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1 SIAM J. SCI. COMPUT. Vol. 39, No. 1, pp. B76 B11 c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES IN CONTINUOUS TIME MARKOV CHAINS AND DYNAMIC DATA MIHAI CUCURINGU AND RADEK ERBAN Abtract. A method for detecting intrinic low variable in tochatic chemical reaction network i developed and analyzed. It combine aniotropic diffuion map (ADM) with approximation baed on the chemical Langevin equation (CLE). The reulting approach, called ADM-CLE, ha the potential of being more efficient than the ADM method for a large cla of chemical reaction ytem, becaue it replace the computationally mot expenive tep of ADM (running local hort burt of imulation) by uing an approximation baed on the CLE. The ADM-CLE approach can be ued to etimate the tationary ditribution of the detected low variable, without any a priori knowledge of it. If the conditional ditribution of the fat variable can be obtained analytically, then the reulting ADM-CLE approach doe not make any ue of Monte Carlo imulation to etimate the ditribution of both low and fat variable. Key word. diffuion map, tochatic chemical reaction network, low variable, tationary ditribution AMS ubject claification. 65C2, 65C4 DOI /15M Introduction. The time evolution of a complex chemical reaction network often occur at different time cale, and the oberver i intereted in tracking the evolution of the lowly evolving quantitie (i.e., of the o-called low variable), a oppoed to recording each and every ingle reaction that take place in the ytem. Whenever a eparation of cale exit, one ha to imulate a large number of reaction in the ytem in order to capture the evolution of the lowly evolving variable. With thi obervation in mind, it become crucial to be able to detect and parametrize the underlying low manifold correponding to the low variable intrinic to the ytem. In thi paper, we introduce an unupervied method of dicovering the underlying hidden low variable in chemical reaction network, and of their tationary ditribution, uing the aniotropic diffuion map (ADM) framework [41]. The ADM i a pecial cla of diffuion map which ha gained tremendou popularity in machine learning and tatitical analyi, a a robut nonlinear dimenionality reduction technique, in recent year [37, 2, 1, 6]. Diffuion map have been uccefully ued a a manifold learning tool, where it i aumed that the high-dimenional data lie on a lower-dimenional manifold, and one trie to capture the underlying Submitted to the journal Computational Method in Science and Engineering ection April 16, 215; accepted for publication (in revied form) October 14, 216; publihed electronically February 22, Funding: Thi work wa upported by the European Reearch Council under the European Community Seventh Framework Programme (FP7/27-213)/ERC grant agreement It i baed on work upported in part by award KUK-C1-13-4, made by King Abdullah Univerity of Science and Technology (KAUST). The firt author reearch wa upported by Amit Singer of PACM, by AFOSR MURI grant FA , by award R1GM92 from the NIGMS, and by award FA from AFOSR. The econd author reearch wa upported by a Univerity Reearch Fellowhip from the Royal Society, by a Nichola Kurti Junior Fellowhip from Braenoe College, Univerity of Oxford, and by a Philip Leverhulme Prize from the Leverhulme Trut. Thi prize money wa ued to upport reearch viit of Mihai Cucuringu in Oxford. Department of Mathematic, UCLA, Lo Angele, CA (mihai@math.ucla.edu). Mathematical Intitute, Univerity of Oxford, Oxford, OX2 6GG, United Kingdom (erban@ math.ox.ac.uk). B76 c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

2 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B77 geometric tructure of the data, a etup where the traditional linear dimenionality reduction technique (uch a principal component analyi) have been hown to fail. In the diffuion map etup, one contruct or i given a pare weighted connected graph (uually in the form of a weighted k-nearet-neighbor graph, with each node connected only to it k nearet or mot imilar neighbor) and ue it to build the aociated combinatorial Laplacian L = D W,whereW denote the matrix of weight and D denote a diagonal matrix with D ii equal to the um of all weight of the node i. Next, one conider the generalized eigenvalue problem Lx = λdx, whoe olution are related to the olution of the eigenvalue problem Lx = λx, wherel = D 1 W i a row-tochatic matrix often dubbed a the random walk normalized Laplacian. Whenever the pair (λ, x) i an eigenvalue-eigenvector olution to Lx = λx, thenoi (1 λ, x) for Lx = λdx. The (nonymmetric) matrix L can alo be interpreted a a tranition probability matrix of a Markov chain with tate pace given by the node of the graph, and with entrie L ij denoting the one-tep tranition probability from node i to j. In the diffuion map framework, one exploit a property of the top nontrivial eigenvector of the graph Laplacian of being piecewie contant on ubet of node in the domain that correpond to the ame tate aociated to the underlying low variable. We make thi tatement precie in ection 4, and further ue the reulting claification in ection 5 to propoe an unupervied method for computing the tationary ditribution of the hidden low variable, without uing any prior information on it tructure. Since the top eigenvector of the above Laplacian define the coaret mode of variation in the data, and have a natural interpretation in term of diffuion and random walk, they have been ued in a very wide range of application, including but not limited to partitioning [43, 42], clutering and community detection [35, 47, 34], image egmentation [39], ranking [49, 18], and data viualization and learning from data [7, 37]. The main application area tudied in thi paper involve tochatic model of chemical reaction network. They are written in term of tochatic imulation algorithm (SSA) [22, 23] which have been ued to model a number of biological ytem, including the phage λ lyi-lyogeny deciion circuit [1], circadian rhythm [46], and the cell cycle [28]. The Gillepie SSA [22] i an exact tochatic method that imulate every chemical reaction, ampling from the olution of the correponding chemical mater equation (CME). To characterize the behavior of a chemical ytem, one need to imulate a large number of reaction and realization, which lead to very computationally intenive algorithm. For uitable clae of chemically reacting ytem, one can ometime ue exact algorithm which are equivalent to the Gillepie SSA, but are le computationally intenive, uch a the Gibon Bruck SSA [2] and the optimized direct method [5]. However, thee method alo tochatically imulate the occurrence of every chemical reaction, which can be a computationally challenging tak for ytem with a very large number of pecie. One way to tackle thi problem i to ue parallel tochatic imulation [29]. In thi work, we dicu an alternative approach which doe not make ue of parallel tochatic imulation, but at the ame time, the propoed approach can alo benefit from large proceing power and parallel computing, a many tep of our propoed algorithm are highly parallelizable. An alternative approach to treating the molecular population a dicrete random variable i to decribe them in term of their continuouly changing concentration, which can be done via the chemical Langevin equation (CLE), a tochatic differential equation that link the dicrete SSA with the determinitic reaction rate equation [21]. Although uch an approach can be le computationally expenive, it c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

