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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
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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
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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)
# ( + # * ' = * ! ( * '+ * !+ $ & + + # + + + $ + # = , , & #$ * '+ + #& * '+ = * '+!+ * '+ * '+ * '+!+ + + + + + +
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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
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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 ⎦ $⎣ #N"⎦ X Φ Y
(Computational Techniques: Module-5 Part-2: http://nptel.ac.in/courses/103106074/16)
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Example
x 0.8 1.4 2.7 3.8 4.8 4.9 y 0.69 1.00 2.02 2.39 2.34 2.83
Using MATLAB lsqcurvefit
• Standard syntax: phi=lsqcurvefit(@(p,xData) fName(p,xData),p0,xData,yData);
• phi parameter vector
• p0 vector of initial guesses
• xData, yData data arrays with ( rows
• fName provides !:;<=> = ? '; Φ
• lsqcurvefit minimizes the error between !
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End of Lecture 6.2
REGRESSION AND INTERPOLATION Lec. 6.3: Functional and Nonlinear Regression
Dr. Niket Kaisare Department of Chemical Engineering IIT–Madras
NPTEL Course: MATLAB Programming for Numerical Computations — Week -6
7 2/15/16
Example: Reaction Rate
• Arrhenius model for reaction rate: 2F/HI 1 C = D$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
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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)
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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
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End of Lecture 6.4
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