Causal discovery from conditionally stationary time series
Authors: Carles Balsells Rodas
Ruibo Tu
Hedvig Kjellstrom
Yingzhen Li
Venue: Submitted to ICLR 2023 (2023)
Keywords: causal discovery
temporal data
graph neural network
time series
non-stationary
probabilistic modelling

Abstract: Causal discovery, i.e., inferring underlying causal relationships from observational data, has been shown to be highly challenging for AI systems. In time series modeling context, traditional causal discovery methods mainly consider constrained scenarios with fully observed variables and/or data from stationary time-series. We develop a causal discovery approach to handle a wide class of non-stationary time-series that are conditionally stationary, where the non-stationary behaviour is modeled as stationarity conditioned on a set of (possibly hidden) state variables. Named state-dependent causal inference (SDCI), our approach is able to recover the underlying causal dependencies, provably with fully-observed states and empirically with hidden states. The latter is confirmed by experiments on synthetic linear system and nonlinear particle interaction data, where SDCI achieves superior performance over baseline causal discovery methods. Improved results over non-causal RNNs on modeling NBA player movements demonstrate the potential of our method and motivate the use causality-driven methods for forecasting.

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CAUSAL DISCOVERY FROM CONDITIONALLY STATIONARY TIME SERIES
Anonymous authors Paper under double-blind review
ABSTRACT
Causal discovery, i.e., inferring underlying causal relationships from observational data, has been shown to be highly challenging for AI systems. In time series mod- eling context, traditional causal discovery methods mainly consider constrained scenarios with fully observed variables and/or data from stationary time-series. We develop a causal discovery approach to handle a wide class of non-stationary time-series that are conditionally stationary, where the non-stationary behaviour is modeled as stationarity conditioned on a set of (possibly hidden) state variables. Causal Inference (SDCT) our approach is able recover
Named State-Dependde the underlying causal dependencies, provably with fully-observed states and em- pirically with hidden states. The latter is confirmed by experiments on synthetic linear system and nonlinear particle interaction data, where SDCI achieves supe- rior performance over baseline causal discovery methods. Improved results over non-causal RNNs on modeling NBA player movements demonstrate the potential of our method and motivate the use causality-drive methods for forecasting.
INTRODUCTION
Deep learning has achieved profound success in vision and language modelling tasks (Brown al.
2020 Nichol et al. [221]. Still. it remains a grand challenge and a prominent research direction
to enable deep neural networks to perform causal discovery and reasoning (Yi et al. 2020 Gird-
har& & Ramanan 2020 Sauer & Geiger, (02T), which is an inherent mechanism in human cognition
(Spelke & Kinzler 2007). Specifically for analysing time series data, causal discovery involves iden-
tifying the underlying temporal causal structure of the observed sequences. Many existing causal
discovery approaches for time series assume stationarity (Granger 1969 Peters et al.) 2017 Löwe et al. 2020 [i et al. 2020 Tank al. (02T), which is restrictive as sequence data from real-worl world scenarios are often non non-stationary with potential hidden confounders. Recent works introduce a tackle causal discovery for non-stationary time series (Zhang eet al. 2017: Ghassami et al., 2018 Huang et al., 2019), but in general, causal discovery on non- stationary time series under mild and realistic assumptions an open problem.
number of different assumptions
This work aims at addressing this open challenge by proposing causal discovery algorithm for condionally stationary time series, for which the dynamics of the observed system change depend- ing on set of "state" variables. This assumption holds for many real-world scenarios, egg., with people who behave differently and take different decisions depending on underlying factors such as mood, previous experience, and the actions of other agents. The causal discovery task for such conditionally stationary time series poses different challenges depending on the observability of the states, which is classified into 4 different scenarios:
1. Scenario class 1 concerns the simplest case, where the states are observed and their dynamics are independent other observed time series data (Figure Lap.
2. In Scenario class 2, the states are unobserved and directly dependent on observed variables. Figure Ib shows an example, where the states of the variables change according to their po- sitions (pink vs purple regions). Another example is to consider an agent moving in a room where different behaviors are observed depending on their location.
3. Scenario class 3 is more challenging: the state depends earlier events, and thus cannot be directly inferred from observations. E.g., in Figure/Ic particles that change state upon collision. Also in football game player acts differently depending on earlier actions of the others.

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Xe collides
# Independent from
П 0 1 1 1111 2 0 1
0 122 2
0 01002 2
1
202 2022022
(a) sccnario class 1
(b) scenario class (c) scenario class
(d) other scenarios
Figure 1: Graphical representations of the data generation processes considered in this work. x represents the observations of a time series sequence, and s' denotes the state variables. The state affects the future observations by changing the causal structure (denoted as f*) for different state values. The representations are examples of (a) scenario class 1, (b) scenario class 2, (c) scenario class 3, and (d) other scenarios (image adapted from Oh al. (2011)).
4. Finally, many real-world scenarios (e.g.. Figure cch are governed by underlying states that are not fully identifiable from the observations over time. Here the states can be unknown confounders to the observed time series, thus the causal discovery task ill-defined.
Our approach, named State- Dependent Causal Inference (SDCI), is based on discovering the sum- mary graphs (Peters et al. 2017) conditioned on states given observed sequences. It fits graph neural network based variational auto-encoder (Löwe et al. 2020) the non-stationary time se- ries data, which enables efficient amortization for causal discovery across multiple observation se- quences. We prove identifiability results for cases with fully-observed states; empirically SDCI also applies to cases with hidden states, which is confirmed by experiments on both synthetic linear datasets and spring data (See Figures Tbb & Tc), covering scenario classes 1-3. Compared to base- lines including non-causal RNN- -aseb approach, SDCI achieves better accuracy in identifying the underlying causal graph and forecasting future trajectories from historical observations, on both simulated and real-world data such particle interactions and player trajectories in NBA games.
RELATED WORK
Causal discovery aims to identify causal relationships over a set of variables from observational data Glymour et al., 2019). Constraint-buld based methods rely on conditional independence tests to recover the underlying DAG structure of the data. Representative approaches include the PC algorithm
Spirtes al.) 2000 and Fast Causal Inference (FCI) (Spirtes 200T), and their extension to the time
series data (Entner& & Hoyer 2010 Runge 2018).
Score-based methods, such Greedy Equivalence Search (GES) (Chickering 2002). define and optimize score functions of causal graphs to identify the underlying causal structure. Regarding time series data, these methods are reformulated learning Dynamic Bayesian Networks (DBNs) from data (Murphy et al., 2002). A recent approach in this line is DYNOTEARS (Pamfil et al, 2020), which aims to estimate both instantaneous and time-lagged relationships between variables in time series without performing combinatorial search in the space of all possible graphs.
Functional causal model-based methods represent the effect as a function of its direct causes and their independent immeasurable noise (Shimizu et al. 2006 Zhang & Hyvärinen, 2009 Peters et al. 2014 Glymour et al., 2019). For time series, these approaches constrained functional forms and connection sparsity in favor of identifiability (Peters et al. 2013).
dynamic model, often with
Our work is concerned modelling non-stationary time series using state variables as entities re- sponsible for changing the dynamics along the sequence. Most relevant to ours is Amortized Causal Discovery (ACD) (Löwe et al. 2220), which assumes stationary time series and amortizes summary graph extraction process from samples with different graphs but shared dynamics. Similar ideas are also proposed in Li et al. (200)) for video applications. We extend ACD by allowing the underlying causal structure vary depending
state variables. For other works, Huang et al. (2015) ex- tended Gaussian Process regression to identify time-varying functional causal models; Zhang et al. 2017) used kerne embeddings to detect distribution shifts in heterogeneous data, and Ghassami
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et al. 2018 & Huang et al. (2019) estimated the time-varying causal effects. However, the latter two methods are limited by their linear causal model assumptions and fixed causal structures. (aa- gioro al. (020) considered causal discovery for regime-dependent dynamics. Our approach is more generic as we allow each variable in the multi-variate time series to have its own states, while regimes be viewed as "global states" applied to all the variables.
