Tuesday, June 28, 2016

Unizor - Linear Regression - Problem 1

Notes to a video lecture on http://www.unizor.com

Linear Regression - Problem 1

Consider a linear regression model described in the previous lecture:
Y = a·X + b + ε
where independent variable X is represented by sample data
x1x2 ...xn
and observed values of dependent variable Y are
y1y2 ...yn.

We have introduced two averages of the sample data:
Ave(x)=(x1+x2+...+xn)/n = U
Ave(y)=(y1+y2+...+yn)/n = V

Using new variables
Xk = xk − U and Yk = yk − V
(where index k is from 1 to n)
we came up with the best possible value for a coefficienta in the formula for linear regression as
a = ΣXk·Yk / ΣXk²


Using the sample averaging function Ave() applied as
prove that the expression for a coefficient a in the formula for linear regression in terms of original sample data xk and yklooks like
a = [Ave(xy)−Ave(x)Ave(y)] / [Ave(x²)−Ave²(x)]

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