Enhancing the accuracy of a data-driven reconstruction of bivariate jump-diffusion models with corrections for higher orders of the sampling interval
aa r X i v : . [ phy s i c s . d a t a - a n ] J a n Enhancing the accuracy of a data-drivenreconstruction of bivariate jump-diffusion models withcorrections for higher orders of the sampling interval
Esra Aslim , , Thorsten Rings , , Lina Zabawa , and KlausLehnertz , , Department of Epileptology, University of Bonn Medical Centre, VenusbergCampus 1, 53127 Bonn, Germany Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn,Nussallee 14–16, 53115 Bonn, Germany Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße7, 53175 Bonn, GermanyE-mail: [email protected]
26 January 2021
Abstract.
We evaluate the significance of a recently proposed bivariate jump-diffusion model for a data-driven characterization of interactions between complexdynamical systems. For various coupled and non-coupled jump-diffusion processes, wefind that the inevitably finite sampling interval of time-series data negatively affectsthe reconstruction accuracy of higher-order conditional moments that are required toreconstruct the underlying jump-diffusion equations. We derive correction terms forconditional moments in higher orders of the sampling interval and demonstrate theirsuitability to strongly enhance the data-driven reconstruction accuracy.
Keywords : nonlinear dynamics, stochastic processes, interactions, higher-ordercorrections
1. Introduction
The problem of reliably characterizing interactions between complex dynamical systemspervades many scientific fields. Since real-world systems quite often impose restrictionsto standard approaches, linear and nonlinear time-series-analysis techniques have beendeveloped that allow one to estimate the strength, the direction, and the functionalform of an interaction from pairs of time series of appropriate system observables. Giventhat interactions can manifest themselves in various aspects of the dynamics, analysistechniques have been developed in diverse fields such as statistics, synchronizationtheory, nonlinear dynamics, information theory, and statistical physics (for an overview,see [1–6]). Most of these techniques specifically concentrate on the (low-dimensional) ata-driven reconstruction of bivariate jump-diffusion models d t , which not only influences the first- and second-order KM coefficients [24] but also causes non-vanishing higher-order ( > ) ones. For(one-dimensional) jump-diffusion processes, additional influences need to be taken intoaccount [13]: jump events induce terms of order O (d t ) in the conditional moments of evenorders and the jump rate and amplitude induce terms of order O (d t ) in all conditionalmoments. We here extend these studies and investigate the data-driven reconstructionof stochastic dynamical equations underlying interacting jump-diffusion processes withfinite sampling interval. We will show that in these cases, corrections for higher ordersof the sampling interval strongly enhance reconstruction accuracy.The outline of this paper is as follows. In section 2, we recall the definition of abivariate jump-diffusion model, and we define our scale-independent measure to assessthe accuracy of the reconstruction of conditional moments from time-series data. Insection 3, we first illustrate the reconstruction of conditional moments of various jump-diffusion models with and without couplings from time-series data with finite samplinginterval, thereby emphasizing the necessity for corrections for higher orders of thesampling interval. We then derive these corrections and demonstrate their suitabilityto enhance reconstruction accuracy. Finally, in section 4 we draw our conclusions. ata-driven reconstruction of bivariate jump-diffusion models
