What Would We Predict Before Fitting Anything?

Botie records how many litres the greenhouse uses each day. He needs to prepare tomorrow's supply before tomorrow's measurement exists. The order of these observations now matters: a good explanation of yesterday is not necessarily a useful prediction made the day before.

Let's begin with a rule that requires no fitted coefficients. Predict the most recently observed demand. We can make that prediction, wait for the next day, and record the difference. That small procedure gives every later model a fair starting comparison.

Day 4 prediction: 11 litres
Day 4 observation: 14 litres
Absolute error = |14 − 11| = 3 litres

A saved prediction must stay fixed when its target arrives. Change an observed total below and compare the error with the forecast that was already recorded for that day.

2

The forecast origin is when we make a prediction. The horizon is how far ahead its target lies. A daily one-step forecast predicts the next day's value using information available now.

Keep timestamps, measurement units, and sampling intervals with the series. Missing days are not automatically zero-demand days, and a list of irregular readings is not automatically a daily series.

Origin: end of Tuesday
Horizon: one day
Target: Wednesday's total demand

The information boundary matters more than whether a rule is simple or learned.

Persistence copies the latest observation. A historical-mean forecast averages a chosen training history. The first follows recent level changes quickly; the second smooths fluctuations but can lag behind a real change.

Choose the history before seeing the target. For a multiple-day forecast made at one origin, persistence repeats the same last known value for every horizon; it cannot copy future observations.

Known values = [10,12,11]
Persistence forecast = 11
Historical-mean forecast = (10 + 12 + 11) / 3 = 11

Agreement on this short record does not make the rules equivalent.

Compare each recorded prediction with its later observation. Absolute error keeps the size of a mistake in the original units. Average it over the same target days for every competing method.

Do not confuse fitted residuals with genuine forecast errors. A model fitted using the target's own date has received information unavailable at the earlier origin.

Observations = [10,12,11,14]
One-step persistence predictions = [10,12,11]
Targets = [12,11,14]
MAE = (2 + 1 + 3) / 3 = 2

The first observation has no preceding value here, so it is not scored.

import numpy as np
history = np.array([10., 12., 11.])
forecast = history[-1]
observed_next_day = 14.0
print(float(forecast))
print(float(abs(observed_next_day - forecast)))

11.0
3.0

Weather, opening hours, and planting schedules might improve the prediction. We will add those kinds of structure gradually, retaining a baseline on the same held-out dates.

A forecast is not a stock decision by itself. Preparing too little water may have a different cost from preparing too much. This lesson establishes point predictions; later evaluation can account for those consequences.

Botie's construction compares the preceding-day predictions and target days supplied separately.

If greenhouse demand follows a watering schedule, the most recent day in the same schedule position can be a stronger baseline than yesterday. A seasonal-naive forecast repeats that earlier observation.

The seasonal period must describe the actual calendar or process. Seven array positions mean one week only when the record really contains one position per day. Missing days need explicit treatment before we use a positional lag.

At a single forecast origin, none of the future observations are available. A multi-step seasonal baseline repeats values from the most recent observed cycle, wrapping around that cycle when necessary.

Keep those forecasts together as one issued forecast. Updating them after tomorrow's reading arrives creates a forecast from a later origin, which should be scored separately.

import numpy as np
last_cycle = np.array([10.,14.,8.])
horizon = 5
forecast = last_cycle[np.arange(horizon) % len(last_cycle)]
print(forecast.tolist())

[10.0, 14.0, 8.0, 10.0, 14.0]

A method that is useful tomorrow may be poor a week ahead. Retain origin, target date, horizon, forecast and eventual observation for every prediction. Compare methods only where those keys match.

Inspect peak-demand days and under-prediction as well as overall absolute error. If running out of water is much more costly than carrying excess, the decision may require a quantile or a stock policy rather than a mean forecast alone.