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From Deterministic ODEs to Dynamic Structural Causal Models

From Deterministic ODEs to Dynamic Structural Causal Models Paul K. Rubenstein Stephan Bongers Bernhard Schölkopf Joris M. Mooij Department of Engineering Informatics Institute Max-Planck Institute for Informatics Institute University of Cambridge University of Amsterdam Intelligent Systems, Tübingen University of Amsterdam United Kingdom The Netherlands Germany The Netherlands [email protected] [email protected] [email protected] [email protected] Abstract describing causal relations and interventions and have been widely applied in the social sciences, economics, genetics and neuroscience (Pearl, 2009; Bollen, 2014). Structural Causal Models are widely used in One of the successes of SCMs over other causal frame- causal modelling, but how they relate to other works such as causal Bayesian networks, for instance, has modelling tools is poorly understood. In this been their ability to express cyclic causal models (Spirtes, paper we provide a novel perspective on the re- 1995; Mooij et al., 2011; Hyttinen et al., 2012; Voortman lationship between Ordinary Differential Equa- et al., 2010; Lacerda et al., 2008; Bongers et al., 2018). tions and Structural Causal Models. We show how, under certain conditions, the asymptotic We view SCMs as an intermediate level of description be- behaviour of an Ordinary Differential Equation tween the highly expressive differential equation models under non-constant interventions can be mod- and the probabilistic, non-causal models typically used in elled using Dynamic Structural Causal Models. machine learning and statistics. This intermediate level In contrast to earlier work, we study not only of description ideally retains the benefits of a data-driven the effect of interventions on equilibrium states; statistical approach while still allowing a limited set of rather, we model asymptotic behaviour that is causal statements about the effect of interventions. While dynamic under interventions that vary in time, it is well understood how an SCM induces a statistical and include as a special case the study of static model (Bongers et al., 2018), much less is known about equilibria. how a differential equation model—our most fundamen- tal level of modelling—can imply an SCM in the first place. This is an important question because if we are to 1 INTRODUCTION have models of a system on different levels of complexity, we should understand how they relate and the conditions Ordinary Differential Equations (ODEs) provide a univer- under which they are consistent with one another. sal language to describe deterministic systems via equa- Indeed, recent work has begun to address the question of tions that determine how variables change in time as a how SCMs arise naturally from more fundamental models function of other variables. They provide an immensely by showing how, under strong assumptions, SCMs can popular and highly successful modelling framework, with be derived from an underlying discrete time difference applications in many diverse disciplines, such as physics, equation or continuous time ODE (Iwasaki and Simon, chemistry, biology, and economy. They are causal in 1994; Dash, 2005; Lacerda et al., 2008; Voortman et al., the sense that at least in principle they allow us to rea- 2010; Mooij et al., 2013; Sokol and Hansen, 2014). With son about interventions: any external intervention in a the exception of (Voortman et al., 2010) and (Sokol and system—e.g., moving an object by applying a force—can Hansen, 2014), each of these methods assume that the be modelled using modified differential equations by, for dynamical system comes to a static equilibrium that is instance, including suitable forcing terms. In practice, of independent of initial conditions, with the derived SCM course, this may be arbitrarily difficult. describing how this equilibrium changes under interven- Structural Causal Models (SCMs, also known as Struc- tion. More recently, the more general case in which the tural Equation Models) are another language capable of equilibrium state may depend on the initial conditions has been addressed (Bongers and Mooij, 2018; Blom and Also affiliated with Max Planck Institute for Intelligent Systems, Tübingen. Mooij, 2018). arXiv:1608.08028v2 [cs.AI] 9 Jul 2018 If the assumption that the system reaches a static equi- sults as ‘orthogonal’ to methods such as Granger causality librium is reasonable for a particular system under study, (Granger, 1969) and difference-in-differences (Card and the SCM framework can be useful. Although the derived Krueger, 1993) which aim to infer causal effects given SCM then lacks information about the (possibly rich) tran- time-series observations of a system. We envision that sient dynamics of the system, if the system equilibrates DSCMs may be used for causal analysis of dynamical quickly then the description of the system as an SCM may systems that undergo periodic motion. Although these be a more convenient and compact representation of the systems have been mostly ignored so far in the field of causal structure of interest. By making assumptions on causal discovery, they have been studied extensively in the dynamical system and the interventions being made, the field of control theory. Some examples of systems that the SCM effectively allows us to reason about a ‘higher naturally exhibit oscillatory stationary states and where level’ qualitative description of the dynamics—in this our framework may be applicable are EEG signals, circa- case, the equilibrium states. dian signals, seasonal influences, chemical oscillations, electric circuits, aerospace vehicles, and satellite control. There are, however, two major limitations that stem from We refer the reader to (Bittanti and Colaneri, 2009) for the equilibrium assumption. First, for many dynamical more details on these application areas from the perspec- systems the assumption that the system settles to a unique tive of periodic control theory. equilibrium, either in its observational state or under inter- vention, may be a bad approximation of the actual system Since the DSCM derived for a simple harmonic oscilla- dynamics. Second, this framework is only capable of tor (see Example 4) is already quite complex, we leave modelling interventions in which a subset of variables are the task of deriving methods that estimate the parame- clamped to fixed values (constant interventions). Even for ters from data for future work. Rather, our current work rather simple physical systems such as a forced damped presents a first necessary theoretical step that needs to be simple harmonic oscillator, these assumptions are vio- done before applications of this theory can be developed, lated. enabling the development of data-driven causal discov- ery and prediction methods for oscillatory systems, and Motivated by these observations, the work presented in possibly even more general systems, down the road. this paper tries to answer the following questions: (i) Can the SCM framework be extended to model systems that The remainder of this paper is organised as follows. In do not converge to an equilibrium? (ii) If so, what assump- Section 2, we introduce notation to describe ODEs. In tions need to be made on the ODE and interventions so Section 3, we describe how to apply the notion of an inter- that this is possible? Since SCMs are used in a variety of vention on an ODE to the dynamic case. In Section 4, we situations in which the equilibrium assumption does not define regularity conditions on the asymptotic behaviour necessarily hold, we view these questions as important of an ODE under a set of interventions. In Section 5, in order to understand when they are indeed theoretically we present our main result: subject to conditions on the grounded as modelling tools. The main contribution of dynamical system and interventions being modelled, a Dy- this paper is to show that the answer to the first question namic SCM can be derived that allows one to reason about is ‘Yes’ and to provide sufficient conditions for the sec- how the asymptotic dynamics change under interventions ond. We do this by extending the SCM framework to on variables in the system. We conclude in Section 6. encompass time-dependent dynamics and interventions and studying how such objects can arise from ODEs. We 2 ORDINARY DIFFERENTIAL refer to this as a Dynamic SCM (DSCM) to distinguish EQUATIONS it from the static equilibrium case for the purpose of ex- position, but note that this is conceptually the same as Let I = f1; : : : ; Dg be a set of variable labels. Con- an SCM on a fundamental level. Our construction draws sider time-indexed variables X (t) 2 R for i 2 I , where i i inspiration from the approach of Mooij et al. (2013), that R  R and t 2 R = [0;1). For I  I , we write i 0 was recently generalized to also incorporate the stochas- X (t) 2 R for the tuple of variables (X (t)) . I i i i2I i2I tic setting (Bongers and Mooij, 2018). Here, we adapt By an ODED, we mean a collection of D coupled ordi- the approach by replacing the static equilibrium states by (k) nary differential equations with initial conditions X : continuous-time trajectories, considering two trajectories as equivalent if they do not differ asymptotically. (k) (k) f (X ;X )(t) = 0; X (0) = (X ) ; i i pa(i) i i 0 D : Note that whilst this paper applies a causal perspective to 0  k  n 1; i 2 I; the study of dynamical systems, the goal of this paper is where the ith differential equation determines the evo- not to derive a learning algorithm which can be applied lution of the variable X in terms of X , where to time series data. In this sense, we view our main re- i pa(i) pa(i)  I are the parents of i, and X itself, and where i (k) edge between X and X iff X is a direct cause of X n is the order of the highest derivative X of X that i i i j i j (in the context of all variables X ). In this section, we appears in equation i. Here, f is a functional that can i I will formalize this causal interpretation by studying inter- include time-derivatives of its arguments. We think of the ventions on the system. ith differential equation as modelling the causal mech- anism that determines the dynamics of the effect X in 3.1 TIME-DEPENDENT PERFECT terms of its direct causes X . pa(i) INTERVENTIONS One possible way to write down an ODE is to canonically decompose it into a collection of first order differential Usually in the causality literature, by a perfect interven- equations, such as is done in Mooij et al. (2013). We tion it is meant that a variable is clamped to take a spe- choose to present our ODEs as “one equation per vari- cific given value. The natural analogue of this in the able” rather than splitting up the equations due to com- time-dependent case is a perfect intervention that forces plications that would otherwise occur when considering a variable to take a particular trajectory. That is, given a time-dependent interventions (cf. Section 3.3). subset I  I and a function  : R ! R , we I 0 i i2I can intervene on the subset of variables X by forcing Example 1. Consider a one-dimensional system of D X (t) =  (t)8t 2 R . Using Pearl’s do-calculus nota- particles of mass m (i = 1; : : : ; D) with positions X I I 0 i i tion (Pearl, 2009) and for brevity omitting the t, we write coupled by springs with natural lengths l and spring do(X =  ) for this intervention. Such interventions I I constants k , where the ith spring connects the ith and are more general objects than those of the equilibrium or (i + 1)th masses and the outermost springs have fixed time-independent case, but in the specific case that we ends (see Figure 1a). Assume further that the ith mass restrict ourselves to constant trajectories the two notions undergoes linear damping with coefficient b . coincide. _  Denoting by X and X the first and second time deriva- i i tives of X respectively, the equation of motion for the ith i 3.2 SETS OF INTERVENTIONS variable is given by Recall that when modelling equilibrating dynamical sys- m X (t) =k [X (t) X (t) l ] i i i i+1 i i tems under constant interventions, the set of interven- tions modelled coincides with the asymptotic behaviour k [X (t) X (t) l ] b X (t) i1 i i1 i1 i i of the system. We will generalise this relation to non- equilibrating behaviour. where we take X = 0 and X = L to be the fixed posi- 0 D tions of the end springs. For the case that D = 2, we can The Dynamic SCMs that we will derive will describe the write the system of equations as: asymptotic dynamics of the ODE and how they change 8 under different interventions. If we want to model ‘all  _ 0 = m X (t) + b X (t) + (k + k )X (t) > 1 1 1 1 1 0 1 > possible interventions’, then the resulting asymptotic dy- k X (t) k l + k l ; > 1 2 0 0 1 1 > namics that can occur are arbitrarily complicated. The idea is to fix a simpler set of interventions and derive an  _ D : 0 = m X (t) + b X (t) + (k + k )X (t) 2 2 2 2 2 1 2 SCM that models only these interventions, resulting in k L k X (t) k l + k l ; > 2 1 1 2 1 2 2 > a model that is simpler than the original ODE but still > allows us to reason about interventions we are interested (k) (k) X (0) = (X ) k 2 f0; 1g; i 2 f1; 2g : i 0 in. In the examples in this paper, we restrict ourselves to periodic or quasi-periodic interventions, but the results hold for more general sets of interventions that satisfy the We can represent the functional dependence structure be- stability definitions presented later. tween variables implied by the functions f with a graph, in which variables are nodes and arrows point X ! X j i We need to define some notation to express the sets of (k) if j 2 pa(i). Self loops X ! X exist if X appears i i i interventions and the set of system responses to these in the expression of f for more than one value of k. This interventions that we will model. Since interventions is illustrated for the system described in Example 1 in correspond to forcing variables to take some trajectory, Figure 1b. we describe notation for defining sets of trajectories: For I  I , let Dyn be a set of trajectories in R . Let i2I 3 INTERVENTIONS ON ODES Dyn = [ Dyn (where P (I ) is the power set of I2P (I) I i.e., the set of all subsets of I ). Thus, an element We interpret ODEs as causal models. In particular, we  2 Dyn is a function R ! R , and Dyn con- I 0 i i2I consider the graph expressing the functional dependence sists of such functions for different I  I . The main structure to be the causal graph of the system, with an idea is that we want both the interventions and the system X = 0 X = L 0 3 X X 1 2 X X X X 1 2 1 2 k k k 0 1 2 (c)D (a) Mass-spring system (b)D do(X = ) 1 1 Figure 1: (a) The mass-spring system of Example 1 with D = 2; (b–c) graphs representing the causal structure of the mass-spring system for (b) the observational system, (c) after the intervention on variable X described in Example 2. As a result of the intervention, X is not causally influenced by any variable, while the causal mechanism of X 1 2 remains unchanged. responses to be elements of Dyn; in other words, the set sets of trajectories we are considering are for the purposes of possible system responses should be large enough to of constructing the Dynamic SCMs. Some examples of contain all interventions that we would like to model, and trivially modular sets of trajectories are: (i) all static (i.e., in addition, all responses of the system to those interven- time-independent) trajectories, corresponding to (Mooij tions. The reader might wonder why we do not simply et al., 2013); (ii) all continuously-differentiable trajecto- take the set of all possible trajectories, but that set would ries that differ asymptotically; (iii) all periodic motions. be so large that it would not be practical for modeling The latter is the running example in this paper. purposes. 3.3 DESCRIBING INTERVENTIONS ON ODEs Since our goal will be to derive a causal model that de- scribes the relations between components (variables) of We can realise a perfect intervention by replacing the the system, we will need the following definition in Sec- equations of the intervened variables with new equations tion 5. that fix them to take the specified trajectories: Definition 1. A set of trajectories Dyn is modular if, for D : do(X = ) I I any fi ; : : : ; i g = I  I , 8 1 n (k) (k) > f (X ;X )(t) = 0 ; X (0) = (X ) ; i i pa(i) i > i 0 0  k  n 1 ; i 2 I n I ; 2 Dyn ()  2 Dyn 8k 2 f1; : : : ; ng: I i X (t)  (t) = 0 ; i 2 I : i i This should be interpreted as saying that admitted tra- jectories of single variables can be combined arbitrarily This procedure is analogous to the notion of intervention into admitted trajectories of the whole system (and vice in an SCM. In reality, this corresponds to decoupling the versa, admitted system trajectories can be decomposed intervened variables from their usual causal mechanism into trajectores of individual variables), and in addition, by forcing them to take a particular value, while leaving that interventions on each variable can be made indepen- the non-intervened variables’ causal mechanisms unaf- dently and combined in any way. This is not to say fected. that all such interventions must be physically possible to Perfect interventions will not generally be realisable in implement in practice. Rather, this means that the mathe- the real world. In practice, an intervention on a variable matical model we derive should allow one to reason about would correspond to altering the differential equation all such interventions. Not all sets of trajectories Dyn are governing its evolution by adding extra forcing terms; modular; in the following sections we will assume that the perfect interventions could be realised by adding forcing terms that push the variable towards its target value at For example, one might want to parameterize the set of trajectories in order to learn the model from data. Without any each instant in time, and considering the limit as these restriction on the smoothness of the trajectories, the problem of forcing terms become infinitely strong so as to dominate estimating a trajectory from data becomes ill-posed. Secondly, the usual causal mechanism determining the evolution of since we would like to identify trajectories that are asymptot- the variable. ically identical in order to focus the modeling efforts on the asymptotic behaviour of the system, we will only put a single Example 2 (continued). Consider the mass-spring sys- trajectory into Dyn to represent all trajectories that are asymptot- tem described in Example 1. If we were to intervene on ically identical to that trajectory, but whose transient dynamics may differ. Note that in the intervened ODE, the initial conditions of This is related to notions that have been discussed in the the intervened variables do not need to be specified explicitly literature under various headings, for instance autonomy and as for the other variables, since they are implied by considering invariance (Pearl, 2009). t = 0. the system to force the mass X to undergo simple har- of Mooij et al. (2013) to allow for non-constant trajec- monic motion, we could express this as a change to the tories in Dyn , and coincide with them in the case that system of differential equations as: Dyn consists of all constant trajectories inR . Definition 2. The ODE D is dynamically stable with D : do(X (t)=l +A cos(!t)) 1 1 reference to Dyn if there exists a unique  2 Dyn 8 ; I I 0 = X (t) l A cos(!t) ; > 1 1 such that X (t) =  (t)8t is a solution to D and that > I ; for any initial condition, the solution X (t) !  (t) as > ;  _ 0 = m X (t) + b X (t) + (k + k )X (t) 2 2 2 2 2 1 2 t ! 1. k L k X (t) k l + k l ; > 2 1 1 2 1 2 2 We use a subscript ; to emphasise that  describes the > ; (k) (k) asymptotic dynamics ofD without any intervention. Ob- X (0) = (X ) k 2 f0; 1g: 2 0 serve that Dyn could consist of the single element in this case. The requirement that this hold for all initial This induces a change to the graphical description of the conditions can be relaxed to hold for all initial conditions causal relationships between the variables. We break any except on a set of measure zero, but that would mean that incoming arrows to any intervened variable, including self the proofs later on require some more technical details. loops, as the intervened variables are no longer causally For the purpose of exposition, we stick to this simpler influenced by any other variable in the system. See Figure case. 1c for the graph corresponding to the intervened ODE in Example 3. Consider a single mass on a spring that is Example 2. undergoing simple periodic forcing and is underdamped. Such a system could be expressed as a single (parent-less) 4 DYNAMIC STABILITY variable with ODE description: A crucial assumption of Mooij et al. (2013) was that the 8  _ mX (t) + bX (t) + k(X (t) l) > 1 1 1 systems considered were stable in the sense that they > = F cos(!t + ) ; would converge to unique stable equilibria (if necessary, D : also after performing a constant intervention). This made > (k) (k) X (0) = (X ) k 2 f0; 1g : them amenable to study by considering the t ! 1 limit 1 0 in which any complex but transient dynamical behaviour would have decayed. The SCMs derived would allow one The solution to this differential equation is to reason about the asymptotic equilibrium states of the X (t) = r(t) + l + A cos(!t +  ) (1) systems after interventions. Since we want to consider 1 non-constant asymptotic dynamics, this is not a notion of where r(t) decays exponentially quickly (and is dependent stability that is fit for our purposes. on the initial conditions) and A and  depend on the Instead, we define our stability with reference to a set of parameters of the equation of motion (but not on the trajectories. We will use Dyn for this purpose. Recall initial conditions). that elements of Dyn are trajectories for all variables Therefore such a system would be dynamically stable with in the system. To be totally explicit, we can think of an reference to (for example) element  2 Dyn as a function 0 0 Dyn = fl + A cos(!t +  ) : A 2 R;  2 [0; 2)g: : R ! R 0 I t 7! ( (t);  (t); : : : ;  (t)) 1 2 D Remark 1. We use a subscript  to emphasise that describes the asymptotic dynamics ofD after performing where  (t) 2 R is the state of the ith variable X at time i i i the intervention do(X =  ). Observe that Dyn could I I t. Note that Dyn is not a single fixed set, independent consist only of the single element  and the above of the situation we are considering. We can choose Dyn definition would be satisfied. But then the original ODE depending on the ODE D under consideration, and the wouldn’t be dynamically stable with reference to Dyn , interventions that we may wish to make on it. nor would other intervened versions of D. This motivates Informally, stability in this paper means that the asymp- the following definition, extending dynamic stability to totic dynamics of the dynamical system converge to a sets of intervened systems. unique element of Dyn , independent of initial condition. The convergence we refer to here is the usual asymptotic If Dyn is in some sense simple, we can simply char- I convergence of real-valued functions, i.e., for f : [0;1) ! R , acterise the asymptotic dynamics of the system under g : [0;1) ! R we have that f ! g iff for every  > 0 there study. The following definitions of stability extend those is a T 2 [0;1) such thatjf(t) g(t)j <  for all t 2 [T;1). Definition 3. Let Traj be a set of trajectories. We say in terms of its parents. We extend this to the case that that the pair (D; Traj) is dynamically stable with ref- our variables do not take fixed values but rather represent erence to Dyn if, for any  2 Traj , D is entire trajectories. I do(X = ) I I I dynamically stable with reference to Dyn . Definition 4. Let Dyn = Dyn be a modular set II Example 3 (continued). Suppose we are interested in of trajectories, where Dyn  R . A deterministic I I modelling the effect of changing the forcing term, either Dynamic Structural Causal Model (DSCM) on the time- in amplitude, phase or frequency. We introduce a second indexed variables X taking values in Dyn is a collection variable X to model the forcing term: of structural equations 0 = f (X ; X )(t) > 1 1 2 M : X = F (X ) i 2 I ; i i pa(i) >  _ = mX (t) + bX (t) + k(X (t) l) X (t) ; 1 1 1 2 > where pa(i)  I nfig and each F is a map Dyn ! Dyn that gives the trajectory of an 0 = f (X )(t) D : pa(i) i 2 2 > effect variable in terms of the trajectories of its direct = X (t) F cos(! t +  ) ; > 2 0 0 0 > causes. (k) (k) X (0) = (X ) ; k 2 f0; 1g : 1 0 The point of this paper is to show that, subject to restric- tions on D and Dyn, we can derive a DSCM that allows If we want to change the forcing term that we apply to the us to reason about the effect on the asymptotic dynamics mass, we can interpret this as performing an intervention of interventions using trajectories in Dyn. ‘Traditional’ on X . We could represent this using the notation we deterministic SCMs arise as a special case, where all have developed as trajectories are constant over time. In an ODE, the equations f determine the causal relation- Dyn = f (t) = F cos(!t +  ) : 2 2 2 f2g ship between the variable X (t) and its parents X (t) pa(i) F ; ! 2 R;  2 [0; 2)g: 2 2 at each instant in time. In contrast, we think of the function F of the DSCM as a causal mechanism that For any intervention  2 Dyn , the dynamics of X in 2 1 f2g determines the entire trajectory of X in terms of the D will be of the form (1). Therefore (D; Dyn ) do(X = ) 2 2 f2g trajectories of the variables X , integrating over the pa(i) will be dynamically stable with reference to instantaneous causal effects over all time. In the case that Dyn consists of constant trajectories (and thus the instan- Dyn = (t) = (l + F cos(!t +  ); F cos(!t +  )) 1 1 2 2 taneous causal effects are constant over time), a DSCM reduces to a traditional deterministic SCM. : F ; F ; ! 