We show how intensive, large and accurate time series can allow us to see through time. Many phenomena have aperiodic and periodic components. An ideal time series analysis method would detect such trend and signal(-s) combinations. The widely-used Discrete Fourier Transform (DFT) and other frequency-domain parametric time series analysis methods have many application limitations constraining the trend and signal(-s) detection. We show that none of those limitations constrains our Discrete 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 (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 DCM can forecast complex time series. 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 DCM can model and forecast El Niño.

翻译：本文论证了密集、大规模且高精度的时间序列如何使我们能够透视时间演化规律。众多现象同时包含非周期与周期成分。理想的时间序列分析方法应能检测此类趋势与信号（或多重信号）的组合。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，制约着趋势与信号（或多重信号）的检测。我们证明这些限制均不适用于本文提出的离散卡方方法（DCM），该方法能够检测叠加在未知趋势上的信号（或多重信号）。通过模拟时间序列分析，我们确认了具有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$h(t)$（或多重信号）。”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优无偏估计量。DCM不会失效，因为这一简洁方法基于对海量线性模型LS拟合的计算。通过F检验可评估信号显著性，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。DCM能够预测复杂时间序列：若最优DCM模型通过我们提出的预测检验，则该模型必然正确。DCM特别适用于预测任务，因其WDE前沿机制对短采样窗口与复杂时间序列具有强鲁棒性。附录部分展示了DCM对厄尔尼诺现象的建模与预测能力。