2. Methods
A bivariate jump-diffusion process consists of two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. It can be modeled via [8, 12, 15] d x ( t )d x ( t ) ! = h h ! d t + g , g , g , g , ! d W d W ! + ξ , ξ , ξ , ξ , ! d J d J ! . (1)The drift is a two-dimensional vector h = ( h , h ) with h ∈ R , where each dimensionof h , i.e., h i , may depend on state variables x ( t ) and x ( t ) . The diffusion takes amatrix g ∈ R × with the diagonal elements of g comprise the diffusion coefficientsof self-contained stochastic diffusive processes. The off-diagonal elements representinterdependencies between the two Wiener processes W = ( W , W ) , i.e., they resultfrom an interaction between the two processes. The Wiener processes act as independentBrownian noises for the state variables with h d W i i = 0 , h d W i i = d t, ∀ i . Thediscontinuous jump terms are contained in ξ ∈ R × and d J ∈ N , where d J representsa two-dimensional Poisson process. These are Poisson-distributed jumps with an averagejump rate λ ∈ R in unit time t . The average expected number of jumps of each jumpprocess J i in a timespan t is λ i t . The jump amplitudes ξ are Gaussian distributed withzero mean and standard deviation (or size) s i,j . We note that elements of vectors h and d J as well as of matrices g and ξ may, in general, be state- and time-dependent; forconvenience of notation, we omit these dependencies.The two-dimensional KM coefficients of orders ( ℓ, m ) of a bivariate process read D ( ℓ,m ) ( x , t, d t ) = 1( ℓ + m )! lim d t → t K ( ℓ,m ) ( x , t, d t ) (2)with the conditional moments K ( ℓ,m ) ( x , t, d t ) = (cid:10) [ x ( t + d t ) − x ( t )] ℓ [ x ( t + d t ) − x ( t )] m (cid:11)(cid:12)(cid:12)(cid:12) x ( t )= x x ( t )= x (3)which can be directly estimated from time-series data [15]. They are related to theelements of the drift vector, the elements of the diffusion matrix, and the jumpcomponents via: K (1 , ( x , t, d t ) = h d t (4) K (0 , ( x , t, d t ) = h d tK (1 , ( x , t, d t ) = h g g + g g i d tK (2 , ( x , t, d t ) = h g + s λ + g + s λ i d tK (0 , ( x , t, d t ) = h g + s λ + g + s λ i d tK (2 ℓ, m ) ( x , t, d t ) = h s ℓ s m λ + s ℓ s m λ i (2 ℓ )!2 ℓ ℓ ! (2 m )!2 m m ! d t, with ( ℓ, m ) ∈ N + . We omitted the state- and time-dependencies in the drift and diffusionfunctions and jump components to enhance readability. If such processes are coupled, ata-driven reconstruction of bivariate jump-diffusion models Following Refs. [9,15], we employ a distance measure to relate theoretical and numericalresults and to quantify the deviation of the obtained conditional moments from thefunctions employed. In order to allow a comparison of the accuracy of the reconstructionof conditional moments of order ( ℓ, m ) , we here use a scale-independent distancemeasure, U ( ℓ,m ) , which is based on the bounded relative error of the difference betweenestimated and theoretical conditional moments (see Appendix A for details). U ( ℓ,m ) < indicates a sufficient accuracy of the reconstruction of the conditional moment of order ( ℓ, m ) .We estimate conditional moments up to orders ℓ = m = 6 from normalizedtime series (zero mean and unit variance) that we obtain from numerically integratingbivariate jump-diffusion equations (Euler-Maruyama scheme [25] with a samplinginterval d t = 10 − ). Time series consist of n = 10 data points (after eliminating × transients), and we ensure that individual jump numbers n j ≃ λn varied by at most10 % for a constant jump rate.
