Time-Series Analysis

A point outlier is one observation far from expectation, after which the series returns to its old level. A level shift moves the series to a new baseline that persists. The difference decides whether you drop a point or re-baseline.

A shuffled split trains on future days and tests on past ones, so the model interpolates between neighbouring dates instead of forecasting. Because adjacent days are highly correlated, the score looks excellent and says nothing about future performance.

Decomposition splits a series into three parts: trend-cycle, the slow movement of the level; seasonal, the pattern that repeats at a fixed known period such as 12 months; and remainder, the variation left after removing both.

Alpha sets how much weight the update puts on the newest observation versus the accumulated past. Alpha near 1 tracks recent data and reacts fast; alpha near 0 averages over a long history and smooths noise.

A lag-7 feature is the target value from seven days earlier, placed on today's row as an input column. It hands the model last week's same-weekday demand, capturing weekly seasonality that a lag-1 feature alone misses.

Weak stationarity means the mean is constant over time, the variance is constant and finite, and the covariance between two observations depends only on the gap between them, not on where in the series they sit.

MASE divides the forecast's mean absolute error by the mean absolute error of a one-step naive rule computed on the training data. The result is unit-free, and a value above 1 means the model averaged larger errors than that naive benchmark.

Score the residual against a seasonal baseline instead of the raw value: compare each day with the same weekday one cycle earlier, then flag unusually large residuals. A Monday is only anomalous relative to other Mondays.

In ARIMA(p,d,q), p is how many of the series' own past values the model regresses on, d is how many times the series is differenced before fitting, and q is how many past forecast errors enter the equation.

The ACF measures the correlation between a series and its own values k steps earlier, including everything routed through the intermediate lags. The PACF measures that same lag-k correlation after the effect of lags 1 through k-1 has been removed.

An expanding window keeps its start fixed so the training set grows at every origin; a sliding window keeps a fixed length and drops the oldest data. Expanding uses more history, sliding adapts faster after a regime change.

A random walk never returns to a mean, so over any window it looks trended and least squares lines up the two drifts. The residuals stay non-stationary, so standard errors are far too small and the fit statistics are meaningless.

Look at whether the seasonal swing grows with the level. Constant-size swings mean additive; swings that widen as the level rises mean multiplicative. Taking logs turns a multiplicative series into an additive one.

Choose multiplicative seasonality when the size of the seasonal swing grows with the level of the series, and additive when the swing stays roughly the same absolute size no matter how high the level is. Multiplicative needs strictly positive data.

A random walk accumulates every past shock, so its variance grows with time and its covariance depends on the date rather than the lag. Differencing once strips the accumulation away and leaves the shock itself, which is stationary noise.

It means the residuals still carry autocorrelation somewhere in the first ten lags, so the model has left predictable structure behind. The Ljung-Box null is that all those residual autocorrelations are zero; a small p-value rejects it.

A centred window straddles the row: a 7-day centred mean at day t averages t-3 through t+3, so three future values enter that row. The model then trains on inputs that will not exist at prediction time.

They mark how large a sample autocorrelation can get by chance if the series were white noise. The band is plus or minus 1.96 over the square root of n, so a bar inside it is indistinguishable from noise.

The two series align best at a two-week offset. Under the usual convention, where the lag shifts ad spend, spend from 14 days earlier tracks today's sign-ups, so spend leads. Confirm the sign convention before claiming a direction.

The seasonal orders repeat the autoregressive, differencing and moving-average structure at multiples of the season length m. On monthly data with m = 12, P uses lag 12, D differences values a year apart, and Q carries last year's shock.

Spikes at lags 12 and 24 on monthly data mean each month resembles the same month one and two years earlier: an annual cycle of period 12. A seasonal period shows up at its multiples.

Past oil prices improve the forecast of airline returns beyond what the returns' own history explains. It is predictive precedence, established by testing whether lagged oil terms jointly add explanatory power, not evidence of a causal mechanism.

Average each point over a window exactly one seasonal period long, so every season appears once and the seasonal effect cancels, leaving the local level. Even periods need a two-step centred average, and the first and last half-window get no trend.

Rolling aggregates compress recent history into stable columns. A 28-day trailing mean gives the current level with daily noise averaged out; the rolling standard deviation gives recent volatility. A single lag carries one noisy day and can express neither.

The augmented Dickey-Fuller null is that the series has a unit root, so failing to reject only means a unit root could not be ruled out. That is weak evidence of non-stationarity, never proof of it.

Seasonal differencing is needed when a repeating annual pattern, not a drifting level, is what moves the mean. Subtracting the value twelve months earlier removes a stable yearly pattern that a lag-1 difference leaves intact.

Empirical coverage far below the nominal level means the intervals are too narrow. Recompute coverage per forecast horizon, check whether the interval width grows with horizon, and confirm the shortfall is larger than the sampling noise in the count of covered points.

Test whether the drop appears in an independent measurement. Instrumentation breaks confine themselves to one SDK version, platform or event pipeline; a real behaviour change also shows in server-side or revenue counts the SDK never touched.

Future values of every exogenous regressor, for every period you forecast. A SARIMAX forecast is conditional on those inputs, so calendar or planned values are safe while values you must forecast yourself add error the intervals never show.