Conditional sum

edit

Emmanuel thanks. You often tell me to learn Python. I have decided to program somes statistical formula I can find easily in some softwares. Other are not.

Emmanuel I need An explanation. As I read it spline need as input a list composed of couples of the type (c,y) . But when when it is called you used only one coordinate. Hopw such a function can be writen ?

Supposing x and y being lists (or other iterables) of the same length,

• zip(x, y) returns a generator of two-elements tuples composed of correspondingx and y elements.

• spline(zip(x, y))returns an interpolation function of these correspondances by piecewise cubic polynomials, minimizing

  • the sum of squares between the function and the orignal points, and

  • the total curvature (integral of squared second derivatives) of the curve.

Try this :

sage: set_random_seed(1789)
sage: M=matrix(RDF, 10,1, range(10))
sage: M=M.augment(vector(map(lambda u:u^2/5-u+0.5-random(), M.column(0))))
sage: points(M)+plot(g, (0, 9))

and explore the resulting objects...

There are many types of splines ; R has implementations for a bunch of them.

HTH,

The spline interpolation may give unreasonable values at points of strong curvature variation ; you may prefer a piecewise interpolation, such as the one given bu R's approxfun function. Compare :

sage: rf=r.approxfun(MD.column(2).list(), MD.column(0).list())
sage: f=lambda x:rf(x).sage()
sage: rf=r.approxfun(MD.column(2).list(), MD.column(0).list())
sage: f=lambda x:rf(x).sage()
sage: plot([f, g], (0, 1), legend_label=["linear", "spline"])