{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [], "source": ["#r \"nuget: Plotly.NET, 4.0.0\"\n", "#r \"nuget: Plotly.NET.Interactive, 4.0.0\"\n", "#r \"nuget: FSharp.Stats\"\n", "\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["# Fitting\n", "\n", "[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/fslaborg/FSharp.Stats/gh-pages?urlpath=/tree/home/jovyan/Fitting.ipynb)\n", "[![Notebook](https://fslab.org/FSharp.Stats/img/badge-notebook.svg)](https://fslab.org/FSharp.Stats/Fitting.ipynb)\n", "\n", "**Summary:** this tutorial will walk through several ways of fitting data with FSharp.Stats.\n", "\n", "### Table of contents\n", "\n", "* [Linear Regression](#Linear-Regression)\n", "\n", " * [Summary](#Summary)\n", " \n", " * [Simple Linear Regression](#Simple-Linear-Regression)\n", " \n", " * [Univariable](#Univariable)\n", " \n", " * [Multivariable](#Multivariable)\n", " \n", " \n", "\n", "* [Polynomial Regression](#Polynomial-Regression)\n", "\n", "* [Nonlinear Regression](#Nonlinear-Regression)\n", "\n", "* [LevenbergMarquardtConstrained](#LevenbergMarquardtConstrained)\n", "\n", "* [Smoothing spline](#Smoothing-spline)\n", "\n", "## Linear Regression\n", "\n", "In Linear Regression a linear system of equations is generated. The coefficients obtained by the solution to this equation\n", "system minimize the squared distances from the regression curve to the data points. These distances are also known as residuals (or least squares).\n", "\n", "### Summary\n", "\n", "With the `FSharp.Stats.Fitting.LinearRegression` module you can apply various regression methods. A `LinearRegression` type provides many common methods for fitting two- or multi dimensional data. These include\n", "\n", "* Simple linear regression (fitting a straight line to the data)\n", "\n", "* Polynomial regression (fitting a polynomial of specified order/degree) to the data\n", "\n", "* Robust regression (fitting a straight line to the data and ignoring outliers)\n", "\n", "The following code snippet summarizes many regression methods. In the following sections, every method is discussed in detail!\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 3, "outputs": [], "source": ["open Plotly.NET\n", "open FSharp.Stats\n", "open FSharp.Stats.Fitting\n", "\n", "let testDataX = vector [|1. .. 10.|]\n", "\n", "let testDataY = vector [|0.;-1.;0.;0.;0.;0.;1.;1.;3.;3.5|]\n", "\n", "\n", "let coefSimpL = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.SimpleLinear) // simple linear regression with a straight line through the data\n", "let coefSimpLRTO = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.SimpleLinear,Constraint = Constraint.RegressionThroughOrigin) // simple linear regression with a straight line through the origin (0,0)\n", "let coefSimpLXY = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.SimpleLinear,Constraint = Constraint.RegressionThroughXY (9.,1.)) // simple linear regression with a straight line through the coordinate 9,1.\n", "let coefPoly_1 = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Polynomial 1) // fits a polynomial of degree 1 (equivalent to simple linear regression)\n", "let coefPoly_2 = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Polynomial 2) // fits a quadratic polynomial\n", "let coefPoly_3 = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Polynomial 3) // fits a cubic polynomial\n", "let coefPoly_3w = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Polynomial 3, Weighting = (vector [1.;1.;1.;1.;2.;2.;2.;10.;1.;1.])) // fits a cubic polynomial with weighted points\n", "let coefRobTh = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Robust RobustEstimator.Theil) // fits a straight line that is insensitive to outliers\n", "let coefRobThSen = LinearRegression.fit(testDataX, testDataY, FittingMethod = Method.Robust RobustEstimator.TheilSen) // fits a straight line that is insensitive to outliers\n", "\n", "// f(x) = -0.167 + -0.096x + -0.001x^2 + 0.005x^3\n", "let functionStringPoly3 = coefPoly_3.ToString()\n", "\n", "let regressionComparison =\n", " [\n", " Chart.Point(testDataX,testDataY,Name=\"data\")\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefSimpL ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"SimpL\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefSimpLRTO) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"SimpL Origin\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefSimpLXY ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"SimpL 9,1\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefPoly_1 ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Poly 1\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefPoly_2 ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Poly 2\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefPoly_3 ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Poly 3\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefPoly_3w ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Poly 3 weight\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefRobTh ) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Robust Theil\"\n", " [1. .. 