## 1 Introduction
### Threshold-Based Approach  
In change detection, using a threshold is known as a send-on-delta (SoD) technique [18; 19; 20; 21; 22; 23]. After a SoD-enabled system uses the predefined thresholds, referred to as delta, to detect significant changes for measurement, it reports the detected event. In this regard, it is called an event-triggered method. This SoD technique is designed to efficiently report events in particular environments where available resources are limited and redundant data are likely to be created at multiple nodes, such as sensor networks [19]. The SoD technology is also applied to determine a criterion for starting the process of selecting a meaningful sample [20]. In order to generate an event to be detected and reported, the accumulative sum of differences between the current signal and the preceding signal is used. However, it is suitable only for a specific domain where the environment uncertainty is bounded. In SoD, if a threshold is given low, the event reporting (occurrence) rate will increase over the periodic reporting rate. Nevertheless, the sensitivity of the system is good to detect the event. On the contrary, a higher threshold indicates an increase in adaptability to changes, but the detectability and the reporting rate of the system may decrease. According to the average event reporting rate acceptable to the system, the threshold is dynamically adjusted in order to provide the higher sensitivity and satisfy the performance level (i.e., the average event reporting rate) required [21]. The threshold is determined by using the relationship between the performance level and the average variation of the continuous signal. For that reason, during a certain period, the difference between the magnitudes of the signals needs to be constantly observed in order to calculate the average amount of change in the magnitudes of the consecutive data. Hence this method requires the consumption of a certain amount of time for change detection, and so change detection is discrete. To address signal uncertainty related to environment uncertainty, the SoD method combining with a linear predictor is developed [22]. In this system, the difference between the real measurement and the predicted value derived from a series of consecutive data is compared with a given constant threshold. This method is useful because it uses only one discrete-time data at a given time for detection, but it does not consider modeling errors. To cope with both the nonlinearity caused by the environment uncertainty and the attacks injected through the network, the SoD method is employed to detect attacks for the generated residual after a detection filter is designed with the residual weight and the filter gain [23]. Since it is assumed that noise follows the Gaussian distribution due to network environments, it is difficult to guarantee performance in an environment where the distribution is unknown. To tackle the SoD of vulnerability such as false alarms, SoD is exploited for accurate fall detection using machine learning method [18]. In this research, however, we see that the threshold is varied only with the values selected from the existing experimental data sets of sensors. From the SoD-enabled systems as mentioned above, we find out that they assume that a threshold is given as a design parameter or they target the event detection under the particular environment where the quantity of uncertainty can be assumed. Rather than optimizing the thresholds to provide an adequate level to allow the high adaptability and the high sensitivity to changes, they try to change the thresholds experimentally and do not take into account model and environment uncertainties at the same time. The main novelty of the presented research is the formulation by non-linear programming for determining an adaptive threshold as an equilibrium point effectively and accurately in response to a given environment.