# Ratio of norms as goodness of fit measure for least square

Let $$x,y\in \mathbb{R}^n$$ be two vectors.

One way to think about a linear regression on $$(x,y)$$ is that there is a random variable $$X$$ and two real numbers $$\beta_0$$ and $$\beta_1$$ such that $$Y = \beta_0 + \beta_1 X+ \epsilon$$ where $$\epsilon \sim N(0,\sigma^2)$$ is a random Gaussian error. We assume that $$(x,y)$$ are $$n$$ samples of the random variables $$(X,Y)$$. We would like to use the sample $$(x,y)$$ to best estimate the two real numbers $$\beta_0,\beta_1$$. The best estimate from the sample $$(x,y)$$ would be $$\hat{\beta_0} = \overline{y}-\hat{\beta_1} \overline{x},\quad \hat{\beta_1} = \frac{s_{xy}}{s_{xx}}$$ where $$\overline{y} = (\sum_{i = 1}^n y_i)/n$$, $$\overline{x} = \sum_{i = 1}^n x_i)/n$$, $$s_{xy} = \sum_{i = 1}^n (x_i-\overline{x})(y_i-\overline{x})$$ and $$s_{xx} = \sum_{i = 1}^n (x_i-\overline{x})^2$$.

The other way to think about the problem is to consider a linear map $$\Phi:\mathbb{R}^2\to \mathbb{R}^n$$ given by $$\Phi(b_0,b_1) = b_0\mathbf{1}+b_1x,$$ and we would like to find $$(b_0,b_1) \in \mathbb{R}^2$$ such that $$\|\Phi(b_0,b_1)-y\|_2$$ is minimized. Assuming $$\mathbf{1}$$ and $$x$$ are linearly independent, it follows that $$\Phi$$ is injective. This assumption means that the input data $$x$$ are not all repeats of the same value. By standard inner product spaces argument, there is a unique vector $$\hat{y} \in \Phi(\mathbb{R}^2)$$ such that $$\|y-\hat{y}\|_2$$ is minimized. It turns out that $$\hat{y} = \hat{\beta_0} \mathbf{1} + \hat{\beta_1} x$$, with the same $$\hat{\beta_0},\hat{\beta_1}$$ as above.

My question is the following: In the first perspective, the natural measure of goodness of fit is given by $$r^2 =\frac{s_{xy}^2}{s_{xx}s_{yy}} = \left(\frac{\|\hat{y}-\overline{y}\cdot \mathbf{1}\|_2}{\|y-\overline{y}\cdot\mathbf{1}\|_2}\right)^2 = 1 - \frac{\|y-\hat{y}\|_2^2}{\|y-\overline{y}\mathbf{1}\|_2^2},$$ which can be thought of as percentage of explained variance. Here the third equality is due to the fact that $$\hat{y}-\overline{y}\cdot \mathbf{1}$$ is orthogonal to $$y-\hat{y}$$.
However, in the second perspective, the natural measure of goodness of fit is given by $$\rho^2 = \frac{\|y\|^2_2-\|y-\hat{y}\|_2^2} {\|y\|_2^2} = \left(\frac{\|\hat{y}\|_2}{\|y\|_2}\right)^2 = 1 - \frac{\|y-\hat{y}\|_2^2} {\|y\|_2^2}.$$ Here the second equality holds because $$\hat{y}$$ and $$y-\hat{y}$$ are orthogonal.
Why do people often use $$r^2$$ and not $$\rho^2$$? What is the name for $$\rho^2$$ in the literature?