SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Shanyun Gao
Purdue University gao565@purdue.edu
Raghavendra Addanki
Adobe Research raddanki@adobe.com
Tong Yu
Adobe Research tyu@adobe.com
Ryan A. Rossi
Adobe Research  
ryrossi@adobe.com
Discovering causal relations from observational time series without making the stationary assumption is a significant challenge. In practice, this challenge is common in many areas, such as retail sales, transportation systems, and medical science. Here, we consider this problem for a class of non-stationary time series. The structural causal model (SCM) of this type of time series, called the semistationary time series, exhibits that a finite number of different causal mechanisms occur sequentially and periodically across time. This model holds considerable practical utility because it can represent periodicity, including common occurrences such as seasonality and diurnal variation. We propose a constraint-based, non-parametric algorithm for discovering causal relations in this setting. The resulting algorithm,  $\mathrm{PCMCl}_{\Omega}$ , can capture the alternating and recurring changes in the causal mechanisms and then identify the underlying causal graph with conditional independence (CI) tests. We show that this algorithm is sound in identifying causal relations on discrete-valued time series. We validate the algorithm with extensive experiments on continuous and discrete simulated data. We also apply our algorithm to a real-world climate dataset.
In modern sciences, causal discovery aims to identify the collection of causal relations from observational data, as in [Pearl][1980], [Peters et al.][2017] and [Spirtes et al.][2000]. One of the most popular causal discovery approaches is the so-called constraint-based method. Constraint-based approaches assume that the probability distribution of variables is causal Markov and faithful to a directed acyclic graph called the causal graph. Given large enough data, they can then recover the corresponding Markov equivalence class by exploiting conditional independence relationships of the variables. See [Peters et al.][2017]. There are many constraint-based algorithms such as PC and FCI algorithms [Spirtes et al.][2000]. The standard assumption of these approaches is that data samples are independent and identically distributed, which makes it possible to perform CI tests. See [Bergsma][2004], [Zhang et al.][2012] and [Shah and Peters][2020].
Recently, there have been numerous efforts to extend such constraint-based algorithms to accommodate time series data. For instance, PCMCI in Runge et al. [2019] and LPCMCI in Gerhardus and Runge [2020] are the PC-based algorithms for time series. Inspired by FCI algorithms, approaches designed for time series include ANLSTM in Chu and Glymour [2008], tsFCI in Entner and Hoyer [2010] and SVAR-FCI in Malinsky and Spirtes [2018]. This setup is relevant in several industrial applications since many data points have an associated time-point, such as root-cause analysis in Ikram et al. [2022]. Most of the existing causal discovery algorithms make the stationary assumption.
See Chu and Glymour [2008], Hyvarinen et al. [2010], Entner and Hoyer [2010], Peters et al. [2013], Malinsky and Spirtes [2018], Runge et al. [2019], Pamfil et al. [2020] and Assaad et al. [2022].
Non-stationary temporal data makes causal discovery more challenging since the statistics are time-variant, and it is unreasonable to expect that the underlying causal structure is time-invariant. Identifying causal relations from non-stationary time series without imposing any restriction on the data is difficult. Here, we focus on a specific class of non-stationary time series, called the semistationary time series, whose structural causal model (SCM) exhibits that a finite number of different causal mechanisms occur sequentially and periodically across time. One example is illustrated in Fig.1 where the time series  $\mathbf{X}^1$  has three different causal mechanisms across time, shown as red edges, green edges, and blue edges, respectively. Similarly, time series  $\mathbf{X}^2$  has two alternative causal mechanisms. This setting holds considerable practical utility. Periodic nature is commonly observed in many real-world time series data. See Han et al. [2002], Nakamura et al. [2003], Carskadon et al. [2005] and Komarzynski et al. [2018]. Here are a few additional intuitive examples: poor traffic conditions often coincide with commute time and weekends; household electricity consumption typically follows a pattern of being higher at night and lower during the daytime. Consequently, it is reasonable to expect periodic changes in the causal relations underlying this type of time series without assuming stationarity. Here, the constraint-based methods in Chu and Glymour [2008], Entner and Hoyer [2010], Malinsky and Spirtes [2018] and Runge et al. [2019], designed for stationary time series, may fail. Given observational data with periodically changing causal structures, it is hard to apply CI tests directly. Most of the other algorithms designed for non-stationary time series rely heavily on model assumptions, as in Gong et al. [2015], Pamfil et al. [2020], and Huang et al. [2019]. These algorithms are discussed further in the related work.