STATE-DEPENDENT CAUSAL INFERENCE (SDCT)
We introduce SDCI to extract causal graphs from time series data where their dynamics are altered by means of a set of categorical variables, referred to as their states.
3.1 PROBLEM FORMULATION
We consider a dataset D, where each sample D consists of N non-stationary time series X = (x1....,Xy' X } of length T. We denote element i at time-step i as X, E R..
Stationary time series. We assume that the data generation process obeys a structural causal model (SCM) (Pearl 2009), where all the observed variables x E y1:77 are its vertices, for each time series at each time-step t, and that there are no instantancous effects and no hidden con- founders. Moreover, same as the definitions of Granger causality (Granger, 1969). we assume that edges of a causal graph cannot go back in time. For the sake of clarity, the underlying causal struc- ture satisfies the first-order Markov property in this work; however, one could directly extend it to the higher order. When considering causality in time series, the corresponding causal graph of SCM called full time graph (Peters et al. 2017). G1=2", Note that our treatment stationarity refers to the structure of the causal mechanisms, i.e., edges are invariant in time.
Assumptions similar to the previous ones have been introduced in the past for the analysis of causal- ity considering stationary time series data (Löwe et al.) 2020 Li et al. 2020), where the task is to extract the summary graph, G = {V,E), where V ( XN) and an edge from X; to x, is included in E if there exists 1 < time graph. Both full time graph and summary graph are identifiable from observational data as shown by Peters et al. (2013) if the generative process satisfies Time Series Models with Indepen- dent Noise (TiMINo). We further assume first-order Markov property and an additive noise model (ANM) without instantancous effects for the generative process:
t' < T such that there is an edge from x to x in the full
x 53 (PPA)) c
where PA C V denotes the parents of x, and eý denotes independent noise. The direct causes of x are (x)))
(P vi:r As here the time series is stationary, we can regard this as first querying the summary graph g to extract the parents, and then using the correct temporal index
(t -t- 1) as a superscript of each parent (PA)).
Conditionally stationary time series. For non-stationary time series considered this work, we assume that at each time-step we have access to state variables st
s where 1++1 In other words, when the sf {1....,') K} controls the causal effects of x² future variables Xi-N- state sy changes, so do the causal effects for x(. We further assume that the time series is stationary the states are held constant along the sequence, i.c., s² = ... = =T For the general case where s changes through time, even though the time series is non-stationary, it is conditionally stationary given the state variables S = (s (ss,....sTT] s
Figure 2a illustrates the full time graph of conditionally stationary time series example following our assumptions. In this case a variable affects different sets future variables depending its state, and the time series is non-stationary as s' + 5² + s². Note that the states are not included explicitly in the SCM of the generative process (otherwise the SCM requires additional variables with their own vertices for S). This is valid simplification since assume access to the states all times. Our thcoretical results presented below do not apply when considering hidden states, since one must then explicitly model its causal structure the SCM.
Conditional summary graph. Our goal for causal discovery is recover the full time graph, where for stationary time series this can be achieved by identifying the summary graph under the first-order Markov assumption. However, for conditionally stationary time series, the summary
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1+1
1-2
1+3
[-4
X1
GLx (G., Gz}
X;
(a) Full time graph
(b) Conditional summary graph
(c) Summary graph
Figure 2: (a) Full time graph G1 of sample considering our problem setting, (b) conditional summary graph 9i.k. and (c) summary graph 9 of the corresponding sample. Different colors (red and blue) denote effects caused by different states.
graph can be non-informative (probably close to fully connected graph), due to different causal effects induced by variables in different states. As an example, Figure the summary graph extracted from the full time graph of Figure 2a Being dense, the summary graph defined for stationary time series is less useful in non-stationary settings. To address this issue, for conditionally stationary time series we define the conditional summary graph.
Definition (Conditional summary graph, first-order Markov setting). Given a full time graph GV:T, its conditional summary graph is a set of K summary graphs, G1.K = (9x : 11 k I K), where K is the number of possible state values. Each summary graph GR = (V; E has the same vertices V = (x1,...,** ..I,,..,,,,., and an edge from x; to x is added to Ex if there exists time-step 1 < 1 T such that s: = k and in G1:T, xx is connected to ++1
Note that both summary and conditional summary graphs do not include auto auto-regressive connections
x² + x x+1 Figure 2b shows the conditional summary graph extracted from the full time graph of
Figure 2a For A: = we have og = 1 and there is a "red edge" connecting x3 and xz. therefore for 91 there is "red edge" in the edge set & Similar reasoning applies for G2. Compared to the summary graph [c] the conditional summary graph contains a compact, yet valid representation of the causal structure, and thus being more informative. Conditional summary graph is more widely applicable beyond regime- deeendedd dynamics (see e.g., (Saggioro et al. 2020)): we do not restrict the entire time series to be in one of the K states, but allow each variable have own states.
State-dependent TIMINo. We extend TiMINo conditionally stationary time series with ob- served states and show the identifiability of both full time graph and conditional summary graph. Consider the update on x as in Eq. First for non-stationary time series the direct causes of x (x,)),. ). are no longer constant in time. However, for conditionally stationary time series satisfying first-order Markov property and our assumptions specified above, PA (x
) is determined by the states at the previous time-step t --1 1. Thus we can write PA(x* = (PA) -1'= and the state dependenn TiMINo as (with first-order Markov assumption, an ANM model and no instantaneous effect):
-3 ((PA) (S + e): PAJIs- (x x si
Si
(2)
where of E (I....,K] K} and (k)) V denotes the children of variable x, when its associated
state equals to K. To illustrate, in Figure 2a we have that PA (x²) {x] x), x!} because x2 E (s})
and X2 E (s)); however (A(x3) {x7, x because now X E (s but X2 & C3($3)
Identifiability of state-dependent TIMINO. As assume access to the state variables at all times, the original properties of Markov assumption and causal minimality are maintained, which are satisfied by TIMINo. Consequently, the conditional summary graph becomes identifiable by extending the indentifiability proof of |Ptters et al. (2013) to our setting and further assuming that all the states are visited at least once. Notice that failing to observe data corresponding to par- ticular state of one variable makes it practically impossible to capture the underlying effect of that variable conditioned on the unvisited state. More details for the preservation of causal minimality and Markov condition in the state-dependent TiMINo, the identifiability of both the full time graph and conditional summary graph, and extensions to incorporate other time-lagged or instantaneous effects can be found in Appendix and B
Considering the previous results, one could directly extend TiMINo causality (Peters et al., 2013
to conditionally stationary time series. The algorithm uses fitting method (to estimate fj in Eq
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and tests for independence using residuals. We notice that in our setting, the direct causes of x] depend on the state values s s which represent K models in total (f) '). This renders such extension impractical for efficient causal discovery in such scenarios. However, our key observation is that in real-world non-stationary data, many of the underlying function components are shared not only across data but also across variables, i.e., f = [] for some variables i, and state sets k, This benefits the use of deep learning as effectively more data is available to identify the individual effects (where the total number of them is significantly less than K~ Therefore, we propose a deep learning-based approach in similar spirit as Löwe et al. (2020) amortise the discovery of causal relationships across data samples within the same dataset. We leave the proof of our method regarding consistent parameter estimates (Geffner et al. (2022) to future work.
State-dependees causal inference. We mainly focus on non-stationary causal graphs which may have different edge-types different times. Based on our assumptions, the interaction (i.e., edge- type) X; + x, can change according to the state of the variable x,. Following Kipf et al. (2018); Li cct al. 2020); Löwe et al. (020). we consider this edge-type for x; - Xj at time t as a categorical variable Ziy {o,.... The If which can represent n. interaction types between pairs of variables.
Specifically we use sdgge-type& O to denote "no causal effect" between two variables. Notice that the edge-type differs from the state: the former controls the functional form of the causal effect and the latter allows the variables to affect others differently along time. For example, one can have x; + Xj whenever zly * O, but the functional form of the causal relationship can differ.
