways to control the number of iterations. The first way is to choose arbitrarily the desired number of measurements ($N$) and to find a scheduling for these $N$ measurements that is optimized with respect to a given criterion. The second way is to stop the iterations when we reach a given objective (for example when the uncertainty on an estimated parameter is below a given threshold). In this second case, the number of measurements $N$ is free as well.

The algorithm consists in the following steps.

1. Fit the model $M(m_i, t_i; \mathbf{p})$ to the existing measurements $O_{existing}$ in order to determine the optimal parameters $\mathbf{p}_{opt}$ (in the case of a comparison of models, both models need to be fitted). For this step, the actual data need to be known, not just their uncertainties.

2. Compute the partial derivatives of the model (of both models in the case of a comparison):

$$\left. \frac{1}{\sigma_i} \frac{\partial M(m_i, t_i; \mathbf{p})}{\partial p_k} \right|_{\mathbf{p}_{opt}}$$

for all the existing and possible future measurements. This evaluation can be computationally intensive (depending on the model) but is required only once.

3. Compute the starting covariance matrix $\mathbf{\Sigma}^{(0)}$ from the set of existing measurements $O_{existing}$ by inverting the Fisher matrix, see Equation (3).
4. Iteratively determine the set of $N$ optimal measurements by using the following procedure (starting with $n = 0$ and with an empty list of optimized measurements $O_{\text{opt}} = \{\}$):

(a) loop on all the possible future measurements ($m_k, t_k, \sigma_k \in O_{\text{possible}}$ and for each of them compute the quantity $\varepsilon_k$ from Equations (15) depending on the scientific objective considered.

(b) find the measurement ($m_k, t_k, \sigma_k$) that maximizes $\varepsilon_k$, add it to the list of optimal measurements $O_{\text{opt}}$, and remove it from the list of future possible measurements $O_{\text{possible}}$.

(c) update the covariance matrix by adding the measurement ($m_k, t_k, \sigma_k$) using Equation (13), increase $n$ and iterate by going back to step (a) as long as the number of measurements in the optimal set of future observations is smaller than $N$, or as long as we have not reached the given threshold.

This algorithm is very efficient and allows one to prioritize quickly measurements based on a criterion related to a scientific objective.

### 3. Discussion

The algorithm presented in the previous section can be applied to a wide class of measurements. Our algorithm presents several differences compared to existing ones (see, for example, Ford 2008 and Loredo et al. 2012). First of all, our algorithm is based on the Fisher matrix and not on a full Bayesian approach. This has the drawback that our algorithm is not guaranteed to work when posteriors are extremely non-Gaussian or when they present multiple modes. The main advantage of using the Fisher matrix lies in the efficiency and in the computational cost of the method. Indeed, we have shown in the previous section that there exists a method to efficiently update the Fisher matrix, its determinant, and its inverse. This allows one to consider a huge number of possible future measurements without any problem (optimization over sets of $10^7$ possible measurements can easily be considered).

Another difference between this work and that presented in Ford (2008) and in Loredo et al. (2012) lies in the criterion that we are optimizing for. In Ford (2008) and in Loredo et al. (2012), the scheduling is performed to optimize the expected global information entropy (maximum entropy sampling). Here, we also consider two other options. The first alternative is to prioritize measurements to minimize the uncertainty of a scientific parameter. The second alternative is to prioritize measurements to maximize the chances of discriminating one model from another. The description of the algorithm presented in the previous section includes the three options.

The algorithm presented in Section 2 is a greedy optimizer (similar to the one used in Section 3 of Loredo et al. 2012): it chooses the measurement that leads to the best one-time increase in the metric optimized for and proceeds iteratively. While the resulting measurement sequence is optimized for a particular criterion, this algorithm is not guaranteed to converge toward the global optimal solution. Still, the superiority of adaptive scheduling over random sampling has been shown in Loredo (2004) and Loredo et al. (2012) using a similar optimization algorithm. This superiority will also be quantified in the three examples developed in Section 4 below. The global optimum of the nonlinear and discrete optimization problem

related to the problem presented in Section 2.1 can be determined by a brute-force method whose complexity grows dramatically as $O(n_p^N)$<fnref n="6" /> where $n_p$ is the number of possible future measurements and $N$ is the length of the optimal sequence. In practice, $n_p$ can be huge (as high as $10^7$), which justifies the use of an $O(n_p)$ algorithm. In the future, it would be interesting to study more complex algorithms that could provide a better optimization than the greedy algorithm from Section 2 with a reasonable computational time complexity.