3. Results
For our investigations, we consider various interacting jump-diffusion processes. Webegin with reconstructing conditional moments of a bivariate jump-diffusion model withuni-directional couplings in the drift and in the diffusion from time-series data. Sinceinteractions between jump-diffusion processes may lead to over- and underrepresentedparts of the dynamics [15], we then reconstruct conditional moments of a bivariate jump-diffusion model with a disproportionally weighted drift, diffusion, and jump part fromtime-series data. ata-driven reconstruction of bivariate jump-diffusion models Figure 1. a) Reconstruction accuracy ( U ( ℓ,m ) ) for conditional moments up to order ℓ = m = 6 of a bivariate jump-diffusion model with a uni-directional coupling in thediffusion part (see equation (5)) for various values of coupling strength c . U ( ℓ,m ) < (horizontal dotted line) indicates a sufficient accuracy. Medians and interquartileranges (shaded area) derived from 50 time series generated with the respective modelsusing random initial conditions. Lines are for eye guidance only. b) Two-dimensionaltheoretical conditional moments (equation (4)) up to order ℓ = m = 6 (black grid)and reconstructed ones (gray surface) from exemplary time series generated usingequation (5) with c = 0 and c = 100 . Reconstruction accuracy is insufficient forconditional moments of orders (2 , , (0 , , (4 , , (0 , , and (6 , ( U (2 , ≈ , U (0 , ≈ , U (4 , ≈ , U (0 , ≈ , and U (6 , ≈ ). ata-driven reconstruction of bivariate jump-diffusion models We examine uni-directional couplings in the drift and in the diffusion of an exemplarybivariate jump-diffusion process modeled via: d x ( t )d x ( t ) ! = − x + x − x + f ( x ) ! d t + . . . . f ( x ) ! d W d W ! (5) + ξ , ξ , ξ , ξ , ! d J d J ! , with s = . . . . ! , λ = . . ! and with the coupling terms f ( x ) = c x and f ( x ) = c x . We vary the couplingstrengths c and c each over four orders of magnitude and reconstruct conditionalmoments from 50 time series with random initial conditions and with finite samplinginterval. For couplings in the drift part ( c ∈ [0 . , ; c = 0 ), almost all conditionalmoments up to order ℓ = m = 6 can be reconstructed with sufficient accuracy (data notshown). An exception builds the conditional moment K (0 , with diffusion contributionsof process x . For this moment, we observe an insufficient reconstruction accuracy( U (0 , > ) in case of strong couplings ( c > ). For couplings in the diffusionpart ( c ∈ [0 . , ; c = 0 ; see figure 1), we observe at large values of the couplingstrength ( c > ) rather strong inaccuracies for some conditional moments. These includeconditional moments with jump contributions of process x , namely K (0 , , K (0 , , and K ( i,i ) with i ∈ { , , } (figure 1a). A visual inspection of these conditional moments fora coupling in the diffusion part with c = 100 (figure 1b) indicates that the theoreticalmoments (equation (4)) fail to account for a dependency on process x . This dependencyon process x appears similar to the dependency of K (0 , on x , where we note that K (0 , contains information about the coupling in