In this paper, we propose an algorithm to address this problem, namely non-parametric causal discovery in time series data with semi-stationary SCMs. The key contributions of our work are:
- We develop an algorithm to discover the causal structure from semi-stationary time series data where the underlying causal structures change periodically. Our algorithm systematically uses the PCMCI algorithm proposed for the stationary setting in [Runge et al. 2019]. The resulting algorithm is hence named  $\mathrm{PCMCI}_{\Omega}$  where  $\Omega$  denotes periodicity.
- We validate our method with synthetic simulations on both continuous-valued and discrete-valued time series, showing that our method can correctly learn the periodicity and causal mechanism of the synthetic time series.
- We utilize our method in a real-world climate application. The result reveals the potential existence of periodicity in those time series, and the stationary assumption made by previous works could be relaxed in some practical situations.
PCMCI has been applied in diverse domains to investigate atmospheric interactions in the biosphere, as demonstrated in [Krich et al. 2020], global wildfires as explored in [Qu et al. 2021], water usage as studied in [Zou et al. 2022], ultra-processed food manufacturing as examined in [Menegozzo et al. 2020], and causal feature selection as discussed in [Peterson 2022], among other applications. See [Arvind et al. 2021], [Gerhardus and Runge 2020], [Castri et al. 2023a,b]. While PCMCI has achieved considerable success, it is not without its limitations. One notable assumption that can be challenged is the concept of causal stationary, that is, causal relations are time-invariant. PCMCI exhibits robustness when applied to linear models with an added non-stationary trend. See also [Runge et al. 2019]. However, there is an ongoing exploration to enhance its performance in a wider range of non-stationary settings.
Although not as extensively as the stationary case, causal discovery in non-stationary time series has been studied by some authors. However, many of those algorithms rely on parametric assumptions such as the vector autoregressive model in Gong et al. [2015] and Malinsky and Spirtes [2019]; linear and nonlinear state-space model in Huang et al. [2019]. One non-parametric algorithm in the literature is CD-NOD proposed by Huang et al. [2020], which has been extended to recover time-varying instantaneous and lagged causal relationships. In very recent work, Fujiwara et al. [2023] proposed an algorithm JIT-LiNGAM to obtain a local approximated linear causal model combining algorithm LiNGAM and JIT framework for non-linear and non-stationary data. To the best of our knowledge, no other non-parametric approaches can discover causal relations underlying time series without
Figure 1: Partial causal graph for 3-variate time series  $V = \{\mathbf{X}^1, \mathbf{X}^2, \mathbf{X}^3\}$  with a Semi-Stationary SCM where  $\tau_{\mathrm{max}} = 3$ ,  $\omega_1 = 3$ ,  $\omega_2 = 2$ ,  $\omega_3 = 1$ ,  $\Omega = 6$  and  $\delta = 6$ . The first  $3(= \tau_{\mathrm{max}})$  time slices  $\{\mathbf{X}_t\}_{1 \leq t \leq 3}$  are the starting points. The same color edges represent the same causal mechanism. E.g. for  $\mathbf{X}^1$ : there are 3  $(= \omega_1)$  time partition subsets  $\{\Pi_k^1\}_{1 \leq k \leq 3}$ . The time points  $t$  of nodes  $X_t^1$  sharing the same filling color are in the same time partition subsets. The time points  $t$  of nodes  $X_t^1$  sharing both the same filling color and the same outline shape are in the same homogenous time partition subsets (the definitions are in the supplementary material). There are 6  $(= \delta)$  different Markov chains in this multivariate time series  $V$ , and the first element of these 6 Markov chains is shown as  $\{Z_1^q\}_{1 \leq q \leq 6}$  and are tinted with a gradient of blue hues. The superscript  $q$  of  $Z_i^q$  is the index of different Markov chains, whereas the subscript  $i$  denotes the running index of that specific Markov chain. For instance,  $Z_1^1$  and  $Z_2^1$  denote the first two elements of the first Markov chain, while  $Z_1^2$  and  $Z_2^2$  denote the first two elements of the second Markov chain.
assuming stationarity and can also allow for sudden changes in causal mechanisms. Our proposed approach does not directly enforce the stationary assumption on the time series. The SCM also integrates a finite set of causal mechanisms that exhibit periodic variations over time.
