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Regression
Take a look at the last term on the left-hand side; it just sums up a constant N time.
Thus, we can rewrite it as:
Reordering the terms, we get:
NN
NN
∑ YY ii − WW ∑ AA ii − NNNN = 0
ii=1
ii=1
NN
NN
bb = 1 NN ∑ YY ii − WW NN ∑ AA ii
ii=1
The two terms on the right-hand side can be replaced by YY̅ , the average price
(output), and AA̅ , the average area (the input), respectively, and thus we get:
bb = YY̅ − WWAA̅
In a similar fashion, we expand the partial differential equation of S with respect
to weight W:
NN
ii=1
∑(YY ii AA ii − WWAA ii 2 − bbAA ii ) = 0
ii=1
Substitute the expression for the bias term b:
Reordering:
NN
NN
∑(YY ii AA ii − WWAA ii 2 − (YY̅ − WWAA̅)AA ii ) = 0
ii=1
∑(YY ii AA ii − YY̅AA ii ) − WW ∑(AA ii 2 − AA̅AA ii ) = 0
ii=1
Playing around with the mean definition, we can get from this the value of
weight W as:
NN
NN
ii=1
WW = ∑ ii=1 YY ii(AA ii − AA̅)
∑ (AA ii − AA̅) 2
NN ii=1
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