Latent Convergent Cross Mapping
Authors: Edward De Brouwer
Adam Arany
Jaak Simm
Yves Moreau
Venue: ICLR 2021 Poster (2021)
Keywords: Causality
Time Series
Chaos
Neural ODE
Missing Values

Abstract: Discovering causal structures of temporal processes is a major tool of scientific inquiry because it helps us better understand and explain the mechanisms driving a phenomenon of interest, thereby facilitating analysis, reasoning, and synthesis for such systems. 
However, accurately inferring causal structures within a phenomenon based on observational data only is still an open problem. Indeed, this type of data usually consists in short time series with missing or noisy values for which causal inference is increasingly difficult. In this work, we propose a method to uncover causal relations in chaotic dynamical systems from short, noisy and sporadic time series (that is, incomplete observations at infrequent and irregular intervals) where the classical convergent cross mapping (CCM) fails. Our method works by learning a Neural ODE latent process modeling the state-space dynamics of the time series and by checking the existence of a continuous map between the resulting processes. We provide theoretical analysis and show empirically that Latent-CCM can reliably uncover the true causal pattern, unlike traditional methods.

Published as a conference paper at ICLR 2021
LATENT CONVERGENT CROSS MAPPING
Edward De Brouwer"
Adam Arany* ESAT-STADIUS KU LEUVEN
ESAT-STADIUS KU LEUVEN
Leuven, 3001, Belgium edward. debrouwer@esat www.:/wwweeeett....eeeeeeee
Leuven, 3001, Belgium
adam. artnyaaaantttt.. http://eesaaaklleeeeeeeeee kuleuven be
Jaak Simm
Yves Moreau
ESAT-STADIUS KU LEUVEN
ESAT-STADIUS KU LEUVEN
Leuven, 3001, Belgium jaak. s simm@esat kuleuven. be
Leuven, 3001, Belgium
waw.auuesat..kuluuueeeeeeae be
ABSTRACT
Discovering causal structures of temporal processes is a major tool of scientific inquiry because it helps us better understand and explain the mechanisms driving a phenomenon of interest, thereby facilitating analysis, reasoning, and synthesis for such systems. However, accurately inferring causal structures within phe- nomenon based on observational data only still an open problem. Indeed, this type of data usually consists in short time series with missing or noisy values for which causal inference is increasingly difficult. In this work, we propose a method uncover causal relations in chaotic dynamical systems from short, noisy and sporadic time series (that is, incomplete observations at infrequent and irregular intervals) where the classical convergent cross mapping (CCM) fails. Our method works by learning a Neural ODE latent process modeling the state-space dynam- ics of the time series and by checking the existence of continuous map between the resulting processes. We provide theoretical analysis and show empirically that Latent- CCM can reliably uncover the true causal pattern, unlike traditional methods.
INTRODUCTION
Inferring right causal model of a physical phenomenon is at the heart of scientific inquiry. It is fundamental to how we understand the world around us and to predict the impact of future interven- tions (Pearl 2009. Correctly inferring causal pathways helps reason about physical system, anticipate its behavior previously unseen conditions, design changes to achieve some objective, or synthesize new systems with desirable behaviors. As an example, in medicine, causality inference could allow predicting whether a drug will be effective for a specific patient, or in climatology, to assess human activity causal factor in climate change. Causal mechanisms are best uncovered by making use of interventions because this framework leads to an intuitive and robust notion of causality. However, there significant need identify causal dependencies when only observa- tional data is available, because such data is more readily available as it is more practical and less costly to collect (e.g., relying on observational studies when interventional clinical trials are not yet available).
However, real-world data arising from less controlled environment than, for instance, clinical trials poses many challenges for analysis. Confounding and selection bias come into play, which bias standard statistical estimators. If no intervention possible, some causal configurations cannot be identified. Importantly, with real-world data comes the major issue of missing values. In particular, when collecting longitudinal data, the resulting time series are often sporadic: sampling is irregular
Both authors contributed equally Corresponding author

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Published as a conference paper at ICLR 2021
in time and across dimensions leading varying time intervals between observations of given variable and typically multiple missing observations at any given time. This problem is ubiquitous in various fields, such healthcare (De Brouwer al. 2019). climate science (Thomson 1990), or astronomy (Cuevas- Tello et al. 2010).
A key problem in causal inference is to assess whether one temporal variable is causing another or is merely correlated with it. From assessing causal pathways for neural activity (Roebroeck et all 2005) ecology (Sggihara al. 2012 or healthcare, is a necessary step unravel underlying generating mechanisms. A common way to infer causal direction between two temporal variables to use Granger causality (Granger 1969p. which defines "predictive causality" in terms of the predictability of one time series from the other. A key requirement of Granger causality is then separability (i.e., that information about causes are not contained in the caused variable itself). This assumption holds in purely stochastic linear systems, but fails in more general cases (such as weakly coupled nonlinear dynamical systems) (Sugihara et al. 2012). To address this nonseparability issue, Sugihara et al. (Sugihara et al. 2012) introduced the Convergent Cross Mapping (CCM) method, which is based on the theory of chaotic dynamical systems, particularly on Takens' theorem. This method has been applied successfully in various fields such as ecology, climatology (Wang al.) 2018), and neuroscience (Schiecke et al., 2013). However, as the method relies on embedding the time series under study with time lags, it highly sensitive to missing values and usually requires long uninterrupted time series. This method is thus not applicable in settings with repeated short sporadic time series, despite their occurrence in many practical situations.