Our method focuses on extracting a conditional summary graph 91:K (assuming K states). Previous approaches aiming for this task assume stationary time series data for which 91 For conditionally stationary time series, we extract the k-th summary graph including edge-types, (V, }. where & and & can differ for k + k.. We further define & {wijk E (0......7 nre 1):X, Xy V) as the collection of edge types for variable pairs in V. Note here that edge is visualised in the conditional summary graph visualisation only when weyk * 0 (see Figure 2b Then the edge-type interaction X; + X; can be queried at each time-step t follows:
GK-
zlj
Wigh if s = k.
Therefore, causal discovery for conditionally stationary time series requires extracting the (un- known) conditional summary graph given observations of X (and perhaps also S). It not only re- quires to design a parametrizable function to infer the causal structure, but also to evaluate how this inference fits to the input observations.
3.2 IMPLEMENTATION
We introduce probabilistic approach which models the distribution of the edge-types {zf,} given observed data. Following our assumptions for conditionally stationary time series, this task can be solved by learning the distribution of W {wijk < i,j < N, N,14 < k < K} given data, after inferring W one can then query the edge-types zf, as in Eq. Inspired by previous approaches implementation is based on a variational auto-encoder ( (VAE) (Kingma & Welling 2014) and graph neural networks. We first discuss our approach for the case with fully observed states, then extend the method to the hidden state regime.
(Li et al. 2020 Löwe et al. 2020 Kipf al. 2018).
A diagram of the proposed approach is visualised in Figure 3
Encoder. Similar to Löwe et al. 2020). we use factorized distribution gø WXX =
State
XL
K I I1 No(Wjj X, The encoder re- ceives both X and S the input, and extracts an embedding that represents the causal inter- action conditioning on the state for every pos- sible edge x; + x,. We compute approximate posterior qo (wigk X, S) as follows,
(W]X.
pe at
(4) Figure 3:
SDCI extracts a conditional summury graph that describes the edge-type interaction for every pair of edges conditioned on the states.
Φ fo (X, S) E KK..
(w X, S) = @(O.gk/T).
(5)
where (X, is a neural network that returns outputs (œ., 1 < i,j < N} given the input as X concatenated with one-hot representation of the state variable S. O(.) denotes softmax activation
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with temperature Note here the softmax activation is taken over the k--th row vector Oiyk E R" in Φi5, which is the edge embedding for x; - X, at state k. As this returns a categorical distribution for (ijj X S), apply the Gumble-softmat trick (Maddison et al. 2017 Jang et al. 2016 to enable direct differentiation for back-propagation during training.
The construction of (X) is based on graph neural networks and MLPs. See Appendix C.2 for more details.
Decoder for X. The decoder for the observations X given edge-types W (e.g., sampled as wiyk qo Woikk X, S)) and the states S are defined following the first-order Markov assumption (with x", as dummy variables):
T-1
T-1
Pe (X
II
W)
II II N .o21). t=0 j=1
(6)
1=0 j=1
The mean x5= is defined as follows. Given sampled edge-types W = [wiste,, the decoder first queries the edge-type for element j at time + 1 as z/ş = Wijk' for of = k'' (also see Eq. 3 Then the information along the predicted edge-type interactions is retrieved and aggregated as follows T
h
xt+1 = x3 + fo 1 x))
(7)
eS0
where {fe]a=] set of parametrizable functions, one defined for each edge type excluding the no-edge interaction. fp is a neural network that aggregates the information from the previous time- step, and models the dynamics for each variable xf++
Objective. The encoder 4a (W and the decoder & state dynamic model py ()) trained using a modified VAE objective:
T-1
C=)
(w]ss))))) log pa px 'x',s, w)] - KL (WX.
(8)
where we use À to balance the accuracy in learning the decoder for X and the state dynamics. Here used factorised prior p(W) I Ily (wWyi))) which acts as a regularizer over the inferred edge-type distribution q. In our experiments we set this prior to be Uniform( (0....: Th - 1}), although in certain applications it may be useful to use different pk to encourage different sparsity levels for Gk.
Hidden state regime. Often in practice only observations of X are available, i.e., the states S are hidden variables. In this case we make a factorised approximation qo To(W (W. S|X) = go (WJX)qs qo (SJX), where q (W)) X) is defined in similar way as in the fully-observed case (see above) except for using network f (X) for the softmax logits. For the states, we consider go (S) = I I go (s |x{),
with each g (s) x defined as
(x(s)(x)) = 0 &f fx(t,,,,
with 7 < 1 as temperature factor and ʃs as a neural network. For the decoding process given W [ X). we also sample states S qs (S(X) using the categorical reparameterization (Maddison cet al., 2017 Jang et al. 2016). Therefore, the decoding step can be performed as in Eq. and resulting message passing operation becomes
Σ 1 [ (x,,
(10)
k=1 and we apply similar Gumbel-softmax tricks as in the fully-observed state case to replace (((==)) and 1 (mase ma) during training. The VAE training objective in this case is as follows
e>00
T-1
L=
D t=
E (w
(())
log Px
(x" x'.s', w)
(11)
O. Note that the objective is still valid lower bound of p(X). See Appendix C. for an alternate formulation. In this hidden state regime the previous theoretical guarantees on identifiability no longer hold due to violations of our assumptions. However, this setting can still be useful showcase for analysis of non-stationary time series and possible directions of future work.
where we set (S/X) PS)), and thus (S
In training, (xt)==) is replaced by the e-th dimension of waysr which sampled from da with Gumbel- softmax relaxation.

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4 EXPERIMENTS
We evaluate SDCI on two synthetic non-stationary time series datasets, and realistic data based on NBA in in-game player movements. Our results are compared to ACD (Löwe et al.) [2000) TdCM Huang et al. (2015), CD-NOD (Zhang al. 2017), SAEM (Huang et al. 2019), and a non-causal deep learning method called variational RNN (VRNN) (Chung et al. 2015).
4.1 EXPERIMENTS ON LINEAR DATA
We start from linear message passing operations between different time series. Conditioned on the states, the variables x/. y R are connected by an edge of n. different types. Each edge- type is captured by the linear coefficients {3r, E R with the convention that fo = 0 (no connection). Notice that the effect of x on x changes during time according to the state value of and the underlying conditional summary graph Gi.k. We report experiments scenario class in Appendix F.T and study scenario class 2 below. Details on data generation, training settings, and additional experiments can be found in Appendices D.1 C.3 and F. respectively.
Results for scenario class 2. Scenario class 2 considers unobserved states (only X is given, see Figure [bb, enabling com- parisons with CD-NOD, SAEM and TdCM which perform causal discovery based on X only. As these baselines consider constant causal connections (i.e., single summary graph), for a fair comparison we evaluate the identification accuracy of the summary graph such that it only considers the exis- tence of an edge rather than capturing the edge type. We simulate this setting with 2 edge-types (2-EDGE), 3 edge-types with constant summary graphs (3-EDGR CONST) and 3 edge-types with no constraints on the conditional summary graph (3-RDGE FREE).
Table 1: Summary graph and conditional summary graph accuracy for linear data in scenario class 2.
SG ACCURACY
METROD
2-EDGE
3-EDGE
CONST
FREE
TDCM (T=100)
65.17 2.65
63.67
1.61
63.50+.0
CD-NOD (T=100)
39.33 2.59
35.25 = 2.51
28.58 1.58+225 2.68
SAEM (T=100)
47.75 3.67
5..41
TDCM (1=1000)
68.25 = 2.29
61.17
2,28
02.00:111111
CD NOD (T=1000)
50.08 2.59
42.08
2.17
41.58
2.02
SAEM (T=1000)
47.38 4.10
2.82 25.93
28.49
+
ACD (1=50)
60.45 1.60
87,00 2000=22.0 2.56
19.25
SDCI (T=50)
97.08 = 1.05
90.17
2.22
64.00 1: 2.93
ACCURACY
2-EDGE
3-EDGE
CONST
FREE
(DC)()
98.08
76.04
65.45
1.99
We report the (conditional) summary graph identification accuracies in Table It is clear that SDCI performs the best in extracting both the summary graph and conditional summary graph. Since ACD assumes stationary time series, it achieves comparable results when considering a constant causal graph. On the other hand, TdCM, CD-NOD and SAEM perform significantly worse, as they are designed for -tatioatio time series with distribution shifts and smooth time-varying coefficients, which are not suited for modeling discrete changes in dynamics. Furthermore, they conduct causal discovery on each multivariate time series separately, while ACD and SCDI benefit from amortized causal graph inference that utilises shared information across different time series.