One input for the algorithm from Section 2 is the uncertainties related to possible future measurements. It is important to note that these uncertainties can vary from measurement to measurement to take into account practicalities such as, for example, visibility from a telescope, different integration times at different observing epochs, confusion events, geometrical considerations (in the case of range measurements for example), etc. Such variations for the uncertainties are taken into account and automatically considered in the presented algorithm. On the other hand, as presented in Section 2, the algorithm cannot take into account the more general case where the uncertainty for a given future measurement is given as a distribution instead of a particular value. One can overcome this drawback by implementing a Monte Carlo algorithm on top of the adaptive scheduling tool. For each instance of the Monte Carlo, the uncertainties considered in the adaptive scheduling would be drawn randomly. More pragmatically and more efficiently, one can consider several cases of measurement uncertainties (e.g., a “realistic” one, an “optimistic” one, and a “pessimistic” one) and run the adaptive scheduling tool on that limited number of cases to assess the impact of the future uncertainties on the scheduling.

We would like to emphasize that practicalities such as observational window, maintenance time, etc., can easily be taken into account in our approach. Indeed, our algorithm is based on a brute-force exploration of all the possible future measurements. This set of possible measurements can therefore include gaps or other constraints.

It is important to note that the algorithm presented in this communication is purely based on statistical considerations. In practice, systematic effects usually play an important role when analyzing measurements, and experimental losses (losses due to weather, to unexpected downtime of a telescope or of experimental facilities, ...) can prevent one from achieving the expected scientific goal. While the results that can be obtained from a scheduling tool are extremely interesting for identifying optimized measurements, it is important to take into account systematics and losses when planning real measurements. When two types of measurement are considered (such as the case of stellar orbits in the GC, where the two types are the astrometric and the spectroscopic measurements), we suggest to plan both measurements at roughly the same epochs even though only one type of measurement is required from the adaptive scheduling tool. Using both types of measurement at the same time will help to control for systematics.

In addition, we suggest to consider measurement losses (atmospheric losses, unexpected maintenance, etc.) by using a very simple statistical model for the losses and by running a Monte Carlo algorithm and determining the distribution of the expected scientific results given this statistical model. This can

<footnote n="6">⁶ This is valid when $N \ll n_p$.</footnote>
help to estimate the number of measurements required to reach a given objective (see Section 4.2 below for a practical example).

Finally, in Equation (2), we implicitly assumed that the measurement error is a white noise. This can easily be generalized to the case where the measurement errors are correlated. In that case, the measurement noise is characterized by a covariance matrix *C*, which is assumed to be diagonal in Equation (2). A covariance matrix is a positive definite matrix, so it is always possible to find a decomposition of the form

$$C = S^T \cdot S, \quad (16)$$

using, e.g., a Cholesky decomposition. The algorithm presented in the previous section can directly be applied by replacing Equation (2) by

$$[\mathbf{P}]_{ij} = [\mathbf{S}^{-1}]_{ik} \frac{\partial \mathbf{M}(m_k, t_k; \mathbf{p})}{\partial p_j}. \quad (17)$$

