Predicting Percentiles

1. What are Percentiles?

• A percentile is a quantile expressed in percentage terms.
• Example:
  • 25th percentile = value below which 25% of data fall.
  • 50th percentile = median.
  • 90th percentile = value below which 90% of data fall.

So predicting percentiles = estimating these cut-points of the outcome distribution, possibly conditional on predictors.

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2. Why Predict Percentiles?

• Beyond the mean: Standard regression predicts the average (mean).
• Percentiles tell us about distributional behavior:
  • Lower percentiles = pessimistic scenarios.
  • Higher percentiles = optimistic scenarios.
• Useful in risk assessment, demand forecasting, and fairness analysis.

Examples:

• Retail: predict 90th percentile demand (to set stock buffers).
• Finance: predict 5th percentile return (Value-at-Risk).
• Medicine: predict 95th percentile of wait times (worst-case planning).

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3. How to Predict Percentiles

A. Quantile Regression

• Directly models conditional quantiles: $Q_\alpha(y|x) = x^\top \beta_\alpha$​
• Choose different α\alphaα levels (e.g., 0.1, 0.5, 0.9) to predict multiple percentiles.

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B. Distributional Forecasting

• Fit a parametric distribution (e.g., Normal, Lognormal) to predictions.
• Then compute percentiles from the fitted CDF.
  • Example: If $y \sim N(\mu,\sigma^2)$, the 95th percentile is $\mu + 1.645\sigma$.

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C. Empirical / Simulation-Based

• Use bootstrapping, Bayesian posterior samples, or ensembles.
• Collect predictive samples and compute empirical percentiles.

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4. Example

Suppose we want to forecast daily demand.

• Model outputs:
  • 10th percentile = 85 units
  • 50th percentile = 100 units (median forecast)
  • 90th percentile = 120 units

Interpretation:

• Most likely demand ≈ 100
• In a pessimistic scenario (low demand), ≈ 85
• In a high-demand scenario, ≈ 120

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5. Relation to Prediction Intervals

• A prediction interval is defined by two percentiles.
• Example: A 90% prediction interval = [5th percentile, 95th percentile].

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6. Applications

• Finance: Value-at-Risk = a lower percentile of return distribution.
• Forecasting: supply chain, electricity load, weather extremes.
• Medicine: survival analysis (percentile life expectancy).
• Recommender Systems: estimate distribution of rating or engagement percentiles.

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Summary:
Predicting percentiles = estimating conditional quantiles of the outcome (e.g., 10th, 50th, 90th) instead of just the mean. Methods include quantile regression, distributional modeling, and simulation/bootstrapping. Percentile predictions provide richer information, enabling risk management, uncertainty quantification, and scenario planning.