Time Series Exercises in R: 30 Real-World Practice Problems

Explanation: ts() takes a numeric vector plus start and frequency and labels each observation with a calendar position. Frequency 4 means quarterly: c(2018, 1) says start in 2018 Q1. A common slip is passing start = 2018.1: that resolves to 2018 + 0.1 of a year and shifts the index. Use the two-element vector form to be explicit.

Explanation: window() is the time-aware counterpart to [ and respects frequency. Passing year only (e.g. start = 1955) anchors to the first cycle position; supplying c(year, cycle) is more explicit and prevents off-by-one errors when the series doesn't start at cycle 1. Avoid integer subscripts on a ts: they discard the time index.

Explanation: Base ts() ignores the Date column entirely: you tell it where the series starts via start = c(year, cycle) and how many observations per cycle via frequency. The Date column is only useful as a sanity check or for a tsibble workflow. For irregular dates or daily data, prefer xts or tsibble, since ts assumes evenly spaced observations.

Explanation: aggregate.ts() collapses high-frequency data to a coarser frequency using FUN. Default is sum; you can pass mean, max, median, or any reducer. The default nfrequency = 1 produces annual output: pass nfrequency = 4 on a monthly series to roll up to quarterly. This is the fastest base-R path before reaching for dplyr or xts.

Explanation: Annual aggregation strips the seasonal swing (CO2 swings about 6 ppm per year) so the diff reveals the underlying climate signal. diff() defaults to lag 1 and order 1, producing first differences: pass lag = 12 on the monthly series instead to get a seasonal difference. Annual means are typically the cleanest unit for talking about secular growth.

Explanation: Classical decomposition uses a centred 12-month moving average for the trend, so the first and last 6 observations of trend and random are NA by construction. Multiplicative mode interprets seasonal indices as ratios around 1 (so 1.11 in June means June runs 11% above the trend). Use additive when the seasonal swing is independent of the level.

Explanation: s.window = "periodic" forces a fixed seasonal pattern repeated every cycle, which is the right call when the seasonality is climatic and stable across years. An integer s.window such as 13 allows the seasonal pattern to drift slowly. STL is more robust to outliers than classical decompose() and handles unequal trend windows via t.window.

Explanation: With s.window = "periodic" the seasonal component is identical from year to year, so the first 12 values describe the entire seasonal cycle. May is the spring peak (Northern Hemisphere photosynthesis hasn't drawn down CO2 yet); October is the trough after the growing season. The seasonal column sums to zero by construction.

Explanation: When seasonal amplitude scales with the level (passenger swings are bigger in 1960 than 1949), additive decomposition has to absorb that into the remainder. Taking logs converts the multiplicative model into an additive one on the log scale, which stl can then fit cleanly. The remainder sd plummets, which is the classic signal that the log transform is warranted. BoxCox.lambda() in forecast generalises this choice.

Explanation: Under a multiplicative model y = trend * seasonal * remainder, dividing by the seasonal component yields trend * remainder, which is by definition seasonally adjusted. This is exactly how the X-13ARIMA-SEATS family produces SA series, just with a more sophisticated seasonal estimate. The series is now interpretable as level changes net of calendar effects, which is what business reporting wants.

Explanation: p = 0.065 means we fail to reject the unit root at the 5% level, so Nile is borderline non-stationary. The known 1898 break (Aswan Low Dam construction) drives this: the series has a level shift that ADF reads as a near-unit-root. A break-aware test (Zivot-Andrews via urca) or first differencing is the typical next step. KPSS is the complementary test with stationarity as the null.

Explanation: First differencing wipes out the level shift and the trend, leaving year-on-year flow changes which fluctuate around zero. The reported p of 0.01 is adf.test's lower bound (it caps the printed p at 0.01); the test statistic is what tells you how confident the rejection is. A series stationary after d = 1 differences is called I(1), which is the diagnosis that justifies an ARIMA(p, 1, q) form.

Explanation: Lag 1 is largest because the series has a strong trend; ignoring lag 0, the secondary peak at lag 12 (0.76) is the seasonal echo. A flat-decaying ACF with a seasonal hump is the textbook signature of a series that needs both regular and seasonal differencing before modelling. Set plot = FALSE to get a list back for programmatic access; the values live in ex_3_3$acf.

Explanation: PACF cuts off after lag p for a pure AR(p) process, which is exactly what's visible here: significant at lags 1 and 2, then noise. ACF for an AR(p) decays geometrically instead. The complementary rule of thumb is that MA(q) processes show an ACF cutoff at lag q and a decaying PACF. Use both functions together for textbook Box-Jenkins identification.

Explanation: Seasonal differencing first removes the annual cycle (each value minus value 12 months ago); the second diff() then removes any residual linear trend in the seasonal differences. Order matters less mathematically (diffs commute) than computationally for keeping the time index aligned. After this transform co2 is stationary, which is what justifies the canonical airline-model spec ARIMA(0,1,1)(0,1,1)[12].

Explanation: SES is the simplest member of the ETS family: ETS(A,N,N), additive errors with no trend and no seasonality. Every forecast horizon receives the same point value (the last smoothed level) but the prediction intervals widen with the square root of horizon. alpha = 0.24 means the level reacts moderately to new shocks: closer to 1 means the level chases recent observations, closer to 0 means it averages over the long run.

