Why encode a pickup hour as a sine-cosine pair instead of the integer 0-23?

## The problem: an ordering that wraps Hour of day is cyclical. 23:00 is one hour from midnight, and midnight is one hour from 01:00. An integer encoding 0-23 represents the ordering correctly *within* the day and then breaks at exactly one point: the distance from 23 to 0 is 23, the largest gap in the column, when the true gap is the smallest one there is. Whether that matters depends on how the model reads the column. - A **linear model** multiplies the hour by one coefficient, so it can only say "later in the day means more" or "means less". It cannot represent a bump at 08:00 and another at 18:00, and it certainly cannot make 23 and 0 behave alike. - A **distance-based** model — k-nearest-neighbours, k-means, anything using Euclidean distance — computes gaps directly. With raw hours, a 23:00 ride and a midnight ride are the least similar pair of rides in the dataset on that dimension. - A **tree** is less affected, because it only ever asks "is hour < t", which is invariant to the scale. But it still cannot express a window that crosses midnight in one split: 22:00-02:00 requires two branches, `hour >= 22` and `hour <= 2`, and it must find both. ## The fix: put the clock on a circle Map the hour to an angle and take its two coordinates: ``` angle = 2 * pi * h / 24 h_sin = sin(angle) h_cos = cos(angle) ``` Now every hour is a point on the unit circle, evenly spaced, and Euclidean distance between consecutive hours is identical everywhere — including between 23 and 0. The chord between any two adjacent hours is `2 * sin(pi / 24)`, about 0.261, whether the pair is 22 and 23 or 23 and 0. The wrap-around has disappeared into the geometry. ## Why both coordinates Sine alone is not one-to-one over a cycle. With period 24, `sin(2*pi*3/24)` and `sin(2*pi*9/24)` are both about 0.707 — 03:00 and 09:00 collapse onto the same value, and any model reading only that column treats them as identical. Cosine distinguishes them (about +0.707 versus -0.707). Two coordinates are the minimum needed to name a point on a circle, and dropping either one folds the day in half. ## Getting the period right Divide by the **period** — the number of distinct positions in the cycle — not by the maximum observed value. For hour of day the period is 24, not 23. Dividing by 23 sends hour 23 to an angle of 2*pi, which is the same point as hour 0, so two genuinely different hours become indistinguishable. The same rule gives period 12 for calendar month (so December and January are neighbours), period 7 for day of week, and period 365 or 366 for day of year. ## What the encoding assumes Sine-cosine encoding buys smoothness and periodicity, and those are assumptions. For a linear model, one sine-cosine pair spans exactly one smooth wave per cycle — a single peak and a single trough per day. If demand has a morning peak *and* an evening peak, one pair cannot express both; adding a second pair at half the period gives the extra shape. It also assumes neighbouring hours behave similarly. If your domain has a hard discontinuity — a tariff that switches at 07:00 sharp — smoothness is the wrong prior, and giving each hour its own indicator column will fit the step better, at the cost of 24 columns and no sharing between neighbours. With plenty of rows, the indicator version usually wins on flexibility; with few rows, the two-column circular version is far cheaper. For tree ensembles the gain is usually modest, because a tree can already isolate arbitrary hour intervals given enough splits. It is worth trying — the wrap-around window becomes a single contiguous region rather than two branches — but it is a candidate to validate, not a rule. ## Where it does not apply Only use it on genuinely cyclical quantities. Wind direction in degrees (period 360) and compass bearings qualify. An unordered category such as product type does not — there is no cycle, and imposing one invents adjacency that is not there. And an ordinary bounded numeric column, like age, is not cyclical either: the oldest customer is not adjacent to the youngest. ## Summary The integer hour is right about ordering and wrong about the one place the clock wraps. Two columns, sine and cosine of `2 * pi * h / period`, make every adjacent pair equidistant, keep every hour uniquely identified, and cost two columns instead of twenty-four — as long as you divide by the period and are willing to assume the effect varies smoothly around the cycle.