A $p$-dimensional space of features

We limit ourselves to two classes of outputs $y_i$ and assign these classes the values $y_i = \pm 1$. In a $p$-dimensional space of say $p$ features we have a hyperplane defines as $$b+wx_1+w_2x_2+\dots +w_px_p=0.$$ If we define a matrix $\boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right]$ of dimension $n\times p$, where $n$ represents the observations for each feature and each vector $x_i$ is a column vector of the matrix $\boldsymbol{X}$, $$\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}.$$ If the above condition is not met for a given vector $\boldsymbol{x}_i$ we have $$b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0,$$ if our output $y_i=1$. In this case we say that $\boldsymbol{x}_i$ lies on one of the sides of the hyperplane and if $$b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0,$$ for the class of observations $y_i=-1$, then $\boldsymbol{x}_i$ lies on the other side.

Equivalently, for the two classes of observations we have $$y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0.$$

When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.

• «
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12
• 13
• 14
• ...
• 28
• »