{ "openapi": "3.1.1", "info": { "title": "Curve Fitter & Regression Calculator API", "version": "1.0.0", "summary": "Fit lines, polynomials, exponentials or your own model: parameters, R², errors, residuals.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflinecurvefit.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Fit lines, polynomials, exponentials or your own model: parameters, R², errors, residuals.", "description": "Paste x/y data (CSV, tabs or spaces) and fit a straight line, polynomial, exponential, power-law, logarithmic, multiple linear or your own nonlinear model, or interpolate with a natural cubic spline. You get the parameters with standard errors, t and p values and confidence intervals, R² and adjusted R², an ANOVA table, a residual plot and predictions with confidence intervals. Verified against the NIST StRD reference datasets; part of the Proofline math notebook.\n\nMethodology:\n- Models that are linear in their parameters (line, polynomial, multiple, logarithmic) are solved by Householder QR of the column-scaled design matrix — never by the normal equations, which square the condition number.\n- Exponential, power and custom models use Levenberg–Marquardt: each damped step solves [J; √λ·D]·δ ≅ [r; 0] by QR with D the column norms of the central-difference Jacobian. Exponential and power fits start from the log-linear fit but minimise the residuals of y itself.\n- Standard errors are √diag(s²·(RᵀR)⁻¹) with s² = SSE/(n − p); t and p values and confidence intervals use Student’s t with n − p degrees of freedom. For nonlinear models these are the usual linearised (asymptotic) values.\n- R² = 1 − SSE/SST (centred); without an intercept R² is uncentred, as in the NIST datasets. The F test compares the model with a constant.\n- Verified against NIST StRD certified values: Norris, Pontius, NoInt1, Longley, Wampler1 and Filip (linear) and Misra1a, Thurber, DanWood, Chwirut2 and MGH09 (nonlinear, both start points) — at least 7 matching digits for the parameters.\n- Splines: natural cubic spline (second derivative zero at both ends) from a tridiagonal system; outside the data the end cubics are continued and flagged as extrapolation.\n\nRegression statistics assume independent errors with constant variance. A high R² does not prove that the model is right — check the residual plot for patterns.", "tags": [ "apps" ], "x-mcp-tool": "proofline_curve_fit", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineCurveFitInput" }, "examples": { "example1": { "summary": "Straight line with prediction", "value": { "data": "x,y\n1,2.1\n2,3.9\n3,6.2\n4,7.8\n5,10.1", "model": "linear", "evaluateAt": "6" } }, "example2": { "summary": "NIST Norris calibration data (certified line)", "value": { "data": "x,y\n0.2,0.1\n337.4,338.8\n118.2,118.1\n884.6,888.0\n10.1,9.2\n226.5,228.1\n666.3,668.5\n996.3,998.5\n448.6,449.1\n777.0,778.9\n558.2,559.2\n0.4,0.3\n0.6,0.1\n775.5,778.1\n666.9,668.8\n338.0,339.3\n447.5,448.9\n11.6,10.8\n556.0,557.7\n228.1,228.3\n995.8,998.0\n887.6,888.8\n120.2,119.6\n0.3,0.3\n0.3,0.6\n556.8,557.6\n339.1,339.3\n887.2,888.0\n999.0,998.5\n779.0,778.9\n11.1,10.2\n118.3,117.6\n229.2,228.9\n669.1,668.4\n448.9,449.2\n0.5,0.2", "model": "linear" } }, "example3": { "summary": "NIST Misra1a: y = b1·(1 − e^(−b2·x))", "value": { "data": "x y\n77.6 10.07\n114.9 14.73\n141.1 17.94\n190.8 23.93\n239.9 29.61\n289.0 35.18\n332.8 40.02\n378.4 44.82\n434.8 50.76\n477.3 55.05\n536.8 61.01\n593.1 66.40\n689.1 75.47\n760.0 81.78", "model": "custom", "expression": "b1*(1 - exp(-b2*x))", "parameters": "b1 = 500, b2 = 0.0001" } }, "example4": { "summary": "Exponential decay", "value": { "data": "t,activity\n0,5.0\n1,3.1\n2,1.8\n3,1.1\n4,0.68\n5,0.40", "model": "exponential", "evaluateAt": "6" } }, "example5": { "summary": "Quadratic polynomial", "value": { "data": "x y\n-2 4.2\n-1 0.9\n0 0.1\n1 1.2\n2 3.8\n3 9.1", "model": "polynomial", "degree": 2 } }, "example6": { "summary": "Natural cubic spline interpolation", "value": { "data": "x,y\n0,0\n1,0.84\n2,0.91\n3,0.14\n4,-0.76", "model": "spline", "evaluateAt": "0.5, 2.5" } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineCurveFitResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineCurveFitInput": { "type": "object", "additionalProperties": false, "properties": { "data": { "title": "Data", "description": "One