4.2 EXPERIMENTS ON NONLINEAR SPRING DATA
We evaluate SCDI on spring data adapted from Kipf et al. 2018 Löwe et al. (2200), which consists of particles (or balls) connected by springs with directed impact- meaning that e.g. particle i could affect particle j with force through connecting spring, but leaving particle unaffected by this spring force. We consider 2 edge-types (presence/absence of directed spring) and scenario classes and 2. We report results with scenario class 3 in Appendix F.2 Details of data generation and model hyper-parameters can be found in Appendices D.2 and respectively.
Results for scenario class 1. In this experiment the states are known and independent from the observations. For the ground truth dynamics, the state transitions incrementally into the next one every 10 time-steps. We report edge identification accuracy, which regards the conditional summary graph including edge-types (91.x). Figure 4a shows results with increasing variables and increasing states, where we compare with ACD (dashed lines) as a stationary baseline. Although performance drops as K increases, SDCI is able to maintain reasonable accuracy in edge-type identification in comparison to the stationary baseline. With increasing number of variables both approaches see accuracy drops; hypothesis is that this can be addressed by increasing model capacity. The next
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1.0
Variables Variables
Variables ACD
1.0
Variables 10 Variables Variables
1.0
Edge accuracy
0.9
20 Variables
Edge accuracy
Edge accuracy
0.9
0.8
ACD
0.8
0.7
0.6
8
5 10
20
50 100
10
20
State values (K)
Train data percentage (%)
Num. variables (data)
(a) Increasing variables and states.
(b) Data efficiency.
(c) Generalisation.
Figure 4: Results spring data with scenario class for (a) increasing variables and state values, (b) data efficiency, and (c) generalisation, where the x-axis indicates the variables in the test data
and the legend indicates the model used. SDCI results are shown with solid lines.
0.9
0.8
56 accuracy
0.7
0.6
SDCI
0.5
ACD
CD-NOD
0.4
(a) ground-truth
(b) SDCI
(c) ACD
0.3
Figure 6: Time series forecasting (dotted lines) of both (b) SCDI and (c) ACD for 50 time-steps along with (a) the ground- truth. The trajectories in solid lines are used as the input and the back- ground color represents the state value.
10
20
Num. variables
Figure- 5: Results on spring data with scenario class 2.
test considers data efficiency of SDCI with results reported in Figure 4b We see that both SDCI and ACD are data efficient in this scenario, where training on 10% of the data returns good performance already. Finally, we also report in Figure 4c on how these methods generalise to unseen data with different number of variables. Here both methods generalise better to settings where the number of variables is similar to the ones they were trained.
Results for scenario class 2. We consider the hidden state regime, where the underlying state of particle changes depending on its location in the box (K = 2, see Figure [bb. Results in Figure S show a clear advantage of SDCI over baselines in terms of summary graph identification accuracy. Again, ACD is limited by its stationary dynamics assumption, and CD-NOD fails due to its inability in handling discrete changes in the causal effects of the full time graph. We visualize the 50 time- steps forecasts with SDCI and ACD given an input sequence of T 80 time-steps Figure The conditional stationary assumption of SDCI allows it to extract more accurate graph structures, which results in better predictions than ACD. Overall, SDCI successfully decomposes the non-stationary dynamics into the conditional stationary ones while capturing the state transition dynamics.
4.3 EXPERIMENTS ON NBA PLAYER TRAJECTORIES
The last experiment considers modeling NBA player movements (Linou 2016), which is real- world multi-agent trajectory dataset with highly non-linear & non- non-stationary dynamics. Details of this dataset can be found in Appendix D.3] including our design of states (as ground-truth is unavailable), which is dependent on the player positions in the court. We evaluate SDCI in both scenarios (states obsserved/nnosservel), To simplify the task, we only model the trajectories of the players (position and velocity), and condition the predictions by modifying Eq. 7 to include the ball features in the message passing aggregation the decoder. Apart from ACD, we further include non-causal baseline VRNN (Chung et al.) 2015), which is popular deep learning approach for forecasting. All the models are trained on sequences of length T = 100 (see Appendix C.4 for VRNN training details).
the position and velocity of the ball,
Figure (7a shows forecast error (in MSE) of the player positions (without velocities) for T = 100 steps computed on held out dataset. In general, the causal discovery methods benefit from using
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0.35
0.35
Mean value
Mean
ACD
500 CSS
SBCI cbs (TOR)
SmetU (K+4]
Unobs
0.23
VBV (TCR)
SBCI Cbn (805)
W 0.2
0.20
0.20
VIV (505)
2.15
0.25
20%
2076
0.10
0.30
50%
3.05
0.25
100%
20
40
60
90
100
Time
Time
Time
(a) Baseline comparison.
(b) Data efficiency.
(c) Generalisation.
Figure 7: Results on NBA data for (a) forecasting on player positions on test data, (b) data efficiency (% indicates the proportion of train data used), and (c) generalisation between NBA team matches. The dashed black line represents a baseline that always predicts the mean value over trajectories.
graph-based approach and outperform VRNN. In addition, SDCI produces more accurate long-term forecasts than ACD which is restricted by the stationarity assumption. For the hidden state setting, we report forecasts using and states, and observe comparable results to the observed case.
We evaluate and report in Figure 7b the data efficiency of SDCI in realistic settings. Again SDCI outperforms VRNN even in low-data regime. But more interestingly, SDCI better exploits the in- creasing amount of data as it continues to improve, especially on long-term prediction. These ob- servations are in line with data efficiency test results springs data (see Figure(4b.
Lastly we explore the generalisation ability of SDCI and evaluate the transferability of learned dy- namics between different teams. Figure 7c shows forecast error for Boston Celtics matches, where the models are trained either the same data (dashed line) or on Toronto Raptors matches (solid line). Both methods produce reasonable out-of-distribution forecasts, suggesting that they are able to learn shared dynamics between samples, and thus allowing generalisation unseen data. Still SDCI outperforms VRNN considerably, showing its strong competence in forecasting for multi- agent scenarios with dynamical changes along the sequences.
qp(1(x))
qg(2[x/)
To better understand the learned states from SDCI in the unobserved states scenario, we visualise in Figure
(a)
1.0
0.5
0.0
9p(1(x))
(2(2xx))
(,(3)x))
g,(4[x))
8 the state posterior as a function of the player's position x in the court. For states, we observe that the state changes in regions close to the mid- court line. This consistent with typical strategies in an NBA match, where the players first run to one side of the court, and then change their behaviour from "defence" to "offence" mode, or vice-versa. Similar arguments can be made for the 4-state case, where the court is segmented such that the second and third region boundaries are close to the 3-point line. Therefore, in addition to better forecast results, SDCI further provides better interpretability of the learned dynamics from data, and returns great promises in applications to other realistic scenarios.
(b)
1.0
0.5
0.0
Figure 8: Learned regimes from SDCI on the NBA dataset using (a) K 2 and (b) K = 4 state values. The dashed line indicates the mid court line, and the colour maps refer the state posterior distribution go (k|x()),
(l...., K).
5 CONCLUSIONS
We have developed SDCI for amortized causal discovery for conditionally stationary time series. Key to our development is the state-dependent TiMINo as an extension of [Pters et al. (2013) to conditionally stationary time series, the new concept of conditional summary graph, and their iden- tifiability under observed states. Evaluations on synthetic linear and nonlinear data show SDCI's improved accuracy in extracting the underlying causal graph and forecasting. Importantly, the im- provement of SDCI over VRNN on modeling NBA player movements demonstrate the promise of -arsality-driven methods for forecasting and data interpretability.
A number of research directions are to be explored in the future. For theoretical studies, identi- fiability for hidden states will be derived with additional assumptions. For practical applications, SDCI will be scaled up and extended to video understanding, where e.g. the interacting objects partially and noisily observed as semantically segmented regions. This would enable efficient and effective of neural networks for causal reasoning in challenging real- world scenarios.

