Forecasting the amplitude, the location and the width of the peak of a time series using the weighted median

In many signal processing applications, defining and finding peaks is an important part of the pipeline. Peak prediction can be a very challenging endeavour, especially when there is a lot of noise. In this work, the peak region is considered as a combination of a Gaussian trend with additive and multiplicative noise, and the peak is modelled as the peak of the Gaussian trend. In the case of free noise, the prediction of the amplitude, location, and width of the peak is exact with only three observations. We propose a criterion based on the weighted median which predicts the optimal number of observations necessary for the exact prediction of the peak's location. We analyse the effects of the width, location, and signal-to-noise ratio on our prediction. We show that for small width or a small signal-to-noise ratio (a lot of noise), the prediction becomes inaccurate. In this case, we propose a new idea called the entry time into the region of the peak and the exit time from the region of the peak. We demonstrate how to apply our new idea to real data and we analyse the effect of the choice of the objective function. We also construct confidence intervals around the estimated peak parameters and predicted infection counts. As a numerical application, we consider the peak prediction of the daily infections during the first wave of COVID-19 in China, France, and Germany.