row per line, values separated by commas, tabs, semicolons or spaces; an optional header row names the columns. Use a decimal point (1.5).", "type": "string", "maxLength": 400000 }, "model": { "title": "Model", "type": "string", "enum": [ "linear", "polynomial", "exponential", "power", "logarithmic", "multiple", "custom", "spline", "interpolation" ], "default": "linear" }, "xColumn": { "title": "x column", "description": "Column number (1 = first) or header name; default the first column. Multiple regression: a comma-separated list (default: every column except y).", "type": "string", "maxLength": 200 }, "yColumn": { "title": "y column", "description": "Column number or header name; default the second column.", "type": "string", "maxLength": 100 }, "degree": { "title": "Polynomial degree", "type": "integer", "minimum": 1, "maximum": 10, "default": 2 }, "intercept": { "title": "Include an intercept (constant term)", "description": "Straight line, polynomial and multiple regression. Without it the fit goes through the origin and R² is uncentred.", "type": "boolean", "default": true }, "expression": { "title": "Custom model y =", "description": "Formula in x and your parameters. Use + − * / ^, implicit products (2x), sin, cos, tan, exp, ln (natural log), log (base 10), sqrt, abs, pi and e. Angles are in radians.", "type": "string", "maxLength": 2000 }, "parameters": { "title": "Parameters with start values", "description": "Custom model: every parameter with a start value. Start values near the solution help Levenberg–Marquardt.", "type": "string", "maxLength": 1000 }, "evaluateAt": { "title": "Evaluate the fit at x =", "description": "Optional list of x values: predicted y with standard error and confidence interval (interpolated value for splines).", "type": "string", "maxLength": 4000 }, "confidence": { "title": "Confidence level", "type": "number", "minimum": 0.5, "maximum": 0.9999, "default": 0.95 } }, "required": [ "data" ] }, "ProoflineCurveFitResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "model", "equation", "n", "xColumns", "yColumn", "parameters", "r2", "adjR2", "residualSd", "sse", "dof", "fStatistic", "fP", "confidence", "converged", "iterations", "points", "predictions", "segments" ], "properties": { "model": { "type": "string", "enum": [ "linear", "polynomial", "exponential", "power", "logarithmic", "multiple", "custom", "spline", "interpolation" ] }, "equation": { "type": "string" }, "n": { "type": "integer" }, "xColumns": { "type": "array", "items": { "type": "string" } }, "yColumn": { "type": "string" }, "parameters": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "name", "value", "se", "t", "p", "ciLow", "ciHigh" ], "properties": { "name": { "type": "string" }, "value": { "type": "number" }, "se": { "type": [ "number", "null" ] }, "t": { "type": [ "number", "null" ] }, "p": { "type": [ "number", "null" ] }, "ciLow": { "type": [ "number", "null" ] }, "ciHigh": { "type": [ "number", "null" ] } } } }, "r2": { "type": [ "number", "null" ] }, "adjR2": { "type": [ "number", "null" ] }, "residualSd": { "type": [ "number", "null" ] }, "sse": { "type": [ "number", "null" ] }, "dof": { "type": [ "integer", "null" ] }, "fStatistic": { "type": [ "number", "null" ] }, "fP": { "type": [ "number", "null" ] }, "confidence": { "type": "number" }, "converged": { "type": "boolean" }, "iterations": { "type": [ "integer", "null" ] }, "points": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "x", "y", "fitted", "residual" ], "properties": { "x": { "type": "number" }, "y": { "type": "number" }, "fitted": { "type": "number" }, "residual": { "type": "number" } } } }, "predictions": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "x", "y", "se", "ciLow", "ciHigh", "extrapolated" ], "properties": { "x": { "type": "number" }, "y": { "type": "number" }, "se": { "type": [ "number", "null" ] }, "ciLow": { "type": [ "number", "null" ] }, "ciHigh": { "type": [ "number", "null" ] }, "extrapolated": { "type": "boolean" } } } }, "segments": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "x0", "x1", "a", "b", "c", "d" ], "properties": { "x0": { "type": "number" }, "x1": { "type": "number" }, "a": { "type": "number" }, "b": { "type": "number" }, "c": { "type": "number" }, "d": { "type": "number" } } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }