Feature transformation. So far, we have been talking about using the raw features$/$predictors$/$input variables X$1,$ X$2,,$ Xp from the data to make prediction about a target variable Y$.$ In most cases, however, it is not the best idea to use the original inputs: a process called feature transformation T is usually performed on the raw data such that we learn a model between T $($X$)$ and Y instead. For the following dataset, let$$s try different feature transformation techniques.

$x_1=1,3,2,4,6$

$x_2=1,2,5,7,9$

$y=7\mathrm{.}5,13\mathrm{.}8,20\mathrm{.}5,33\mathrm{.}6,40\mathrm{.}7$

$$ Normalization: transform both $x_1$ and $x_2$ so that they are zero mean and unit variance.

$$ Quantization: transform both $x_1$ and $x_2$ so that they contain only zero and ones. You can choose any threshold. There are more than one solution here.

$$ Log transformation: transform both $x_1$ and $x_2$ with log function.

$$ Polynomial expansion: transform $x_1$ and $x_2$ together into one polynomial

feature array such that

$[1,x_1,x_2,x_1^2,x_1\backslash$times $x_2,x_2^2$

You don$$t need to compute all the data points, you can choose any row from the dataset above.

$$ Trigonometric functions: transform $x_1$ and $x_2$ together into one feature array with sine and cosine such that

$[$sin $x_1,$ sin $x_2,$ cos $x_1,$ cos $x_2,$ sin $2x_1,$ sin $2x_2,$ sin $3x_1,$ sin $3x_2$