3 B78 MIHAI CUCURINGU AND RADEK ERBAN come with the diadvantage that, for certain chemical ytem, it can lead to negative population [48]. In addition, note that none of the above approache take explicit advantage of the eparation of cale, if one exit, omething we will exploit in thi paper, a detailed in ection 4 and 5. It i often the cae that a modeler i not intereted in every ingle reaction which take place in the ytem but only in the lowly evolving quantitie. Certain ytem poe multiple time cale, meaning that one ha to imulate a large number of reaction to reveal the low dynamic. Several algorithm for chemical network with fat and low variable have already been developed in the literature. The author of [26] propoed imulating the fat reaction uing Langevin dynamic and the low reaction uing the Gillepie algorithm. Thi approach require both the time cale eparation and a ufficiently large ytem volume; however, the latter contraint can be avoided uing probability denitie of the fat pecie conditioned on the low pecie and etimating the effective propenity function of the low pecie [3, 4, 14, 38, 45]. An alternative approach to imulating the evolution of the low variable while avoiding doing o for the fat variable i to etimate the probability ditribution of the low variable [19]. The key point in thi approach i to ue hort burt of appropriately initialized tochatic imulation to etimate the drift and diffuion coefficient for an approximating Fokker Planck equation written in term of the low variable [16]. The ucce of thi approach ha already been demontrated in a range of application, including material cience [25], cell motility [15], and ocial behavior of inect [44]. Reference [8] introduce the conditional tochatic imulation algorithm (CSSA) that allow one to ample efficiently from the ditribution of the fat variable conditioned on the low one [8] and to etimate the coefficient of the effective tochatic differential equation (SDE) on the fly via a propoed contrained multicale algorithm (CMA). The CMA can be further modified by etimating the drift and diffuion coefficient in the form given by the CLE for the low ubytem, which require the etimation of effective propenity function of low reaction [9]. The main quetion we plan to addre in thi paper build on and combine two already exiting idea invetigated in [8] and [41] and bring everal computational and algorithmic improvement. The above-mentioned CSSA algorithm explicitly make ue of the knowledge of the low variable (often unavailable in many real application), a drawback we plan to addre a explained later in ection 4, where, driven by the top eigenvector of an appropriately contructed Laplacian, we dicover the underlying low variable. In doing o, we make ue of the ADM framework [41], which modifie the traditional diffuion map approach to take into account the time dependence of the data, i.e., the time tamp of each of the data point under conideration. By integrating local imilaritie at different cale, the ADM give a global decription of the data et. The ret of thi paper i organized a follow. In ection 2 we provide a mathematical framework for multicale modeling of tochatic chemical reaction network and detail the two chemical ytem via which we illutrate our approach. In ection 3 we introduce the ADM-CLE framework and highlight it difference from the approache which were previouly introduced in the literature. In ection 4 we propoe a robut mapping from the obervable pace to the dynamically meaningful inacceible pace, which allow u to recover the hidden low variable. In ection 5 we introduce a Markov-baed approach for approximating the teady ditribution of the low variable and compare our reult with another recently propoed approach. We conclude with a ummary and dicuion of future work in ection 6. c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

4 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B79 2. Problem formulation. A multicale modeling framework for tochatic chemical reaction network can be formulated a follow. We conider a well-mixed ytem of l chemical pecie, denoted by X 1,X 2,...,X l, that interact through m reaction channel R 1,R 2,...,R m in a reactor of volume V. We denote the tate of the ytem by X(t) =[X 1 (t),x 2,...,X l (t)], where X i (t), i =1, 2,...,l,repreent the number of molecule of type X i in the ytem at time t. With a light abue of notation, we interchangeably ue X i to denote the type i of the molecule. In certain cenario, one may aume that the reaction can be claified a either fat or low, depending on the time cale of occurrence [3]. A expected, the fat reaction occur many time on a time cale for which the low reaction occur with very mall probability. A defined in [3], the fat pecie denoted by F are thoe pecie whoe population get changed by a fat reaction. Slow pecie (denoted by S) arenot changed by fat reaction. Conidering that low pecie are not only pecie from the et {X 1,X 2,...,X l }, but alo that their function are not changed by fat reaction, the component of the fat and low pecie can be ued a a bai for the tate pace of the ytem, whoe dimenion equal the number of linearly independent pecie. For each reaction channel R j, j =1, 2,...,m, there exit a correponding propenity function α j α j (x), uch that α j dt denote the probability that, given X(t) =x, reaction R j occur within the infiniteimal time interval [t, t +dt). We denote by ν the tochiometric matrix of ize m l, withentryν ji denoting the change in the number of molecule of type X i caued by one occurrence of reaction channel R j. The continuou time dicrete in pace Markov chain can be further approximated by the CLE for a multivariate continuou Markov proce [21]. Uing time tep Δt, the Euler Maruyama dicretization of the CLE i given by m m X i (t +Δt) =X i (t)+δt ν ji α j (X(t)) + ν ji α j (X(t)) N j (t) Δt (2.1) for all i =1, 2,...,l, j=1 where X i, with another light abue of notation, denote a real-valued approximation of the number of molecule of the ith chemical pecie, i =1, 2,...,l. Here, N j (t), j = 1, 2,..., m, denote the et of m independent normally ditributed random variable with zero mean and unit variance Illutrative example CS-I. A the firt illutrative example, we conider the following imple two-dimenional chemical ytem, with the two chemical pecie denoted by X 1 and X 2 (i.e., l = 2) which are ubject to four reaction channel R j, j =1, 2, 3, 4 (i.e., m =4),givenby k 2 (2.2) k1 X 1 k3 X 2 j=1 k 4. Throughout the ret of thi paper, we hall refer to the chemical ytem (2.2) a CS-I (i.e., chemical ytem I ). We label R j the reaction correponding to the reaction rate ubcript k j, j = 1, 2, 3, 4, and note that each reaction R j ha a propenity function α j (t) given by [22] aociated to it: (2.3) α 1 (t) =k 1 V, α 2 (t) =k 2 X 1 (t), α 3 (t) =k 3 X 2 (t), α 4 (t) =k 4 X 2 (t), where V denote the volume of the reactor. We conider the ytem with the following dimenionle parameter: (2.4) k 1 V = 1, k 2 = k 3 = 2, and k 4 =1. c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

