Curve Fitting and Interpolation

Transcription

1 Chapter 5 Curve Fitting and Interpolation 5.1 Basic Concepts Consider a set of (x, y) data pairs (points) collected during an experiment, Curve fitting: is a procedure to develop or evaluate mathematical formulas (functions or equations) that best fit the relationship between x and y. The fitted function & data may not exactly agree but fit well overall. Interpolation: is a procedure to use a mathematical formula to represent a set of measured data points, such that the formula gives exact value at all given points and estimate a value between known values of data points, i.e., a technique to estimate y at some x where there s no measured data. Thus, interpolated function passes exactly through known data. 5.2 Curve Fitting Suppose you sampled a set of (x, y) data pairs, curve fitting is to find a BEST function (curve) which minimizes the sum of error between the sampled and the predicated by the function (curve) for all data. Curve fitting includes: (1). Linear regress (2). Polynomial regress (3). Multiple linear regress Linear Regress Linear Regress: is to find a line that will best fit the sampled data. 47

2 Because the sampled data points may not all fall on the line found, the fitted is not an exact solution but an approximated solution. There are errors between the fitted model and the observation. The BEST line is the one which minimizes the sum of error for all data. The error may be defined as: But, the best line is found when the sum of squared error is minimized, the curve found by this way is called least-square fit. For linear fitting, to minimize the sum of the square of the errors equals to minimize Differentiate with respect to each coefficient: Setting derivatives = 0 : From Σa 0 = n a 0, express equations as set of 2 unknowns (a 0, a 1 ) Solve equations simultaneously: 48

3 Example 5.1 Fit a straight line to x and y values listed below xi yi xi 2 xi*yi Error analysis: Sum of the square of the errors: Standard errors of the estimation: Sum of the square around the mean: Standard deviation: 49

4 Coefficient of determination: It is easily found that for perfect fit E = 0 and r = Example 5.1cont. Analyze the error of the linear fit found above. xi yi Since Sy/x < Sy, linear regression has merit. r = Linear model explains 86.8%of original uncertainty. Linear regress is only good for those problems for which relationship between the quantities is linear. However, for most of the problems in science and engineering, the relationship between the quantities being considered is not linear. Some nonlinear problems can be converted into linear form, e.g. Power function Exponential function Reciprocal function then linear regress still can be used. For other nonlinear problems, polynomial regress should be used Polynomial Regress Polynomial Regress: is a procedure to determine the coefficients of a polynomial of a second or higher order such that the polynomial curve will best fit the sampled data. e.g. 2nd-order polynomial: Sum of the squares of the errors: 50

5 Take derivative with respect to each coefficient (unknown): Then, we can get 3 linear equations: Solve above equations, we can get the coefficients of the 2nd-order polynomial function. For mth-order polynomial: y = a 0 + a 1 x + a 2 x a m x m We have to solve m+1 simultaneous linear equations. Example 5.2 Fit a second-order polynomial to the data listed below. xi yi MATLAB Function for Polynomial Regress 51

6 The most widely used MATLAB functions for polynomial regress is: polyfit() C = polyfit(x, Y, n) where X = vector of independent variable values of the data points Y = vector of dependent variable values of the data points n = degree of polynomial C = coeff. of polynomial in descending power y = c 1 *x n + c 2 *x n c n *x + c n+1 For example: y = x x C = [ ] You can evaluate polynomial at the points defined by the input vector >> y = polyval(c, X) where X = Input vector y = Value of polynomial evaluated at x Example 5.3 Fit a second-order polynomial to the data in Example 5.2 and calculate the coefficient of determination by MATLAB Multiple Linear Regress Multiple Linear Regress: is to find a linear function of multiple variables (x 1,x 2, x n ) that will fit the sampled data. y = c 0 + c 1 x 1 + c 2 x c p x p For example: two independent variables y = c 0 + c 1 x 1 + c 2 x 2 52

7 Sum of squares of the error: E = Differentiate with respect to unknowns: Setting partial derivatives = 0 and expressing result in matrix form: Example 5.4 Fit a linear polynomial with two variables to the data listed. x1 x2 y Interpolation Interpolation estimates the data point value between the known points. 53