0.01 .. 10.] |\u003e List.map (fun x -\u003e x,LinearRegression.predict(coefRobThSen) x) |\u003e Chart.Line |\u003e Chart.withTraceInfo \"Robust TheilSen\"\n", " ]\n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored\n", " |\u003e Chart.withAnnotation(LayoutObjects.Annotation.init(X = 9.5,Y = 3.1,XAnchor = StyleParam.XAnchorPosition.Right,Text = functionStringPoly3)) \n", " |\u003e Chart.withXAxisStyle(\"x data\")\n", " |\u003e Chart.withYAxisStyle(\"y data\")\n", " |\u003e Chart.withSize(800.,600.)\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"5abbfd44-e04f-4fd3-92c7-afaaa1e917c3\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_5abbfd44e04f4fd392c7afaaa1e917c3 = function() {", "", " var data = 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TheilSen\"}];", "", " var layout = {\"width\":800,\"height\":600,\"template\":{\"layout\":{\"paper_bgcolor\":\"white\",\"plot_bgcolor\":\"white\",\"xaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true},\"yaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true}},\"data\":{}},\"annotations\":[{\"x\":9.5,\"y\":3.1,\"text\":\"f(x) = -0.167 + -0.096x + -0.001x^2 + 0.005x^3\",\"xanchor\":\"right\"}],\"xaxis\":{\"title\":{\"text\":\"x data\"}},\"yaxis\":{\"title\":{\"text\":\"y data\"}}};", "", " var config = {\"responsive\":true};", "", " Plotly.newPlot(\u00275abbfd44-e04f-4fd3-92c7-afaaa1e917c3\u0027, data, layout, config);", "", "};", "", "renderPlotly_5abbfd44e04f4fd392c7afaaa1e917c3();", "", "\u003c/script\u003e\u003c/div\u003e"] }, "execution_count": null, "metadata": {}, "output_type": "execute_result" }], "source": ["regressionComparison\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["### Simple Linear Regression\n", "\n", "Simple linear regression aims to fit a straight regression line to the data. While the least squares approach efficiently minimizes the sum of squared residuals it is prone to outliers.\n", "An alternative is a robust simple linear regression like Theil\u0027s incomplete method or the Theil-Sen estimator, that are outlier resistant.\n", "\n", "#### Univariable\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 5, "outputs": [], "source": ["open Plotly.NET\n", "open FSharp.Stats\n", "open FSharp.Stats.Fitting.LinearRegression\n", "\n", "let xData = vector [|1. .. 10.|]\n", "let yData = vector [|4.;7.;9.;12.;15.;17.;16.;23.;5.;30.|]\n", "\n", "//Least squares simple linear regression\n", "let coefficientsLinearLS = \n", " OLS.Linear.Univariable.fit xData yData\n", "let predictionFunctionLinearLS x = \n", " OLS.Linear.Univariable.predict coefficientsLinearLS x\n", "\n", "//Robust simple linear regression\n", "let coefficientsLinearRobust = \n", " RobustRegression.Linear.theilSenEstimator xData yData \n", "let predictionFunctionLinearRobust x = \n", " RobustRegression.Linear.predict coefficientsLinearRobust x\n", "\n", "//least squares simple linear regression through the origin\n", "let coefficientsLinearRTO = \n", " OLS.Linear.RTO.fitOfVector xData yData \n", "let predictionFunctionLinearRTO x = \n", " OLS.Linear.RTO.predict coefficientsLinearRTO x\n", "\n", "\n", "\n", "let rawChart = \n", " Chart.Point(xData,yData)\n", " |\u003e Chart.withTraceInfo \"raw data\"\n", " \n", "let predictionLS = \n", " let fit = \n", " [|0. .. 11.|] \n", " |\u003e Array.map (fun x -\u003e x,predictionFunctionLinearLS x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"least squares (LS)\"\n", "\n", "let predictionRobust = \n", " let fit = \n", " [|0. .. 11.|] \n", " |\u003e Array.map (fun x -\u003e x,predictionFunctionLinearRobust x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"TheilSen estimator\"\n", "\n", "let predictionRTO = \n", " let fit = \n", " [|0. .. 11.