Analysis of an Ill-posed Problem of Estimating the Trend Derivative Using Maximum Likelihood Estimation and the Cramér-Rao Lower Bound

Abstract: The amount of carbon dioxide in the Earth’s atmosphere has significantly increased in the last few decades as compared to the last 80,000 years approximately. The increase in carbon dioxide levels are affecting the temperature and therefore need to be understood better. In order to study the effects of global events on the carbon dioxide levels, one need to properly estimate the trends in carbon dioxide in the previous years. In this project, we will perform the task of estimating the trend in carbon dioxide measurements taken in Mauna Loa for the last 46 years, also known as the Keeling Curve, using estimation techniques based on a Taylor and Fourier series model equation. To perform the estimation, we will employ Maximum Likelihood Estimation (MLE) and the Cramér-Rao Lower Bound (CRLB) and review our results by comparing it to other estimation techniques. The estimation of the trend in Keeling Curve is well-posed however, the estimation for the first derivative of the trend is an ill-posed problem. We will further calculate if the estimation error is under a suitable limit and conduct statistical analyses for our estimated results.