5 B8 MIHAI CUCURINGU AND RADEK ERBAN Number of molecule Trajectory of X 1, X 2, and (X 1 + X 2 )/ X 1 X 2 (X 1 + X 2 )/ Time t (a) Number of molecule Trajectory of X 1, X 2, and S/3 = (X X 2 )/3 X 1 X 2 (X X 2 )/ Time t (b) Fig (a) Trajectorie of CS-I conidered in (2.2) howing the behavior of the low variable S =(X 1 + X 2 )/2 in contrat to the behavior of the fat variable X 1 and X 2, where the ytem propenity function and parameter are given by (2.3) and (2.4). (b) Trajectorie of CS-II conidered in (2.7), howing the low behavior of the variable S = X 1 +2X 2 in contrat to the fat behavior of variable X 1 and X 2, where the ytem parameter are given by (2.8). We plot in Figure 2.1(a) the time evolution of the two different pecie in ytem (2.2), together with the low variable S =(X 1 + X 2 )/2, tarting from initial condition X 1 () = X 2 () = 1. A the figure how, the ytem variable X 1 and X 2 are changing very frequently (thu we label them a fat variable), while the newly defined variable S change very infrequently and can be conidered to be a low variable. Following [27], for the chemical ytem in (2.2) compried only of monomolecular reaction, it i poible to compute analytically the tationary ditribution of the low variable S, ince the joint probability ditribution of the two variable X 1 and X 2 i a multivariate Poion ditribution (2.5) P(X 1 = n 1,X 2 = n 2 )= with parameter given by n1 λ 1 λ n2 2 n 1! n 2! exp( λ 1 λ ) 2 (2.6) λ1 = k 1V (k 3 + k 4 ) = 1.5 and λ2 = k 1V = 1. k 2 k 4 k Illutrative example CS-II. The econd example i taken from [8]. We hall refer to it a CS-II from now on. We conider the following ytem: k 1 (2.7) X 2 X 1 + X 2, k3 X 1, X 1 + X 1 k2 k4 k 5 X 2, k6 involving two molecular pecie X 1 and X 2, whoe reaction R 1,R 2,...,R 6 have the propenity function given by α 1 (t) =k 1 X 2 (t), α 2 (t) =k 2 X 1 (t)x 2 (t)/v, α 3 (t) =k 3 V, X 1 (t)(x 1 (t) 1) α 4 (t) =k 4 X 1 (t), α 5 (t) =k 5, α 6 (t) =k 6 X 2 (t), V where V denote the ytem volume. Figure 2.1(b) how a imulated trajectory of thi chemical ytem uing the Gillepie algorithm for the following dimenionle parameter [8]: (2.8) k 1 =32, k 2 =.4V, k 3 V = 1475, k 4 =19.75, k 5 =1V, k 6 = 4, c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

6 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B81 whereweuev = 8. Note that in thi econd example, reaction R 5 and R 6 are occurring on a much fater time cale than the other four reaction R 1, R 2, R 3,and R 4. A natural choice for the low variable i S = X 1 +2X 2, which i invariant with repect to all fat reaction [8], a we illutrate in Figure 2.1(b) Main problem. Our end goal in thi paper i to propoe an algorithm that efficiently and accurately etimate the tationary probability denity of the hidden low variable S, without any prior knowledge of it. The approach we propoe build on the aniotropic diffuion map (ADM) framework to implicitly dicover the mapping from the obervable tate pace to the dynamically meaningful coordinate of the fat and low variable, a previouly introduced in [41], and on the CLE approximation (2.1). 3. ADM-CLE approach. Let u conider example CS-II, and aume that = (x 1,x 2 )=x 1 +2x 2 and f = f(x 1,x 2 )=x 1 are the lowly and rapidly changing variable, repectively. They together define a mapping g :(x 1,x 2 ) (, f) fromthe obervable tate variable x 1 and x 2 in the acceible pace O to the dynamically meaningful (but in more complicated example inacceible) low variable and the fat acceible variable f, bothinpaceh. In other word, g map (x 1,x 2 ) (x 1 +2x 2,x 1 ) and, converely, it invere h := g 1 :(, f) (f, f 2 ). The approach introduced in [41] exploit the local point cloud generated by many local burt of imulation at each point (x 1,x 2 ) in the obervable pace O. Such obervable local point cloud are the image under h of imilar local point cloud in the inacceible pace H (at correponding coordinate (, f) uch that h(, f) =(x 1,x 2 )), which, due to the eparation of cale between the fat and low variable f and, have the appearance of thin elongated ellipe. It i preciely thi eparation of cale that we leverage into building a pare aniotropic graph Laplacian L in the obervable pace, and we ue it a an approximation of the iotropic graph Laplacian in the inacceible pace H. A we hall ee, the top nontrivial eigenvector of L will robutly indicate all pair of original tate (x 1,x 2 ) that correpond to the ame low variable S = (where = x 1 +2x 2 for CS-II). In other word, we dicover on the fly the tructure of the low variable S, and further integrate thi information into a Markov-baed method for etimating it tationary ditribution P(S = ), while alo computing along the way an analytical expreion for the conditional ditribution of the fat variable given the low variable P(F = f S = ). Singer et al. [41] run many local burt of imulation for a hort time tep δt tarting at (x 1,x 2 ). Such trajectorie end up at random location, forming a cloud of point in the obervable plane O, with a bivariate normal ditribution with 2 2 covariance matrix Σ. The hape of the reulting point cloud i an ellipe, whoe axe reflect the dynamic of the data point. In other word, when there i a eparation of cale, the ellipe are thin and elongated, with the ratio between the axi of the ellipe given by the ratio (3.1) τ = λ 1 λ 2 of the two eigenvalue of Σ. The firt eigenvector correponding to λ 1 point in the direction of the fat dynamic on the line x 1 +2x 2 =, while the econd one point in the direction of the low dynamic. In particular, τ i a mall parameter, i.e., <τ 1. In general, we wih to piece together locally defined component into a globally conitent framework, a nontrivial tak when the underlying unobervable c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

