Causal Inference in Time Series Via Supervised Learning

Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

Causal Inference in Time Series via Supervised Learning

Yoichi Chikahara and Akinori Fujino NTT Communication Science Laboratories, Kyoto 619-0237, Japan [email protected], [email protected]

Abstract only on its past values are significantly reduced by addition- ally using the past values of X. When the regression model Causal inference in time series is an important can be well fitted to the data, we can infer correct causal di- problem in many fields. Traditional methods use rections. However, in practice, selecting an appropriate re- regression models for this problem. The infer- gression model for each time series data is difficult and re- ence accuracies of these methods depend greatly quires a deep understanding of the data analysis. Therefore, on whether or not the model can be well fitted to it is not easy to identify correct causal directions with these the data, and therefore we are required to select an model-based methods. appropriate regression model, which is difficult in practice. This paper proposes a supervised learn- The goal of this paper is to build an approach to causal ing framework that utilizes a classifier instead of inference in time series that does not require a deep under- regression models. We present a feature represen- standing of the data analysis. To realize this goal, we pro- tation that employs the distance between the con- pose a supervised learning framework that utilizes a classifier ditional distributions given past variable values and instead of regression models. Specifically, we propose solv- show experimentally that the feature representation ing the problem of Granger causality identification by ternary provides sufficiently different feature vectors for classification, in other words, by training a classifier that as- time series with different causal relationships. Fur- signs ternary causal labels (X → Y , X ← Y , or No Cau- thermore, we extend our framework to multivariate sation) to time series. In fact, several methods have already time series and present experimental results where been proposed that perform classification to infer causal re- our method outperformed the model-based meth- lationships from i.i.d. data, which have worked well ex- ods and the supervised learning method for i.i.d. perimentally [Bontempi and Flauder, 2015; Guyon, 2013; data. Lopez-Paz et al., 2015; 2017]. To solve causal inference in time series via classification, we formulate a feature representation that provides sufficiently different feature vectors for 1 Introduction time series with different causal relationships. The idea for Discovering temporal causal directions is an important task obtaining such feature vectors is founded on the definition of in time series analysis and has key applications in various Granger causality: X is the cause of Y if the following two fields. For instance, finding the causal direction that indi- conditional distributions of the future value of Y are differ- cates that the research and development (R&D) expenditure ent; one is given the past values of Y and the other is given X influences the total sales Y , but not vice versa, is helpful the past values of X and Y . To build the classifier for Granger for decision making in companies. In addition, identifying causality identification, we utilize the distance between these causal (regulatory) relationships between genes from time se- distributions when preparing feature vectors. To compute the ries gene expression data is one of the most important topics distance, by using kernel mean embedding, we map each dis- in bioinformatics. tribution to a point in the feature space called the reproduc- As a definition of temporal causality, Granger causality ing kernel Hilbert space (RKHS) and measure the distance [Granger, 1969] is widely used [Kar et al., 2011; Yao et al., between the points, which is termed the maximum mean dis- 2015]. According to its definition, the variable X is the cause crepancy (MMD) [Gretton et al., 2007]. of the variable Y if the past values of X are helpful in pre- In experiments, our method sufficiently outperformed the dicting the future value of Y . model-based Granger causality methods and the supervised Traditional methods for identifying Granger causality use learning method for i.i.d. data by using the same feature rep- regression models [Bell et al., 1996; Cheng et al., 2014; resentation and the same classifier. Furthermore, we describe Granger, 1969; Marinazzo et al., 2008; Sun, 2008] such as the how our approach can be extended to multivariate time series vector autoregressive (VAR) model and the generalized addi- and show experimentally that feature vectors have a sufficient tive models (GAM). With these methods, we can determine difference that depends on Granger causality, which demon- that X is the cause of Y if the prediction errors of Y based strates the effectivity of our proposed framework.