## 4. Application to the Detection of Relativistic Effects on Stellar Orbits in the GC

To illustrate the utility of the algorithm presented in the previous section, we use the case of the measurements of stellar orbits in the GC as an example. Of particular interest is the star S0-2 (also named S2), which has been followed astrometrically and spectroscopically for nearly 25 yr (Eckart & Genzel 1997; Ghez et al. 1998, 2000, 2003, 2005a, 2005b, 2008; Eckart et al. 2002; Schödel et al. 2002; Eisenhauer et al. 2003, 2005; Gillessen et al. 2009, 2017; Boehle et al. 2016). One of the long-standing scientific objectives pursued by measuring the motion of this star is to measure relativistic effects around the SMBH and to test the gravitational theory of general relativity (Jaroszynski 1998; Rubilar & Eckart 2001; Alexander 2005; Weinberg et al. 2005; Zucker et al. 2006; Psaltis 2008; Will 2008; Angélil & Saha 2010; Angélil et al. 2010; Iorio 2011; Johannsen 2016; Johannsen et al. 2016; Parsa et al. 2017). Detection of relativistic effects using measurements of S0-2 relies on the long-standing data set available and on new high-accuracy measurements. The first relativistic effects measurable using S0-2's measurements are the relativistic redshift and the relativistic advance of the periastron. Recently, the impact of the relativistic redshift on S0-2's radial velocity has been measured successfully (GRAVITY Collaboration et al. 2018; Do et al. 2019). In this section, we will show how the adaptive scheduling tool presented in the previous section has helped the GCOI team in planning the 2018 campaign of observations that led to a successful detection of the relativistic redshift (Do et al. 2019). In addition, we will show how the same tool can be used to identify suitable measurements in order to enhance the detection of the relativistic advance of the periastron in the near future. Finally, we will also prioritize measurements to improve the knowledge of $R_0$, our distance to the GC.

### 4.1. Data and Model

The set of existing measurements used in this analysis consists in astrometric measurements performed at the Keck observatory between 1995 and 2017 and spectroscopic measurements performed at the Keck observatory and at the VLT between 2000 and 2017. These measurements are reported in the literature (Gillessen et al. 2017; Do et al.

<table><caption>**Table 1**
Sequence of Measurements in 2018 Optimized in Order to Measure the Relativistic Redshift Given Existing Past Measurements</caption><thead><tr><th>Order of Importance</th><th>Epoch</th><th>Type of Observation</th><th>Stat. S/N</th></tr></thead><tbody><tr><td>1</td><td>2018.286</td><td>RV</td><td>4.50</td></tr><tr><td>2</td><td>2018.283</td><td>RV</td><td>4.97</td></tr><tr><td>3</td><td>2018.700</td><td>RV</td><td>5.47</td></tr><tr><td>4</td><td>2018.264</td><td>RV</td><td>6.09</td></tr><tr><td>5</td><td>2018.275</td><td>RV</td><td>6.33</td></tr><tr><td>6</td><td>2018.699</td><td>RV</td><td>6.53</td></tr><tr><td>7</td><td>2018.272</td><td>RV</td><td>6.71</td></tr><tr><td>8</td><td>2018.700</td><td>astro</td><td>6.84</td></tr><tr><td>9</td><td>2018.288</td><td>RV</td><td>6.98</td></tr><tr><td>10</td><td>2018.696</td><td>RV</td><td>7.10</td></tr></tbody></table>

**Note.** Column 1: *i*, the order of importance of the measurement. Column 2: epoch of the measurement. Column 3: type of measurement. Column 4: estimation of the statistical signal-to-noise ratio of the relativistic redshift at the end of 2018 given the existing measurements and including *i* new measurements. Systematics are not included here.

2019), and here we consider them to be independent. The model $M(\{m_k, t_k\}; \mathbf{p})$ used for this analysis relies on the integration of the first post-Newtonian equations of motion and includes the Römer time delay and the first-order relativistic redshift. This model is fully described in the appendix of Do et al. (2019). There are 15 parameters that characterize this model: the six orbital parameters for the star S0-2, the SMBH mass parameter GM, the distance to the GC $R_0$, the 2D position and 3D velocity of the SMBH, a parameter that encodes a deviation from the relativistic redshift, and a parameter that encodes a deviation from the first post-Newtonian order in the equations of motion (or in other words that encodes a deviation from the relativistic advance of the periastron).

### 4.2. An Example of Scheduling Including Systematics and Weather Losses to Detect the Relativistic Redshift on S0-2

In this section, we give a full example of telescope time scheduling using the tool presented in Section 2, including considerations related to systematics and to weather and instrumental losses. We focus on the measurement of the relativistic redshift of S0-2 and therefore fix the parameter that encodes a deviation from the first post-Newtonian equations of motion to its relativistic value (i.e., to unity). Therefore, the model contains 14 parameters. The objective is to find a measurement sequence in 2018 that would optimize the chances of a successful detection of the relativistic redshift given the set of existing measurements (up to 2017). The set of future possible measurements considered comprises two distinct sets: the first one consists of astrometric measurements for every day between 2018 March 1 and September 15 and the second one consists of radial velocity measurements for every day in the same time period. The uncertainty for the future measurements has been chosen as $20 \text{ km s}^{-1}$ for the radial velocity and 0.9 mas for the astrometry.

