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Explained sum of squares: Difference between revisions

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Thus,
:<math>
:<math> TSS = \|y - \bar y\|^2 = \|y - {\hat y} + {\hat y} - \bar y\|^2 </math>
\begin{align}
:<math> TSS = \|y - {\hat y}\|^2 + \|{\hat y} - \bar y\|^2 + 2 <y - {\hat y}, {\hat y} - {\bar y}> </math>▼
:<math> TSS & = RSS\|y +- ESS + 2\bar y\|^T2 = {\hat |y} -2 {\hat y}^T + {\hat y} - 2 y^T {\bar y} + \|^2 {\hat y}^T{\bar y} </math>
:<math> TSS& = RSS + ESS \|y - 2{\hat y}\|^T2 + \|{\barhat y} - \bar y\|^2 + 2 <y - {\hat y}^T, {\barhat y} </math- {\bar y}> \\
▲:<math> TSS& = RSS + ESS + 2 y^T {\|hat y} -2 {\hat y}\|^2T + \|{\hat y} - \bar2 y\|^2 + 2 <y -T {\hatbar y}, + 2 {\hat y} - ^T{\bar y}> </math>\\
& = RSS + ESS - 2 y^T {\bar y} + 2 {\hat y}^T{\bar y}
\end{align}
</math>
which again gives the result that ''TSS'' = ''ESS'' + ''RSS'', since <math>(y-\hat y)^T \bar y = 0</math>.
 
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