🤖 machine learning

Adaptive Conformal Anomaly Detection with Time Series Foundation Models for Signal Monitoring

This paper proposes a post-hoc, model-agnostic adaptive conformal anomaly detection framework that leverages pre-trained time series foundation models to generate interpretable, calibrated anomaly scores with controlled false alarm rates, enabling robust and rapid deployment in resource-constrained industrial settings without requiring additional fine-tuning.

Original authors: Natalia Martinez Gil, Fearghal O'Donncha, Wesley M. Gifford, Nianjun Zhou, Dhaval C. Patel, Roman Vaculin

Published 2026-04-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Natalia Martinez Gil, Fearghal O'Donncha, Wesley M. Gifford, Nianjun Zhou, Dhaval C. Patel, Roman Vaculin

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 massive ship, and your job is to keep the engine running smoothly. You have a very smart, pre-trained AI assistant (a "Foundation Model") that has read every engine manual ever written. This assistant can predict what the engine should be doing next based on its vast knowledge.

However, there's a problem: You don't have enough data from your specific ship to teach the AI how your engine behaves. Also, engines are tricky; they change over time (they get hot, they wear out, the fuel changes). If you just set a rigid alarm that goes off when the engine gets 5 degrees hotter than usual, you might get flooded with false alarms every time the weather changes.

This paper introduces a clever solution called W1-ACAS. Think of it as a "Smart, Self-Adjusting Alarm System" that works with your AI assistant without needing to retrain it.

Here is how it works, broken down into simple concepts:

1. The "Crystal Ball" and the "Surprise Meter"

First, your AI assistant looks at the engine's past behavior and predicts what will happen next.

The Prediction: "I think the temperature will be 100°C."
The Reality: The temperature is actually 105°C.
The "Surprise Meter" (Non-conformity Score): The difference (5°C) is how "surprised" the system is. A small surprise is normal; a huge surprise might mean something is broken.

2. The Problem with Fixed Rules

Usually, if the surprise is bigger than a fixed number (say, 3°C), the alarm rings.

The Flaw: What if the engine is just naturally a bit "noisy" today? A 3°C jump might be normal. If you ring the alarm, you waste time (a False Alarm).
The Shift: What if the engine suddenly starts running hotter due to a new type of fuel? A 3°C jump might be tiny now, and you miss a real disaster.

3. The Solution: The "Adaptive Judge" (Conformal Prediction)

Instead of a fixed rule, this paper proposes a Judge that looks at the "Surprise Meter" and asks: "How rare is this surprise compared to what we've seen recently?"

This is where the magic happens. The system doesn't just look at the raw number; it calculates a p-value.

Think of a p-value as a "Rarity Score."
If the p-value is 0.01, it means: "This surprise is so weird that it only happens 1% of the time under normal conditions."
If the p-value is 0.5, it means: "This is totally normal; we see this kind of surprise half the time."

The Benefit: You can tell the system, "I only want to be alarmed if something is in the top 1% of weirdness." The system translates this directly into a probability, so you know exactly how often you'll get a false alarm.

4. The "Weighted Memory" (The Adaptive Part)

Here is the paper's biggest innovation. The system doesn't treat all past memories equally.

Old Memory: "Remember last year when the engine was cold?" -> Maybe not relevant now.
Recent Memory: "Remember yesterday when the engine was hot?" -> Very relevant!

The system uses a mathematical trick (called Wasserstein distance) to decide which past memories to trust.

It looks at the current engine state and asks: "Which past days did the engine behave most like it does right now?"
It gives high weight to those similar days and low weight to the dissimilar ones.
Analogy: Imagine you are trying to guess the weather tomorrow. You wouldn't look at what the weather was like in July to predict December. You'd look at what it was like last week. This system does that automatically. It "learns" which past data is relevant to the present moment.

5. Why This is a Game-Changer for Industry

No Re-training Needed: You can plug this into any pre-trained AI (like the ones IBM or others have built) immediately. You don't need a team of data scientists to re-teach the AI your specific machine.
Handles "Drift": Machines change. Seasons change. This system adapts on the fly, so it doesn't get confused when the "normal" baseline shifts.
Trustworthy: Because it gives you a "Rarity Score" (p-value), you know exactly what the risk is. If you set the alarm for 1% rarity, you know you will get a false alarm roughly 1% of the time. No guessing.

Summary Analogy

Imagine you are a parent monitoring a child's temperature.

Old Way: "If the temperature is above 100°F, call the doctor." (Problem: The child might just be running a fever from playing outside, or the thermometer might be slightly off).
New Way (W1-ACAS): You look at the child's temperature relative to how they usually feel right now.
- If they usually run hot when playing, a 100°F reading is ignored (low weight).
- If they are usually cool and suddenly hit 100°F, the alarm rings (high weight).
- You set a rule: "Only call the doctor if the temperature is in the top 1% of weirdness for this specific child at this specific time."

This paper gives us the mathematical toolkit to build that perfect, self-adjusting, trustworthy alarm system for everything from factory machines to stock markets, using powerful AI that we don't have to retrain from scratch.

