Interpolation and extrapolation

Transcription

1 Interpolation and extrapolation Alexander Khanov PHYS6260: Experimental Methods is HEP Oklahoma State University October 30, 207

2 Interpolation/extrapolation vs fitting Formulation of the problem: there are two quantities x and y related by some (generally unknown) functional dependence y = ϕ(x) we have a set of N independent measurements of y: y = y,..., y N taken at N values of x: x = x,..., x N we want to evaluate y at arbitrary x Data fit: y = y,..., y N have known variances σ 2,..., σ2 N we want to find a function f (x, p) which depends on n < N parameters p = p,..., p n where p are chosen in some optimal way, e.g. minimize the overall difference between y i and f (x i ) methods: least squares, maximum likelihood Data interpolation/extrapolation: we want to find a function f (x) such that for all i =,..., N: f (x i ) = y i the function must be smooth and not far away from ϕ(x) A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 2 /

3 Linear interpolation For each i, define F i (x) = (x x i)y i+ + (x i+ x)y i x i+ x i Fi (x) is linear Fi (x i ) = y i, F i (x i+ ) = y i+ Simple, easy to calculate, well behaved No good if need to estimate derivatives A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 3 /

4 Lagrange interpolation Since all good functions can be expanded in a Taylor series, it makes sense to take f (x) as a polynomial unless it s a degenerate case, having N coefficients is necessary and sufficient to satisfy N conditions f (x i ) = y i How to construct such a polynomial? (x x )... (x x k )(x x k+ )... (x x N ) is a polynomial of degree N which is equal to zero at all x = x i x k F i (x) = (x x )... (x x k )(x x k+ )... (x x N ) (x k x )... (x k x k )(x k x k+ )... (x k x N ) is a polynomial of degree N which is equal to zero at all x = x i x k and equal to one at x = x k N f (x) = y i F i (x) is a polynomial of degree N which satisfies f (x i ) = y i for all i i= A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 4 /

5 Lagrange interpolation (2) A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 5 /

6 How good is polynomial interpolation? Since we don t know ϕ(x) we can t say for certain, but in general for any x there is ξ such that ϕ(x) f (x) = ϕ(n) (ξ) N! N (x x i ) If we want to optimize the interpolation by adjusting the points x i where we take the measurements, then we should try to minimize the N maximum (x x i ) (so called minimax principle) i= i= A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 6 /

7 Problems of polynomial interpolation Depending on ϕ(x), the polynomial interpolation can exhibit oscillating behavior as the number of points increases ( Runge s phenomenon ) ϕ = + (0x 5) 2 A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 7 /

8 Chebyshev polynomials If the choice of points x i is up to us, one can use Chebyshev polynomials T n (x) = cos(n cos x) x T (x) = T 2 (x) = x T 3 (x) = 2x 2 T 4 (x) = 4x 3 3x... T n+ (x) = 2xT n(x) T n (x) and tons of other funny properties The trick is to take the roots of Nth Chebyshev polynomial as the measurement points: ( x i = cos π 2i ), i =,..., N 2N One can show that this choice of x i is optimal in the minimax sense A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 8 /

9 Cubic splines The polynomial interpolation provides a single formula solution with nice mathematical properties but it s often an overkill a simpler solution: a piecewise low degree polynomial which provides a smooth curve Cubic spline: for each interval x i < x < x i+, define a cubic polynomial F i (x) = a 0 + a x + a 2 x 2 + a 3 x 3 four unknowns need four equations Two conditions per point come from F i (x i ) = y i, F i (x i+ ) = y i+ Two more conditions per point make sure that first and second derivatives are continuous at join points One also needs two more conditions at the end points e.g. natural spline F (x ) = 0, F N (x N) = 0 Given the x i, y i, the spline coefficients are found from a system of linear equations It s also possible to do a combination of splines and fitting when the spline join points ( knots ) are chosen different from the actual measurement points A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 9 /

10 Extrapolation Extrapolation: want to know ϕ(x) outside the data range (for x < x or x > x N ) Can use polynomial extrapolation, but no error estimate is available! A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 0 /

11 Data smoothing Fitting data without using any explicit fitting function The idea is to get rid of noise while preserving the important pattern properties Usual smoothing techniques: running medians, e.g. zi = median(y i, y i, y i+ ) running means, e.g. zi = (y i + y i + y i+ )/3, z i = y i /4 + y i /2 + y i+ /4 popular algorithm (back to 70 s): 353QH Usually smoothing is neither desirable nor permissible A. Khanov (PHYS6260, OSU) PHYS6260 0/30/7 /