2042 Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

2 Granger Causality Basic Ideas for Granger Causality Identification Granger causality defines X as the cause of Y if the past val- Simply by using the definition of Granger causality (Defini- tion 1)1, for instance, we regard the causal label as X → Y if ues of X contain helpful information for predicting the future value of Y . Formally, it is defined as follows: X is the cause of Y , and if Y is not the cause of X. Formally, we regard the causal label as 2 Definition 1 (Granger causality[Granger, 1969]) Suppose P (X |S ,S ) = P (X |S ) we have a stationary sequence of random variables {(Xt, X → Y if t+1 X Y t+1 X (2) Yt)} (t ∈ N), where Xt and Yt are on X and Y, respectively. P (Yt+1|SX ,SY ) 6= P (Yt+1|SY ) Let SX and SY be observations of {X1, ··· , Xt} and {Y1, P (Xt+1|SX ,SY ) 6= P (Xt+1|SX ) ··· , Yt}, respectively. X ← Y if (3) P (Y |S ,S ) = P (Y |S ) Granger causality defines {Xt} as the cause of {Yt} if t+1 X Y t+1 Y P (Xt+1|SX ,SY ) = P (Xt+1|SX ) P (Yt+1|SX ,SY ) 6= P (Yt+1|SY ) No Causation if (4) P (Yt+1|SX ,SY ) = P (Yt+1|SY ) and states that {Xt} is not the cause of {Yt} if To assign causal labels to time series based on (2), (3), P (Yt+1|SX ,SY ) = P (Yt+1|SY ) (1) and (4), it is necessary to determine whether or not the two conditional distributions are identical. To represent information about conditional distributions, instead of using regres- To see if the two conditional distributions P (Y |S ,S ) sion models, we utilize kernel mean embedding. Kernel mean t+1 X Y embedding maps a distribution to a point in the feature space and P (Yt+1|SY ) are identical, traditional methods [Bell et al., 1996; Granger, 1969; Marinazzo et al., 2008; Sun, 2008] called the RKHS. Interestingly, when a characteristic kernel use statistical testing to determine whether or not the two con- (e.g., a Gaussian kernel) is used, the mapping is injective: different distributions are not mapped to the same point [Sripe- ditional means E[Yt+1|SX ,SY ] and E[Yt+1|SY ] are equal, which is a much simpler problem than (1). For instance, in rumbudur et al., 2010]. [Granger, 1969], the conditional means are represented by us- Suppose that kernel mean embedding maps condi- ing the (V)AR model to compute the test statistic based on the tional distributions P (Xt+1|SX ,SY ), P (Xt+1|SX ) and P (Y |S ,S ) P (Y |S ) µ prediction errors. t+1 X Y , t+1 Y to the points Xt+1|SX ,SY , µ ∈ H and µ , µ ∈ H , respec- To represent the conditional means, these methods require Xt+1|SY X Yt+1|SX ,SY Yt+1|SY Y us to use an appropriate regression model that can be well tively, where HX and HY are the RKHSs. Then, when using fitted to the data; however, such a model is difficult to select a characteristic kernel, (2), (3), and (4) can be written as in practice. For this problem, we propose a novel approach µ = µ that utilizes a classifier instead of regression models. X → Y if Xt+1|SX ,SY Xt+1|SX (5) µYt+1|SX ,SY 6= µYt+1|SY µ 6= µ 3 Proposed Method X ← Y if Xt+1|SX ,SY Xt+1|SX (6) µYt+1|SX ,SY = µYt+1|SY 3.1 Classification Setup µX |S ,S = µX |S Suppose the training data are N pairs of bivariate time se- No Causation if t+1 X Y t+1 X (7) µ = µ ries S1, ··· , SN , where each time series Sj with the fixed Yt+1|SX ,SY Yt+1|SY length Tj consists of the observations of random variables To assign labels based on (5), (6), and (7), we only have {(Xj,Y j), ··· , (Xj ,Y j )} (j ∈ {1, ··· , N}). Here, each 1 1 Tj Tj to determine whether or not two points in the RKHS are the time series Sj has a causal label lj ∈ {+1, −1, 0} that indi- same over time t or, equivalently, to determine whether or not cates Xj → Y j, Xj ← Y j, or No Causation, where Xj = the distance between the two points in the RKHS, which is j j j j j termed the MMD [Gretton , 2007], is zero over time t. (X1 , ··· , XT ) and Y = (Y1 , ··· , YT ). et al. j j For this reason, to develop the classifier for Granger causal- Using a function ν(·) that maps Sj to a single feature vec- ity identification, in our feature representation, we utilize the tor, we first train a classifier with {(ν(Sj), lj)}N . Then, the j=1 MMDs, which are defined and estimated as follows. task of inferring the causal relationship in another time series Definition: Let k and k be kernels on X and Y, and S0 (test data) can be rephrased as assigning the label to ν(S0) X Y by using the trained classifier. 1Note that since our approach is founded on Definition 1, which Such a classification task can be extended to multivariate cannot address the case where there are latent confounders (i.e., un- time series as detailed in Section 3.3. observed variables that influence both X and Y ), as with the existing methods [Bell et al., 1996; Cheng et al., 2014; Granger, 1969; 3.2 Classifier Design Marinazzo et al., 2008; Sun, 2008], this paper does not deal with such a case. To build a classifier that assigns causal labels to time series, 2 Although we do not consider the case where P (Xt+1|SX ,SY ) we formulate the feature representation ν(·). In what follows, 6= P (Xt+1|SX ) and P (Yt+1|SX ,SY ) 6= P (Yt+1|SY ) (i.e., X is we describe our ideas for obtaining feature vectors that are the cause of Y , and Y is also the cause of X), we can straightfor- sufficiently different depending on Granger causality. wardly address such a case by adding an extra label.

