Modeling discretely observed time series is a challenging problem that arises in quantitative sciences and many fields of engineering. While it is possible to tackle such problems with difference equations or ordinary differential equations (ODEs), both approaches suffer from serious drawbacks. Difference equations are difficult to apply if the observation times are unevenly distributed. Furthermore, they do not generalize well across observation frequencies, while natural laws do not care about this artificial construct. ODEs can deal with these challenges, but fail to incorporate the inherent stochastic behavior present in many physical, chemical or biological systems. These effects can be captured by introducing stochasticity in the dynamics model, which brings to stochastic differential equations (SDEs). In this paper, we will use exclusively the Itô-form
is the time-dependent vector of states we would like to model,collects the parameters of the model, is the drift term, is the vector-valued diffusion function and is a Wiener-process of the same dimension as the state vector .
While SDEs can efficiently capture stochasticity and deal with unevenly spaced observation times or observation frequency, inference is rather challenging. Due to the stochasticity of , the state vector
is itself a random variable. Except for few special cases, it is not possible to find an analytic solution for the statistics offor general drift and diffusion terms. The problem is even more challenging if we were to condition on or state-estimate some discrete time observations (filtering/smoothing) or infer some statistics for the parameters (parameter inference). It is well known that the parameter inference problem is a difficult task, with most approaches either being very sensitive to initialization (Picchini, 2007)
, strongly dependent on the choice of hyperparameters like the spacing of the integration grid(Bhat et al., 2015) or using excessive amount of computational resources even for small scale systems and state-of-the-art implementation (Ryder et al., 2018).
The difficulty of the parameter estimation problem of estimating parameters of drift and diffusion under observational noise is readily exemplified by the fact that even major scientific programming environment providers like are still lack an established toolbox for practical use. In this paper, we will take a step into a novel direction tackling this open and exciting research question.
1.1 Related Work
While it is impossible to cover all related research efforts, we would like to give a quick overview by mentioning some of the most relevant. For a more in-depth discussion, we recommend Tronarp and Särkkä (2018), who provide an excellent review of the current state-of-the-art smoothing schemes. Moreover, Sørensen (2004) and Nielsen et al. (2000) provide extensive explanations of the more traditional approaches.
Most classical methods rely on calculating . Since is usually analytically intractable, approximation schemes are necessary. Elerian et al. (2001) and Eraker (2001) used the Euler-Maruyama discretization to approximate on a fixed, fine grid of artificial observation times later to be leveraged in a MCMC sampling scheme. Pieschner and Fuchs (2018) and van der Meulen et al. (2017) subsequently refined this approach with improved bridge constructs and incorporated partial observability. Ryder et al. (2018) recently followed up on this idea by combining discretization procedures with variational inference. Särkkä et al. (2015) investigate different approximation methods based on Kalman filtering, while Archambeau et al. (2007) and Vrettas et al. (2015) use a variational Gaussian Process-based approximation. Finally, it should also be mentioned that can be inferred by solving the Fokker-Planck-Kolmogorov equations using standard numerical methods for PDEs (Hurn and Lindsay, 1999; Aït-Sahalia, 2002).
Instead of approximating in a variational fashion, Gaussian Processes can as well be used to directly model and , ignoring in this way any prior knowledge about their parametric form. This approach was investigated by Ruttor et al. (2013), whose linearization and discretization assumptions which were later relaxed by Yildiz et al. (2018). While we will show in our experiments that these methods can be used for parameter estimation if the parametric form of drift and diffusion are known, it should be noted that parameter inference was not the original goal of their work.
1.2 Our Work
To the best of our knowledge, there are only very few works that try to circumvent calculating
at all. Our approach is probably most closely related to the ideas presented byRiesinger et al. (2016). Our proposal relies on the Doss-Sussman transformation (Doss, 1977; Sussmann, 1978) to reduce the parameter inference problem to parameter inference in an ensemble of random ordinary differential equations (RODEs). These equations can then be solved path-wise using either standard numerical schemes or using the computationally efficient gradient matching scheme of Gorbach et al. (2017) as proposed by Bauer et al. (2017).
The path-wise method of Bauer et al. (2017) has natural parallelization properties, but there is still an inherent approximation error due to the Monte Carlo estimation of the expectation over the stochastic element in the RODE. Furthermore, their framework imposes severe linearity restrictions on the functional form of the drift , while it is unable to estimate the diffusion matrix .
