C# implementation of nonlinear least squares fitting including calculation of t-profiles and pairwise profile plots (see [1]). The t-profiles allow to calculate exact confidence intervals for nonlinear parameters and approximate pairwise confidence regions.
Implementation is based on:
[1] Douglas Bates and Donald Watts, Nonlinear Regression and Its Applications, John Wiley and Sons, 1988
git clone https://github.com/heal-research/HEAL.NonlinearRegression
cd HEAL.NonlinearRegression
dotnet build
Run the tests for fitting nonlinear models:
dotnet test --filter "FullyQualifiedName~Fit"
Starting test execution, please wait...
p_opt: 1.10421e+002 1.03488e+002
Successful: True, NumIters: 1, NumFuncEvals: 11, NumJacEvals: 11
SSR: 9.5471e+003 s: 3.0898e+001 RMSE: 3.0898e+001 AICc: 123.4 BIC: 121.8 DL: 67.33 DL (lattice): 58.69 neg. Evidence: 41.93
Para Estimate Std. error z Score Lower Upper Correlation matrix
0 1.1042e+002 2.3371e+001 4.72e+000 5.8347e+001 1.6249e+002 1.00
1 1.0349e+002 1.2024e+001 8.61e+000 7.6697e+001 1.3028e+002 -0.67 1.00
Optimized: 110.42108 * x0 + 103.48806
p_opt: 1.38378e+000 4.84833e-002 5.24299e-001 3.52511e-001 -6.84851e-002 -1.11809e+001
Successful: True, NumIters: 3, NumFuncEvals: 43, NumJacEvals: 43
Deviance: 7.8438e+002 AICc: 796.5 BIC: 825.6 DL: 458 DL (lattice): 456 neg. Evidence: 389.84
Para Estimate Std. error z Score Lower Upper Correlation matrix
0 1.3838e+000 1.7042e-001 8.12e+000 1.0493e+000 1.7182e+000 1.00
1 4.8483e-002 7.5999e-003 6.38e+000 3.3569e-002 6.3398e-002 -0.04 1.00
2 5.2430e-001 9.7525e-002 5.38e+000 3.3291e-001 7.1569e-001 -0.08 0.02 1.00
3 3.5251e-001 8.0993e-002 4.35e+000 1.9357e-001 5.1145e-001 -0.16 -0.08 -0.55 1.00
4 -6.8485e-002 2.4032e-001 -2.85e-001 -5.4009e-001 4.0312e-001 -0.04 0.00 0.02 -0.10 1.00
5 -1.1181e+001 1.0533e+000 -1.06e+001 -1.3248e+001 -9.1140e+000 -0.60 -0.38 -0.11 0.12 -0.62 1.00
Optimized: logistic(1.3837835 * BI_RADS + 0.048483267 * Age + 0.5242994 * Shape + 0.35251072 * Margin + -0.068485134 * Density + -11.180916)
p_opt: 6.41213e-002 2.12684e+002
Successful: True, NumIters: 1, NumFuncEvals: 21, NumJacEvals: 21
SSR: 1.1954e+003 s: 1.0934e+001 RMSE: 1.0934e+001 AICc: 98.5 BIC: 96.9 DL: 61.23 DL (lattice): 59.14 neg. Evidence: 33.06
Para Estimate Std. error z Score Lower Upper Correlation matrix
0 6.4121e-002 8.7112e-003 7.36e+000 4.4711e-002 8.3531e-002 1.00
1 2.1268e+002 7.1607e+000 2.97e+001 1.9673e+002 2.2864e+002 0.78 1.00
Optimized: x0 / (0.064121282 + x0) * 212.68374
Passed! - Failed: 0, Passed: 5, Skipped: 0, Total: 5, Duration: 705 ms
Run the tests for profile likelihood confidence intervals:
dotnet test --filter "(FullyQualifiedName~ProfilePuromycin|FullyQualifiedName~ProfileMammography)" -v detailed
Starting test execution, please wait...
profile-based marginal confidence intervals (alpha=0.05)
p0 1.3838e+000 1.0586e+000 1.7270e+000
p1 4.8483e-002 3.3810e-002 6.3677e-002
p2 5.2430e-001 3.3325e-001 7.1665e-001
p3 3.5251e-001 1.9415e-001 5.1249e-001
p4 -6.8485e-002 -5.3640e-001 4.0944e-001
p5 -1.1181e+001 -1.3314e+001 -9.1744e+000
Passed ProfileMammography [8 s]
profile-based marginal confidence intervals (alpha=0.05)
p0 6.4121e-002 4.6920e-002 8.6157e-002
p1 2.1268e+002 1.9730e+002 2.2929e+002
NUnit Adapter 4.5.0.0: Test execution complete
Passed ProfilePuromycin [22 ms]
Test Run Successful.
Total tests: 2
Passed: 2
Total time: 9,2968 Seconds
To call the library you have to provide an expression for the model as well as a dataset to fit to.
var x = new double[,] { { 0.02 }, { 0.02 }, { 0.06 }, { 0.06 }, { 0.11 }, { 0.11 }, { 0.22 }, { 0.22 }, { 0.56 }, { 0.56 }, { 1.10 }, { 1.10 } };
var y = new double[] {76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200 };
var nlr = new NonlinearRegression();
var modelExpr = "0.1 * x0 / (1.0f + 0.1 * x0)";
var parser = new HEAL.Expressions.Parser.ExprParser(modelExpr,
new[] { "x0" },
System.Linq.Expressions.Expression.Parameter(typeof(double[]), "x"),
System.Linq.Expressions.Expression.Parameter(typeof(double[]), "p"));
var likelihood = new SimpleGaussianLikelihood(x, y, parser.Parse());
nlr.Fit(parser.ParameterValues, likelihood);
nlr.WriteStatistics();Output:
SSR: 1.1954e+003 s: 1.0934e+001 RMSE: 1.0934e+001 AICc: 98.5 BIC: 96.9 DL: 66.34 DL (lattice): 54.81 neg. Evidence: 33.06
Para Estimate Std. error z Score Lower Upper Correlation matrix
0 3.3169e+003 3.7006e+002 8.96e+000 2.4924e+003 4.1414e+003 1.00
1 1.5595e+001 2.1187e+000 7.36e+000 1.0875e+001 2.0316e+001 0.98 1.00
The implementation uses alglib (https://alglib.net) for linear algebra and nonlinear least squares fitting. Alglib is licensed under GPL2+ and includes code from other projects. Commercial licenses for alglib are available.
The code is licensed under the conditions of the GPL version 3.