Versie van 5 dec 2008 00:42 bewerken Pjetter (overleg \| bijdragen) 10.553 bewerkingen →‎SSR ← Vorige wijziging		Versie van 5 dec 2008 01:09 bewerken ongedaan maken Pjetter (overleg \| bijdragen) 10.553 bewerkingen →‎regression Volgende wijziging →
Regel 208: :<math>\hat{y} = 49.48 + 1.96{x}</math> : <math> y_i = \beta x_i + \varepsilon_i. \, </math> The sum of squares to be minimized is :<math> S = \sum_{i=1}^{n} \left(~~y_i~~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: :<math> \frac {\delta} {\delta {\beta}} \sum_{i=1}^{n} \left(~~y_i~~Y_i - \hat\beta x_i\right)^2 = -2 \sum_{i=1}^{n} \left(~~y_i~~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~~Y_i - \hat\beta x_i\right) = 0</math> This you can rewrite as: :<math> \sum_{i=1}^{n} ~~y_i~~Y_i = \hat\beta \sum_{i=1}^{n} x_i </math> Let's multiply with: Regel 231: and you get: :<math> \sum_{i=1}^{n} x_i \sum_{i=1}^{n} ~~y_i~~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~~Y_i = \hat\beta (\sum_{i=1}^{n} x_i)^2 </math> And therefore the least squares estimate for ''β'', is given by :<math>\hat \beta=\frac{\sum_i^{n} x_i ~~y_i~~Y_i}{\sum_i^{n} ~~x_i~~Y_i^2}.</math> :<math>\hat \beta=\frac{\sum_i^{n} x_i y_i}{\sum_i^{n} x_i^2}</math> ==SSR==

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