the diffusion part (see equation (4)). Next, we examine processes derived from an exemplary bivariate jump-diffusion modelwith disproportionally weighted parts, which we obtain by rescaling drift, diffusion, andjump dynamics. For the latter two, we allow for a mixing of the Wiener processes anda mixing of the Poisson processes (non-vanishing off-diagonal elements in diffusion andjump size matrix): d x ( t )d x ( t ) ! = − x + x − αx ! d t + . . β . ! d W d W ! + ξ , ξ , ξ , ξ , ! d J d J ! , (6)with s = . . γ . ! , λ = . . ! . ata-driven reconstruction of bivariate jump-diffusion models α denotes the drift-, β the diffusion- and γ the jump-scaling parameter. We rescalethe jump size since it equals the variance of the Gaussian-distributed jump amplitudes Figure 2.
Reconstruction accuracy ( U ( ℓ,m ) ) of conditional moments up to order ℓ = m = 6 of a bivariate jump-diffusion model with disproportionally weighted parts(see equation (6)) for various values of drift-, diffusion-, and jump-scaling parameters( α , β , γ ). The vertical dotted line at β = g , = 0 . and γ = 0 . indicates thepoint where the diffusion-scaling parameter is equal to the jump-scaling parameter.The horizontal dotted line indicates a sufficient accuracy ( U ( ℓ,m ) < ). Medians andinterquartile ranges (shaded area) derived from 50 time series generated with therespective models using random initial conditions. Lines are for eye guidance only. ata-driven reconstruction of bivariate jump-diffusion models β = 0 . and γ = 0 . ), almost all conditionalmoments up to order ℓ = m = 6 can be reconstructed with sufficient accuracy (figure 2).The inaccuracy seen for K (0 , for α < is to be expected given our chosen bivariatejump-diffusion model. If we rescale the diffusion part (with α = 1 and γ = 0 . ), weobserve at large values of the scaling parameter ( β > ) rather strong inaccuracies forconditional moments K (0 , , K (0 , , and K ( i,i ) with i ∈ { , , } (figure 2). As with jump-diffusion models with uni-directional couplings (section 3.1), these conditional momentscontain jump contributions of process x . Rescaling the jump part (with α = 1 and β = 0 . ) has no effect for the considered range of values here ( γ ∈ [0 . , ; seefigure 2). Our findings presented above demonstrate that uni-directional couplings in the diffusionmay have a similar impact on the accuracy of the reconstruction of conditional momentsas rescaling the diffusion part. Estimated conditional moments differ from the respectivetheoretical moments with jump contributions of process x at large values of either thecoupling strength or the scaling parameter. Given these observations and since we knowthat a finite sampling interval may have a non-negligible impact on the reconstructionof higher-order conditional moments of one-dimensional jump-diffusion models [13] fromtime-series data, we conjecture that a similar impact can be expected for bivariate jump-diffusion models.To test this conjecture, we derive the theoretical conditional moments for differentorders of the sampling interval d t of a bivariate jump-diffusion model (the derivationand further expressions of conditional moments can be found in Appendix B). With theabbreviations A (1 , = h A (0 , = h B (1 , = 12 h g g + g g i B (2 , = 12 h g + s λ + g + s λ i B (0 , = 12 h g + s λ + g + s λ i C (2 ℓ, m ) = 1(2 ℓ + 2 m )! h s ℓ s m λ + s ℓ s m λ i (2 ℓ )!2 ℓ ℓ ! (2 m )!2 m m ! , where ( ℓ, m ) ∈ N + , the theoretical conditional moments of orders (0 , and (2 , with ata-driven reconstruction of bivariate jump-diffusion models O (d t ) read K (0 , ( x , t, d t ) = 4! C (0 , d t + 12 h (cid:0) B (0 , (cid:1) + 4! (cid:0) A (0 , ∂ x C (0 , + A (1 , ∂ x C (0 , (cid:1) + 4 · C (0 , ∂ x A (0 , + 4! (cid:0) B (0 , ∂ x C (0 , + B (2 , ∂ x C (0 , + 2 B (1 , ∂ x ∂ x C (0 , (cid:1) + 6 · (cid:0) C (0 , ∂ x B (0 , + C (2 , ∂ x B (0 , (cid:1) + 4 · (cid:0) C (0 , ∂ x A (0 , + 3 C (2 , ∂ x ∂ x A (0 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (0 , + 3 · C (2 , ∂ x ∂ x B (0 , + O ( δ ) i d t + O (d t ) and K (2 , ( x , t, d t ) = 4! C (2 , d t + 12 h (cid:0) B (2 , B (0 , + 2 (cid:0) B (1 , (cid:1) (cid:1) + 4! (cid:0) A (1 , ∂ x C (2 , + A (0 , ∂ x C (2 , (cid:1) + 2 · (cid:0) C (2 , ∂ x A (1 , + C (2 , ∂ x A (0 , (cid:1) + 4! (cid:0) B (2 , ∂ x C (2 , + B (0 , ∂ x C (2 , + 2 B (1 , ∂ x ∂ x C (2 , (cid:1) + 4! (cid:0) C (4 , ∂ x B (0 , + C (2 , ∂ x B (0 , + C (2 , ∂ x B (2 , + C (0 , ∂ x B (2 , + 8 C (2 , ∂ x ∂ x B (1 , (cid:1) + 2 · (cid:0) C (4 , ∂ x A (1 , + 3 C (2 , ∂ x ∂ x A (1 , + C (2 , ∂ x A (0 , + 3 C (4 , ∂ x ∂ x A (0 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (2 , + 5!2! (cid:0) (cid:0) C (4 , ∂ x ∂ x B (0 , + C (2 , ∂ x ∂ x B (2 , (cid:1) + 16 (cid:0) C (4 , ∂ x ∂ x B (1 , + C (2 , ∂ x ∂ x B (1 , (cid:1)(cid:1) + O ( δ ) i d t + O (d t ) . For the differential operator, we use the short notation ∂ x i = ∂∂x i . With O ( δ ) , we indicateall terms that contain C ( ℓ,m ) of higher-order or derivatives ∂ jx i , j > , and with O (d t ) all terms that contain higher orders ( ≥ of the sampling interval d t . We note thatthe terms of order O (d t ) can introduce drift, diffusion and jump contributions to eachconditional moment.With these correction terms, we now focus on conditional moments, which areaffected by a uni-directional coupling in the diffusion or a more weighted diffusion part.Already a visual inspection reveals that considering terms of order O (d t ) can clearlyimprove the reconstruction of some conditional moments (see figure 3; U ( ℓ,m ) corr indicatesthe distance measure for which correction terms were considered). For the data shownin figure 3a, the correction terms of order O (d t ) can be of the same or even greatermagnitude than terms of order O (d t ) and thus have a non-negligible effect on theaccuracy of the reconstruction of conditional moments. Particularly the accuracy ofthe reconstruction of moments K (0 , and K (2 , is considerably improved ( U (0 , corr < and U (2 , corr < ; see figure 4) even at large values of the coupling strength or at largevalues of the diffusion-scaling parameter. However, we still observe inaccuracies in thereconstruction of conditional moments K (0 , , K (4 , , and K (6 , , and we expect that ata-driven reconstruction of bivariate jump-diffusion models d t ( O (d t i ) , i ≥ ) will further improve the accuracyof the reconstruction of these conditional moments. Figure 3. a) Same as figure 1b) but in addition we present theoretical conditionalmoments, for which we considered terms of O (d t ) (red grid). In this case, we obtaina sufficient quality of the reconstruction of conditional moments of all shown ordersexcept of (0 , , (4 , and (6 , ( U (0 , corr ≈ , U (4 , corr ≈ and U (6 , corr ≈ ). b) Sameas a) but for a bivariate jump-diffusion model with a more weighted diffusion part(see equation (6) with α = 1 , β = 100 and γ = 0 . ). By considering terms of order O (d t ) in the theoretical conditional moments, we obtain a sufficient quality of thereconstruction of conditional moments of all orders except of (0 , , (4 , and (6 , ( U (0 , corr ≈ , U (4 , corr ≈ , U (6 , corr ≈ ). ata-driven reconstruction of bivariate jump-diffusion models Figure 4.