2 R;  ;  2 [0; 2) : 1 2 1 2 The rest of this section is laid out as follows. In Section 5.1 we define what it means to make an intervention in a The independence of initial conditions for Example 3 is DSCM. In Section 5.2 we show how, subject to certain illustrated in Figure 2. conditions, a DSCM can be derived from a pair (D; Dyn). Note that if (D; Traj) is dynamically stable with refer- The procedure for doing this relies on intervening on all ence to Dyn , and Dyn  Dyn is a larger set of trajec- I I I but one variable at a time. In Section 5.3, Theorem 2 tories that still satisfies the uniqueness condition in the states that the DSCM thus derived is capable of modelling definition of dynamic stability, then (D; Traj) is dynam- the effect of intervening on arbitrary subsets of variables, ically stable with reference to Dyn . I even though it was constructed by considering the case that we consider interventions on exactly D 1 variables. 5 DYNAMIC STRUCTURAL CAUSAL Theorem 3 and Corollary 1 in Section 5.4 prove that the MODELS notions of intervention in ODE and the derived DSCM coincide. Collectively, these theorems tell us that we can A deterministic SCMM is a collection of structural equa- derive a DSCM that allows us to reason about the effects tions, the ith of which defines the value of variable X of interventions on the asymptotic dynamics of the ODE. Proofs of these theorems are provided in Section A of the 5 0 Namely: 8 2 Traj; 9!  2 Dyn such that under I I Supplementary Material. D and for any initial condition, X (t) !  (t) as do(X = ) I I I I t ! 1. Assuming that (D; Traj) is dynamically stable with 5.1 INTERVENTIONS IN A DSCM reference to Dyn , a sufficient condition for this is that none of the elements in Dyn n Dyn are asymptotically equal to any of I I 0 0 Interventions in (D)SCMs are realized by replacing the the elements of Dyn . That is: 8 2 Dyn ; 8 2 Dyn nDyn , I I I I (t) 9  (t) as t ! 1 . structural equations of the intervened variables. Given 10 10 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 t t (a) (b) Figure 2: Simulations from the forced simple harmonic oscillator in Example 3 showing the evolution of X with different initial conditions for different forcing terms (interventions on X ). The parameters used were m = 1; k = 1; l = 2; F = 2; b = 0:1, with (a) ! = 3 and (b) ! = 2. Dynamic stability means that asymptotic dynamics are independent of initial conditions, and the purpose of the DSCM is to quantify how the asymptotic dynamics change under intervention. 2 Dyn for some I  I , the intervened DSCM It should be noted that (D; Dyn) being structurally dy- M can be written: namically stable is a strong assumption in general. If do(X = ) I I Dyn is too small, then it may be possible to find a larger 0 0 set Dyn  Dyn such that (D; Dyn ) is structurally dy- X = F (X ) i 2 I n I ; i i pa(i) M : do(X = ) I I namically stable. The procedure described in this section X =  i 2 I : i i describes how to derive a DSCM capable of modelling all The causal mechanisms determining the non-intervened interventions in Dyn , which can thus be used to model variables are unaffected, so their structural equations re- interventions in Dyn. main the same. The intervened variables are decoupled Henceforth, we use the notation I = I n fig for from their usual causal mechanisms and are forced to take brevity. Suppose that (D; Dyn) is structurally dynam- the specified trajectory. ically stable. We can derive structural equations F : Dyn ! Dyn to describe the asymptotic dynam- pa(i) i 5.2 DERIVING DSCMs FROM ODEs ics of children variables as functions of their parents as follows. Pick i 2 I . The variable X has parents X . In order to derive a DSCM from an ODE, we require the pa(i) Since Dyn is modular, for any configuration of parent dy- following consistency property between the asymptotic namics  2 Dyn there exists  2 Dyn such dynamics of the ODE and the set of interventions. pa(i) pa(i) i I that ( ) =  . i pa(i) pa(i) Definition 5 (Structural dynamic stability). Let Dyn be By structural dynamic stability, the system D modular. The pair (D; Dyn) is structurally dynamically do(X = ) I I i i has asymptotic dynamics specified by a unique element stable if (D; Dyn ) is dynamically stable with refer- Infig 2 Dyn , which in turn defines a unique element  2 ence to Dyn for all i. i Dyn specifying the asymptotic dynamics of variable X This means that for any intervention trajectory since Dyn is modular. 2 Dyn , the asymptotic dynamics of the inter- Infig Infig Theorem 1. Suppose that (D; Dyn) is structurally dynam- vened ODED are expressible uniquely do(X = ) Infig Infig ically stable. Then the functions as an element of Dyn . Since Dyn is modular, the asymp- totic dynamics of the non-intervened variable can be re- F : Dyn ! Dyn :  7! i pa(i) i pa(i) i alised as the trajectory  2 Dyn , and thus Dyn is rich enough to allow us to make an intervention which forces constructed as above are well-defined. the non-intervened variable to take this trajectory. This is a crucial property that allows the construction of the struc- Given the structurally dynamically stable pair (D; Dyn) tural equations. In the particular case that Dyn consists we define the derived DSCM of all constant trajectories, structural dynamic stability means that after any intervention on all-but-one-variable, M : X = F (X ) i 2 I ; i i D pa(i) the non-intervened variable settles to a unique equilib- rium. In the language of Mooij et al. (2013), this would For example, if Dyn is not modular or represents interven- imply that the ODE is structurally stable. tions on only a subset of the variables. X where the F : Dyn ! Dyn are defined as above. coincide, and hence that DSCMs provide a representation pa(i) i Note that structural dynamic stability was a crucial prop- to reason about the asymptotic behaviour of the ODE un- erty that ensured F (Dyn )  Dyn . If (D; Dyn) is not der interventions in Dyn. A consequence of these results pa(i) i structurally dynamically stable, we cannot build structural is that the diagram in Figure 3 commutes. equations in this way. Theorem 3. Suppose that (D; Dyn) is structurally dy- namically stable. Let I  I and let  2 Dyn . Then We provide next an example of a DSCM for the mass- I M = (M ) . (D ) D do(X = ) spring system of Example 1 with D = 2. The derivation I I do(X = ) I I of this for the general case of arbitrarily many masses is Corollary 1. Suppose additionally that J  I n I and included in the Supplementary Material. let  2 Dyn . Then Example 4. Consider the systemD governed by the dif- M = (M ) : (D ) D do(X = ;X = ) ferential equation of Example 1 with D = 2. Let Dyn do(X = ) I I J J f1;2g I I do(X = ) J J be the modular set of trajectories with To summarise, Theorems 1–3 and Corollary 1 collectively j j j state that if (D; Dyn) is dynamically structurally stable Dyn = A cos(! t +  ) : fig i i i then it is possible to derive a DSCM that allows us to j=1 reason about the asymptotic dynamics of the ODE under j j j j any possible intervention in Dyn. w ;  ; A 2 R; jA j < 1 i i i i j=1 5.5 RELATION TO ODEs AND DYNAMIC BAYESIAN NETWORKS for i = 1; 2, where for each i it holds that jA j < j=1 i 1 (so that the series is absolutely convergent). Then An ODE is capable of modelling arbitrary interventions (D; Dyn ) is structurally dynamically stable and ad- f1;2g on the system it describes. At the cost of only modelling mits the following DSCM. a restricted set of interventions, a DSCM can be derived which describes the asymptotic behaviour of the system X = F (X ) 1 1 2 M : under these interventions. This may be desirable in cases X = F (X ) 2 2 1 for which transient behaviour is not important. j j j 2 2 where, writing C = [k + k m (! ) ] and C = 1 2 1 1 2 2 We now compare DSCMs to Dynamic Bayesian Net- 2 2 [k + k m (! ) ] , the functionals F and F are 1 2 2 1 2 works (DBNs), an existing popular method for causal given by Equations 2 and 3 overleaf. modelling of dynamical systems (Koller and Friedman, 2009). DBNs are essentially Markov chains, and thus are appropriate for discrete-time systems. When the discrete- 5.3 SOLUTIONS OF A DSCM time Markov assumption holds, DBNs are a powerful tool Theorem 1 states that we can construct a DSCM by the capable of modelling arbitrary interventions. However, described procedure. We constructed each equation by approximations must be made whenever these assump- intervening on D 1 variables at a time. The result of tions do not hold. In particular, a continuous system must this section states that the DSCM can be used to cor- be approximately discretised in order to be modelled by a rectly model interventions on arbitrary subsets of vari- DBN (Sokol and Hansen, 2014). ables. We say that  2 Dyn is a solution of M if = F ( )8i 2 I . By using the Euler method for numerically solving ODEs, i i pa(i) we can make such an approximation to derive a DBN de- Theorem 2. Suppose that (D; Dyn) is structurally dy- scribing the system in Example 1, leading to the discrete namically stable. Let I  I , and let  2 Dyn . Then time equation given in (8) the Supplementary Material. D is dynamically stable if and only if the inter- do(X = ) I I For DBNs, the main choice to be made is how fine the vened SCMM has a unique solution. If there (D ) do(X = ) I I temporal discretisation should be. The smaller the value is a unique solution, it coincides with the element of Dyn of , the better the discrete approximation will be. Even describing the asymptotic dynamics ofD . do(X = ) I I if there is a natural time-scale on which measurements Remark 2. We could also take I = ;, in which case the can be made, choosing a finer discretisation than this will above theorem applies to justD. provide a better approximation to the behaviour of the true system. The choice of  should reflect the natural 5.4 CAUSAL REASONING IS PRESERVED timescales of the interventions to be considered too; for We have defined ways to model interventions in both example, it is not clear how one would model the interven- 2t ODEs and DSCMs. The following theorem and its imme- tion do X (t) = cos with a discretisation length diate corollary proves that these notions of intervention . Another notable disadvantage of DBNs is that the Sec. 3.3 Sec. 3.3 Intervened ODE Intervened ODE ODE D D D do(X = ) do(X = ;X = ) I I I I J J Sec. 5.2 Sec. 5.2 Sec. 5.2 Sec. 5.1 Sec. 5.1 Intervened DSCM Intervened DSCM DSCM M M D D D do(X = ) do(X = ;X = ) I I I I J J Figure 3: Top-to-bottom arrows: Theorems 1 and 2 together state that if (D; Dyn) is structurally dynamically stable then we can construct a DSCM to describe the asymptotic behaviour of D under different interventions in the set Dyn. Left-to-right arrows: Both ODEs and DSCMs are equipped with notions of intervention. Theorem 3 and Corollary 1 say that these two notions of intervention coincide, and thus the diagram commutes. 0 1 " #! 1 1 j j X X k l k A b ! j j j 1 1 1 j j 1 2 2 @ A F A cos(! t +  ) = + q cos ! t +  arctan 2 2 2 2 2 j (2) k + k j j C 1 0 2 j=1 j=1 1 C + b m (! ) 1 1 1 2 0 1 " #! 1 1 j j X X k l k l k L k A b ! 1 1 2 2 2 1 2 j j j 1 j j 1 @ A F A cos(! t +  ) = + + q cos ! t +  arctan (3) 1 1 1 1 1 k + k k + k j j 1 2 2 3 C j=1 j=1 C + b m (! ) 2 2 2 1 Figure 4: Equations giving the structural equations for the DSCM describing the mass-spring system of Example 4 computational cost of learning and inference increases for how they change under intervention. smaller , where computational cost becomes infinitely We identify three possible directions in which to extend large in the limit  ! 0. this work in the future. The first is to properly understand In contrast, the starting point for DSCMs is to fix a conve- how learning DSCMs from data could be performed. This nient set of interventions we are interested in modelling. is important if DSCMs are to be used in practical applica- If a DSCM containing these interventions exists, it will tions. Challenges to be addressed include finding practical model the asymptotic behaviour of the system under each parameterizations of DSCMs, the presence of measure- of these interventions exactly, rather than approximately ment noise in the data and the fact that time-series data are modelling the transient and asymptotic behaviour as in usually sampled at a finite number of points in time. The the case of a DBN. Computational cost does not relate second is to relax the assumption that the asymptotic dy- inversely to accuracy as for DBNs, but depends on the namics are independent of initial conditions, as was done chosen representation of the set of admitted interventions. recently for the static equilibrium scenario by Blom and Mooij (2018). The third extension is to move away from deterministic systems and consider Random Differential 6 DISCUSSION AND FUTURE WORK Equations (Bongers and Mooij, 2018), thereby allowing The main contribution of this paper is to show that the to take into account model uncertainty, but also to include SCM framework can be applied to reason about time- systems that may be inherently stochastic. dependent interventions on an ODE in a dynamic setting. ACKNOWLEDGEMENTS In particular, we showed that if an ODE is sufficiently well-behaved under a set of interventions, a DSCM can Stephan Bongers was supported by NWO, the Nether- be derived that captures how the asymptotic dynamics lands Organization for Scientific Research (VIDI grant change under these interventions. This is in contrast to 639.072.410). This project has received funding from previous approaches to connecting the language of ODEs the European Research Council (ERC) under the Euro- with the SCM framework, which used SCMs to describe pean Union’s Horizon 2020 research and innovation pro- the stable (constant-in-time) equilibria of the ODE and gramme (grant agreement n 639466). References stochastic case. arXiv.org preprint, arXiv:1803.08784 [cs.AI], March 2018. URL https://arxiv.org/ Judea Pearl. Causality: Models, Reasoning, and Infer- abs/1803.08784. ence. Cambridge University Press, New York, NY, 2nd Tineke Blom and Joris M. Mooij. Generalized edition, 2009. structural causal models. arXiv.org preprint, Kenneth A. Bollen. Structural equations with latent vari- https://arxiv.org/abs/1805.06539 [cs.AI], May ables. John Wiley & Sons, 2014. 