Fittind data with shifted Chebyshev polynomials

Question

I am trying to fit data from a simulation to a particular class of polynomials, according to least squares approach.

data = Import["Mathematica file.txt", "Table"]

The polynomials are the shifted Chebyshev polynomials, that is: $${\displaystyle T_{n}^{*}(x)=T_{n}(2x-1)~.}$$ How could one use Mathematica to find the coefficients $a_0$, $a_1$ and so on in: $$\sum_{n=0}^N a_n T_{n}(2x-1)$$ which fits the imported data? Is there a way in Mathematica to find the goodness of fit in Mathematica?

Answer 1

Clear["Global`*"]

f[x_] = x*Sin[x];

data = Table[{x, f[x]}, {x, 0, 10, 0.5}];

model[m_, x_] := Sum[a[n] ChebyshevT[n, 2 x - 1], {n, 0, m}];

nlm[m_Integer?Positive, x_] :=
 NonlinearModelFit[data, model[m, x], Array[a, m + 1, 0], x]

Manipulate[
 Module[{model = nlm[m, x]},
  Column[{
    Plot[Evaluate@{f[x], model // Normal}, {x, 0, 10},
     PlotRange -> Full,
     ImageSize -> 300,
     Epilog -> {Red, AbsolutePointSize[4], Point[data]},
     PlotLegends -> Placed[{"f[x]", "approx"}, {.475, .7}]],
    model["ParameterTable"]},
   Alignment -> Center,
   ItemSize -> {35, Automatic}]],
 {{m, 4}, 1, 8, 1, ControlType -> SetterBar}]

EDIT: The unformatted array of values from the table are

nlm[4, x]["ParameterTableEntries"]

(* {{0.983793, 0.783451, 1.25572, 0.227247}, 
 {2.24593, 0.592713, 3.78923, 0.00160888}, 
 {-0.506845, 0.0754447, -6.7181, 4.93792*10^-6}, 
 {0.0272154, 0.00327794, 8.3026, 3.41441*10^-7}, 
 {-0.000413407, 0.0000450602, -9.17455, 8.98221*10^-8}} *)

Answer 2

Shifted Chebyshev polynomials are orthogonal on {0,1} with the weight function 1/Sqrt[x-x^2].

As the data: dat is not known, but we can create an interpolating function:

fun[x_]=Interpolation[dat]

For an example, we choose funx[x_]=Sin[2Pi x]. We can now expand this function in a series of shifted Chebyshev polynomials. The expansion coefficients are given by:

Clear[sT, fun, c, approx]
sT[n_, x_] := ChebyshevT[n, 2 x - 1]
fun[x_] := Sin[2 Pi x];
c[n_] := NIntegrate[fun[x] sT[n, x]/Sqrt[x - x^2], {x, 0, 1}]/
    NIntegrate[sT[n, x]^2/Sqrt[x - x^2], {x, 0, 1}] // Simplify // Chop

The partial sum tp to n:

approx[n_, x_] := Sum[c[i] sT[i, x], {i, 0, n}] // Quiet

I put a quiet at the end, because this will eventually give some warnings about precision and convergence if, due to symmetry, some coefficients are zero. But these can mostly be ignored. If you want exact results, you may use "Integrate", but this may take a long time.

We can now create the first few partial sums and plot them against the original function:

Do[
 f[x_] = approx[i, x];
 Plot[{f[x], fun[x]}, {x, 0, 1}] // Print
 , {i, 1, 5, 2}]