Reconstruction accuracies with ( U ( ℓ,m ) corr ; red) and without ( U ( ℓ,m ) ;black) integrating correction terms of order O (d t ) for various conditional moments(cf. figure 1a and figure 2). a) Bivariate jump-diffusion model with uni-directionalcouplings (see equation (5)) for various values of coupling strength c . b) Bivariatejump-diffusion model (see equation (6)) for various values of the diffusion-scalingparameter β . The horizontal dotted line indicates a sufficient accuracy. Medians andinterquartile ranges (shaded area) derived from 50 time series generated with therespective models using random initial conditions. Lines are for eye guidance only.
4. Concluding remarks
We evaluate the significance of a bivariate jump-diffusion model for a data-drivencharacterization of interactions between complex dynamical systems. Investigatingvarious coupled and non-coupled jump-diffusion processes, we observed strong deviationsbetween conditional moments of the underlying jump-diffusion model and thoseestimated from time-series data and conjectured that these deviations result from thefiniteness of the sampling interval. We derived correction terms for conditional moments ata-driven reconstruction of bivariate jump-diffusion models O (d t ) in all conditional moments and are most pronounced in conditional momentswith jump contributions (orders ≥ ). A blending of all parts of the dynamics shouldthus be taken into account when investigating interacting jump-diffusion processes. Tofurther enhance the significance of the bivariate jump-diffusion model for the analysis ofempirical data, future studies should investigate other possible influencing factors suchas measurement noise, limited observation time (finite number of data points), or theimpact of indirect interactions mediated by observed/unobserved additional processes. Acknowledgments
We are grateful to M. Reza Rahimi Tabar for constructive discussions and valuablecomments. ata-driven reconstruction of bivariate jump-diffusion models AppendixAppendix A. Scale-independent measure to assess the accuracy of adata-driven reconstruction of conditional moments
For each two-dimensional conditional moments of order ( ℓ, m ) , we consider the unscaledmean bounded relative absolute error [26] U ( ℓ,m ) = R ( ℓ,m ) b − R ( ℓ,m ) b , with the weighted average of bounded relative errors R ( ℓ,m ) b = B X i,j =1 p ( x ,i , x ,j ) (cid:12)(cid:12)(cid:12) ∆ ( ℓ,m ) ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ( ℓ,m ) ij (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) K ( ℓ,m ) ( x ,i , x ,j , d t ) (cid:12)(cid:12)(cid:12) , where ∆ ( ℓ,m ) ij = ˆ K ( ℓ,m ) ( x ,i , x ,j , d t ) − K ( ℓ,m ) ( x ,i , x ,j , d t ) is the difference between theestimated and theoretical conditional moment, p ( · ) the estimated probability densitythat is used as normalized weight, and B the number of bins for each dimension. U ( ℓ,m ) < indicates a sufficient accuracy of the reconstruction of conditionalmoments of order ( ℓ, m ) in the sense that ∆ ( ℓ,m ) ij is smaller than K ( ℓ,m ) ( x ,i , x ,j , d t ) on average. We note that the value of U ( ℓ,m ) becomes large or undefined if the value ofthe theoretical conditional moments tends to , which can lead to a