2018. URL https://arxiv.org/abs/1805. Peter Spirtes. Directed cyclic graphical representations of feedback models. In Proceedings of the Eleventh con- Clive W.J. Granger. Investigating causal relations ference on Uncertainty in Artificial Intelligence (UAI by econometric models and cross-spectral methods. 1995), pages 491–498, 1995. Econometrica: Journal of the Econometric Society, Joris M. Mooij, Dominik Janzing, Tom Heskes, and Bern- pages 424–438, 1969. hard Schölkopf. On causal discovery with cyclic addi- David Card and Alan B. Krueger. Minimum wages and tive noise models. In Advances in Neural Information employment: A case study of the fast food industry Processing Systems (NIPS 2011), pages 639–647, 2011. in New Jersey and Pennsylvania. Technical report, Antti Hyttinen, Frederick Eberhardt, and Patrik O. Hoyer. National Bureau of Economic Research, 1993. Learning linear cyclic causal models with latent vari- Sergio Bittanti and Patrizio Colaneri. Periodic systems: ables. The Journal of Machine Learning Research, 13 filtering and control, volume 5108985. Springer Sci- (1):3387–3439, 2012. ence & Business Media, 2009. Mark Voortman, Denver Dash, and Marek J. Druzdzel. Daphne Koller and Nir Friedman. Probabilistic Graph- Learning why things change: the difference-based ical Models: Principles and Techniques - Adaptive causality learner. In Proceedings of the 26th Con- Computation and Machine Learning. The MIT Press, ference on Uncertainty in Artificial Intelligence (UAI 2010), 2010. Gustavo Lacerda, Peter L. Spirtes, Joseph Ramsey, and Patrik O. Hoyer. Discovering cyclic causal models by independent components analysis. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence (UAI 2008), 2008. Stephan Bongers, Jonas Peters, Bernhard Schölkopf, and Joris M. Mooij. Theoretical aspects of cyclic structural causal models. arXiv.org preprint, arXiv:1611.06221v2 [stat.ME], 2018. Yumi Iwasaki and Herbert A. Simon. Causality and model abstraction. Artificial Intelligence, 67(1):143– 194, 1994. Denver Dash. Restructuring dynamic causal systems in equilibrium. In Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics (AIS- TATS 2005), 2005. Joris M. Mooij, Dominik Janzing, and Bernhard Schölkopf. From ordinary differential equations to structural causal models: the deterministic case. In Proceedings of the Twenty-Ninth Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI 2013), pages 440–448, 2013. Alexander Sokol and Niels Richard Hansen. Causal inter- pretation of stochastic differential equations. Electronic Journal of Probability, 19(100):1–24, 2014. Stephan Bongers and Joris M. Mooij. From random dif- ferential equations to structural causal models: the SUPPLEMENTARY MATERIAL A PROOFS A.1 PROOF OF THEOREM 1 0 0 0 Proof. We need to show that if  and  are such that ( ) = ( ) =  , then  =  . To see that this I I i i i pa(i) pa(i) pa(i) I I i i i is the case, observe that the system of equations forD is given by: do(X = ) I I i i X (t) =  (t) j 2 I n (pa(i)[fig) ; < j j X (t) =  (t) j 2 pa(i) ; D : j j do(X = ) I I i i (k) (k) f (X ;X )(t) = 0 X (0) = (X ) ; 0  k  n 1 : i i pa(i) i i i 0 The equations for D are similar, except with X (t) =  (t) for j 2 I n (pa(i)[fig). In both cases, the do(X = ) j I j i I equations for all variables except X are solved already. The equation for X in both cases reduces to the same quantity i i by substituting in the values of the parents, namely f (X ;  )(t) = 0 : i i pa(i) The solution to this equation in Dyn must be unique and independent of initial conditions, else the dynamic stability of the intervened systemsD and D would not hold, contradicting the dynamic structural stability of do(X = ) do(X = ) I I I i i i I (D; Dyn). It follows that  =  . A.2 PROOF OF THEOREM 2 Proof. By construction of the SCM,  2 Dyn is a solution ofM if and only if the following two conditions (D ) I do(X = ) I I hold: for i 2 I n I , X (t) =  (t) 8t is a solution to the differential equation f (X ;  )(t) = 0; i i i i pa(i) for i 2 I ,  (t) =  (t) for all t. i i which is true if and only if X =  is a solution to D in Dyn . Thus, by definition of dynamic stability, do(X = ) I I I D is dynamically stable with asymptotic dynamics describable by  2 Dyn if and only if X =  uniquely do(X = ) I I solves M . (D ) do(X = ) I I A.3 PROOF OF THEOREM 3 Proof. We need to show that the structural equations of M and (M ) are equal. Observe that (D ) D do(X = ) I I do(X = ) I I the equations forD are given by: do(X = ) I I X =  ; i 2 I ; i i D : do(X = ) (k) (k) I I f (X ;X ) = 0; X (0) = (X ) ; 0  k  n 1; i 2 I n I : i i i i pa(i) 0 Therefore, when we perform the procedure to derive the structural equations forD , we see that: do(X = ) I I if i 2 I , the ith structural equation will simply be X =  since intervening on I does not affect variable X . i i i i if i 2 I n I , the ith structural equation will be the same as for M , since the dependence of X on the other D i variables is unchanged. Hence the structural equations forM are given by: (D ) do(X = ) I I X =  ; i 2 I ; i i M : (D ) do(X = ) I I X = F (X ); i 2 I n I : i i pa(i) and thereforeM = (M ) . (D ) D do(X = ) I I do(X = ) I I A.4 PROOF OF COROLLARY 1 Proof. Corollary 1 follows very simply from the observation that if (D; Dyn) is structurally dynamically stable then so is (D ; Dyn ). The result then follows by application of Theorem 3. do(X = ) I I InI B DERIVING THE DSCM FOR THE MASS-SPRING SYSTEM Consider the mass-spring system of Example 1, but with D  1 an arbitrary integer. We repeat the setup: We have D masses attached together on springs. The location of the ith mass at time t is X (t), and its mass is m . i i For notational ease, we denote by X = 0 and X = L the locations of where the ends of the springs attached to 0 D+1 the edge masses meet the walls to which they are affixed. X and X are constant. The natural length and spring 0 D+1 constant of the spring connecting masses i and i + 1 are l and k respectively. The ith mass undergoes linear damping i i with coefficient b , where b is small to ensure that the system is underdamped. The equation of motion for the ith mass i i (1  i  D) is given by:  _ m X (t) = k [X (t) X (t) l ] k [X (t) X (t) l ] b X (t) i i i i+1 i i i1 i i1 i1 i i so, defining  _ f (X ; X ; X )(t) = m X (t) k [X (t) X (t) l ] + k [X (t) X (t) l ] + b X (t) i i i1 i+1 i i i i+1 i i i1 i i1 i1 i i we can write the system of equationsD for our mass-spring system as D : f (X ; X ; X )(t) = 0 i 2 I : i i i1 i+1 In the rest of this section we will explicitly calculate the structural equations for the DSCM derived from D with two different sets of interventions. First, we will derive the structural equations for the case that Dyn consists of all constant trajectories, corresponding to constant interventions that fix variables to constant values for all time. This illustrates the correspondence between the theory in this paper and that of Mooij et al. (2013). Next, we will derive the structural equations for the case that Dyn consists of interventions corresponding to sums of periodic forcing terms. B.1 MASS-SPRING WITH CONSTANT INTERVENTIONS In order to derive the structural equations we only need to consider, for each variable, the influence of its parents on it. (Formally, this is because of Theorem 1). Consider variable i. If we intervene to fix its parents to have locations X (t) =  and X (t) =  for all t, then the equation of motion for variable i is given by i1 i1 i+1 i+1  _ m X (t) + b X (t) + (k + k )X (t) = k [ l ] + k [ + l ] : i i i i i i1 i i i+1 i i1 i1 i1 There may be some complicated transient dynamics that depend on the initial conditions X (0) and X (0) but provided i i that b > 0, we know that the X (t) will converge to a constant and therefore the asymptotic solution to this equation i i  _ can be found by setting X and X to zero. Note that in general, we could explicitly find the solution to this differential i i equation (and indeed, in the next example we will) but for now there is a shortcut to deriving the structural equations. The asymptotic solution is: k [ l ] + k [ + l ] i i+1 i i1 i1 i1 X = : k + k i i1 Therefore the ith structural equation is: k [X l ] + k [X + l ] i i+1 i i1 i1 i1 F (X ; X ) = : i i1 i+1 k + k i i1 This is analogous to the approach taken in Mooij et al. (2013) in which the authors first define the Labelled Equilibrium Equations and from these derive the SCM. Hence the SCM for (D; Dyn ) is: k [X l ] + k [X + l ] i i+1 i i1 i1 i1 M : X = i 2 I : D i k + k i i1 We can thus use this model to reason about the effect of constant interventions on the asymptotic equilibrium states of the system. B.2 SUMS OF PERIODIC INTERVENTIONS Suppose now we want to be able to make interventions of the form: do X (t) = A cos(!t + ) : (4) Such interventions cannot be described by the DSCM derived in Section B.1. In this section we will explicitly derive a DSCM capable of reasoning about the effects of such interventions. It will also illustrate why we need dynamic structural stability. By Theorem 1, to derive the structural equation for each variable we only need to consider the effect on the child of intervening on the parents according to interventions of the form (4). Consider the following linear differential equation:  _ mX (t) + bX (t) + kX (t) = g(t) : (5) In general, the solution to this equation will consist of two parts—the homogeneous solution and the particular solution. The homogeneous solution is one of a family of solutions to the equation  _ mX (t) + bX (t) + kX (t) = 0 (6) and this family of solutions is parametrised by the initial conditions. If b > 0 then all of the homogeneous solutions decay to zero as t ! 1. The particular solution is any solution to the original equation with arbitrary initial conditions. The particular solution captures the asymptotic dynamics due to the forcing term g. Equation 5 is a linear differential equation. This means that if X = X is a particular solution for g = g and X = X is a particular solution for g = g , 1 1 2 2 then X = X + X is a particular solution for g = g + g . 1 2 1 2 In order to derive the structural equations, the final ingredient we need is an explicit representation for a particular solution to (5) in the case that g(t) = A cos(!t +). We state the solution for the case that the system is underdamped— this is a standard result and can be verified by checking that the following satisfies (5): 0 0 X (t) = A cos(!t +  ) where A b! 0 0 A = p ;  =  arctan : (7) 2 2 2 k m! [k m! ] + bm! Therefore if we go back to our original equation of motion for variable X  _ m X (t) + b X (t) + (k + k )X (t) = k [X (t) l ] + k [X (t) + l ] i i i i i i1 i i i+1 i i1 i1 i1 and perform the intervention do(X (t) = A cos(! t +  ); X (t) = A cos(! t +  )) i1 i1 i1 i1 i+1 i+1 i+1 i+1 we see that we can write the RHS of the above equation as the sum of the three terms g (t) = k l k l ; 1 i1 i1 i i g (t) = k A cos(! t +  ) ; 2 i1 i1 i1 i1 g (t) = k A cos(! t +  ) : 3 i i+1 i+1 i+1 Using the fact that linear differential equation have superposable solutions and (7), we can write down the resulting asymptotic dynamics of X : k l k l i1 i1 i i X (t) = k + k i i1 k A b ! i1 i1 i i1 + cos ! t +  arctan i1 i1 k + k m ! 2 2 2 i i1 i i1 [k + k m ! ] + b m ! i i1 i i i i1 i1 k A b ! i i+1 i i+1 +q cos ! t +  arctan : i+1 i+1 k + k m ! 2 2 2 i i1 i i+1 [k + k m ! ] + b m ! i i1 i i i i+1 i+1 However, note that if we were using Dyn consisting of interventions of the form of equation (4), then we have just shown that the mass-spring system would not be structurally dynamically stable with respect to this Dyn, since we need two periodic terms and a constant term to describe the motion of a child under legal interventions of the parents. This illustrates the fact that we may sometimes be only interested in a particular set of interventions that may not itself satisfy structural dynamic stability, and that in this case we must consider a larger set of interventions that does. In this case, we can consider the modular set of trajectories generated by trajectories of the following form for each variable: j j j X (t) = A cos(! t +  ) i i i j=1 where for each i it holds that jA j < 1 (so that the series is absolutely convergent and thus does not depend on j=1 i the ordering of the terms in the sum). Call this set Dyn (“quasi-periodic”). By equation (7), we can write down the qp structural equations 0 1 1 1 X X j j j j j j @ A F A cos(! t +  ); A cos(! t +  ) i1 i1 i1 i+1 i+1 i+1 j=1 j=1 k l k l i1 i1 i i k + k i i1 " #! 1 j j k A b ! i1 i i1 j j i1 + q cos ! t +  arctan i1 i1 j j 2 2 2 k + k m (! ) i i1 i i1 j=1 [k + k m (! ) ] + b m (! ) i i1 i i i i1 i1 " #! 1 j j k A b ! i i i+1 j j i+1 + q cos ! t +  arctan : i+1 i+1 j j k + k m (! ) 2 2 2 i i+1 i i+1 j=1 [k + k m (! ) ] + b m (! ) i i+1 i i i i+1 i+1 Since this is also a member of Dyn , the mass-spring system is dynamically structurally stable with respect to Dyn qp qp and so the equations F define the Dynamic Structural Causal Model for asymptotic dynamics. C DYNAMIC BAYESIAN NETWORK REPRESENTATION By using Euler’s method, we can obtain a (deterministic) Dynamic Bayesian Network representation of the mass-spring system. For D = 2, this yields (t+1) > X = X (t) + X (t) 1 1 > 1 h i (t+1) _ _ _ X = X (t) + k X (t) b X (t) (k + k )X (t) + k l k l > 1 1 1 2 1 1 0 1 1 0 0 1 1 > 1 (t+1) DBN : (8) X = X (t) + X (t) 2 2 h i (t+1) _ _ _ X = X (t) + k X (t) b X (t) (k + k )X (t) + k l k l + k L > 2 2 1 1 2 2 1 2 2 1 1 2 2 2 > 2 (k) (k) X (0) = (X ) k 2 f0; 1g; i 2 f1; 2g : i 0 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computing Research Repository arXiv (Cornell University)

