Time Series Decomposition Exercises in R: 50 Problems

Explanation: frequency = 12 is the load-bearing argument. It tells R that twelve observations complete one seasonal cycle, which is exactly the number every decomposition function reads to know how long a season is. Leave it out and you get frequency = 1, and decompose() will refuse to run because a series with no repeating cycle has no seasonal component to extract. The printed month-by-year grid is your confirmation that the calendar landed where you intended.

Explanation: start() and end() return the cycle position as a pair, so c(1949, 1) means January 1949 rather than the decimal year 1949.0. Checking length() against the span matters: twelve years of monthly data should be 144 rows, and if it is not, someone has dropped observations and your seasonal indices will silently shift. tsp(AirPassengers) returns the same information as a compact three-number vector if you prefer one call.

Explanation: window() keeps the series a series. Compare it with AirPassengers[73:120], which returns the same 48 numbers as a plain vector with no start date and no frequency, so a later decompose() call on it fails. Both ends are inclusive, and you can omit either one to mean "from the beginning" or "to the end". Every later exercise that reports a single year uses this same function.

Explanation: July and August stand far above November and February, which is the summer travel peak showing through. Treat this as a sketch, not a seasonal index: because the series trends steeply upward, later years pull every month's average up by roughly the same amount, so the raw month means confound season with growth. Removing the trend first is exactly what decompose() does, and Section 3 rebuilds that step by hand.

Explanation: aggregate() dispatches to a time-series method here, not the data-frame one, so it slices the series into whole cycles and applies FUN to each. Annual totals are one legitimate way to see trend without seasonality, but they cost you eleven twelfths of your resolution and only work when the cycle divides evenly into the span. Seasonal adjustment, covered in Section 2, gives you a trend view that keeps every monthly observation.

Explanation: The standard deviation grows from 13.7 to 77.7, a factor of nearly six, while the mean grows by a factor of under four. Absolute swings clearly widen with the level, which rules out a pure additive split. The ratio column, the coefficient of variation, is the sharper diagnostic: it drifts up only mildly, from about 0.11 to 0.16, meaning the swings are roughly proportional to the level. That proportionality is the definition of a multiplicative series.

Explanation: Raw spread multiplies by 5.67 across the twelve years; log spread multiplies by only 1.44. Taking logs converts multiplication into addition, so a seasonal factor that scales the level becomes a seasonal offset that shifts it, and a constant-width additive model suddenly fits. This is why stl(), which only knows how to do additive splits, is almost always applied to log(x) for series like this. Note the log yearly standard deviations, 0.108 and 0.157, match the coefficients of variation from Exercise 1.6 closely, because for small variation the log spread approximates the relative spread.

Explanation: A subseries plot separates the two effects that a raw time plot tangles together. Read across the mean bars and you see the seasonal shape; read up each individual segment and you see the trend within that month. Both are visible at once, which is what makes this more useful than a plain plot() for deciding whether seasonality is stable. If one month's segment sloped differently from the rest, the seasonal pattern would be evolving, and you would want stl() with a numeric s.window rather than a fixed index.

Explanation: Six slots, and the distinction between two of them matters. seasonal is a full-length series that repeats the pattern for every observation, whereas figure holds just the twelve distinct monthly values once. Reach for figure when you want to report or plot the seasonal profile, and for seasonal when you want to subtract it from the data. The values say May runs about 3.0 ppm above the trend and October about 3.25 ppm below it, the annual breathing cycle of Northern Hemisphere plant growth.

Explanation: Read these as percentages of a normal month. July at 1.227 means July traffic runs about 23% above trend, and November at 0.801 means November runs about 20% below it. Multiplicative factors average to 1 by construction, whereas the additive offsets in Exercise 2.1 sum to roughly 0. The practical advantage is portability: a 23% July uplift stays meaningful as the airline grows, while an additive offset measured in 1949 passenger counts does not.

Explanation: Six months are lost at each end, exactly half the seasonal period, because a centred 12-month average needs six observations on either side of the point it is estimating. That is a real cost when the most recent months are the ones a stakeholder cares about, and it is the single strongest argument for stl(), which uses local regression and returns a trend for every observation including the last. Any accuracy metric you compute on the components must handle these NAs with na.rm = TRUE or it returns NA.

