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Linear Least Squares
Consider the linear regression model, [ y_i=f_i(mathbf{x}|boldsymbol{beta})+varepsilon_i,quadmathbf{x}_i=left[ begin{array}{cccc} 1&x_{11}&cdots&x_{1p} end{array}right],quadboldsymbol{beta}=left[begin{array}{c}beta_0\beta_1\vdots\beta_pend{array}right], ] where $y_i$ is the response or the dependent variable at the $i$th case, $i=1,cdots, N$. The $f_i(mathbf{x}|boldsymbol{beta})$ is the deterministic part of the model that depends on both the parameters $boldsymbol{beta}inmathbb{R}^{p+1}$ and the predictor variable $mathbf{x}_i$, which in matrix form, say $mathbf{X}$, is represented as follows [ mathbf{X}=left[ begin{array}{cccccc} 1&x_{11}&cdots&x_{1p}\ 1&x_{21}&cdots&x_{2p}\ vdots&vdots&ddots&vdots\ 1&x_{N1}&cdots&x_{Np}\ end{array} right]. ] $varepsilon_i$ is the error term at the $i$th case which we assumed to be Gaussian distributed with mean 0 and variance $sigma^2$. So that [ mathbb{E}y_i=f_i(mathbf{x}|boldsymbol{beta}), ] i.e. $f_i(mathbf{x}|boldsymbol{beta})$ is the expectation function. The uncertainty around the response variable is also modelled by Gaussian distribution. Specifically, if $Y=f(mathbf{x}|boldsymbol{beta})+varepsilon$ and $yin Y$ such that $y>0$, then begin{align*} mathbb{P}[Yleq y]&=mathbb{P}[f(x|beta)+varepsilonleq y]\ &=mathbb{P}[varepsilonleq y-f(mathbf{x}|boldsymbol{beta})]=mathbb{P}left[frac{varepsilon}{sigma}leq frac{y-f(mathbf{x}|boldsymbol{beta})}{sigma}right]\ &=Phileft[frac{y-f(mathbf{x}|boldsymbol{beta})}{sigma}right], end{align*} where $Phi$ denotes the Gaussian distribution with density denoted by $phi$ below. Hence $Ysimmathcal{N}(f(mathbf{x}|boldsymbol{beta}),sigma^2)$. That is, begin{align*} frac{operatorname{d}}{operatorname{d}y}Phileft[frac{y-f(mathbf{x}|boldsymbol{beta})}{sigma}right]&=phileft[frac{y-f(mathbf{x}|boldsymbol{beta})}{sigma}right]frac{1}{sigma}=mathbb{P}[y|f(mathbf{x}|boldsymbol{beta}),sigma^2]\ &=frac{1}{sqrt{2pi}sigma}expleft{-frac{1}{2}left[frac{y-f(mathbf{x}|boldsymbol{beta})}{sigma}right]^2right}. end{align*} If the data are independent and identically distributed, then the log-likelihood function of $y$ is, begin{align*} mathcal{L}[boldsymbol{beta}|mathbf{y},mathbf{X},sigma]&=mathbb{P}[mathbf{y}|mathbf{X},boldsymbol{beta},sigma]=prod_{i=1}^Nfrac{1}{sqrt{2pi}sigma}expleft{-frac{1}{2}left[frac{y_i-f_i(mathbf{x}|boldsymbol{beta})}{sigma}right]^2right}\ &=frac{1}{(2pi)^{frac{n}{2}}sigma^n}expleft{-frac{1}{2}sum_{i=1}^Nleft[frac{y_i-f_i(mathbf{x}|boldsymbol{beta})}{sigma}right]^2right}\ logmathcal{L}[boldsymbol{beta}|mathbf{y},mathbf{X},sigma]&=-frac{n}{2}log2pi-nlogsigma-frac{1}{2sigma^2}sum_{i=1}^Nleft[y_i-f_i(mathbf{x}|boldsymbol{beta})right]^2. end{align*} And because the likelihood function tells us about the plausibility of the parameter $boldsymbol{beta}$ in explaining the sample data. We therefore want to find the best estimate of $boldsymbol{beta}$ that likely generated the sample. Thus our goal is to maximize the likelihood function which is equivalent to maximizing the log-likelihood with respect to $boldsymbol{beta}$. And that’s simply done by taking the partial derivative with respect to the parameter $boldsymbol{beta}$. Therefore, the first two terms in the right hand side of the equation above can be disregarded since it does not depend on $boldsymbol{beta}$. Also, the location of the maximum log-likelihood with respect to $boldsymbol{beta}$ is not affected by arbitrary positive scalar multiplication, so the factor $frac{1}{2sigma^2}$ can be omitted. And we are left with the following equation, begin{equation}label{eq:1} -sum_{i=1}^Nleft[y_i-f_i(mathbf{x}|boldsymbol{beta})right]^2. end{equation} One last thing is that, instead of maximizing the log-likelihood function we can do minimization on the negative log-likelihood. Hence we are interested on minimizing the negative of Equation (ref{eq:1}) which is begin{equation}label{eq:2} sum_{i=1}^Nleft[y_i-f_i(mathbf{x}|boldsymbol{beta})right]^2, end{equation} popularly known as the residual sum of squares (RSS). So RSS is a consequence of maximum log-likelihood under the Gaussian assumption of the uncertainty around the response variable $y$. For models with two parameters, say $beta_0$ and $beta_1$ the RSS can be visualized like the one in my previous article, that is
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