From Deterministic ODEs to Dynamic Structural Causal Models

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Paul K. Rubenstein Stephan Bongers Bernhard Schölkopf Joris M. Mooij Department of Engineering Informatics Institute Max-Planck Institute for Informatics Institute University of Cambridge University of Amsterdam Intelligent Systems, Tübingen University of Amsterdam United Kingdom The Netherlands Germany The Netherlands [email protected] [email protected] [email protected] [email protected] Abstract describing causal relations and interventions and have been widely applied in the social sciences, economics, genetics and neuroscience (Pearl, 2009; Bollen, 2014). Structural Causal Models are widely used in One of the successes of SCMs over other causal frame- causal modelling, but how they relate to other works such as causal Bayesian networks, for instance, has modelling tools is poorly understood. In this been their ability to express cyclic causal models (Spirtes, paper we provide a novel perspective on the re- 1995; Mooij et al., 2011; Hyttinen et al., 2012; Voortman lationship between Ordinary Differential Equa- et al., 2010; Lacerda et al., 2008; Bongers et al., 2018). tions and Structural Causal Models. We show how, under certain conditions, the asymptotic We view SCMs as an intermediate level of description be- behaviour of an Ordinary Differential Equation tween the highly expressive differential equation models under non-constant interventions can be mod- and the probabilistic, non-causal models typically used in elled using Dynamic Structural Causal Models. machine learning and statistics. This intermediate level In contrast to earlier work, we study not only of description ideally retains the benefits of a data-driven the effect of interventions on equilibrium states; statistical approach while still allowing a limited set of rather, we model asymptotic behaviour that is causal statements about the effect of interventions. While dynamic under interventions that vary in time, it is well understood how an SCM induces a statistical and include as a special case the study of static model (Bongers et al., 2018), much less is known about equilibria. how a differential equation model—our most fundamen- tal level of modelling—can imply an SCM in the first place. This is an important question because if we are to 1 INTRODUCTION have models of a system on different levels of complexity, we should understand how they relate and the conditions Ordinary Differential Equations (ODEs) provide a univer- under which they are consistent with one another. sal language to describe deterministic systems via equa- Indeed, recent work has begun to address the question of tions that determine how variables change in time as a how SCMs arise naturally from more fundamental models function of other variables. They provide an immensely by showing how, under strong assumptions, SCMs can popular and highly successful modelling framework, with be derived from an underlying discrete time difference applications in many diverse disciplines, such as physics, equation or continuous time ODE (Iwasaki and Simon, chemistry, biology, and economy. They are causal in 1994; Dash, 2005; Lacerda et al., 2008; Voortman et al., the sense that at least in principle they allow us to rea- 2010; Mooij et al., 2013; Sokol and Hansen, 2014). With son about interventions: any external intervention in a the exception of (Voortman et al., 2010) and (Sokol and system—e.g., moving an object by applying a force—can Hansen, 2014), each of these methods assume that the be modelled using modified differential equations by, for dynamical system comes to a static equilibrium that is instance, including suitable forcing terms. In practice, of independent of initial conditions, with the derived SCM course, this may be arbitrarily difficult. describing how this equilibrium changes under interven- Structural Causal Models (SCMs, also known as Struc- tion. More recently, the more general case in which the tural Equation Models) are another language capable of equilibrium state may depend on the initial conditions has been addressed (Bongers and Mooij, 2018; Blom and Also affiliated with Max Planck Institute for Intelligent Systems, Tübingen. Mooij, 2018). arXiv:1608.08028v2 [cs.AI] 9 Jul 2018 If the assumption that the system reaches a static equi- sults as ‘orthogonal’ to methods such as Granger causality librium is reasonable for a particular system under study, (Granger, 1969) and difference-in-differences (Card and the SCM framework can be useful. Although the derived Krueger, 1993) which aim to infer causal effects given SCM then lacks information about the (possibly rich) tran- time-series observations of a system. We envision that sient dynamics of the system, if the system equilibrates DSCMs may be used for causal analysis of dynamical quickly then the description of the system as an SCM may systems that undergo periodic motion. Although these be a more convenient and compact representation of the systems have been mostly ignored so far in the field of causal structure of interest. By making assumptions on causal discovery, they have been studied extensively in the dynamical system and the interventions being made, the field of control theory. Some examples of systems that the SCM effectively allows us to reason about a ‘higher naturally exhibit oscillatory stationary states and where level’ qualitative description of the dynamics—in this our framework may be applicable are EEG signals, circa- case, the equilibrium states. dian signals, seasonal influences, chemical oscillations, electric circuits, aerospace vehicles, and satellite control. There are, however, two major limitations that stem from We refer the reader to (Bittanti and Colaneri, 2009) for the equilibrium assumption. First, for many dynamical more details on these application areas from the perspec- systems the assumption that the system settles to a unique tive of periodic control theory. equilibrium, either in its observational state or under inter- vention, may be a bad approximation of the actual system Since the DSCM derived for a simple harmonic oscilla- dynamics. Second, this framework is only capable of tor (see Example 4) is already quite complex, we leave modelling interventions in which a subset of variables are the task of deriving methods that estimate the parame- clamped to fixed values (constant interventions). Even for ters from data for future work. Rather, our current work rather simple physical systems such as a forced damped presents a first necessary theoretical step that needs to be simple harmonic oscillator, these assumptions are vio- done before applications of this theory can be developed, lated. enabling the development of data-driven causal discov- ery and prediction methods for oscillatory systems, and Motivated by these observations, the work presented in possibly even more general systems, down the road. this paper tries to answer the following questions: (i) Can the SCM framework be extended to model systems that The remainder of this paper is organised as follows. In do not converge to an equilibrium? (ii) If so, what assump- Section 2, we introduce notation to describe ODEs. In tions need to be made on the ODE and interventions so Section 3, we describe how to apply the notion of an inter- that this is possible? Since SCMs are used in a variety of vention on an ODE to the dynamic case. In Section 4, we situations in which the equilibrium assumption does not define regularity conditions on the asymptotic behaviour necessarily hold, we view these questions as important of an ODE under a set of interventions. In Section 5, in order to understand when they are indeed theoretically we present our main result: subject to conditions on the grounded as modelling tools. The main contribution of dynamical system and interventions being modelled, a Dy- this paper is to show that the answer to the first question namic SCM can be derived that allows one to reason about is ‘Yes’ and to provide sufficient conditions for the sec- how the asymptotic dynamics change under interventions ond. We do this by extending the SCM framework to on variables in the system. We conclude in Section 6. encompass time-dependent dynamics and interventions and studying how such objects can arise from ODEs. We 2 ORDINARY DIFFERENTIAL refer to this as a Dynamic SCM (DSCM) to distinguish EQUATIONS it from the static equilibrium case for the purpose of ex- position, but note that this is conceptually the same as Let I = f1; : : : ; Dg be a set of variable labels. Con- an SCM on a fundamental level. Our construction draws sider time-indexed variables X (t) 2 R for i 2 I , where i i inspiration from the approach of Mooij et al. (2013), that R  R and t 2 R = [0;1). For I  I , we write i 0 was recently generalized to also incorporate the stochas- X (t) 2 R for the tuple of variables (X (t)) . I i i i2I i2I tic setting (Bongers and Mooij, 2018). Here, we adapt By an ODED, we mean a collection of D coupled ordi- the approach by replacing the static equilibrium states by (k) nary differential equations with initial conditions X : continuous-time trajectories, considering two trajectories as equivalent if they do not differ asymptotically. (k) (k) f (X ;X )(t) = 0; X (0) = (X ) ; i i pa(i) i i 0 D : Note that whilst this paper applies a causal perspective to 0  k  n 1; i 2 I; the study of dynamical systems, the goal of this paper is where the ith differential equation determines the evo- not to derive a learning algorithm which can be applied lution of the variable X in terms of X , where to time series data. In this sense, we view our main re- i pa(i) pa(i)  I are the parents of i, and X itself, and where i (k) edge between X and X iff X is a direct cause of X n is the order of the highest derivative X of X that i i i j i j (in the context of all variables X ). In this section, we appears in equation i. Here, f is a functional that can i I will formalize this causal interpretation by studying inter- include time-derivatives of its arguments. We think of the ventions on the system. ith differential equation as modelling the causal mech- anism that determines the dynamics of the effect X in 3.1 TIME-DEPENDENT PERFECT terms of its direct causes X . pa(i) INTERVENTIONS One possible way to write down an ODE is to canonically decompose it into a collection of first order differential Usually in the causality literature, by a perfect interven- equations, such as is done in Mooij et al. (2013). We tion it is meant that a variable is clamped to take a spe- choose to present our ODEs as “one equation per vari- cific given value. The natural analogue of this in the able” rather than splitting up the equations due to com- time-dependent case is a perfect intervention that forces plications that would otherwise occur when considering a variable to take a particular trajectory. That is, given a time-dependent interventions (cf. Section 3.3). subset I  I and a function  : R ! R , we I 0 i i2I can intervene on the subset of variables X by forcing Example 1. Consider a one-dimensional system of D X (t) =  (t)8t 2 R . Using Pearl’s do-calculus nota- particles of mass m (i = 1; : : : ; D) with positions X I I 0 i i tion (Pearl, 2009) and for brevity omitting the t, we write coupled by springs with natural lengths l and spring do(X =  ) for this intervention. Such interventions I I constants k , where the ith spring connects the ith and are more general objects than those of the equilibrium or (i + 1)th masses and the outermost springs have fixed time-independent case, but in the specific case that we ends (see Figure 1a). Assume further that the ith mass restrict ourselves to constant trajectories the two notions undergoes linear damping with coefficient b . coincide. _  Denoting by X and X the first