L2GTX: From Local to Global Time Series Explanations

Imagine you have a super-smart robot chef (a Deep Learning model) that can taste a cup of coffee and instantly tell you if it's Arabica or Robusta. It gets this right 99% of the time. But here's the problem: the robot is a "black box." It gives you the answer, but it won't tell you why. Did it taste the sweetness? The bitterness? The acidity?

If you ask the robot, "Why did you pick Robusta?" it might just point to a specific second on the clock and say, "Because of the flavor at 2:14 PM." That's not very helpful. You need to know the pattern, not just the timestamp.

This is where the paper L2GTX comes in. It's a new tool designed to translate the robot's secret thoughts into human language.

The Problem: Too Many Clues, No Story

Currently, there are tools that can explain the robot's decision for one single cup of coffee (Local Explanation). They might say, "For this cup, the robot liked the bitter spike at the end."

But what if you want to know the robot's general rule for all Robusta coffees?

Existing tools are like trying to understand a whole forest by looking at one leaf at a time.
They are often tied to specific robot "brains" (model-specific), so if you change the robot, the tool breaks.
They struggle to find the common patterns that repeat across thousands of cups.

The Solution: L2GTX (Local-to-Global Time Series eXplanations)

Think of L2GTX as a detective who interviews a few key witnesses to write a summary report for the whole case.

Here is how it works, step-by-step, using a simple analogy:

1. The "Detective" (LOMATCE)

First, the system picks a few representative cups of coffee (instances) from the dataset. It uses a tool called LOMATCE to interview the robot about each cup individually.

Instead of saying "flavor at 2:14," LOMATCE translates the robot's thoughts into Event Primitives.
Analogy: Imagine the robot doesn't say "Time 2:14." Instead, it says, "I saw a sharp spike (Local Max) here," or "I saw a slow rise (Increasing Trend) there." These are the "events."

2. The "Grouping" (Clustering)

Now, the detective has a pile of notes from 15 different cups.

Cup 1 had a spike at 2:10.
Cup 2 had a spike at 2:12.
Cup 3 had a spike at 2:09.
The Magic: L2GTX realizes these are all the same type of event: "A spike near the 2-minute mark." It groups them together. It ignores the tiny differences (2:09 vs 2:12) and focuses on the pattern.
It creates a "Global Cluster" of events. Think of this as creating a folder labeled "Important Spikes."

3. The "Smart Selection" (Budgeting)

The detective can't interview every single cup in the world (there are too many!). So, L2GTX uses a smart strategy to pick the best cups to interview.

It asks: "Which cups, if I interview them, will give me the most information about the 'Important Spikes' folder?"
It picks a small, diverse group of cups that covers all the major patterns without repeating the same thing over and over.

4. The "Final Report" (Global Explanation)

Finally, L2GTX writes the summary. Instead of a messy list of 1,000 timestamps, it gives you a clean, readable story:

"To identify Robusta, the robot looks for strong, high peaks in the middle of the taste profile, followed by a steady decline. To identify Arabica, it looks for gentler dips and flatter lines."

Why is this a Big Deal?

It's Universal (Model-Agnostic): It doesn't matter if the robot chef is a "Convolutional Network" or an "LSTM." L2GTX works with any of them. It's like a translator that works for any language.
It's Trustworthy: The paper tested this on medical data (heartbeats) and food data (coffee).
- Heartbeats: It correctly identified that "Normal" hearts have a specific wave pattern, while "Heart Attack" hearts have a distinct, sharp dip. This matches what real doctors look for!
- Coffee: It found that Robusta has stronger peaks, which matches what coffee experts know about the beans.
It's Simple: It turns complex math into "shapes" (spikes, dips, rises) that humans can actually visualize and understand.

The Bottom Line

L2GTX takes the robot's scattered, confusing local thoughts and weaves them into a single, coherent story. It tells us not just when the robot is looking, but what it is looking for (the shape of the data).

It's the difference between a robot saying, "I chose this because of the data point at index 452," and a human saying, "I chose this because the heart rate showed a dangerous, sudden drop." That is the power of moving from Local to Global explanations.

1. Problem Statement

Deep learning models achieve high accuracy in time series classification but operate as "black boxes," making it difficult to understand their decision-making logic. While Explainable AI (XAI) has matured for static data (images, tabular), extending these methods to time series is challenging due to:

Temporal Dependencies: Time steps are not independent; explanations must preserve ordering, duration, and recurrence.
Local vs. Global Gap: Existing methods primarily offer local explanations (explaining a single prediction) or are model-specific (relying on internal activations).
Lack of Model-Agnostic Global Synthesis: There is a scarcity of methods that can aggregate local insights into global, class-level explanations without relying on specific model architectures. Current global approaches often fail to capture recurring patterns across instances or lack faithfulness to the underlying model.

2. Methodology: L2GTX

The authors propose L2GTX (Local-to-Global Time-series eXplanations), a fully model-agnostic framework that synthesizes class-wise global explanations by aggregating local explanations from a representative subset of instances. The method operates in five sequential steps:

Step 1: Local Attribution via LOMATCE

The process begins by generating local explanations for a batch of time series instances using LOMATCE (LOcal Model-Agnostic Time Series Classification Explanations).

Mechanism: LOMATCE perturbs temporal segments to create a local neighborhood, extracting Parameterised Event Primitives (PEPs).
PEP Types: These include monotonic trends (increasing/decreasing) parameterized by start time, duration, and gradient, and local extrema (maxima/minima) parameterized by time and value.
Output: For each instance, the top- $n$ PEP clusters are identified based on surrogate model importance scores, creating a compact local explanation.

Step 2: Merge Similar Clusters & Construct Matrix

To reduce redundancy across the dataset, local clusters from different instances are merged into Global Clusters.

Agglomerative Clustering: Centroids of PEPs of the same type (e.g., all "Increasing" trends) are clustered using Euclidean distance. A user-defined merge percentile ( $p$ ) determines the cut-off distance, controlling the granularity of the global clusters.
Instance-Cluster Matrix ( $M$ ): A matrix is constructed where rows represent instances and columns represent global clusters. Entries contain the importance scores of the local clusters mapped to their corresponding global cluster.

Step 3: Compute Global Cluster Importance

Global importance scores ( $I_j$ ) for each cluster are calculated by aggregating the absolute importance values across all instances (inspired by SP-LIME). This identifies which temporal patterns are most influential across the entire dataset.

Step 4: Select Top $B$ Representative Instances

Under a user-defined budget ( $B$ ), L2GTX selects a diverse subset of instances that maximizes the coverage of the most important global clusters.

Greedy Selection: An iterative algorithm selects instances that cover the highest-weighted, previously uncovered clusters. This ensures the final explanation is based on the most representative data points without redundancy.

Step 5: Extract & Aggregate Events

Finally, the PEPs from the selected instances are flattened and aggregated into the global clusters.

Statistical Summarization: For each global cluster, the method computes the mean and standard deviation of event attributes (e.g., start time, duration, value).
Output: This results in a concise, class-wise global explanation describing typical event timing, magnitude, and variability (e.g., "An increasing trend typically occurs between $t=20$ and $t=30$ with a gradient of $0.5 \pm 0.1$ ").

3. Key Contributions

Model-Agnostic Global Synthesis: L2GTX is the first method to generate faithful, class-level global explanations for time series classifiers without accessing internal model weights or architectures.
Event-Based Representation: Unlike methods that highlight raw time steps, L2GTX uses Parameterised Event Primitives (PEPs), providing semantically meaningful explanations (e.g., "peaks," "trends") that align with human reasoning.
Principled Instance Selection: The method introduces a submodular optimization approach to select a representative subset of instances, ensuring global explanations are derived from the most informative data points while controlling computational cost.
Faithfulness Preservation: The framework demonstrates that aggregating local explanations does not necessarily degrade global faithfulness to the underlying black-box model.

4. Experimental Results

The method was evaluated on six UCR Archive datasets (including ECG200, GunPoint, Coffee, FordA/B, and CBF) using two distinct deep learning architectures: Fully Convolutional Networks (FCN) and LSTM-FCN.

Global Faithfulness (GF): Measured by the mean local surrogate fidelity ( $R^2$ $R^{2}$ ) across selected instances.
- Results showed stable GF scores (ranging from ~0.5 to ~0.8 depending on the dataset) across different merge percentiles ( $p=25$ to $95$).
- This indicates that L2GTX can compress the explanation space into a compact set of global clusters without losing fidelity to the model's behavior.
Interpretability Case Studies:
- ECG200: Successfully distinguished between Normal and Myocardial Infarction classes by highlighting distinct configurations of extrema and short trends, aligning with clinical morphology.
- Coffee: Differentiated Arabica and Robusta beans by identifying specific spectral regions and event types (e.g., high-magnitude maxima for Robusta vs. distributed minima for Arabica), consistent with domain literature.
Robustness: The method produced structurally consistent explanations across different model architectures (FCN vs. LSTM-FCN), suggesting it captures invariant decision-relevant cues rather than model-specific artifacts.

5. Significance and Conclusion

L2GTX addresses a critical gap in time series XAI by bridging the divide between local instance-level insights and global class-level understanding.

Trust and Accountability: By providing human-readable, event-based global summaries, it enhances trust in safety-critical applications like healthcare and finance.
Efficiency: The instance selection budget allows for scalable explanation synthesis, avoiding the need to process the entire dataset for global insights.
Future Directions: The authors plan to extend L2GTX to multivariate time series (handling inter-channel dependencies) and conduct human-centered evaluations with domain experts to assess the practical utility of these global summaries in real-world decision-making.

In summary, L2GTX offers a robust, architecture-neutral framework for transforming opaque time series predictions into transparent, actionable global knowledge.