# Adaptive CUSUM Chart for Real-Time Monitoring of Bivariate Event Data

## TL;DR

This paper introduces an adaptive CUSUM chart for real-time monitoring of bivariate time-between-events data, effectively detecting small to moderate changes by updating with each new observation and incorporating historical information.

## Contribution

We develop a novel adaptive CUSUM chart that overcomes delays and limitations of existing Shewhart-type methods for bivariate TBE data, improving detection performance.

## Key findings

- Outperforms existing Shewhart chart in simulations
- Effectively detects small to moderate process changes
- Demonstrated success on real-world data

## Abstract

Monitoring time-between-events (TBE) data, where the goal is to track the time between consecutive events, has important applications across various fields. Many existing schemes for monitoring multivariate TBE data suffer from inherent delays, as they require waiting until all components of the observation vector are available before making inferences about the process state. In practice, however, these components are rarely recorded simultaneously. To address this issue, Zwetsloot et al. proposed a Shewhart chart for bivariate TBE data that updates the process status as individual observations arrive. However, like most Shewhart-type charts, their method evaluates the process based solely on the most recent observation and does not incorporate historical information. As a result, it is ineffective in detecting small to moderate changes. To overcome this limitation, we develop an adaptive CUSUM chart that updates with each incoming observation while also accumulating information over time. Simulation studies and real-data applications demonstrate that our method substantially outperforms the Shewhart chart of Zwetsloot et al., offering a robust and effective tool for real-time monitoring of bivariate TBE data.