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:

    “Wonderful “Force of Public Opinion!” We must act and walk in all points as it prescribes; follow the traffic it bids us, realise the sum of money, the degree of “influence” it expects of us, or we shall be lightly esteemed; certain mouthfuls of articulate wind will be blown at us, and this what mortal courage can front?”
    —Thomas Carlyle (1795–1881)

    “And New York is the most beautiful city in the world? It is not far from it. No urban night is like the night there.... Squares after squares of flame, set up and cut into the aether. Here is our poetry, for we have pulled down the stars to our will.”
    —Ezra Pound (1885–1972)