⚡ electrical engineering

Diffusion-Based Scenario Tree Generation for Multivariate Time Series Prediction and Multistage Stochastic Optimization

This paper introduces Diffusion Scenario Tree (DST), a novel framework that leverages diffusion-based probabilistic forecasting to generate non-anticipative scenario trees for multivariate time series, demonstrating superior performance in multistage stochastic optimization and Model Predictive Control tasks compared to conventional forecasting and reinforcement learning baselines.

Original authors: Stelios Zarifis, Ioannis Kordonis, Petros Maragos

Published 2026-02-16

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Stelios Zarifis, Ioannis Kordonis, Petros Maragos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a ship trying to navigate through a stormy sea to reach a profitable port. You have a map, but the weather is unpredictable. You can't just guess the weather for the next week; you need to prepare for every possible storm, wind shift, and calm patch that might happen, so you can make the best decision right now.

This paper introduces a new, super-smart way to create that "weather map" for decision-making, specifically for things like trading electricity or managing batteries. They call it Diffusion Scenario Tree (DST).

Here is the breakdown using simple analogies:

1. The Problem: The "Crystal Ball" vs. The "Fork in the Road"

Traditionally, when computers try to predict the future (like electricity prices), they act like a crystal ball that gives you one single answer: "Tomorrow, the price will be $50."

The Flaw: The real world isn't that simple. Sometimes the price is $50, sometimes it's $100, and sometimes it's $10. If you only plan for $50, you might lose money if the price spikes to $100.

Old methods tried to fix this by drawing a tree of possibilities (a "Scenario Tree"). Imagine a tree where the trunk is "Today," and the branches are "Tomorrow could be High, Medium, or Low."

The Old Tree Problem: The branches were often drawn using simple math (like guessing based on average past weather). These simple models were bad at predicting weird, complex, or "two-faced" weather patterns (multimodal dynamics). They missed the rare but expensive storms.

2. The Solution: The "Diffusion" Artist

The authors use a new type of AI called a Diffusion Model (think of it like a highly advanced artist who learns to paint by starting with a messy scribble and slowly refining it into a perfect picture).

How it works: Instead of guessing one number, this AI learns the entire shape of the future. It knows that the price could be low, or it could be high, or it could be in a weird middle zone. It captures the full "mood" of the market.

3. The Innovation: Building the "Decision Tree"

Here is where the paper gets clever. Just having a smart artist isn't enough; you need to organize their paintings into a usable map.

The Diffusion Scenario Tree (DST) does this in three steps:

Painting the Future: The AI generates hundreds of possible future paths (trajectories) for the next few days.
Grouping the Chaos: It takes all those hundreds of messy paths and groups them into clusters (like sorting a pile of mixed-up socks into pairs).
Building the Tree: It turns these groups into the branches of a tree.
- Crucial Rule: The tree respects Non-Anticipativity. This is a fancy way of saying: "You can't make a decision for tomorrow based on information you don't have yet." If two branches look the same today, you must make the same decision for both. You only split your decision when the paths actually diverge.

4. The Real-World Test: The Energy Arbitrage Game

The authors tested this on a Battery Energy Storage System (BESS) in New York.

The Goal: Buy electricity when it's cheap, store it, and sell it when it's expensive (like buying apples at the farmer's market and selling them at the grocery store).
The Challenge: Electricity prices are wild. They can spike unexpectedly.

The Results:

The "Old Way" (Simple Math Trees): Made decent money, but missed the big spikes.
The "Super Simple" Way (Just one prediction): Lost a lot of money because it was too confident.
The "Reinforcement Learning" Way (AI that learns by trial and error): Struggled because there wasn't enough historical data to learn from, and it got confused easily.
The DST Way (The New Method): Won big. By understanding the full range of possibilities and organizing them into a structured tree, the battery knew exactly when to buy and sell to maximize profit, even when the market acted weirdly.

The Takeaway

Think of DST as a super-organized travel planner.

Old planners said: "You will likely arrive at 5 PM." (You miss the flight if traffic is bad).
The new DST planner says: "Here are 50 possible traffic scenarios. If traffic is light, take Route A. If it's heavy, take Route B. If it's a disaster, take Route C. But right now, we all start on the same road until we see the traffic report."

By combining a powerful AI that understands complex uncertainty with a structured tree that forces smart, non-guessing decisions, this method helps machines make better choices in a chaotic world. It's not just about predicting the future; it's about preparing for all possible futures at once.

1. Problem Definition

The paper addresses the challenge of robust decision-making under uncertainty in multistage stochastic systems, such as energy markets, finance, and robotics.

Core Challenge: Traditional decision-making often relies on point forecasts (e.g., from VAR or LSTM models) which fail to capture the full distribution of future outcomes, particularly complex, multimodal dynamics.
Limitations of Current Methods:
- Standard Forecasting: Models like ARIMA, GARCH, or standard LSTMs often assume normal error distributions and provide single-point estimates, lacking the richness needed for risk-aware optimization.
- Unstructured Scenarios: While diffusion models (like TimeGrad) can generate probabilistic forecasts, they typically produce unstructured sets of scenarios. These lack the hierarchical structure and non-anticipativity guarantees required for rigorous multistage stochastic optimization.
- Reinforcement Learning (RL): Model-free RL baselines often struggle in low-data regimes and partial observability settings, exhibiting unstable training and poor convergence.
Goal: To bridge the gap between advanced probabilistic forecasting (Diffusion Models) and stochastic optimization by creating a structured Scenario Tree that respects non-anticipativity (decisions depend only on observed history) while capturing complex uncertainty distributions.