|] \n", " |\u003e Array.map (fun x -\u003e x,predictionFunctionLinearRTO x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"LS through origin\"\n", "\n", "let simpleLinearChart =\n", " [rawChart;predictionLS;predictionRTO;predictionRobust;] \n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"346e32fa-b333-4490-ab75-a1a6e6813c8a\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_346e32fab3334490ab75a1a6e6813c8a = function() {", "", " var data = [{\"type\":\"scatter\",\"mode\":\"markers\",\"x\":[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0],\"y\":[4.0,7.0,9.0,12.0,15.0,17.0,16.0,23.0,5.0,30.0],\"marker\":{},\"line\":{},\"name\":\"raw data\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0],\"y\":[3.6666666666666576,5.509090909090901,7.3515151515151445,9.193939393939388,11.036363636363632,12.878787878787875,14.721212121212119,16.563636363636363,18.406060606060606,20.24848484848485,22.090909090909093,23.933333333333337],\"marker\":{},\"line\":{},\"name\":\"least squares (LS)\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0],\"y\":[0.0,2.366233766233766,4.732467532467532,7.0987012987012985,9.464935064935064,11.83116883116883,14.197402597402597,16.563636363636363,18.929870129870128,21.296103896103894,23.66233766233766,26.028571428571425],\"marker\":{},\"line\":{},\"name\":\"LS through origin\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0],\"y\":[1.4999999999999998,4.1,6.7,9.3,11.9,14.5,17.1,19.7,22.3,24.900000000000002,27.5,30.1],\"marker\":{},\"line\":{},\"name\":\"TheilSen estimator\"}];", "", " var layout = {\"width\":600,\"height\":600,\"template\":{\"layout\":{\"paper_bgcolor\":\"white\",\"plot_bgcolor\":\"white\",\"xaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true},\"yaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true}},\"data\":{}}};", "", " var config = {\"responsive\":true};", "", " Plotly.newPlot(\u0027346e32fa-b333-4490-ab75-a1a6e6813c8a\u0027, data, layout, config);", "", "};", "", "renderPlotly_346e32fab3334490ab75a1a6e6813c8a();", "", "\u003c/script\u003e\u003c/div\u003e"] }, "execution_count": null, "metadata": {}, "output_type": "execute_result" }], "source": ["simpleLinearChart\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["#### Multivariable\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 7, "outputs": [], "source": ["//Multivariate simple linear regression\n", "let xVectorMulti =\n", " [\n", " [1.; 1. ;2. ]\n", " [2.; 0.5;6. ]\n", " [3.; 0.8;10. ]\n", " [4.; 2. ;14. ]\n", " [5.; 4. ;18. ]\n", " [6.; 3. ;22. ]\n", " ]\n", " |\u003e Matrix.ofJaggedSeq\n", "\n", "let yVectorMulti = \n", " let transformX (x:Matrix\u003cfloat\u003e) =\n", " x\n", " |\u003e Matrix.mapiRows (fun _ v -\u003e 100. + (v.[0] * 2.5) + (v.[1] * 4.) + (v.[2] * 0.5))\n", " xVectorMulti\n", " |\u003e transformX\n", " |\u003e vector\n", "\n", "let coefficientsMV = \n", " OLS.Linear.Multivariable.fit xVectorMulti yVectorMulti\n", "let predictionFunctionMV x = \n", " OLS.Linear.Multivariable.predict coefficientsMV x\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["### Polynomial Regression\n", "\n", "In polynomial regression a higher degree (d \u0026gt; 1) polynomial is fitted to the data. The coefficients are chosen that the sum of squared residuals is minimized.\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 8, "outputs": [], "source": ["open FSharp.Stats\n", "open FSharp.Stats.Fitting.LinearRegression\n", "\n", "let xDataP = vector [|1. .. 10.|]\n", "let yDataP = vector [|4.;7.;9.;8.;6.;3.;2.;5.;6.;8.;|]\n", "\n", "//Least squares polynomial regression\n", "\n", "//define the order the polynomial should have (order 3: f(x) = ax^3 + bx^2 + cx + d)\n", "let order = 3\n", "let coefficientsPol = \n", " OLS.Polynomial.fit order xDataP yDataP \n", "let predictionFunctionPol x = \n", " OLS.Polynomial.predict coefficientsPol x\n", "\n", "//weighted least squares polynomial regression\n", "//If heteroscedasticity is assumed or the impact of single datapoints should be \n", "//increased/decreased you can use a weighted version of the polynomial regression.\n", "\n", "//define the order the polynomial should have (order 3: f(x) = ax^3 + bx^2 + cx + d)\n", "let orderP = 3\n", "\n", "//define the weighting vector\n", "let weights = yDataP |\u003e Vector.map (fun y -\u003e 1. / y)\n", "let coefficientsPolW = \n", " OLS.Polynomial.fitWithWeighting orderP weights xDataP yDataP \n", "let predictionFunctionPolW x = \n", " OLS.Polynomial.predict coefficientsPolW x\n", "\n", "let rawChartP = \n", " Chart.Point(xDataP,yDataP)\n", " |\u003e Chart.withTraceInfo \"raw data\"\n", " \n", "let fittingPol = \n", " let fit = \n", " [|1. .. 0.1 .. 10.|] \n", " |\u003e Array.map (fun x -\u003e x,predictionFunctionPol x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"order = 3\"\n", "\n", "let fittingPolW = \n", " let fit = \n", " [|1. .. 