misinterpretation.In our histogram-based investigations, we used B = 20 bins for each dimensionand considered a range of ± σ i for each x i . We refer to Ref. [27] for a discussion on theoptimal choice of the number of bins and to Refs. [28, 29] for other, e.g., kernel-basedestimation techniques. Appendix B. Derivation of conditional moments of bivariate jump-diffusionmodels for different orders of d t We follow Refs. [8, 13] to derive conditional moments of bivariate jump-diffusionmodels for different orders of d t using the Kramers-Moyal adjoint operator. With theabbreviations A (1 , = h A (0 , = h B (1 , = 12 h g g + g g i B (2 , = 12 h g + s λ + g + s λ i B (0 , = 12 h g + s λ + g + s λ i C (2 ℓ, m ) = 1(2 ℓ + 2 m )! h s ℓ s m λ + s ℓ s m λ i (2 ℓ )!2 ℓ ℓ ! (2 m )!2 m m ! , ata-driven reconstruction of bivariate jump-diffusion models ( ℓ, m ) ∈ N + , one can find the following corrections for conditional moments oforders ℓ = m = 6 of a bivariate jump-diffusion model: K (1 , ( x , t, d t ) = A (1 , d t + 12 h A (1 , ∂ x A (1 , + A (0 , ∂ x A (1 , + B (2 , ∂ x A (1 , + B (0 , ∂ x A (1 , + 2 B (1 , ∂ x ∂ x A (1 , + 6 C (2 , ∂ x ∂ x A (1 , + O ( δ ) i d t + O (d t ) K (2 , ( x , t, d t ) = 2 B (2 , d t + 12 h (cid:0) A (1 , (cid:1) + 2 (cid:0) A (1 , ∂ x B (2 , + A (0 , ∂ x B (2 , (cid:1) + 4 (cid:0) B (2 , ∂ x A (1 , + B (1 , ∂ x A (1 , (cid:1) + 2 (cid:0) B (2 , ∂ x B (2 , + B (0 , ∂ x B (2 , + 2 B (1 , ∂ x ∂ x B (2 , (cid:1) + 8 (cid:0) C (4 , ∂ x A (1 , + 3 C (2 , ∂ x ∂ x A (1 , (cid:1) + 12 C (2 , ∂ x ∂ x B (2 , + 5! C (4 , ∂ x ∂ x A (1 , + O ( δ ) i d t + O (d t ) K (4 , ( x , t, d t ) = 4! C (4 , d t + 12 h (cid:0) B (2 , (cid:1) + 4! (cid:0) A (1 , ∂ x C (4 , + A (0 , ∂ x C (4 , (cid:1) + 4 · C (4 , ∂ x A (1 , + 4! (cid:0) B (2 , ∂ x C (4 , + B (0 , ∂ x C (4 , + 2 B (1 , ∂ x ∂ x C (4 , (cid:1) + 6 · (cid:0) C (4 , ∂ x B (2 , + C (2 , ∂ x B (2 , (cid:1) + 4 · (cid:0) C (6 , ∂ x A (1 , + 3 C (4 , ∂ x ∂ x A (1 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (4 , + 3 · C (4 , ∂ x ∂ x B (2 , + O ( δ ) i d t + O (d t ) K (6 , ( x , t, d t ) = 6! C (6 , d t + 12 h · B (2 , C (4 , + 6! (cid:0) A (1 , ∂ x C (6 , + A (0 , ∂ x C (6 , (cid:1) + 6 · C (6 , ∂ x A (1 , + 6! (cid:0) B (2 , ∂ x C (6 , + B (0 , ∂ x C (6 , + 2 B (1 , ∂ x ∂ x C (6 , (cid:1) + 6 · (cid:0) C (4 , ∂ x C (4 , + C (2 , ∂ x C (4 , (cid:1) + (6!) (cid:0) C (6 , ∂ x B (2 , + C (4 , ∂ x B (2 , (cid:1) + 6 · (cid:0) C (8 , ∂ x A (1 , + 3 C (6 , ∂ x ∂ x A (1 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (6 , + 6 · (6!) C (4 , ∂ x ∂ x C (4 , + 30 · C (6 , ∂ x ∂ x B (2 , + O ( δ ) i d t + O (d t ) ata-driven reconstruction of bivariate jump-diffusion models K (1 , ( x , t, d t ) = 2 B (1 , d t + 12 h A (1 , A (0 , + 2 (cid:0) A (1 , ∂ x B (1 , + A (0 , ∂ x B (1 , (cid:1) + 2 (cid:0) B (2 , ∂ x A (0 , + B (1 , ∂ x A (0 , + B (1 , ∂ x A (1 , + B (0 , ∂ x A (1 , (cid:1) + 2 (cid:0) B (2 , ∂ x B (1 , + B (0 , ∂ x B (1 , + 2 B (1 , ∂ x ∂ x B (1 , (cid:1) + 4 (cid:0) C (4 , ∂ x A (0 , + 3 C (2 , ∂ x ∂ x A (0 , + C (0 , ∂ x A (1 , + 3 C (2 , ∂ x ∂ x A (1 , (cid:1) + 12 C (2 , ∂ x ∂ x B (1 , + 5!2! (cid:0) C (4 , ∂ x ∂ x A (0 , + C (2 , ∂ x ∂ x A (1 , (cid:1) + O ( δ ) i d t + O (d t ) K (2 , ( x , t, d t ) = 4! C (2 , d t + 12 h (cid:0) B (2 , B (0 , + 2 (cid:0) B (1 , (cid:1) (cid:1) + 4! (cid:0) A (1 , ∂ x C (2 , + A (0 , ∂ x C (2 , (cid:1) + 2 · (cid:0) C (2 , ∂ x A (1 , + C (2 , ∂ x A (0 , (cid:1) + 4! (cid:0) B (2 , ∂ x C (2 , + B (0 , ∂ x C (2 , + 2 B (1 , ∂ x ∂ x C (2 , (cid:1) + 4! (cid:0) C (4 , ∂ x B (0 , + C (2 , ∂ x B (0 , + C (2 , ∂ x B (2 , + C (0 , ∂ x B (2 , + 8 C (2 , ∂ x ∂ x B (1 , (cid:1) + 2 · (cid:0) C (4 , ∂ x A (1 , + 3 C (2 , ∂ x ∂ x A (1 , + C (2 , ∂ x A (0 , + 3 C (4 , ∂ x ∂ x A (0 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (2 , + 5!2! (cid:0) (cid:0) C (4 , ∂ x ∂ x B (0 , + C (2 , ∂ x ∂ x B (2 , (cid:1) + 16 (cid:0) C (4 , ∂ x ∂ x B (1 , + C (2 , ∂ x ∂ x B (1 , (cid:1)(cid:1) + O ( δ ) i d t + O (d t ) K (4 , ( x , t, d t ) = 8! C (4 , d t + 12 h · (cid:0) B (2 , C (2 , + B (0 , C (4 , (cid:1) + 2 · (cid:0) C (4 , C (0 , + 18 (cid:0) C (2 , (cid:1) (cid:1) + 12 · (cid:0) C (2 , B (2 , + C (4 , B (0 , (cid:1) + 8! (cid:0) A (1 , ∂ x C (4 , + A (0 , ∂ x C (4 , (cid:1) + 4 · (cid:0) C (4 , ∂ x A (1 , + C (4 , ∂ x A (0 , (cid:1) + 8! (cid:0) B (2 , ∂ x C (4 , + B (0 , ∂ x C (4 , + 2 B (1 , ∂ x ∂ x C (4 , (cid:1) + 3 · (cid:0) C (4 , ∂ x C (2 , + C (2 , ∂ x C (2 , + C (0 , ∂ x C (4 , + C (2 , ∂ x C (4 , (cid:1) + 12 · (cid:0) C (6 , ∂ x C (0 , + C (4 , ∂ x C (0 , + C (2 , ∂ x C (4 , + C (0 , ∂ x C (4 , + 36 (cid:0) C (4 , ∂ x C (2 , + C (2 , ∂ x C (2 , (cid:1)(cid:1) + 6 · (cid:0) C (4 , ∂ x B (2 , + C (2 , ∂ x B (2 , + C (6 , ∂ x B (0 , + C (4 , ∂ x B (0 , + 16 · C (4 , ∂ x ∂ x B (1 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (4 , + 6 · (cid:0) C (6 , ∂ x ∂ x C (0 , + C (2 , ∂ x ∂ x C (4 , + 36 C (4 , ∂ x ∂ x C (2 , (cid:1) + O ( δ ) i d t + O (d t ) ata-driven reconstruction of bivariate jump-diffusion models K (6 , ( x , t, d t ) = 12! C (6 , d t + 12 h · (cid:0) B (2 , C (4 , + B (0 , C (6 , (cid:1) + 6!8!2! (cid:0) C (4 , C (2 , + C (0 , C (6 , + 15 C (2 , C (4 , (cid:1) + 2 · (cid:0) C (6 , C (0 , + (6!) (2!4!) C (2 , C (4 , (cid:1) + 6!8!2! (cid:0) C (2 , C (4 , + C (6 , C (0 , + 15 C (4 , C (2 , (cid:1) + 30 · (cid:0) C (4 , B (2 , + C (6 , B (0 , (cid:1) + 12! (cid:0) A (1 , ∂ x C (6 , + A (0 , ∂ x C (6 , (cid:1) + 6 · (cid:0) C (6 , ∂ x A (1 , + C (6 , ∂ x A (0 , (cid:1) + 12! (cid:0) B (2 , ∂ x C (6 , + B (0 , ∂ x C (6 , + 2 B (1 , ∂ x ∂ x C (6 , (cid:1) + 6!10!2!2! (cid:0) C (4 , ∂ x C (4 , + C (2 , ∂ x C (4 , + C (2 , ∂ x C (6 , + C (0 , ∂ x C (6 , (cid:1) + 30 · (cid:0) C (6 , ∂ x C (2 , + C (4 , ∂ x C (2 , + C (2 , ∂ x C (6 , + C (0 , ∂ x C (6 , + 15 (cid:0) C (4 , ∂ x C (4 , + C (2 , ∂ x C (4 , (cid:1)(cid:1) + 6!8!2! (cid:16) C (8 , ∂ x C (0 , + C (6 , ∂ x C (0 , + C (2 , ∂ x C (6 , + C (0 , ∂ x C (6 , + (6!) (4!2!) (cid:0) C (4 , ∂ x C (4 , + C (2 , ∂ x C (4 , + C (6 , ∂ x C (2 , + C (4 , ∂ x C (2 , (cid:1)(cid:17) + 6!10!2!2! (cid:0) C (4 , ∂ x C (4 , + C (2 , ∂ x C (4 , + C (8 , ∂ x C (0 , + C (6 , ∂ x C (0 , + 15 (cid:0) C (6 , ∂ x C (2 , + C (4 , ∂ x C (2 , (cid:1)(cid:1) + 15 · (cid:0) C (8 , ∂ x B (0 , + C (6 , ∂ x B (0 , + C (6 , ∂ x B (2 , + C (4 , ∂ x B (2 , + 4!4!5! C (6 , ∂ x ∂ x B (1 , (cid:1) + 6 · C (2 , ∂ x ∂ x C (6 , + 30 · (cid:0) C (6 , ∂ x ∂ x C (2 , + C (2 , ∂ x ∂ x C (6 , + 15 C (4 , ∂ x ∂ x C (4 , (cid:1) + (6!) (4!2) (cid:0) C (4 , ∂ x ∂ x C (4 , + C (6 , ∂ x ∂ x C (2 , (cid:1) + 90 · (cid:0) C (4 , ∂ x ∂ x C (4 , + C (8 , ∂ x ∂ x C (0 , + 15 C (6 , ∂ x ∂ x C (2 , (cid:1) + O ( δ ) i d t + O (d t ) For the differential operator, we use the short notation ∂ x i = ∂∂x i . With O ( δ ) , weindicate all terms that contain C ( ℓ,m ) of higher-order or derivatives ∂ jx i , j > , andwith O (d t ) all terms that contain higher orders ( ≥ of the sampling interval d t . 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