When the End Is Beside the Beginning

The hour after 23 is zero, yet ordinary subtraction puts those hour numbers far apart. A repeating feature needs a representation that keeps its beginning beside its end.

10 sections · 3 questions · 2 coding challenges

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Representing a Repeating Cycle as a Position

BOTIE

Time-related measurements create a practical modelling problem: day, week, and seasonal cycles have boundaries in their written numbering that are not boundaries in the phenomenon. A representation that ignores repetition can make neighbouring times look unrelated.

Sine and cosine coordinates use the familiar geometry of a circle to preserve a cycle's continuity. Scikit-learn's time-feature example documents this representation among several modelling choices. Our greenhouse clock is an invented teaching example, and we are not presenting the circle representation as a newly invented algorithm.

Sources for this history

Scikit-learn: Time-related feature engineering

Why Does Midnight Look So Far From Eleven?

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.

The issue is the shape of the information. An hour of the day repeats after a full turn. We can preserve that relationship by placing each hour around a circle and describing its position with two coordinates.

The written hour wraps around

The order continues through midnight. Zero belongs immediately after 23 on the next day.

Late evening to morning

An invented example for this lesson.

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What we can now calculate

Dividing identifies the fraction of a cycle. Sine and cosine locate that fraction on a circle, so a full turn returns to the start.

Explore the full example

Choose an example, then change a value. Follow the calculation below to see which result changes and why. These are small teaching examples; the construction later asks you to choose the operations yourself.

BOTIE

Botie's hour feature should preserve that the day repeats. We will return the position around a circle, with cosine first and sine second.

Choose an example

Hour or cycle position

Length of one cycle

Full turn in radians

Result from these values

Follow the operations

1. Hour or cycle position
```
0
```
2. Length of one cycle
```
24
```
3. Divide
```
0
```
4. Full turn in radians
```
6.2832
```
5. Multiply
```
0
```
6. Cosine of an angle
```
1
```
7. Sine of an angle
```
0
```
8. Collect two results
1. 01
2. 10
9. Result
1. 01
2. 10

This encodes a fixed-period feature. It does not fit a temperature curve or handle calendar irregularities. Editing this example does not change the workshop tests.

1. Decide What Repeats

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.

2. Turn the Fraction of a Cycle Into an Angle

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.

3. Use Both Coordinates

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.

4. The Representation Is Not the Whole Prediction Rule

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.