Discrete Fourier Transform versus Discrete Chi-Square Method

We compare two time series analysis methods, the Discrete Fourier Transform (DFT) and our Discrete Chi-square Method (DCM). DCM is designed for detecting many signals superimposed on an unknown trend. The solution for the non-linear DCM model is an ill-posed problem. The backbone of DCM is the Gauss-Markov theorem that the least squares fit is the best unbiased estimator for linear regression models. DCM is a simple numerical time series analysis method that performs a massive number of linear least squares fits. Hence, the data spacing, even or uneven, is irrelevant. We show that our numerical solution for the DCM model fulfils the three conditions of a well-posed problem: existence, uniqueness and stability.The Fisher-test is used to identify the best DCM model from all alternative tested DCM models. The correct DCM model must also pass our Predictivity-test. Our analyses of seven different simulated data samples expose the weaknesses of DFT and the efficiency of DCM. The DCM signal and trend detection depend only on the sample size and the accuracy of data. DCM is an ideal forecasting method because the time span of observations is irrelevant. If the Gauss-Markov theorem is valid, DCM can not fail. We recommend fast sampling of large high quality datasets and the analysis of those datasets using numerical DCM parallel computation Python code.