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 $ΔT$, 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's weaknesses and the DCM's efficiency. The DCM's backbone is the Gauß-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.

翻译：预报厄尔尼诺是科学界最具挑战性的难题之一。我们展示了密集、大规模且高精度的时间序列如何让我们透视时间。所提出的离散卡方方法（DCM）能检测任意趋势与信号组合，并可预报复杂时间序列。广泛使用的离散傅里叶变换（DFT）及其他频域参数化时间序列分析方法存在诸多应用局限，而DCM不受这些限制。模拟时间序列分析验证了革命性的窗口维度效应（WDE）：“对于任意采样窗口ΔT，当样本容量n和/或数据精度σ增大时，DCM必然能检测出正确的趋势p(t)与信号h(t)。”模拟结果同时揭示了DFT的弱点与DCM的效率。DCM的数学基础是高斯-马尔可夫定理——最小二乘法（LS）是线性回归模型的最优无偏估计。由于该方法基于大规模线性模型的最小二乘拟合计算，因此DCM不会失效。费舍尔检验提供信号显著性估计，并从所有备选DCM模型中识别最优模型。非线性DCM模型的解析解属于不适定问题，我们提出了计算上的适定解决方案。若最优DCM模型通过我们的预报检验，则该模型必然正确。DCM因其WDE前沿性对短采样窗口与复杂时间序列均具有鲁棒性，因此是理想的预报工具。附录表明，DCM可对1870至2026年厄尔尼诺数据进行建模与预报。对本研究进行即时、独立且客观的有效性验证或许能节约部分资金。