0.1 .. 10.|] \n", " |\u003e Array.map (fun x -\u003e x,predictionFunctionPolW x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"order = 3 weigthed\"\n", "\n", "let polRegressionChart =\n", " [rawChartP;fittingPol;fittingPolW] \n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"a816f56d-cc3d-4c9d-a4a7-1c151575b885\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_a816f56dcc3d4c9da4a71c151575b885 = function() {", "", " var data = [{\"type\":\"scatter\",\"mode\":\"markers\",\"x\":[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0],\"y\":[4.0,7.0,9.0,8.0,6.0,3.0,2.0,5.0,6.0,8.0],\"marker\":{},\"line\":{},\"name\":\"raw data\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7000000000000002,1.8,1.9,2.0,2.1,2.2,2.3,2.4000000000000004,2.5,2.6,2.7,2.8,2.9000000000000004,3.0,3.1,3.2,3.3000000000000003,3.4000000000000004,3.5,3.6,3.7,3.8000000000000003,3.9000000000000004,4.0,4.1,4.2,4.300000000000001,4.4,4.5,4.6,4.7,4.800000000000001,4.9,5.0,5.1000000000000005,5.2,5.3,5.4,5.5,5.6000000000000005,5.7,5.800000000000001,5.9,6.0,6.1000000000000005,6.2,6.300000000000001,6.4,6.5,6.6000000000000005,6.7,6.800000000000001,6.9,7.0,7.1000000000000005,7.2,7.300000000000001,7.4,7.5,7.6000000000000005,7.7,7.800000000000001,7.9,8.0,8.100000000000001,8.2,8.3,8.4,8.5,8.600000000000001,8.7,8.8,8.9,9.0,9.1,9.200000000000001,9.3,9.4,9.5,9.6,9.700000000000001,9.8,9.9,10.0],\"y\":[4.282517482508914,4.6875512820436125,5.066051282044467,5.4186515151455135,5.745986013980782,6.048688811184309,6.327393939390127,6.582735431232271,6.815347319344769,7.025863636361662,7.214918414916979,7.383145687644755,7.531179487179024,7.659653846153818,7.769202797203169,7.860460372961115,7.934060606061688,7.990637529138915,8.030825174826841,8.055257575759493,8.064568764570904,8.059392773895109,8.040363636366141,8.008115384618035,7.963282051284827,7.90649766900054,7.838396270399217,7.759611888114891,7.670778554781593,7.572530303033351,7.465501165504212,7.350325174828199,7.22763636363935,7.0980687645716944,6.962256410259275,6.820833333336113,6.674433566436248,6.523691142193705,6.369240093242537,6.211714452216762,6.051748251750425,5.889975524477542,5.727030303032167,5.563546620048305,5.400158508160022,5.23750000000134,5.076205128206286,4.9169079254089,4.760242424243206,4.606842657343243,4.457342657343055,4.31237645687666,4.172578088578096,4.038581585081413,3.911020979020609,3.790530303029758,3.677743589742864,3.5732948717939728,3.4778181818171205,3.391947552446318,3.3163170163156295,3.25156060605908,3.198312354310694,3.1572062937044976,3.128876456874565,3.1139568764548784,3.113081585079506,3.126884615382451,3.155999999997775,3.201061771559516,3.2627039627016785,3.341560606058316,3.438265734263453,3.553453379951115,3.687757575755384,3.841812354310221,4.016251748249715,4.211709790207877,4.42882051281876,4.668217948716347,4.930536130534719,5.216409090907888,5.526470862469893,5.861355477854801,6.2216969696965805,6.608129370629328,7.02128671328704,7.461803030303727,7.930312354313486,8.427448717950327,8.95384615384826],\"marker\":{},\"line\":{},\"name\":\"order = 3\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7000000000000002,1.8,1.9,2.0,2.1,2.2,2.3,2.4000000000000004,2.5,2.6,2.7,2.8,2.9000000000000004,3.0,3.1,3.2,3.3000000000000003,3.4000000000000004,3.5,3.6,3.7,3.8000000000000003,3.9000000000000004,4.0,4.1,4.2,4.300000000000001,4.4,4.5,4.6,4.7,4.800000000000001,4.9,5.0,5.1000000000000005,5.2,5.3,5.4,5.5,5.6000000000000005,5.7,5.800000000000001,5.9,6.0,6.1000000000000005,6.2,6.300000000000001,6.4,6.5,6.6000000000000005,6.7,6.800000000000001,6.9,7.0,7.1000000000000005,7.2,7.300000000000001,7.4,7.5,7.6000000000000005,7.7,7.800000000000001,7.9,8.0,8.100000000000001,8.2,8.3,8.4,8.5,8.600000000000001,8.7,8.8,8.9,9.0,9.1,9.200000000000001,9.3,9.4,9.5,9.6,9.700000000000001,9.8,9.9,10.0],\"y\":[4.2183133626518705,4.62787859214931,5.009516841065428,5.363911333661949,5.69174529420059,5.993701946943079,6.2704645161511365,6.5227162260864855,6.751140301010847,6.956419965185945,7.139238442873504,7.300278958335244,7.440224735832885,7.559758999628157,7.659564973982775,7.740325883158468,7.802724951416955,7.84744540301996,7.875170462229207,7.886583353306415,7.882367300513308,7.8632055281116084,7.82978126036304,7.782777721529324,7.722878135872187,7.650765727653345,7.567123721134528,7.47263534057745,7.3679838102438415,7.25385235439542,7.130924197293911,6.9998825632010355,6.8614106763785205,6.716191761088078,6.564909041591436,6.40824574215033,6.2468850870264685,6.08151030048157,5.912804606777367,