Separating Repetition From Change

Botie sees high demand every time a scheduled irrigation cycle begins. A model that treats each rise as a new upward trend will repeatedly overreact. We need to compare today's demand with a relevant earlier position.

An ordinary difference compares neighbouring observations. A seasonal difference compares observations one full cycle apart. Let's see which pattern each comparison removes and what information we must keep to return to the original units.

Day 2 minus day 1: 12 − 10 = 2
Day 3 minus day 2: 14 − 12 = 2
Day 4 minus day 3: 16 − 14 = 2

Cycle 1: 10, 20
Cycle 2: 12, 22
First position: 12 − 10 = 2
Second position: 22 − 20 = 2

The reference is the same position in an earlier cycle. Change the period and check which observations now get compared before interpreting their differences.

A trend is a persistent movement in level. Seasonality is a pattern recurring at a known calendar or sampling period. Irregular variation includes changes these descriptions do not explain.

Do not infer a reliable seasonal pattern from one repetition. Inspect several cycles and understand the schedule that might produce it. A changing sampling interval can also make an apparent period misleading.

Example period = 2 observations
Matching positions: 0 with 2, 1 with 3

The period counts observations here, not an unspecified number of days.

An ordinary first difference removes the previous observation. A seasonal difference removes the observation one period earlier. The operations ask different questions and can sometimes be combined.

A stable-looking difference series is evidence to inspect, not proof that all distributional properties remain constant. Stationarity is stronger than simply having no obvious trend in one short plot.

Ordinary difference: d[t] = y[t] - y[t-1]
Seasonal difference: d[t] = y[t] - y[t-period]

The transformation discards initial levels from its output. Keep them for inversion.

If a model predicts tomorrow's ordinary difference, add it to the last observed level to obtain tomorrow's level forecast. For later horizons, use earlier predicted levels when the actual future levels are still unavailable.

For seasonal differences, use the matching seasonal reference. Depending on the horizon, that reference may itself be a forecast. Record which quantities are measured and which are predicted.

Last observed level = 16
Predicted next change = 2
Next level forecast = 16 + 2 = 18

A forecast in differenced units cannot be compared directly with an observed level.

import numpy as np
earlier_cycle = np.array([8, 12])
later_cycle = np.array([11, 13])
change = later_cycle - earlier_cycle
print(change.tolist())
print((earlier_cycle + change).tolist())

[3, 1]
[11, 13]

Differencing a series that was already stable can amplify noise and create unnecessary dependence between adjacent changes. Decide the transformation from training history and diagnostics, then evaluate the complete forecasting procedure.

The workshop calculates seasonal changes with a supplied period. Try a period of one to recover ordinary differencing, and a repeating cycle to see zero seasonal changes.

Seasonal naive forecasting instead copies the previous matching level directly. It provides a useful baseline without fitting a model to the differences.

Predicting a change removes the level from the model's target. To recover demand, add the predicted change to the last known level. At later horizons, keep adding changes to the level reconstructed so far.

Seasonal differences use a different reference: the corresponding earlier season position. If that reference lies in the forecast future, use the previously reconstructed forecast, not a secretly available observation. Save the initial history required to undo every transformation.

The first forecast begins from the last observed demand. Each later forecast begins from the preceding forecast. Cumulative sums express this recurrence once the changes have been predicted.

The example supplies changes to isolate the inversion. It does not fit a model or establish that these changes are predictable.

import numpy as np
last = 20.
predicted_change = np.array([2., -1., 3.])
level = last + np.cumsum(predicted_change)
print(level.tolist())
print(np.diff(np.r_[last, level]).tolist())

[22.0, 21.0, 24.0]
[2.0, -1.0, 3.0]

Subtracting adjacent noisy readings includes noise from both readings. Consecutive differences share one of those noise terms with opposite signs, so differencing can introduce negative short-lag dependence even when the original noise was independent.

Do not repeatedly difference until a plot merely looks flat. Compare suitable models and inspect residuals. A trend, a seasonal pattern, a level shift and a measurement fault can require different explanations despite all producing visible changes over time.