2  $\mathbf{PCMCI}_{\Omega}$ : Capturing Periodicity of the Causal Structure
In this section, we formulate the problem of learning the causal graph on multi-variate time series data when the SCM exhibits periodicity in causal mechanisms. In section 2.1, we present the necessary definitions and provide an overview of the problem setting. In section 2.2, we introduce the required assumption.
Let  $\mathcal{G}(V,E)$  denote the underlying causal graph, and for each variable  $X\in V$  , we denote the set of all incoming neighbors as parents, denoted by  $\operatorname {Pa}(X)$
For any two variables  $X, Y \in V$  and  $S \subset V$ , we denote the CI relation  $X$  is independent of  $Y$  conditioned on  $S$ , by  $X \perp Y \mid S$ .
For simplicity's sake, define sets:  $[b] := \{1, 2, \dots, b\}$  and  $[a, b] := \{a, a + 1, \dots, b\}$ , where  $a, b \in \mathbb{N}$ .
In the time series setting, let  $X_{t}^{j}\in \mathbb{R}^{1}$  denote the variable of  $j$ th time series at time  $t$ ,  $\mathbf{X}^j = \{X_t^j\}_{t\in [T]}\in \mathbb{R}^T$  denote a univariate time series and  $\mathbf{X}_t = \{X_t^j\}_{j\in [n]}\in \mathbb{R}^n$  denote a slice of all variables at time point  $t$ .  $V = \{\mathbf{X}^j\}_{j\in [n]} = \{\mathbf{X}_t\}_{t\in [T]}\in \mathbb{R}^{n\times T}$  denotes a  $n$ -variate time series. By default, we assume  $n > 1$  and hence  $\mathbf{X}^j\subsetneq V$ , and  $p(V)\neq 0$ , where  $p(.)$  denotes the probability or probability density.
Definition 2.1 (Non-Stationary SCM). A Non-Stationary Structural Causal Model (SCM) is a tuple  $\mathcal{M} = \langle V, \mathcal{F}, \mathcal{E}, \mathbb{P} \rangle$  where there exists a  $\tau_{\max} \in \mathbb{N}^{+}$ , defined as:
$$
\tau_ {\max } := \arg \max  _ {\tau} \left\{\tau : X _ {t - \tau} ^ {i} \in \operatorname {P a} \left(X _ {t} ^ {j}\right), i, j \in [ n ] \right\}, \tag {1}
$$
such that with this  $\tau_{\mathrm{max}}$ , each variable  $X_{t > \tau_{\mathrm{max}}}^j \in V$  is a deterministic function of its parent set  $\mathrm{Pa}(X_{t > \tau_{\mathrm{max}}}^j) \in V$  and an unobserved (exogenous) variable  $\epsilon_{t > \tau_{\mathrm{max}}}^j \in \mathcal{E}$ :
$$
X _ {t} ^ {j} = f _ {j, t} \left(\operatorname {P a} \left(X _ {t} ^ {j}\right), \epsilon_ {t} ^ {j}\right), j \in [ n ], t \in \left[ \tau_ {\max } + 1, T \right], \tag {2}
$$
and there exist at least two different time points  $t_0, t_1 \in [\tau_{\max} + 1, T]$  satisfying
$$
f _ {j, t _ {0}} \neq f _ {j, t _ {1}}, \exists j \in [ n ], \left\{t _ {0}, t _ {1} \right\} \subset \left[ \tau_ {\max } + 1, T \right]. \tag {3}
$$
where  $f_{j,t}, f_{j,t_0}, f_{j,t_1} \in \mathcal{F}$  and  $\{\epsilon_t^j\}_{t \in [T]}$  are jointly independent with probability measure  $\mathbb{P}$ .  $\tau_{\max}$  is the finite maximal lag in terms of the causal graph  $\mathcal{G}$ .