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A IDENTIFIABILITY PROOFS
This section presents the detailed proofs for the theoretical results of our paper. Again we note here that the results only apply when the states are fully observed. Our theory extend the theoretical results presented in Peters et al. (2013) and Peters et al. (2017), which are presented first before our proofs.
Proposition (Prop. 6.36 in Peters et al. (2017)). ) Causal minimality is satisfied if and only if Vx,; Vy E PA (xz) we have that x; H PPA (x, }
Proposition 2 (Prop. 7.4 in Peters al. 2017)). Consider distribution given by the following additive noise model
xj (PA(()) + +Gj j 1..,..,,N
(12)
if the functions f are not constant in any of their arguments, then the joint distribution satisfies causal minimality with respect to the corresponding graph.
Recall from the main text that state-dependent TiMINo is defined as (with first-order Markov as- sumption, additive noise model, and no instantancous effect):
x ʃg ((PA) (() + c/
(13)
PA {x, Xy E Ci(s:- s siiiing
(14)
Below derive the identifiability results for state-dependent TiMINo given fully observed states
A. STATE-DEPENDENT TIMINO PRESERVES MARKOV CONDITION
We show that the state-dependent TiMINo is Markov.
Lemma 1. Given the states S, X is generated by a state-dependent TiMINo, then each variable is conditionally independent of each non-descendants given its parents.
Proof. As the states S are given,
retrieve the direct causes of
P (x(x))
{x] xx E (, N).
and use state-dependent TIMINo compute x3 given its parents:
xy|p=p> = fs(p) + e/.
Given its parents P, variable x² is therefore independent of its non-descendants.
A.2 STATE-DEPENDENT TIMINO PRESERVES CAUSAL MINIMALITY
We show that the state-dependens TiMINo model satisfies causal minimality given observed states. Lemma 2. Assume all the functions f3 in state-dependent TiMINo are not constant in any of their arguments. Then causal minimality is preserved given observed states.
Proof. Again as the states S are observed, then for any x² the direct causes of it can be retrieved by:
(A (x :x, E (,
Assume causal minimality is not satisfied, then following Proposition i there exists x², and x e P such that
x) IL x
Denote P((x as PA. Then, if B[e[] = 0, there exists function e(-) such that
E
= fy x = c(PA), Vx!-1.
This implies that f, is constant with respect x It contradicts with the assumption that f, is not constant in any of its arguments. Therefore, causal minimality in state-dependent TiMINo is preserved given observed states.
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A.3 IDENTIFAABILTYY IN DEPEEDEEEN TIMINO
Using the Markov and causal minimality results, we derived the following identifiability result for state-dependent TiMINo, and the proof is inspired by Peters et al. (013)).
Theorem 1. Consider the state-dependent TiMINo, where only first-order Markov interactions are present, and there no instantaneous effects. Then the full time graph G identifiable from the distribution of data given states.
Proof. Conditioned S, assume that X, can be computed from two state-dependent TiMINo with two different full time graphs, G and 923². Suppose there an edge x; 1/-1 x5 which 9 but not in G227
1. From Gb.T" and the Markov condition (Lemma 1. we have x3. 11 x 1 s k
N, k + it.
2. From G12 and causal minimality (Lemma)) 2 we have x 44 x; [x t-1 < k S N k i}.
Since we have contradiction, the full time graphs G and 93dT must be equal. Therefore, the full time graph is identifiable from the distribution of data given states.
Given the identifiability results, in theory, with maximum likelihood (Zhang et al. 2015). In our implementation, we train the model with the modified VAE objective g The empirical results show that infers causal structures correctly; however, it is nontrivial to prove that the identifiability holds with the modified VAE objective, which has been discussed in Geffner et al. (2022) well and will be our future work.
estimate the state-dependent TiMiNo in Eq.
A.4 IDENTIFIABLLTYY OF CONDITIONAL SUMMARY GRAPH
Assuming the full time graph has been extracted successfully, conditional summary graph by observing the edges of pairs of variables conditioned on the state variable which the cause of the interaction. In particular, the precise mathematical statement for this identifiability result is as follows, assuming that all the states of each element have been visited least once in order to condition on each possible state.
deduce the structure of the
Corollary 1. Consider the state-dependent TiMINo, where only first-order Markov interactions
present, and there is instantaneous effects. Then the conditional summary graph G1:K is identifiable given the states S. for any f & N and any 1 S S K there exists S t S T such that s k.
Proof. Notice that Gx = (V,Ex) with V (xI...., XN therefore identifying Ga is equivalent to identifying the edge set E From the assumptions, the full time graph G is identifiable (Theorem
1h. Then for each variable x we can retrieve from GI.T the parents (1 Then for each xj to the edge set E if sf k.
(A we can query its state st and add edge Now for any i, k consider the i-th node in the k-th summary graph Gr. since we assume that there exists < t < such that sf k. this makes sure that the edges coming out of node x, state I are added to & Therefore this procedure captures all possible edge interactions, and G1-K is identifiable given identifiability of Gt:T.
x
Notice that the assumption on each variable visiting all possible states is required if one aims to obtain the full structure of the conditional summary graph. Otherwise, if the state & of a variable x; not visited, this leaves the outgoing edges of the i-th node in G, undetermined, since there is no information to extract from the full time graph.
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B IDENTIFIABILITY EXTENSIONS FOR STATE-DEPENDENT TIMINO
The first-order Markov assumption makes our definitions and identifiability analysis simpler. How- argue this can still be very useful because (i) it usually suffices to describe particle trajecto- ries (position, velocity, acceleration, etc), (ii) it allows us to present the conditional summary graph a compact representation of the causal structure, and (iii) we present method (SDCI) which leverages this assumption for efficient causal discovery. One could easily extend our identifiability results and definitions higher order time-lagged effects. Incorporating contemporaneous effects is also possible, but requires further assumptions to consider an identifiable functional model class
(IFMOC) (Peters al. 2011) (e.g. linear with non-Gaussian disturbances).
C IMPLEMENTATION DETAILS
All the experiments are implemented in Pytorch (Paszke et al. 2019) and carried out on NVIDIA
RTX 2080Ti GPUs.
C.I PROBABILISTIC FRAMEWORK FOR SDCI
This section aims to clarify our implementation in terms of the probabilistic modelling perspective. Note that some text from the main paper is re-used in this section. The underlying graphical model for conditionally stationary time series differs among scenarios. Here we illustrate the graphical models for scenario class and 2.
C.1.1 SCENARIO CLASS 1
Generative model. Since the states S are observed at all times from our assumptions, we represent the joint distribution given some deterministic observed states S.
p W(S $) Pa X W. S (W)
(15)
where p(W) = IT k= II Pa (wirk) represents the prior over conditional summary graphs (includ- ing edge-types). We note that W represents the set of graphs denoting the state-dependent interaction types among elements, and thus p(W) is categorical distribution. p(W) can be used provide do- main knowledge by e.g. encouraging sparsity. The generative process of sequence X, factorises
as
T-1 N
T-1 N
Ps ( = II Ps (x
X W)
II t= ビ= j=1
N .o
(16)
1= 1=1
given some input states S and conditional summary graph W (W). The mean x is defined as follows. Given sampled conditional sumary graph W = -Wwikh the decoder first queries the edge-type for element j at time t + as 2iy = Wifk for sf = k (also see Eq. 3 in main paper). Then the information along the predicted edge-type interactions is retrieved and aggregated as follows x
h
Le 62-0
x = x
x
(17)
(g
where {fe)"=7 is set of parametrizable functions, defined for each edge type excluding the no-edge interaction. fp is a neural network that aggregates the information from the previous time- step, and models the dynamics for each variable x
Inference. Note that we wish to compute the posterior over the conditional summary graphs, which might be intractable. We introduce variational distribution 4o X. that approxi- mates the posterior for tractable inference and learning. The corresponding evidence lower bound (ELBO) is
(KL(gkLL(9 log S) KKL (W ()]lp(W)) +I (wXxs ( = S)} (18)
"In training, ((xt_=e) is replaced by the e-th dimension of waysr which sampled from da with Gumbel- softmax relaxation.
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Similar
to Löwe et al. 2020
use factorized q distribution 96 (W
I ILs 9s (Wigk S) The encoder receives both X and as the input, and extracts an embed-
ding that represents the causal interaction conditioning on the state for every possible edge X; + Xy- We compute approximate posterior 2s (Wizk X, S) as follows,
Φi - fo(X E RXX" go(witt (. = ((Oisk/T).
(19)
(20)
where (X,S) is neural network that returns outputs {Øi : -1si1 S N) given the input as X concatenated with a one-hot representation of the state variable S. ((-) denotes a softmax
activation with temperature т. Note here the softmax activation is taken over the k-th row vector
Øizk Rdr in Øsj, which is the edge embedding for x; x, at state k. As this returns a categorical
distribution for 96 (u(Wssk X, S), we apply the Gumble-softmax trick (Maddison et al. 2017 to enable
direct differentiation for pack-propagation during training.
The construction of SX is based on graph neural networks and ML.ss. See Appendix C.2 for more details.
C.1.2 SCENARIO CLASS 2
Generative model. Note that now the states not observed, but could directly inferred from observed data (as illustrated in Figure Tb.. The corresponding joint distribution is as follows.
PX, W - Pa ( SWW))) - P(W) II p (x s W)p(s'sx') (x'
(21)
where we represent the time series structure for clarity purposes. pl (), and the generative process defined similarly as above, except now also define the generation for S.
Inference. Again, we
variational approximation over the latent variables for tractable
inference. We make a factorised approximation 2o (W, S|X) = 26 [V where 2o (W[X)
defined in similar way as the fully-observed case (see above) except for using network (X)
for the softmax logits. For the states, we consider 2s (S|X) = II= Il (o(s) xx), with each (x(xx) defined as
(s())) (st/7), (x)),
(22)
with y < 1 as temperature factor and f. neural network. The ELBO is
(X) log / p(X, W. S)dWds (W.S(i)
(X,S W))
log
(23)
Lo (W
T
p(x* p(x*|x x stt
W) pp (x))
2
+ (W.Ss)))
log
(24)
(xt)
Bys B. [x' s111 w)]
KL (Z
(25)
In our implementation, we set qs (S p(S]X), and thus KL
Note
that we also sample states S go (S)X) using the categorical reparameterization (Maddison al. 2017). Therefore, the decoding step can be performed as in Eq. 7 and the resulting message passing operation becomes
K
(4f k)
Σ
(26)
e50
and we apply similar Gumbel-softmax tricks as in the fully-observed state case to replace (()=)) and 1 (wrsz=e) during training.
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9 ( [X, S) go (105a X, 8)
Pairwise Embedding
Input
Aggr
Embedding
concatenation
Pairwise
Aggregate
Edge distribution
Figure 9: Illustration of the implementation of the SDCI encoder which is adapted from ACD (Löwe et al. 2020 and allow for conditioning on states. In the example, we consider states.
C.2 ENCODER ARCHITECCTRRE
Below we provide details of the encoder architectures.
SDCI encoder The first design of the architecture extends directly from ACD (Löwe al. 2020)
and we refer to this model as SDCI. We discuss the network construction for hidden state case as an example, for which the logits Øig for the distribution qo (W) X) are obtained follows. First, the model computes a latent embedding for each node using the whole sequence:
h
Sen
(27)
Then each embedding is updated using graph neural network (GNN) that captures the correlations between nodes. Specifically the message passing procedure follows the two equations below:
h/ foz (h/. h1)),
(28)
h fos
(29)
Finally, obtain the softmax logit Φij I Knn, for every possible edge - Xy and every possible state < < K:
Φ for for h²)
(30)
The above network architecture design is visualised in Figure according to equation The details of the architecture settings follows the design Löwe al. (2020). Each embedding step Je: uses two-layers of 256 dimensions and ELU (Clevert et al. 2016) activations followed by batch normalization. Sea uses skip connections and we modify its output size to generate a pairwise embedding for each of the K states. For fully-observed state case, the architecture for qo (W|X,) S) follows similar structure, except that for the first layer use h fè, (x s s!!T)) where !!T is set of one-hot vectors representing the states {s/
C.3 TRAINING SPECIFICATIONS (SDCI AND ACD)
All SDCI and ACD models have been trained using the following training scheme, including ACD (öwwe al. 2020).
Customized decoder (linear data only) One of our objectives the linear data experiments is
to recover the underlying world parameters { [Bx] 1 Therefore the decoder design imitates the message passing operation presented in Eq. [1] which allows to initialize the decoder using the underlying world parameters and analyse the performance of the encoder as a separate entity from the whole model.
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Hyper-parameters Following Kipf et al. (2018), the models trained using ADAM optimizer Kingma & Ba 2015). The learning rate of the encoder is 5- 100 the learning rate of the decoder is 1.10 for the synthetie linear data experiments and 5 10 for spring data experiments. Learning rate decay is use with factor of 0.5 every 200 epochs. We train for 1000 epochs in the linear experiments and 500 epochs in the springs experiments, using batch size of 128. The decoder is trained with teacher forcing every 10 time-step,, i.c., it receives the ground-truth as input every 10 time-steps. The temperature is set to 0.5 and the variance of the Gaussian distribution of the decoder is o² = 5 10-6. When considering the setting where we make the state dependent on the dynamics of the objects (scenarios and 3), we À 10 For inferring the hidden states, we set the temperature 7 0.1 in the linear experiments and 0.05 in the spring data experiments.
C.4 TRAINING SPECIFICATIONS (VRNN)
The experiments with NBA player trajectories consider VRNN non-causal baseline to compare forecasting performance. Below we specify the network architecture and training scheme. To allow fair comparison between SDDC, ACD, and VRNN, modify the decoder defined in inChung al. (2015) to condition the player positions on the ball features, similarly as we did for the previous models: p=(x"xx<3 z b'), where b' represents the ball features at time t. The architecture of the model follows the original work: 3-layer LSTM networks with 256 dimensions and a latent space size of 128 dimensions. The encoder and decoder architectures use two-layer MLPs of 256 dimensions. The models are trained using ADAM (Kingma & Ba 2013) for 350K iterations with learning rate of 10-4 and batch size 32.
D DATASETS
In this section we provide detailed information about the datasets used in this work. For linear and spring data, we generate 50000 samples of each setting for training the models. Regarding testing, we compute all the metrics using 10000 samples, except when comparing with CD-NOD, SAEM, and TdCM, where we use 200 samples since they require retraining the whole model for each sample. Moreover, notice that for these baselines (e.g., T = 100 or T = 1000) as these methods rely on longer sequences to get accurate results.
sequences of much longer length
D.1 LINEAR DATA
The ground tround-truth structural equation for the data generation process at time t is
x5+
ax} + Σ
Baxt + ej. k= (Ex),
(31)
iny'
where a controls the self-connection, and et denotes independent noise sampled at each time- step.
Below we provide details of the data generation process for the linear data. First, we set the edge-type interactions. In our experiments we set ef ~ ((-0.2,00.2) 0.2). To generate each sample, we need to sample the initial values of the contin- uous variable for each element, xP ~ (N, 2). and the underlying causal structure dependent on the state, S1.k. At each time-step, it suffices to query the edge-type for each pair of variables and
0.9, B1 0.5. and B2 = -055 and
apply the corresponding causal effect 3g following Equation 31 The edge-type is k -E&:)
where (E+),; denotes the causal effect from j to i at state s, which has been defined at the beginning of the sequence. For all our experiments with this dataset, we simulate N variables. When considering hidden states in scenario class 2, we set sf ((x))22) (2 states).
There are potential concerns that the generated samples produced in the linear data may be unstable. Still we use this data for one of the evaluations with the following reasons. First, they define a simulated environment where one has the ground truth and debug and control simulation emros with ease. Furthermore. for -ie-dimesioonl variables x; E R (which is our case), this dataset reduces to first order Vector Autoregressive (VAR) model (Sims) 1980), which is widely used
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in works related to causal discovery for time series data (Gong al. 2015). The evolution of sequence in this case can be expressed as follows:
= Ax
te
(32)
where A is the causal transition matrix and et is an independent noise process.
