fit: export Func1D.Hessian

Signed-off-by: Sebastien Binet <[email protected]>

Author: sbinet

fit/curve1d_example_test.go

@@ -5,6 +5,7 @@
 package fit_test
 
 import (
+	"fmt"
 	"image/color"
 	"log"
 	"math"
@@ -13,7 +14,10 @@ import (
 	"go-hep.org/x/hep/hbook"
 	"go-hep.org/x/hep/hplot"
 	"gonum.org/v1/gonum/floats"
+	"gonum.org/v1/gonum/mat"
 	"gonum.org/v1/gonum/optimize"
+	"gonum.org/v1/gonum/stat"
+	"gonum.org/v1/gonum/stat/distuv"
 	"gonum.org/v1/plot/plotter"
 	"gonum.org/v1/plot/vg"
 )
@@ -289,3 +293,130 @@ func ExampleCurve1D_powerlaw() {
 		}
 	}
 }
+
+func ExampleCurve1D_hessian() {
+	var (
+		cst   = 3.0
+		mean  = 30.0
+		sigma = 20.0
+		want  = []float64{cst, mean, sigma}
+	)
+
+	xdata, ydata, err := readXY("testdata/gauss-data.txt")
+	if err != nil {
+		log.Fatal(err)
+	}
+
+	// use a small sample
+	xdata = xdata[:min(25, len(xdata))]
+	ydata = ydata[:min(25, len(ydata))]
+
+	gauss := func(x, cst, mu, sigma float64) float64 {
+		v := (x - mu)
+		return cst * math.Exp(-v*v/sigma)
+	}
+
+	f1d := fit.Func1D{
+		F: func(x float64, ps []float64) float64 {
+			return gauss(x, ps[0], ps[1], ps[2])
+		},
+		X:  xdata,
+		Y:  ydata,
+		Ps: []float64{10, 10, 10},
+	}
+	res, err := fit.Curve1D(f1d, nil, &optimize.NelderMead{})
+	if err != nil {
+		log.Fatal(err)
+	}
+
+	if err := res.Status.Err(); err != nil {
+		log.Fatal(err)
+	}
+	if got := res.X; !floats.EqualApprox(got, want, 1e-3) {
+		log.Fatalf("got= %v\nwant=%v\n", got, want)
+	}
+
+	inv := mat.NewSymDense(len(res.Location.X), nil)
+	f1d.Hessian(inv, res.Location.X)
+	// fmt.Printf("hessian: %1.2e\n", mat.Formatted(inv, mat.Prefix("         ")))
+
+	popt := res.Location.X
+	pcov := mat.NewDense(len(popt), len(popt), nil)
+	{
+		var chol mat.Cholesky
+		if ok := chol.Factorize(inv); !ok {
+			log.Fatalf("cov-matrix not positive semi-definite")
+		}
+
+		err := chol.InverseTo(inv)
+		if err != nil {
+			log.Fatalf("could not inverse matrix: %+v", err)
+		}
+		pcov.Copy(inv)
+	}
+
+	// compute goodness-of-fit.
+	gof := newGoF(f1d.X, f1d.Y, popt, func(x float64) float64 {
+		return f1d.F(x, popt)
+	})
+
+	pcov.Scale(gof.SSE/float64(len(f1d.X)-len(popt)), pcov)
+
+	// fmt.Printf("pcov: %1.2e\n", mat.Formatted(pcov, mat.Prefix("      ")))
+
+	var (
+		n   = float64(len(f1d.X))    // number of data points
+		ndf = n - float64(len(popt)) // number of degrees of freedom
+		t   = distuv.StudentsT{
+			Mu:    0,
+			Sigma: 1,
+			Nu:    ndf,
+		}.Quantile(0.5 * (1 + 0.95))
+	)
+
+	for i, p := range popt {
+		sigma := math.Sqrt(pcov.At(i, i))
+		fmt.Printf("c%d: %1.5e [%1.5e, %1.5e] -- truth: %g\n", i, p, p-sigma*t, p+sigma*t, want[i])
+	}
+	// Output:
+	//c0: 2.99999e+00 [2.99999e+00, 3.00000e+00] -- truth: 3
+	//c1: 3.00000e+01 [3.00000e+01, 3.00000e+01] -- truth: 30
+	//c2: 2.00000e+01 [2.00000e+01, 2.00000e+01] -- truth: 20
+}
+
+type GoF struct {
+	SSE        float64 // Sum of squares due to error
+	Rsquare    float64 // R-Square is the square of the correlation between the response values and the predicted response values
+	NdF        int     // Number of degrees of freedom
+	AdjRsquare float64 // Degrees of freedom adjusted R-Square
+	RMSE       float64 // Root mean squared error
+}
+
+func newGoF(xs, ys, ps []float64, f func(float64) float64) GoF {
+	switch {
+	case len(xs) != len(ys):
+		panic("invalid lengths")
+	}
+
+	var gof GoF
+
+	var (
+		ye = make([]float64, len(ys))
+		nn = float64(len(xs) - 1)
+		vv = float64(len(xs) - len(ps))
+	)
+
+	for i, x := range xs {
+		ye[i] = f(x)
+		dy := ys[i] - ye[i]
+		gof.SSE += dy * dy
+		gof.RMSE += dy * dy
+	}
+
+	gof.Rsquare = stat.RSquaredFrom(ye, ys, nil)
+	gof.AdjRsquare = 1 - ((1 - gof.Rsquare) * nn / vv)
+	gof.RMSE = math.Sqrt(gof.RMSE / float64(len(ys)-len(ps)))
+	gof.NdF = len(ys) - len(ps)
+
+	return gof
+}

fit/fit.go

@@ -87,6 +87,14 @@ func (f *Func1D) init() {
 	}
 }
 
+// Hessian computes the hessian matrix at the provided x point.
+func (f *Func1D) Hessian(hess *mat.SymDense, x []float64) {
+	if f.hess == nil {
+		f.init()
+	}
+	f.hess(hess, x)
+}
+
 // FuncND describes a multivariate function F(x0, x1... xn; p0, p1... pn)
 // for which the parameters ps can be found with a fit.
 type FuncND struct {