ARIMA(p, d, q) models a differenced series as autoregression plus moving average; seasonal orders add the same structure at multiples of the period m. This tool fits by conditional sum of squares — the exact likelihood conditional on the first observations — with every AR/MA parameter constrained inside the stable and invertible region, so a forecast can never come back explosive.
Forecasts extend the recursion with future innovations set to zero; the 95% intervals come from the ψ-weight accumulation of the estimated innovation variance. AIC/AICc/BIC allow honest comparison across (p, d, q) — prefer the smallest model within about 2 AIC points of the best.
Frequently asked questions
How do I pick p, d, q?
Differencing follows the stationarity tests; then compare a few small candidates by AICc. The Stationarity and ARIMA Forecast tools are designed to be used in that order.
Why are the intervals for d > 0 "approximate"?
They accumulate the differenced-scale variance without adding uncertainty about the level the differences hang from. For long horizons with d ≥ 1, treat the bands as a rough guide.
中文说明
ARIMA(p, d, q) 把差分后的序列建模为自回归加移动平均；季节阶数在周期 m 的倍数上叠加同样结构。本工具用条件平方和拟合——给定首批观测下的精确似然——且所有 AR/MA 参数被约束在稳定可逆区域内，预测永远不会发散。
预测将递推外推、未来扰动置零；95% 区间由 ψ 权重累积估计的新息方差给出。AIC/AICc/BIC 支持跨 (p, d, q) 的诚实比较——在最优模型约 2 个 AIC 点以内选最简者。
常见问题
怎么选 p、d、q？
差分阶数跟着平稳性检验走；再用 AICc 比较几个小候选。平稳性检验与 ARIMA 预测两个工具就是设计成按此顺序使用的。
为什么 d > 0 的区间标为“近似”？
区间累积的是差分尺度方差，未叠加“差分所挂的水平”本身的不确定性。d ≥ 1 且预测较长时，应把区间当作粗略参考。