Continuous-Time Structure Learning from Observational Time Series Data using Stochastic Differential Equations

Core Concepts

A novel structure learning framework, SCOTCH, that combines stochastic differential equations (SDEs) and variational inference to infer the underlying causal structure from continuous-time observational data, including irregularly sampled time series.

Abstract

The paper introduces a novel structure learning framework called SCOTCH (Structure learning with COntinuous-Time stoCHastic models) that combines stochastic differential equations (SDEs) and variational inference to infer the underlying causal structure from continuous-time observational data.
Key highlights:

SCOTCH uses a latent SDE formulation to model the underlying temporal process, which can naturally handle irregularly sampled time series data.
The authors provide a rigorous theoretical analysis, proving the structural identifiability of the SDE model and the consistency of the variational inference approach.
Empirically, SCOTCH outperforms existing discrete-time and ODE-based structure learning methods on both synthetic and real-world datasets, especially when dealing with irregularly sampled data.
SCOTCH can also be used for intervention analysis by modifying the simulated SDE trajectories.
The paper first introduces the necessary background on Bayesian structure learning, SDEs, and Euler discretization. It then presents the SCOTCH framework, including the latent SDE formulation, the graph prior, the likelihood, and the variational inference procedure.
The theoretical analysis section establishes the structural identifiability of the SDE model and the consistency of the variational inference approach. The experimental section compares SCOTCH to various baselines on synthetic (Lorenz-96, Glycolysis) and real-world (DREAM3, Netsim) datasets, demonstrating the advantages of the continuous-time SDE formulation.

Stats

Lorenz-96 dataset:

The Lorenz-96 model is a well-known example of a chaotic system observed in biology.
The dataset consists of N=10 time series with sequence length I=100 (before random drops) and dimensionality 10.
Glycolysis dataset:

The Glycolysis dataset models metabolic iterations that break down glucose in cells, described by an SDE with 7 variables.
The dataset consists of N=10 time series with sequence length I=100 (before random drops) and dimensionality 7.
DREAM3 dataset:

The DREAM3 datasets contain in silico measurements of gene expression levels for 5 different gene expression networks.
Each dataset contains N=46 time series of 100 dimensional variables, with I=21 observations per series.
Netsim dataset:

The Netsim dataset consists of blood oxygenation level dependent imaging data.
It contains 5 time series, each with 15 dimensional observations and I=200 time points.

Quotes

"Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science."
"Unfortunately, most existing structure learning approaches assume that the underlying process evolves in discrete-time and/or observations occur at regular time intervals. These mismatched assumptions can often lead to incorrect learned structures and models."

Key Insights Distilled From

Neural Structure Learning with Stochastic Differential Equations

by Benjie Wang,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2311.03309.pdf

Deeper Inquiries

How can SCOTCH be extended to handle instantaneous effects, which can arise due to data aggregation

To handle instantaneous effects that may arise due to data aggregation, SCOTCH can be extended by incorporating additional terms in the latent stochastic differential equation (SDE) formulation. Specifically, the drift function $f_{\theta}$ and diffusion function $g_{\theta}$ can be modified to include components that capture the instantaneous effects. These additional terms can model the immediate impact of external interventions or changes in the system, allowing SCOTCH to account for sudden shifts in the dynamics of the variables. By including these components in the SDE, SCOTCH can effectively capture and represent instantaneous effects in the continuous-time structure learning process.

Can SCOTCH be made more computationally efficient, for example by incorporating an encoder network to infer latent states at arbitrary time points

To enhance the computational efficiency of SCOTCH, an encoder network can be integrated into the model architecture to infer latent states at arbitrary time points. By incorporating an encoder network, SCOTCH can learn a more compact and informative representation of the latent variables, reducing the computational burden associated with processing and analyzing the time series data. The encoder network can efficiently encode the temporal information and extract relevant features from the data, enabling SCOTCH to make accurate predictions and inferences while optimizing computational resources.

What other real-world applications, beyond the ones discussed in the paper, could benefit from the continuous-time structure learning capabilities of SCOTCH

Beyond the applications discussed in the paper, SCOTCH's continuous-time structure learning capabilities can benefit various real-world scenarios. One such application is in the field of cybersecurity, where SCOTCH can be used to analyze network traffic data and detect anomalous behavior or cyber threats in real-time. By modeling the temporal relationships between network variables using SDEs, SCOTCH can identify patterns of malicious activity and enhance cybersecurity measures. Additionally, in healthcare, SCOTCH can be applied to medical time series data to predict patient outcomes, optimize treatment strategies, and improve healthcare decision-making processes. By leveraging continuous-time structure learning, SCOTCH can provide valuable insights into complex systems and support data-driven decision-making across diverse domains.