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Lecture 10

ENG 06 – Lecture 10 notes.docx

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ENG 06 – Lecture 10; 2/7/2013 Interpolation and Curve Fitting  Least squares regression: minimizes area between data and line you predict  polyfit: fits line to data  polyval: takes coefficients of polynomial (for example, from polyfit) and a vector of x points to evaluate at that polynomial  Regression example: linear fit o Torque needed to turn torsion spring of mousetrap through an angle is given in data points o Find constants for model given by T = k1 + k2x  >> xp=[0.698132, 0.959931, 1.134464, 1.570796, 1.919862];  >> yp=[0.188224, 0.209138, 0.230052, 0.250965, 0.313707];  >> coeffs = polyfit(xp,yp,1);
  >> xfit=[0.6:0.01:2];yfit = polyval(coeffs,xfit);  >> plot(xp,yp,'O'); hold on; plot(xfit,yfit); hold off;  >> coeffs  coeffs = 0.0961 0.1177  Interpolation or Regression? o For acceleration/velocity/distance problems, need to differentiate and integrate o Best to fit a known function that can be “easily differentiated or integrated o Use a spline fit  physics of gravity tells us that it should follow a smooth curve  Gives us just one function to work with  Fitting to More complicated functions o Linearizing: making a few substitutions for variables which result in an equation for a line  Linear vs. Nonlinear fitting o Linear in terms of fitting constants:  Y = a + bt  Y = a + bt + ct^2  y = asin(t) + bcos(t)  y = asin(3t)  y = ae^-t o Not linear in terms of all the fitting constants  y = asin(bt) + ccos(dt)  y = asing(bt)  y = ae^-bt  Linearizing Commonly Found Nonlinear functions o Log functions:
 y = Klog(x) + c  Looks like a straight line when you plot y values vs. log of x values  plot(log(x),y)  also semilogx(x,y)  creates a logarithmic x axis o Exponential functions: y = Ce^Kx  Plot log of y values vs. x values to make it look like a straight line  plot(x,log(y))  can also use semilogy(x,y)  creates an exponential y axis
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