1. Problem Statement

Industrial signal monitoring (e.g., predictive maintenance) faces three critical challenges:

Data Scarcity: Lack of sufficient historical data or labeled anomalies to train robust, specialized deep learning models.
Distribution Shifts: Real-world time series data is non-stationary; statistical properties change over time, causing static models to fail or produce excessive false alarms.
Interpretability and Calibration: Existing anomaly detection methods often rely on arbitrary thresholds or lack probabilistic guarantees. They struggle to provide a direct link between an anomaly score and a desired false alarm rate (p-value), making it difficult for operators to trust the system.

While Time Series Foundation Models (TSFMs) offer powerful zero-shot forecasting capabilities, they require a post-hoc method to convert their prediction errors into reliable, calibrated anomaly scores that adapt to distribution shifts without retraining.

2. Methodology: W1-ACAS

The authors propose W1-ACAS (1-Wasserstein Adaptive Conformal Anomaly Score), a post-hoc framework that integrates pre-trained TSFMs with Weighted Conformal Prediction.

Core Components:

Non-Conformity Scores: The method uses the absolute error between the TSFM's forecast and the actual observation as the non-conformity score ( $S_t = |Y_t - \hat{Y}_t|$ ).
Conformal P-Values: Instead of using raw errors, the method transforms scores into conformal p-values ( $\beta_t$ ). A p-value represents the probability of observing a score as extreme as the current one under the assumption of normality. If $\beta_t < \alpha$ , an anomaly is flagged.
Adaptive Weighting (The Innovation): Standard conformal prediction assumes exchangeability (data points are i.i.d.), which is violated in time series. W1-ACAS addresses this by learning a weight vector ( $w$ ) for past non-conformity scores.
- Objective: The weights are optimized to minimize the 1-Wasserstein distance ( $W_1$ ) between the cumulative distribution function (CDF) of the adaptive p-values and the uniform distribution $U[0,1]$ .
- Mechanism: By minimizing this distance, the method ensures that the p-values are well-calibrated (i.e., $P(\beta_t \leq \alpha) \approx \alpha$ ) even when the underlying data distribution shifts.
- Optimization: The weights are updated online using projected gradient descent (ADAM) on batches of recent scores. This allows the system to "forget" outdated distribution patterns and emphasize recent, relevant data.
Multi-Horizon Aggregation: The method runs parallel instances for multiple forecast horizons ( $d=1$ to $D$ ). The final anomaly score is the median of the p-values across these horizons, ensuring robustness against single-point errors.

Algorithm Flow:

Input: Pre-trained TSFM (e.g., Chronos, TTM, TiRex) and a stream of time series data.
Forecast: Generate predictions for multiple horizons.
Score: Calculate absolute errors (non-conformity scores).
Calibration: Compute adaptive p-values using the current weight vector $w$ .
Update: Periodically update $w$ via gradient descent to minimize the $W_1$ distance between the empirical p-value distribution and the uniform distribution.
Decision: Flag an anomaly if the aggregated p-value falls below a user-defined threshold $\alpha$ .

3. Key Contributions

Post-Hoc Adaptivity: A novel framework that adapts conformal prediction to non-exchangeable time series data without requiring fine-tuning of the underlying foundation model.
Interpretability: The anomaly score is a direct p-value, offering a transparent, probabilistic basis for decision-making. Users can set a threshold (e.g., 0.01) knowing it corresponds to a 1% false alarm rate.
Distribution-Agnostic Robustness: By leveraging weighted quantile conformal prediction and Wasserstein distance minimization, the method remains robust to heavy-tailed errors and complex distribution shifts.
Model Agnosticism: The approach is compatible with any pre-trained forecaster (demonstrated with TTM, Chronos, and TiRex) and requires no retraining, making it ideal for resource-constrained environments.

4. Experimental Results

The authors evaluated W1-ACAS on 7 univariate and 4 multivariate benchmark datasets (including Yahoo, NAB, MSL, and GECCO) against classical methods (KShape, Sub-PCA), deep learning baselines (USAD, OmniAnomaly), and other conformal approaches.

Performance: W1-ACAS consistently achieved the highest PA-F1 and Affiliation-F scores (threshold-dependent metrics) while maintaining the lowest False Positive Rates (FPR) and Calibration Errors (CalErr).
Comparison: It outperformed semi-supervised deep learning models (which require training on clean data) and standard Gaussian/Conformal baselines.
Calibration: In low-FPR regimes, W1-ACAS produced the most conservative and stable thresholds, staying closest to the ideal identity line (where FPR equals the threshold), whereas other methods were often over-confident.
Adaptivity: Visualizations of the learned weights showed that the model successfully captured periodic patterns and adapted to distribution shifts (e.g., sudden jumps in error variance), reducing false alarms during transitions.

5. Significance

This work bridges the gap between the powerful predictive capabilities of Time Series Foundation Models and the rigorous statistical guarantees of Conformal Prediction.

Industrial Impact: It solves the "cold start" problem in industrial monitoring by allowing immediate deployment of pre-trained models with calibrated anomaly detection, eliminating the need for extensive data collection and model retraining.
Trustworthy AI: By providing interpretable p-values and controlling false alarm rates, it fosters trust in automated monitoring systems, preventing "alert fatigue" for human operators.
Theoretical Advancement: It extends conformal prediction theory to non-exchangeable time series by introducing an adaptive weighting mechanism based on Wasserstein distance, offering a principled solution to distribution shift in streaming data.

In summary, W1-ACAS offers a simple, robust, and highly interpretable solution for real-time anomaly detection in data-scarce, non-stationary industrial environments.