What You Can Do Now

You’ve been staring at the same handful of series since chapter 1, on and off: Bellingham’s temperature record, Lake Huron’s water level, Race Rocks sea surface temperature, the Nooksack in flood and in trickle, tree rings from the Colorado watershed, hare and lynx counts pulled from the Hudson’s Bay Company’s fur-trading ledgers. Each of those series was doing the same essential thing this whole time: waiting for someone to ask what its dependence structure was telling them.

Now you can ask. Given a series, you can look at its ACF and PACF and have some idea whether an AR or MA process built it. You can tell the difference between a trend you should subtract and a trend that’s actually a random walk in disguise, and you know why that distinction matters more than it looks. You can decompose a series into trend, season, and noise, and you know the noise itself deserves a second look before you call it noise. You can fit a model, check whether its residuals are still carrying structure, and hold off on trusting a p-value until you’ve made sure the standard error underneath it wasn’t fooled by autocorrelation. You can forecast, and, more importantly, you know to withhold data and test the forecast before you believe it.

Those skills fall into three bins.

The first: don’t fool yourself, because R will hand you a number whether it means anything or not. decompose() will find a seasonal cycle in pure noise if you ask it to. ar() will fit an indefensible AR(24) to a river’s monthly flow before a plain climatology baseline embarrasses it. Mann-Kendall will print a trend on a random walk. None of these functions know when they’re lying to you. That’s your job, and the whole point of a Ljung-Box test, a backtest, or a Monte Carlo null is to catch it before you publish it.

The second: stationarity is an assumption you earn, not one you assume. Almost every method in this book, the ACF, ARMA, GLS, needs a series to behave the same way throughout its length, and almost no environmental series does that on its own. The whole detrend-versus-difference decision, the entire reason ADF and KPSS exist, comes down to figuring out which kind of trend you’re actually looking at before you touch the data.

The third: pattern is a clue, process is the prize. An ACF that decays slowly, a periodogram with a peak at eleven years, a cross-correlation that leads by three months: none of that is the finding. It’s a pointer toward the finding. The sockeye’s four-year cycle is ocean conditions during a specific life stage. The eleven-year sunspot peak is the solar dynamo. The lynx-hare cycle is a predator and its prey chasing each other through a shared, lagged feedback. Pattern narrows down what could be generating a series. Process is what’s actually doing it, and getting from one to the other is most of what makes this interesting.

There’s a thread underneath all three convictions: temporal dependence is structure to be modeled, not error to be ignored. Sometimes that dependence is the signal you’re after, the ACF’s shape is the fingerprint of the mechanism that built the series. Sometimes it’s the saboteur working against you, shrinking your standard errors and handing you a p-value you shouldn’t trust. Either way the fix is the same: model the structure instead of pretending it isn’t there.

A few things this book didn’t get to. State-space models and the Kalman filter handle multiple interacting series and missing data more gracefully than anything covered here; the MARSS package (Holmes et al. 2012) is the place to start if your field is ecology or fisheries, where it’s close to standard practice. Generalized additive models extend the loess-based smoothing from the filtering chapter into full inferential territory, fitting a smooth trend with a proper standard error instead of just eyeballing it. And everything here assumed one series, or at most a pair. Most environmental systems are driven by many series moving together, which is its own field, vector autoregression and beyond.

For deeper reading: Cowpertwait and Metcalfe (2009) is the book, and it’s worth owning if any of this stuck. Shumway and Stoffer (2017) is the standard graduate treatment, denser but the reference you’ll reach for once you’re past the basics. Hyndman and Athanasopoulos (2021) is the place to go for forecasting specifically, freely available and constantly updated. Zuur et al. (2009) is the ecology-facing route into GLS and mixed models if the regression chapter left you wanting more.

Back to Box, one more time. In chapter 1 I told you his line: all models are wrong, but some are useful. Every model in this book was wrong, the decomposition, the ARMA fit, the GLS regression, the forecast, all of it. What you’ve actually learned is how to tell whether a wrong model earns your trust anyway, whether it’s useful enough to hand a decision to. That’s the skill this whole book was building toward.

Go find some structure in time. Embrace it!