7 B82 MIHAI CUCURINGU AND RADEK ERBAN low variable (or the propenity function of the ytem) are complicated nonlinear function of the obervable variable in O. The contruction of the ADM framework in [41] relate the aniotropic graph Laplacian in the obervable pace O with the iotropic graph Laplacian in the inacceible pace H. In that etup, each of the N data point x (i), i =1, 2,...,N, live in an l-dimenional data pace. For both CS-I and CS-II, the data i two-dimenional, and thu l = 2. For the former ytem, we conider each lattice point in the domain [5, 15] [5, 15], and hence there are N = 11 2 =1, 21 tate, while for the latter one we conider the domain [1, 11] [1, 11], i.e., N = 11 2 =12, 1. Throughout thi paper, we will often refer to the N data point x (i) =(x 1,x 2 ) (i), i =1, 2,...,N, a O-tate of the chemical ytem. The ADM [41] then generate enemble of hort imulation burt at each of the N point in the data et, compute the averaged poition after tatitically averaging over the many imulated trajectorie, and obtain an etimate of the local 2 2 covariance matrix Σ (i). For each data point x (i), the invere of Σ (i) i computed and ymmetric Σ-dependent quared ditance between pair of data point in the two-dimenional obervable pace R 2 (given by (3.3) below) i defined. The ADM framework then ue thi dynamic ditance meaure to approximate the Laplacian on the underlying hidden low manifold. We provide further detail on the ADM framework in ection 3.2. We now highlight the firt difference between the approach taken in thi paper and in [41] Replacing hort imulation burt by the CLE approximation. The local burt of imulation initiated at each data point in order to etimate the local covariance may be computationally expenive to etimate. In thi paper, we bypa thee hort burt of imulation by uing an approximation given by the CLE (2.1), which allow for a theoretical derivation of the local 2 2 covariance matrice. Uing (2.1), we obtain Cov(X i (t +Δt),X k (t +Δt)) = E[X i (t +Δt)X k (t +Δt)] (3.2) E[X i (t +Δt)]E[X k (t +Δt)] m =Δt ν ji ν jk α j (X). Computing the eigendecompoition of a local covariance matrix i analogou to performing principal component analyi on the local cloud of point, generated by the hort imulation burt. The advantage of (3.2) over the computational approach ued in [41] i that Σ (i) can be computed at each data point without running (computationally intenive) hort burt of imulation. The error of the CLE approximation depend on the value of coordinate of the data point x (i), i.e., on the ytem volume V [24, 13]. However, for the chemical ytem CS-II, the domain doe include tate with mall molecule number, and the CLE approximation i le well jutified. From looking at the etimated ditribution of the low variable S = X 1 +2X 2 hown in Figure 5.4(b), the mot probable tate contain a lot le than 1 molecule of each chemical pecie; however, the fact that we are able to robutly recover the ditribution even in thi cenario i perhap a good indication that the CLE approximation wa jutified. To circumvent the aumption needed for the CLE approximation, one could alternatively conider etimating, for mall molecule number, the local covariance matrice via direct imulation, a oppoed to relying on the CLE approximation. A potential improvement in thi direction may alo alleviate the iue encountered in the j=1 c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

8 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B83 low variable detection tep via our bin denoiing procedure detailed later in ection 4, which lead u to truncate at the boundary of the low variable (perhap due to unreliable CLE approximation). Combining the CLE approximation for large number of molecule with direct imulation for the few tate that involve mall molecule number i an intereting reearch direction to explore [12] Aniotropic diffuion kernel. The next tak i the integration of all local principal component into a global framework, with the purpoe of identifying the hidden low variable. We etimate the ditance (and hence the imilarity meaure) between the low variable in the underlying inacceible manifold uing the aniotropic graph Laplacian [41]. We derive a ymmetrized econd order approximation of the (unknown) ditance in the inacceible pace H, baed on the Jacobian of the unknown mapping from the inacceible to the obervable pace. The Σ-dependent ditance between two O-tate i given by ( d 2 Σ (x 1,x 2 ) (i), (x 1,x 2 ) (j)) = 1 ( (x 1,x 2 ) (i) (x 1,x 2 ) (j))( Σ 1 2 (x 1,x 2) +Σ 1 (i) ((x 1,x 2 ) (i) (x 1,x 2 ) (j)) T (3.3) (x 1,x 2) (j) ) and repreent a econd order approximation 1 of the Euclidean ditance in the inacceible (, τf)-pace (3.4) d 2 Σ [(x 1,x 2 ) (i), (x 1,x 2 ) (j) ] ( (i) (j) ) 2 + τ 2 (f (i) f (j) ) 2 ( (i) (j) ) 2, where the lat approximation i due to the fact that τ i a mall parameter; ee (3.1). Note that it i alo poible to extend (3.4) to higher dimenion, a long a there exit a eparation of cale between the et of low variable and the et of fat variable [41]. Uing approximation (3.3) (3.4) of the ditance between tate of the low variable, we next contruct (an approximation of) the Laplacian on the underlying hidden low manifold, uing the Gauian kernel a a imilarity meaure between the low variable tate. We build an N N imilarity matrix W with entrie (3.5) { d 2 W ij =exp Σ [(x 1,x 2 ) (i), (x 1,x 2 ) (j) } { ] ( (i) (j) ) 2 } exp, i, j =1, 2,...,N, ε 2 where the ingle moothing parameter ε (the kernel cale) ha a two-fold interpretation. On one hand, ε denote the quared radiu of the neighborhood ued to infer local geometric information; in particular, W ij i O(1) when (i) and (j) are in a ball of radiu ε, and thu cloe on the underlying low manifold, but it i exponentially mall for tate that are more than ε apart. On the other hand, ε repreent the dicrete time tep at which the underlying random walker jump from one point to another. We refer the reader to [32] for a detailed urvey of random walk on graph ε 2 1 The principal component of Σ are the local direction of the rapidly changing variable at a particular tate of the ytem, wherea component with mall eigenvalue correpond to the low variable. We remark that a ingular covariance matrix would correpond to having only change of the fat variable, and almot no change in the low variable, which i unlikely but poible, depending on the ytem under conideration. We have not run into uch problem in the example conidered. Furthermore, we remark that in the initial work of Singer and Coifman [4], which introduced the above econd order approximation of the Euclidean ditance in the inacceible pace, the author rely on the peudoinvere Σ. c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

9 B84 MIHAI CUCURINGU AND RADEK ERBAN and their application. We normalize W uing the diagonal matrix D to define the row-tochatic matrix L by (3.6) D ii = N W ij, L = D 1 W. j=1 Since L i a row-tochatic matrix, it ha eigenvalue λ = 1 with trivial eigenvector Φ =(1, 1,...,1) T. The remaining eigenvalue can be ordered a 1=λ λ 1 λ 2 λ N 1. We denote by Φ i the correponding eigenvector, i.e., LΦ i = λ i Φ i.thetopd nontrivial eigenvector of the random walk aniotropic Laplacian L decribe the geometry of the underlying d-dimenional manifold [17]; i.e., the ith data point x (i) i repreented by the following diffuion map: (3.7) (Φ 1 (i), Φ 2 (i),...,φ d (i)), i =1, 2,...,N, where Φ j (i) denote the ith component of the eigenvector Φ j. However, note that ome of the conidered eigenvector can be higher harmonic of the ame principal direction along the manifold; thu in practice one compute the correlation between the computed eigenvector before electing the above d eigenvector choen to parametrize the underlying manifold. For the two chemical ytem conidered in thi paper, we how in the remainder of thi ection how the top (i.e., d = 1) nontrivial eigenvector of L can be ued to uccefully recover the underlying low variable. Uing the tochaticity of L, we can interpret it a a random walk matrix on the weighted graph G =(V,E), where the et of node correpond to the original obervable tate (x 1,x 2 ) (i), i =1, 2,...,N (and implicitly to tate (i) of the low variable), and there i an edge between node i and j if and only if W ij >. The aociated combinatorial Laplacian i given by L = D W. Whenever the pair (λ i, Φ i ) i an eigenvalue-eigenvector olution to LΦ i = λ i Φ i,thenoi(1 λ i, Φ i )forthe generalized eigenvalue problem LΦ i = λ i DΦ i. We plot in Figure 3.1(a) and 3.2(a) the pectrum of the combinatorial Laplacian L = D W for the chemical ytem CS-I and CS-II. In Figure 3.1(b) and 3.2(b) we color the tate of the network with the top nontrivial eigenvector Φ 1. Before conidering the top eigenvector of L for determining the underlying low variable and etimating it tationary ditribution, we propoe uing a pare graph Laplacian which differ from the ADM method in [41], where the Laplacian matrix i aociated to a complete weighted graph. However, uing a complete graph lead to computing the Σ-dependent quared ditance in (3.3) for any pair of node; thu an O(N 2 ) number of computation i ued. In light of the approximation (3.4), a pair of point which are far away in the obervable pace (i.e., for which d 2 Σ ((x 1,x 2 ) (i), (x 1,x 2 ) (j) ) i large) denote a pair of correponding tate of the low variable which are alo far away in the inacceible pace. Thu we do not have to do uch computation, becaue point far away in the unobervable pace will have an exponentially mall imilarity W ij cloe to. The fact that the hape of the local point cloud i an ellipe provide ome inight in thi direction. Thu we will build a pare graph of pairwie meaurement and no longer compute the Σ-dependent ditance between all point of the data et but only between a very mall ubet of the point. The pectrum of the covariance matrix Σ i, in particular the ratio τ of it two eigenvalue given by (3.1), guide u in building locally at each point a pare c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

10 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B85 1 λ count Top 5 eigenvalue rank larget eigenvalue (a) Hitogram of weighted node degree in G weighted degree (c) x (b) Weighted degree x 1 Fig Illutrative example CS-I. (a) The top 5 eigenvalue of the aociated combinatorial Laplacian, i.e., (1 λ i ) for i =1, 2,...,5. (b) The coloring of the node of G (tate of the obervable pace) according to their correponding entry in the top eigenvector Φ 1 of L given by (3.6). (c) The weighted degree ditribution of the ground tate graph G. (d) A catterplot of the tate of the ytem, colored by their weighted degree. ellipoid-like neighborhood graph. However, we cautiouly remark that the econd order approximation (3.3) of the ditance in the inacceible pace i not a monotonically increaing function of the ditance in the acceible pace and depend on the particular covariance matrice ued and potentially the eparation of cale. In the example we have conidered, we have indeed oberved a monotonically decreaing Gauian imilarity (3.5) a a function of the ditance between the point in the acceible pace, which motivated our approach to conider ellipoid-like neighborhood graph. For each obervable tate (x 1,x 2 ) (i), we build a local adjacency graph, denoted by G i, in the hape of an ellipe pointing in the direction of the fat dynamic, and whoe mall axi point in the direction of the low dynamic. In our computation, we fix the length of the emiminor axi of any ellipe to be equal to 3, while the length of it correponding emimajor axi i given by 3τ, whereτ i given by the ratio of the two eigenvalue of the covariance matrix a in (3.1). Figure 3.3 how an example of uch a local 1-hop neighborhood graph G i, where the central node (x 1,x 2 ) (i) i connected to all point contained within the boundarie of an appropriately caled ellipe centered at (x 1,x 2 ) (i). Finally, we define the pare graph G of ize N N aociated to the (d) c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

11 B86 MIHAI CUCURINGU AND RADEK ERBAN 1 λ Top 5 eigenvalue count rank larget eigenvalue (a) Hitogram of weighted node degree in G weighted degree (c) x (b) Weighted degree x 1 Fig Illutrative example CS-II. (a) The top 5 eigenvalue of the aociated combinatorial Laplacian, i.e., (1 λ i ) for i =1, 2,...,5. (b) The coloring of the node of G (tate of the obervable pace) according to their correponding entry in the top eigenvector Φ 1 of L given by (3.6). (c) The weighted degree ditribution of the ground tate graph G. (d) A catterplot of the tate of the ytem, colored by their weighted degree (d) Fig The local neighborhood graph G i at a given node (x 1,x 2 ) (i) ; the hape i an ellipoid whoe axi ratio i given by the ratio of the eigenvalue τ of the local covariance matrix d 2 Σ ((x 1,x 2 ) (i), (x 1,x 2 ) (j) ), i.e., by (3.1). The correponding eigenvector are ued to calculate the orientation of the ellipe. c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

