// <copyright file="Fit.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2018 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using System.Linq;
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using IStation.Numerics.LinearAlgebra;
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using IStation.Numerics.LinearRegression;
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using IStation.Numerics.Providers.LinearAlgebra;
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namespace IStation.Numerics
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{
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/// <summary>
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/// Least-Squares Curve Fitting Routines
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/// </summary>
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public static class Fit
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{
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a line y : x -> a+b*x,
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/// returning its best fitting parameters as [a, b] array,
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/// where a is the intercept and b the slope.
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/// </summary>
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public static Tuple<double, double> Line(double[] x, double[] y)
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{
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return SimpleRegression.Fit(x, y);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a line y : x -> a+b*x,
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/// returning a function y' for the best fitting line.
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/// </summary>
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public static Func<double, double> LineFunc(double[] x, double[] y)
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{
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var parameters = SimpleRegression.Fit(x, y);
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double intercept = parameters.Item1, slope = parameters.Item2;
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return z => intercept + slope*z;
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a line through origin y : x -> b*x,
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/// returning its best fitting parameter b,
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/// where the intercept is zero and b the slope.
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/// </summary>
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public static double LineThroughOrigin(double[] x, double[] y)
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{
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return SimpleRegression.FitThroughOrigin(x, y);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a line through origin y : x -> b*x,
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/// returning a function y' for the best fitting line.
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/// </summary>
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public static Func<double, double> LineThroughOriginFunc(double[] x, double[] y)
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{
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double slope = SimpleRegression.FitThroughOrigin(x, y);
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return z => slope * z;
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an exponential y : x -> a*exp(r*x),
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/// returning its best fitting parameters as (a, r) tuple.
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/// </summary>
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public static Tuple<double, double> Exponential(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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// Transformation: y_h := ln(y) ~> y_h : x -> ln(a) + r*x;
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double[] lny = Generate.Map(y, Math.Log);
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double[] p = LinearCombination(x, lny, method, t => 1.0, t => t);
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return Tuple.Create(Math.Exp(p[0]), p[1]);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an exponential y : x -> a*exp(r*x),
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/// returning a function y' for the best fitting line.
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/// </summary>
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public static Func<double, double> ExponentialFunc(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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var parameters = Exponential(x, y, method);
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var a = parameters.Item1;
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var r = parameters.Item2;
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return z => a * Math.Exp(r * z);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a logarithm y : x -> a + b*ln(x),
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/// returning its best fitting parameters as (a, b) tuple.
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/// </summary>
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public static Tuple<double, double> Logarithm(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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double[] lnx = Generate.Map(x, Math.Log);
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double[] p = LinearCombination(lnx, y, method, t => 1.0, t => t);
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return Tuple.Create(p[0], p[1]);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a logarithm y : x -> a + b*ln(x),
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/// returning a function y' for the best fitting line.
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/// </summary>
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public static Func<double, double> LogarithmFunc(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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var parameters = Logarithm(x, y, method);
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var a = parameters.Item1;
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var b = parameters.Item2;
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return z => a + b * Math.Log(z);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a power y : x -> a*x^b,
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/// returning its best fitting parameters as (a, b) tuple.
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/// </summary>
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public static Tuple<double, double> Power(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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// Transformation: y_h := ln(y) ~> y_h : x -> ln(a) + b*ln(x);
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double[] lny = Generate.Map(y, Math.Log);
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double[] p = LinearCombination(x, lny, method, t => 1.0, Math.Log);
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return Tuple.Create(Math.Exp(p[0]), p[1]);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a power y : x -> a*x^b,
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/// returning a function y' for the best fitting line.
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/// </summary>
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public static Func<double, double> PowerFunc(double[] x, double[] y, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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var parameters = Power(x, y, method);
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var a = parameters.Item1;
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var b = parameters.Item2;
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return z => a * Math.Pow(z, b);
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k,
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array, compatible with Polynomial.Evaluate.
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/// A polynomial with order/degree k has (k+1) coefficients and thus requires at least (k+1) samples.
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/// </summary>
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public static double[] Polynomial(double[] x, double[] y, int order, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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var design = Matrix<double>.Build.Dense(x.Length, order + 1, (i, j) => Math.Pow(x[i], j));
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return MultipleRegression.DirectMethod(design, Vector<double>.Build.Dense(y), method).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k,
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/// returning a function y' for the best fitting polynomial.
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/// A polynomial with order/degree k has (k+1) coefficients and thus requires at least (k+1) samples.