While we will keep their assumption of a constant diffusion matrix, i.e. , our approach gets rid of the linearity assumptions on the drift . Furthermore, we substitute the Monte Carlo approximation by embedding the SDE into a fully statistical framework, allowing for efficient estimation of both and using state-of-the-art statistical inference methods.
Even though a constant diffusion assumption might seem restrictive at first, such SDE models are widely used, e.g. in chemical engineering (Karimi and McAuley, 2018), civil engineering (Jiménez et al., 2008), pharmacology (Donnet and Samson, 2013) and of course in signal processing, control and econometrics. While we believe that this approach could be extended approximately to systems with general diffusion matrices, we leave this for future work.
The contributions of our framework are the following:
We derive a new statistical framework for diffusion and drift parameter estimation of SDEs using the Doss-Sussmann transformation and Gaussian Processes.
We introduce a grid-free, computationally efficient and robust parameter inference scheme that combines a non-parametric Gaussian Process model with adversarial loss functions.
We demonstrate that our method is able to estimate constant but non-diagonal diffusion terms of stochastic differential equations without any functional form assumption on the drift.
We show that our method significantly outperforms the state-of-the-art algorithms for SDEs with multi-modal state posteriors, both in terms of diffusion and drift parameter estimation.
We share and publish our code to facilitate future research.111Code available at https://github.com/gabb7/AReS-MaRS
In this section, we formalize our problem and introduce the necessary notation and background drawn from Gaussian process-based gradient matching for ODEs.
2.1 Problem Setting
We consider SDEs of the form
where is the -dimensional state vector at time ; are the increments of a standard Wiener process; is an arbitrary, potentially highly nonlinear function whose parametric form is known, save for the unknown parameter vector ; is the unknown but constant diffusion matrix. Without loss of generality, we can assume to be a lower-diagonal, positive semi-definite matrix.
The system is observed at arbitrarily spaced time points , subjected to Gaussian observation noise:
where we assume the noise variances to be state-dependent but time-independent, i.e.
for and .
2.2 Deterministic ODE Case
In the context of Bayesian parameter inference for deterministic ordinary differential equations, Calderhead et al. (2009) identify numerical integration as the main culprit for bad computational performance. Thus, they propose to turn the parameter estimation procedure on its head: instead of calculating using numerical integration, they extract two probabilistic estimates for the derivatives, one using only the noisy observations and one using the differential equations. The main challenge is then to combine these two distributions, such that more information about can guide towards good parameter estimates . For this purpose, Calderhead et al. (2009)
propose a product of experts heuristics that was accepted and reused until recentlyWenk et al. (2018)
showed that this heuristic leads to severe theoretical issues. They instead propose an alternative graphical model, forcing equality between the data based and the ODE based model save for a Gaussian distributed slack variable.
In this paper, we use another interpretation of gradient matching, which is aimed at finding parameters such that the two distributions over match as closely as possible. Acknowledging the fact that standard methods like minimizing the KL divergence are not tractable, we use robust moment matching techniques while solving a much harder problem with . However, it should be clear that our methodology could easily be applied to the special case of deterministic ODEs and thus provides an additional contribution towards parameter estimation for systems of ODEs.
Throughout this paper, bold, capital letters describe matrices. Values of a time-dependent quantities such as the state vector can be collected in the matrix of dimensions , where the -th row collect the single-state values at times for the state .
The matrix can be vectorized by concatenating its rows and defining in this way the vector . This vector should not be confused with , which is still a time-dependent vector of dimension .
As we work with Gaussian processes, it is useful to standardize the state observations by subtracting the mean and dividing by the standard deviation, in a state-wise fashion. We define the vector of the data standard deviation, and the matrix as:
where indicates the Kronecker product and
is the identity matrix of size. Similarly for the means, we can define the vector that contains the state-wise means of the observations, each repeated times. Thus the standardize vector can be defined as:
For the sake of clarity, we omit the normalization in the following sections. It should be noted however that in a finite sample setting, standardization strongly improves the performance of GP regression. In both our implementation and all the experiments shown in section 4, we assume a GP prior on that were standardized using the state-wise mean and standard deviation of the observations .
In the deterministic case with , the GP regression model can be directly applied to the states . However, if , the derivatives of the states with respect to time do no longer exist due to the contributions of the Wiener process. Thus, performing direct gradient matching on the states is not feasible.
3.1 Latent States Representation
We propose to tackle this problem by introducing a latent variable , defined via the linear coordinate transformation
where is the solution of the following SDE:
Without loss of generality, we set and thus . While in principle the framework supports any initial condition as long as , the reasons for this choice will become clear in section 3.2.3.