Explanation: Raw 1960 traffic runs from 390 in November to 622 in July, a spread that tells you nothing except that summer exists. Adjusted, the same year sits between 416 and 507, and the real story appears: March was the genuinely weak month, not November. Note that adjustment uses only the seasonal component, so unlike the trend it is defined for every observation including the final month. Use division for multiplicative models and subtraction for additive ones; mixing them up leaves a residual seasonal wave.

Explanation: The difference is zero to ten decimal places, and it must be, because the random component is defined as whatever is left over: random = x / (trend * seasonal). That makes reassembly a tautology rather than evidence the model is good. The count of 132 is the informative half of this answer, since 12 of the 144 observations have no trend estimate and therefore no residual. For an additive decomposition run the same check with + in place of *.

Explanation: The multiplicative model leaves residuals roughly half the size, 9.91 passengers against 19.34, measured in the same units, which is the decisive comparison. Both models still show residual spread rising with the level, and the correlations are noisy over only ten yearly points, so treat them as a weaker secondary signal than the raw standard deviations. Comparing residuals in original units is the trick worth keeping: the multiplicative random component is a ratio around 1, so its standard deviation is not comparable to the additive one until you convert it back.

Explanation: Sorting turns a wall of numbers into a finding: the summer block of June through September carries everything above trend, and November is the deepest trough at 20% below. month.abb and month.name are built-in character constants, which saves typing the labels and guarantees the order matches cycle(). The one trap is that sort() reorders the values away from calendar order, so never feed a sorted index back into a reconstruction; keep it for reporting only.

Explanation: The message is precise about the requirement: decompose() needs at least two complete cycles, so with frequency 1 there is nothing to average across. This is the single most common decomposition error, and it usually means someone built their ts() without a frequency argument rather than that the data is genuinely aseasonal. Check frequency(x) first. An annual series like this still has a trend, which you can estimate with a moving average or loess(), but it has no seasonal component by definition.

Explanation: Trend accounts for 210.6 of the 216.0 total, so the steady rise in atmospheric carbon dioxide dwarfs both the annual breathing cycle and the noise. Two cautions on this arithmetic. The component variances sum to 214.9 rather than exactly 216.0, because the components are not perfectly uncorrelated, so treat these as shares rather than an exact partition. And a var_random of 0.07 does not mean the model is near-perfect; it means the remainder is small relative to a very strong trend.

Explanation: Write stats::filter() rather than plain filter(). Once dplyr is loaded, an unqualified filter() resolves to the data-frame verb and throws a confusing error about an unused argument, and this collision catches almost everyone at least once. sides = 2 centres the window so each average sits at the middle of its five years, which is why the first two and last two values are NA: there is no complete window there. Weights rep(1/5, 5) sum to 1, keeping the smoothed series on the original scale.

Explanation: An exact match, which tells you decompose() is not doing anything mysterious in its first step. The half weights exist because averaging twelve consecutive months leaves the result sitting between two months; averaging two such windows shifts it back onto a month and gives January and the following January half a vote each, so no month is double counted. Thirteen weights summing to 1 is the 2x12 moving average, written as 2x4 with weights c(0.5, 1, 1, 1, 0.5)/4 for quarterly data.

Explanation: These are one year's worth of season-plus-noise ratios: July 1955 ran 27.4% above its local trend. They are close to the final seasonal factors from Exercise 2.2 but not identical, because each year's detrended value still carries that year's noise. Averaging these ratios across years is what removes the noise, and that is the next exercise. For an additive model you would subtract the trend instead, leaving values scattered around 0 rather than around 1.

Explanation: Forgetting na.rm = TRUE returns NA for the six months at each end of the cycle, since their first or last year has no trend estimate. Note the asymmetry this creates: January through June average over eleven years while July through December average over twelve, which is a small imbalance classical decomposition simply accepts. Compare these against Exercise 1.4's raw month means, which were dominated by growth; detrending first is what turns them into a usable seasonal profile.