To address this important limitation, we propose to learn the causal dependencies between time series by checking the existence of convergent cross mappings between latent processes of those time series. Using a joint model across all segments of sporadically observed time series and forcing the model to learn the inherent dynamic of the data, we show that our method can detect causal relationship from short and sporadic time series, without computing delay embeddings. To learn a continuous time latent representation of the system's 's state-space, we leverage GRU-ODE-Bay -ayes (De Brouwer et al. 2019), recently introduced filtering method that extends the Neural ODE model Chen et al. 2018). Importantly for causal inference, the filtering nature of the model makes sure no future information can leak into the past. We then check the existence of continuous maps between the learnt latent representations and infer the causal direction accordingly.
In series of increasingly challenging test cases, our method accurately detects the correct causal de- pendencies with high confidence, even when fed very few observations, and outperforms competing methods such multi-spatial CCM or CCM with multivariate Gaussian process interpolation.
Hx
Hy
X(t]
Y[t]
causes YIJ"
Figure 1: Schematic of the Latent-CCM rationale. If X[t] causes Y[O, there exists continuous map (dotted line) from the latent process of (Hy) to the latent process of X (Hx).
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2 RELATED WORK
provided the first significant framework to infer causal dependencies from time series. Relying predictability between dynamical systems, was extended to account for different limitations, such as nonlinearity (Chen et al. 2004) instantaneous relationships (Schiatti al. 2015). However, the assumption of separability of information between causative and caused variables leads to the failure of the Granger paradigm for significant number of time series coupling scenarios (Sugihara et al. 2012) (see Appendix D for a revealing worked out example). Convergent Cross Mapping, a technique based on nonlinear state space reconstruction was introduced to tackle this issue (Sugihara et al., 2012). Recently, several works have proposed extensions of CCM, such as the extended CCM, to address issues such as synchrony (Ye et al., 2015) or to improve the discrimination of the confounding case (Benkō et al.
CCM to address failure of Granger causality. Granger causality (Granger 1969)
2018). Synchrony occurs when one time series can be expressed as a function of
the other (e.g. (t)= = (t) and attractors of both dynamical systems become homeomorphic to each other (Rulkov et al. 1995). This occurs when coupling between two chaotic system is too strong. Confounding, on the other hand, occurs when two variables are causally driven by third one. In general say that X confounds the relation between Y and Z if X causes both Y and Z.
Huang et al. (2020) also proposed to predict directly the driving time series from the driven one with reservoir computing, bypassing the delay embedding step, making it more robust to noise. However, those methods still require long regularly sampled time series.
Causality for short sporadic time series. Short time series are very common in practice and there has been some work proposing to learn causality from short time series relying on state space reconstruction. Ma et al. (2014) proposed method for short, fully observed, unique time series. Multi-spatial CCM (Clark et al. 2013), considered the problem of inferring causality from several short fully observed snippets of the same dynamical system by computing delay embeddings com- patible with the lengths of the time series and aggregating them. In comparison, on top of addressing irregular sampling, our approach computes more informative state-space representations by sharing model across all segments. Techniques to infer causal direction from incomplete time series have also been proposed, but all are relying on the Granger causality framework, which limits their appli- regularly sampled data (but with missing values) (Elsegai) 2019) or generalizations of similarity measures for sporadic time series (Bahadori & Liu 2012). To the best of our knowledge, this is the first work investi- gating the identification of causal dependencies from short sporadic time series using state-space reconstruction.
cability to separable dynamical systems. They use direct partial correlations
3 METHOD
We consider the problem of inferring a causal dependency between two temporal variables from several segments of their multivariate time series X(t] E ædx and [t] E gdr We assume that X and [(] have been generated by an unknown dynamical system. In this work, we refer to the dynamical system of time varying variable X as the smallest dynamical system that fully describes the dynamics of X. As an example, let's consider the following system of ODEs representing the dynamics of X and Y:
dX(() f(X(I)
(1)
dl
d 9(x(()) +(Y()))
(2)
dt
The dynamical system of X is given by Equation @.. On the other hand, the dynamical system of Y is Equation + as Equation M is required to describe the dynamics of Y.
To account for the more general and most frequent case, we consider those time series are only observed in segments of finite duration. X(t) and Y[t) then consist of collections of N short time series (X [t] Y is observed concomitantly. To proceed with a lighter notation, we'll 'lll drop the superscript when referring to segment of time series.
(tt)) and (Y [t.......y [t])) respectively. Importantly, each segment of X and
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Each of those time series is also sporadic namely the regularly sampled and not all dimen- sions are observed each time.
In this work, we define the notion of causality by considering the equations of the dynamical system as a structural causal model. In this framework, X causes Y if p(Y [do(X)) & P(Y) where do(X) an intervention on X (Pearl 2009). Then, if X causes Y, X is part of the dynamical system of Y (X is required to describe the dynamics of ). In the case of the example described by Equations and X causes Y if g(-) is not constant function.
3.1 CONVERGENT CROSS MAPPING AND TAKENS' THEOREM
CCM aims at discovering the causal direction between temporal variables in dynamical systems by checking if the state-space dynamics of their time series can be recovered from one another. As shown above, if X causes Y, X is then contained in the dynamical system of Y and it should be possible to recover a representation of the dynamical system of X from the dynamical system of Y.