and second time deriva- i i tives of X respectively, the equation of motion for the ith i 3.2 SETS OF INTERVENTIONS variable is given by Recall that when modelling equilibrating dynamical sys- m X (t) =k [X (t) X (t) l ] i i i i+1 i i tems under constant interventions, the set of interven- tions modelled coincides with the asymptotic behaviour k [X (t) X (t) l ] b X (t) i1 i i1 i1 i i of the system. We will generalise this relation to non- equilibrating behaviour. where we take X = 0 and X = L to be the fixed posi- 0 D tions of the end springs. For the case that D = 2, we can The Dynamic SCMs that we will derive will describe the write the system of equations as: asymptotic dynamics of the ODE and how they change 8 under different interventions. If we want to model ‘all  _ 0 = m X (t) + b X (t) + (k + k )X (t) > 1 1 1 1 1 0 1 > possible interventions’, then the resulting asymptotic dy- k X (t) k l + k l ; > 1 2 0 0 1 1 > namics that can occur are arbitrarily complicated. The idea is to fix a simpler set of interventions and derive an  _ D : 0 = m X (t) + b X (t) + (k + k )X (t) 2 2 2 2 2 1 2 SCM that models only these interventions, resulting in k L k X (t) k l + k l ; > 2 1 1 2 1 2 2 > a model that is simpler than the original ODE but still > allows us to reason about interventions we are interested (k) (k) X (0) = (X ) k 2 f0; 1g; i 2 f1; 2g : i 0 in. In the examples in this paper, we restrict ourselves to periodic or quasi-periodic interventions, but the results hold for more general sets of interventions that satisfy the We can represent the functional dependence structure be- stability definitions presented later. tween variables implied by the functions f with a graph, in which variables are nodes and arrows point X ! X j i We need to define some notation to express the sets of (k) if j 2 pa(i). Self loops X ! X exist if X appears i i i interventions and the set of system responses to these in the expression of f for more than one value of k. This interventions that we will model. Since interventions is illustrated for the system described in Example 1 in correspond to forcing variables to take some trajectory, Figure 1b. we describe notation for defining sets of trajectories: For I  I , let Dyn be a set of trajectories in R . Let i2I 3 INTERVENTIONS ON ODES Dyn = [ Dyn (where P (I ) is the power set of I2P (I) I i.e., the set of all subsets of I ). Thus, an element We interpret ODEs as causal models. In particular, we  2 Dyn is a function R ! R , and Dyn con- I 0 i i2I consider the graph expressing the functional dependence sists of such functions for different I  I . The main structure to be the causal graph of the system, with an idea is that we want both the interventions and the system X = 0 X = L 0 3 X X 1 2 X X X X 1 2 1 2 k k k 0 1 2 (c)D (a) Mass-spring system (b)D do(X = ) 1 1 Figure 1: (a) The mass-spring system of Example 1 with D = 2; (b–c) graphs representing the causal structure of the mass-spring system for (b) the observational system, (c) after the intervention on variable X described in Example 2. As a result of the intervention, X is not causally influenced by any variable, while the causal mechanism of X 1 2 remains unchanged. responses to be elements of Dyn; in other words, the set sets of trajectories we are considering are for the purposes of possible system responses should be large enough to of constructing the Dynamic SCMs. Some examples of contain all interventions that we would like to model, and trivially modular sets of trajectories are: (i) all static (i.e., in addition, all responses of the system to those interven- time-independent) trajectories, corresponding to (Mooij tions. The reader might wonder why we do not simply et al., 2013); (ii) all continuously-differentiable trajecto- take the set of all possible trajectories, but that set would ries that differ asymptotically; (iii) all periodic motions. be so large that it would not be practical for modeling The latter is the running example in this paper. purposes. 3.3 DESCRIBING INTERVENTIONS ON ODEs Since our goal will be to derive a causal model that de- scribes the relations between components (variables) of We can realise a perfect intervention by replacing the the system, we will need the following definition in Sec- equations of the intervened variables with new equations tion 5. that fix them to take the specified trajectories: Definition 1. A set of trajectories Dyn is modular if, for D : do(X = ) I I any fi ; : : : ; i g = I  I , 8 1 n (k) (k) > f (X ;X )(t) = 0 ; X (0) = (X ) ; i i pa(i) i > i 0 0  k  n 1 ; i 2 I n I ; 2 Dyn ()  2 Dyn 8k 2 f1; : : : ; ng: I i X (t)  (t) = 0 ; i 2 I : i i This should be interpreted as saying that admitted tra- jectories of single variables can be combined arbitrarily This procedure is analogous to the notion of intervention into admitted trajectories of the whole system (and vice in an SCM. In reality, this corresponds to decoupling the versa, admitted system trajectories can be decomposed intervened variables from their usual causal mechanism into trajectores of individual variables), and in addition, by forcing them to take a particular value, while leaving that interventions on each variable can be made indepen- the non-intervened variables’ causal mechanisms unaf- dently and combined in any way. This is not to say fected. that all such interventions must be physically possible to Perfect interventions will not generally be realisable in implement in practice. Rather, this means that the mathe- the real world. In practice, an intervention on a variable matical model we derive should allow one to reason about would correspond to altering the differential equation all such interventions. Not all sets of trajectories Dyn are governing its evolution by adding extra forcing terms; modular; in the following sections we will assume that the perfect interventions could be realised by adding forcing terms that push the variable towards its target value at For example, one might want to parameterize the set of trajectories in order to learn the model from data. Without any each instant in time, and considering the limit as these restriction on the smoothness of the trajectories, the problem of forcing terms become infinitely strong so as to dominate estimating a trajectory from data becomes ill-posed. Secondly, the usual causal mechanism determining the evolution of since we would like to identify trajectories that are asymptot- the variable. ically identical in order to focus the modeling efforts on the asymptotic behaviour of the system, we will only put a single Example 2 (continued). Consider the mass-spring sys- trajectory into Dyn to represent all trajectories that are asymptot- tem described in Example 1. If we were to intervene on ically identical to that trajectory, but whose transient dynamics may differ. Note that in the intervened ODE, the initial conditions of This is related to notions that have been discussed in the the intervened variables do not need to be specified explicitly literature under various headings, for instance autonomy and as for the other variables, since they are implied by considering invariance (Pearl, 2009). t = 0. the system to force the mass X to undergo simple har- of Mooij et al. (2013) to allow for non-constant trajec- monic motion, we could express this as a change to the tories in Dyn , and coincide with them in the case that system of differential equations as: Dyn consists of all constant trajectories inR . Definition 2. The ODE D is dynamically stable with D : do(X (t)=l +A cos(!t)) 1 1 reference to Dyn if there exists a unique  2 Dyn 8 ; I I 0 = X (t) l A cos(!t) ; > 1 1 such that X (t) =  (t)8t is a solution to D and that > I ; for any initial condition, the solution X (t) !  (t) as > ;  _ 0 = m X (t) + b X (t) + (k + k )X (t) 2 2 2 2 2 1 2 t ! 1. k L k X (t) k l + k l ; > 2 1 1 2 1 2 2 We use a subscript ; to emphasise that  describes the > ; (k) (k) asymptotic dynamics ofD without any intervention. Ob- X (0) = (X ) k 2 f0; 1g: 2 0 serve that Dyn could consist of the single element in this case. The requirement that this hold for all initial This induces a change to the graphical description of the conditions can be relaxed to hold for all initial conditions causal relationships between the variables. We break any except on a set of measure zero, but that would mean that incoming arrows to any intervened variable, including self the proofs later on require some more technical details. loops, as the intervened variables are no longer causally For the purpose of exposition, we stick to this simpler influenced by any other variable in the system. See Figure case. 1c for the graph corresponding to the intervened ODE in Example 3. Consider a single mass on a spring that is Example 2. undergoing simple periodic forcing and is underdamped. Such a system could be expressed as a single (parent-less) 4 DYNAMIC STABILITY variable with ODE description: A crucial assumption of Mooij et al. (2013) was that the 8  _ mX (t) + bX (t) + k(X (t) l) > 1 1 1 systems considered were stable in the sense that they > = F cos(!t + ) ; would converge to unique stable equilibria (if necessary, D : also after performing a constant intervention). This made > (k) (k) X (0) = (X ) k 2 f0; 1g : them amenable to study by considering the t ! 1 limit 1 0 in which any complex but transient dynamical behaviour would have decayed. The SCMs derived would allow one The solution to this differential equation is to reason about the asymptotic equilibrium states of the X (t) = r(t) + l + A cos(!t +  ) (1) systems after interventions. Since we want to consider 1 non-constant asymptotic dynamics, this is not a notion of where r(t) decays exponentially quickly (and is dependent stability that is fit for our purposes. on the initial conditions) and A and  depend on the Instead, we define our stability with reference to a set of parameters of the equation of motion (but not on the trajectories. We will use Dyn for this purpose. Recall initial conditions). that elements of Dyn are trajectories for all variables Therefore such a system would be dynamically stable with in the system. To be totally explicit, we can think of an reference to (for example) element  2 Dyn as a function 0 0 Dyn = fl + A cos(!t +  ) : A 2 R;  2 [0; 2)g: : R ! R 0 I t 7! ( (t);  (t); : : : ;  (t)) 1 2 D Remark 1. We use a subscript  to emphasise that describes the asymptotic dynamics ofD after performing where  (t) 2 R is the state of the ith variable X at time i i i the intervention do(X =  ). Observe that Dyn could I I t. Note that Dyn is not a single fixed set, independent consist only of the single element  and the above of the situation we are considering. We can choose Dyn definition would be satisfied. But then the original ODE depending on the ODE D under consideration, and the wouldn’t be dynamically stable with reference to Dyn , interventions that we may wish to make on it. nor would other intervened versions of D. This motivates Informally, stability in this paper means that the asymp- the following definition, extending dynamic stability to totic dynamics of the dynamical system converge to a sets of intervened systems. unique element of Dyn , independent of initial condition. The convergence we refer to here is the usual asymptotic If Dyn is in some sense simple, we can simply char- I convergence of real-valued functions, i.e., for f : [0;1) ! R , acterise the asymptotic dynamics of the system under g : [0;1) ! R we have that f ! g iff for every  > 0 there study. The following definitions of stability extend those is a T 2 [0;1) such thatjf(t) g(t)j <  for all t 2 [T;1). Definition 3. Let Traj be a set of trajectories. We say in terms of its parents. We extend this to the case that that the pair (D; Traj) is dynamically stable with ref- our variables do not take fixed values but rather represent erence to Dyn if, for any  2 Traj , D is entire trajectories. I do(X = ) I I I dynamically stable with reference to Dyn . Definition 4. Let Dyn = Dyn be a modular set II Example 3 (continued). Suppose we are interested in of trajectories, where Dyn  R . A deterministic I I modelling the effect of changing the forcing term, either Dynamic Structural Causal Model (DSCM) on the time- in amplitude, phase or frequency. We introduce a second indexed variables X taking values in Dyn is a collection variable X to model the forcing term: of structural equations 0 = f (X ; X )(t) > 1 1 2 M : X = F (X ) i 2 I ; i i pa(i) >  _ = mX (t) + bX (t) + k(X (t) l) X (t) ; 1 1 1 2 > where pa(i)  I nfig and each F is a map Dyn ! Dyn that gives the trajectory of an 0 = f (X )(t) D : pa(i) i 2 2 > effect variable in terms of the trajectories of its direct = X (t) F cos(! t +  ) ; > 2 0 0 0 > causes. (k) (k) X (0) = (X ) ; k 2 f0; 1g : 1 0 The point of this paper is to show that, subject to restric- tions on D and Dyn, we can derive a DSCM that allows If we want to change the forcing term that we apply to the us to reason about the effect on the asymptotic dynamics mass, we can interpret this as performing an intervention of interventions using trajectories in Dyn. ‘Traditional’ on X . We could represent this using the notation we deterministic SCMs arise as a special case, where all have developed as trajectories are constant over time. In an ODE, the equations f determine the causal relation- Dyn = f (t) = F cos(!t +  ) : 2 2 2 f2g ship between the variable X (t) and its parents X (t) pa(i) F ; ! 