Explanation: Identical to three decimals, so you have now rebuilt decompose() end to end from a filter and a group mean. Normalisation matters because unnormalised factors would quietly inflate or deflate the reconstructed series: if the twelve factors averaged 1.02, dividing the data by them would shrink the seasonally adjusted level by 2% everywhere. For an additive decomposition the equivalent step subtracts the mean instead, forcing the offsets to sum to zero.

Explanation: Both routes give 0.03339, meaning the residual noise is about 3.3% of the local level. The rep(idx, length.out = n) idiom works here only because the series starts in January, so the recycling lines up with the calendar; a series starting in April would need the index vector rotated first, and getting that wrong shifts the entire seasonal pattern by a few months without any error message. That silent failure mode is the main reason to let decompose() or stl() handle the bookkeeping in production code.

Explanation: The 3-term average barely helps, cutting month-to-month movement from 1.21 to 1.01, because a three-month window cannot span an annual cycle and so leaves the seasonal wave almost intact. The 13-term average drops it to 0.10, an order of magnitude, because its window covers a full year and averages the cycle out. The lesson is that the window must match the seasonal period to remove seasonality: use 13 terms for monthly data and 5 for quarterly, and pay for it with lost observations at each end.

Explanation: Five weights for a four-quarter average, exactly the pattern from Exercise 3.2 scaled down. The raw series swings violently between a weak Q1 and a strong Q4, while the smoothed version climbs steadily from 240.75 to 298.12: the seasonality is gone and only growth is left. Two quarters are lost at each end, half the period, and with only sixteen observations that is a quarter of your data. Short series are where classical decomposition hurts most and where stl() earns its keep.

Explanation: Three differences from decompose() matter here. The components live in a single multiple time series called time.series, so you reach them with fit$time.series[, "trend"] rather than fit$trend. The third is named remainder, not random. And there are no NAs: STL estimates trend at January 1959, the very first observation, because local regression does not need a symmetric window. STL is always additive, so decompose logs when the series is multiplicative.

Explanation: Taking the first twelve rows is only valid under s.window = "periodic"; with a numeric seasonal window the pattern evolves and row 1 differs from row 13, so you would be reporting one particular year as if it were the whole story. Compare these with the decompose() figures from Exercise 2.1: May is 2.96 here against 3.00 there, close but not identical, because STL fits the seasonal component with weighted local regression rather than a plain average of detrended values.

Explanation: July comes out at 1.242 against decompose()'s 1.227 from Exercise 2.2, and the ordering of months is identical. Both are defensible estimates of the same quantity, arrived at differently, and a gap of that size is normal. The reason this works is that log(T * S * R) = log(T) + log(S) + log(R), so an additive method on logs is a multiplicative method on the original scale. Do not exponentiate the trend and multiply it back if you need a forecast in original units without a bias correction, since exp() of a mean is not the mean of the exponential.

Explanation: January moved from 0.14 ppm below trend to 0.06 above it across 38 years, and July fell from 0.90 to 0.77, so the annual cycle has genuinely changed shape. s.window = 7 means the seasonal smoother looks at roughly seven cycles when estimating each month, so smaller values track faster drift at the cost of absorbing noise into the seasonal component. Always use an odd number, and treat "periodic" as the extreme where the window is infinite and the pattern is frozen.

Explanation: The wide window moves less than half as much month to month, giving a nearly straight line, while the narrow one wiggles and picks up short-run variation that arguably belongs in the remainder. Neither is correct in the abstract; the choice depends on whether a two-year dip counts as a change in trend or as noise, which is a subject-matter judgement rather than a statistical one. Left unset, STL picks t.window from the frequency, a sensible default worth keeping unless you have a reason not to.

Explanation: The ordinary fit routes only 202.5 of the injected 250 into the remainder, meaning about 47.5 passengers of pure error were absorbed into trend and seasonal, permanently distorting them. The robust fit captures 253.4, essentially the entire spike, leaving the components clean. Robust fitting works by running extra iterations that down-weight observations with large remainders, which costs computation and can under-react when a large move is genuine rather than erroneous. Use it when you suspect data errors, not when you suspect real shocks you want modelled.