The sequence of 10 measurements optimized in order to detect the relativistic redshift (we are optimizing on the redshift parameter) using the algorithm presented in Section 2 is presented in Table 1 and Figure 1. It is important to keep in mind that this result depends on the existing measurements: different existing measurements could have led to another sequence. This sequence also depends on the expected
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Representation of the sequence of radial velocity measurements in 2018 optimized to detect the relativistic redshift (see Table 1). The top panel represents the radial velocity measurement optimized using the algorithm from Section 2 in order to detect the relativistic redshift. The solid curve is the radial velocity model. The bottom panel shows the relativistic contribution from the same measurements with the redshift signal. In these simulations, all measurements prior to 2018 are used (black points). The blue points show the optimized sequence of measurements under the assumption that all the model parameters are fitted simultaneously. The critical radial velocity measurements are located at the turning points of the radial velocity curve and not at the maximum of the redshift because of correlations with *T*₀ and *GM*. The observations in orange represent the optimized sequence of measurements obtained under the assumption that *T*₀ and *GM* are not fitted. These measurements are located at the maximum of the redshift signal.</figcaption></figure>

accuracy of future possible measurements and on the
parameters that are included in the fit.

Two interesting facts need to be noticed about the optimized
sequence of measurements obtained in this analysis. First, 9 out
of the 10 measurements are spectroscopic observations. This
makes sense considering that the redshift impacts directly on
the radial velocity but not the astrometry. A more surprising
fact is that the adaptive scheduling tool does not favor
measurements at the maximum of the redshift signal. Rather,
it favors measurements that are at the radial velocity turning
points (i.e., the maximum and minimum of the radial velocity
curve, see Figure 1). This is due to correlations between the
redshift parameter and the other model parameters, in particular
with T₀ (the time of closest approach for S0-2) and with the
SMBH mass parameter GM. These correlations are maximal
exactly when the redshift signal is maximal, making that epoch
not optimal in order to measure the redshift signal. In other
words, the redshift signal can easily be absorbed by a small
change in T₀ or GM. To demonstrate the effect of correlations,
we run a test case and search for the optimized sequence of
measurements if we fit for all the model parameters except for
T₀ and GM. The orange measurements from Figure 1 are the
radial velocity measurements optimized for that case using the
algorithm from Section 2. When T₀ and GM are not included in
the fit, these measurements are all located close to the
maximum of the redshift signal, as expected.

It is interesting to quantify the impact of an optimized
scheduling compared to a “naive” one. For this, let us consider
the gain in the signal-to-noise ratio (S/N) of the relativistic
redshift due to one RV measurement in 2018. The redshift S/N
using measurements existing up to 2017 is 1.16. A naive
scheduling consisting in performing one additional

measurement at the time when the redshift signal is maximal
induces an increase in S/N of 4.5% (the S/N being increased
up to 1.205), while an optimized scheduling consisting in
performing the measurement at the time when the RV is
maximal increases the S/N by 335% (the S/N is now 4.5).
This illustrates clearly the benefit of an optimized scheduling
and how such a scheduling tool can help to increase the
outcome of a project.