2 R;  2 [0; 2)g: 2 2 at each instant in time. In contrast, we think of the function F of the DSCM as a causal mechanism that For any intervention  2 Dyn , the dynamics of X in 2 1 f2g determines the entire trajectory of X in terms of the D will be of the form (1). Therefore (D; Dyn ) do(X = ) 2 2 f2g trajectories of the variables X , integrating over the pa(i) will be dynamically stable with reference to instantaneous causal effects over all time. In the case that Dyn consists of constant trajectories (and thus the instan- Dyn = (t) = (l + F cos(!t +  ); F cos(!t +  )) 1 1 2 2 taneous causal effects are constant over time), a DSCM reduces to a traditional deterministic SCM. : F ; F ; ! 2 R;  ;  2 [0; 2) : 1 2 1 2 The rest of this section is laid out as follows. In Section 5.1 we define what it means to make an intervention in a The independence of initial conditions for Example 3 is DSCM. In Section 5.2 we show how, subject to certain illustrated in Figure 2. conditions, a DSCM can be derived from a pair (D; Dyn). Note that if (D; Traj) is dynamically stable with refer- The procedure for doing this relies on intervening on all ence to Dyn , and Dyn  Dyn is a larger set of trajec- I I I but one variable at a time. In Section 5.3, Theorem 2 tories that still satisfies the uniqueness condition in the states that the DSCM thus derived is capable of modelling definition of dynamic stability, then (D; Traj) is dynam- the effect of intervening on arbitrary subsets of variables, ically stable with reference to Dyn . I even though it was constructed by considering the case that we consider interventions on exactly D 1 variables. 5 DYNAMIC STRUCTURAL CAUSAL Theorem 3 and Corollary 1 in Section 5.4 prove that the MODELS notions of intervention in ODE and the derived DSCM coincide. Collectively, these theorems tell us that we can A deterministic SCMM is a collection of structural equa- derive a DSCM that allows us to reason about the effects tions, the ith of which defines the value of variable X of interventions on the asymptotic dynamics of the ODE. Proofs of these theorems are provided in Section A of the 5 0 Namely: 8 2 Traj; 9!  2 Dyn such that under I I Supplementary Material. D and for any initial condition, X (t) !  (t) as do(X = ) I I I I t ! 1. Assuming that (D; Traj) is dynamically stable with 5.1 INTERVENTIONS IN A DSCM reference to Dyn , a sufficient condition for this is that none of the elements in Dyn n Dyn are asymptotically equal to any of I I 0 0 Interventions in (D)SCMs are realized by replacing the the elements of Dyn . That is: 8 2 Dyn ; 8 2 Dyn nDyn , I I I I (t) 9  (t) as t ! 1 . structural equations of the intervened variables. Given 10 10 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 t t (a) (b) Figure 2: Simulations from the forced simple harmonic oscillator in Example 3 showing the evolution of X with different initial conditions for different forcing terms (interventions on X ). The parameters used were m = 1; k = 1; l = 2; F = 2; b = 0:1, with (a) ! = 3 and (b) ! = 2. Dynamic stability means that asymptotic dynamics are independent of initial conditions, and the purpose of the DSCM is to quantify how the asymptotic dynamics change under intervention. 2 Dyn for some I  I , the intervened DSCM It should be noted that (D; Dyn) being structurally dy- M can be written: namically stable is a strong assumption in general. If do(X = ) I I Dyn is too small, then it may be possible to find a larger 0 0 set Dyn  Dyn such that (D; Dyn ) is structurally dy- X = F (X ) i 2 I n I ; i i pa(i) M : do(X = ) I I namically stable. The procedure described in this section X =  i 2 I : i i describes how to derive a DSCM capable of modelling all The causal mechanisms determining the non-intervened interventions in Dyn , which can thus be used to model variables are unaffected, so their structural equations re- interventions in Dyn. main the same. The intervened variables are decoupled Henceforth, we use the notation I = I n fig for from their usual causal mechanisms and are forced to take brevity. Suppose that (D; Dyn) is structurally dynam- the specified trajectory. ically stable. We can derive structural equations F : Dyn ! Dyn to describe the asymptotic dynam- pa(i) i 5.2 DERIVING DSCMs FROM ODEs ics of children variables as functions of their parents as follows. Pick i 2 I . The variable X has parents X . In order to derive a DSCM from an ODE, we require the pa(i) Since Dyn is modular, for any configuration of parent dy- following consistency property between the asymptotic namics  2 Dyn there exists  2 Dyn such dynamics of the ODE and the set of interventions. pa(i) pa(i) i I that ( ) =  . i pa(i) pa(i) Definition 5 (Structural dynamic stability). Let Dyn be By structural dynamic stability, the system D modular. The pair (D; Dyn) is structurally dynamically do(X = ) I I i i has asymptotic dynamics specified by a unique element stable if (D; Dyn ) is dynamically stable with refer- Infig 2 Dyn , which in turn defines a unique element  2 ence to Dyn for all i. i Dyn specifying the asymptotic dynamics of variable X This means that for any intervention trajectory since Dyn is modular. 2 Dyn , the asymptotic dynamics of the inter- Infig Infig Theorem 1. Suppose that (D; Dyn) is structurally dynam- vened ODED are expressible uniquely do(X = ) Infig Infig ically stable. Then the functions as an element of Dyn . Since Dyn is modular, the asymp- totic dynamics of the non-intervened variable can be re- F : Dyn ! Dyn :  7! i pa(i) i pa(i) i alised as the trajectory  2 Dyn , and thus Dyn is rich enough to allow us to make an intervention which forces constructed as above are well-defined. the non-intervened variable to take this trajectory. This is a crucial property that allows the construction of the struc- Given the structurally dynamically stable pair (D; Dyn) tural equations. In the particular case that Dyn consists we define the derived DSCM of all constant trajectories, structural dynamic stability means that after any intervention on all-but-one-variable, M : X = F (X ) i 2 I ; i i D pa(i) the non-intervened variable settles to a unique equilib- rium. In the language of Mooij et al. (2013), this would For example, if Dyn is not modular or represents interven- imply that the ODE is structurally stable. tions on only a subset of the variables. X where the F : Dyn ! Dyn are defined as above. coincide, and hence that DSCMs provide a representation pa(i) i Note that structural dynamic stability was a crucial prop- to reason about the asymptotic behaviour of the ODE un- erty that ensured F (Dyn )  Dyn . If (D; Dyn) is not der interventions in Dyn. A consequence of these results pa(i) i structurally dynamically stable, we cannot build structural is that the diagram in Figure 3 commutes. equations in this way. Theorem 3. Suppose that (D; Dyn) is structurally dy- namically stable. Let I  I and let  2 Dyn . Then We provide next an example of a DSCM for the mass- I M = (M ) . (D ) D do(X = ) spring system of Example 1 with D = 2. The derivation I I do(X = ) I I of this for the general case of arbitrarily many masses is Corollary 1. Suppose additionally that J  I n I and included in the Supplementary Material. let  2 Dyn . Then Example 4. Consider the systemD governed by the dif- M = (M ) : (D ) D do(X = ;X = ) ferential equation of Example 1 with D = 2. Let Dyn do(X = ) I I J J f1;2g I I do(X = ) J J be the modular set of trajectories with To summarise, Theorems 1–3 and Corollary 1 collectively j j j state that if (D; Dyn) is dynamically structurally stable Dyn = A cos(! t +  ) : fig i i i then it is possible to derive a DSCM that allows us to j=1 reason about the asymptotic dynamics of the ODE under j j j j any possible intervention in Dyn. w ;  ; A 2 R; jA j < 1 i i i i j=1 5.5 RELATION TO ODEs AND DYNAMIC BAYESIAN NETWORKS for i = 1; 2, where for each i it holds that jA j < j=1 i 1 (so that the series is absolutely convergent). Then An ODE is capable of modelling arbitrary interventions (D; Dyn ) is structurally dynamically stable and ad- f1;2g on the system it describes. At the cost of only modelling mits the following DSCM. a restricted set of interventions, a DSCM can be derived which describes the asymptotic behaviour of the system X = F (X ) 1 1 2 M : under these interventions. This may be desirable in cases X = F (X ) 2 2 1 for which transient behaviour is not important. j j j 2 2 where, writing C = [k + k m (! ) ] and C = 1 2 1 1 2 2 We now compare DSCMs to Dynamic Bayesian Net- 2 2 [k + k m (! ) ] , the functionals F and F are 1 2 2 1 2 works (DBNs), an existing popular method for causal given by Equations 2 and 3 overleaf. modelling of dynamical systems (Koller and Friedman, 2009). DBNs are essentially Markov chains, and thus are appropriate for discrete-time systems. When the discrete- 5.3 SOLUTIONS OF A DSCM time Markov assumption holds, DBNs are a powerful tool Theorem 1 states that we can construct a DSCM by the capable of modelling arbitrary interventions. However, described procedure. We constructed each equation by approximations must be made whenever these assump- intervening on D 1 variables at a time. The result of tions do not hold. In particular, a continuous system must this section states that the DSCM can be used to cor- be approximately discretised in order to be modelled by a rectly model interventions on arbitrary subsets of vari- DBN (Sokol and Hansen, 2014). ables. We say that  2 Dyn is a solution of M if = F ( )8i 2 I . By using the Euler method for numerically solving ODEs, i i pa(i) we can make such an approximation to derive a DBN de- Theorem 2. Suppose that (D; Dyn) is structurally dy- scribing the system in Example 1, leading to the discrete namically stable. Let I  I , and let  2 Dyn . Then time equation given in (8) the Supplementary Material. D is dynamically stable if and only if the inter- do(X = ) I I For DBNs, the main choice to be made is how fine the vened SCMM has a unique solution. If there (D ) do(X = ) I I temporal discretisation should be. The smaller the value is a unique solution, it coincides with the element of Dyn of , the better the discrete approximation will be. Even describing the asymptotic dynamics ofD . do(X = ) I I if there is a natural time-scale on which measurements Remark 2. We could also take I = ;, in which case the can be made, choosing a finer discretisation than this will above theorem applies to justD. provide a better approximation to the behaviour of the true system. The choice of  should reflect the natural 5.4 CAUSAL REASONING IS PRESERVED timescales of the interventions to be considered too; for We have defined ways to model interventions in both example, it is not clear how one would model the interven- 2t ODEs and DSCMs. The following theorem and its imme- tion do X (t) = cos with a discretisation length diate corollary proves that these notions of intervention . Another notable disadvantage of DBNs is that the Sec. 3.3 Sec. 3.3 Intervened ODE Intervened ODE ODE D D D do(X = ) do(X = ;X = ) I I I I J J Sec. 5.2 Sec. 5.2 Sec. 5.2 Sec. 5.1 Sec. 5.1 Intervened DSCM Intervened DSCM DSCM M M D D D do(X = ) do(X = ;X = ) I I I I J J Figure 3: Top-to-bottom arrows: Theorems 1 and 2 together state that if (D; Dyn) is structurally dynamically stable then we can construct a DSCM to describe the asymptotic behaviour of D under different interventions in the set Dyn. Left-to-right arrows: Both ODEs and DSCMs are equipped with notions of intervention. Theorem 3 and Corollary 1 say that these two notions of intervention coincide, and thus the diagram commutes. 0 1 " #! 1 1 j j X X k l k A b ! j j j 1 1 1 j j 1 2 2 @ A F A cos(! t +  ) = + q cos ! t +  arctan 2 2 2 2 2 j (2) k + k j j C 1 0 2 j=1 j=1 1 C + b m (! ) 1 1 1 2 0 1 " #! 1 1 j j X X k l k l k L k A b ! 1 1 2 2 2 1 2 j j j 1 j j 1 @ A F A cos(! t +  ) = + + q cos ! t +  arctan (3) 1 1 1 1 1 k + k k + k j j 1 2 2 3 C j=1 j=1 C + b m (! ) 2 2 2 1 Figure 4: Equations giving the structural equations for the DSCM describing the mass-spring system of Example 4 computational cost of learning and inference increases for how they change under intervention. smaller , where computational cost becomes infinitely We identify three possible directions in which to extend large in the limit  ! 