Question

We would like to fit data to the equation y -a, that is, a horizontal line. (a) For the equation above, define a function ф(a) which is the sum of the square residuals for the vertical offsets. (b) Taking the appropriate derivative, derive an expression for a in terms sums over data points (c) For the data below, calculate the value of the parameter a. 4.524 0.886 10.880 2.573 8.532 -0.455 -14.339 1901 -12.9980.224 15.231 -0.109 19.297 0.794 26.111 2.681 21.0140.638 25.6300.288

Answer 1

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Given y = a phi (a) = Sigma^n_i = 1 (y_i - a)^2 Rightarrow Sum of squares of vertical offsets d phi (a)/da = 0 Rightarrow 2 Sigma^n_i = 1 (y_i - a) = 0 Rightarrow a = Sigma^n_i = 1 y_i Thus, a is the mean of the data points

For part (c), the data given is a 2d data, for which the variable 'a' will be a vector of 2 values, representing the slope and the y-intercept of the best fit line. The value of a is found using Matlab:

**************** Code ***********

y = [-0.886 -2.573 -0.455 -1.901 -0.224 -0.109 -0.794 -2.681 0.638 -0.288]';                    % y data
x1 = [-4.524 -10.880 -8.532 -14.339 -12.998 -15.231 -19.297 -26.111 -21.014 -25.630]';          % x data
n = ones(size(x1));
x = [n x1];                                                                                     % formatting x data
a = x\y     

*************** End of Code ***********

Output:

a =

   -0.9278
   -0.0000

However, here the second value is zero, which means a = -0.9278

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