At this stage, the important point learned from our adaptive
scheduling tool is that, from a statistical point of view, nearly
all the power to measure the relativistic redshift comes from
spectroscopic measurements around the two RV turning points.
This result relies purely on statistical considerations, and
practicalities and systematics need to be considered as well.
First of all, other longer-term scientific objectives will be
pursued with S0-2's measurements. It is important to realize
that measurements around closest approach, while not of prime
importance for the redshift, are important for measuring other
scientific parameters such as, e.g., R₀ (see Section 4.4) and the
relativistic advance of the periastron (see Section 4.3).
Therefore, we included measurements at closest approach, an
important event as well. In addition, two strategies have been
used to control systematics for the relativistic redshift
measurement: (i) we decided to use different telescopes (Keck
and Gemini) to measure the radial velocities in order to assess
hypothetical biases related to one instrument and (ii) we
decided to also take astrometric measurements at the same
epochs as radial velocities and vice versa (see the discussion in
Section 3). Finally, the results from Table 1 have been obtained
assuming a perfect knowledge of S0-2's orbital parameters. In
practice, there is an uncertainty associated with these
parameters. In particular, at the end of 2017, the uncertainty
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Top: representation of the scheduling for S0-2's measurements in 2018 as planned by the GCOI considering the analysis presented in this section. Bottom: actual measurements performed in 2018, see Do et al. (2019). Left: radial velocity (spectroscopic measurements). Right: astrometric measurements presented as a 2D radial projection between S0-2 and the center of the reference frame (Sakai et al. 2019).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Left: distribution of the relativistic redshift S/N after 2018 using the scheduling described in this section and assuming that each measurement has a 50% probability of being lost. This S/N corresponds to the statistical uncertainty of the relativistic redshift; no systematics are considered here. Right: evolution of the relativistic redshift S/N as a function of time. The blue curve represents the case corresponding to our original scheduling assuming all the measurements have been taken. The orange curve represents the evolution of the S/N obtained with the real measurements from 2018. These S/Ns correspond to the statistical uncertainty of the redshift parameter; no systematics are considered in this analysis. Systematics have been thoroughly analyzed in Do et al. (2019).</figcaption></figure>

related to the 2018 time of closest approach $T_0$ was of the order of five days. In order to control for a possible bias in the estimate of $T_0$, we included measurements very early in 2018.

In addition, it is important to consider instrumental losses and weather losses. Using different telescopes (Keck and Gemini) for the capture of the important measurements is one way to mitigate the risk of missing important measurements because of instrumental losses (in addition to allowing us to check for systematics). In addition, we increased the number of requested measurements in 2018 in order to take into account possible measurements losses. A careful analysis of the Keck Observatory statistics shows that roughly 50% of GC measurement nights are lost because of weather or instrumental problems. This has been included in our scheduling by making a large number ($10^5$) of simulations where each measurement has a probability of 50% of being lost and by assessing the distribution of the relativistic redshift S/N at the end of 2018.

To summarize, the scheduling of the 2018 GC measurement campaign for the GCOI team was based on: (i) measurements of radial velocities very early in 2018 to control our estimate of $T_0$; (ii) radial velocities at the two radial velocity turning points, the most important measurements to detect the redshift, using at least two different instruments to control for systematics; (iii) measurements at closest approach important for other scientific objectives; (iv) dual spectroscopic-astrometric measurements

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Distribution of an optimized sequence of measurements in order to detect the relativistic advance of periastron on S0-2's orbit. This sequence of 30 measurements has been obtained considering possible observational windows that span each year from March to September from 2018 to 2058. This figure shows three interesting features: (i) measuring the advance of the periastron relies mainly on astrometry measurements, (ii) late measurements are favored because it is a secular effect, and (iii) measurements around closest approach are very sensitive to this effect.</figcaption></figure>

at the important epochs to control for systematics; (v) the total number of measurements being determined by requiring that the relativistic redshift S/N at the end of 2018 has a probability of at least 99.9% of being above 5, considering that each night has a 50% chance of being lost, and (vi) the exact scheduling being determined by considering a weather pattern lasting three days. The scheduling resulting from this analysis is presented at the top of Figure 2 and consists of 36 spectroscopic
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Distribution of an optimized sequence of measurements in order to detect the relativistic advance of periastron on S0-2's orbit. Top left: distribution of the optimized measurements over one orbital period, each bin corresponding to half a year. The blue bins correspond to astrometric measurements while the orange bins correspond to the RV measurements. Bottom left: representation of the astrometric measurements (the same sequence as the top left panel) plotted in the plane of the sky. Right: representation of the measurements as a function of time: top: R.A., middle: decl., bottom: RV. The numbers represent different sets of astrometric measurements. 1: astrometric measurements bracketing closest approach, 2: astrometric measurements around apastron, 3: astrometric measurements around the maximal R.A. turning point, 4: astrometric measurements between the minimal R.A. turning point and closest approach.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Distribution of the sequence of measurements optimized in order to improve our knowledge of the distance to the Galactic Center $R_0$ on S0-2's orbit. Top left: distribution of the optimized measurements over one orbital period, each bin corresponding to half a year. The blue bins correspond to astrometric measurements while the orange bins correspond to the RV measurements. Bottom left: representation of the astrometric measurements plotted in the plane of the sky. Right: representation of the measurements as a function of time: top: R.A., middle: decl., bottom: RV. The numbers represent different sets of measurements. 1: astrometric and RV measurements just before closest approach, around RV max, 2: RV measurements around closest approach, 3: astrometric and RV measurements around the maximal R.A. turning point, 4: astrometric measurements after apastron.</figcaption></figure>