A common way to obtain a representation of a dynamical system from its time series relies on Takens' embedding theorem [Takens] 1981).
Let [ E Rdxx be issued from a chaotic dynamical system that has a strange attractor M with box-counting dimension dms where we define an attractor as the manifold toward which the state of a chaotic dynamical system tends to evolve. The dynamics of this system are specified by flow on M, (+)())) R M + M, where é- (MM)) _M+++ and Mr stands for the point on the manifold time index This flow is encoded in the ODE of the system. The observed time series X [t] is then obtained through an observation function foba(-): : [t] foro(MM) Takens' theorem then
states that a delay embedding & with delay and embedding dimension k
t (.. = (a(d)(
all (M)))
an embedding of the strange attractor M if 2dar and : Rdd R is twice-diiferretiable observation function. More specifically, the embedding map & is a diffeomorphism between the original strange attractor manifold M and a shadow attractor manifold M generated by the delay embeddings. Under these assumptions, one can then theoretically reconstruct the original time series from the delay embedding.
The simplest observation function a consists in simply taking one of the dimensions of observa- tions of the dynamical system. In this case, writing X₃ as the i-th dimension of X(t), Tak- ens' theorem ensures that there is a diffeomorphism between the original attractor manifold of full dynamical system and the shadow manifold M' that would be generated by X'[1] krT)). To see how this theorem can be used to infer the causal direction, let us consider the manifold Mz of the joint dynamical system resulting of the concatenation of X [t]] and Y(t). We then generate two shadow manifolds M X and My from the delay embeddings kr))). Now, X unidirectionally causes Y (i.e., Y does not cause X). it means that X is part of an autonomous dynamical system and that Y is part of a larger one, containing X. The attractor of Y is then the same the one of the joint dynamical system Z. By contrast, the attractor of X is only subset of it. From Taken's 's theorem, it is theoretically possible to recover the original M2 from My and hence, by extension, recover M'x from My. However, the contrary not true and it in general not possible to recover My from M'x-
X,
Xt [fft-rr)..
X = (X, Xitt r)..... X,(t - kr)) and ( (Y, [t]. Yt - -l.....,Y, Yj
The CCM algorithm uses this property to infer causal dependency. embeds both dynamical sys- tems X and Y and use k-nearest neighbors to predict points on M'x from M' and inversely. The result then consists in the correlation of the predictions with the true values. We write Ccm(X,Y) the Pearson correlation for the task of reconstructing M'x from My.
Cem Ccm(X,Y) (X,Y) = Corr(M'x
where M'x stands fro the prediction of Mzx obtained from My. Importantly, this measure is non- symmetric as an non- coj-injective map between M'y and My would lead to an accurate reconstruction being possible in direction only.
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To infer that there is a causal link between the predictor dynamical system and the predicted one, this correlation should be high and, importantly, increase with the length of the observed time series, as the observed manifolds become denser.
The potential results are then interpreted in the following way (1) X causes Y if one can reconstruct with high accuracy MMxx from My; (2) X and Y are not causally related (but not necessarily statistically independent) if nor MMx nor Myy can be reconstructed from the other; (3) and Y in circular causal relation if both My and M'x can be reconstructed from the other. In the extreme case of strong coupling, the two systems are said to be in synchrony, and it becomes hard distinguish between unidirectional or bidirectional coupling (Ye et al. 2015).
3.2 NEURAL ODES
Many continuous-time deterministic dynamical systems are usefully described ODEs. But in general, not all dimensions of the dynamical system will be observed so that the system is better described as an ODE on a continuous latent process (t), conditioned on which the observations X [[4] are generated. For instance, when observing only one dimension of a 2 dimensional dynamical system, we cannot find flow (X) on that single dimension variable, but we can find one on the latent process H (t). We then have the following description of the dynamics:
X[1 = ((((() with dH Io(H()),)
dt
where 9 represents the parameters of the ODE, fo(-) is a uniformly Lipschitz continuous function and g()) is continuous function. Learning the dynamics of the system then consists in learning those parameters 0 from a finite set of (potentially noisy) observations of the process X. Neural ODEs (Chen et al., 2018) parametrize this function by neural network. Learning the weights of this network can be done using the adjoint method or by simply backpropagating through the numerical integrator. Note that one usually allows X[t] be stochastic (e.g. observation noise). In that case, the mean of X (rather than X [t] itself) follows Equation 3
3.3 CAUSAL INFERENCE WITH LATENT CCM
A key step in the CCM methodology is to compute the delay embedding of both time series: ( and ( However, when the data is only sporadically observed at irregular intervals, the prob- ability of observing the delayed samples X; [t), X,{L -]...., Xi(L- kr] is vanishing for any t. ['] and ['[]] are then never fully observed (in fact, only dimension is observed) and nearest neighbor prediction cannot be performed. What is more, short time series usually do not allow to compute a delay embedding of sufficient dimension (k) and lag (7) (Clark all 2015).