Definition 2.2 (Semi-Stationary SCM). A Semi-Stationary SCM is a Non-Stationary SCM that additionally satisfies the following conditions. For each  $j \in [n]$ , there exists an  $\omega \in \mathbb{N}^{+}$  such that:
$$
a) f _ {j, t} = f _ {j, t + N \omega}, \tag {4}
$$
$$
b) \operatorname {P a} \left(X _ {t + N \omega} ^ {j}\right) = \left\{X _ {s + N \omega} ^ {i}: X _ {s} ^ {i} \in \operatorname {P a} \left(X _ {t} ^ {j}\right), i \in [ n ] \right\}, \tag {5}
$$
$$
c) \epsilon_ {t} ^ {j}, \epsilon_ {t + N \omega} ^ {j} \text {a r e i . i . d}. \tag {6}
$$
are satisfied for all  $t \in [\tau_{\max} + 1, T]$ ,  $N \in \{N : N \in \mathbb{N}, t + N\omega \leq T\}$ . This means that a finite number of causal mechanisms are repeated periodically for every univariate time series  $\mathbf{X}^j$  in  $V$ . One example of this model is illustrated in Fig.1. For  $\mathbf{X}^1$  in Fig.1, three causal mechanisms are reiterated periodically with  $\omega_1 = 3$ , represented by red, green, and blue edges, respectively.
The minimum value that satisfies the above conditions for  $\mathbf{X}^j$  is defined as the periodicity of  $\mathbf{X}^j$ , denoted by  $\omega_{j}$ . Furthermore, for an  $n$ -variate time series  $V$ ,  $\Omega$  denotes the minimum periodicity across all time series  $\mathbf{X}^j, j \in [n]$ . The number of causal mechanisms occurring sequentially and periodically of univariate time series  $\mathbf{X}^j$  is  $\omega_{j}$ , and that number of causal mechanisms of multivariate time series  $V$  is  $\Omega$ . For  $\mathbf{X}^j$ , the causal mechanisms are associated with each variable  $X_{t}^{j}$ . However, for  $V$ , the causal mechanisms are related to each time slice vector  $\mathbf{X}_t$  as a whole. The relationship between  $\Omega$  and  $\omega_{j}$  can be captured by:
$$
\Omega = \operatorname {L C M} \left(\left\{\omega_ {j}: j \in [ n ] \right\}\right) \tag {7}
$$
where  $\mathrm{LCM}(.)$  is an operation to find the least common multiple between any two or more numbers. Here,  $\Omega$  is the smallest common multiple among  $\{\omega_1,\dots \omega_n\}$ . In Fig 1,  $\Omega = \mathrm{LCM}(3,2,1) = 6$ .
Definition 2.3 (Time Partition). A time partition  $\Pi^j (T)$  of a univariate time series  $\mathbf{X}^j$  in a Semi-Stationary SCM with periodicity  $\omega_{j}$  is a way of dividing all time points  $t\in [T]$  into a collection of non-overlapping non-empty subsets  $\{\Pi_k^j (T)\}_{k\in [\omega_j]}$  such that:
$$
\Pi_ {k} ^ {j} (T) := \left\{t: \tau_ {\max } + 1 \leq t \leq T, (t \bmod \omega_ {j}) + 1 = k \right\}. \tag {8}
$$
where mod denotes the modulo operation. For instance,  $5 \mod 3 = 2$ .
We can observe that the variables in  $\{X_{t}^{j}\}_{t\in \Pi_{k}^{j}(T)}$  share the same causal mechanism. Since the number of potentially different causal mechanisms of variables in  $\mathbf{X}^j$  is  $\omega_{j}$ , the number of such time partition subsets is  $\omega_{j}$ . For simplicity, notations  $\Pi^j$  and  $\Pi_k^j$  are used instead of  $\Pi^j (T)$  and  $\Pi_k^j (T)$ . In Fig.1,  $\Pi_1^1 = \{4,7,10,13,\dots ,4 + 3N,\dots \}$ ,  $\Pi_2^1 = \{5,8,11,14,\dots ,5 + 3N,\dots \}$  and  $\Pi_3^1 = \{6,9,12,15,\dots ,6 + 3N,\dots \}$  where  $N\in \mathbb{N}^{+}$ . The nodes  $X_{t}^{1}$  are classified into their associated time partition subsets by the matching colors.
Definition 2.4 (Illusory Parent Sets). For a univariate time series  $\mathbf{X}^j\in V$  with Semi-Stationary SCM having periodicity  $\omega_{j} > 1$ , parent set index  $\mathrm{pInd}_{k\in [\omega_j]}^j$  is defined as:
$$
\operatorname {p I n d} _ {k} ^ {j} := \left\{\left(y _ {i}, \tau_ {i}\right) \right\} _ {i \in [ n ^ {\prime} ]}, \text {g i v e n} \operatorname {P a} \left(X _ {t} ^ {j}\right) = \left\{X _ {t - \tau_ {1}} ^ {y _ {1}}, X _ {t - \tau_ {2}} ^ {y _ {2}}, \dots , X _ {t - \tau_ {n ^ {\prime}}} ^ {y _ {n ^ {\prime}}} \right\}, \forall t \in \Pi_ {k} ^ {j} \tag {9}
$$
where  $n' = |\mathrm{Pa}(X_t^j))|$ ,  $\tau_i$  is the time lag and  $y_i$  is the variable index. Given  $\mathrm{pInd}_k^j$ , Illusory Parent Sets are defined as:
$$
\operatorname {P a} _ {k} \left(X _ {t} ^ {j}\right) = \left\{X _ {t - \tau_ {i}} ^ {y _ {i}}: \left(y _ {i}, \tau_ {i}\right) \in \mathrm {p I n d} _ {k} ^ {j} \right\}, \forall k \in \{k: t \notin \Pi_ {k} ^ {j} \} \tag {10}
$$
Put simply, the illusory parent sets of  $X_{t}^{j}$  are the time-shift version of the parent set of other variables in  $\mathbf{X}^j$  that have a different causal mechanism from  $X_{t}^{j}$ . Note that the illusory parent sets are constructed specifically in the Semi-Stationary SCM. For stationary SCM, there is no illusory parent set needed. To maintain consistency in notation, for time points  $t \in \Pi_k^j$ , the notation  $\mathrm{Pa}_k(X_t^j)$  can also be extended to encompass the true parent set of  $X_{t}^{j}$ :
$$
\operatorname {P a} _ {k} \left(X _ {t} ^ {j}\right) := \operatorname {P a} \left(X _ {t} ^ {j}\right), \forall t \in \Pi_ {k} ^ {j} \tag {11}
$$
By doing so,  $\mathrm{Pa}(X_t^j) \subset \cup_{k \in [\omega_j]} \mathrm{Pa}_k(X_t^j)$ . In Fig.1, by observing  $\mathrm{Pa}(X_7^1)$ , we have  $\mathrm{pInd}_1^1 = \{(1,1),(2,2)\}$ ; by observing  $\mathrm{Pa}(X_8^1)$ , we have  $\mathrm{pInd}_2^1 = \{(1,1),(3,1)\}$  and finally by observing  $\mathrm{Pa}(X_9^1)$ ,  $\mathrm{pInd}_3^1 = \{(1,1),(1,2)\}$ . Based on those indexes,  $P a_{1}(X_{7}^{1}) = \{X_{7 - 1}^{1}, X_{7 - 2}^{2}\} = \{X_{6}^{1}, X_{5}^{2}\}$ ,  $P a_{2}(X_{7}^{1}) = \{X_{7 - 1}^{1}, X_{7 - 1}^{3}\} = \{X_{6}^{1}, X_{6}^{3}\}$  and  $P a_{3}(X_{7}^{1}) = \{X_{7 - 1}^{1}, X_{7 - 2}^{1}\} = \{X_{6}^{1}, X_{5}^{1}\}$ . The first one is the true parent set of  $X_{7}^{1}$  and the latter two are the illusory parent sets. The order of those parent sets is not important.
At last, we need to further define a series of Markov chains that are associated tightly with Semi-Stationary SCM. The presence and characteristics of these Markov chains are thoroughly examined in the supplementary materials. The motivation behind creating such Markov chains is to introduce assumptions on them rather than directly on the original data  $V$ .
