# hyperplane in svm

What is the correlation between finding hyperplane and use it in prediction process of svm ?

I still don't get it, after finding hyperplane, then what ? how it helps to find correct class from test data

The prediction function $f(\mathbf{z})$ for an SVM model is exactly the signed distance of $\mathbf{z}$ to the separating hyperplane. The separating hyperplane itself is the geometric place $f(\mathbf{z}) = 0$.

For a linear SVM, the separating hyperplane's normal vector $\mathbf{w}$ can be written in input space, and we get:

$$f(\mathbf{z}) = \langle \mathbf{w}, \mathbf{z} \rangle + \rho = \mathbf{w}^T\mathbf{z} + \rho,$$

with $\rho$ the model's bias term.

If a kernel function $\kappa(\mathbf{u},\mathbf{v})=\langle \varphi(\mathbf{u}), \varphi(\mathbf{v})\rangle$ is used, $\mathbf{w}$ typically can no longer be expressed in input space, but only in the space spanned by the embedding function $\varphi(\cdot)$. Then we obtain the following:

\begin{align} f(\mathbf{z}) &= \langle \mathbf{w}, \varphi(\mathbf{z})\rangle + \rho = \mathbf{w}^T\varphi(\mathbf{z}) + \rho, \\ &= \sum_{i\in SV} y_i\alpha_i \kappa(\mathbf{x}_i,\mathbf{z}) + \rho, \end{align} with $y$ the vector of labels, $\alpha$ the vector of support values, $\mathbf{x}$'s the support vectors.

you seem a little bit confuse. First of all try to read this tutorial that in my opinion is a good introduction. Anyway we want to find an hyperplane because we want to find a rule to discriminate different classes. So at the end you put your test set in the hyperspace and see where every sample is located respect the hyperplane. For example if an element of test set is on the "right" of the hyperplane, you label it "class1", if the sample is on the "left" you label it as "class2". Obviously the stuff is more complex, but this is the basic idea behind svm concept.