2043 Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

... ...No Causation HX and HY be the RKHSs deﬁned by kX and kY , respec- ... tively. The MMD for the two distributions P (Xt+1|SX ,SY ) and P (Xt+1|SX ) is simply deﬁned as the distance between

µXt+1|SX ,SY , µXt+1|SX ∈ HX as follows

2 ≡ kµ − µ k2 MMDXt+1 Xt+1|SX ,SY Xt+1|SX HX (8)

2 Similarly, MMDYt+1 is deﬁned as the distance between

µYt+1|SX ,SY , µYt+1|SY ∈ HY . Estimation: The MMD can be estimated without using re- Figure 1: Different MMD pairs are estimated from time series with gression models and without performing a density estima- different causal labels. Each dot represents the MMD pair estimated tion. At this point, the MMD is much more attractive than the from each time series. Kolmogorov-Smirnov statistic [Chen and An, 1997] and the Kullback-Leibler divergence [Kullback and Leibler, 1951] Feature Representation since the former requires us to select regression models and the latter requires a density estimation, which is difficult when To build a classifier for Granger causality identification, we obtain the feature vectors by using the MMD pairs, where there are insufficient samples. 2 2 d = [\ \ ]> To estimate the MMD (8), we estimate the kernel mean each pair t MMDXt+1 , MMDYt+1 is estimated by (11). embeddings of conditional distributions µXt+1|SX ,SY and By using the MMD pairs, we can expect sufficiently dif- [ ] µXt+1|SX . As detailed in e.g., Muandet et al., 2017 , in ferent feature vectors to be obtained from time series with general, the kernel mean embedding of the distribution is different causal labels. This is because whether or not the estimated by taking the weighted sum of the so-called fea- MMD becomes zero depends on the causal label as indicated ture mapping function. Specifically, when using the existing by (5), (6), and (7). Although each MMD in dt cannot be- method called the kernel Kalman filter based on a conditional come exactly zero since it is a finite sample estimate, we can embedding operator (KKF-CEO) [Zhu et al., 2014], we can expect sufficiently different MMD pairs to be estimated from estimate µXt+1|SX ,SY and µXT +1|SX by the weighted sum of time series with different causal labels as intuitively shown in the feature mapping ΦX : Fig. 1, which we confirm experimentally in Section 5.2. To prepare dt for each t, given a time series with the length t−1 T , S = {(x1, y1), ··· , (xT , yT )}, we use its subsequence X XY µˆXt+1|SX ,SY = wτ ΦX (xτ ) (9) with the length W (W < T ), i.e., {(xt−(W −1), yt−(W −1)), τ=2 4 ··· , (xt, yt)} (t = W , ··· , T ) . As a result, we obtain the t−1 MMD pairs {d , ··· , d }. X X W T µˆXt+1|SX = wτ ΦX (xτ ) (10) Although we can directly use these MMD pairs as a single τ=2 feature vector, such a feature vector has the dimensionality

3 2(T − W + 1), which depends on the time series length T . where ΦX (xτ ) ≡ kX (xτ , ·) is a feature mapping function , XY XY XY > X X As feature vectors whose dimensionalities are the same for and w = [w2 , ··· , wt−1 ] and w = [w2 , ··· , time series with different lengths, we utilize the mean of the X > wt−1] (t > 3) are the real-valued weight vectors. MMD pairs. However, when simply using the mean (dW + X XY To compute the weight vectors w and w , we em- ··· dT ) / (T − W + 1) as a feature vector, the feature vec- ployed KKF-CEO. In fact, KKF-CEO provides the algorithm tors take the same value for the two sets of the MMD pairs X needed to estimate w from the observations SX for time se- whose empirical means are the same, but whose empirical ries prediction. Therefore, we can compute wX by directly distributions are different. XY employing KKF-CEO. To estimate w from SX and SY , For this reason, to avoid mapping different distributions of we simply used KKF-CEO with the product kernel kX · kY . the MMD pairs to the same feature vector, we again utilize Although computing weight vectors by KKF-CEO requires kernel mean embedding. By using a different kernel function the setting of several hyperparameters, they can be appropri- kD from kX and kY , we deﬁne our feature representation as ately set for each time series by minimizing the squared errors T between observations and the values predicted by KKF-CEO. 1 X ν(S) ≡ ΦD(dt) (12) Applying (9) and (10) to (8), MMD2 is estimated as T − W + 1 Xt+1 t=W 2 2 t−1 t−1 where d = [MMD\ , MMD\ ]> 2 t Xt+1 Yt+1 \ X X XY XY X X = (w w 0 + w w 0 MMDXt+1 τ τ τ τ τ=2 τ 0=2 which is the mean over the feature mappings ΦD(dt) ≡ 5 XY X kD(dt, ·) . − 2wτ wτ 0 )kX (xτ , xτ 0 ) (11) 4By using shorter time series, we can reduce the time complexity 3 0 3 For instance, when using the Gaussian kernel kX (x, x ) = exp when computing weight vectors by KKF-CEO (i.e., O(T ) [Zhu et (−γkx−x0k2) (γ > 0 is a parameter), the feature mapping becomes al., 2014]). 2 p p 2 2 > 5 ΦX (x) = exp (−γx ) [1, 2γ/1! x, (2γ) /2! x , ··· ] . When using samples that are drawn directly from a distribution,

