Learning From a Series' Own Past

Botie's moving mean treats the previous two days equally. But demand may respond more strongly to the most recent day, or alternate with a schedule. We can let labelled historical input-target pairs determine those contributions.

The machinery is familiar regression. What changes is the source of its features and the care needed when using a prediction as the input to another prediction.

Known window: [8, 12]
Next prediction = (0.25 × 8) + (0.75 × 12) + 1 = 12
Next window: [12 observed, 12 predicted]

The coefficients belong to particular lag positions. Change one past reading and check its contribution before combining it with the other lags and the intercept.

12

Each row contains a fixed number of preceding observations and one later target. Add a column of ones to fit an intercept. Least squares then finds coefficients that minimise squared training residuals.

Neighbouring rows overlap in their histories. Randomly splitting them can give an unrealistic evaluation of forecasting into a later period.

prediction = intercept + older_weight × older + newer_weight × newer

The column order is part of the fitted model and must travel with it.

Use a least-squares solver rather than explicitly inverting a matrix. The solver can handle an overdetermined design and report rank information. Perfectly correlated lag columns can make individual coefficients poorly determined even when predictions look reasonable.

Choose the lag count using time-respecting validation. More lags add parameters and can overfit a short record.

design = np.column_stack([np.ones(len(X)), X])
coefficients = np.linalg.lstsq(design, targets, rcond=None)[0]

Check predictions and held-out behaviour rather than treating every fitted coefficient as a causal explanation.

For a one-step forecast, every lag in the current window is observed. For later horizons from the same origin, some lag values are not yet known. A recursive procedure substitutes its previous predictions.

An alternative fits separate direct models for different horizons. These are different forecasting procedures and should be evaluated under the horizon they will actually serve.

History = [8,12]
Rule = 1 + 0.25 × older + 0.75 × newer
Next = 1 + 0.25 × 8 + 0.75 × 12 = 12
Following = 1 + 0.25 × 12 + 0.75 × 12 = 13

The second calculation uses a prediction, not a secretly revealed observation.

import numpy as np
coefficients = np.array([1.0, 0.5])
latest_reading = 8.0
row = np.array([1.0, latest_reading])
contributions = row * coefficients
print(contributions.tolist())
print(float(contributions.sum()))

[1.0, 4.0]
5.0

Repeated application can magnify errors when a fitted recurrence is unstable. Even a stable fitted rule leaves unexplained variation, so a point forecast should not be read as a guarantee.

Plot residuals in time order, compare with simple baselines, and test on later origins. Uncertainty intervals need assumptions about future disturbances and parameter uncertainty, beyond the deterministic workshop calculation.

Botie's construction applies supplied lag coefficients. The Python task also fits a one-lag model from a small historical sequence.

When yesterday and the day before are almost identical across the training record, their columns contain nearly the same information. Many pairs of coefficients can give similar predictions. Individual coefficient interpretations become fragile even when the fitted curve looks stable.

Inspect the design rank and sensitivity to the training period. Regularisation may stabilise estimation, but coefficients in an observational forecasting rule do not become causal effects merely because a penalty was added.

For a one-lag rule, a difference in the current value is multiplied by the lag coefficient at each recursive step. A magnitude above one amplifies that difference. More complicated lag models need a corresponding recurrence analysis.

This does not mean every growing series is invalid. It means a model intended to describe stable fluctuations must be checked against the behaviour of its fitted recurrence.

import numpy as np
for coefficient in [.8, 1.2]:
    discrepancy = 1.
    path = []
    for _ in range(4):
        discrepancy *= coefficient
        path.append(round(discrepancy, 4))
    print(path)

[0.8, 0.64, 0.512, 0.4096]
[1.2, 1.44, 1.728, 2.0736]

Place residuals in time order. Repeated positive residuals at the same schedule position suggest structure that the fitted features did not explain. Compare residuals with lagged residuals and with known operating conditions.

Use those patterns to form a new modelling hypothesis, then test it on later data. Selecting many lags after repeatedly inspecting the final test period turns that period into model-development data.