Definition 2.5 (Time Series as a Markov Chain). For time series  $V$  with Semi-Stationary SCM, there are (potentially)  $\delta$  different Markov chains  $\{Z_n^q\}_{n \in \mathcal{N}}, q \in [\delta]$ :
$$
Z _ {n} ^ {q} = \left\{\mathbf {X} _ {\tau_ {\max } + q + (n - 1) \delta}, \mathbf {X} _ {\tau_ {\max } + q + 1 + (n - 1) \delta}, \dots , \mathbf {X} _ {\tau_ {\max } + q - 1 + n \delta} \right\},
$$
where  $\mathcal{N} \coloneqq \{n \in \mathbb{N}^+ : \tau_{\max} + q - 1 + n\delta \leq T\}$ ,  $\delta = \lceil \frac{\tau_{\max} + 1}{\Omega} \rceil \Omega$ . Note that in  $\{Z_n^q\}$ ,  $q$  is used to indicate a specific Markov chain, while  $n$  serves as the running index for that particular Markov chain. Such a Markov chain  $\{Z_n^q\}$ ,  $q \in [\delta]$  exists as long as  $\mathrm{Pa}(Z_n^q) \subset Z_n^q \cup Z_{n-1}^q$  for all  $n$ . This is a finite state Markov Chain if all time series in  $V$  are discrete-valued time series. The state space of  $\{Z_n^q\}$  is the set containing all possible realization of  $\{\mathbf{X}_{\tau_{\max} + q + (i-1) + (n-1)\delta}\}_{i \in [\delta], n \in \mathbb{N}}$ . The transition probabilities between the states are the product of associated causal mechanisms based on Assumption A2, which is elaborated by an example in section C.1 (Eq.(10)-(11)) of the supplementary material.
2.2 Assumptions for  $\mathbf{PCM}\mathbf{C}\mathbf{I}_{\Omega}$
A1. Sufficiency: There are no unobserved confounders.
A2. Causal Markov Condition: Each variable  $X$  is independent of all its non-descendants, given its parents  $\operatorname{Pa}(X)$  in  $\mathcal{G}$ .
A3. Faithfulness Condition (Pearl [1980]): Let  $P$  be a probability distribution generated by  $\mathcal{G}$ .  $\langle \mathcal{G}, P \rangle$  satisfies the Faithfulness Condition if and only if every conditional independence relation true in  $P$  is entailed by the Causal Markov Condition applied to  $\mathcal{G}$ .
A4. No Contemporaneous Causal Effects: Edges between variables at the same time are not allowed.
A5. Temporal Priority: Causal relations that point from the future to the past are not permitted.
A6. Hard Mechanism Change: If at time points  $t_1$  and  $t_2$ , the causal mechanisms of  $X_{t_1}^j$  and  $X_{t_2}^j$  are different, then their corresponding parent sets can not be transformed to each other by time shifts:
$$
f _ {j, t _ {1}} \neq f _ {j, t _ {2}} \Rightarrow \mathrm {P a} (X _ {t _ {2}} ^ {j}) \neq \{X _ {s + (t _ {2} - t _ {1})} ^ {i}: X _ {s} ^ {i} \in \mathrm {P a} (X _ {t _ {1}} ^ {j}), i \in [ n ] \}.
$$
A7. Irreducible and Aperiodic Markov Chain: The Markov chains  $\{Z_n\}$  of  $V$  are assumed to be irreducible (Serfzo [2009]): for all states  $i$  and  $j$  of  $\{Z_n\}$ ,  $\exists n$  so that
$$
p _ {i j} ^ {(n)} := p \left(Z _ {n + 1} = j \mid Z _ {1} = i\right) > 0 \tag {12}
$$
and aperiodic(Karlin [2014]): for every state  $i$  of  $\{Z_{n}\}$ ,  $d(i) = 1$ , where the period  $d(i)$  of the state  $i$  is the greatest common divisor of all integers  $n$  for which  $p_{ii}^{(n)} > 0$ .
Assumptions A1-A5 are conventional and commonly employed in causal discovery methods for time series data. On the other hand, our approach requires additional Assumptions A6-A7 to be in place. To clarify, A6 is essential because our method may encounter challenges in distinguishing distinct causal mechanisms for variables in  $\{X_{n}^{j}\}_{n\in [T]}$  if they share identical parent sets after time shifts. As for A7, it serves a crucial role in establishing the soundness of our algorithm.
3  $\mathbf{PCMCI}_{\Omega}$  Algorithm
In this section, we propose an algorithm called  $\mathrm{PCMCl}_{\Omega}$ , and in section 3.1, we present a theorem demonstrating the soundness of  $\mathrm{PCMCl}_{\Omega}$  and its ability to recover the causal graph. Our algorithm
Algorithm 1  $\mathrm{PCMCl}_{\Omega}$