measurements (for Keck and Gemini) and 10 astrometric measurements. For the sake of comparison, the actual measurements performed in 2018 are presented in the lower panel of Figure 2. They consist of 19 radial velocity measurements and five astrometric measurements. The main sources of losses for 2018 are related to bad weather and to eruptions from the volcano in Hawaii, one of them being accompanied by an earthquake that damaged the instruments

for a couple of days. The distribution of the relativistic redshift S/N at the end of 2018 using the scheduling described in this section and assuming that each measurement has a 50% chance of being lost is presented in the right panel of Figure 3. The probability of having an S/N below 5 is 0.03%. The S/N presented in this figure is a statistical S/N. No systematics are considered in this analysis and systematics will diminish the S/N calculated here. The analysis of the real data has shown that
including systematics such as offsets between instruments,
correlations in the astrometric positional uncertainty, or
additional systematics uncertainties will produce a systematic
uncertainty of the same order of magnitude as the statistical
uncertainty (see Do et al. 2019 for the analysis of the
measurements and a detailed study of systematics). The right
panel of Figure 3 shows the evolution of the S/N with time in
2018. The blue curve corresponds to the evolution expected
from the original scheduling while the orange curve corre-
sponds to the evolution of the real measurements. Although
nearly half of the measurements in 2018 have been lost, the
final S/N obtained in the analysis remains far above 5. This is
due to the careful planning and to the number of measurements
scheduled but also to the fact that some of the measurements
were actually better than anticipated.

This first example shows the utility of the adaptive
scheduling tool presented in Section 2 and how it can lead to
surprising results. As a result of this analysis, the GCOI team
decided to put more effort into measuring the radial velocity of
S0-2 around the two turning points in order to maximize the
chances of a successful detection of the relativistic redshift (Do
et al. 2019). We also developed the details behind the GCOI's
decision to include systematics and possible weather/instru-
mental losses in the scheduling of the 2018 measurement
campaign.

4.3. *Measurements on S0-2's Orbit Optimized to Detect the Relativistic Advance of Periastron*

The relativistic redshift is only the first relativistic effect measurable using stellar orbits in the GC. The next effect expected to be measured is the well-known advance of periastron, which is due to the first post-Newtonian relativistic correction to the equations of motion. In this section, we are interested in prioritizing measurements in order to measure this effect. For this reason, we fix the redshift parameter to its relativistic value (i.e., to unity) but we let the parameter that encodes a deviation from the first post-Newtonian equation of motion be free. As a result, the model again contains 14 parameters. The set of “existing measurements” consists of astrometric and radial velocity measurements of S0-2 up to 2018. The set of future possible measurements consists of astrometric and spectroscopic measurements for each year between March and September (the yearly window for which GC measurements are feasible from Earth) on a daily basis. The uncertainty considered for the future measurements is 0.2 mas for the astrometry and 20 km s⁻¹ for the radial velocity.

As a first step, we run the adaptive scheduling tool considering possible measurements between 2018 and 2058. The distribution of the measurements from an optimized sequence of 30 observations is presented in Figure 4. While of limited interest, this figure shows three interesting features. First, it shows that, as anticipated, the detection of the advance of the periastron relies mainly on the astrometry. Second, it shows that the optimal strategy to measure the advance of the periastron given a certain measurement window is to take measurements as late as possible. This is easily explained considering that this effect is a secular effect that increases with time. Finally, we can notice that measurements around periastron (2050 corresponds to a time of closest approach) are of prime importance. This last point will be developed further below. Changing the possible measurement window does not change these three results qualitatively.

