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1 Homework set 1: posted on website 1

2 Last time: Interpolation, fitting and smoothing Interpolated curve passes through all data points Fitted curve passes closest (by some criterion) to all data points for a given functional form (Here: f(x)=1+0.1x 2 ) Smoothed curve (here same as fitted curve) is a simple numerical representation that displays the trend of the data 2

3 Interpolation Want f(x i )=y i and f(x) reasonable for x!x i when x 1 <x<x N! Not extrapolation which is always dangerous! Basis functions! Weierstrass theorem: for g(x) continuous on [x 1 x N ], a polynomial f N exists such that May need a giant N to achieve very small!! Basic idea: unique polynomial of degree N-1 will pass through N points:! N data points (x i ) gives us N equations in N unknowns (a i )

4 Polynomial Interpolation We can put this system of equations into matrix form: x i =x values for measurements y i =vector of observations at x s, a i =unknowns - coefficients in polynomial! This is a Vandermonde matrix - special form that can be solved very efficiently: Solve for a using methods of Ch.2! But: may be ill-conditioned or near-singular Power series can produce large range of values!! Always check to see if results make sense! Generally foolish to use N>5 Unless underlying behavior is known to be consistent with large N (eg, trig functions)

5 Lagrange Interpolation Suppose b ij = 1 for i=j, =0 otherwise (so B=I)! Then y=ba! a=y How can we achieve this convenient situation?! Very easily! If N=3, for example P N!1 (x) = P 2 (x) = b 11 y 1 + b 22 y 2 + b 33 y 3 We must have P 2 (x = x 1 ) = y 1, P 2 (x 2 ) = y 2, P 2 (x 3 ) = y 3 so, if x = y 1 : b 11 = 1, b 22 = 0, b 33 = 0 if x = y 2 : b 11 = 0, b 22 = 1, b 33 = 0 if x = y 3 : b 11 = 0, b 22 = 0, b 33 = 1! This is true for the following coefficients b ij :! Easy to remember! desert-island interpolator! Lagrange interpolation gives the polynomial interpolator of degree N-1 for N points

6 Newton s method for polynomial interpolation Alternate way to get the same unique polynomial passing through N points Popular before computers: easy, relatively foolproof procedure for hand calculation! Estimate derivatives using divided differences:! Make a difference table:

7 Newton's method: simple example Quadratic interpolation using Newton s method x data 1 st diff. 2 nd diff. 3 rd diff. Tableau describing this method! More modern recursive methods: Aitken's, Neville s (see N.R. for implementations) 7

8 Neville method: successive approximations Neville s algorithm uses a tableau that looks like the difference table in Newton s, but has different significance:! For M subscripts, P ijk (M) = unique polynomial of deg M-1 which passes through points i, j, k, (M) Examples: P 1 = constant (deg 0) through point 1 = y 1 P 123 = parabola (deg 2) through y 1, y 2 and y 3 x 1 y 1 = P 1 P 12 x 2 y 2 = P 2 P 123 P 23 P 1234 x 3 y 3 = P 3 P 234 P 34 x 4 y 4 = P 4!!!!! Then ( P i(i+1)...i+ M = x! x i+ M )P i(i+1)...(i+ M!1) + ( x i! x)p (i+1)...(i+ M ) x i! x i+ M Child P is simple interpolation between parent P s, which match at points (i+1) (i+m-1) 8

9 Neville method for interpolation Then, if we define differences between generations (columns) in the tableau as C M, i = P i(i+1)...(i+ M )! P i(i+1)...(i+ M!1) x 1 y 1 = P 1 P 12 D M, i = P i(i+1)...(i+ M )! P (i+1)...(i+ M ) x 2 y 2 = P 2 P 123 C 3,1 P 23 P 1234 x 3 y 3 = P 3 P 234 P 34 x 4 y 4 = P 4 D 3,1 the recurrence relation for P s gives a recurrence relation for C s and D s: ( C M +1, i = x! x i+ m+1 )( C M, i+1! D M, i ) ( x i! x i+ m+1 ) D M +1, i = x! x i ( x i! x i+ m+1 ) So at any level in the tableau, C and D give the corrections that take the interpolation one order further the last C or D used = error estimate dy Can obtain P(x) by adding up C s or Ds for steps along path from any y to tableau s apex: e.g. P(x)=y 1 + C 2,1 + C 3,1 ( ) ( ) C M, i+1! D M, i try it at home 9