Using Itô’s formula, we obtain the following SDE for
This means that for a given realization of , we obtain a differentiable latent state . In principle, we could sample realizations of and solve the corresponding deterministic problems, which is equivalent to approximately marginalizing over . However, it is actually possible to treat this problem statistically, completely bypassing such marginalization. We do this by creating probabilistic generative models for observations and derivatives analytically. The equations are derived in this section, while the final models are shown in Figure 1.
3.2 Generative Model for Observations
Let us define as the Gaussian observation error at time . Using the matrix notation introduced in section 2.3, we can write
where and are the matrices corresponding to the lower-case variables introduced in the previous section. In contrast to standard GP regression, we have an additional additive noise term , which is the result of the stochastic process described by equation (8). As in standard GP regression, it is possible to recover a closed form Gaussian distribution for each term.
3.2.1 GP Prior
We assume a zero-mean Gaussian prior over the latent states , whose covariance matrix is given by a kernel function , in turn parameterized by the hyperparameter vector :
We treat all state dimensions as independent, meaning that we put independent GP priors with separate hyperparameters on the time evolution of each state. Consequently, will be a block diagonal matrix with blocks each of dimension . The blocks model the correlation over time introduced by the GP prior.
3.2.2 Error Model
3.2.3 Ornstein-Uhlenbeck Process
Through the coordinate transformation in equation (7), all stochasticity is captured by the stochastic process described by equation (8). Such mathematical construct has a closed-form, Gaussian solution and is called Ornstein-Uhlenbeck process. For the one-dimensional case with zero initial condition and unit diffusion
we get the following mean and covariance:
Sampling at the discrete sample points yields the vector , which is Gaussian distributed:
where as given by equation (16). In the case of a -dimensional process with identity diffusion, i.e.
we can just treat each state dimension as an independent, one-dimensional OU process. Thus, after sampling times at the time points in and unrolling the resulting matrix as described in section 2.3, we get
where is a block diagonal matrix with one for each state dimension.
Using Itô’s formula, we can show that the samples of the original Ornstein-Uhlenbeck process at each time point can be obtained via the linear coordinate transformation
be defined as the matrix that performs this linear transformation for the unrolled vectors. We can then write the density of the original Ornstein-Uhlenbeck process as
3.2.4 Marginals of the Observations
Using equation (10), the marginal distribution of can be computed as the sum of three independent Gaussian-distributed random variables with zero mean, described respectively by equations (11), (13) and (21). Thus, is again Gaussian-distributed, according to
It is important to note that due to the latent state representation, the diffusion matrix is now a part of the hyperparameters of the observation model. It will later be inferred alongside the hyperparameters of our GP using maximum evidence (Rasmussen, 2004). Using a stationary kernel , captures the stationary part of as in standard GP regression, while the parameters in describe the non-stationary part due deriving from . This ultimately leads to an identifiable problem.
3.3 Generative Model for Derivatives
Similarly to gradient matching approaches, we create two generative models for our derivatives, one based on the data and one based on the SDE model.
3.3.1 Data-based Model
3.3.2 SDE-based Model
There also is a second way of obtaining an expression for the derivatives of , namely using equation (9):
where represents the dirac delta.
Combined with the modeling paradigms introduced in the previous sections, this yields two generative models for the observations, as pictured in Figure 1. The graphical model in Figure 0(a) represents the derivatives we get via the generative process described by the SDEs via the nonlinear drift function . The model in Figure 0(b) represents the derivatives yielded by the generative process described by the Gaussian process. Assuming a perfect GP fit and access to the true parameters , intuitively these two distributions should be equal. We thus want to find parameters that minimizes the difference between these two distributions.
Compared to the deterministic ODE case, the graphical models in Figure 1 contain additional dependencies on the contribution of the Ornstein-Uhlenbeck process
. Furthermore, the SDE-driven probability distribution ofin Figure 0(a) depends on , , and through a potentially highly nonlinear drift function . Thus, it is not possible to do analytical inference without making restrictive assumptions on the functional form of .
However, as shown in the appendix in section A.2, it is possible to derive computationally efficient ancestral sampling schemes for both models, as summarized in Algorithm 1. While this rules out classical approaches like analytically minimizing the KL divergence, we are now able to deploy likelihood-free algorithms that were designed for matching two probability densities based on samples.
3.5 Adversarial Sample-based Inference
Arguably, among the most popular algorithms of this kind, one can find the generative adversarial network (GAN), where a parametric neural network is trained to match the unknown likelihood of our data(Goodfellow et al., 2014). The basic GAN setup consists of a fixed data set, a generator that tries to create realistic samples of said dataset and a discriminator that tries to tell the fake samples from the true ones. As recently shown by Yang et al. (2018)
, GANs have a huge potential for solving stochastic partial differential equations (SPDEs).Yang et al. (2018) assume a fixed data set consisting of observations (similar to our ) and use an SPDE-inspired neural network as a generator for realistic observations. In the case of SDEs however, this would still involve a lot of numerical integration. Thus, we modify the GAN setup by leaving behind the idea of having a fixed data set. Instead of relying on bootstrapped samples of our observations, we sample the derivatives of our data based model shown in Figure 0(b), assuming a sufficiently good model fit, thus representing the true derivatives of our latent variable . We then use the SDE-based model shown in Figure 0(a) as a generator. To avoid standard GAN problems such as training instability and to improve robustness, we choose to replace the discriminator with a critic, trained to estimate the Wasserstein distance between samples as proposed by Arjovsky et al. (2017). The resulting algorithm, summarized in Algorithm 2, can be interpreted as performing Adversarial Regression for SDEs and will thus be called AReS. In Figure 2, we show the derivatives sampled from the two models both before and after training for one example run of the Lotka Volterra system (cf. Section 4.4). While not perfect, the GAN is clearly able to push the SDE gradients towards the gradients of the observed data.
3.6 Maximum Mean Discrepancy
Even though GANs work well in practical settings, they need careful balancing between their generator and discriminator. Dziugaite et al. (2015) propose to solve this problem using Maximum Mean Discrepancy (MMD) (Gretton et al., 2012) as a metric to substitute the discriminator. As proposed by Li et al. (2015), we choose a rational quadratic kernel to obtain a robust discriminator that can be deployed without fine-tuning on a variety of problems. The resulting algorithm, summarized in Algorithm 3, can be interpreted as performing Maximum mean discrepancy-minimizing Regression for SDEs and will thus be called MaRS.
To evaluate the empirical performance of our method, we conduct several experiments on simulated data, using four standard benchmark systems and comparing against the EKF-based approach of Särkkä et al. (2015) and the two GP-based approaches by Vrettas et al. (2015) and Yildiz et al. (2018).
The first system is a simple Ornstein-Uhlenbeck process shown in Figure 2(a), given by the SDE
As mentioned in Section 3.2.3, this system has an analytical Gaussian process solution and serves thus more academic purposes. We simulate our dataset using , and .
The second system is the Lorenz63 model given by the SDEs
In both systems, the drift function is linear in one state or one parameter conditioned on all the others (cf. Gorbach et al., 2017). Furthermore, there is no coupling across state dimensions in the diffusion matrix. This leads to two more interesting test cases.
To investigate the algorithm’s capability to deal with off-diagonal terms in the diffusion, we introduce the two dimensional Lotka-Volterra system shown in Figure 2(b), given by the SDEs
where is, without loss of generality, assumed to be a lower triangular matrix. The true vector parameter is and the system is simulated starting from . Since its original introduction by Lotka (1932), the Lotka Volterra system has been widely used to model population dynamics in biology. The system is observed at 50 equidistant points in the interval . As it turns out, this problem is significantly challenging for all algorithms, even without the presence of observation noise.
To investigate the effect of strong non-linearities in the drift, we introduce the Ginzburg-Landau double-well potential shown in Figure 2(c), defined by the SDE
Using , and , this system exhibits an interesting bifurcation effect. While there are two stable equilibria at , it is completely up to noise in which one the system will end up. This is thus the perfect system to test how well an algorithm can deal with multi-modal SDEs. The system is observed at 50 equidistant points in the interval , subjected to observational noise with .
Lastly, some implementation details are constant throughout each experiment: the critic in the adversarial parameter estimation is a 2-layer fully connected neural network, with respectively 256 and 128 nodes. Every batch, for both MMD and adversarial training contains 256 elements. While Ornstein-Uhlenbeck and the double-well potential were modeled with a sigmoid kernel, for the Lotka Volterra we used a common RBF (we point at Rasmussen (2004) for more information about kernels and GPs).
For all systems, the parameters turn out to be identifiable. Thus, the parameter value is a good indicator of how well an algorithm is able to infer the drift function. However, since the different dimensions of are independent, there are multiple diffusion matrices that create the same trajectories. We thus directly compare the variance of the increments, i.e. the components of .
To account for statistical fluctuations, we use 100 independent realizations of the SDE systems and compare the mean and standard deviation of and . and are compared against the Gaussian process-based by Vrettas et al. (2015) and by Yildiz et al. (2018) as well as the classic Kalman filter-based recommended by Särkkä et al. (2015).
4.3 Locally Linear Systems
As mentioned in Section 4.1, the functional form of the drift functions of both the Ornstein Uhlenbeck process and the Lorenz 63 system satisfy a local linearity assumption, while their diffusion is kept diagonal. They thus serve as excellent benchmark systems for a wide variety of parameter inference algorithms. The empirical results are shown in Table 3 for the Ornstein-Uhlenbeck. Unfortunately, turns out to be rather unstable if both diffusion and parameters are unknown, even though it takes on average roughly 58 hours to converge. We have thus provided it with the true and show only its empirical parameter estimates. Since both and are using Equation (22) to determine , they share the same values. Due to space restrictions, we moved the results for Lorenz 63 to Table 4 in the appendix. As demonstrated by this systems, and are both able to deal with locally linear systems, occasionally outperforming their competitors, especially in their estimates of the diffusion.
4.4 Non-Diagonal Diffusion
To investigate the effect of off-diagonal entries in , we use the Lotka Volterra dynamics. For the diffusion estimates, and share again the same values. Since was unable to model non-diagonal diffusions, we provide it with the true and only compare parameter estimates. As was already struggling in the lower dimensional cases, we omitted it from this comparison due to limited computational resources. The results are shown in Table 3. and are clearly outperforming their competition in terms of diffusion estimation, while is the only algorithm able to keep up in terms of drift parameter estimation.
4.5 Dealing with Multi-Modality
|222Need to compare its squared value!!|
As a final challenge, we investigate the Ginzburg-Landau double well system. While it is one dimensional, its state distribution is multi-modal even if all parameters are known. As shown in Table 3, this is a huge challenge for all classical approaches. While probably does not have enough data points for its non-parametric proxy for the drift function, the time-dependent Gaussianity assumptions in both and are problematic in this case. In our gradient matching framework, no such assumption is made. Thus, both and are able to deal with the multimodality of the problem.
Parameter and diffusion estimation in stochastic systems arises in quantitative sciences and many fields of engineering. Current estimation techniques based on Kalman filtering or Gaussian processes try to approximate the state distribution conditioned on the parameters and then iteratively optimize the data likelihood. In this work, we propose to turn this procedure on its head by leveraging key ideas from gradient matching algorithms designed for deterministic ODEs. By introducing a novel noise model for Gaussian process regression using the Doss-Sussmann transformation, we are able to reliably estimate the diffusion process and the drift parameters. Our algorithm is able to keep up and occasionally outperform the state-of-the-art on the simpler benchmark systems, while it is also accurately estimating parameters for systems that exhibit multi-modal state densities, a case where traditional methods fail. While our approach is currently restricted to systems with a constant diffusion matrix , it would be interesting to see how it generalizes to other systems by using different or approximate bridge constructs. Unfortunately, this is outside of the scope of this work. However, we hope that our publicly available code will facilitate future research in that direction.
This research was supported by the Max Planck ETH Center for Learning Systems. GA acknowledges funding from Google DeepMind and University of Oxford.
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a.1 Parameter Estimation Lorenz 63
a.2 Densities for Ancestral Sampling of the SDE Based Model
Given the graphical model in Figure 0(a), it is straight forward to calculate the densities used in the ancestral sampling scheme in Algorithm 1. After marginalizing out , the joint density described by the graphical model can be written as
Using variable substitution to simplify notation, we write
This equation is now subsequently modified by observing that the product of two Gaussian densities in the same random variable is again a Gaussian density:
This formula can be further modified using the Woodbury identity, i.e.
which leads to
which leads to
Since we observe , we are interested in calculating the conditional distribution
As is independent of , we can employ ancestral sampling by first obtaining a sample of by sampling and then using that sample to obtain a sample of by sampling .
a.3 Calculating the GP Posterior for Data Based Ancestral Sampling
Given the graphical model in Figure 0(b), it is straight forward to calculate the densities used in the ancestral sampling scheme in Algorithm 1. After marginalizing out and using the variable substitutions introduced in Equation (36), the joint density described by the graphical model can be written as
and , is independent of .
After marginalizing out and then dividing by the marginal of , we get the conditional distribution