Discrete Chi-Square Method can model and forecast complex time series, like El Nino data between 1870 and 2026

Abstract

Forecasting El Nino is one of the greatest challenges of science. We show how intensive, large and accurate time series allow us to see through time. Our Discrete Chi-square Method (DCM) can detect arbitrary trend and signal(-s) combinations. It can forecast complex time series. The widely-used Discrete Fourier Transform (DFT) and other frequency-domain parametric time series analysis methods have many application limitations. None of those limitations constrains the DCM. Our simulated time series analyses ascertain the revolutionary Window Dimension Effect (WDE): "For any sample window $\Delta T$, DCM inevitably detects the correct $p(t)$ trend and $h(t)$ signal(-s) when the sample size $n$ and/or data accuracy $\sigma$ increase." The simulations also expose the DFT's weaknesses and the DCM's efficiency. The DCM's backbone is the Gau{\ss}-Markov theorem that the Least Squares (LS) is the best unbiased estimator for linear regression models. DCM can not fail because this simple method is based on the computation of a massive number of linear model LS fits. The Fisher-test gives the signal significance estimates and identifies the best DCM model from all alternative tested DCM models. The analytical solution for the non-linear DCM model is an ill-posed problem. We present a computational well-posed solution. The best DCM model must be correct if it passes our Forecast-test.Our DCM is ideal for forecasting because its WDE spearhead is robust against short sample windows and complex time series. In our appendix, we show that the DCM can model and forecast El Nino data between 1870 and 2026. An immediate, independent and objective validity check of our analysis may save some money.