Homework set 1: posted on website


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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 N1 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 illconditioned or nearsingular 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! desertisland interpolator! Lagrange interpolation gives the polynomial interpolator of degree N1 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 M1 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+m1) 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 wellbehaved 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 Equallyspaced data For equallyspaced 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 2dimensional ( Ndimensional) 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 1dim 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 degreen 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 highorder polynomials by using several lowdegree 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 3rddeg 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 Notaknot: 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=i1 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 finegrained 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: runningsample methods! Running means, effectively = rebinning data! Running lown polynomial fits or interpolations Least squares = minimumvariance 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 5point 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 5day running average for gps data: Running average 24
25 Running medians Median = central value in sorted list! g k = med(f k1, f k, f k+1 ) = 3s For end points, take! g 1 = med(f 1, g 2, (3g 22g 3 ) )! g N = med(f N1, g N, (3g N12g N2 ) ) For 3s, monotonic sequences are unchanged! Points larger or smaller than both adjacent points are moved inward! g k = med(f k2, f k1, f k, f k+1, f k+2 ) = 5s Treat nexttoend points as 3s of 5s! g 2 = med(g 1, g 2, g 3 ), etc for i=n1 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 3flats Not necessarily desirable! Overclips peaks! Discontinuous slopes 25
26 Smoothing example Points = data set with fluctuations (sunspots) Curve = 34 running average smoother 3median smoother introduces flats, follows data in some places 26
27 Another example of medians vs 34RA DowJones monthly averages for
28 Another smoothing example Time series with fluctuations (sound intensity levels)! Raw data are very scattered Try central 3point running average, 5 point central median filters: g k = average(f k1, f k, f k+1 ) g k = median(f k2, f k1, 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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