The scatter plot of a data set {(x1,y1), (x2,y2), ..., (xn,yn)} : Figure 1: Fitt

Question

The scatter plot of a data set {(x1,y1), (x2,y2), ..., (xn,yn)} :

Figure 1: Fitting a quadratic function to data.

As it shows a pattern of quadratic curve, we want to fit a quadratic curve y = 1

y
0 50 100 150 200 250 300

ax2 + bx + c to the data. The least squares method is to find a, b and c which minimize the following sum of squared errors

n i=1

In this assignment, each student is given a different data set. To know which data set you should use, look at the mapping in another file in this module of VU Collaborate. Your answers to the following questions should be specific to your own data set.

Obtain fa,fb, and fc.

Obtain the stationary point of the function f(a,b,c).

Obtain the Hessian matrix at the stationary point.

By using the second derivative test, determine whether the stationary point is minima, maxima, or a saddle point.

Write down the fitted coefficients aˆ, ˆb, and cˆ.

(Optional) By using a software produce a plot of your data and the fitted curve.

Explanation / Answer

X =

    1.0000         0         0
    1.0000    0.2500    0.0625
    1.0000    0.5000    0.2500
    1.0000    0.7500    0.5625
    1.0000    1.0000    1.0000
    1.0000    1.2500    1.5625
    1.0000    1.5000    2.2500
    1.0000    1.7500    3.0625
    1.0000    2.0000    4.0000

y =

    7.8200
    7.8800
    9.8700
    9.2100
   13.5500
   18.6100
   21.1700
   27.2300
   33.6400

b = inv(X'*X)*X'*y

b =

    7.8504
   -1.4625
    7.1756

hence

y^ = 7.8504 -1.4625 *x + 7.1756*x^2

here c^ = 7.8504 b^ = -1.4625 , a^ = 7.1756

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