Point Forecasts

1. Definition

• A point forecast is a single predicted value for a future observation.
• It represents the model’s “best guess” without explicitly showing uncertainty.

Formally:

$\hat{y}_{t+h} \in \mathbb{R}$

where $\hat{y}_{t+h}$​ is the forecast for horizon $h$.

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2. Examples

• Weather: “Tomorrow’s temperature will be 25°C.”
• Sales: “Expected demand next month = 1,200 units.”
• Energy: “Forecasted load at 6 PM = 30 GW.”

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3. How Point Forecasts Are Produced

• Statistical models: ARIMA, ETS, linear regression → output mean or fitted value.
• Machine learning models: random forest, gradient boosting, neural networks → output a numeric prediction.
• Quantile regression (special case): produces a point forecast for a chosen quantile (e.g., median).

Most commonly, the point forecast corresponds to the expected value (mean):

$\hat{y}_{t+h} = E[Y_{t+h} \mid \text{past data}]$

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4. Advantages

• Simple & clear: easy to communicate (e.g., “GDP will grow by 2%”).
• Useful when uncertainty is low or when only a single decision threshold is needed.

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5. Limitations

• Ignores uncertainty in outcomes.
• Can be misleading if future variability is large (e.g., stock market, weather extremes).
• In decision-making, often insufficient — you may need prediction intervals instead.

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6. Evaluation

Point forecasts are evaluated with error metrics comparing forecast vs observed outcome:

• MAE (Mean Absolute Error)
• RMSE (Root Mean Squared Error)
• MAPE (Mean Absolute Percentage Error)

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7. Contrast

• Point forecast: Single number (best guess).
• Prediction interval: Range of likely values (e.g., 95% chance sales between 1,000–1,400).
• Probabilistic forecast: Full probability distribution of outcomes.

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Summary:
A point forecast is a single predicted value for a future observation — the model’s best guess, often the mean. It is simple and widely used, but it ignores uncertainty. For risk-sensitive decisions, prediction intervals or probabilistic forecasts are more informative.