Least-squares line fitting: Difference between revisions

Revision as of 11:16, 7 November 2010

This little JXSGraph application finds the line - described by homogeneous coordinates [a,b,c] - that minimizes

[math]\displaystyle{ \sum_{i=1}^n (ax_i+by_i+cz_i)^2. }[/math]

Coming soon...

@@ Line 15: / Line 15: @@
 for (i=0;i<5;i++) {
    yr = 10*(Math.random()-0.5);
-   xr = 0.*yr+delta*(Math.random()-0.5) +1.0;
+   xr = 0.*yr+delta*(Math.random()-0.5);
    p.push(brd.create('point',[xr, yr], {withLabel:false}));
 }
 brd.unsuspendUpdate();
+var M = [], y = [], MT, B, c, z, n;
+n = p.length;
+for (i=0;i<n;i++) {
+    M.push([p[i].X(), p[i].Y(), 1.0]);
+    y.push(p[i].X()*p[i].X() + p[i].Y()*p[i].Y());
+}
+// Now, the general linear least-square fitting problem
+//    min_z || M*z - y||_2^2
+// is solved by solving the system of linear equations
+//    (M^T*M) * z = (M^T*y)
+// with Gauss elimination.
+MT = JXG.Math.transpose(M);
+B = JXG.Math.matMatMult(MT, M);
+c = JXG.Math.matVecMult(MT, y);
+z = JXG.Math.Numerics.Gauss(B, c);
+// Finally, we can read from the solution vector z the coordinates [xm, ym] of the center
+// and the radius r and draw the circle.
+var xm = z[0]*0.5;
+var ym = z[1]*0.5;
+var r = Math.sqrt(z[2]+xm*xm+ym*ym);
+brd.create('circle',[ [xm,ym], r]);
 /*
 // Having constructed the points, we can fit a line described