Instead of computing delay embeddings, we learn the dynamics of the process with continuous- time hidden process parametrized by Neural ODE (as in Eq, 3 and use this hidden representation as complete representation of the state-space, therefore eliminating the need for delay-embedding that was limiting the applicability of CCM to long, constant sampling time series. A graphical representation of the method is shown on Figure 1
To infer causality between temporal variables from their time series X[t) and Y(t), the first step to train two GRU-ODE Bayes models (De Brouwer et al. 2019). filtering technique that extends Neural-ODEs. Being filtering approach, GRU-ODE- Bayes ensures no leakage of future informa- tion backward in time, an important requirement for our notion of causality. The continuity of the latent process is also important as it provides more coverage of the attractor of the dynamical sys- tem. Indeed, constant latent process in between observations (such as obtained with classical recurrent neural network such as GRU) would lead to fewer unique latent process observations.
The same model is used for all segments of each time series and is trained to minimize forecasting error. We learn the observation function g. the ODE fo and the continuous-tin time latent process Ht).. We write the resulting space of latent vectors from time series X on all segments as Hx.
Causality is then inferred by checking the existence of a continuous map between Hx and Hy- Analogously to CCM, we consider X causes Y if there exists continuous map between Hy and
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Hxx- This is consistent because Hx. just as the delay embedding 4x. is embedding of the strange attractor the dynamical system as stated in Lemma 3.1 for which we give the proof in the Ap- pendix E
Lemma 3.1. For a sporadic time series X X satisfying the following dynamics.
X|I ((H(() with dH(()
(t(())))
di
with g(-) and Jo continuous functions. If there exists one observation function ou E C² X + R along with a valid couple TK, (in the Takens' embedding theorem sense) such that the map T a(d (t) injective, the latent process (t) is an embedding of the strange attractor of the full dynamical system containing X.
The requirement of & gion)..w being injective is not enforced in our architecture. However, with sufficient regularization of the network, is satisfied in practice as shown by our results in Section
4.5
The same reasoning as in CCM then applies to the latent process and causal direction can be inferred. The existence of continuous map between the latent spaces of both time series is quantitatively assessed with the correlation between the true latents of the driven time series and the reconstructions obtained with k-nearest neighbors model on the latents of the driven time series. For instance, for a direction X + Y. we report the correlation between predictions of Hx obtained from Hy and the actual ones (Hxx). A strong positive correlation suggests an accurate reconstruction and thus a causal link in the studied direction between the variables (e.g., X + Y). By contrast, a weak correlation suggests no causal link in that direction.
4 EXPERIMENTS
We evaluate the performance of our approach on data sets from physical and neurophysiology models, namely double pendulum and neurons activity data. We show that our method de- tects the right causal topology all cases, outperforming multi-spatial CCM, as well as base- lines designed to address the sporadicity of the time series. The code is available at https:
4.1 BASELINE METHODS
To the best of our knowledge, this is the first time CCM is applied short sporadic time series. Indeed, because of missing variables, many standard approaches are simply not applicable. The main baseline consists in multi-spatial CCM (Clark al. 2015) applied regularly sampled data with a sampling rate similar to the one of the sporadic data. We also compare our approach to variants where multi-spatial CCM is applied an interpolation of the sporadic time series using (1) linear interpolation and (2) univariate and multivariate Gaussian Processes (GP and MVGP). For the Gaussian Process, we chose a mixture of RBF and identity kernel and learn the parameters from the data. To model multivariate GPs (MVGP), we used the combination of Matern and periodic Matern kernel for the time dimension and used co-regionalization (Bonilla al. 2008) with full-rank interaction matrix. We then use the mean of the posterior process as the reconstruction subsequentially fed to the classical CCM method. Implementation was done with GPflow (Matthews) et al. 2017).
We also compared our approach to non-CCM causal discovery methods such as PCMCI (ungee al. 2019) and VARLinGAM (Hyvärinen et al., 2010). PCMCI uses conditional independence testing between time series at differrent lags to infer causal dependencies. VARLinGAM lears graphical model of the longitudinal variables and their time lags, using the LinGAM method (Shimizu et al. 2006). These methods do not allow for short sporadic time series as input but a comparison with less challenging non-sporadic variant of our datasets is presented in Appendix F

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Double Pendulums
activity
0.8-
0.33
Afrom B sperodi data
Rersontmmctinn come someattnnn
Bffoma sporadc daia
Reconstmuctinn C011
Reconstructngg X from
0.25
Reconstructing Y from
Reconstructing from
0.2
Reconstructing from
02-
Reconstructing from
Reconsltuctnng from
0.10
-02
0.002
0.4
0.5
0.6
0.002
0.4 Proportion the data used
Proportion of the data used
(a) 3 Double Pendulums
(h) Neural Activity Data
Figure 2: Pearson correlation of the reconstructions function of the proportion of data used for the Latent-CCM. Solid line represent the mean the repeats and the shaded the range of values. For both experiments, only the true causal directions show convergent reconstruction. Cemfull the score when the whole data used. Cemo is the score when only 100 samples of the latent space are used.
4.2 PERFORMANCE METRICS
Our method assesses causality by detecting convergent reconstruction accuracy between the latent processes of different time series. To account for both aspects in a single score, we use the dif- ference between the correlation of the reconstruction and the target latent vector using the whole data (Cem)uul and the correlation using only 100 sample points (Cemo). as shown on Figure 2 and suggested in Clark al. (2015). The score for the causal coupling from [ to ((1) is then dehned as
Sex (Xmmuu(X.)) Como (X,Y). Y).
with higher score implying certainty about the presence of causal edge in the data generation graph, we compare the obtained scores with the ones that would be obtained with CCM fully observed but independent time series. We compute the Mann-Whine Whitney U-statistics (Ma et al. 2014) and provide the corresponding p-value.
confidence in causal relationship. Additionally, quantify the
Nevertheless, in practice. one might not have access to the score of independent time series, making difficult assess from the score only if a causal relationship is present. To address this issue, we visualize the results graphically as shown in Figure 2 Causal directions should then stand out clearly and have the characteristic convergent pattern (Sugihara et 2012).
4.3 DOUBLE PENDULUM
Description. The double pendulum is simple physical system that chaotic and exhibits rich dynamical behavior. It consists of two point masses my and m2 connected to pivot point and to each other by weightless rods of length I and l2, as shown on Figure in Appendix A The trajectories of the double pendulum respect to the vertical (0, and f2), as well as the angular momenta P1 and pa conjugate to these angles. Each trajectory is then collection of d-miennsional vector observations.
described by the time series of the angles of the rods with
To introduce causal dependencies from pendulum X to Y, we include a non non- non-physical asymmetrical coupling term in the update of the momentum conjugate to the first angle:

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DHY
bY
Yee - ex, -0 -)
where cx,y coupling parameter. The term corresponding to quadratic potential incorporated to the Hamiltonian of system Y results in attraction system Y by system X. Depending the values cx,y and Cy . we have different causal relationships between X and Y. Namely, (1) X causes Y iff cx.Y # 0, (2) Y causes X iff cy x 0 and (3) X is not causally related Y if Cx,y = xy, 0.
Data generation. We consider two cases of generating models. The first consists of two double pendulums (X[4) and Y(4) with high observation noise with Y causing X. In this case, we set cx.Y and cy,x 0.2. The second consists of double pendulums (X(4), Y[ and Z((), with one of them causing the other two (cz.x = 0.5, cz.Y = 1). We then infer the causal model relations between those variables in pairwise fashion (i.e. we infer the causal direction between all pairs of variables in the system). Remarkably, X and Y are here correlated but not causally related. Graphical representations of both considered cases are presented in Figure S Parameters of the pendulums (lengths and masses) are presented in Appendix A We generate 5 trajectories with different initial conditions (0, ~ (- 0.05) and 02 (0.5, 0.05)). We simulate observation noise by adding random Gaussian noise to the samples with NN( 0, 0.1) for the first case and n ~ N(u = 0, a = 0.01) ) for the second. To account for the short length of time series usually encountered in the real world, seconds. To simulate sporadicity, we sample observation uniformly at random with an average rate of 4 samples per second. Furthermore, for each of those samples, that keeps each individual dimension with probability 0.3. This whole procedure leads to a sporadic pattern as shown Figure of Appendix A We used 80% of available windows for training and used the remaining 20% for hyperaarameter tuning with the MSE prediction on future samples used as model selection criterion. More details on this procedure is given in Appendix G
randomly split the trajectories in windows of 10
apply observation mask
X
X
Y
Figure 3: Graphical model representation of both cases considered in the main body of the paper.
Left: Case 1. Right: Case 2 (confounding).
4.4 NEURAL ACTIVITY DATA
We also evaluate our approach neural activity data. We generate time series of the average brane potential of two populations of leaky integrate-and-fire neurons with alpha-function shaped synaptic currents (iaf-scc__ psc sacaaphh simulated by NEST-2.20.0 (Fardet al. 2020). Each neuron popu- lation contains 100 units with sparse random excitatory synapses within the population. We consider two cases, one where population A unidirectionally excites population B and another case where both populations fire independently. To account for the short and sporadic nature of real-world data, seconds from which we sample observation every second on average. This leads to 20 samples being available average per time window.
we generate 5,000 windows
4.5 RESULTS
Results over 5 repeats for the double pendulums and the neural activity data are presented in Table and in Figure Because this method uses different metrics for inferring the underlying causal graph, the results for PCMCI are presented in Appendix F where we show that the method cannot reliably infer the generative causal dependencies in data.
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Table 1: Average reconstruction scores Se (and their standard deviations) in all directions for the double pendulum and neural activity experiments. Standard deviations are computed using 5 repe- titions. Significant correlations compared to noncoupled dynamical systems are in bold (p 0.01). Significance is computed using the Mann- Mann-Whitney rank test. Our approach detects the correct causal structure. /and highlight correct wrong direction detection respectively.
