auto.arima()与arima()结果差异：诺丁汉气温时间序列分析

Great question! This is a common point of confusion when using automated ARIMA selection tools like auto.arima()—let’s break down the key reasons this might happen, and what you can investigate next:

1. auto.arima() uses a stepwise (not exhaustive) search by default #

By default, auto.arima() doesn’t test every possible (p,d,q) combination—it relies on a stepwise search algorithm to narrow down candidates efficiently. It starts with an initial model, then makes incremental tweaks (adding/removing AR/MA terms, adjusting d) to find lower-AIC options.

Your ARIMA(3,1,3) model might fall outside the path this algorithm takes. For example, if the tool locks into exploring d=0 models first and finds a strong candidate (like your ARIMA(5,0,1)), it might not fully dig into all d=1 combinations that could have a lower AIC.

To test this, force an exhaustive search with:

auto.arima(x.t, trace=True, stepwise=False, approximation=False)

This will evaluate all possible (p,d,q) combinations within default limits, and it’s likely to pick up your ARIMA(3,1,3) model if it truly has the lowest AIC.

2. The choice of d (differencing order) might be driving the discrepancy #

auto.arima() first uses statistical tests (like KPSS for stationarity, ADF for unit roots) to pick potential d values. It’s possible these tests suggested d=0 was sufficient for your temperature data, so the tool prioritized non-differenced models.

But manual differencing (d=1) might better capture subtle non-stationarity or trends the tests missed. To verify:

• Run stationarity tests on your original series (tseries::kpss.test(x.t)) and differenced series (tseries::kpss.test(diff(x.t))).
• Compare residual plots and Ljung-Box test results for both models—lower AIC is useful, but you also want residuals that behave like white noise (no lingering autocorrelation).

3. Don’t overlook seasonality! #

Your data is monthly average temperatures, which almost certainly has a strong seasonal component. By default, auto.arima() runs in non-seasonal mode. A seasonal ARIMA (SARIMA) model might be a far better fit overall. Try:

auto.arima(x.t, trace=True, seasonal=True)

This will explore seasonal (P,D,Q) terms alongside non-seasonal (p,d,q) terms, potentially leading to an even stronger model than either of your current candidates.

Final checks #

Even if ARIMA(3,1,3) has a lower AIC, make sure to:

• Plot residuals to check for remaining patterns or autocorrelation.
• Compare out-of-sample forecast accuracy (e.g., using forecast::accuracy() on a held-out test set)—AIC is an in-sample metric, and real-world performance matters most.

内容的提问来源于stack exchange，提问作者Thai Phi