Top 20 Time Series Problems to Solve in R

Explanation: ts() does not reshuffle your data; it only records that observation one lands in January 2021 and that 12 observations complete a year. That metadata is what lets window(), aggregate(), decompose(), and every model function reason about seasons. Set frequency = 4 for quarterly data or frequency = 7 for daily-with-weekly cycles.

Explanation: start() and end() return a two-element vector of (year, period), so 1949 1 means January 1949 and 1960 12 means December 1960. Reading the calendar from the object rather than retyping it keeps code correct when the series is later trimmed or extended. frequency() returning 12 confirms monthly seasonality is available to seasonal models.

Explanation: window() is the calendar-aware cousin of [. Passing AirPassengers[133:144] would return a bare numeric vector with no dates, breaking any later seasonal call; window() preserves the frequency and time index. A common mistake is forgetting that end is inclusive, so c(1960, 12) keeps December 1960 rather than stopping before it.

Explanation: aggregate.ts() with nfrequency = 1 reduces 12 monthly points to a single annual point using FUN; switching to FUN = sum would give annual totals instead of averages. The result is a frequency-1 series, so it prints in the plain "Time Series" form rather than a month grid. This roll-up is the standard first move before studying the multi-year solar cycle.

Explanation: decompose() estimates the trend with a centered moving average, subtracts it, then averages the detrended values within each calendar month to get one offset per month; those 12 offsets are the $figure. Negative values mark the cold months (January near -9.3) and positive values the warm ones (July near 13). Because the offsets sum to roughly zero, they add to, rather than scale, the trend, which is what "additive" means.

Explanation: With type = "multiplicative" the model is value = trend times seasonal times remainder, so the seasonal factors are ratios around 1 rather than additive offsets. The trend itself is still a centered moving average and is why the first and last six months of the whole series are NA; 1950 sits safely inside, so every month is defined. Choosing additive here would leave a fan-shaped pattern in the remainder, the classic tell that you picked the wrong type.

Explanation: stl() stores its three components as columns of a multivariate time series in $time.series, so you index the "seasonal" column before slicing. With s.window = "periodic" the seasonal shape is forced to repeat identically every year, which is why 1920's figures nearly match the classical offsets from Exercise 2.1. Using a finite s.window instead would let the seasonal pattern drift slowly across the two decades.

Explanation: The sample autocorrelation at lag $k$ is $r_k = \dfrac{\sum_{t=k+1}^{n}(y_t-\bar y)(y_{t-k}-\bar y)}{\sum_{t=1}^{n}(y_t-\bar y)^2}$, and lag 0 is always 1 because a series correlates perfectly with itself. The slow, smooth decay from 0.83 downward is the signature of a series with persistent memory, typical of an autoregressive process. Setting plot = FALSE returns the object so you can read or test the values instead of eyeballing a chart.

Explanation: The PACF at lag 2 is large and negative (-0.27) while lags 3 and beyond hover near zero, so the partial autocorrelations effectively cut off after lag 2. That pattern (ACF tailing off, PACF cutting off at lag 2) is the classic fingerprint of an AR(2) process, which is exactly the model you will fit in Section 6. Unlike the ACF, the PACF has no lag-0 entry because a value's partial correlation with itself is not defined.

Explanation: A purely damped decay means memory only; an oscillating ACF means periodicity. Here the correlation troughs around lag 5 (values near a decade apart move together, values half a decade apart move oppositely) and peaks again near lags 9 and 10, pinning the cycle length at about ten years. A short lag.max of 5 would show only the descent and hide the periodicity, which is why longer horizons matter for cyclic wildlife and economic series.

Explanation: First differencing computes $\nabla y_t = y_t - y_{t-1}$, which removes a trend but does nothing about a spread that widens over time. The two standard deviations are not directly comparable in scale, but the point is that on the log series each difference is an approximate percentage change, so its size no longer depends on the current level. Taking log() before differencing is the standard recipe for multiplicative series and is exactly what a log-ARIMA does under the hood.

Explanation: The seasonal difference $\nabla_{12} y_t = y_t - y_{t-12}$ subtracts each month from the same month a year earlier, cancelling the repeating pattern; the extra $\nabla$ then removes the leftover trend. Each difference costs observations, so 144 monthly points fall to 131 (12 lost to the seasonal lag, 1 more to the regular lag). The near-zero mean and small stable variance are the hallmarks of a series ready for an ARIMA model, and this exact recipe underlies the "airline model" ARIMA(0,1,1)(0,1,1).

Explanation: The two half-means differ by only about 1.4 feet, yet the p-value is roughly 1.8e-08, so the difference is overwhelmingly significant and the mean level is drifting downward: a red flag for stationarity. Treat this as a screening heuristic, not a formal test, because the t-test assumes independent samples while consecutive lake levels are strongly autocorrelated. It is a fast sanity check that motivates the formal unit-root tests you would reach for next.

Explanation: Time series validation must respect order: shuffling rows would leak future information into training and inflate accuracy. Ending the training set at December 1958 leaves the final two years (24 months) as a clean test window, a common 2-year holdout for monthly data. Every forecast in this section is built only from train and scored only against test, which is what makes the comparison fair.

Explanation: The naive method predicts 337 (December 1958) for every month of 1959 and 1960, ignoring both the upward trend and the summer peaks, so an RMSE near 137 passengers is large. It is deliberately simplistic, and its value is as a benchmark: any real model must beat it clearly to justify its complexity. Because AirPassengers is strongly seasonal, the next method, which reuses a whole year, should do far better.

Explanation: By repeating 1958's seasonal shape, the seasonal naive forecast captures the summer peaks and winter troughs the plain naive method missed, roughly halving the RMSE from 137 to 77. It still ignores the ongoing growth trend, which is why it is not perfect and why a drift or ARIMA model can improve further. For seasonal series this method, not the plain naive, is the benchmark to beat.

Explanation: RMSE $=\sqrt{\tfrac{1}{h}\sum_{t}(y_t-\hat y_t)^2}$ squares errors so it punishes large misses harder than MAE, the mean absolute error; reporting both guards against one metric flattering a model. Seasonal naive wins here on both scores because seasonality dominates this series, while drift beats plain naive by following the trend but cannot capture the seasonal swings. A table like this is the honest scorecard every candidate model, including the ARIMA in the next section, has to beat.

Explanation: The fitted model is $y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \varepsilon_t$, with $\phi_1 = 1.04$, $\phi_2 = -0.25$, and the "intercept" 579.05 being the estimated mean level, not a regression constant. Divide each coefficient by its standard error to gauge significance: both ar terms are several standard errors from zero, so both matter. The sigma^2, log likelihood, and AIC summarise fit quality and are the numbers you compare against a rival model in the next problem.

Explanation: AIC trades goodness of fit against the number of parameters, so a lower value means a better balance, not merely a closer fit. The AR(2) scores about 215.3 versus 219.2 for the AR(1), and a gap of roughly 4 is meaningful evidence in favour of the second autoregressive term, matching the PACF signal from Exercise 3.2. Because AIC is only comparable across models fit to the same data, never compare it between the raw and a differenced or transformed series.

Explanation: The Box-Ljung test pools autocorrelation across the first 10 residual lags into one chi-squared statistic; fitdf = 2 subtracts the two estimated ar parameters so the degrees of freedom fall to 8. A p-value of 0.65 is far above any usual threshold, so you fail to reject the null of no residual autocorrelation: the AR(2) has captured the structure and the leftovers resemble white noise. A small p-value here would instead signal an under-fit model that needs more terms.