5.74145123017558,5.568133394937934,5.393534325326144,5.218337245601941,5.043225380027039,4.868881952863163,4.695990188372047,4.525233310815398,4.357294544454945,4.192857113552417,4.032604242369523,3.8772191551679995,3.7273850762095577,3.583785229755925,3.4471028400688297,3.3180211314099815,3.1972233280411118,3.085392654223938,2.9832123342202053,2.8913655922915993,2.810535652699862,2.741405739706721,2.6846590775738903,2.6409788905631046,2.6110484029360492,2.5955508389545017,2.595169422880147,2.610587378974728,2.642487931499943,2.691554304717542,2.758469722889224,2.843917410276731,2.948580591141784,3.0731424897460755,3.2182863303513543,3.3846953372193695,3.573052734611778,3.784041746790365,4.018345598016836,4.276647512552856,4.559630714660216,4.867978428600637,5.202373878635825,5.563500289027488,5.952040884037373,6.36867888792716,6.814097524958626,7.2889800193934775,7.794009595493421,8.329869477520177,8.897242889735494,9.49681305640108],\"marker\":{},\"line\":{},\"name\":\"order = 3 weigthed\"}];", "", " var layout = {\"width\":600,\"height\":600,\"template\":{\"layout\":{\"paper_bgcolor\":\"white\",\"plot_bgcolor\":\"white\",\"xaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true},\"yaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true}},\"data\":{}}};", "", " var config = {\"responsive\":true};", "", " Plotly.newPlot(\u0027a816f56d-cc3d-4c9d-a4a7-1c151575b885\u0027, data, layout, config);", "", "};", "", "renderPlotly_a816f56dcc3d4c9da4a71c151575b885();", "", "\u003c/script\u003e\u003c/div\u003e"] }, "execution_count": null, "metadata": {}, "output_type": "execute_result" }], "source": ["polRegressionChart\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["## Nonlinear Regression\n", "\n", "Nonlinear Regression is used if a known model should be fitted to the data that cannot be represented in a linear system of equations.\n", "Common examples are:\n", "\n", "* gaussian functions\n", " \n", "\n", "* log functions\n", " \n", "\n", "* exponential functions\n", " \n", "\n", "To fit such models to your data the `NonLinearRegression` module can be used. Three solver-methods are available to iteratively converge to a minimal least squares value.\n", "\n", "* GaussNewton\n", " \n", "\n", "* LevenbergMarquardt\n", " \n", "\n", "* LevenbergMarquardtConstrained\n", " \n", "\n", "For solving a nonlinear problem the model function has to be converted to a `NonLinearRegression.Model` type consisting of\n", "\n", "* parameter names,\n", " \n", "\n", "* the function itself, and\n", " \n", "\n", "* partial derivatives of all unknown parameters.\n", " \n", "\n", "For clarification a exponential relationship in the form of `y = a * exp(b * x)` should be solved:\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 10, "outputs": [], "source": ["open System\n", "open FSharp.Stats.Fitting\n", "open FSharp.Stats.Fitting.LinearRegression\n", "open FSharp.Stats.Fitting.NonLinearRegression\n", "\n", "let xDataN = [|1.;2.; 3.; 4.|]\n", "let yDataN = [|5.;14.;65.;100.|]\n", "\n", "//search for: y = a * exp(b * x)\n", "\n", "// 1. create the model\n", "// 1.1 define parameter names\n", "let parameterNames = [|\"a\";\"b\"|]\n", "\n", "// 1.2 define the exponential function that gets a parameter vector containing the \n", "//searched parameters and the x value and gives the corresponding y value\n", "let getFunctionValue = \n", " fun (parameterVector: Vector\u003cfloat\u003e) x -\u003e \n", " parameterVector.[0] * Math.Exp(parameterVector.[1] * x)\n", " //a * exp(b * x)\n", "\n", "// 1.3 Define partial derivatives of the exponential function. \n", "// Take partial derivatives for every unknown parameter and\n", "// insert it into the gradient vector sorted by parameterNames.\n", "let getGradientValues =\n", " fun (parameterVector:Vector\u003cfloat\u003e) (gradientVector: Vector\u003cfloat\u003e) xValueN -\u003e \n", " // partial derivative of y=a*exp(b*x) in respect to the first parameter (a) --\u003e exp(b*x)\n", " gradientVector.[0] \u003c- Math.Exp(parameterVector.[1] * xValueN) \n", " // partial derivative of y=a*exp(b*x) in respect to the second parameter (b) --\u003e a*x*exp(b*x)\n", " gradientVector.[1] \u003c- parameterVector.[0] * xValueN * Math.Exp(parameterVector.