12 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B87 entire network a the union of all locally defined ellipoid-like neighborhood graph G = N i=1 G i. Note that the union graph G i till a imple graph, with no elf-edge and no multiple edge connecting the ame pair of node. We compute the ditance d Σ by (3.3) (and thu the imilarity W ij ) between a pair of node (x 1,x 2 ) (i) and (x 1,x 2 ) (j) if and only if the correponding edge (i, j) exiting. We plot in Figure 3.1(c) and 3.2(c) the hitogram of the weighted degree of the node in the weighted graph W defined in (3.5), while Figure 3.1(d) and 3.2(d) how a catterplot of the tate of the ytem, where each tate i i colored by it weighted degree, i.e., the um of all it outgoing weighted edge W ij, j =1, 2,...,n. Throughout the computational example in thi paper, the moothing parameter ε which appear in (3.5) wa et to ε =.1. In contrat to the approach in [41], which compute all O(N 2 ) pairwie imilaritie, the parity of G (and thu of the aociated graph Laplacian L) in our approach only require the computation of a much maller number of ditance, a low a linear, depending on the dicretization of the domain, and make it computationally feaible to olve problem with thouand or even ten of thouand of node. 4. A robut mapping from the obervable pace O to the dynamically meaningful inacceible pace H. A a firt tep toward partitioning the node of the original graph G and detecting the aociated low variable, we ort the entrie of the top eigenvector Φ 1, which we then denote by Φ 1 with Φ 1 (1) Φ 1 (2) Φ 1 (N). Thi orting proce define permutation σ of the original index et i =1, 2,...,N o that Φ 1 (σ(i)) = Φ 1 (i). We conider the increment between two conecutive (orted) value (4.1) δ i = Φ 1 (i) Φ 1 (i +1), i =1, 2,...,N 1. Next, we ort the vector of uch increment, denote it entrie by δ 1 δ 2 δ N 1, and how in Figure 4.1(a) (rep., Figure 4.2(a)) the top 3 (rep., top 42) larget uch increment δ i for illutrative example CS-I (rep., CS-II). Note that thi already give u an idea about the number of ditinct low tate in the ytem, a et which we denote by S. Ideally, the difference Φ 1 (i) Φ 1 (j) in the entrie of the top eigenvector correponding to two obervable tate (x 1,x 2 ) (i) and (x 1,x 2 ) (j) that belong to the ame low variable (i.e., x (i) 1 +2x(i) 2 = x (j) 1 +2x(j) 2 = for illutrative example CS-II) hould be zero or cloe to zero, in which cae we expect that only approximately S of the N 1incrementδ i are ignificantly larger than zero, while the remaining majority are zero or cloe to zero. In Figure 4.1(b) and 4.2(b) we highlight the correlation between the entrie of the top nontrivial eigenvector Φ 1 and the correponding low variable S. In Figure 4.1(c) and 4.2(c), we zoom on a ubet of tate to make the point that the eigenvector Φ 1 i almot contant on the O-tate that correpond to the ame value of the low variable. The plot in Figure 3.1(b) and 3.2(b) how a coloring of the network generated by the two chemical ytem CS-I and CS-II, baed on the firt nontrivial eigenvector of the aociated pare Laplacian L. Note that the eigenvector look almot piecewie contant along the line that point to the evolution of the fat variable, for a given value of the low variable (S =(X 1 + X 2 )/2 for CS-I and S = X 1 +2X 2 for CS-II), yet nowhere along the way do we have to input thi information into the method. In the next tep we ue thi top eigenvector to identify all node of the graph (original tate of the chemical ytem) that correpond to the ame value of the underlying low variable. In other word, all node whoe correponding eigenvector entrie are between an appropriately choen interval (that we hall refer to a a bin) will be c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

13 B88 MIHAI CUCURINGU AND RADEK ERBAN 7 x 1 8 Top larget jump δ i.15 n=121; Spearman Correlation= x 1 4 zoom in Jump ize Top eigenvector eigenvector value Eigenvector entry True low variable Index (a) (b) (c) Fig Illutrative example CS-I. (a) Jump ize (4.1) of the orted eigenvector Φ 1 of the pare aniotropic graph Laplacian L. (b) The correlation of Φ 1 with the ground truth low variable S =(X 1 + X 2 )/2. On the y-axi we plot the ith entry of the top eigenvector Φ 1 (i) aociated to tate (x 1,x 2 ) (i) veru it correponding true low variable =(x 1 + x 2 )/2. (c) Zoom-inonthe orted top eigenvector Φ 1 (the color denote the correponding low variable) howing that Φ 1 i almot piecewie contant on the bin that correpond to ditinct low variable tate. The kernel cale i et to ε =.1. Jump ize 1.2 x Top larget jump δ i Eigenvector entry (a) Top eigenvector n=121; Spearman Correlation= True low variable (b) eigenvector value x 1 3 zoom in Index Fig Illutrative example CS-II. (a) Jump ize (4.1) of the orted eigenvector Φ 1 of the pare aniotropic graph Laplacian L. (b) The correlation of Φ 1 with the ground truth low variable variable S = X 1 +2X 2.Onthey-axi we plot the ith entry of the top eigenvector Φ 1 (i) aociated to tate (x 1,x 2 ) (i) veru it correponding true low variable = x 1 +2x 2. (c) Zoom-inonthe orted top eigenvector Φ 1 (the color denote the correponding low variable) howing that Φ 1 i almot piecewie contant on the bin that correpond to ditinct low variable tate. The kernel cale i et to ε =.1. labeled a belonging to the ame low variable S. In other word, we eek a partition of the obervable tate in O, i.e., of the node of G, uch that all original tate (x 1,x 2 ) (i) with the ame value of the correponding low variable ((x 1,x 2 ) (i) ) end up in the ame bin. Our goal i to find a partition P = {P 1, P 2,...,P k } of O uch that (4.2) P j = {(x 1,x 2 ) (i) O (x 1,x 2 )=q j } and (c) k P j = O, j=1 where k denote the number of ditinct value q j, j =1, 2,...,k,ofthelowvariable S. A an example, in the cae of CS-I given by (2.2), the partition P j = {(1, 99), (2, 98),...,(99, 1)} correpond to all node in the graph for which the value of the aociated low variable i contant: q j = 5. The key obervation we exploit here i that the top eigenvector of the Laplacian matrix i almot piecewie contant on the bin that partition O, ince the node of G that correpond to the ame value of the low variable have a very high pairwie imilarity, with W ij very cloe to 1. One may alo interpret the above problem a a clutering problem, where the c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

14 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B89 Number of original tate Cardinality eig low var. S = 21, Θ = Slow variable Our recovered 21 low variable Jaccard index heatmap Ground truth 21 low variable Ground truth label of matched low variable Spearman correlation = Order of our low variable (1:21) Jaccard Similarity Index Max matching in the Jaccard index matrix Our recovered low variable (1:21) (a) (b) (c) (d) Fig Illutrative example CS-I. (a) The eigenvector-baed low variable cardinality. The Theta core Θ i the moothne meaure of the bin cardinalitie, defined in (4.5). The algorithm perfectly recover the ground truth partition. (b) The heatmap of the pairwie Jaccard imilarity matrix given by (4.4). (c) The correlation between the ordering of the ground truth low variable and the eigenvector recovered low variable. (d) The Jaccard index of the pairwie matched bin (from the maximum matching). imilarity between pair of point i given by (3.5), and i uch that node that belong to the ame bin have a much higher imilarity compared to node that belong to two different bin, an effect due to the trong eparation of cale. In the cae of illutrative example CS-I, the cluter correpond to line in the two-dimenional plane uch that (x 1 + x 2 )/2 =c for a contant c. We point the intereted reader to the work of [33], where the top eigenvector of the random walk Laplacian are ued for clutering. While in practice one ue the top everal eigenvector a the reduced eigenpace where clutering i ubequently performed, in our cae the top eigenvector alone uffice to capture the many different cluter (i.e., bin), a fact we attribute to the trong eparation of cale exhibited by the illutrative chemical ytem CS-I and CS-II. If everal eigenvector were conidered, then one could ue a clutering algorithm, uch a k-mean or pectral clutering [39, 47], to obtain the partitioning (4.2). However, a impler method ha been uccefully ued for the one-dimenional eigenpacein both example we conidered. It i decribed a follow. Recall the orted vector of increment δ 1 δ 2 δ N 1 defined in (4.1), and conider the et of the k 1 larget uch increment { δ 1, δ 2,..., δ k 1 } where δ 1 δ 2 δ k 1.Next, from the orted eigenvector Φ 1 we extract the poition of the entrie whoe aociated increment (with repect to it right-next neighbor index) belong to { δ 1, δ 2,..., δ k 1 }. In other word, we compute (4.3) b t = arg Φ 1 (i) Φ 1 (i +1)= δ t, where t =1, 2,...,k 1, i=1,2,...,n 1 and b =andb k = N. Finally, we compute an etimated partition ˆP of O by ˆP q = {i O σ(i) (b q 1,b q ]}, whereq =1, 2,...,k,andσ i the permutation of the original index et i =1, 2,...,N, given in the definition of Φ 1, i.e. Φ1 (σ(i)) = Φ 1 (i). We remark here that k i not a parameter, per e, but repreent the number of ditinct tate of the low variable in the ytem, correponding to the current bounded domain. If not known a priori, one can etimate k from the orted barplot of the increment between two conecutive entrie in the orted eigenvector, a hown in Figure 4.1(a) and 4.2(a). To illutrate the correctne of our propoed technique, we compute the Jaccard index between each propoed partition et ˆP j, j =1, 2,...,k, and each ground truth c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

15 B9 MIHAI CUCURINGU AND RADEK ERBAN Cardinality true low var. S = 328, Θ = 18 Cardinality eig low var. S = 328, Θ = 726 Cardinality eig low var. S = 328, Θ = Number of original tate Number of original tate Number of original tate Slow variable Slow variable Slow variable (zoom in) (a) (b) (c) Fig Illutrative example CS-II. (a) The ground truth low variable cardinality. (b) The eigenvector-baed low variable cardinality. Θ capture the moothne of the bin cardinalitie, a introduced in (4.5). (c) Plot of the cardinalitie of a ubet of bin, howing the erroneou bin aignment in the eigenvector-baed partition. Thi i a zoomed-in verion of panel (b). partition et P i, i =1, 2,..., S : (4.4) J ij = P i ˆP j P i ˆP, where i =1, 2,..., S, j =1, 2,...,k, j and we how a heatmap of thi matrix in Figure 4.3(b). Since we are intereted not only in the partition but alo in recovering the ordering of the low variable, we how in Figure 4.3(c) the correlation between the ground truth ordering of the low variable and our recovered ordering. Note that we can only recover the ordering up to a global ign, ince Ψ 1 i alo an eigenvector of L. Finally, we compute the maximum weight matching (uing, for example, the Hungarian method [3]) in the bipartite graph with node et P ˆP and edge acro the two et given by matrix J in (4.4). In Figure 4.3(d) we plot the Jaccard index of the matched partition. For the firt chemical ytem CS-I, note that the algorithm perfectly recover the ground truth partition. In Figure 4.4 we preent the outcome of the binning algorithm for the illutrative example CS-II, which i no longer atifactorily by itelf and require further improvement. Though the bin cardinalitie in the initial olution viually reemble the ground truth, there are numerou mitake being made. To illutrate thi, for a given partition P, we compute the following meaure of continuity of the recovered bin cardinalitie: S 1 (4.5) Θ P =Θ(P 1,...,P S )= ( P i P i+1 ) 2. In other word, Θ P capture the quared difference in the cardinalitie of two conecutive bin. For the chemical ytem CS-II, the ground truth yield a core Θ = 18, while for the eigenvector-recovered olution Θ = 726, thu indicating already that numerou miclaification are being made, without even computing the Jaccard imilarity matrix (4.4) between the two partition. To thi end, we introduce in the next ubection a heuritic denoiing technique followed by a truncation of the domain, which altogether lead to a better partitioning of O into group of tate that correpond to the ame low variable A bin denoiing cheme. While the eigenvector-baed partition procedure detailed above yield accurate reult for CS-I in (2.2), thi procedure alone i not ufficient for obtaining a atifactory partition for the more complex CS-II conidered in (2.7), a illutrated by the high correponding Θ-core (Θ = 726) hown i=1 c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

16 ADM-CLE APPROACH FOR DETECTING SLOW VARIABLES B91 Algorithm 1. Bin-merging algorithm. 1: Initialize FLAG = TRUE 2: while FLAG i TRUE do 3: Compute α i := Θ ( ) P 1,...,P i P i+1,...p S definition (4.5) 4: if min i < Θ(P 1,...,P S ) then i=1,2,..., S 1 5: q = argmin α i i=1,2,..., S 1 6: P := P 1,...,P q P q+1,...p S 7: S = S 1 8: ele 9: FLAG = FALSE 1: end if 11: end while Number of original tate Updated cardinality low var. S = 314, Θ = Slow variable (after truncation and denoiing) (a) Our recovered 314 low variable Jaccard index heatmap Ground truth 314 low variable (b) Ground truth label of matched low variable Spearman correlation = Order of our low variable (1:314) (c) i =1, 2,... S 1uing Jaccard Similarity Index Max matching in the Jaccard index matrix Our recovered low variable (1:314) Fig Illutrative example CS-II. (a) The eigenvector-baed low variable cardinality after truncating and bin denoiing. The Theta core Θ i the moothne meaure of the bin cardinalitie, defined in (4.5). (b) The heatmap of the pairwie Jaccard imilarity matrix given by (4.4). (c) The correlation between the ordering of the ground truth low variable and the eigenvector recovered low variable. (d) The Jaccard index of the pairwie matched bin (from the maximum matching). in Figure 4.4(b). In Figure 4.4(c) we zoom into ome of the recovered bin, howing that the eigenvector-baed recontruction plit ome of the inner bin, which explain the high aociated Θ-core. In other word, tate/bin which in the ground truth olution correpond to the ame value of the low tate variable, are divided into two adjacent bin, and are mitaken for two ditinct tate of the low variable. To olve thi iue, we propoe a bin-denoiing heuritic that robutly aign data point to their repective bin. In hindight, the continuity of the eigenfunction of the Laplacian hould be reflected in the continuity of the hitogram of tate count in bin correponding to adjacent interval. We detail in Algorithm 1 an iterative heuritic procedure which, at each tep, merge two adjacent bin uch that the reulting Θ- core i minimized acro all poible pair of adjacent bin that can be merged. We how in Figure 4.5(a) the reulting bin cardinalitie after the bin-merging heuritic and after truncating at the boundary of the low variable. Note that the new denoied partition yield Θ = 13, and the number of bin (tate of the low variable) decreae from S = 328 to S = 314. Furthermore, in Figure 4.5(b) we compute the Jaccard imilarity matrix between the ground truth and the newly obtained partition, howing in Figure 4.5(d) that we almot perfectly recover the tructure of the ground truth bin. Finally, we remark that our approach extend to higher-dimenional ubpace a well; e.g., if S = l i=1 a ix i, the top eigenvector will take contant, or almot contant, (d) c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene

17 B92 MIHAI CUCURINGU AND RADEK ERBAN Algorithm 2. One iteration of the CSSA for computing the conditional ditribution P(F S = ) ofthefatvariablef given a value of the low variable S. 1: Compute the propenity function α i (t), for i =1, 2,...,m, and their um α (t) = m i=1 α i(t). 2: Generate r 1 and r 2, two uniformly ditributed random number in (, 1). 3: Compute the next reaction time a t + τ, whereτ = log(r 1 )/α (t). 4: Ue r 2 to elect reaction R j which occur at time t + τ (each reaction R i, i =1, 2,...,m, occur with probability α i /α ). 5: If the low variable S change it current tate from to due to reaction R j occurring, reet S = to it previou value while not changing the value of the fat variable F. 6: If any of the variable X i goe outide the boundary of the conidered domain, then revert to the tate of the ytem in tep 4 before reaction R j occurred. value on the tate (x 1,x 2,...,x l ) (i) correponding to the low tate S =, uch that l i=1 a ix i =. In other word, both the above-mentioned methodology that recover the mapping and the bin-denoiing cheme can be thought of in term of bin and urn and are agnotic to the dimenion of the ytem. 5. A Markov approach for computing the teady ditribution of the low variable. In thi ection, we focu on the final tep of the ADM-CLE approach, of etimating the tationary ditribution of the low variable, without any prior knowledge of what the low variable actually i. One of the ingredient needed along the way i an etimation of the conditional ditribution P(F S = ) ofthe fat variable F given a value of the low variable S, which we compute via two approache. A the firt approach, we conider the conditional tochatic imulation algorithm (CSSA) [8] which i given in Algorithm 2. It ample from the ditribution of the fat variable conditioned on the low variable. The econd approach i entirely analytic and free of any tochatic imulation and amount to analytically olving the CME for each et in the partition P = {P 1, P 2,...,P k }. We then compare our reult to the contrained multicale algorithm (CMA) introduced in [8], which approximate the effective dynamic of the low variable a an SDE, after etimating the effective drift and diffuion uing the CSSA (Algorithm 2) A tochatic imulation algorithm for etimating the conditional probability (CSSA). Our next tak i to etimate the conditional ditribution P(F S = ) ofthefatvariablef given a value of the low variable S. One poible approach for doing thi relie on the CSSA algorithm to globally integrate the effective dynamic of the low variable. One iteration of the CSSA i given in Algorithm 2. Ideally, one repeat tep 1 6 of Algorithm 2 and ample value of F until the ditribution P(F S = ) converge. In practice, we run Algorithm 2 until L c change of the low variable S occur. Thi computation i done for each value in the range of the low variable S = { 1, 2,..., S }. A an example, Table 5.1 illutrate the computation of the tranitioning probability from the low variable tate S =7toS = 8, aociated to CS-II. We remark that thi approach i in line with that of previou work [8], in which the running time of CSSA wa implicitly defined by counting up to L c change in S. We conider the ame range of value for L c a in [8], i.e., L c = {1, 1 2, 1 3, 1 4, 1 5 }, a illutrated in Figure 5.5 below. An alternative approach would be to check for convergence in ditribution by comparing the cumulative ditribution function at L c and 2L c. c 217 SIAM. Publihed by SIAM under the term of the Creative Common 4. licene