2044 Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

In (12), to compute the feature mapping ΦD(·), we em- n-variate Time Series (n > 3) ployed random Fourier features (RFF) [Rahimi and Recht, Although it is possible to develop the feature representation 2007], which approximate a feature mapping as a low- for n-variate time series (n > 3) by adding the extra MMDs to dimensional vector of random features that are sampled from dt, it becomes very difficult to prepare enough training data to the Fourier transform of the kernel function. In experiments, train the classifier since the number of possible combinations we set the number of features m = 100 and obtained an m- of the common cause variables of the variable pair X and Y dimensional feature vector for each time series, where we ob- grows super-exponentially in n. served no significant improvements in the inference accuracy For this reason, we used the feature representation for when using a larger m. trivariate time series. From n-variate time series, we infer a causal relationship between each variable pair X and Y in 3.3 Extensions to Multivariate Time Series three steps. First, for each v ∈ {1, ··· , n − 2}, we obtain Finally, we describe how our approach can be extended to the feature vector from the observations of the triplet of the n-variate time series (n ≥ 3). We first present the feature variables (X, Y , Zv). Second, based on each feature vector, representation for trivariate time series (i.e., n = 3) and we we use a trained classifier to compute the probabilities of the then address the case where n > 3. three labels (X → Y , X ← Y , and No Causation). Finally, Trivariate Time Series we assign the label with the highest average probability. Our feature representation for trivariate time series is founded Addressing the case where there are more than one com- on conditional Granger causality [Geweke, 1984], which can mon cause variables is left as our future work. be applied to multivariate time series unlike Definition 1. When using Definition 1 for trivariate time series, we can 4 Related Work wrongly identify Granger causality. For instance, when there It is worth noting that one of the supervised learning methods is no causal relationship between X and Y , if the third vari- for i.i.d. data, the randomized causation coefficient (RCC) able Z is their common cause, we can wrongly conclude that [Lopez-Paz et al., 2015], also uses kernel mean embedding X is the cause of Y , or Y is the cause of X. This is be- to obtain the features of the distribution that differ depending cause P (Yt+1|SX ,SY ) 6= P (Yt+1|SY ) or P (Xt+1|SX ,SY ) on the causal relationships. However, RCC and our method 6= P (Xt+1|SX ) might hold due to the influence of Z. are designed to obtain different features of the distribution. To address the influence of Z, conditional Granger causal- Specifically, via kernel mean embedding, RCC obtains in- ity compares the two conditional distributions given SZ , i.e., formation about the marginal and conditional distributions, the observations of the random variables {Z1, ··· , Zt}, each which are known to differ depending on the causal directions of which is defined on Z. Formally, it defines X as the cause according to an assumption called the independence of cause of Y given Z if P (Yt+1|SX ,SY ,SZ ) 6= P (Yt+1|SY ,SZ ) and mechanism (ICM) [Janzing and Scholkopf, 2010]. In holds; otherwise, X is not the cause of Y given Z. contrast, by using kernel mean embedding, our method mea- We introduce causal labels based on conditional Granger sures the distance between the conditional distributions given causality. For instance, similarly to (2), we regard the causal past variable values since it becomes sufficiently different de- label X → Y as pending on Granger causality. P (X |S ,S ,S ) = P (X |S ,S ) X → Y if t+1 X Y Z t+1 X Z P (Yt+1|SX ,SY ,SZ ) 6= P (Yt+1|SY ,SZ ), 5 Experiments which can be rewritten as 5.1 Experimental Settings µ = µ We compared the performance of our method (hereafter re- X → Y if Xt+1|SX ,SY ,SZ Xt+1|SX ,SZ ferred to as the supervised inference of Granger causality µYt+1|SX ,SY ,SZ 6= µYt+1|SY ,SZ (SIGC)) with the supervised learning method for i.i.d. data where µ , µ , µ , Xt+1|SX ,SY ,SZ Xt+1|SX ,SZ Yt+1|SX ,SY ,SZ RCC [Lopez-Paz et al., 2015]6, with the Granger causal- and µY |S ,S are the kernel mean embed- 7 t+1 Y Z ity methods GCV AR [Granger, 1969] , GCGAM [Bell et al., dings of P (Xt+1|SX ,SY ,SZ ), P (Xt+1|SX ,SZ ), 7 8 1996] , and GCKER [Marinazzo et al., 2008] , which iden- P (Yt+1|SX ,SY ,SZ ), and P (Yt+1|SY ,SZ ), respectively. tify Granger causality by using the VAR model, the GAM, Motivated by these expressions, we address the case where 2 and kernel regression, respectively, and with transfer entropy Z X Y \ TE [Schreiber, 2000]9, which infers causal relationships not might influence and by adding MMDXt+1|Z and 2 by using regression models, but by performing density esti- MMD\ to the feature representation, where they are Yt+1|Z mation. 10 the MMD between µXt+1|SX ,SY ,SZ and µXt+1|SX ,SZ and For our SIGC, we used a random forest classifier to make the MMD between µYt+1|SX ,SY ,SZ and µYt+1|SY ,SZ , respec- a fair comparison with RCC, which has achieved better per- tively. For this reason, we extend the feature representation 6 (12) simply by modifying dt as follows https://github.com/lopezpaz/causation learning theory 7 2 2 2 2 http://people.tuebingen.mpg.de/jpeters/onlineCodeTimino.zip > d = [\ , \ , \ , \ ] 8 t MMDXt+1 MMDYt+1 MMDXt+1|Z MMDYt+1|Z https://github.com/danielemarinazzo/KernelGrangerCausality the kernel mean embedding of the distribution is estimated by using 9https://github.com/Healthcast/TransEnt the same weight values (e.g., 1/(T − W + 1) in (12)) [Muandet et 10The number of trees is selected from {100, 200, 500, 1000, al., 2017]. 2000} via 5-fold cross validation.

2045 Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

Linear Test Data Nonlinear Test Data formance with the random forest classiﬁer than with the SVM 1.0 1.0 [Lopez-Paz et al., 2015]. To prepare feature vectors, we used 0.8 0.8 the Gaussian kernel as kX , kY , and kD and set the kernel parameter using the median heuristic, which is a well-known1.0 0.6 0.6 heuristic for selecting it [Scholkopf and Smola, 2001]. We 0.4 0.4

0.8 Accuracy Test set the parameter W in our method and the parameters in the Accuracy Test existing methods to provide the best performance for each 0.2 0.6 0.2 SIGC GC GC method in our synthetic data experiments. For our method, VAR KER 1.0 RCC GCGAM TE 0GC 0 we selected W = 12. 0.4 20 VAR 40 60 80 100 120 Test Accuracy Test 20 60 100 140 180 240 Time Series Length Time Series Length 0.8 0.2 SIGC GCGAM SIGC GCVAR GCKER 5.2 Experiments on Bivariate Time Series DataRCC GCKER RCC GC Figure 2: Test accuracies for 300 pairsGAM of timeTE series data against 0.6 GCVAR TE GC 0 VAR Classifier Training 20 40 time60 series80 length100 120 (left: linear test data; right: nonlinear test data). Time Series Length We trained a classifier to infer causal0.4 relationships from bi- Means and standard deviations (error bars) are shown for our method Test Accuracy Test variate time series data. and RCC based on 20 runs with different training data. As with the existing supervised0.2 learning methods [Bon- tempi and Flauder, 2015; Lopez-Paz0 et al., 2015; 2017], we 20 40 60 80 100 120 • Nonlinear Test Data: We used 300 pairs of nonlinear used synthetic training data in both syntheticTime Series andLength real-world time series, where there were 100 time series with X data experiments since there are few real-world data where → Y , X → Y , and No Causation in each dataset. We the causal relationships are known. generated nonlinear time series with X → Y by We generated 15, 000 pairs of synthetic time series with the X = 0.2X + 0.9E (14) length T = 42 so that there were 5.000 instances each with t t−1 Xt 2 causal labels X → Y , X ← Y , and No Causation. Here, we Yt = −0.5 + exp(−(Xt−1 + Xt−2) ) used the following linear and nonlinear time series: 2 + 0.7 cos(Yt−1) + 0.3EYt (15) • Linear time series were sampled from the VAR model: where the noise variables EXt ,EYt were sampled from P N (0, 1). Similarly, we prepared nonlinear time series h i 1 h i h i Xt X Xt−τ EXt = Aτ + (13) with X ← Y . To prepare time series with No Causation, Yt P Yt−τ EYt τ=1 we simply ignored the exponential term in (15). Using linear and nonlinear test data, we compared the per- where τ = 1, ··· , P (P ∈ {1, 2, 3}) and EXt , EYt were sampled from the Gaussian distribution N (0, 1). To ob- formance of our method with that of the existing methods. tain time series with X → Y , we used the following Fig. 2 shows the test accuracies. Note that for SIGC and coefficient matrix RCC, we show the means and the standard deviations (error bars) in 20 experiments with different training data since h aτ 0.0 i Aτ = these methods use randomly generated training data. cτ dτ The performance of the Granger causality methods where aτ , dτ were drawn from the uniform distribution (GCV AR, GCGAM , and GCKER) depended on whether or U(−1, 1), and cτ ∈ {−1, 1}. Similarly, we prepared not the regression model could be well fitted to the data. For time series with X ← Y , and No Causation. instance, since the VAR model could be well fitted to linear test data, GC performed well on linear test data although • V AR Nonlinear time series were also similarly generated by it worked badly on nonlinear test data. In addition, with non- using the VAR model with a standard sigmoid function linear test data, GC was less accurate than GC be- g(x) = 1/(1+exp(−x)) KER GAM . For instance, we prepared time cause the time series was too short for us to perform kernel X → Y Y {[g(X ) series with so that t depended on t−τ , regression. Similarly, TE worked poorly since the time series Y ]>}P X {X }P t−τ τ=1 while t depended only on t−τ τ=1. was too short for us to perform density estimation. • Finally, we scaled each time series with mean 0 and vari- In contrast, our method worked sufficiently well on linear ance 1. and nonlinear test data. The main reason for the good performance lies in our feature representation. This can be seen Synthetic Time Series from a comparison with the supervised learning method RCC Next, we tested the performance of our method for inferring since it prepares training data in the same way as our method. X → Y X ← Y causal relationships ( , , and No Causation) To verify our feature representation, we confirmed experi- from synthetic time series data. We used the following linear mentally that feature vectors are sufficiently different depend- and nonlinear test data: ing on causal labels. To do so, we used nonlinear test data to • Linear Test Data: We prepared 300 pairs of linear time plot a histogram of the MMD pairs {dt} that were used to series so that the numbers of time series with X → Y , compute the feature vector for each time series. Fig. 3 shows X → Y , and No Causation were 100. In a similar way the results. Since each MMD in dt is a finite sample esti- to the linear time series in the training data, each time mate, no MMD becomes exactly zero. However, we can see series was sampled from the VAR model (13) although that the MMDs became sufficiently different according to the several parameter settings were different (e.g., the noise causal labels. We obtained similar results with linear test data variance was given as p ∈ {0.5, 1.0, 1.5, 2.0}). although we have omitted them due to space limitations.

2046 Proceedings of the Twenty-Seventh International Joint Conference on Artiﬁcial Intelligence (IJCAI-18)

SIGC GCV AR GCGAM GCKER TE Temperature 400 400 (T = 16382) Radiation 200 200 (T = 8401) Internet 0 0 (T = 498) 0.0 0.0 0.2 0.2 Sun Spots 0.0 0.2 0.4 0.0 0.2 0.4 (T = 1632) 0.4 0.6 0.4 0.6 0.6 0.8 0.6 0.8 River Runoff 0.8 0.8 (T = 432)

SIGC RCC GCV AR GCGAM GCKER TE 400 Temperature 0.961 0.432 No Causation (T = 200)(0.011)(0.242) 0.950 0.848 0.234 0.492 200 Radiation 0.987 0.515 (T = 200)(0.053)(0.345) 0.156 0.0 0.782 0.394 0 Internet 1.0 0.478 0.0 (T = 200)(0.0)(0.222) 0.157 0.387 0.261 0.498 0.2 Sun Spots 1.0 0.435 0.908 0.704 0.076 0.522 0.0 0.4 (T = 200)(0.0)(0.182) 0.2 0.4 0.6 0.6 0.8 River Runoff 0.958 0.399 0.8 (T = 200)(0.058)(0.193) 0.684 0.406 0.155 0.485