[1] * xValueN) \n", "\n", " gradientVector\n", "\n", "// 1.4 create the model\n", "let model = createModel parameterNames getFunctionValue getGradientValues\n", "\n", "// 2. define the solver options\n", "// 2.1 define the stepwidth of the x value change\n", "let deltaX = 0.0001\n", "\n", "// 2.2 define the stepwidth of the parameter change\n", "let deltaP = 0.0001\n", "\n", "// 2.3 define the number of iterations\n", "let k = 1000\n", "\n", "// 2.4 define an initial guess\n", "// For many problems you can set a default value or let the user decide to choose an \n", "// appropriate guess. In the case of an exponential or log model you can use the \n", "// solution of the linearized problem as a first guess.\n", "let initialParamGuess (xData:float []) (yData:float [])=\n", " //gets the linear representation of the problem and solves it by simple linear regression \n", " //(prone to least-squares-deviations at high y_Values)\n", " let yLn = yData |\u003e Array.map (fun x -\u003e Math.Log(x)) |\u003e vector\n", " let linearReg = \n", " LinearRegression.OLS.Linear.Univariable.fit (vector xData) yLn\n", " //calculates the parameters back into the exponential representation\n", " let a = exp linearReg.[0]\n", " let b = linearReg.[1]\n", " [|a;b|]\n", "\n", "// 2.5 create the solverOptions\n", "let solverOptions = \n", " let guess = initialParamGuess xDataN yDataN\n", " NonLinearRegression.createSolverOption 0.0001 0.0001 1000 guess\n", "\n", "// 3. get coefficients\n", "let coefficientsExp = GaussNewton.estimatedParams model solverOptions xDataN yDataN\n", "//val coefficients = vector [|5.68867298; 0.7263428835|]\n", "\n", "// 4. create fitting function\n", "let fittingFunction x = coefficientsExp.[0] * Math.Exp(coefficientsExp.[1] * x)\n", "\n", "let rawChartNLR = \n", " Chart.Point(xDataN,yDataN)\n", " |\u003e Chart.withTraceInfo \"raw data\"\n", "\n", "let fittingNLR = \n", " let fit = \n", " [|1. .. 0.1 .. 10.|] \n", " |\u003e Array.map (fun x -\u003e x,fittingFunction x)\n", " Chart.Line(fit)\n", " |\u003e Chart.withTraceInfo \"NLR\"\n", "\n", "let NLRChart =\n", " [rawChartNLR;fittingNLR] \n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"7f262d11-69fd-4a5e-b246-cf542f6f6d0c\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_7f262d1169fd4a5eb246cf542f6f6d0c = function() {", "", " var data = [{\"type\":\"scatter\",\"mode\":\"markers\",\"x\":[1.0,2.0,3.0,4.0],\"y\":[5.0,14.0,65.0,100.0],\"marker\":{},\"line\":{},\"name\":\"raw data\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7000000000000002,1.8,1.9,2.0,2.1,2.2,2.3,2.4000000000000004,2.5,2.6,2.7,2.8,2.9000000000000004,3.0,3.1,3.2,3.3000000000000003,3.4000000000000004,3.5,3.6,3.7,3.8000000000000003,3.9000000000000004,4.0,4.1,4.2,4.300000000000001,4.4,4.5,4.6,4.7,4.800000000000001,4.9,5.0,5.1000000000000005,5.2,5.3,5.4,5.5,5.6000000000000005,5.7,5.800000000000001,5.9,6.0,6.1000000000000005,6.2,6.300000000000001,6.4,6.5,6.6000000000000005,6.7,6.800000000000001,6.9,7.0,7.1000000000000005,7.2,7.300000000000001,7.4,7.5,7.6000000000000005,7.7,7.800000000000001,7.9,8.0,8.100000000000001,8.2,8.3,8.4,8.5,8.600000000000001,8.7,8.8,8.9,9.0,9.1,9.200000000000001,9.3,9.4,9.5,9.6,9.700000000000001,9.8,9.9,10.0],\"y\":[11.761370744374533,12.647439218383038,13.600261590189321,14.62486691003262,15.72666310260211,16.911465510368533,18.185527587289872,19.555573904894192,21.02883564494757,22.613088766039464,24.316695045530913,26.148646213487194,28.118611411536484,30.2369882271449,32.51495757266988,34.96454269884634,37.59867265418255,40.43125052520761,43.477226817746555,46.75267836653381,50.27489318965371,54.062461735675775,58.13537500509086,62.515130063939864,67.22484350654116,72.28937346617971,77.73545081773719,83.59182026475766,89.88939205561493,96.66140512954738,103.9436025536547,111.77442017682289,120.19518949630329,129.25035580768684,138.9877127896824,149.45865476184915,160.71844794671551,172.8265221680197,185.84678452467222,199.84795669603048,214.90393765879924,231.09419372999986,248.50417799467462,267.2257813320914,287.35781742098936,309.0065442837502,332.2862251222361,357.31973140541567,384.23919139191105,413.18668751039985,444.3150062786929,477.7884447195946,513.7836775298583,552.490689579207,594.1137786611919,638.8726337884766,687.0034947238411,738.7603988669756,794.4165220782154,854.2656205161387,918.6235810991518,987.8300887744649,1062.2504193944073,1142.2773676629915,1228.3333203285174,1320.8724855646662,1420.3832903068692,1527.3909581972252,1642.460281744528,1766.1986033309834,1899.2590207995809,2042.343834541357,2196.2082542764665,2361.6643850937385,2539.5855137871026,2730.9107181126037,2936.649823293766,3157.8887319360924,3395.7951554823153,3651.624777459229,3926.7278810462412,4222.556475946156,4540.67196217432,4882.7533712159775,5250.60622804923,5646.172080807705,6071.538748381174,6528.951340041444,7020.824105255769,7549.753176231746,8118.530270449623],\"marker\":{},\"line\":{},\"name\":\"NLR\"}];", "", " var