When the End Is Beside the Beginning

Botie records greenhouse temperatures throughout the day. If we encode the hour as a number from zero to 23, late evening and early morning sit at opposite ends of the number line. On a clock they are neighbours.

A clock gives us a useful picture here. The hour moves around the face and returns to its starting position each day. We can put our hour values on a circle too, then use two coordinates to describe each position. Hours on either side of midnight will stay close together.

An hour number jumps at midnight even though very little time has passed. Change the hour around that boundary and compare the two coordinates of its position on the cycle.

An hour of day repeats every 24 hours. Elapsed time since an experiment began does not. We should use a circular representation only for the repeating feature, while keeping nonrepeating information elsewhere.

For this lesson, the input is an hour measured from midnight and the period is 24 hours. Fractions such as half past six can be represented too.

The period is a modelling assumption. A weekly pattern needs a different period, and calendar months do not all contain the same number of days.

Divide the hour by the period to find how far around the cycle we are. A full circle is two pi radians. Multiplying by that full-turn angle places the hour around the circle.

Hour 6:  fraction = 6 / 24 = 1/4
         angle = (1/4) × 2π = π/2
Hour 12: fraction = 12 / 24 = 1/2
         angle = (1/2) × 2π = π

Radians are a way to measure an angle. The important fact here is that a full period brings us back to the same position.

Cosine gives the horizontal coordinate of our point on a unit circle. Sine gives its vertical coordinate. We need both because a single coordinate is shared by different positions.

Hour 0:  [cos(0), sin(0)]       = [1, 0]
Hour 6:  [cos(π/2), sin(π/2)]   = [0, 1]
Hour 12: [cos(π), sin(π)]       = [-1, 0]
Hour 18: [cos(3π/2), sin(3π/2)] = [0, -1]

Sine alone would give zero at both midnight and noon. The cosine coordinate distinguishes them. Near midnight, both coordinates change smoothly across the written day boundary.

Two coordinates tell a model where we are in the cycle. They do not guarantee that the daily temperature follows a simple sinusoid. Weather, heating controls, and the season can still matter.

A linear combination of this single sine-cosine pair describes one smooth daily oscillation. More complicated recurring shapes may need additional harmonics, splines, or a flexible model. Choose that complexity through appropriate evaluation.

The workshop builds the pair for one hour. The Python practice builds it for several times and then changes the period to a week.

Putting hours on a circle makes late night and early morning neighbours. A linear model using one sine and one cosine can then describe one smooth rise and fall per cycle. It cannot independently place several sharp peaks wherever we choose.

If demand rises at two watering times, additional harmonics or another representation may help. Each additional harmonic introduces a faster repeating pattern. Choose that flexibility with validation rather than assuming a circular representation solves every time-related prediction problem.

The first harmonic completes one turn over a day. The second completes two. They are separate features, so a model can combine broad daily variation with a repeated morning-and-evening pattern.

Hour-of-day does not identify the date, season or timezone. Preserve those when they matter, and handle irregular or repeated clock times before treating array positions as equal time intervals.

import numpy as np
hour = np.array([0., 6., 12., 18.])
angle = 2 * np.pi * hour / 24
print(np.round(np.cos(angle), 3).tolist())
print(np.round(np.cos(2 * angle), 3).tolist())

[1.0, 0.0, -1.0, -0.0]
[1.0, -1.0, 1.0, -1.0]