Updating a Forecast Without Starting Over

Yesterday's demand may be useful, but it can also be unusually high or low. Averaging the entire history has the opposite problem: old conditions may keep influencing the estimate after the greenhouse has changed.

We can retain a current estimate of demand and adjust it toward each new observation. The question becomes how much of the new discrepancy to use. That single choice produces a sequence of estimates we can inspect.

Stored level: 10
New reading: 14
Fraction to use: 0.25
Change = 0.25 × (14 − 10) = 1

Updated level = 10 + 1 = 11
Next forecast = 11

A new reading disagrees with the retained level. Try a small response rate and then a larger one, watching how much of that surprise enters the updated level.

11

Before today's reading arrives, Botie has yesterday's retained level estimate. He uses that to forecast today. Once the actual demand is known, he can update the level for tomorrow.

Keep those two moments separate in the code. If we update first and then score the result against today's reading, we are using part of the answer in what we call a forecast.

Before observation: forecast[t] = level[t-1]
After observation: update level[t]

The order of operations prevents tomorrow's information from changing yesterday's saved prediction.

The smoothing parameter alpha lies between zero and one. A small value changes the estimate slowly; a large value responds quickly. Alpha zero ignores new data and alpha one replaces the level with the latest observation.

The error-correction form and weighted-average form describe the same update.

level_new = level_old + alpha × (reading - level_old)
= 10 + 0.25 × (14 - 10)
= 11

The two weights in the equivalent average sum to one.

Repeated substitution shows that each older contribution is multiplied by another factor of one minus alpha at every update. Its influence decreases geometrically. The initial state also retains some influence until enough updates have occurred.

This explains the name exponential smoothing. It does not mean that the data itself is assumed to grow exponentially.

Weight on latest reading: alpha
Weight on preceding reading: alpha × (1-alpha)
Weight one step earlier: alpha × (1-alpha)^2

Choose an initial state and record it when reproducing a calculation.

import numpy as np
level = 10.0
reading = 14.0
alpha = 0.25
forecast_before_reading = level
surprise = reading - level
level = level + alpha * surprise
print(f'{forecast_before_reading:.2f}')
print(f'{level:.2f}')

10.00
11.00

Simple exponential smoothing gives a flat forecast beyond the current level. A persistent trend or seasonal pattern calls for additional structure or a different method.

Choose alpha using a training and validation procedure that respects time. Do not select it on final test outcomes. In the workshop, build one update and inspect how the old estimate and new reading each contribute.

The Python loop repeats the same update over a record, saving predictions before each new reading.

A candidate alpha defines an entire sequence of states. For each candidate, reset to the same initial state, save a forecast, observe the target and only then update. Score all candidates on matching validation targets.

Updating before saving the prediction evaluates how well the smoother explains a value it has already seen. That produces an artificially favourable result, especially at a high alpha.

import numpy as np
observed = np.array([12.,14.,13.])
for alpha in [.2,.8]:
    level = 10.
    errors = []
    for value in observed:
        errors.append(abs(value-level))
        level += alpha*(value-level)
    print(alpha, round(float(np.mean(errors)), 3))

0.2 2.493
0.8 1.64

A level-only smoother keeps one current estimate and predicts that level at every future horizon. A steadily rising process needs another quantity: the estimated change per interval. A trend model updates both level and trend, then extends the level using the trend.

The extra state adds flexibility and another estimation problem. A temporary jump can be mistaken for persistent growth. Compare with the level-only and seasonal baselines rather than assuming an extra state must improve forecasts.