Discrete Chi-Square Method can model and forecast complex time series

We show how intensive, large and accurate time series can allow us to see through time. Manyphenomena have aperiodic and periodic components. An ideal time series analysis method woulddetect such trend and signal(-s) combinations. The widely-used Discrete Fourier Transform (DFT) andother frequency-domain parametric time series analysis methods have many application limitationsconstraining the trend and signal(-s) detection. We show that none of those limitations constrains ourDiscrete Chi-square Method (DCM) which can detect signal(-s) superimposed on an unknown trend.Our simulated time series analyses ascertain the revolutionary Window Dimension Effect (WD-effect):“For any sample window ∆T , DCM inevitably detects the correct p(t) trend and h(t) signal(-s)when the sample size n and/or data accuracy σ increase.” The simulations also expose the DFT’sweaknesses and the DCM’s efficiency. The DCM’s backbone is the Gauß-Markov theorem that theLeast Squares (LS) is the best unbiased estimator for linear regression models. DCM can not failbecause 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 allalternative tested DCM models. The analytical solution for the non-linear DCM model is an ill-posedproblem. We present a computational well-posed solution. The DCM can forecast complex time series.The best DCM model must be correct if it passes our Forecast-test. Our DCM is ideal for forecastingbecause its WD-effect spearhead is robust against short sample windows and complex time series. Inour appendix, we show that DCM can model and forecast El Niño.