Regarding stability, the samples in this dataset are described by a causal transition matrix A where the diagonal elements are and the off-diagonal elements are 3x where k is the edge-type interac- tion. For a first-order VAR to be stable, the singular values A need to be smaller than one. Taking obey different underlying causal graph, one needs to check
into the account that each sample this condition for all the possible arrangements of the off-diagonal elements (since the diagonal ele- ments are always (x). The number of matrices that needs check grows rapidly for increasing number of variables, which makes the verification of this condition computationally infeasible (re- call that computing the eigenvalues random samples and keep them if the magnitude of the last observation is comparable to the initial one. We also require that the states are visited in similar proportions to ease the causal discovery task.
matrix has cubic cost )(N))). In practice, we generate
D.2 SPRING DATA
When considering springs with directed connections, we follow the generation procedure described Kipf et al. (2018) with a small modification where the spring interaction between pair of particles can change over time (depending on the state).
In this dataset, N particles are simulated inside 2D box where they can collide elastically with its walls. Each pair of variables is connected by a spring with uniform probability. To allow for identification of causal connections (directed edges), the connection is made unidirectional. The springs interact via the Hooke's law and this setting yield the following equations:
fg - (r. ry);
X; (r, r.)
(33)
where fij is the unidirectional interaction from particle to particle i, òş denotes the edge-type for each pair variables, and r, and it denote the 2D position and velocity of each particle. The continuous variable x; is constructed by concatenating the position and the velocity measurements.
Notice that the above equation defines the evolution of the continuous variable for a single time-step. In our setting, we have that A: (E::) Thus, f.j will change over time, contrary to Kipf et al.
2018). Since we consider two edge-types, we define d, = 0 and 81 = 0.1. To generate samples, we first generate random conditional summary graph G1.K and the initial location and velocity. Then, trajectories are simulated by solving the previous differential equations using leapfrog integration. The step size used is 0.001 and the trajectories are obtained by sub-sampling each 100 steps. In our experiments, we set = 80. When considering hidden states in scenario class 2, we set s = (x4>0) (2 states).
D.3 NBA DATA
The NBA dataset (Linou 2016) consists of recordings from 632 NBA games played during Winter 2015-2016. Each game is composed by approximately 400 600 events, which represents sequences of plays. In each tra- jectory, we find information about the ball location and 5 players of the 2 different teams (10 in total). The coor- dinates of the ball and players are represented in 3D and the length of each sequence can vary from 100 to 600. In our experiments, we consider a sequence up to 200 time- steps (T' 100 for reconstruction and the rest 100 steps
20
10
Figure 10: Hand-crafted ground-truth state map on NBA data which is for- warded to SDCI with observed states. Colours indicate different states.
Data extracted from the following code repository Mok-Player-Moon
https: :/tps:///tt thub. com inmm//nnoukk2.
last accessed 2022-09-28.
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for prediction), which gives total training dataset of 150K samples and test of 6380 samples. Data inspection shows that the court size 100 x 50, and we use this information to standardise the data. For experiments with SDCI with observed states, design some ground-truth states which depend on different locations of the court. We set = 4 and our choice is shown figure TO
E COMMENTS ON EVALUATION METRICS
E.1 SUMMARY GRAPH IN TDCM
Huang et al. 2015) do not specify explicitly the computation of the summary graph (or an equivalent object). To allow a fair comparison, we take the same approach as in Huang et al. (2019), where and edge from to j not incorporated in the summary graph if the corresponding estimated time- varying coefficient has mean and variance lower than theshold.
E.2 ACCURACY OF SUMMARY GRAPH
To clarify, the evaluation of the summary graph considers correctly classifying the interaction be- tween all pairs of nodes in both directions. To exemplify, for N=3 variables, our method needs to produce 6 predictions (one for every pair of edges in both directions). We then compute the accuracy of these predictions and average across all the samples in the test set.
E.3 COMPUTING THE SUMMARY GRAPH IN SDCI
Notice that SDCI can extract the conditional summary graph (CSG) whereas the baselines we com- pare with only consider the summary graph (SG). Consequently, the only immediate way to compare the performance capturing the causal structure among the methods we consider to evaluate the latter. From the definition of summary graph, we deduce that of the graphs in the CSG. This is used to compute the summary graphs of both SDCI and the ground truth structure of the generative process.
estimate it by taking the union
F ADDITIONAL RESULTS
In this section we report additional experiments and qualitative visualisations, which can be helpful to complement the main results from Section in the main text.
F.1 LINEAR DATA
Scenario class 1 We con-
sider the case of scenario class I (see Section [ where the states S are ob- served and their dynam- ies are independent from X. We compare SCDI- Static with ACD, and eval- uate the effect of explic- itly modeling the underly- ing state. We further con-
Table 2: Test edge-type accuracy (in %) and MSE for linear data gen- erated with scenario class 1 settings (fully-observed states) for states and 2 edge-types.
METHOD
EDGE ACCURACY
TEST MSE
ACD FIXED DECODER
66,02 i 0.29
0.49 + 1.89. 10-2
ACD
66.44 = 0.29
0.47 4: 1.98 10-2
SDCI FIXED DEC. SDCI
90.43 0.23
2.64 2 4.55 10
93.84 = 0.19
1.57 10 4.03 10-5
sider the case where the decoder fixed and uses the ground-truth 8g values, for which we denote FIXED DECODER or FIXED DEC. for short.
Table shows the edge-type identification accuracy and test data reconstruction mean-squared error (MSE) for a simulation with N = 3 variables, K states, and n. 2 edge-types (no-edge and B1). SDCI successfully performs the task of identifying the edges. In terms of the & error for estimating (.x). ACD scores worse (~ 10-3) compared to SDCI (~ 10-5).
We repeat the same experiments with edge-types and states, and report the results Table As before, SCDI shows superior performance in comparison to ACD. In terms of the l error for
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Table 3: Test edge-type accuracy (in %) and MSE for linear data generated with scenario class settings (fully-observed states) for 2 states and 3 edge-types.
METHOD
EDGE ACCURACY
TEST MSE
ACD FIXED DECODER
49.29 0.31
0.50 1.56 10
ACD
34.11 0.28
0.66 + 2.32 10-2
SDCI FIXED DEC.
92.75 0.22
1.00 10- 1.77 10
SDCI
84.36 0.28
3.64-10-2 2.19 10-3
estimating {Bk], ). we observe that SDCI provides more accurate estimations (~ 10 respectively) compared to ACD (~ 10-1).
Scenario class The results considering lin- ear data with hidden states in scenario class 2 show that the CSG accuracy decreases when considering an additional edge-type (see Ta- ble [T. However, when leaving the summary graph constant (3 (3-EDGE CONST) we obtain bet- ter CSG estimations. Figure [1 shows the ap- proximate posterior go (s.] xx)) inferred by SDCI in each of the 3 settings studied in this experi- ment along with the true underlying state func- tion, of ((x)>>))- We notice that SDCI achieves better results when the quality of the estimated state function is higher. This is ex- pected, since querying the correct edge-type from the conditional summary graph strongly relies on accurate state estimations. Therefore, the model has to learn the state distribution and only then will be able to infer accurate causal structures. Although they can be challenging in general, our results show that SDCI success- ful in performing both learning steps in simple settings. This offers promising direction of work towards the conditional stationary setting with hidden states.
Ground tround-truth
(b) edges
(c) 3-edges const
(d) 3-edges free
Figure 11: Underlying state function for scenario class 2 in the linear dataset, sf =1
Figure 12 shows samples for the linear data considering hidden states and 2 edge-types along with the corresponding causal summary graphs and summary graphs inferred by both SDCI and ACD. Since SDCI achieves high accuracy, the majority of the graph estimations match the true causal summary graph, which results in good forecasts. We also show two cases (two last rows) where the graph estimations do match the ground-truth, and in these cases the model does predict the future trajectories accurately. Similar to what we show in the main text, the trajectories obtained with ACD drop rapidly and the summary graph estimates are considerably worse.