2. Methodology: Diffusion Scenario Tree (DST)

The authors propose the Diffusion Scenario Tree (DST) framework, which integrates a diffusion-based forecaster (specifically TimeGrad) into a Model Predictive Control (MPC) loop via a recursive tree construction algorithm.

A. The Forecaster (TimeGrad)

The system uses TimeGrad, an autoregressive diffusion model. It learns the conditional distribution $p_\theta(x_{k0:N} | x_{1:k0-1})$ of future multivariate time series given historical data. Unlike standard models, it captures long-range dependencies and multimodal patterns.

B. The DST Algorithm (Algorithm 1)

The framework constructs a scenario tree stage-by-stage:

Sampling: At each node of the tree (representing a specific history), the diffusion model samples $M$ future trajectories for the stochastic component (e.g., electricity prices) over a horizon $H$ .
Clustering: These $M$ trajectories are clustered (using K-means) into $K$ groups. The centroids of these clusters serve as the representative future states for the child nodes.
Probability Assignment: Branch probabilities are calculated based on the proportion of samples falling into each cluster.
Pruning & Selection: To manage tree complexity, only the top- $L$ child nodes (based on cumulative path probabilities from the root) are retained.
Non-Anticipativity: The tree is constructed recursively. Decisions at any stage $t$ are made based only on the information available up to stage $t$ (the path from the root), ensuring the non-anticipativity constraint is satisfied by construction.

C. Stochastic Optimization (MPC)

The generated tree is fed into a Multistage Stochastic MPC solver. The optimizer maximizes the expected cumulative discounted reward over the tree, enforcing that actions are identical for all scenarios sharing the same history up to the current decision point.

3. Key Contributions

Novel Framework (DST): Introduction of a general methodology to convert unstructured diffusion-based probabilistic forecasts into structured, non-anticipative scenario trees.
Integration of ML and OR: Successfully bridges Machine Learning (Diffusion Models) and Operations Research (Stochastic Programming) by providing a structured uncertainty representation that is directly usable in optimization solvers.
Superior Handling of Multimodality: Demonstrates that discretizing complex, multimodal distributions via clustering within a tree structure yields better decision policies than using the same forecaster for direct Monte Carlo sampling or point forecasts.
Empirical Validation: Extensive evaluation on a real-world energy arbitrage problem (Battery Energy Storage System in NY electricity markets), outperforming conventional baselines.

4. Experimental Results

The framework was evaluated on energy arbitrage using 10 months of hourly electricity price data from New York State. A Battery Energy Storage System (BESS) was tasked with optimizing charge/discharge actions based on price forecasts.

Performance Comparison (Average Reward):

DST-MPC: 85.81
Monte Carlo SMPC (Same Diffusion Forecaster): 85.56 (DST is slightly better, +0.29%).
Oracle MPC (Perfect Forecasts): 88.93 (The theoretical upper bound).
LSTM Scenario Tree: 72.73 (DST outperforms by 18.0%).
VAR Scenario Tree: 68.46 (DST outperforms by 25.4%).
Model-Free RL (Deep Q-Network): 26.92 (Significantly lower due to data scarcity and partial observability).
Perfect MPC (Full Knowledge): 94.26.

Key Findings:

Structure Matters: Even when using the same underlying diffusion forecaster, organizing predictions into a scenario tree (DST) yields better results than direct Monte Carlo sampling, highlighting the value of hierarchical discretization.
Diffusion vs. Traditional: DST significantly outperforms scenario trees built from traditional models (LSTM, VAR), proving the superiority of diffusion models in capturing complex market dynamics.
Stability: The diffusion forecaster training was highly stable, whereas the RL baselines were unstable and failed to converge to effective policies in this low-data, partial-observability setting.

5. Significance and Future Work

Practical Impact: The DST framework provides a robust tool for decision-makers in uncertain environments (energy, finance) where understanding the full distribution of risks is critical. It enables more efficient policies than deterministic or simple stochastic approaches.
Interpretability: The resulting scenario tree offers interpretability, as each branch represents a distinct, probable future scenario with associated optimal actions.
Future Directions: The authors suggest exploring problem-driven scenario tree generation, where the tree structure is explicitly aligned with the specific constraints and structural characteristics of the optimization problem, rather than being purely data-driven. They also note that for real-time applications, faster diffusion inference variants (e.g., DDIM) should be investigated to reduce computational costs.

In summary, this paper demonstrates that structuring probabilistic forecasts into non-anticipative trees is a critical step for leveraging advanced generative models in real-world stochastic optimization, significantly outperforming both traditional statistical methods and model-free reinforcement learning.