# Conformal Test Martingale-Based Change-Point Detection for Geospatial Object Detectors
## 1 Introduction  
Object detection for geospatial images is a fundamental task in remote sensing [1]. Nowadays, many deep learning-based object detectors have been proposed to improve the performance of geospatial object detectors drastically when the training and test data are independent and identically distributed (IID assumption) [2, 3, 4, 5, 6]. Since the test images may be distributed differently from the training images because of the variation of illumination, image quality, and background, the domain shift problem has been considered for object detection [7]. To overcome the large drop in detection precision, the strategy of unsupervised domain adaptation has been utilized in many works [8, 9, 10, 11], and which improves the cross-domain robustness of object detection from label-rich to label-poor domains in an unsupervised way. The existing literature builds on the assumption that the domain shift is known in advance and the research mainly focuses on how to make the trained object detector adapt to a new domain. In practice, however, the domain shift phenomenon may not appear immediately after the deployment of an object detector, and the time when the domain changes should be detected automatically. In other words, one important issue of detecting domain shift and deciding when to call domain adaptation algorithms intelligently has not been fully addressed. Figure 1 shows the flowchart of deploying object detectors in a situation where the domain of the test images changes, and our focus in this paper is to design the change-point detector. This is essential for geospatial object detectors, which are deployed in satellites in military or civilian areas without human intervention. In this case, automatically detecting the change point of two domains is important for timely avoiding the considerable drop in detection precision caused by domain shift. Detecting the change point of data streams is important and challenging for both statistics and machine learning areas [12; 13]. Without the detailed assumptions of underlying distributions, many statistical methods can fail to effectively and efficiently detect the changes [14; 15]. Also, the mainstream work in statistics focuses on the batch setting where the observations are real numbers, which limits usability. The existing algorithms in machine learning, on the other hand, do not test IID assumption directly, which lacks theoretical bases and interpretability [16; 17]. Based on the idea of conformal prediction [18; 19], one promising non-parametric framework of testing IID assumption online named conformal test martingale has been proposed, which does not require any detailed assumptions of the data-generating process [20; 21]. The conformal prediction-based method can construct exchangeability martingales with the assumption of the test data stream, satisfying IID assumptions, and makes use of the martingale inequalities to test this null assumption [22]. Each martingale inequality can define a corresponding little probability event, and the IID assumption can be rejected if the event happens. These martingales are built on the \(p\)-values calculated by conformal prediction, which are uniformly distributed when the IID assumption holds. Thus, these conformal prediction-based martingales for testing IID assumption are called conformal test martingales (CTMs), and have been successfully used for change-point detection in the area of classification or regression problems. The first work of using CTMs to detect change-point was proposed in [14], and more explorations [17; 23; 24] were conducted. The recently proposed CTMs with Simple Jumper [17; 25] were successfully applied to change-point detection for regression problems, and further extended to probabilistic prediction for classification problems to protect the predictors from significant deterioration after the change point [26]. Despite their success in detecting a change point, CTMs have only been tested for data with a relatively simple structure, such as vectors or handwritten digits.