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 2024. An immediate, independent and objective validity check of our analysis may save some money.

翻译：厄尔尼诺预测是科学界最重大的挑战之一。本文展示了密集、大规模且精确的时间序列如何使我们能够透视时间演变规律。我们提出的离散卡方方法（DCM）能够检测任意趋势与信号组合，并可对复杂时间序列进行预测。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限制，而DCM不受任何此类限制约束。通过模拟时间序列分析，我们确定了革命性的窗口维度效应（WDE）：“对于任意采样窗口$ΔT$，当样本量$n$和/或数据精度$σ$提升时，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的缺陷与DCM的效能。DCM的理论基础是高斯-马尔可夫定理，即最小二乘法（LS）是线性回归模型的最佳无偏估计量。该方法通过计算海量线性模型的最小二乘拟合，从根本上保证了其可靠性。Fisher检验为信号显著性提供估计，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，我们提出了计算层面的适定解法。若最优DCM模型通过我们设计的预测检验，则其必然正确。DCM因其WDE特性对短样本窗口和复杂时间序列具有强鲁棒性，成为理想预测工具。在附录中，我们演示了DCM对1870至2024年间厄尔尼诺数据的建模与预测能力。对本研究进行即时、独立且客观的有效性验证，可能产生显著的经济效益。