10 Error bounds for polynomial interpolation Theorem: if y i =g(x i ) where g is a continuous function with N continuous derivatives g (1) (N)! For most well-behaved functions, g (N) grows faster than N! so error grows with N! Not true for functions like e x which are defined by power series Example:! Given x y Find f(0.6) Numbers here are from f(x)=ln(x): true f(0.6)= Let s compare different methods to interpolate 10

11 Interpolation example 1.! Get P(x) by solving linear equations: 2.! Get P(x) using Lagrange formula: 11

12 x 1 y 1 f [x1,x2 ] Interpolation example 3. Get P(x) using Newton s method For f [x i, x j,x k ] = f [x j,x k ]! f [x i, x j ] x k! x i and a n = f [x 1, x 2...x n ] P(x) = f [x 1 ] + f [x 1,x 2 ](x! x 1 ) + f [x 1,x 2,x 3 ](x! x 1 )(x! x 2 ) +... (Nth order forward differences) Construct the difference table: x 2 y 2 f [x 1,x 2,x 3 ] f [x 2, x 3 ] f [x 1,x 2,x 3,x 4 ] x 3 y 3 f [x 2,x 3,x 4 ] f [x 3,x 4 ] x 4 y 4 12

13 Equally-spaced data For equally-spaced x i, the Newton formulae become very simple: x i+1 -x i = h, for all i! Define difference operator! :! Divided difference is now! Difference table becomes 13

14 Equally spaced example: Use data from previous example: 14

15 Rational function and 2D interpolation Instead of polynomial, use ratio of polynomials:! Constant a 1 or b 1 can be arbitrary, so we have (" +#+1) unknowns = N data points required! Can handle functions having poles, eg 2-dimensional ( N-dimensional) interpolation! Data set defined on 2 (or more) variables, eg: y=y(x 1,x 2 )! N values of x 1, M of x 2 (perhaps on a regular grid)! Simple approach Treat as M 1-dim interpolants in x 1 (one for each row of grid), then interpolate in x 2 X 24 X 23 X 22 X 21 X11 X12 X13 X14! Fancy approach: bicubic spline - demand continuity of slopes between grid squares! Or: apply simple approach using cubic splines (later tonight) 15

16 Chebyshev (Tchebycheff) Polynomials Important family of orthogonal polynomials with many useful features! Recall:! Recursion relation gives T n (x) has n zeroes in [-1,+1], and T n (x) < 1! Symmetry: T n (-x)= (-1) n T n (x)! Orthogonality:! T i (x k ) T j (x k ) = 0, for i " j j = m / 2, for i = j (= m, if i = 1) Where x k are the zeroes of T m (x) 16

17 Minimax property Most important feature of Chebyshev polynomials is their Minimax property:! Recall max norm definition:! Theorem: of all degree-n polynomials, is smallest on [-1,+1] Recall Weierstrass thm:! This property means Chebyshev interpolator has minimum maximum deviation for given N Interpolating with Chebyshev polynomials! First map [a, b] onto [-1,+1]! Then Shift and scale x! Orthogonality property guarantees that if the N data points are {x i }={zeroes of T n } then there will be zero error at the {x i } and thus the error E= f(x)-p N (x) oscillates within [-1,+1], in contrast to monotone increase characteristic of Taylor series (polynomial interp) 17

18 Chebyshev interpolator properties Is minimax the Holy Grail?! Maybe not: we often need to minimize another norm (eg: f 2 = least squares)! We must be free to choose {x i }={zeroes of T n } May be possible if we are using interpolation to represent a much larger set of data Important application: numerical integration and are simply related to the C k 18

