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TailedTS: Benchmark Dataset for Heavy-Tailed Time Series Prediction and Periodicity Quantification

This paper introduces TailedTS, a large-scale benchmark dataset of Wikipedia page views characterized by heavy-tailed and zero-inflated distributions, designed to evaluate time series forecasting models under non-Gaussian conditions and to reveal insights into the periodicity of high-traffic digital platforms.

Original authors: Xinyu Chen, HanQin Cai, Lijun Ding, Jinhua Zhao

Published 2026-05-19

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CC BY 4.0

Original authors: Xinyu Chen, HanQin Cai, Lijun Ding, Jinhua Zhao

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 trying to predict the weather. Most of the time, the weather is predictable: it's sunny, then cloudy, then maybe a light rain. You can build a model based on "average" days, and it works pretty well. This is like the old time-series datasets scientists have used for years (like electricity usage or traffic flow), which mostly follow a "bell curve" where extreme events are rare.

But what if you are trying to predict something that behaves like a viral internet meme? Most of the time, nobody looks at it. Then, suddenly, a celebrity tweets about it, and millions of people flood the page in an hour. Then, silence again. This is "heavy-tailed" data: it's full of boring zeros and occasional, massive spikes that break the rules of "average."

This paper introduces TailedTS, a massive new dataset designed specifically to test computer models on this kind of chaotic, "spiky" data. Here is a breakdown of what they did, using simple analogies:

1. The Dataset: A Library of Viral Moments

The authors built a giant dataset using Wikipedia page views from 2024.

The Scale: They collected about 24.7 billion data points (like counting every single visitor to every single page, every hour, for a whole year).
The "Heavy Tail": In this library, a tiny handful of pages (about 5%) get the vast majority of the attention (over 70% of all views). The rest of the millions of pages get very few views.
The Problem: Most computer models are trained on "calm" data. If you throw this "viral" Wikipedia data at them, they get confused because they expect the traffic to be steady. They don't know how to handle the sudden, massive surges.

2. The Discovery: Popular Pages are Chaotic

The researchers asked a simple question: "Do popular pages follow a predictable schedule, like a train timetable?"

The Analogy: Think of a quiet neighborhood street (less popular pages). It has a predictable rhythm: kids walk to school at 8 AM, people come home at 5 PM. It's very periodic.
The Finding: Now think of a busy city intersection (popular pages). It's chaotic. A celebrity might post a photo, or a news event might happen, causing a massive, unpredictable crowd.
The Result: The team found that popular Wikipedia pages are actually less predictable than the quiet ones. They don't follow a strict daily or weekly cycle as closely because they are constantly reacting to random, real-world events. This is a big deal for anyone trying to manage server traffic for big websites.

3. The Solution: Changing the "Scorecard"

To predict these chaotic numbers, the researchers tested different ways to measure "error" (how wrong the model was).

The Old Way (The "Squeaky Wheel"): Traditional models use a method called "Least Squares" (ℓ2-norm). Imagine a teacher grading papers where one tiny mistake is fine, but if a student gets one question wildly wrong, the teacher screams and fails the whole test. This method gets obsessed with the biggest mistakes (the viral spikes) and ruins the prediction for everyone else.
The New Way (The "Tough Coach"): The researchers tested "Robust" methods (like Huber loss or ℓp-norm). Imagine a coach who says, "Okay, you missed that one huge spike, but let's look at the rest of the game." These methods ignore the extreme outliers or treat them gently, so the model learns the general pattern without going crazy.
The Result: When they used these "tough coach" methods, the models became much better at predicting the traffic, especially for the most popular pages. The old methods failed miserably on the big spikes; the new methods handled them with grace.

4. Why This Matters

The paper isn't just about Wikipedia; it's about stress-testing our AI.

The Benchmark: They created a "stress test" (a benchmark) to see if AI models can handle real-world chaos.
The Lesson: If you build a model using only "calm" data, it will break when it hits the real world, where viral events and extreme volatility happen.
The Takeaway: To predict the future of things like internet traffic, stock markets, or emergency calls, we need to stop assuming everything follows a nice, neat bell curve. We need models that are ready for the "heavy tails"—the rare, massive events that change everything.

In short: The authors gave us a giant, messy dataset of Wikipedia traffic to prove that our current AI models are too fragile for the real world. They showed that by changing how we measure mistakes (ignoring the extreme outliers), we can build models that are much more resilient and accurate when things get crazy.

Technical Summary: TailedTS Benchmark Dataset for Heavy-Tailed Time Series Prediction

Problem Statement

Time series forecasting is a critical component of machine learning applications across economics, engineering, and environmental science. However, traditional predictive models predominantly rely on Gaussian assumptions for residuals, which are frequently violated by the heavy-tailed, zero-inflated, and non-Gaussian behaviors observed in real-world empirical datasets. Existing benchmarks, such as the M-Competitions (M4, M5) and UCI electricity datasets, capture various temporal dynamics but often fail to represent the extreme, event-driven volatility and high-dimensional sparsity inherent in complex human interest patterns. Specifically, these datasets do not adequately exhibit the power-law distributions where a minute fraction of events accounts for the majority of observations, creating a gap in evaluating model robustness against stochastic bursts and extreme outliers.