8 Extrapolation predicts the data point value outside the range over which the given data points are measured. For any number of points n there is a polynomial of order n-1 that passes through all of the points. The interpolation methods can be divided into the following types: (1). Lagrange Interpolation (2). Newton Interpolation (3). Piecewise (spline) Interpolation Lagrange Polynomial Interpolation Lagrange polynomial interpolation: uses a polynomial of n-1 order consisting of Lagrange functions to fit n given data points, and apply this function to determine the y value of any point with provided x value. The polynomial is Where are called Lagrange functions. For two points, (x 1,y 1 ) and (x 2,y 2 ), the 1 st -order Lagrange polynomial that passes through the points is For three points, (x 1,y 1 ), (x 2,y 2 ), and (x 3,y 3 ),the 2 nd -order Lagrange polynomial that passes through the points is 54

9 Example 5.5 For the data listed below, 1) determine the 4 th -order Lagrange polynomial that passes through the points, 2) use the polynomial obtained in 1) to estimate the value at x=3. xi yi Newton Polynomial Interpolation Newton polynomial interpolation: uses a polynomial of n-1 order to fit n given data points, and apply this function to determine the y value of any point with provided x value. The polynomial is in the form of Where the coefficients a k are calculated by k-th divided difference f[x k,x k-1,,x 2,x 1 ] f[x k,x k-1,,x 2,x 1 ]= The divided differences between two points (x i,y i ),(x j,y j ) are 55

10 The divided differences for 5 data points can be calculated by the table below, Example 5.6 Repeat Example 5.5 using Newton polynomial interpolation. Note: Both Lagrange and Newton polynomial interpolation methods are using a single polynomial passing through all given points and estimating other points between. These 56

11 methods work well when the number of given points is small. When the number of given points is large, the order of polynomial is high, and the error will be large Piecewise Polynomial Interpolation Piecewise Polynomial Interpolation: When a large number of points is involved, a better interpolation can be obtained by using many low-order polynomials instead of a single high-order polynomial. Normally, all the polynomials are in same order but with different coefficients, and each polynomial is only valid in one interval between two or several given points. Interpolation in this way is called piecewise or spline interpolation. Typical spline interpolation includes linear (1 st -order polynomial), quadratic (2 nd -order) and cubic (3 rd -order). Linear spline, for n given points, there are n-1 intervals. For each interval, using Lagrange form, the straight line that connects the two points (x i, y i ) and (x i+1, y i+1) in the interval i is Example 5.7 For the data listed below, 1) determine all the linear splines that fit the points, 2) Estimate the interpolated value at x=12.7. xi yi Quadratic spline, for n given points, there are n-1 intervals. For each interval, using a 2 nd - order polynomial to interpolate, the 2 nd -order (quadratic) polynomial that connects the two points (x i, y i )and (x i+1, y i+1 )in the interval i is Overall, there are 3(n-1) polynomial coefficients to be determined, need 3(n-1) equations: 1). Each polynomial passes through the two endpoints of its corresponding interval 2). Except at two outside endpoints, the slopes at both sides of each point are equal. 57

12 3). The 2 nd -order derivative at the 1 st point is 0. Example 5.8 For the data listed below, 1) determine all the quadratic splines that fit the points, 2) Estimate the interpolated value at x=12.7. xi yi

13 5.4 MATLAB Functions for Curve Fitting and Interpolation The MATLAB built-in function for polynomial curve fitting is C=polyfit(x,y,m) which has been explained in Section The MATLAB function for interpolation is yi=interpl(x, Y, xi, 'method') in which X: vector of independent variable values of the data points Y: vector of dependent variable values of the data points xi: is the value of x at which y is to be interpolated method: can be 1) nearest---returns the value of the data point that is nearest to the interpolated point; 2) linear---uses linear spline interpolation; 3) spline--- uses cubic spline interpolation Example 5.9 Use MATLAB built-in function with linear spline to repeat Example 5.8, and sketch the interpolated curve