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/// </summary>
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public static Func<double, double> PolynomialFunc(double[] x, double[] y, int order, DirectRegressionMethod method = DirectRegressionMethod.QR)
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{
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var parameters = Polynomial(x, y, order, method);
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return z => Numerics.Polynomial.Evaluate(z, parameters);
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}
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/// <summary>
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/// Weighted Least-Squares fitting the points (x,y) and weights w to a k-order polynomial y : x -> p0 + p1*x + p2*x^2 + ... + pk*x^k,
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array, compatible with Polynomial.Evaluate.
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/// A polynomial with order/degree k has (k+1) coefficients and thus requires at least (k+1) samples.
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/// </summary>
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public static double[] PolynomialWeighted(double[] x, double[] y, double[] w, int order)
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{
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var design = Matrix<double>.Build.Dense(x.Length, order + 1, (i, j) => Math.Pow(x[i], j));
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return WeightedRegression.Weighted(design, Vector<double>.Build.Dense(y), Matrix<double>.Build.Diagonal(w)).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an arbitrary linear combination y : x -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearCombination(double[] x, double[] y, params Func<double,double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.QR(design, Vector<double>.Build.Dense(y)).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an arbitrary linear combination y : x -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<double, double> LinearCombinationFunc(double[] x, double[] y, params Func<double, double>[] functions)
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{
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var parameters = LinearCombination(x, y, functions);
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return z => functions.Zip(parameters, (f, p) => p*f(z)).Sum();
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an arbitrary linear combination y : x -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearCombination(double[] x, double[] y, DirectRegressionMethod method, params Func<double, double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.DirectMethod(design, Vector<double>.Build.Dense(y), method).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (x,y) to an arbitrary linear combination y : x -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<double, double> LinearCombinationFunc(double[] x, double[] y, DirectRegressionMethod method, params Func<double, double>[] functions)
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{
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var parameters = LinearCombination(x, y, method, functions);
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return z => functions.Zip(parameters, (f, p) => p*f(z)).Sum();
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to a linear surface y : X -> p0*x0 + p1*x1 + ... + pk*xk,
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// If an intercept is added, its coefficient will be prepended to the resulting parameters.
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/// </summary>
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public static double[] MultiDim(double[][] x, double[] y, bool intercept = false, DirectRegressionMethod method = DirectRegressionMethod.NormalEquations)
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{
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return MultipleRegression.DirectMethod(x, y, intercept, method);
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to a linear surface y : X -> p0*x0 + p1*x1 + ... + pk*xk,
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/// returning a function y' for the best fitting combination.
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/// If an intercept is added, its coefficient will be prepended to the resulting parameters.
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/// </summary>
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public static Func<double[], double> MultiDimFunc(double[][] x, double[] y, bool intercept = false, DirectRegressionMethod method = DirectRegressionMethod.NormalEquations)
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{
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var parameters = MultipleRegression.DirectMethod(x, y, intercept, method);
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return z => LinearAlgebraControl.Provider.DotProduct(parameters, z);
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}
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/// <summary>
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/// Weighted Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) and weights w to a linear surface y : X -> p0*x0 + p1*x1 + ... + pk*xk,
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] MultiDimWeighted(double[][] x, double[] y, double[] w)
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{
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return WeightedRegression.Weighted(x, y, w);
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to an arbitrary linear combination y : X -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearMultiDim(double[][] x, double[] y, params Func<double[], double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.QR(design, Vector<double>.Build.Dense(y)).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to an arbitrary linear combination y : X -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<double[], double> LinearMultiDimFunc(double[][] x, double[] y, params Func<double[], double>[] functions)
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{
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var parameters = LinearMultiDim(x, y, functions);
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return z => functions.Zip(parameters, (f, p) => p * f(z)).Sum();
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to an arbitrary linear combination y : X -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearMultiDim(double[][] x, double[] y, DirectRegressionMethod method, params Func<double[], double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.DirectMethod(design, Vector<double>.Build.Dense(y), method).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (X,y) = ((x0,x1,..,xk),y) to an arbitrary linear combination y : X -> p0*f0(x) + p1*f1(x) + ... + pk*fk(x),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<double[], double> LinearMultiDimFunc(double[][] x, double[] y, DirectRegressionMethod method, params Func<double[], double>[] functions)
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{
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var parameters = LinearMultiDim(x, y, method, functions);
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return z => functions.Zip(parameters, (f, p) => p * f(z)).Sum();
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}
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/// <summary>
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/// Least-Squares fitting the points (T,y) = (T,y) to an arbitrary linear combination y : X -> p0*f0(T) + p1*f1(T) + ... + pk*fk(T),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearGeneric<T>(T[] x, double[] y, params Func<T, double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.QR(design, Vector<double>.Build.Dense(y)).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (T,y) = (T,y) to an arbitrary linear combination y : X -> p0*f0(T) + p1*f1(T) + ... + pk*fk(T),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<T, double> LinearGenericFunc<T>(T[] x, double[] y, params Func<T, double>[] functions)
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{
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var parameters = LinearGeneric(x, y, functions);
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return z => functions.Zip(parameters, (f, p) => p * f(z)).Sum();
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}
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/// <summary>
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/// Least-Squares fitting the points (T,y) = (T,y) to an arbitrary linear combination y : X -> p0*f0(T) + p1*f1(T) + ... + pk*fk(T),
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/// returning its best fitting parameters as [p0, p1, p2, ..., pk] array.
