2/15/16 REGRESSION AND INTERPOLATION Lec. 6.1: Introduction Dr. Niket Kaisare Department of Chemical Engineering IIT–Madras NPTEL Course: MATLAB Programming for Numerical Computations — Week -6 Example: Regression • Given the following data: x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83 Regression: Obtain a straight line that best fits the data 1 2/15/16 Example: Interpolation • Given the following data: x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83 Interpolation: “Join the dots” and find a curve passing through the data. Regression vs. Interpolation • Given the following data: x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83 • In regression, we are interested in fitting a chosen function to data y = 0.45 + 0.47x • In interpolation, given finite amount of data, we are interested in obtaining new data-points within this range. At x = 2.0, y = 1.87 2 2/15/16 What Comes Next • This lecture (for demo): • Linear Regression: Fit a straight line to the given data • Newton’s Interpolation: For values at intermediate points • L–6.2: “Curve fitting” in multiple parameters & lsqcurvefit • L–6.3: “Parameter Estimation” (using these concepts) • L–6.4: Interpolation (using spline and pchip) Linear Least Squares • Fit a straight line ! = #$ + #&' to the data: x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83 • Parameters #$ and #& satisfy the following equations: (Computational Techniques, Module 5: http://nptel.ac.in/courses/103106074/15) # ( + # * ' = * ! ( * '+ * !+ $ & + + # + + + $ + # = , , & #$ * '+ + #& * '+ = * '+!+ * '+ * '+ * '+!+ + + + + + + 3 2/15/16 Newton’s Divided Difference Formula x y D D2 0.8 0.69 !, − !& -, − -& ', − '& ' − ' 1.4 1.00 / & !/ − !, -/ − -, 2.7 2.02 '/ − ', '0 − ', 3.8 2.39 ⋮ - − - 4.8 2.34 ⋮ 12& 12, !1 − !12& '1 − '12, 4.9 2.83 '1 − '12& End of Lecture 6.1 4 2/15/16 REGRESSION AND INTERPOLATION Lec. 6.2: Linear Least Squares Regression Dr. Niket Kaisare Department of Chemical Engineering IIT–Madras NPTEL Course: MATLAB Programming for Numerical Computations — Week -6 Linear Regression for Multiple Parameters • Data: '&,5&, 6&; !& , ',,5,, 6,; !, , ⋯ , '9 ,59 ,69 ; !9 • Model to fit: ! = #$ + #&' + #,5 + #/6 ⎡1 x1 u1 w1 ⎤⎡a0 ⎤ ⎡ y1 ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ 1 1 x2 u2 w2 a1 y2 T − T ⎢ ⎥⎢ ⎥ = ⎢ ⎥ Φ = X X X Y ⎢% % % % ⎥⎢a ⎥ ⎢ % ⎥ ( ) 2 Least Squares Solution ⎢1 x u w ⎥⎢a ⎥ ⎢y ⎥ $⎣ &&N&#N&&&N"⎦⎣!3 ⎦ $⎣ #N"⎦ X Φ Y (Computational Techniques: Module-5 Part-2: http://nptel.ac.in/courses/103106074/16) 5 2/15/16 Example x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83 Using MATLAB lsqcurvefit • Standard syntax: phi=lsqcurvefit(@(p,xData) fName(p,xData),p0,xData,yData); • phi parameter vector • p0 vector of initial guesses • xData, yData data arrays with ( rows • fName provides !:;<=> = ? '; Φ • lsqcurvefit minimizes the error between !<ABA and !:;<=> 6 2/15/16 End of Lecture 6.2 REGRESSION AND INTERPOLATION Lec. 6.3: Functional and Nonlinear Regression Dr. Niket Kaisare Department of Chemical Engineering IIT–Madras NPTEL Course: MATLAB Programming for Numerical Computations — Week -6 7 2/15/16 Example: Reaction Rate • Arrhenius model for reaction rate: 2F/HI 1 C = D$E J • We will solve it in two ways: 1. Linear least squares regression taking logarithm M 1 ln C = ln D + − + S ln J $ N P⏟ T R 2. Using MATLAB function lsqnonlin Parameter Estimation using Matrix Method (OLS) Reaction rate (in mol/(l.s)) for various C and T values 400 K 450 K 500 K 550 K 600 K 1 mol/l 1.48 1.67 1.86 1.96 2.16 2 mol/l 2.35 2.79 3.07 3.37 3.62 3 mol/l 3.28 3.78 4.24 4.48 5.00 4 mol/l 4.12 4.64 5.15 5.76 6.08 2F/HI 1 ! = # + # ' + # 5 C = D$E J log $ & , ln C ln D$ 1/P ln J 8 2/15/16 Using MATLAB lsqnonlin • Standard syntax: phi=lsqnonlin(@(p) fName(p),p0); • phi parameter vector • p0 vector of initial guesses • fName provides vector of errors, E+ = !+ − ? '+;Φ • lsqcurvefit minimizes sum of square errors End of Lecture 6.3 9 2/15/16 REGRESSION AND INTERPOLATION Lec. 6.4: Interpolation Options in MATLAB Dr. Niket Kaisare Department of Chemical Engineering IIT–Madras NPTEL Course: MATLAB Programming for Numerical Computations — Week -6 Interpolation in MATLAB • Most popular interpolation techniques: • spline Cubic spline interpolation • pchip Piecewise Cubic Hermite Polynomial • Syntax: yInterpolated = spline(xData,yData,xval); • xData,yData S×1 data vectors • yInterpolated values interpolated at xVal (can be vectors) 10 2/15/16 Example: Temperature variation in a day time 00 01 02 03 04 05 06 07 08 09 10 11 12 T 25.6 25.4 25.1 24.9 24.9 25.2 25.9 26.3 27.1 29.3 30.8 31.2 32.1 time 13 14 15 16 17 18 19 20 21 22 23 24 T 31.0 30.3 31.4 30.6 31.8 29.6 28.4 28.1 28.2 27.4 26.8 26.1 We are interested in finding temperature at various times during the day, in addition to the ones where data is available. We interpolate or “fill in” the missing data Example: Vehicle speed in front of Govt. Hospital time (s) 0 10 20 30 40 50 60 70 80 90 speed 45 32 0 0 7 12 20 15 29 55 11 2/15/16 End of Lecture 6.4 12.