Removing systematic errors for exoplanet search via latent causes

Transcription

1 Bernhar Schölkopf Max lanck Institute for Intelligent Systems, Tübingen, GERMANY Davi W. Hogg Dun Wang Daniel Foreman-Mackey Center for Cosmology an article hysics, New York University, New York, NY 0003, USA Dominik Janzing Carl-Johann Simon-Gabriel Jonas eters Max lanck Institute for Intelligent Systems, Tübingen, GERMANY Abstract We escribe a metho for removing the effect of confouners in orer to reconstruct a latent quantity of interest. The metho, referre to as halfsibling regression, is inspire by recent work in causal inference using aitive noise moels. We provie a theoretical justification an illustrate the potential of the metho in a challenging astronomy application.. Introuction The present paper proposes an analyzes a metho for removing the effect of confouning noise. The analysis is base on a hypothetical unerlying causal structure. The metho oes not infer causal structures; rather, it is influence by a recent thrust to try to unerstan how causal structures facilitate machine learning tasks (Schölkopf et al., 202). Causal graphical moels as pioneere by earl (2000); Spirtes et al. (993) are joint probability istributions over a set of variables X,..., X n, along with irecte graphs (usually, acyclicity is assume) with vertices X i, an arrows inicating irect causal influences. By the causal Markov assumption, each vertex X i is inepenent of its non-escenants, given its parents. There is an alternative view of causal moels, which oes not start from a joint istribution. Instea, it assumes a set roceeings of the 32 n International Conference on Machine Learning, Lille, France, 205. JMLR: W&C volume 37. Copyright 205 by the author(s). of jointly inepenent noise variables, one for each vertex, an a structural equation for each variable that escribes how the latter is compute by evaluating a eterministic function of its noise variable an its parents. This view, referre to as a functional causal moel (or nonlinear structural equation moel), leas to the same class of joint istributions over all variables (earl, 2000; eters et al., 204), an we may thus choose either representation. The functional point of view is useful in that it often makes it easier to come up with assumptions on the causal mechanisms that are at work, i.e., on the functions associate with the variables. For instance, it was recently shown (Hoyer et al., 2009) that assuming nonlinear functions with aitive noise reners the two variable case ientifiable i.e., a case where conitional inepenence tests o not provie any information, an it was thus previously believe that it is impossible to infer the structure of the graph base on observational ata. In this work we start from the functional point of view an assume the unerlying causal graph shown in Fig.. Here, N, Q, X, Y are jointly ranom variables (RVs) (i.e., RVs efine on the same unerlying probability space), taking values enote by n, q, x, y. We o not require the ranges of the ranom variables to be R, in particular, they may be vectorial. All equalities regaring ranom variables shoul be interprete to hol with probability one. We further (implicitly) assume the existence of conitional expectations. 2. Half-Sibling Regression Suppose we are intereste in the quantity Q, but unfortunately we cannot observe it irectly. Instea, we observe Y, which we think of as a egrae version of Q that is

2 unobserve observe Y X Figure. We are intereste in reconstructing the quantity Q base on the observables X an Y affecte by noise N, using the knowlege that (N, X) Q. Note that the involve quantities nee not be scalars, which makes the moel more general than it seems at first glance. For instance, we can think of N as a multiimensional vector, some components of which affect only X, some only Y, an some both X an Y. affecte by noise N. Clearly, without knowlege of N, there is no way to recover Q. However, we assume that N also affects another observable quantity (or a collection of quantities) X. By the graph structure, conitional on Y, the variables Q an X are epenent (in the generic case), thus X contains information about Q. This situation is quite common if X an Y are measurements performe with the same apparatus, introucing the noise N. In the physical sciences, this is often referre to as systematics, to convey the intuition that these errors are not simply ue to ranom fluctuations, but cause by systematic influences of the measuring evice. In our application below, both types of errors occur, but we will not try to tease them apart. How can we use this information in practice? Unfortunately, without further restrictions, this problem is still too har. Suppose that N ranomly switches between {,..., v}, where v N (Schölkopf et al., 202). Define the structural equation f Y for the variable Y as follows: y = f Y (n, q) := f n (q), where f,..., f v are v istinct functions that compute Y from Q in other wors, we ranomly switch between v ifferent mechanisms. Clearly, no matter how many pairs (x, y) we observe, we can choose a sufficiently large v along with functions f,..., f v such that there is no way of gleaning any reliable information on Q from the f i (Q) e.g., there may be more f i than there were ata points. Things coul get even worse: for instance, N coul be real value, an switch between an uncountable number of functions. To prevent this kin of behavior, we nee to simplify the way in which Y is allowe to epen on N. Before we o so, we nee to point out a funamental limitation. The above example shows that it can be arbitrarily har to get information about Q from finite ata. However, even from infinite ata, only partial information is available an certain gauge egrees of freeom remain. In particular, given a reconstructe Q, we can always construct another one by applying an invertible transformation to it, an incorporating its inverse into the function computing Y Q N from Q an N. This inclues the possibility of aing an offset, which we will see below. We next propose an assumption which allows for a practical metho to solve the problem of reconstructing Q up to the above gauge freeom. The metho is surprisingly simple, an while we have not seen it in the same form elsewhere, we o not want to claim originality for it. Relate tricks are occasionally applie in practice, often employing factor analysis to account for confouning effects (rice et al., 2006; Yu et al., 2006; Johnson & Li, 2007; Kang et al., 2008; Stegle et al., 2008; Gagnon-Bartsch & Spee, 20). We will also present a theoretical analysis that provies insight into why an when these methos work. 2.. Complete Information Inspire by recent work in causal inference, we use nonlinear aitive noise moels (Hoyer et al., 2009). Specifically, we assume that there exists a function f such that Y = Q + f(n). () Note that we coul equally well assume the more general form Y = g(q) + f(n), an the following analysis woul look the same. However, in view of the above remark about the gauge freeom, this is not necessary since Q can at most be ientifie up to a (nonlinear) reparametrization anyway. Note, moreover, that while for Hoyer et al. (2009), the input of f is observe an we want to ecie if it is a cause of Y, in the present setting the input of f is unobserve (Janzing et al., 2009), an the goal is to recover Q, which for Hoyer et al. (2009) playe the role of the noise. The intuition behin our approach is as follows. Since X Q, X cannot preict Q, an thus neither Q s influence on Y. It may contain information, however, about the influence of N on Y, since X is also influence by N. Now suppose we try to preict Y from X. As argue above, whatever comes from Q cannot be preicte, hence only the component coming from N will be picke up. Trying to preict Y from X is thus a vehicle to selectively capture N s influence on Y, with the goal of subsequently removing it, to obtain an estimate of Q referre to as ˆQ: Definition ˆQ := Y E[Y X] (2) For an aitive moel (), our intuition can be formalize: in this case, we can preict the aitive component in Y coming from N which is exactly what we want to remove to cancel the confouning effect of N an thus reconstruct Q (up to an offset): roposition Suppose N, X are jointly ranom variables, an f is a measurable function. If there exists a

3 function ψ such that f(n) = ψ(x), (3) i.e., f(n) can in principle be preicte from X perfectly, then we have f(n) = E[f(N) X]. (4) If, moreover, the aitive moel assumption () hols, with Q, Y RVs on the same unerlying probability space, an Q X, then ˆQ = Q E[Q]. (5) In our main application below, N will be systematic errors from an astronomical spacecraft an telescope, Y will be a star uner analysis, an X will be a large set of other stars. In this case, the assumption that f(n) = ψ(x) has a concrete interpretation: it means that the evice can be self-calibrate base on measure science ata only (amanabhan et al., 2008). roof. Due to (3), we have E[f(N) X] = E[ψ(X) X] = ψ(x) = f(n). (6) To show the secon statement, consier the conitional expectation E[Y X] = E[Q + f(n) X] (7) Using Q X an (4), we get E[Y X] = E[Q] + f(n) = E[Q] + Y Q. (8) Recalling Definition completes the proof. roposition provies us with a principle recommenation how to remove the effect of the noise an reconstruct the unobserve Q up to its mean E[Q]: we nee to subtract the conitional expectation (i.e., the regression) E[Y X] from the observe Y (Definition ). The regression E[Y X] can be estimate from observations (x i, y i ) using (linear or nonlinear) off-the-shelf