19 Piecewise interpolation Avoid unnecessary reversals in high-order polynomials by using several low-degree functions for subsets of points! Linear: just connect the dots with straight lines Unique, but has corners at each point (knot)! Cubic: pin a 2nd or 3rd-deg polynomial to sets of 3 or 4 points, then use only part of the interval (1st 2 or 3 point range) Not unique, and still has corners at knots! Cubic spline: set of cubics, with continuity conditions across knots Spline = draftsman s flexible metal ruler Constrained to pass through knots Unique; continuous 1st and 2nd derivatives across knots Cubic spline 19

20 Cubic splines Natural spline: set f =0 at ends! Matches many physical situations (eg beam bending) Spline in tension :! piecewise linear as tension is increased General case: cubic has 4 coefficients, so we need 4 constraints! 2 points = 2 constraints! Choice of additional constraints: Complete: specify f at both ends Not-a-knot: require f continuous at one knot Natural: f =0 at ends Spline solution: given {x i,y i }=set of N points, and h i =x i+1 - x i (not necessarily all equal) 20

21 Spline solutions; Hermite polynomials S i (x)=a i1 +a i2 x i +a i3 x i2 +a i4 x i3, i=1 N Note that the resulting coefficient matrix will be tridiagonal! Each data point is only coupled to its nearest neighbors, n=i-1 and n=i+1! easy to store for large N! LU decomposition only takes O(N) operations Cubic Hermite interpolation! Piecewise cubic with continuous first derivatives! Specify {x,y} and slopes {d} at each knot! {d} may be available from data If not, must define: for example, use linear estimate, or pin a quadratic to 3 points! Simpler than splines, adequate and faster for many cases; may know slopes at knots 21

22 Smoothing of data Want to extract features and trends from a large data set with fine-grained fluctuations! eg: sunspots, stock market, robot positions vs t! Assume observed data = true values plus random fluctuations In Fourier language: low and high frequency components No a priori model for trends (on some scale)! Eg, sunspots this month, stock price this week Need a robust (model independent) method Smoothing = lowpass filtering Obvious method: Fourier analysis! Too cumbersome for many applications! Data may arrive in realtime, don t have all at once Simpler / faster: running-sample methods! Running means, effectively = rebinning data! Running low-n polynomial fits or interpolations Least squares = minimum-variance Chebyshev = minimax! Running medians: clip peaks, reject outliers In general: given {x i, y i =f(x i )} N, replace {y i } with {g i } which is smoother 22

23 Running means and fits Running means:! (g k is central, h k is forward ; can be backward)! Equivalent to rebinning histogram from bin size $ to bin size (2m+1)$ Running fits We ll discuss fitting next but for now:! Fit parabola to a 5-point window of data! Take central value of parabola to replace y i *Ref. at end of Ch 14 in NR 23

24 Running average example Raw 5-day running average for gps data: Running average 24

25 Running medians Median = central value in sorted list! g k = med(f k-1, f k, f k+1 ) = 3s For end points, take! g 1 = med(f 1, g 2, (3g 2-2g 3 ) )! g N = med(f N-1, g N, (3g N-1-2g N-2 ) ) For 3s, monotonic sequences are unchanged! Points larger or smaller than both adjacent points are moved inward! g k = med(f k-2, f k-1, f k, f k+1, f k+2 ) = 5s Treat next-to-end points as 3s of 5s! g 2 = med(g 1, g 2, g 3 ), etc for i=n-1 Use 3s result for ends! Monotonic sequences unchanged! Flats of length >3 unchanged! Flats of length <3 move inward! Typical procedure: 353 smooth =3(5(3)) Clips peaks to leave 3-flats Not necessarily desirable! Overclips peaks! Discontinuous slopes 25

26 Smoothing example Points = data set with fluctuations (sunspots) Curve = 3-4 running average smoother 3-median smoother introduces flats, follows data in some places 26

27 Another example of medians vs 3-4RA Dow-Jones monthly averages for

28 Another smoothing example Time series with fluctuations (sound intensity levels)! Raw data are very scattered Try central 3-point running average, 5 point central median filters: g k = average(f k-1, f k, f k+1 ) g k = median(f k-2, f k-1, f k, f k+1, f k+2 )! Smoothing removes outliers, displays trends May lose valuable information! For some noise studies, outliers are the critical data! 28

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