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Curve Fitting and Regression EEE 244 Descriptive Statistics in MATLAB • MATLAB has several built-in commands to compute and display descriptive statistics. Assuming some column vector s: – mean(s), median(s), mode(s) • Calculate the mean, median, and mode of s. mode is a part of the statistics toolbox. – min(s), max(s) • Calculate the minimum and maximum value in s. – var(s), std(s) • Calculate the variance (square of standard deviation) and standard deviation of s • Note - if a matrix is given, the statistics will be returned for each column. Measurements of a Voltage Drain Current in mA at intervals t=10am-10pm 6.5 6.3 6.2 6.5 6.2 6.7 6.4 6.4 6.8 Find mean, median, mode , min, max, variance and standard deviation using appropriate Matlab functions. Histograms in MATLAB • [n, x] = hist(s, x) – Determine the number of elements in each bin of data in s. x is a vector containing the center values of the bins. – Open Matlab help and run example with 1000 randomly generated values for s. • [n, x] = hist(s, m) – Determine the number of elements in each bin of data in s using m bins. x will contain the centers of the bins. The default case is m=10 – Repeat the previous example with setting m=5 • Hist(s) ->histogram plot EEE 244 REGRESSION Linear Regression • Fitting a straight line to a set of paired observations: (x1, y1), (x2, y2),…,(xn, yn). y=a0+a1x+e a1- slope a0- intercept e- error, or residual, between the model and the observations 6 Linear Least-Squares Regression • Linear least-squares regression is a method to determine the “best” coefficients in a linear model for given data set. • “Best” for least-squares regression means minimizing the sum of the squares of the estimate residuals. For a straight line model, this gives: n n Sr e yi a0 a1 xi 2 i i1 i1 2 Least-Squares Fit of a Straight Line • Using the model: y a0 a1x the slope and intercept producing the best fit using: can be found a1 n xi yi xi yi n x a0 y a1 x 2 i x 2 i Example V (m/s) F (N) a1 i xi yi (xi)2 x iy i 1 10 25 100 250 2 20 70 400 3 30 380 900 1400 11400 4 40 550 1600 22000 5 50 610 2500 30500 6 60 1220 3600 73200 7 70 830 4900 58100 8 80 1450 6400 116000 360 5135 20400 312850 n xi yi xi yi n x 2 i x 2 i 8312850 3605135 820400 360 2 19.47024 a0 y a1 x 641.875 19.47024 45 234.2857 Fest 234.2857 19.47024v Standard Error of the Estimate • Regression data showing (a) the spread of data around the mean of the dependent data and (b) the spread of the data around the best fit line: • The reduction in spread represents the improvement due to linear regression. MATLAB Functions • MATLAB has a built-in function polyfit that fits a least-squares nth order polynomial to data: – p = polyfit(x, y, n) • • • • x: independent data y: dependent data n: order of polynomial to fit p: coefficients of polynomial f(x)=p1xn+p2xn-1+…+pnx+pn+1 • MATLAB’s polyval command can be used to compute a value using the coefficients. – y = polyval(p, x) Polyfit function • Can be used to perform REGRESSION if the number of data points is a lot larger than the number of coefficients – p = polyfit(x, y, n) • • • • x: independent data (Vce, 10 data points) y: dependent data (Ic) n: order of polynomial to fit n=1 (linear fit) p: coefficients of polynomial (two coefficients) f(x)=p1xn+p2xn-1+…+pnx+pn+1 Polynomial Regression • • The least-squares procedure can be readily extended to fit data to a higher-order polynomial. Again, the idea is to minimize the sum of the squares of the estimate residuals. The figure shows the same data fit with: a) A first order polynomial b) A second order polynomial Process and Measures of Fit • For a second order polynomial, the best fit would mean minimizing: n n Sr e yi a0 a1 xi a x i1 2 2 2 i 2 i i1 • In general, this would mean minimizing: n n Sr e yi a0 a1 xi a x 2 i i1 2 2 i i1 m 2 m i a x EEE 244 INTERPOLATION Polynomial Interpolation • You will frequently have occasions to estimate intermediate values between precise data points. • The function you use to interpolate must pass through the actual data points - this makes interpolation more restrictive than fitting. • The most common method for this purpose is polynomial interpolation, where an (n-1)th order polynomial is solved that passes through n data 2 n1 points: f (x) a1 a2 x a3 x an x MATLAB version : f (x) p1 x n1 p2 x n2 pn1 x pn Determining Coefficients using Polyfit • MATLAB’s built in polyfit and polyval commands can also be used - all that is required is making sure the order of the fit for n data points is n-1. Newton Interpolating Polynomials • Another way to express a polynomial interpolation is to use Newton’s interpolating polynomial. • The differences between a simple polynomial and Newton’s interpolating polynomial for first and second order interpolations are: Order 1st 2nd Simple f1 (x) a1 a2 x f2 (x) a1 a2 x a3 x 2 Newton f1 (x) b1 b2 (x x1 ) f2 (x) b1 b2 (x x1 ) b3(x x1 )(x x2 ) Newton Interpolating Polynomials (contd.) • The first-order Newton interpolating polynomial may be obtained from linear interpolation and similar triangles, as shown. • The resulting formula based on known points x1 and x2 and the values of the dependent function at those points is: f x2 f x1 f1 x f x1 x x1 x2 x1 Newton Interpolating Polynomials (contd.) • The second-order Newton interpolating polynomial introduces some curvature to the line connecting the points, but still goes through the first two points. • The resulting formula based on known points x1, x2, and x3 and the values of the dependent function at those points is: f x3 f x2 f x2 f x1 f x2 f x1 x 3 x2 x2 x1 f2 x f x1 x x 1 x x1 x x2 x2 x1 x3 x1 Lagrange Interpolating Polynomials • Another method that uses shifted value to express an interpolating polynomial is the Lagrange interpolating polynomial. • The differences between a simply polynomial and Lagrange interpolating polynomials for first and second order polynomials is: Order 1st 2nd Simple f1 (x) a1 a2 x f2 (x) a1 a2 x a3 x 2 Lagrange f1 (x) L1 f x1 L2 f x2 f2 (x) L1 f x1 L2 f x2 L3 f x3 where the Li are weighting coefficients that are functions of x. Lagrange Interpolating Polynomials (contd.) • The first-order Lagrange interpolating polynomial may be obtained from a weighted combination of two linear interpolations, as shown. • The resulting formula based on known points x1 and x2 and the values of the dependent function at those points is: f1 (x) L1 f x1 L2 f x2 x x2 x x1 L1 , L2 x1 x2 x2 x1 x x2 x x1 f1 (x) f x1 f x2 x1 x2 x2 x1 x x0 x x1 f1 ( x) f ( x0 ) f ( x1 ) x0 x1 x1 x0 x x0 x x2 x x1 x x2 f 2 ( x) f ( x0 ) f ( x1 ) x0 x1 x0 x 2 x1 x0 x1 x 2 x x0 x x1 f ( x2 ) x2 x0 x2 x1 •As with Newton’s method, the Lagrange version has an estimated error of: n Rn f [ x, xn , xn 1 , , x0 ] ( x xi ) i 0 23 Figure 18.10 24 Lagrange Interpolating Polynomials (contd.) • In general, the Lagrange polynomial interpolation for n points is: n fn1 xi Li x f xi i1 where Li is given by: n Li x j1 ji x xj xi x j