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:

    “To sum up:
    1. The cosmos is a gigantic fly-wheel making 10,000 revolutions a minute.
    2. Man is a sick fly taking a dizzy ride on it.
    3. Religion is the theory that the wheel was designed and set spinning to give him the ride.”
    —H.L. (Henry Lewis)

    “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)