layout = {\"width\":600,\"height\":600,\"template\":{\"layout\":{\"paper_bgcolor\":\"white\",\"plot_bgcolor\":\"white\",\"xaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true},\"yaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true}},\"data\":{}}};", "", " var config = {\"responsive\":true};", "", " Plotly.newPlot(\u00277f262d11-69fd-4a5e-b246-cf542f6f6d0c\u0027, data, layout, config);", "", "};", "", "renderPlotly_7f262d1169fd4a5eb246cf542f6f6d0c();", "", "\u003c/script\u003e\u003c/div\u003e"] }, "execution_count": null, "metadata": {}, "output_type": "execute_result" }], "source": ["NLRChart\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["### LevenbergMarquardtConstrained\n", "\n", "For nonlinear regression using the LevenbergMarquardtConstrained module, you have to follow similar steps as in the example shown above.\n", "In this example, a logistic function of the form `y = L/(1+e^(-k(t-x)))` should be fitted to count data:\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 12, "outputs": [], "source": ["open FSharp.Stats.Fitting.NonLinearRegression\n", "\n", "let xHours = [|0.; 19.5; 25.5; 30.; 43.; 48.5; 67.75|]\n", "\n", "let yCount = [|0.0; 2510000.0; 4926400.0; 9802600.0; 14949400.0; 15598800.0; 16382000.0|]\n", "\n", "// 1. Choose a model\n", "// The model we need already exists in FSharp.Stats and can be taken from the \"Table\" module.\n", "let model\u0027 = Table.LogisticFunctionAscending\n", "\n", "// 2. Define the solver options\n", "// 2.1 Initial parameter guess\n", "// The solver needs an initial parameter guess. This can be done by the user or with an estimator function.\n", "// The cutoffPercentage says at which percentage of the y-Range the lower part of the slope is. \n", "// Manual curation of parameter guesses can be performed in this step by editing the param array.\n", "let initialParamGuess\u0027 = LevenbergMarquardtConstrained.initialParam xHours yCount 0.1\n", "\n", "// 2.2 Create the solver options\n", "let solverOptions\u0027 = Table.lineSolverOptions initialParamGuess\u0027\n", "\n", "// 3. Estimate parameters for a possible solution based on residual sum of squares\n", "// Besides the solverOptions, an upper and lower bound for the parameters are required.\n", "// It is recommended to define them depending on the initial param guess\n", "let lowerBound =\n", " initialParamGuess\u0027\n", " |\u003e Array.map (fun param -\u003e\n", " // Initial paramters -20%\n", " param - (abs param) * 0.2\n", " )\n", " |\u003e vector\n", "\n", "let upperBound =\n", " initialParamGuess\u0027\n", " |\u003e Array.map (fun param -\u003e\n", " param + (abs param) * 0.2\n", " )\n", " |\u003e vector\n", "\n", "let estParams =\n", " LevenbergMarquardtConstrained.estimatedParams model\u0027 solverOptions\u0027 0.001 10. lowerBound upperBound xHours yCount\n", "\n", "// 3.1 Estimate multiple parameters and pick the one with the least residual sum of squares (RSS)\n", "// For a more \"global\" minimum. it is possible to estimate multiple possible parameters for different initial guesses.\n", "\n", "//3.1.1 Generation of parameters with varying steepnesses\n", "let multipleSolverOptions =\n", " LevenbergMarquardtConstrained.initialParamsOverRange xHours yCount [|0. .. 0.1 .. 2.|]\n", " |\u003e Array.map Table.lineSolverOptions\n", "\n", "let estParamsRSS =\n", " multipleSolverOptions\n", " |\u003e Array.map (fun solvO -\u003e\n", " let lowerBound =\n", " solvO.InitialParamGuess\n", " |\u003e Array.map (fun param -\u003e param - (abs param) * 0.2)\n", " |\u003e vector\n", " let upperBound =\n", " solvO.InitialParamGuess\n", " |\u003e Array.map (fun param -\u003e param + (abs param) * 0.2)\n", " |\u003e vector\n", " LevenbergMarquardtConstrained.estimatedParamsWithRSS \n", " model\u0027 solvO 0.001 10. lowerBound upperBound xHours yCount\n", " )\n", " |\u003e Array.minBy snd\n", " |\u003e fst\n", "\n", "// The result is the same as for \u0027estParams\u0027, but tupled with the corresponding RSS, which can be taken\n", "// as a measure of quality for the estimate.\n", "\n", "// 4. Create fitting function\n", "let fittingFunction\u0027 = Table.LogisticFunctionAscending.GetFunctionValue estParamsRSS\n", "\n", "let fittedY = Array.zip [|1. .. 68.|] ([|1. .. 68.|] |\u003e Array.map fittingFunction\u0027)\n", "\n", "let fittedLogisticFunc =\n", " [\n", " Chart.Point (Array.zip xHours yCount)\n", " |\u003e Chart.withTraceInfo\"Data Points\"\n", " Chart.Line fittedY\n", " |\u003e Chart.withTraceInfo \"Fit\"\n", " ]\n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored \n", " |\u003e Chart.withXAxisStyle \"Time\"\n", " |\u003e Chart.withYAxisStyle \"Count\"\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"44a4442f-f94d-4eb6-95a3-be81fddb9a7b\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_44a4442ff94d4eb695a3be81fddb9a7b = function() {", "", " var data = [{\"type\":\"scatter\",\"mode\":\"markers\",\"x\":[0.0,19.5,25.5,30.0,43.0,48.5,67.75],\"y\":[0.0,2510000.0,4926400.0,9802600.0,14949400.0,15598800.0,16382000.0],\"marker\":{},\"line\":{},\"name\":\"Data Points\"},{\"type\":\"scatter\",\"mode\":\"lines\",\"x\":[1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0,12.0,13.0,14.0,15.0,16.0,17.0,18.0,19.0,20.0,21.0,22.0,23.0,24.0,25.0,26.0,27.0,28.0,29.0,30.0,31.0,32.0,33.0,34.0,35.0,36.0,37.0,38.0,39.0,40.0,41.0,42.0,43.0,44.0,45.0,46.0,47.0,48.0,49.0,50.0,51.0,52.0,53.0,54.0,55.0,56.0,57.0,58.0,59.0,60.0,61.0,62.0,63.0,64.0,65.0,66.0,67.0,68.0],\"y\":[\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\",\"NaN\"],\"marker\":{},\"line\":{},\"name\":\"Fit\"}];", "", " var layout = {\"width\":600,\"height\":600,\"template\":{\"layout\":{\"paper_bgcolor\":\"white\",\"plot_bgcolor\":\"white\",\"xaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true},\"yaxis\":{\"ticks\":\"inside\",\"mirror\":\"all\",\"showline\":true,\"zeroline\":true}},\"data\":{}},\"xaxis\":{\"title\":{\"text\":\"Time\"}},\"yaxis\":{\"title\":{\"text\":\"Count\"}}};", "", " var config = {\"responsive\":true};", "", " Plotly.newPlot(\u002744a4442f-f94d-4eb6-95a3-be81fddb9a7b\u0027, data, layout, config);", "", "};", "", "renderPlotly_44a4442ff94d4eb695a3be81fddb9a7b();", "", "\u003c/script\u003e\u003c/div\u003e"] }, "execution_count": null, "metadata": {}, "output_type": "execute_result" }], "source": ["fittedLogisticFunc\n"] } , { "cell_type": "markdown", "metadata": {}, "source": ["## Smoothing spline\n", "\n", "A smoothing spline aims to minimize a function consisting of two error terms:\n", "\n", "* error1: sum of squared residuals\n", " \n", "\n", " * Similar to the OrdinaryLeastSquares regression this error term ensures the fidelity to the data.\n", " \n", " \n", "\n", "* error2: integral of the second derivative of the fitting function\n", " \n", "\n", " * This error term ensures the smoothness of the resulting curve.\n", " \n", " \n", "\n", "A smoothing parameter (lambda) mediates between the two error terms.\n", "\n", "* E = error1 + (lambda * error2)\n", " \n", "\n", " * If lambda = 0, the resulting curve minimizes the sum of squared residuals and results in an interpolating curve.\n", " \n", " \n", " * If lambda = infinity, the resulting curve is punished by the smoothness measurement and results in a straight regression line.\n", " \n", " \n", "\n", "The spline is constructed out of piecewise cubic polynomials that meet at knots. In the defined knots the function is continuous.\n", "Depending on the used smoothing factor and the defined knots the smoothing spline has a unique solution. The resulting curve is just defined within the interval defined in the x values of the data.\n", "\n", "The right amount of smoothing can be determined by cross validation or generalized cross validation.\n", "\n"] } , { "cell_type": "code", "metadata": {}, "execution_count": 14, "outputs": [], "source": ["open FSharp.Stats.Fitting\n", "\n", "let xDataS = [|1.;2.; 3.; 4.|]\n", "let yDataS = [|5.;14.;65.;75.|]\n", "\n", "let data = Array.zip xDataS yDataS\n", "\n", "//in every x position a knot should be located\n", "let knots = xDataS\n", "\n", "let spline lambda x = (Spline.smoothingSpline data knots) lambda x\n", "\n", "let fit lambda = \n", " [|1. .. 0.1 .. 4.|]\n", " |\u003e Array.map (fun x -\u003e x,spline lambda x)\n", " |\u003e Chart.Line\n", " |\u003e Chart.withTraceInfo (sprintf \"lambda: %.3f\" lambda)\n", "\n", "let rawChartS = Chart.Point(data)\n", "\n", "let smoothingSplines =\n", " [\n", " rawChartS\n", " fit 0.001\n", " fit 0.02\n", " fit 1.\n", " ]\n", " |\u003e Chart.combine\n", " |\u003e Chart.withTemplate ChartTemplates.lightMirrored \n"] } , { "cell_type": "code", "metadata": {}, "execution_count": null, "outputs": [ { "data": { "text/html": ["\u003cdiv\u003e\u003cdiv id=\"4eef3af4-90c7-484f-9f14-5356316a86d9\"\u003e\u003c!-- Plotly chart will be drawn inside this DIV --\u003e\u003c/div\u003e\u003cscript type=\"text/javascript\"\u003evar renderPlotly_4eef3af490c7484f9f145356316a86d9 = function() {", "", " var data = 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