Methodology

The paper introduces TailedTS, a large-scale benchmark dataset derived from Wikipedia hourly page view observations throughout 2024. The methodology encompasses three primary components:

1. Dataset Construction and Alignment

The dataset comprises approximately 24.69 billion data points spanning roughly 3 million unique Wikipedia pages per month. To ensure data quality and focus on frequently viewed content, the authors filtered the raw data to include only pages maintaining a minimum threshold of 10 views per day for every day of the month. This filtering results in a dataset where roughly 5% of pages account for over 70% of total page views, adhering to a pronounced power-law distribution. The data is stored in high-efficiency Apache Parquet format.

2. Periodicity Quantification Framework

To analyze the periodic structure of the time series, the authors propose a sparse autoregression framework with sparsity and non-negativity constraints. The optimization problem is formulated to identify dominant auto-correlations (e.g., daily and weekly cycles) automatically.

Formulation: The problem minimizes the squared error subject to an $\ell_0$ -norm constraint on the coefficient vector $w_\gamma$ and a non-negativity constraint ( $w_\gamma \in \mathbb{R}^d_+$ ).
Implementation: The problem is reformulated as a Mixed-Integer Quadratic Programming (MIQP) problem, solved using the CPLEX algorithm. The model assumes consistent support sets across different categories of page popularity.

3. Robust Predictive Modeling

The paper evaluates autoregressive prediction models using a suite of non-Gaussian loss functions to address heavy-tailed noise. The optimization objective minimizes the sum of a loss function $\rho(\varepsilon_t)$ over residuals. The study compares:

$\ell_2$ -norm loss: Standard Least Squares (Gaussian assumption).
$\ell_1$ -norm loss: Least Absolute Deviations (LAD).
Huber loss: Interpolates between $\ell_2$ and $\ell_1$ , applying quadratic penalization to small residuals and linear penalization to large ones.
Quantile loss: Enables estimation of conditional quantiles using asymmetric linear penalties.
$\ell_p$ -norm loss ( $0 < p < 1$ ): Provides aggressive robustness by penalizing residuals sublinearly, though it introduces nonconvex optimization challenges.

Key Contributions

The paper outlines three-fold contributions:

Benchmark Dataset: TailedTS provides a high-dimensional, heavy-tailed, and zero-inflated time series dataset containing millions of time series over a long period (8,784 hourly time steps in 2024). It allows the research community to evaluate models against non-Gaussian residuals and stochastic bursts, a scenario where standard Gaussian-based estimators often fail.
Periodicity Quantification: The authors utilize sparse autoregression to reveal that frequently-viewed pages exhibit significantly weaker periodic structure than less-viewed counterparts. This finding suggests that popular pages are more susceptible to special events and anomalies, complicating their prediction despite their high volume.
Robust Predictive Modeling: The study demonstrates that standard Gaussian-based estimators degrade substantially on high-volume page categories. In contrast, robust alternatives (Huber, $\ell_p$ -norm, and Quantile losses) provide consistent gains across all traffic scales by mitigating the influence of extreme outliers.

Experimental Results

The authors conducted extensive experiments on the TailedTS dataset and four external benchmark datasets (UCI electricity, SF traffic, Air quality, and Weather data).

Seasonality Analysis: Under sparsity levels $\tau=8$ and $\tau=10$ , the dominant auto-correlations for daily (24h), bi-daily (48h), and weekly (168h) cycles were identified. The analysis confirmed that high-volume page categories (e.g., $O(10^4)$ views) showed lower seasonality coefficients compared to lower-volume categories, indicating higher volatility.
Prediction Performance on TailedTS:
- For high-volume categories ( $O(10^4)$ ), the standard $\ell_2$ -norm loss yielded an RMSE of 1163.08.
- Robust estimators significantly outperformed the baseline: $\ell_p$ -norm loss achieved an RMSE of 129.28, and Huber loss achieved 139.87.
- The $\ell_2$ -norm loss showed competitive results only on lower-magnitude categories ( $O(10^2)$ ), but performance degraded sharply as volume increased.
Performance on External Benchmarks: The study verified that residuals in standard benchmarks (UCI electricity, SF traffic, etc.) also exhibit heavy-tailed behavior via log-log histograms. Non-Gaussian loss functions consistently reduced MAPE and RMSE across these datasets. For instance, on UCI electricity data, the $\ell_1$ and Huber losses reduced MAPE from 0.095 ( $\ell_2$ ) to 0.083, a roughly 13% relative reduction.

Significance and Claims

The paper positions TailedTS as a rigorous stress test for time series forecasting models, specifically targeting the limitations of current State-of-the-Art (SOTA) methods that rely on Gaussian assumptions. The authors claim that:

Robustness is Critical: Standard estimators are ill-suited for real-world digital traffic characterized by heavy-tailed noise and extreme volatility.
Loss Function Selection Matters: The choice of loss function is a primary determinant of model resilience. Robust losses (Huber, $\ell_p$ , Quantile) are necessary to capture irregular, interconnected human interest patterns that deviate from physical constraints found in transportation or climate systems.
Scalability and Generalizability: By integrating TailedTS with existing meteorological benchmarks, researchers can better assess the scalability of GPU-native frameworks and the efficacy of robust loss functions.

The authors conclude that while robust estimators offer significant error reduction, the challenges posed by extreme scale and pervasive zero-inflation remain a formidable frontier. They aim for this benchmark to catalyze the development of next-generation, distribution-aware models that bridge the gap between theoretical optimization and the heavy-tailed realities of large-scale time series data.

Limitations: The authors note that the benchmark is derived solely from Wikipedia data and may not capture all forms of real-world nonstationarities, such as abrupt regime shifts or adversarial manipulation. Consequently, conclusions drawn may not fully generalize to time series governed by different data generating mechanisms.