F.2 SPRINGS DATA
Scenario class 2 We provide additional visualisations for the case with hidden states in Figure 13 where we show the predictions of both SDCI and ACD well as the corresponding conditional summary graphs and summary graphs extracted by both methods respectively. SDCI produces accurate causal graph estimates. Regarding time series forecasting, our method is able make reasonable predictions. Notice that to train the models, we use teacher forcing every 10 time-steps, which means that the learned models are less suited for long-term dynamics modelling. However, one can expect to obtain more accurate predictions by progressively reducing the teacher forcing frequency during training. Considering ACD, despite being restricted by assuming stationary time series, it still infers graph structures that allow the model to produce decent forecasts.
in the linear case,
We consider the case where the particles are contained in a box, and the state of partiele transitions when it collides with the wall of the box (see Figure Icp. For simplicity, only consider K = 2 states that transition alternatively on wall collision. Due to the complexity of the state transitions, method cannot identify accurate causal structures for hidden states. Instead,
Scenario class
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Table 4: Test Edge-type accuracy (in %) and test MSE using spring data with state transitions on wall collision.
METHOD ACD
EDGE ACCURACY
TEST MSE
STATE
68.63 0.14
1.46 10
1.35 10
98.21 0.02
SDCI
79.19
10 1.3--11-000 10
98.53 0.02
we use the state information to learn the transitions and leave the hidden setting for future work. We report SDCT's performance along with ACD in Table Regarding edge accuracy SDCI per- forms significantly better, and ACD is limited by considering stationary dynamics only. However, both methods achieve comparable test MSE metrics, indicating that that ACD can still make decent predictions even when it fails in identifying the edge-type interactions.
Predictions
Conditional summary graph
Summary graph
W
W W
M
MW
MW
W
ACD
300
W
ALD
500
Mtaie
QT
ACD
$2.0
Figure 12: time series forecasting (left) of both SDCI (solid line) and ACD (dashed line) for 50 time-steps along with the ground-truth (GT, transparent line). The first 50 time-steps are given to the models as input and the background color represents the state value. We show the associated conditional summary graph (center) and summary graph (right) of SCDI (red) and ACD (blue) along with the ground-truth (GT, black) for each sample. Each row represents different sample.
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Predictions SDCI
Ground-truth
ACD
Conditional summary graph
Summary graph
ACD
ACD
ACD
Figure 13: time series forecasting (left, dotted lines) of SDCI and ACD for 50 time-steps along with the ground-truth. We use solid lines to denote the input to the models and the background color represents the state value. We show the associated conditional summary graph (center) and summary graph (right) of SCDI (red) and ACD (blue) respectively along with the ground-truth (GT, black) for each sample. Each row represents a different sample.
23


Reviews:
[{"review_id": "hqcnES2rjtX", "reviewer": "Reviewer_3oxu", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "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": "This paper proposes a state-dependent causal learning framework for multivariate time series. The goal seems to learn a state-dependent causal graph between multivariate time series. Since causal structures can change dynamically in practice, the targeted task is practically useful if the state is automatically inferred. In this paper, the authors assume that the states are fully observable. Under this setting, the task is to learn a causal graph for each of the K distinct states.\n \nThe authors introduce a VAE-based algorithm to learn the probability of getting an edge between a node pair. The model seems to follow an existing work, but the detail is not clearly given. \n ", "summary_of_the_review": "From what is written above, I cannot endorse the technical validity of the proposed framework. However, I encourage the authors to continue this line of research. If they significantly improve the presentation, in particular how the GNN part and the generative model are designed, it is going to be a good paper. ", "strength_and_weaknesses": "Strength\n- Addresses a general problem of state-dependent causal graph learning.\n- Proposed a generative model to probabilistically learn the causality graph.\n- Conducted empirical evaluation to show a good performance. \n \nWeakness\n- The assumption of being able to observe the state variable sounds very strong. The relative novelty to the existing works might not be very significant.\n- Presentation can be improved. The main probabilistic model is not fully explained. It is not clear how the authors extend the existing work. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "YbOZk8TPk3z", "reviewer": "Reviewer_F9nk", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "This paper considers a particular setting for causal discovery from non-stationary time series. In particular, the non-stationary behavior is modeled as stationarity conditioned on a set of state variables.  The authors show the identifiability of their model, under the assumption that state variables are observed and there is no instantaneous causal effect. They further propose a VAE-based estimation method to estimate the causal graph, but without a consistency guarantee.", "summary_of_the_review": "The paper is well presented, but the novelty is limited, especially in theoretical identifiability, and from the experimental results, the performance of the proposed method seems to decrease quickly with increasing variables and state values.", "strength_and_weaknesses": "Strength:\n\n1. This paper considers an interesting and practically useful problem: time-delayed causal discovery from non-stationary data, and models the non-stationary behavior to be stationary conditioned on a set of state variables, which I think is a reasonable assumption.\n\n2. The reported experimental results show that the proposed method has a better performance compared to other methods.\n\n3. The organization and writing are good.\n\nWeakness:\n\n1. The authors only give the identifiability results when the state variables are observed (Scenario 1), which is trivial. Because of this, I fail to find significant contributions of this paper.  \n\n2. From Figure 4, it seems the practical performance of the proposed method decreases quickly with increasing variables and state values. \n\n Also, is the proposed method able to handle different time lags?\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "NxuPv0-SGk", "reviewer": "Reviewer_QqiG", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "The authors extend the existing work in time series causal discovery from stationary data to conditionally stationary time series data given the state. ", "summary_of_the_review": "Comments on experiments:\n\"for a fair comparison we evaluate the identification accuracy of the summary\ngraph such that it only considers the exis tence of an edge rather than capturing the edge type.\"\nDoes this mean you are simply checking the graph skeleton? Could you explain why this is more fair?\n\nIn the other experiments also, it's not clear if the proposed method shows promise compared to the baseline. \n\nIn Section 4.3, could you explain why causal discovery is important for forecasting, which is a predictive task?", "strength_and_weaknesses": "The authors assume access to the states at all times. Then one can simply run the existing stationary causal discovery algorithms conditioned on each realization of the state. \n\nIt turns out this is exactly what the authors propose as well. \n\nUnfortunately, this simple extension might not be a sufficient contribution for an ICLR publication. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "5qnqDttdG7", "reviewer": "Reviewer_Hjuv", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "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": "This paper proposes to discovery causal graphs in term of conditional summary graphs with conditional stationary time series data. Authors first present the framework for learning conditional time series data and then presented a VAE framework to instantiate the framework. Empirical study shows the superior performance over existing baselines. ", "summary_of_the_review": "it is a decent work to address a new problem with decent empirical performance, but technically it is not new and theoretical results are not consistent with the algorithm. ", "strength_and_weaknesses": "Strength:\nThe empirical evaluation is convincing with strong performance.\n\nWeakness:\nPresentation could be improved, with many design choices left poorly explained.\nIdentification result is decoupled from the deep learning framework  with VAE, which makes the theoretical results not really useful. \nThe proposed method seem to be adopted from many existing related work. While being applied to a new problem, it is unclear what difficulties in just combing them. \n\n\nComments:\n-Figure 1 Scenarios: I found these scenarios seems arbitrary set. For example, in S2, s_t only depends on x_t, but in S3, s_t depends on x_t and x_{t-1} (and s_{t-1}). What about any subset of the dependence in Senario 3? Similarly, in S1 and S3, s_t depends on s_{t-1}  but not in S2. These inconsistency should be explained, if there is a specific choice behind it. \n- Stationarity: the description is not totally clear. If one treat the joint distribution as a mixture distribution (with the states being the mixture variables), would it be stationary as well? If not, do you only assume the non-stationarity comes from s? \n- is there a reason why auto-regressive connections are not considered?\n- identiability: it would be good to have a formal statement on this in the paper. In addition, \"at in this hidden state regime the\nprevious theoretical guarantees on identifiability no longer hold due to violations of our assumption\" I'm not sure if the identifiability holds in the proposed VAE structure where you find the lower bounds. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject