2. Predictive Analytics

Regression and Forecasting

Estimate quantities and future demand with baselines, residuals and decision-relevant loss

Regression and Forecasting

Conditional mean, not automatic causality

A regression model may write daily orders as

Y_t=\beta_0+\beta_1\text{temperature}_t+\beta_2\text{promotion}_t+\varepsilon_t.

$\beta_1$ describes a conditional relationship under the model. Calling it the effect of temperature or promotion requires assumptions about omitted causes, selection, timing and functional form.

Small fitted example

The following five-day dataset is a calculation exercise, not empirical evidence:

Temperature °C	Orders
10	120
12	126
14	133
16	141
18	150

For simple least squares,

\hat\beta_1=\frac{\sum_i(x_i-\bar x)(y_i-\bar y)} {\sum_i(x_i-\bar x)^2},\qquad \hat\beta_0=\bar y-\hat\beta_1\bar x.

Fit and audit a simple regression

The fitted slope is about 3.75 orders per degree within this narrow sample. Extrapolating to 35°C or claiming a causal response is unsupported.

Forecast origin and horizon

A forecast is indexed by both origin $t$ and horizon $h$ :

\hat y_{t+h\mid t}=E(Y_{t+h}\mid\mathcal I_t).

The information set $\mathcal I_t$ must be frozen. A Monday replenishment forecast may use weather forecasts issued by Monday, not realised Thursday weather.

Choose loss from consequences

Metric	Formula idea	Boundary
MAE	mean $	y-\hat y
RMSE	square-root mean squared error	emphasises large misses
MAPE	mean percentage error	unstable near zero
weighted absolute percentage	$\sum	e
pinball loss	asymmetric quantile error	useful for service/stock trade-offs

If under-forecasting fresh milk costs more than over-forecasting canned goods, a single unweighted RMSE can select the wrong operational model.

Point and distribution

Inventory and staffing require uncertainty, not just a point. Report quantiles or prediction intervals and test their empirical coverage.

A nominal 90% interval should contain roughly 90% of comparable future outcomes. Coverage alone is not enough: an interval covering everything is uselessly wide. Compare calibration and sharpness.

Residual audit

Plot or tabulate residuals against:

fitted value and key inputs;
time and forecast horizon;
store, product and channel;
promotion and disruption periods;
missingness patterns.

Systematic residuals are information the model failed to encode or a sign that the data-generating process changed.

Quick check

Model A has lower daily RMSE; Model B has lower shortage cost in a replenishment simulation. Which wins?

Answer

For that replenishment decision, B is preferable if the simulation and costs are credible and constraints match deployment. Report the statistical trade-off and test sensitivity rather than claiming B is universally more accurate.

For deeper dynamic modelling, continue to the dedicated Time Series course.

Next: Classification and Calibration

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Validation, Baselines and Leakage

Simulate future use before comparing predictive models

Classification and Calibration

Evaluate risk scores through confusion, ranking, probability accuracy and prevalence