methos. We refer to this proceure as half-sibling regression to reflect the fact that we are trying to explain aspects of the chil Y by regression on its half-sibling(s) X in orer to reconstruct properties of its unobserve parent Q. Note that m(x) := E[f(N) X = x] is a function of x, an E[f(N) X] is the ranom variable m(x). Corresponingly, (4) is an equality of RVs. By assumption, all RVs live on the same unerlying probability space. If we perform the associate ranom experiment, we obtain values for X an N, an (4) tells us that if we substitute them into m an f, respectively, we get the same value with probability. Eq. (5) is also an equality of RVs, an the above proceure therefore not only reconstructs some properties of the unobservable RV Q it reconstructs, up to the mean E[Q], an with probability, the RV itself. This may soun too goo to be true in practice, of course its accuracy will epen on how well the assumptions of roposition hol. If the following conitions are met, we may expect that the proceure shoul work well in practice: (i) X shoul be (almost) inepenent of Q otherwise, our metho coul possibly remove parts of Q itself, an thus throw out the baby with the bathtub. A sufficient conition for this to be the case is that N be (almost) inepenent of Q, which often makes sense in practice, e.g., if N is introuce by a measuring evice in a way inepenent of the unerlying object being measure. Clearly, we can only hope to remove noise that is inepenent of the signal, otherwise it woul be unclear what is noise an what is signal. A sufficient conition for N Q, finally, is that the causal DAG in Fig. correctly escribes the unerlying causal structure. Note, however, that roposition an thus our metho also applies if N Q, as long as X Q. (ii) The observable X is chosen such that Y can be preicte as well as possible from it; i.e., X contains enough information about f(n) an, ieally, N acts on both X an Y in similar ways such that a simple function class suffices for solving the regression problem in practice. This may soun like a rather strong requirement, but we will see that in our astronomy application, it is not unrealistic: X will be a large vector of pixels of other stars, an we will use them to preict a pixel Y of a star of interest. In this kin of problem, the main variability of Y will often be ue to the systematic effects ue to the instrument N also affecting other stars, an thus a large set of other stars will inee allow a goo preiction of the measure Y. Note that it is not require that the unerlying structural equation moel be linear N can act on X an Y in nonlinear ways, as an aitive term f(n). In practice, we never observe N irectly, an thus it is har to tell whether the assumption of perfect preictability of f(n) from X hols true. We now relax this assumption Incomplete Information First we observe that E[f(N) X] is a goo approximation for f(n) whenever f(n) is almost etermine by X: Lemma 2 For any two jointly ranom variables Z, X, we have E[(Z E[Z X]) 2 ] = E[Var[Z X]]. (9) Here, E[Z X] is the ranom variable g(x) with g(x) = E[Z X = x], an Var[Z X] is the ranom variable h(x)

4 with h(x) = Var[Z X = x]. Then (9) turns into E[(Z g(x)) 2 ] = E[h(X)]. (0) roof. Note that for any ranom variable Z we have Var[Z X = x] = E[(Z E[Z X = x]) 2 X = x], by the efinition of variance, applie to the variable Z X=x. Hence Var[Z X] = E[(Z E[Z X]) 2 X], where both sies are functions of X. Taking the expectation w.r.t. X on both sies yiels E[Var[Z X]] = E[(Z E[Z X]) 2 ], where we have use the law of total expectation E[E[W X]] = E[W ] on the right han sie. This leas to a stronger result for our estimator ˆQ (2): roposition 3 Let f be measurable, N, Q, X, Y jointly ranom variables with Q X, an Y = Q + f(n). The expecte square eviation between ˆQ an Q E[Q] satisfies E[( ˆQ (Q E[Q])) 2 ] = E[Var[f(N) X]]. () roof. We rewrite the argument of the square in () as ˆQ (Q E[Q]) = Y E[Y X] Q + E[Q] = f(n) + Q E[f(N) X] E[Q X] Q + E[Q] = f(n) E[f(N) X]. The result follows using Lemma 2 with Z := f(n). Note that roposition is a special case of roposition 3: if there exists a function ψ such that ψ(x) = f(n), then the r.h.s. of () vanishes. roposition 3 rops this assumption, which is more realistic: consier the case where X = g(n) + R, where R is another ranom variable. In this case, we cannot expect to reconstruct the variable f(n) from X exactly. There are, however, two settings where we woul still expect goo approximate recovery of Q: (i) If the stanar eviation of R goes to zero, the signal of N in X becomes strong an we can approximately estimate f(n) from X, see roposition 4. (ii) Alternatively, we observe many ifferent effects of N. In the astronomy application below, Q an R are stars, from which we get noisy observations Y an X. roposition 5 below shows that observing many ifferent X i helps reconstructing Q, even if all X i epen on N through ifferent functions g i an their unerlying (inepenent) signals R i o not follow the same istribution. roposition 4 Assume that Y = Q + f(n) an let X s := g(n) + s R, where R, N an Q are jointly inepenent, f C b (R), g C (R), s R, an g is invertible. Then ˆQ s L 2 Q E[Q] as s, where ˆQ s := Y E[Y X s ]. The proof is skippe since it is analogous to the more complicate proof of roposition 5. roposition 5 Assume that Y = Q+f(N) an that X := (X,..., X ) satisfies X i := g i (N) + R i, i =,...,, where all R i, N an Q are jointly inepenent, i= i 2 var(r i ) <, f C b (R), g i C (R) for all i, an g := j= is invertible with ( g ) uniformly equicontinuous. Then g j ˆQ L 2 Q E[Q] as, where we efine ˆQ := Y E[Y X ]. roof. By Kolmogorov s strong law, we have for µ := i= E[R i] that ( g g ( i= R i µ i= (g i(n) + R i ) µ g (N) ) i= (g i(n) + R i ) µ g ( g (N)) ) i= X i µ N ( f g ( i= X i µ )) f(n) ψ (X ) f(n) for some ψ that are uniformly boune in (the implication follows from uniform equicontinuity, implication by the continuous mapping theorem). This implies E[f(N) X ] f(n) L2 0 The notation enotes convergence in probability with respect to the measure of the unerlying probability space.

5 unobserve Q N observe Y X Figure 2. Causal structure from Fig. when relaxing the assumption that X is an effect of N. because E[(f(N) E[f(N) X ]) 2 ] E[(f(N) ψ (X )) 2 ] 0 (The convergence of the right han sie follows from ψ (X ) f(n) an bouneness of ψ (X ) f(n)). But then Q E[Q] ˆQ = f(n) E[Q] + E[f(N) + Q X ] = E[f(N) X ] f(n) L2 0 The next subsection mentions an optional extension of our approach. Another extension, to the case of timeepenent ata, is briefly iscusse elsewhere (Schölkopf et al., 205) reiction from Non-Effects of the Noise Variable While Fig. shows the causal structure motivating our work, our metho oes not require a irecte arrow from N to X it only requires that N X, to ensure that X contains information about N. We can represent this by an unirecte connection between the two (Fig. 2), an note that such a epenence may arise from an arrow irecte in either irection, an/or another confouner that influences both N an X. This confouner nee not act eterministically on N, hence effectively removing our earlier requirement of a eterministic effect, cf. (). 3. Applications 3.. Synthetic Data We analyze two simulate ata sets that illustrate the ientifiability statements from Sections 2. an 2.2. Increasing relative strength of N in a single X. We consier 20 instances (each time we sample 200 i.i.. ata points) of the moel Y = f(n) + Q an X = g(n) + R, where f an g are ranomly chosen sigmoi functions an the variables N, Q an R are normally istribute. The stanar eviation for R is chosen uniformly between 0.05 an, the stanar eviation for N is between 0.5 an. Because Q can be recovere only up to a shift in the mean, we set its sample mean to zero. The istribution for R, Figure 3. Left: we observe a variable X = g(n)+r with invertible function g. If the variance of R ecreases, the reconstruction of Q improves because it becomes easier to remove the influence f(n) of the noise N from the variable Y = f(n) + Q by using X, see roposition 4. Right: a similar behavior occurs with increasing the number p of preictor variables X i = g i(n) + R i, see roposition 5. Both plots show 20 scenarios, each connecte by a thin line. however, has a mean that is chosen uniformly between an an its stanar eviation is chosen from the vector (, 0.5, 0.25, 0.25, , , 0). roposition 4 shows that with ecreasing stanar eviation of R we can recover the signal Q. Stanar eviation zero correspons to the case of complete information (Section 2.). For regressing Y on X, we use the function gam (penalize regression splines) from the R-package mgcv; Figure 3 shows that this asymptotic behavior can be seen on finite ata sets. Increasing number of observe X i variables. Here, we consier the same simulation setting as before, this time simulating X i = g i (N) + R i for i =,..., p. We have shown in roposition 5 that if the number of variables X i tens to infinity, we are able to reconstruct the signal Q. In this experiment, the stanar eviation for R i an Q is chosen uniformly between 0.05 an ; The istribution of N is the same as above. It is interesting to note that even aitive moels (in the preictor variables) work as a regression metho (we use the function gam from the R-package mgcv on all variables X,..., X p an its sum X X p ). Figure 3 shows that with increasing p the reconstruction of Q improves Exoplanet Light Curves The fiel of exoplanet search has recently become one of the most popular areas of astronomy research. This is largely ue to the Kepler space observatory launche in Kepler observe a tiny fraction of the Milky Way in search of exoplanets. The telescope was pointe at same patch of sky for more than four years (Fig. 4). In that

6 patch, it monitore the brightness of stars (selecte from among 3.5 million stars in the search fiel), taking a stream of half-hour exposures using a set of CCD (Charge- Couple Device) imaging chips arrange in its focal plane using the layout visible in Fig. 4. Figure 4. Kepler search fiel as seen from Earth, locate close to the Milky Way plane, in a star-rich area near the constellation Cygnus (image creit: NASA). Figure 5. Sketch of the transit metho for exoplanet etection. As a planet passes in front of its host star, we can observe a small ip in the apparent star brightness (image creit: NASA Ames). Kepler etects exoplanets using the transit metho. Whenever a planet passes in front of their host star(s), we observe a tiny ip in the light curve (Fig. 5). This signal is rather faint, an for our own planet as seen from space, it woul amount to a brightness change smaller than 0 4, lasting less than half a ay, taking place once a year, an visible from about half a percent of all irections. The level of require photometric precision to etect such transits is one of the main motivations for performing these observations in space, where they are not isturbe by atmospheric effects, an it is possible to observe the same patch almost continuously using the same instrument. For planets orbiting stars in the habitable zone (allowing for liqui water) of stars similar to the sun, we woul expect the signal to be observable at most every few months. We thus have very few observations of each transit. However, it has become clear that there is a number of confouners introuce by spacecraft an telescope that lea to systematic changes in the light curves which are of the same magnitue or larger than the require accuracy. The ominant error is pointing jitter: if the camera fiel moves by a tiny fraction of a pixel (for Kepler, the orer of magnitue is 0.0 pixels), then the light istribution on the pixels will change. Each star affects a set of pixels (Fig. 9 in (Schölkopf et al., 205)), an we integrate their measurements to get an estimate of the star s overall brightness. Unfortunately, the pixel sensitivities are not precisely ientical, an even though one can try to correct for this, we are left with significant systematic errors. Overall, although Kepler is highly optimize for stable photometric measurements, its accuracy falls short of what is require for reliably etecting earth-like planets in habitable zones of sun-like stars. We obtaine the ata from the Mikulski Archive for Space Telescopes (MAST) (see eu/inex.html). Our system, which we abbreviate as CM (Causal ixel Moel), is base on the assumption that stars on the same CCD share systematic errors. If we pick two stars on the same CCD that are far away from each other, they will be light years apart in space an no physical interaction between them can take place. As Fig. 9 in (Schölkopf et al., 205) shows, the light curves nevertheless have similar trens, which is cause by systematics. In CM, we use linear regression to preict the light curve of each pixel belonging to the target star as a linear combination of a set of preictor pixels. Specifically, we use 4000 preictor pixels from about 50 stars, which are selecte to be closest in magnitue to the target star. 2 This is one since the systematic effects of the instruments epen somewhat on the star brightness; e.g., when a star saturates a pixel, blooming takes place an the signal leaks to neighboring pixels. To rule out any irect optical cross-talk by stray light, we require that the preictor pixels are from stars sufficiently far away from the target star (at least 20 pixels istance on the CCD), but we always take them from the same CCD (note that Kepler has a number of CCDs, an we expect that systematic errors epen on the CCD). We train the moel separately for each month, which contains about 300 ata points. 3 Stanar L2 regularization is employe to avoi overfitting, an parameters (regularization strength an number of input pixels) were optimize using cross-valiation. Nonlinear kernel regression was also evaluate, but i not lea to better results. This may be ue to the fact that the set of preictor pixels is relatively large (compare to the training set size); an among this large set, it seems that there are sufficiently many pixels who are affecte by the systematics in a rather 2 The exact number of stars varies with brightness, as brighter stars have larger images on the CCD an thus more pixels. 3 The ata come in batches which are separate by larger errors, since the spacecraft nees to perioically re-irect its antenna to sen the ata back to earth.

7 similar way as the target. We have observe in our results that the metho removes some of the intrinsic variability of the target star. This is ue to the fact that the signals are not i.i.. an time acts as a confouner. If among the preictor stars, there exists one whose intrinsic variability is very similar to the target star, then the regression can attenuate variability in the latter. This is unlikely to work exactly, but given the limite observation winow, an approximate match (e.g., stars varying at slightly ifferent frequencies) will alreay lea to some amount of attenuation. Since exoplanet transits are very rare, it is extremely unlikely (but not impossible) that the same mechanism will remove some transits. Note that for the purpose of exoplanet search, the stellar variability can be consiere a confouner as well, inepenent of the planet positions which are causal for transits. In orer to remove this, we use as aitional regression inputs also past an future of the target star. This as an autoregressive (AR) component to our moel, removing more of the stellar variability an thus increasing the sensitivity for transits. In this case, we select an exclusion winow aroun the point of time being correcte, to ensure that we o not remove the transit itself. Below, we report results where the AR component uses as inputs the three closest future an the three closest past time points, subject to the constraint that a winow of ±9 hours aroun the consiere time point is exclue. Choosing this winow correspons to the assumption that time points earlier than -9 hours or later than +9 hours are not informative for the transit itself. Smaller winows allow more accurate preiction, at the risk of amaging slow transit signals. Our coe is available at jvc2688/keplerixelmoel. To give a view on how our metho performs, CM is applie on several stars with known transit signals. After that, we compare them with the Kepler re-search Data Conitioning (DC) metho (see nasa.gov/ipelinedc.shtml). The first version of DC remove systematic errors base on correlations with a set of ancillary engineering ata, incluing temperatures at the etector electronics below the CCD array, an polynomials escribing centroi motions of stars. The current DC (Stumpe et al., 202) performs CA on filtere light curves of stars, projects the light curve of the target star on a CA subspace, an subsequently removes this projection. The CA is performe on a set of relatively quiet stars close in position an magnitue. For non-i.i.. ata, this proceure coul remove temporal structure of interest. To prevent this, the CA subspace is restricte to eight imensions, strongly limiting the capacity of the moel (cf. Foreman-Mackey et al., 205). In Fig. 6, we present correcte light curves for three typical stars of ifferent magnitues, using both CM an DC. Note that in our theoretical analysis, we ealt with aitive noise, an coul eal with multiplicative noise, e.g., by log transforming. In practice, none of the two moels is correct for our application. If we are intereste in the transit (an not the stellar variability), then the variability is a multiplicative confouner. At the same time, other noises may better be moele as aitive (e.g., CCD noise). In practice, we calibrate the ata by iviing by the regression estimate an then subtracting, i.e., Y E[Y x] = Y E[Y x] E[Y x] E[Y x] Y E[Y x] =. E[Y x] Effectively, we thus perform a subtractive normalization, followe by a ivisive one. This worke well, taking care of both types of contaminations. The results illustrate that our approach removes a major part of the variability present in the DC light curves, while preserving the transit signals. To provie a quantitative comparison, we ran CM on 000 stars from the whole Kepler input catalog (500 chosen ranomly from the whole list, an 500 ranom G-type sun-like stars), an estimate the Combine Differential hotometric recision (CD) for CM an DC. CD is an estimate of the relative precision in a time winow, inicating the noise level seen by a transit signal with a given uration. The uration is typically chosen to be 3, 6, or 2 hours (Christiansen et al., 202). Shorter urations are appropriate for planets close to their host stars, which are the ones that are easier to etect. We use the 2-hours CD metric, since the transit uration of an earth-like planet is roughly 0 hours. Fig. 7 presents our CD comparison of CM an DC, showing that our metho outperforms DC. This is no small feat, since DC is highly optimize for the task at han, incorporating substantial astronomical knowlege (e.g., it attempts to remove stellar variability as well as systematic trens). 4. Conclusion We have assaye a simple yet effective metho for removing the effect of systematic noise from observations. It utilizes the information containe in a set of other observations affecte by the same noise source. The main motivation for the metho was its application to exoplanet ata processing, which we iscusse in some etail, with rather promising results. However, we expect that it will have a large range of applications in other omains as well. We expect that our metho may enable astronomical iscoveries at higher sensitivity on the existing Kepler satellite ata. Moreover, we anticipate that methos to remove systematic errors will further increase in importance: by May 203, two of the four reaction wheels use to control the Kepler spacecraft were isfunctional, an in May 204,

8 (a) (b) (c) Figure 6. Correcte fluxes using our metho, for three example stars, spanning the main magnitue (brightness) range encountere. In (a), we consier a bright star, in (b), a star of moerate brightness, an in (c), a relatively faint star. SA stans for Simple Aperture hotometry (in our case, a relative flux measure compute from summing over the pixels belonging to a star). In all three panels, the top plot shows the SA flux (black) an the CM regression (re), i.e., our preiction of the star from other stars. The mile panel shows the CM flux correcte using the regression (etails see text), an the bottom shows the DC flux (i.e., the efault metho). The CM flux curve preserves the exoplanet transits (little ownwar spikes), while removing a substantial part of the variability present in the DC flux. All x-axes show time, measure in ays since //2009. (a) (b) (c) Figure 7. Comparison of the propose metho (CM) to the Kepler DC metho in terms of Combine Differential hotometric recision (CD) (see text). lot (a) shows our performance (re) vs. the DC performance in a scatter plot, as a function of star magnitue (note that larger magnitue means fainter stars, an smaller values of CD inicate a higher quality as measure by CD. lot (b) bins the same ataset an shows box plots within each bin, inicating meian, top quartile an bottom quartile. The re box correspons to CM, while the black box refers to DC. lot (c), finally, shows a histogram of CD values. Note that the re histogram has more mass towars the left, i.e., smaller values of CD, inicating that our metho overall outperforms DC, the Kepler gol stanar. NASA announce the K2 mission, using the remaining two wheels in combination with thrusters to control the spacecraft an continue the search for exoplanets in other star fiels. Systematic errors in K2 ata are significantly larger since the spacecraft has become harer to control. In aition, NASA is planning the launch of another space telescope for 207. TESS (Transiting Exoplanet Survey Satellite) 4 will perform an all-sky survey for small (earth-like) planets of nearby stars. To ate, no earth-like planets orbiting sun-like stars in the habitable zone have been foun. This is likely to change in the years to come, which woul be a major scientific iscovery. 5 Machine learning can con Decaes, or even centuries after the TESS survey is complete, the new planetary systems it iscovers will continue to be tribute significantly towars the analysis of these atasets, an the present paper is only a start. Acknowlegments We thank Stefan Harmeling, James McMurray, Oliver Stegle an Kun Zhang for helpful iscussion, an the anonymous reviewers for helpful suggestions an references. C-J S-G was supporte by a Google Europe Doctoral Fellowship in Causal Inference. stuie because they are both nearby an bright. In fact, when starships transporting colonists first epart the solar system, they may well be heae towar a TESS-iscovere planet as their new home. (Haswell, 200)

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