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/// </summary>
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public static double[] LinearGeneric<T>(T[] x, double[] y, DirectRegressionMethod method, params Func<T, double>[] functions)
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{
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var design = Matrix<double>.Build.Dense(x.Length, functions.Length, (i, j) => functions[j](x[i]));
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return MultipleRegression.DirectMethod(design, Vector<double>.Build.Dense(y), method).ToArray();
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}
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/// <summary>
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/// Least-Squares fitting the points (T,y) = (T,y) to an arbitrary linear combination y : X -> p0*f0(T) + p1*f1(T) + ... + pk*fk(T),
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/// returning a function y' for the best fitting combination.
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/// </summary>
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public static Func<T, double> LinearGenericFunc<T>(T[] x, double[] y, DirectRegressionMethod method, params Func<T, double>[] functions)
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{
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var parameters = LinearGeneric(x, y, method, functions);
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return z => functions.Zip(parameters, (f, p) => p * f(z)).Sum();
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p, x),
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/// returning its best fitting parameter p.
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/// </summary>
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public static double Curve(double[] x, double[] y, Func<double, double, double> f, double initialGuess, double tolerance = 1e-8, int maxIterations = 1000)
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{
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return FindMinimum.OfScalarFunction(p => Distance.Euclidean(Generate.Map(x, t => f(p, t)), y), initialGuess, tolerance, maxIterations);
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p0, p1, x),
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/// returning its best fitting parameter p0 and p1.
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/// </summary>
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public static Tuple<double, double> Curve(double[] x, double[] y, Func<double, double, double, double> f, double initialGuess0, double initialGuess1, double tolerance = 1e-8, int maxIterations = 1000)
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{
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return FindMinimum.OfFunction((p0, p1) => Distance.Euclidean(Generate.Map(x, t => f(p0, p1, t)), y), initialGuess0, initialGuess1, tolerance, maxIterations);
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p0, p1, p2, x),
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/// returning its best fitting parameter p0, p1 and p2.
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/// </summary>
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public static Tuple<double, double, double> Curve(double[] x, double[] y, Func<double, double, double, double, double> f, double initialGuess0, double initialGuess1, double initialGuess2, double tolerance = 1e-8, int maxIterations = 1000)
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{
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return FindMinimum.OfFunction((p0, p1, p2) => Distance.Euclidean(Generate.Map(x, t => f(p0, p1, p2, t)), y), initialGuess0, initialGuess1, initialGuess2, tolerance, maxIterations);
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p, x),
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/// returning a function y' for the best fitting curve.
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/// </summary>
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public static Func<double, double> CurveFunc(double[] x, double[] y, Func<double, double, double> f, double initialGuess, double tolerance = 1e-8, int maxIterations = 1000)
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{
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var parameters = Curve(x, y, f, initialGuess, tolerance, maxIterations);
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return z => f(parameters, z);
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p0, p1, x),
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/// returning a function y' for the best fitting curve.
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/// </summary>
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public static Func<double, double> CurveFunc(double[] x, double[] y, Func<double, double, double, double> f, double initialGuess0, double initialGuess1, double tolerance = 1e-8, int maxIterations = 1000)
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{
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var parameters = Curve(x, y, f, initialGuess0, initialGuess1, tolerance, maxIterations);
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return z => f(parameters.Item1, parameters.Item2, z);
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}
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/// <summary>
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/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p0, p1, p2, x),
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/// returning a function y' for the best fitting curve.
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/// </summary>
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public static Func<double, double> CurveFunc(double[] x, double[] y, Func<double, double, double, double, double> f, double initialGuess0, double initialGuess1, double initialGuess2, double tolerance = 1e-8, int maxIterations = 1000)
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{
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var parameters = Curve(x, y, f, initialGuess0, initialGuess1, initialGuess2, tolerance, maxIterations);
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return z => f(parameters.Item1, parameters.Item2, parameters.Item3, z);
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}
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}
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}
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