DATA
MULTI-SPATIAL CCM
LINEAR
MVGP
LATENI-CCM
PENDULUM
CASB
X +
-0.017 0.037
0.001 0.006
0.009 = 0.014
0.001 + 0.013
X K+Y
0.018 + 0.0056
-0.011 + 0.01
-0.001 + 0.021
0.055 + 0.001
AUC1-2
0.2 (P=0.98)<
0.49 (P=0.52) /
0.66 (P=0.11)
0.55 (P=0.35) /
AUC2-11
0.6 (P=0.23)X
0.17 (P=0.99)
0.44 (P=0.67)
(P<:0.001)
CASE 2
X +
0.488 + 0.074
-0.006 = 0.005
-0.055 0.009
0.001 0.005
X +
0.181 0.119
-0.01 = 0.01
-0.007 0.012
0.009 0.014
X +
0.054 0.021
-0.002 1 0.007
-..022
0.019 1 1.017 0.017
Z > X
1.199
0.003 0.004
0.012
0.657 +: 0.105
Y ZZ
-0.071 0.078
-0.005 + 0.012
-..003
0.005 0.011
-0.002 = 0.006
-0,003
0.555 0.109
AUC1-22
1.00 (P<0..01)X
0.27 (P=0.95)V
0.31 (P=0.91)
0.78 (P=0.02)
AUC2-1
1.00 (P<O..01)XX
0.2 (P=0.98)V
0.31 (P=0.92)
0.67 (P-0.09). /
AUC1-43
0.98 (P<0.001)X
0.41 (P=0.74)V
0.61 (P=0.19)
0.79 (P=0.02) /
AUC
0.93 (P<0.001)
0.25 (P=0.97)
0.81 (P=0.01)
1.00 (P<0.001)
AUC2-33
0.26 (P=0.97>V
(P=0.76VV
0.45 (P=0.63) /
0.46 (P=0.62)
AUC
0.79 (P=0.02)<
0.2(P=0.98)
0.43 (P=0.69) X
1.00 (P<00.00))
NEURONS
COUPLED
A +
0.267 + 0.001
0.045 0.014
0.006
0.295 0.012
A + B
0.055 + 0.003 1.00 (r=0.006)> 1.00(P=0.006)
0.014 - 0.009
0.026 = 0.010
0.033 + 0.012
AUCA-B
1.00(<=0.006) /
1.00 (r=0.006)
/ 1.00(P=0.006)
1.00 (P=0.006) /
AUCB-A
1.00(F=0.006)
1.00(r=0.006) X
INDEPENDENT
A
-0.012 0.001
0.001 0.003
-0.002 + 0.008
-0.006 0.007
A
-0.001 0.001
-0.001 + 0.003
-0.003 0.005
-0.002 + 0.008
Double Pendulum. Our approach the only one to recover the right causal direction from sporadic data. The other baselines do not detect any significant correlation and thus causal link between double pendulums, Despite having access to constant sampling data, multi-spatial CCM is also not able detect the right data structure. We argue this is caused by the short length of time series window, and thus the low number and quality of delay embeddings that can be computed. In contrast, our method shares the same model across all time windows, represents more reliably the (hidden) state-space at any point in time. Importantly, the perfect reconstruction for Case 2 shows that we can distinguish confounding from correlation between time series. Indeed, when inferring causal directions between X and Y, variable Z is not used and thus hidden. Yet, our methods detects no causal relation between and Y. Figure 2a graphically presents the results of our method for the second case, where it is obvious that the only two convergent mappings are the ones corresponding the true directions (solid blue and green lines), providing a strong signal for the right underlying causal mechanism.
Neural activity For the neuron activity data, we observe that our method delivers the largest effect size towards the true data generating model (S, 0.295). Baselines methods relying on imputation do not provide any clear signal for a causal coupling (score 10 times lower). Multi-spatial CCM with the regularly sampled original data provides similar signal than approach but dampened. Inter- estingly, we observe a small but significant correlation in the wrong direction (A +- B) suggesting a small coupling in this direction. An inspection of Figure 2N however, will convince the reader that

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Published as a conference paper at ICLR 2021
the main causal effect is indeed from A to B. This small correlation in the direction A <- B is also observed in the fully observed data as shown in Figure in Appendix
CONCLUSION AND FUTURE WORK
In this work, we propose a novel way to detect causal structure linking chaotic dynamical sys- tems that are sporadically observed using reconstruction of underlying latent processes learnt with Neural-ODE models. We show that our method correctly detects the causal directions between tem- poral variables in low and irregular sampling regime, when time series observed in only short noncontiguous time windows and even in the case of hidden confounders, which are characteristics of weal-world data. Despite the apparent limitation of our method chaotic systems, it has been shown that CCM is broadly applicable in practice as many real dynamical systems are either chaotic or empirically allow Takens' -like embeddings. As our work builds upon CCM theoretically, we ex- pect the range of application to be at least as large and leave the application to other real-world data for future work.
ACKNOWLEDGEMENTS
YM funded by the Research Council of KU Leuven through projects SymBioSys3 (C14/18/092); Federated cloud- Artificial Intelligence-driven platform for liquid biopsy analyses (C3/20/100) and CELSA-HIDUCTION
(CELSA/17/032). YM also acknowledges the FWO Elixir Belgium (I002819N) and Elixir Infrastructure (1002919N. This research received funding from the Flem- ish Government (AI Research Program). YM is affiliated to Leuven.AI KU Leuven institute for AI, B-3000, Leuven, Belgium. YM received funding from VLAIO PM: Augmanting Ther- apeutic Effectiveness through Novel Analytics (HBC.2019.2522 Mart (HBC.2018.22) 8.2277) EU. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 956832. We also thank Nvidia for supporting this research by donating GPUs. EDB is funded by SB grant from FWO.
and Industrial Project MaDeS-
10

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Published as a conference paper at ICLR 2021
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12


Reviews:
[{"review_id": "kWq5gD9gBby", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work proposes latent CCM, a causal discovery method for short, noisy time series with missing values. The method checks whether there exists CCM between latent processes of the time series without computing delay embeddings. Empirical results show the proposed method is more accurate in finding the right causal direction in various datasets.\n\nOne of my major concerns is: Is it necessary to learn continuous time latent representation? As time series are not continuous. It would not be useful to go beyond the granularity of the original time series. And solving the problem of missing value does not necessarily require “continuous time latent representation”. Also in the Double Pendulum results, the authors claim the improvement over the multispatial CCM is because the proposed model shares the same parameters across time windows, not because of the continuity of the model. So, it would be interesting to see an ablation study with a model which is based on GRU but without Neural ODEs.\n\nIn 4.3.2, the authors mentioned, “We used 80% of available windows for training and used the remaining 20% for hyperparameter tuning”. It would be better to specify what criterion is used for model selection. In addition, one good practice in this kind of causal direction discovery is to anonymize the two time series in training and validation (avoid using the ground truth in model selection), it is not clear if the authors followed this routine.\n\nIt would be interesting to report the computational complexity (time and space) of the proposed method. In addition, it would be appreciated if the authors can also report the runtime comparison between the algorithms.\n\nIt would also be better if the choice of baselines can be more comprehensive. For example, the authors can try to include algorithms beyond the CCM family such as PCMCI [1] and VARLINGAM [2].\n\nIn terms of writing and presentation, It would be better to make the content self-contained and make the use of terminologies consistent. For example, the authors did not explain terms like \"synchrony\" and \"confounding case\" etc.\n\n\n[1] Runge, Jakob, Peer Nowack, Marlene Kretschmer, Seth Flaxman, and Dino Sejdinovic. \"Detecting and quantifying causal associations in large nonlinear time series datasets.\" Science Advances 5, no. 11 (2019): eaau4996.\n\n[2] Hyvärinen, Aapo, Kun Zhang, Shohei Shimizu, and Patrik O. Hoyer. \"Estimation of a structural vector autoregression model using non-gaussianity.\" Journal of Machine Learning Research 11, no. 5 (2010).\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "S-ixVIJlkjp", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "### Summary\nThis paper studies short, chaotic time series and uses the Taken's theorem to discover the causality between two time series. The main challenge is that for short time series, the delay embedding is not possible. Thus, the authors propose to fit a latent neural ODE and theoretically argue that they can use the Neural ODE embeddings in place of the delay maps. The authors provide two sets of experiments, both on simulation data. Unfortunately, they never tested the algorithm on real data.\n\n### Feedback\n* The main confusion about this paper is the correlation function. $Corr(X, Y)$ is a symmetric function, likewise, the $S_{c X\\to Y}$  defined on page 6 is symmetric too. How can we infer the direction of causality using a symmetric metric?\n* The authors also do not describe why the $S_{c X\\to Y}$ is an appropriate metric because it just finds the small sample bias in the estimation of $Corr(X, Y)$. Thus, it should be susceptible to the choice of the 100 data points. What is the expected value of this metric?\n* The authors argue that their method works for short time series. But they use 100 data points as the initial point in Figure 2. Time series with length 100 is considered long in many applications.\n* In Lemma 3.1, can you elaborate on how strong the following assumption is? How likely is it to hold for time series data in domains such as healthcare, finance, and natural sciences? \n    > “If there exists one observation function $\\alpha_H$ along with a valid couple $(k; \\tau)$ (in the Takens’ embedding theorem sense) such that the map $\\Phi(H(t))$ is injective,”\n* The neural activity data is a point process. The ODE described in the paper is not an appropriate model for it. For example, associating the activity pattern with the missing values is inappropriate. Please use another ODE for this case.\n* It would be nice to have a non-chaotic synthetic dataset and show the results of the proposed algorithm on it.\n* The data generation in Section 4.3.2 uses missing totally at random. In reality, this seldom happens. The authors should use more realistic models for missing data.\n* Please add markers to Figure 2.\n* The authors are recommended to comment on how to apply their method to multivariate time series.\n* Generally, sticking to the causality literature, I suggest the authors put the assumptions under which the results are causal in a clear statement. See for example the following paper for the statement of the assumptions:\n    * Kennedy, E. H., Ma, Z., McHugh, M. D., & Small, D. S. (2015). Nonparametric methods for doubly robust estimation of continuous treatment effects. _Journal of the Royal Statistical Society. Series B: Statistical Methodology_, 79(4), 1229–1245.\n\n-----\n### Post-Response Update\nAfter reading the authors' response and given the changes made in the paper, I increase my rating by one point.", "extra_scores": {}, "extra_text": {}}, {"review_id": "6mn5YPJOy2m", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper introduces Latent Convergent Cross Mapping (Latent CCM) that combines Neural ODEs with ideas from dynamical systems to detect causation between time series. The latent spaces learnt using Neural ODEs, and more specifically GRU-ODE-Bayes, are used to infer the direction of the causal links between two time series X[t] and Y[t]. \n\nQuality, clarity, originality and significance:\nPro: The paper is very well written, and the idea is interesting and quite novel. Inferring causation in time series is a significant problem in all areas involving dynamical systems.  Cons: As described below, there are a few critical conceptual aspects about the purpose of the method that need to be clarified, mainly related to the distinction between dynamical systems and time series. \n\nOne aspect that I find confusing is the lack of a clear distinction between the concepts of “dynamical systems” and “time series”. They seem to be used interchangeably, but there is a difference as time series are generated from dynamical systems. Thus, two different time series can still describe the same dynamical system, for example its different dimensions. The original paper for CCM states: “CCM uses Takens embedding to detect if two variables belong to the same dynamical system.”  