Explanation: Raw 1997 carbon dioxide swings by roughly 6 ppm across the year purely from the seasonal cycle; adjusted, it rises quietly from 363.29 to 365.26, which is the actual annual increase. The adjusted series is trend plus remainder by construction, so it stays bumpy, and those bumps are real information rather than calendar artefacts. Use division instead of subtraction only when the decomposition itself was multiplicative, as in Exercise 2.4.

Explanation: On a clean, strongly trending series the two methods agree almost perfectly, never differing by more than 0.06 ppm out of about 340. The real difference is coverage: STL returns a trend for all 468 months while decompose() returns NA for twelve of them, six at each end. When your stakeholder cares most about the latest month, that gap decides the method for you. On messier series with outliers or evolving seasonality the estimates diverge much more, which is when the robustness options of Exercise 4.6 start to matter.

Explanation: Both series are strongly trended and strongly seasonal, which is why they are the standard teaching examples. The max(0, ...) guard matters: when a component is weaker than the noise around it the ratio exceeds 1 and the raw score goes negative, which is meaningless, so it is clamped to zero. Scores near 1 justify seasonal adjustment and a seasonal forecasting model; a seasonal score near 0, as you will see for daily stock prices in Exercise 5.5, says the seasonal component is noise and should be ignored.

Explanation: Lag 1 sits at 0.371 and lag 4 at -0.354, both well beyond the roughly plus or minus 0.16 band that 144 observations would give for pure noise. The remainder is therefore not white noise: a high month tends to be followed by another high month. That does not invalidate the decomposition, it just means trend and seasonality were never the whole story and short-run dynamics remain, which is exactly what an ARIMA model on the remainder would capture. The [2:7] indexing is the classic off-by-one trap, since acf always reports lag 0 first with a value of 1.

Explanation: Three months out of 144 break the threshold, and March 1960 is the largest miss at 3.2 standard deviations below expectation, roughly 10% fewer passengers than trend and season predicted. Because the fit was on logs, the remainder is a proportional error, so the same threshold means the same relative miss in 1951 and in 1960; on the raw scale a fixed cutoff would flag only late years. One caveat: an extreme outlier inflates sd(rem) and so raises its own threshold, which is why a robust STL fit or a median-based scale makes this more reliable in production.

Explanation: The null hypothesis is that the first twelve autocorrelations are jointly zero, and a p-value of about 7e-16 rejects it emphatically, confirming what Exercise 5.1 showed by eye. The remainder mean of -0.00033 is effectively zero, as it must be, since a decomposition centres the remainder by construction and so tells you nothing about fit. Choosing the lag count is a judgement call: a common rule of thumb is one full seasonal period for seasonal data, which is why 12 is used here.

Explanation: Trend carries 98% of the variation, so any summary of this record that ignores the long-run rise is missing almost everything. Two honest caveats. The denominator here is the sum of component variances rather than the variance of the original series, and those differ slightly because the components are not exactly uncorrelated, so these are shares of a constructed total. And a tiny remainder share is a statement about the size of the trend, not proof of a good model: Exercise 5.3 showed this same remainder is far from white noise.

Explanation: A seasonal score of 0.007 for the DAX is the useful answer: there is no meaningful day-of-week pattern in the index level, so the seasonal component STL dutifully returned is noise, and seasonally adjusting the series would remove nothing real. Its trend score of 0.988 is high but nearly meaningless too, because a random walk drifts and therefore scores as strongly trended without being predictable. Strength scores tell you which components exist, never whether the series can be forecast.

Explanation: The search lands on 1899, matching the historical construction of the Aswan Low Dam, with average annual flow falling from 1097.8 to 850.0. This is a step change in level, not a trend and not a season, which is why decomposition tools have nothing to say about it: a moving-average trend would smear the step across several years and make it look like a gradual decline. Restricting candidates to 1881 onward avoids degenerate splits with only a handful of years on one side, and this brute-force search is the intuition behind formal changepoint methods.

Explanation: Symmetry around zero is what you want, and this remainder delivers it, which supports the decision to work on the log scale. Run the same plot on the untransformed series and the histogram skews right, because large positive misses in the high-traffic later years have no small-scale counterpart early on. A histogram checks the shape of the errors while the Ljung-Box test of Exercise 5.3 checks their independence, and the two answer different questions: this remainder looks well shaped but is definitely not independent.

Explanation: The cycle has widened at both ends: spring peaks climbed by up to 0.36 ppm while the September trough deepened by 0.53 ppm, so the annual amplitude has grown by roughly a fifth. That is a real geophysical finding and it means s.window = "periodic" is the wrong choice for this record; a numeric window as in Exercise 4.4 tracks the drift properly. Splitting a series and fitting each half is the general, assumption-light way to test any modelling choice you are unsure about, at the cost of halving the data behind each estimate.

Explanation: The gap column is what makes this a report rather than a table of numbers: July's 622 passengers overstate underlying demand by 115, and November's 390 understate it by 97. A stakeholder reading only reported figures would conclude demand collapsed after summer, when adjusted demand actually held near 490 all year. This works because d$figure is in calendar order and the 1960 slice starts in January, so the two line up; a slice starting mid-year would need the index rotated to match.

Explanation: This is the seasonal naive forecast done properly: forecast the adjusted series, then reseasonalise. The forecast reproduces the summer peak and November trough without any model of seasonality beyond the twelve factors already estimated. Its weakness is the flat level assumption, which throws away the obvious upward trend and so forecasts 1961 at roughly 1960's level. Extrapolating the adjusted trend, or fitting a drift or damped trend to it, is the natural next step. The forecast is also point-only: no interval comes free from this construction.

Explanation: The decomposition forecast misses by 6.82% on average against the seasonal naive baseline's 9.99%, a genuine improvement of about a third, achieved with no model fitting beyond a moving average and twelve ratios. The reason it wins is that the naive forecast repeats 1959 including its noise, while the decomposition forecast uses a level averaged over the whole series and smoothed seasonal factors. Note the discipline that makes this an honest test: decompose() runs on train, never on the full series, because seasonal factors estimated with 1960 in them would leak the answer.

Explanation: Raw September shows a 16.2% collapse and raw July a 16.3% boom, but both are the calendar talking, not the business: adjusted, they are -3.6% and +5.5%. The most instructive row is March, where raw growth of +7.2% turns into an adjusted decline of -6.0%, a complete sign reversal. Reporting raw month-over-month figures for a seasonal series will eventually put a sign error in front of a stakeholder, which is the practical case for adjustment. Year-over-year change is the other common fix and needs no model, at the cost of a twelve-month lag in spotting turns.

Explanation: April and May 1952 are overestimated by 20 and 27 passengers because straight-line interpolation between March and June ignores the fact that spring months sit below the summer level, so it draws a line straight through the seasonal dip. The isolated June 1956 gap fares much better at -8.5, since a single missing point between two neighbours in the same part of the cycle is an easier problem. The lesson is to interpolate on the seasonally adjusted series and reseasonalise, or to fill from the same month in adjacent years, rather than drawing a naive line through a seasonal series.

Explanation: Raw sales zigzag between a weak Q1 and a strong Q4 in every year, hiding the actual story. Adjusted, the series climbs almost monotonically from 227 to 330, roughly 45% growth over five years, with no zigzag left. The adjusted column has no gaps even though the trend component does, because adjustment uses only the seasonal factors, which are defined everywhere. That is why seasonal adjustment is usually the right deliverable for a business audience and the trend component is not.

Explanation: decompose() always returns a seasonal component, so getting numbers back is never evidence that seasonality exists. The comparison is what matters: the day-of-week effect spans 2.39 index points while the trend spans 372.28, so seasonality is about 0.6% of the movement and is almost certainly sampling noise. This matches the near-zero seasonal strength score in Exercise 5.5. Report the ratio rather than the factors, and resist the temptation to explain why Wednesday looks weak.

Explanation: Returning a one-row data frame rather than a named vector is the design decision that makes this scale: rbind() stacks the rows, a character label survives alongside the numbers, and swapping rbind for lapply plus do.call handles hundreds of series unchanged. Read remainder_sd only within a series, never across, since it carries the units of its own input: 0.0328 is a log ratio, 0.2601 is ppm and 9.4734 is index points. The strength columns are the comparable ones, and they route each series to a different treatment: seasonal models for the first two, none for the third.