Gebruiker:Pjetter/klad: verschil tussen versies
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Regel 212:
The sum of squares to be minimized is
:<math> S = \sum_{i=1}^{n} \left(y_i - \hat\beta x_i\right)^2. </math>
So we do some derivation and you get for minimization the derivative is set to 0:
The least squares estimate for ''β'', is given by ▼
:<math> \frac {\delta} {\delta {\beta}} \sum_{i=1}^{n} \left(y_i - \hat\beta x_i\right)^2 = -2 \sum_{i=1}^{n} \left(y_i - \hat\beta x_i\right) = 0</math>
Divide both sides by -2 and you get:
:<math> \sum_{i=1}^{n} \left(y_i - \hat\beta x_i\right) = 0</math>
This you can rewrite as:
:<math> \sum_{i=1}^{n} y_i = \hat\beta \sum_{i=1}^{n} x_i </math>
Let's multiply with:
:<math>\sum_{i=1}^{n} x_i </math>
and you get:
:<math> \sum_{i=1}^{n} x_i \sum_{i=1}^{n} y_i = \hat\beta (\sum_{i=1}^{n} x_i)^2 </math>
This can be rewritten as:
:<math> \sum_{i=1}^{n} x_i y_i = \hat\beta (\sum_{i=1}^{n} x_i)^2 </math>
:<math>\hat \beta=\frac{\sum_i x_i y_i}{\sum_i x_i^2}.</math>
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