Explanation: Holt extends SES with a second equation that smooths the trend (slope) via the beta parameter. Forecasts ramp linearly off the last level using the last estimated slope. With alpha near 1, the level closely tracks recent observations; beta = 0.23 means the slope is smoothed more aggressively. Add damped = TRUE to introduce a phi parameter that pulls long-horizon forecasts back toward a flat line, which is usually more honest at longer horizons.

Explanation: Multiplicative Holt-Winters models seasonal effects as ratios applied to the level-plus-trend forecast, which is appropriate when peaks grow with the trend (more passengers, bigger summer spikes). Additive would force the absolute July-minus-trend gap to be constant, understating later peaks. The three smoothing parameters alpha, beta, gamma jointly capture level, trend, and seasonal dynamics.

Explanation: ETS(A,N,A) means additive errors, no trend, additive seasonality, which fits a series oscillating around a stable mean. The almost-zero gamma says the seasonal pattern is essentially fixed across years (climatic), so the smoother need barely update it. ets() minimises AICc across the candidate family by default; pass model = "ZZZ" to be explicit, or hard-code letters such as "AAN" to skip the search.

Explanation: Damping adds the phi parameter that flattens long-horizon trend forecasts, which usually helps when horizons are long. Here the undamped model edges out by less than one AICc point, which is well inside the noise floor. The general guidance: pick damped when you forecast more than a year out, since undamped growth assumptions tend to extrapolate too aggressively. Always compare on AICc, not AIC, in finite samples.

Explanation: ARIMA(p,d,q) on Nile uses d = 1 because the level break makes the raw series I(1); the AR(1) captures the mild negative autocorrelation of first differences. Compare AIC across candidates: ARIMA(0,1,1) typically has lower AIC for Nile because differenced flows behave more like a moving-average process. Use forecast::Arima() instead of stats::arima() if you plan to chain a forecast() call.

Explanation: The Box-Jenkins airline model is the canonical fit for log monthly passenger data: one regular MA term plus one seasonal MA term, with both regular and seasonal differencing. auto.arima() uses a stepwise search by default for speed; pass stepwise = FALSE, approximation = FALSE for a full grid that occasionally finds a slightly better model at much higher cost. Always model log(y) rather than y for multiplicative-seasonal data.

Explanation: The seasonal MA coefficient near -0.85 says shocks to the annual cycle decay quickly: each year's seasonal innovation barely persists. The regular MA captures month-to-month residual structure. Whenever you pass seasonal = list(...) you must include period; otherwise R falls back to frequency(x) which is usually right but worth specifying. SARIMA notation is (p,d,q)(P,D,Q)[s] where s is the seasonal period.

Explanation: Ljung-Box pools autocorrelations across many lags into one chi-squared statistic. A high p-value (here 0.79) means we fail to reject the null of no autocorrelation, so the model has captured the time-dependence structure. Common choices for lag: 10 for non-seasonal or twice the seasonal period for seasonal series. checkresiduals() in forecast bundles this test, an ACF plot, and a histogram in one call.

Explanation: Adding xreg turns ARIMA into a regression-with-ARIMA-errors model: the mean is a linear function of the regressors, and the residuals follow an ARIMA process. The estimated xreg coefficient of 18.6 recovers the true 18 lift within sampling error. For forecasting, you must supply a future xreg matrix to forecast(). This is the standard pattern for promotions, holidays, weather, or any known intervention.

Explanation: Holding back the last segment of a time series (never random rows) is essential because temporal order matters: a random train/test split would let the model peek at the future. The convention: train on the past, evaluate forecasts on the most recent block. Use window(start = c(1959, 1)) to grab the test set, or subset(AirPassengers, end = length(AirPassengers) - 24) in the tsibble idiom.

Explanation: Test-set MAPE of 3.7% is competitive for this series; a naive seasonal benchmark (last year repeats) sits around 10%. accuracy() returns both training and test rows so you can spot overfitting (much better train than test). MASE is the scaled measure to prefer because it's unit-free and benchmarks against a naive forecast; MAPE breaks down when values cross zero.

Explanation: ARIMA edges out ETS here, but the gap is small enough that you might pick ETS for its interpretability and faster refits. A robust workflow benchmarks against a seasonal naive (snaive()) and a damped ETS too; if no model beats the naive by a meaningful margin, the series may not be forecastable to better than baseline. Repeat over several holdout windows for stability.

Explanation: tsCV() walks the origin forward one observation at a time, refitting the model and recording the one-step error. The first few entries are NA because the model needs minimum data to fit. This is closer to how a model performs in production, where you retrain daily. Squaring and averaging gives MSE; take the square root for RMSE comparable to a static holdout RMSE.

Explanation: A forecast object stores mean, lower, and upper as ts objects (lower and upper are matrices with one column per confidence level). Pulling them into a tibble with a real date column makes downstream joins, plotting in ggplot2, or export to a BI tool straightforward. The fabletools::forecast() workflow returns a tsibble natively, which is the cleaner modern alternative.