Figure 4 is of somewhat limited interest because “late”
measurements are always favored by the adaptive scheduling
tool. A more interesting result consists in identifying the
regions within one orbit that are particularly sensitive to the
advance of the periastron. One way to derive such a result is to
run the adaptive scheduling tool multiple times by increasing
the measurement window one year at a time and to save all the
optimized sequences of measurements after removing the
measurements that correspond to the last year of the
observational window (to discard the effect of “late” measure-
ments due to the secular effect). The result from such an
analysis is presented in Figure 5 and is particularly interesting.
The most sensitive measurements are astrometric measure-
ments around closest approach. It is interesting to note that the
closest approach itself is not favored by the adaptive scheduling
tool (see the bottom left panel from Figure 5) because of
correlations with other fitted parameters. Rather, it is important
to have measurements bracketing the closest approach. The
second important set of measurements are those around
apastron. This can easily be interpreted since this is the
location where a precession of the orbit will have the largest
impact. The next set of important measurements are those of
the R.A. turning point (astrometric measurements around the
maximum of the R.A.). In addition, while radial velocity
measurements are less crucial for the detection of the
relativistic advance of periastron, the important spectroscopic
measurements are all located around the closest approach and
at the two turning points in the radial velocity curve.

In order to quantify the impact of an optimized scheduling
obtained using the algorithm from Section 2, we compare the
gain in the relativistic (advance of periastron) S/N after one
orbital period for two different scheduling strategies. More
precisely, we use the current existing measurements and
compare two schedulings of 32 astrometric measurements for
the next 16 yr. The first scheduling consists in a “naive”
sampling for which two measurements are taken every year,
and the second scheduling results from our optimization such
that the measurements are distributed in time following the
results presented in the top left panel of Figure 5. The S/N
resulting from a regular sampling is 4.1, while that resulting
from the optimized scheduling is 5.9, a relative increase of
45%. This shows that, although the scheduling resulting from
the algorithm presented in Section 2 is not guaranteed to be the
global optimum (see the discussion in Section 3), it is
nevertheless superior to standard non-optimized scheduling
and it can help scientists to prioritize their measurements for a
given scientific objective.

As a conclusion, the relativistic advance of the periastron is a
secular effect such that a long time baseline increases the
chance of its detection. On top of that, measurements at the
astrometric turning points (closest approach, apastron, and R.A.
turning point) are important. In the mid-term, this means that
2019 will be an important year since it corresponds to the R.A.
turning point, and the years between 2022 and 2026 will be
important for measuring precisely the motion of S0-2 around
the apastron.

4.4. *Measurements on S0-2's Orbit Optimized to Improve Our Knowledge about the Distance to the GC*

In this section, we will focus on the distance to our GC ($R_0$),
and prioritize measurements to improve our knowledge of $R_0$
using the algorithm from Section 2. For this reason, we fix both
parameters of general relativity to their relativistic values (i.e., to unity) so that the model contains 13 free parameters. The set of “existing measurements” is exactly the same as that used in the previous section, i.e., astrometric and radial velocity measurements of S0-2 up to 2018. The set of future possible measurements consists of astrometric and spectroscopic measurements for each year between March and September (the yearly window for which GC measurements are feasible from Earth) on a daily basis. The uncertainty considered for the future measurements is 0.2 mas for the astrometry and $20 \text{ km s}^{-1}$ for the radial velocity.

First, a similar behavior to that for the advance of the periastron is observed: late measurements are always favored by the adaptive scheduling tool and a long time baseline will increase our knowledge of $R_0$. Therefore, we use the same strategy as the one described in the previous section: we run the adaptive scheduling tool by increasing the time baseline one year at a time, record the optimized sequences of measurements but discard the observations related to the last year of the observational window. The result from such an analysis is presented in Figure 6. The estimation of $R_0$ relies on both astrometry and radial velocity. The important RV measurements are located at closest approach and bracket it. The important astrometric measurements are located at the R.A. turning point (maximum of the R.A. curve) and before closest approach. Of particular interest, this analysis shows that the 2019 astrometric measurements will be of prime importance for improving our knowledge about $R_0$.

