A Minimal Bayesian Model for History-Referencing Jump Event Detection:From Single-Law to Mixture-State Paradigm in Time-Series Analysis

This preprint introduces a new, minimal paradigm for interpretable time-series modeling—one that goes far beyond conventional single-law approaches such as ARIMA or hidden Markov models.

Conventional models often assume that all data can be explained by a single smooth process, treating jumps or anomalies as noise. In contrast, our history-referencing Bayesian model explicitly separates smooth trends from jump (event) states, expressing real-world phenomena as a mixture of regimes. Each parameter (trend, jump, event probability, etc.) is fully interpretable, enabling both detection and causal explanation of discontinuities.

Key innovations include:

• Paradigm shift: From single-law smoothing to explicit mixture of trend and event processes.

• Factor decomposition: Quantifies not only "what" happened but "why"—with probabilistic confidence.

• Full interpretability: Each variable (event probability, jump magnitude, trend) is human-understandable.

• Minimal implementation: The entire approach can be realized in a few lines of Bayesian code, demonstrated here with PyMC and Python.

This repository includes:

• Theoretical exposition and "news paper" style preprint (PDF)

• Fully commented minimal Python sample (see [lambda3_jump_event_detector.py])

• Example results and figures

We also present an advanced Lambda³ model that further generalizes the approach:

• Structural variables (progression index, positive/negative jumps, tension density)

• Captures both the directionality and the structural "history" of events

• Universal applicability for physics, biology, economics, and social science

This work aims to provide a universal, open-source “computational microscope” for analyzing and explaining complex, event-driven time series.
See the full preprint for details, model code, and scientific discussion.

Contact

For questions, feedback, or collaboration inquiries, please contact:
 iizumimasamichi@gamil.com