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Paper writing templete

In this section we develop two simple but powerful prediction methods: the linear model fit by least squres and the KNN prediction rule. The linear model makes huge assumptions about structure and yields stable but possibly inaccurate predictions. The method of knn makes very mild structural assumptions: its predictions are often accurate but can be unstable.

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  Given a vector of inputs $X^T=(X_1,X_2,...,X_p)$, we predict the ouput $Y$ via the model
  $$\hat{Y}={\hat{\beta }}_0+\sum _{j=1}^p{X}_j{\hat{\beta }}_j$$
  The term ${\hat{\beta }}_0$ is the intercept, also known as the bias in machine learning. Often it is convenient to include the constant variable 1 in $X$, include ${\hat{\beta }}_0$ in the vector of coefficients $\hat{\beta }$, and then write the linear model in vector form as an inner product
  $$\hat{Y}=X^T\hat{\beta },$$
  where $X^T$ denotes vector or matrix transpose ($X$ being a column vector) .

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In this approach, we pick the coefficients $\beta$ to minimize the residual sum of squares
$$RSS(\beta )=\sum _{i=1}^N(y_i-x_i^T\beta {)}^2$$
$RSS(\beta )$ is a quadratic function of the parameters, and hence its minimum always exists, but may not be unique. The solution is easiest to characterize in matrix notation. We can write
$$RSS(\beta )=(\mathbf{y}-\mathbf{X}{)}^T(\mathbf{y}-\mathbf{X})$$
where $\mathbf{X}$ is an $N\times p$ matrix with each row an input vector, and $\mathbf{y}$ is an $N\text{-vector}$ of the outputs in the training set. Differentiating w.r.t. $\beta$ we get the normal equations
$${\mathbf{X}}^{\mathbf{T}}(\mathbf{y}-\mathbf{X}\beta )=0.$$
If ${\mathbf{T}}^T\mathbf{X}$ is nonsingular, then the unique solution is given by
$$\hat{\beta }=({\mathbf{X}}^{\mathbf{T}}\mathbf{X}{)}^{-1}{\mathbf{X}}^{\mathbf{T}}\mathbf{y},$$
and the fitted value at the $i$th input $x_i$ is ${\hat{y}}_i=\hat{y}(x_i)=x_i^T\hat{\beta }$.