The current paper seems to state that it can detect causation not only between two time series, but also between two different dynamical systems. I believe this is a critical aspect and should be clarified and justified. If it is between two time series, do they need to be generated by the same dynamical system?\n\nThe idea of using the latent space of a neural ODE to replace the delay embeddings in CCM is very nice, but I wonder if we can say that there is a one-on-one mapping between the time series from Neural ODE and the latent space? If this is not case, can we say that we can infer causation? Lemma 3.1 goes in this direction, but it depends on the existence of $\\alpha_H$ and $(k, \\tau)$. How easy/difficult is it to find $\\Phi$ that is injective? This is not clear to me.\n\nDifferent concepts (irregular, sporadic, incomplete, partially observed, missing, non-fully-observed) are used to define incomplete observations. In dynamical systems theory, “partial observations” often refers to the fact that only some of the dimensions of the dynamical system/attractor are observed. Having missing data is different: it might be that all dimensions are observed, but not at every timestep. Using a consistent terminology is needed, otherwise too many concepts to describe the same things (and potentially different things) make reading confusing because this can refer to missing/incomplete information on either the samples or the features/dimensions.\n\nThe plots in Fig. 1 are a bit too crowded, the missing data/points could be plotted with a lighter color, and indicate that the dots are the observed data in the caption (that was not clear to me right away). Going back to the discussion of time series vs. dynamical system, $X[t]$ and $Y[t]$ in Fig. 1 seem to be from the same dynamical system, is it the same Lorenz attractor with the same parameters? \n\nThe paper states that it doesn’t use delay embedding, however by using segments in a way it is doing something very similar to delay embedding because it uses information about the past trajectory of a point as one single input to identify the latent space. This could also potentially explain why using different segments helps to find the latent state space. Could the authors extend on this connection? Do the authors believe that the method would work if only the individual points would be given as input, and not segments?\n\nSect. 3: are $d_X$ and $d_Y$ different? I don’t see clearly the connection with the do operation, as it is not mentioned afterwards? Would be good to clarify what $t_X$ and $t_Y$ are, it is not very intuitive.\n\nSect. 3.3: How could one check or guarantee that the “hidden representation is a complete representation of the state-space”? It would be great to give some details on GRU-ODE-Bayes as this is the method used. \n\nLemma 3.1.: Should it be $dH(t)/dt$? Before, the subscript for $\\Phi$ was $\\phi$ not $g(\\phi)$. What is the difference?\n\nSect. 4.2.: What are the “both aspects”? Why 100 samples for Corr_0, and could the authors explain a bit more the reasoning behind the choice of the score? Intuitively, I would have thought a score that is a percentage between the two correlations would be easier to interpret, but maybe I am wrong?\n\nSect. 4.3.1: Which is the quadratic potential? What happens for noisy observations? Have the authors tried adding noise and see if the causal inference still works?\n\nFig. 3: should be Y -> X instead of X-> Y\n\nSect. 4.5: What do the authors mean by folds here, is there a cross validation step? In the Neural activity paragraph, it should be A and B instead of X and Y.\n\nTable 1: Could the authors name the methods and describe them briefly? The acronym MVGP is only explained in the appendix. In the first case, for Latent CCM, the score is significantly smaller than 1, and very close to zero. Is there a bound/threshold under which we cannot say that we infer causality?\n\nWhat is the connection between this paper and the work presented in Rubanova et al, 2019: Latent ODEs from irregularly-sampled time series?\n\n-----------------------------------\nRebuttal: Thank you to the authors for considering the comments and for the changes. I am happy with the response, and have increased the score accordingly. \n\nFig. 5 could be changed similarly to Fig. 1 for visual clarity. The authors mention in the response that both GRU-ODE-Bayes and Rubanova et al. 2019 can be used interchangeably, but there is no reference to the latter. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "27b4zUE-nEW", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "##################################################\n\nSummary:\nThe paper provides an interesting extension to the convergent cross mapping methods. It proposes a latent process model to learn the directions of causal mechanisms from time series observed at irregular intervals. The method is also developed to be suitable for the situation where the time series are observed at short segments. \n\n##################################################\n\nPros:\nThe Neural ODE method is interesting and novel. It solves the problem of having irregular time series in the applications where the dynamic systems have been extensively studied and represented by ODEs.\n\n##################################################\n\nCons:\n1. The causality established by looking at H_x and H_y is not fully described vigorously. There has been much recent work in the traditional causal inference literature where latent variables are involved. However, the theoretical results are generally difficult to establish. In the context of this paper, more justification is needed to convince readers that you can replace X and Y with their (estimated) latent process counterparts. Do you need to assume there is enough information to exactly reconstruct the latent process? Will you run into model misspecification problem of the latent process as both time series are short?\n\n2. In the conclusion it is claimed the method can detect causal directions even in the case of hidden confounders. I do not see from the main text where this property is discussed. Case 2 of the simulation seems to be correspondong to this statement, but is Z[t] hidden in the simulation? If so it should be made clearer. But in general, I do not quite see how the lack of casual arrows from X to Y can be recovered when there is a hidden confounder Z. Maybe it is possible in some particular cases. Can you clarify? \n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper is concerned with finding causal relations from temporal processes and extends the Convergent Cross Mapping (CCM) method.  It focuses on finding information of chaotic dynamical systems from short, noisy and sporadic time series, and the idea of using the latent space of neural ODEs to replace the delay embeddings in CCM seems interesting. All reviewers like the idea. Please try to make the paper more self-contained and provide some of the justifications suggested by the reviewers.
Decision: Accept (Poster)