In order to assess the impact of the optimized scheduling, we compare the S/N of $R_0$ between two schedulings extending over one orbital period: one regular sampling and one optimized scheduling using the algorithm from Section 2. More precisely, we use the current existing measurements and compare two schedulings of 32 astrometric measurements and 16 RVs for the next 16 yr. The first scheduling consists of a “naive” scheduling for which two astrometric measurements and one radial velocity measurement are performed every year, and the second scheduling results from our optimization such that the measurements are distributed in time following the results presented in the top left panel of Figure 6. The S/N resulting from a regular sampling is 287 while that resulting from the optimized scheduling is 380, a relative increase of 30%. Similarly to what has been discussed in Sections 4.2 and 4.3, this shows the superiority of the scheduling obtained from the algorithm presented in Section 2 over standard non-optimized and uniform scheduling and it can help scientists to prioritize measurements for a given scientific objective.

## 5. Conclusion

In this paper, we have presented a simple but efficient algorithm aiming at prioritizing measurements in order to reach a scientific objective. Identifying an optimized scheduling and the best type of measurements will become more and more important considering the increasing costs of measurements and considering the increasing pressure to obtain telescope time or time in different experimental facilities. The algorithm developed and presented in Section 2 relies on the assumption that the measurement errors and the posterior are Gaussian but can be applied to any type of measurement or observation. This adaptive scheduling tool is based on the computation of the Fisher matrix, on a very efficient technique to update it, and on a greedy algorithm. As discussed in Section 3, a greedy

algorithm is not guaranteed to converge to the global optimal solution but it has the advantage of having a reasonable time complexity, which makes it relatively fast and which allows the user to consider a very large set of possible future measurements. In addition, the three examples discussed in Section 4 clearly show the superiority of the optimized scheduling over non-optimized standard samplings (see also the discussion in Loredo et al. 2012).

We have used the case of measurements of the short-period star S0-2 in our GC as an example to illustrate the type of analysis and results that can be obtained with this adaptive scheduling tool. In particular, we have shown that the measurements that increase the relativistic redshift S/N are radial velocity measurements at the turning points of the radial velocity curve and not at the maximum of the signal, a surprising result that is explained by correlations. We also suggest a method to plan measurements to control systematics and to account for possible weather/instrumental losses. This detailed analysis led to the scheduling for the GCOI 2018 measurement campaign detailed in Section 4.2, which resulted in a successful detection of the relativistic redshift in 2018 (see Do et al. 2019).

In addition, we have identified sets of measurements that are particularly important for detecting the relativistic advance of periastron on S0-2's orbit. They consist of astrometric measurements at the astrometric turning points, in particular around periastron and apastron. Finally, we have shown that both astrometry and radial velocity are important for improving our knowledge about $R_0$, the distance to our GC. The important astrometric measurements are those corresponding to the R.A. turning point and before closest approach, while the radial velocities adapted to measure $R_0$ are at closest approach and bracketing this epoch.

Similar analysis can be performed in various fields of physics and astrophysics including solar system measurements, exoplanets, gravitational waves, and even laboratory measurements in order to help scientists to prioritize the type of measurements and their scheduling so as to increase their desired S/N in order to enhance the scientific return from different instruments and experiments.

Support for this work was provided by the W. M. Keck Foundation (20170668), by the Heising Simons Foundation (2017-282), and by the National Science Foundation (AST-1412615). The authors warmly thank the anonymous referee whose comments helped to improve the manuscript.

## ORCID iDs

A. Hees https://orcid.org/0000-0002-2186-644X

A. Dehghanfar https://orcid.org/0000-0002-7476-2521

T. Do https://orcid.org/0000-0001-9554-6062

A. M. Ghez https://orcid.org/0000-0003-3230-5055

G. D. Martinez https://orcid.org/0000-0002-7476-2521

R. Campbell https://orcid.org/0000-0002-3289-5203

J. R. Lu https://orcid.org/0000-0001-9611-0009

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