Rolling and Expanding Computations

## One answer per position A window computation walks along an ordering — most often a time ordering, but any stable ordering will serve — and emits **one answer per position**, computed from whatever records the window covers while it is sitting at that position. That is the whole shape of the operation. Everything that distinguishes one window computation from another is a rule about where the window's two edges sit and how they move as the position advances. Two rules dominate, and a third sits between them. The market's words for the first two are a *rolling* window — one that walks along with its length held fixed — and an *expanding* window — one whose left edge is pinned to the first record and never moves. ## A moving window of fixed length Both edges advance together, a fixed distance apart. That distance, the window's length, is declared either as a count of records or as a span of time, and which one you declared changes which records the window covers whenever the observations are not evenly spaced. Its defining property is that this window **forgets**. A record enters at the right edge, contributes to a bounded number of answers, then leaves at the left edge and never affects another one. So: - the answer at each position is a *local* statistic — it describes the recent stretch, not the history; - an extreme reading perturbs a bounded number of answers and then vanishes completely; - the answer follows a genuine change of level within roughly one window length; - early positions have fewer records behind them than the length asks for, so the leading edge of the output needs a rule of its own. ## A window anchored at the start The left edge is pinned to the first record and never moves; only the right edge advances. Each answer therefore covers **everything from the beginning up to and including that position**, and each one covers one more record than the last. Its defining property is that this window **never forgets**: - an early extreme reading is folded into every later answer, its influence shrinking only as the record count grows; - the answer becomes progressively harder to move — by the ten-thousandth position, one new record shifts a running mean by roughly one ten-thousandth of its distance from that mean; - a real change of level late in the sequence is nearly invisible, because the accumulated history outweighs it; - there is no length to choose, which is often why people reach for it, and is also why it answers a different question. ## Side by side | | moving window of fixed length | window anchored at the start | |---|---|---| | left edge | advances with the position | pinned to the first record | | records per answer | fixed by count, or variable by span | grows by one each position | | an old record's influence | drops out entirely | never drops out | | response to a level shift | within about one window length | heavily damped, more so over time | | parameter to choose | the length | none | | question it answers | what is happening lately | what has happened so far | ## A third shape, with no hard left edge Between them sits an **exponentially weighted moving statistic**: a moving statistic in which no record is ever dropped, but each earlier observation contributes with a geometrically smaller weight than the one after it. There is no length, only a decay rate. It behaves like a fixed-length window in that recent records dominate, and like an anchored window in that nothing is discarded outright. It is useful when you want recency without the cliff a hard left edge creates, where one record's departure visibly moves the answer. ## What each one actually costs It is tempting to say a window computation is one pass with a cheap incremental update, so its cost is independent of the length. That holds for **additive statistics** — a sum, a count, a mean — where a running accumulator can add the entering record and subtract the leaving one. It does not hold for **order statistics** such as a median or a quantile, which have no cheap removal step, and it does not hold when the per-position computation is a function you wrote yourself: whether that is invoked once per position over the small set of covered records, or recognised and dispatched to a compiled routine, differs by design and by what you passed. The honest answer is that the cost depends on the statistic and on the path the tool took, not that windows are free. ## Choosing 1. Ask what the number is for. An alert threshold, a control limit or a display of current behaviour wants the fixed-length form; a cumulative total, a to-date average or a best estimate of a level assumed stable wants the anchored form. 2. If you take the fixed length, choose it from how fast the underlying process actually changes rather than from a round number, and write the reason down — every consumer of that column inherits it. 3. If you take the anchored form, understand that you have chosen to give recent records almost no weight once the sequence is long, and confirm that this is what you meant.