Partition of Sums of Squares - Partitioning The Sum of Squares in Linear Regression

Partitioning The Sum of Squares in Linear Regression

Theorem. Given a linear regression model including a constant based on a sample containing n observations, the total sum of squares (TSS) can be partitioned as follows into the explained sum of squares (ESS) and the residual sum of squares (RSS):

where this equation is equivalent to each of the following forms:



\begin{align}
\left\| y - \bar{y} \mathbf{1} \right\|^2 &= \left\| \hat{y} - \bar{y} \mathbf{1} \right\|^2 + \left\| \hat{\varepsilon} \right\|^2, \quad \mathbf{1} = (1, 1, \ldots, 1)^T ,\\
\sum_{i = 1}^n (y_i - \bar{y})^2 &= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n (y_i - \hat{y}_i)^2 ,\\
\sum_{i = 1}^n (y_i - \bar{y})^2 &= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 .\\
\end{align}

Read more about this topic:  Partition Of Sums Of Squares

Famous quotes containing the words sum and/or squares:

    I was brought up to believe that the only thing worth doing was to add to the sum of accurate information in the world.
    Margaret Mead (1901–1978)

    An afternoon of nurses and rumours;
    The provinces of his body revolted,
    The squares of his mind were empty,
    Silence invaded the suburbs,
    —W.H. (Wystan Hugh)