Fit monotonic curves from noisy data using averaging. Compare weighted errors, blocks, and residual metrics. Export clean results tables for audits, reports, and teaching.
This sample shows a noisy upward trend where isotonic regression pools adjacent values to enforce monotonicity.
| X | Y | Weight |
|---|---|---|
| 1 | 5.1 | 1 |
| 2 | 4.3 | 1 |
| 3 | 4.8 | 2 |
| 4 | 6.2 | 1 |
| 5 | 6.0 | 1 |
| 6 | 7.4 | 1 |
| 7 | 7.1 | 2 |
| 8 | 8.0 | 1 |
The tool solves a weighted isotonic regression problem using the Pool Adjacent Violators (PAV) algorithm.
When adjacent fitted block means violate monotonicity, the algorithm merges them and replaces both with a pooled weighted mean:
Pooled Mean = (Σ wᵢyᵢ) / (Σ wᵢ)
Isotonic regression fits data when the response should move only one way across X, but measurements are noisy. The calculator enforces monotonic order and replaces local violations with pooled weighted means. This is valuable for calibration, scorecards, dose response, and risk tiers where fitted values must remain consistent with business or scientific direction. It is also effective for ranking systems that require stable, nondecreasing score translation tables across production.
Accurate output starts with aligned X and Y lists and valid positive weights. The tool parses commas, spaces, or line breaks, then checks counts automatically. Sorting by X is usually best because isotonic fitting assumes ordered observations. Optional duplicate aggregation combines repeated X values using weighted averages, reducing clutter while preserving objective function behavior. Assign larger weights to measurements with stronger evidence, exposure, or sample support during preprocessing.
The fitted table reports observed values, fitted values, residuals, absolute residuals, squared errors, and pooled block labels. Blocks show where adjacent points were merged after monotonic violations occurred. Larger blocks often indicate noisy regions or flat relationships. Residual patterns help diagnose bias, scaling issues, or an incorrect direction choice before deploying predictions. Grouped counts are especially useful when duplicate X values were aggregated before fitting in practice.
Summary metrics include RMSE, weighted RMSE, MAE, weighted MAE, SSE, weighted SSE, and weighted R². RMSE highlights larger errors, while MAE is easier to explain in original units. Weighted metrics matter when observations represent different sample sizes or confidence levels. Use these measures together to compare preprocessing options and monitoring thresholds. Weighted R² supports quick benchmarking, but domain review remains essential for interpretability and stakeholder communication.
Prediction X inputs return stepwise estimates from the fitted monotonic curve. The calculator uses edge extrapolation outside the observed range and right continuous steps inside it. Export CSV for analytics workflows and PDF for reviews or audits. The example dataset demonstrates how pooled levels improve consistency while keeping observed trend information usable for decisions. Teams can archive exports to document model assumptions and fitted monotonic levels internally.
It fits a monotonic stepwise curve to your Y values across ordered X values. The algorithm pools adjacent points whenever a monotonic violation occurs, then reports fitted levels, blocks, and error metrics.
Yes, in most cases. Isotonic regression expects observations in X order. Enable sorting unless your data is already ordered and you intentionally want to preserve the current sequence.
Use weights when observations represent different reliability, exposure, or sample sizes. Larger weights give those points more influence in pooled means and weighted error calculations.
You can keep duplicates as separate rows or aggregate them. Aggregation combines duplicate X entries into one weighted mean response, which often simplifies blocks and improves table readability.
Predictions use the fitted monotonic step function. Inside the observed X range, the tool returns a right continuous step estimate. Outside the range, it uses edge extrapolation.
The results panel provides CSV and PDF downloads. CSV is useful for spreadsheets and scripts, while PDF is convenient for sharing summaries, fitted tables, and pooled block outputs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.