0. this work in the future. The first is to properly understand In contrast, the starting point for DSCMs is to fix a conve- how learning DSCMs from data could be performed. This nient set of interventions we are interested in modelling. is important if DSCMs are to be used in practical applica- If a DSCM containing these interventions exists, it will tions. Challenges to be addressed include finding practical model the asymptotic behaviour of the system under each parameterizations of DSCMs, the presence of measure- of these interventions exactly, rather than approximately ment noise in the data and the fact that time-series data are modelling the transient and asymptotic behaviour as in usually sampled at a finite number of points in time. The the case of a DBN. Computational cost does not relate second is to relax the assumption that the asymptotic dy- inversely to accuracy as for DBNs, but depends on the namics are independent of initial conditions, as was done chosen representation of the set of admitted interventions. recently for the static equilibrium scenario by Blom and Mooij (2018). The third extension is to move away from deterministic systems and consider Random Differential 6 DISCUSSION AND FUTURE WORK Equations (Bongers and Mooij, 2018), thereby allowing The main contribution of this paper is to show that the to take into account model uncertainty, but also to include SCM framework can be applied to reason about time- systems that may be inherently stochastic. dependent interventions on an ODE in a dynamic setting. ACKNOWLEDGEMENTS In particular, we showed that if an ODE is sufficiently well-behaved under a set of interventions, a DSCM can Stephan Bongers was supported by NWO, the Nether- be derived that captures how the asymptotic dynamics lands Organization for Scientific Research (VIDI grant change under these interventions. This is in contrast to 639.072.410). This project has received funding from previous approaches to connecting the language of ODEs the European Research Council (ERC) under the Euro- with the SCM framework, which used SCMs to describe pean Union’s Horizon 2020 research and innovation pro- the stable (constant-in-time) equilibria of the ODE and gramme (grant agreement n 639466). References stochastic case. arXiv.org preprint, arXiv:1803.08784 [cs.AI], March 2018. URL https://arxiv.org/ Judea Pearl. Causality: Models, Reasoning, and Infer- abs/1803.08784. ence. Cambridge University Press, New York, NY, 2nd Tineke Blom and Joris M. Mooij. Generalized edition, 2009. structural causal models. arXiv.org preprint, Kenneth A. Bollen. Structural equations with latent vari- https://arxiv.org/abs/1805.06539 [cs.AI], May ables. John Wiley & Sons, 2014. 2018. URL https://arxiv.org/abs/1805. Peter Spirtes. Directed cyclic graphical representations of feedback models. In Proceedings of the Eleventh con- Clive W.J. Granger. 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To see that this I I i i i pa(i) pa(i) pa(i) I I i i i is the case, observe that the system of equations forD is given by: do(X = ) I I i i X (t) =  (t) j 2 I n (pa(i)[fig) ; < j j X (t) =  (t) j 2 pa(i) ; D : j j do(X = ) I I i i (k) (k) f (X ;X )(t) = 0 X (0) = (X ) ; 0  k  n 1 : i i pa(i) i i i 0 The equations for D are similar, except with X (t) =  (t) for j 2 I n (pa(i)[fig). In both cases, the do(X = ) j I j i I equations for all variables except X are solved already. The equation for X in both cases reduces to the same quantity i i by substituting in the values of the parents, namely f (X ;  )(t) = 0 : i i pa(i) The solution to this equation in Dyn must be unique and independent of initial conditions, else the dynamic stability of the intervened systemsD and D would not hold, contradicting the dynamic structural stability of do(X = ) do(X = ) I I I i i i I (D; Dyn). It follows that  =  . A.2 PROOF OF THEOREM 2 Proof. By construction of the SCM,  2 Dyn is a solution ofM if and only if the following two conditions (D ) I do(X = ) I I hold: for i 2 I n I , X (t) =  (t) 8t is a solution to the differential equation f (X ;  )(t) = 0; i i i i pa(i) for i 2 I ,  (t) =  (t) for all t. i i which is true if and only if X =  is a solution to D in Dyn . Thus, by definition of dynamic stability, do(X = ) I I I D is dynamically stable with asymptotic dynamics describable by  2 Dyn if and only if X =  uniquely do(X = ) I I solves M . (D ) do(X = ) I I A.3 PROOF OF THEOREM 3 Proof. We need to show that the structural equations of M and (M ) are equal. Observe that (D ) D do(X = ) I I do(X = ) I I the equations forD are given by: do(X = ) I I X =  ; i 2 I ; i i D : do(X = ) (k) (k) I I f (X ;X ) = 0; X (0) = (X ) ; 0  k  n 1; i 2 I n I : i i i i pa(i) 0 Therefore, when we perform the procedure to derive the structural equations forD , we see that: do(X = ) I I if i 2 I , the ith structural equation will simply be X =  since intervening on I does not affect variable X . i i i i if i 2 I n I , the ith structural equation will be the same as for M , since the dependence of X on the other D i variables is unchanged. Hence the structural equations forM are given by: (D ) do(X = ) I I X =  ; i 2 I ; i i M : (D ) do(X = ) I I X = F (X ); i 2 I n I : i i pa(i) and thereforeM = (M ) . (D ) D do(X = ) I I do(X = ) I I A.4 PROOF OF COROLLARY 1 Proof. Corollary 1 follows very simply from the observation that if (D; Dyn) is structurally dynamically stable then so is (D ; Dyn ). The result then follows by application of Theorem 3. do(X = ) I I InI B DERIVING THE DSCM FOR THE MASS-SPRING SYSTEM Consider the mass-spring system of Example 1, but with D  1 an arbitrary integer. We repeat the setup: We have D masses attached together on springs. The location of the ith mass at time t is X (t), and its mass is m . i i For notational ease, we denote by X = 0 and X = L the locations of where the ends of the springs attached to 0 D+1 the edge masses meet the walls to which they are affixed. X and X are constant. The natural length and spring 0 D+1 constant of the spring connecting masses i and i + 1 are l and k respectively. The ith mass undergoes linear damping i i with coefficient b , where b is small to ensure that the system is underdamped. The equation of motion for the ith mass i i (1  i  D) is given by:  _ m X (t) = k [X (t) X (t) l ] k [X (t) X (t) l ] b X (t) i i i i+1 i i i1 i i1 i1 i i so, defining  _ f (X ; X ; X )(t) = m X (t) k [X (t) X (t) l ] + k [X (t) X (t) l ] + b X (t) i i i1 i+1 i i i i+1 i i i1 i i1 i1 i i we can write the system of equationsD for our mass-spring system as D : f (X ; X ; X )(t) = 0 i 2 I : i i i1 i+1 In the rest of this section we will explicitly calculate the structural equations for the DSCM derived from D with two different sets of interventions. First, we will derive the structural equations for the case that Dyn consists of all constant trajectories, corresponding to constant interventions that fix variables to constant values for all time. This illustrates the correspondence between the theory in this paper and that of Mooij et al. (2013). Next, we will derive the structural equations for the case that Dyn consists of interventions corresponding to sums of periodic forcing terms. B.1 MASS-SPRING WITH CONSTANT INTERVENTIONS In order to derive the structural equations we only need to consider, for each variable, the influence of its parents on it. (Formally, this is because of Theorem 1). Consider variable i. If we intervene to fix its parents to have locations X (t) =  and X (t) =  for all t, then the equation of motion for variable i is given by i1 i1 i+1 i+1  _ m X (t) + b X (t) + (k + k )X (t) = k [ l ] + k [ + l ] : i i i i i i1 i i i+1 i i1 i1 i1 There may be some complicated transient dynamics that depend on the initial conditions X (0) and X (0) but provided i i that b > 0, we know that the X (t) will converge to a constant and therefore the asymptotic solution to this equation i i  _ can be found by setting X and X to zero. Note that in general, we could explicitly find the solution to this differential i i equation (and indeed, in the next example we will) but for now there is a shortcut to deriving the structural equations. The asymptotic solution is: k [ l ] + k [ + l ] i i+1 i i1 i1 i1 X = : k + k i i1 Therefore the ith structural equation is: k [X l ] + k [X + l ] i i+1 i i1 i1 i1 F (X ; X ) = : i i1 i+1 k + k i i1 This is analogous to the approach taken in Mooij et al. (2013) in which the authors first define the Labelled Equilibrium Equations and from these derive the SCM. Hence the SCM for (D; Dyn ) is: k [X l ] + k [X + l ] i i+1 i i1 i1 i1 M : X = i 2 I : D i k + k i i1 We can thus use this model to reason about the effect of constant interventions on the asymptotic equilibrium states of the system. B.2 SUMS OF PERIODIC INTERVENTIONS Suppose now we want to be able to make interventions of the form: do X (t) = A cos(!t + ) : (4) Such interventions cannot be described by the DSCM derived in Section B.1. In this section we will explicitly derive a DSCM capable of reasoning about the effects of such interventions. It will also illustrate why we need dynamic structural stability. By Theorem 1, to derive the structural equation for each variable we only need to consider the effect on the child of intervening on the parents according to interventions of the form (4). Consider the following linear differential equation:  _ mX (t) + bX (t) + kX (t) = g(t) : (5) In general, the solution to this equation will consist of two parts—the homogeneous solution and the particular solution. The homogeneous solution is one of a family of solutions to the equation  _ mX (t) + bX (t) + kX (t) = 0 (6) and this family of solutions is parametrised by the initial conditions. If b > 0 then all of the homogeneous solutions decay to zero as t ! 1. The particular solution is any solution to the original equation with arbitrary initial conditions. The particular solution captures the asymptotic dynamics due to the forcing term g. Equation 5 is a linear differential equation. This means that if X = X is a particular solution for g = g and X = X is a particular solution for g = g , 1 1 2 2 then X = X + X is a particular solution for g = g + g . 1 2 1 2 In order to derive the structural equations, the final ingredient we need is an explicit representation for a particular solution to (5) in the case that g(t) = A cos(!t +). We state the solution for the case that the system is underdamped— this is a standard result and can be verified by checking that the following satisfies (5): 0 0 X (t) = A cos(!t +  ) where A b! 0 0 A = p ;  =  arctan : (7) 2 2 2 k m! [k m! ] + bm! Therefore if we go back to our original equation of motion for variable X  _ m X (t) + b X (t) + (k + k )X (t) = k [X (t) l ] + k [X (t) + l ] i i i i i i1 i i i+1 i i1 i1 i1 and perform the intervention do(X (t) = A cos(! t +  ); X (t) = A cos(! t +  )) i1 i1 i1 i1 i+1 i+1 i+1 i+1 we see that we can write the RHS of the above equation as the sum of the three terms g (t) = k l k l ; 1 i1 i1 i i g (t) = k A cos(! t +  ) ; 2 i1 i1 i1 i1 g (t) = k A cos(! t +  ) : 3 i i+1 i+1 i+1 Using the fact that linear differential equation have superposable solutions and (7), we can write down the resulting asymptotic dynamics of X : k l k l i1 i1 i i X (t) = k + k i i1 k A b ! i1 i1 i i1 + cos ! t +  arctan i1 i1 k + k m ! 2 2 2 i i1 i i1 [k + k m ! ] + b m ! i i1 i i i i1 i1 k A b ! i i+1 i i+1 +q cos ! t +  arctan : i+1 i+1 k + k m ! 2 2 2 i i1 i i+1 [k + k m ! ] + b m ! i i1 i i i i+1 i+1 However, note that if we were using Dyn consisting of interventions of the form of equation (4), then we have just shown that the mass-spring system would not be structurally dynamically stable with respect to this Dyn, since we need two periodic terms and a constant term to describe the motion of a child under legal interventions of the parents. This illustrates the fact that we may sometimes be only interested in a particular set of interventions that may not itself satisfy structural dynamic stability, and that in this case we must consider a larger set of interventions that does. In this case, we can consider the modular set of trajectories generated by trajectories of the following form for each variable: j j j X (t) = A cos(! t +  ) i i i j=1 where for each i it holds that jA j < 1 (so that the series is absolutely convergent and thus does not depend on j=1 i the ordering of the terms in the sum). Call this set Dyn (“quasi-periodic”). By equation (7), we can write down the qp structural equations 0 1 1 1 X X j j j j j j @ A F A cos(! t +  ); A cos(! t +  ) i1 i1 i1 i+1 i+1 i+1 j=1 j=1 k l k l i1 i1 i i k + k i i1 " #! 1 j j k A b ! i1 i i1 j j i1 + q cos ! t +  arctan i1 i1 j j 2 2 2 k + k m (! ) i i1 i i1 j=1 [k + k m (! ) ] + b m (! ) i i1 i i i i1 i1 " #! 1 j j k A b ! i i i+1 j j i+1 + q cos ! t +  arctan : i+1 i+1 j j k + k m (! ) 2 2 2 i i+1 i i+1 j=1 [k + k m (! ) ] + b m (! ) i i+1 i i i i+1 i+1 Since this is also a member of Dyn , the mass-spring system is dynamically structurally stable with respect to Dyn qp qp and so the equations F define the Dynamic Structural Causal Model for asymptotic dynamics. C DYNAMIC BAYESIAN NETWORK REPRESENTATION By using Euler’s method, we can obtain a (deterministic) Dynamic Bayesian Network representation of the mass-spring system. For D = 2, this yields (t+1) > X = X (t) + X (t) 1 1 > 1 h i (t+1) _ _ _ X = X (t) + k X (t) b X (t) (k + k )X (t) + k l k l > 1 1 1 2 1 1 0 1 1 0 0 1 1 > 1 (t+1) DBN : (8) X = X (t) + X (t) 2 2 h i (t+1) _ _ _ X = X (t) + k X (t) b X (t) (k + k )X (t) + k l k l + k L > 2 2 1 1 2 2 1 2 2 1 1 2 2 2 > 2 (k) (k) X (0) = (X ) k 2 f0; 1g; i 2 f1; 2g : i 0

Journal

Computing Research Repository – arXiv (Cornell University)

Published: Aug 29, 2016

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