Figure 3: Histogram of MMDs used to compute the feature vector Table 1: Causal relationships inferred from the first test dataset (top; for each time series in nonlinear test data with X → Y (top left), X and denote correct and incorrect results, respectively) and test ← Y (top right), and No Causation (bottom). accuracies for the second test dataset (bottom; Means and standard deviations are shown for SIGC and RCC based on 20 runs). In fact, since the MMD pairs are sufficiently different, we can assign the causal label by taking another approach, i.e., ply chopped each time series into multiple subsequences an unsupervised approach that uses no training data, which of length T = 200. outputs the causal label in two steps. First, using the MMD As regards training data, we used synthetic time series that pairs, the two statistical tests are performed to determine if we prepared in the same way as those for synthetic data ex- \2 \2 the mean of MMDXt+1 is zero and if the mean of MMDYt+1 periments. is zero. Then, by using the two p-values and some threshold Table 1 shows the result for each test dataset. Note that value (significance level), we can assign a causal label (X → we have omitted RCC from the top table in Table 1 because Y , X ← Y , or No Causation) to each time series. it showed different outputs in 20 experiments where different However, we confirmed experimentally that the perfor- training data were used as in the synthetic data experiments, mance of such an unsupervised approach depended greatly while our SIGC always output the same causal directions. As on the threshold value. Furthermore, it was less accurate than shown in Table 1, our SIGC outperformed the other existing our method (e.g., on nonlinear test data with the length T = methods regardless of the time series length T . 250, its test accuracy was 0.810 (not shown in Fig. 2) while our method achieved 0.966) although we selected the thresh- 5.3 Experiments on Multivariate Time Series Data old value that provided the best performance. These results We tested SIGCtri, which utilizes a feature representation suggest the effectiveness of our supervised learning approach, for trivariate time series. For classifier training, we used syn- which can obtain the decision boundary needed to determine thetic trivariate time series that were generated in a similar the causal label by training a classifier. way to those used in the experiments described in Section Real-world Time Series 5.2. As test data, we used the following time series gene ex- We tested our method by using real-world time series. To im- pression data: prove the reliability of the experiment, we used the following • We used the Saccharomyces cerevisiae (yeast) cell cycle two test datasets: gene expression dataset collected by [Spellman et al., • The first test dataset was composed of five pairs of bi- 1998]. By combining four short time series that were variate time series downloaded from the Cause-Effect measured in different microarray experiments, we pre- Pairs database [Jakob, ], whose true causal relationships pared a time series with the length T = 57, where the are reported in [Jakob, ] as X → Y for three pairs and number of genes was n = 14. To determine the true as X ← Y for the others. For instance, the River Runoff causal relationships between the genes, we used the gene is a bivariate time series concerning average precipita- network database KEGG [KEGG, 1995]. tion X and average river runoff Y , and the true causal Since the number of non-causally-related gene pairs was relationship is regarded as X → Y . much larger than the number of causally-related gene pairs, • Using the above five real-world time series, we prepared we evaluated the performance of each method in terms of the a second test dataset that consisted of subsequences in macro and micro-averaged F1 scores rather than test accu- each time series. To prepare the subsequences, we sim- racy.

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SIGCtri SIGCbi RCC GCV AR GCGAM GCKER TE [Jakob, ] Zscheischler Jakob. Database with cause-effect pairs. 0.483 0.431 0.407 macro F1 (0.0)(0.007)(0.096) 0.457 0.437 0.351 0.430 https://webdav.tuebingen.mpg.de/cause-effect/. 0.567 micro F1 0.637 0.578 0.567 0.513 0.436 0.449 [Janzing and Scholkopf, 2010] Dominik Janzing and Bernhard (0.0)(0.011)(0.161) Scholkopf. Causal inference using the algorithmic Markov con- dition. IEEE Transactions on Information Theory, 56(10):5168– Table 2: Macro and micro-averaged F1 scores. Means and standard 5194, 2010. deviations are shown for our methods and RCC based on 10 runs. [Kar et al., 2011] Muhsin Kar, S¸aban Nazlıoglu,˘ and Huseyin¨ Agır.˘ Financial development and economic growth nexus in the MENA Table 2 shows the results. Since the data were measured countries: Bootstrap panel granger causality analysis. Economic in different microarray experiments, all the methods could modelling, 28(1):685–693, 2011. not sufficiently work well. However, our SIGCtri worked [KEGG, 1995] KEGG: Kyoto Encyclopedia of Genes and better than the existing Granger causality methods. It also Genomes. https://www.genome.jp/kegg/, 1995. performed better than SIGCbi, which uses the feature rep- [Kullback and Leibler, 1951] Solomon Kullback and Richard A. resentation for bivariate time series, thus indicating that it is Leibler. On information and sufficiency. The annals of mathe- important to consider the influence of the common cause vari- matical statistics, 22(1):79–86, 1951. able as described in Section 3.3. [Lopez-Paz et al., 2015] David Lopez-Paz, Krikamol Muandet, Bernhard Scholkopf,¨ and Ilya Tolstikhin. Towards a learning the- 6 Conclusions ory of cause-effect inference. In ICML, pages 1452–1461, 2015. We have proposed a classification approach to Granger [Lopez-Paz et al., 2017] David Lopez-Paz, Robert Nishihara, causality identification. Whereas the performance of the Soumith Chintala, Bernhard Scholkopf,¨ and Leon´ Bottou. model-based methods depended hugely on whether the re- Discovering Causal Signals in Images. In CVPR, 2017. gression model could be well fitted to the data, our method [Marinazzo et al., 2008] Daniele Marinazzo, Mario Pellicoro, and performed sufficiently well by using the same feature repre- Sebastiano Stramaglia. Kernel-Granger causality and the analysis sentation and the same classifier (random forest classifier). of dynamical networks. Physical Review E, 77(5):056215, 2008. Furthermore, we demonstrated experimentally the reason for [Muandet et al., 2017] Krikamol Muandet, Kenji Fukumizu, such good performance by showing a sufficient difference be- Bharath Sriperumbudur, Bernhard Scholkopf,¨ et al. Kernel mean tween the feature vectors that depends on Granger causality. embedding of distributions: A review and beyond. Foundations These results demonstrate the effectiveness of classification and Trends R in Machine Learning, 10(1-2):1–141, 2017. approaches to Granger causality identification. [Rahimi and Recht, 2007] Ali Rahimi and Benjamin Recht. Ran- Addressing complicated real-world scenarios (e.g., infer- dom features for large-scale kernel machines. In NIPS, pages ring the causal directions that change over time) constitutes 1177–1184, 2007. our future work. [Scholkopf and Smola, 2001] Bernhard Scholkopf and Alexan- der J. Smola. Learning with kernels: support vector machines, References regularization, optimization, and beyond. MIT press, 2001. [Bell et al., 1996] David Bell, Jim Kay, and Jim Malley. A non- [Schreiber, 2000] Thomas Schreiber. Measuring information trans- parametric approach to non-linear causality testing. Economics fer. Physical review letters, 85(2):461, 2000. , 51(1):7–18, 1996. Letters [Spellman et al., 1998] Paul T. Spellman, Gavin Sherlock, [Bontempi and Flauder, 2015] Gianluca Bontempi and Maxime Michael Q. Zhang, Vishwanath R. Iyer, Kirk Anders, Michael B. Flauder. From dependency to causality: a machine learning ap- Eisen, Patrick O. Brown, David Botstein, and Bruce Futcher. proach. JMLR, 16:2437–2457, 2015. Comprehensive identification of cell cycle-regulated genes of [Chen and An, 1997] Min Chen and Hong Zhi An. A Kolmogorov- the yeast Saccharomyces cerevisiae by microarray hybridization. Smirnov type test for conditional heteroskedasticity in time se- Molecular biology of the cell, 9(12):3273–3297, 1998. ries. Statistics & probability letters, 33(3):321–331, 1997. [Sriperumbudur et al., 2010] Bharath K. Sriperumbudur, Arthur [Cheng et al., 2014] Dehua Cheng, Mohammad Taha Bahadori, Gretton, Kenji Fukumizu, Bernhard Scholkopf,¨ and Gert R.G. and Yan Liu. FBLG: a simple and effective approach for tempo- Lanckriet. Hilbert space embeddings and metrics on probability ral dependence discovery from time series data. In KDD, pages measures. JMLR, 11:1517–1561, 2010. 382–391, 2014. [Sun, 2008] Xiaohai Sun. Assessing nonlinear Granger causal- [Geweke, 1984] John F. Geweke. Measures of conditional linear ity from multivariate time series. In ECML, pages 440–455. dependence and feedback between time series. Journal of the Springer, 2008. American Statistical Association, 79(388):907–915, 1984. [Yao et al., 2015] Shun Yao, Shinjae Yoo, and Dantong Yu. Prior [Granger, 1969] Clive WJ Granger. Investigating causal relations knowledge driven Granger causality analysis on gene regulatory by econometric models and cross-spectral methods. Economet- network discovery. BMC bioinformatics, 16(1):273, 2015. rica: Journal of the Econometric Society, pages 424–438, 1969. [Zhu et al., 2014] Pingping Zhu, Badong Chen, and Jose C. [Gretton et al., 2007] Arthur Gretton, Karsten M. Borgwardt, Malte Principe. Learning nonlinear generative models of time series Rasch, Bernhard Scholkopf,¨ and Alex J. Smola. A kernel method with a Kalman filter in RKHS. IEEE Transactions on Signal for the two-sample-problem. In NIPS, pages 513–520, 2007. Processing, 62(1):141–155, 2014. [Guyon, 2013] Isabelle Guyon. ChaLearn cause-effect pair chal- lenge. https://www.kaggle.com/c/cause-effect-pairs/, 2013.

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