Each kernel possesses the hyperparameter σ f , the signal variance, which is a scale factor controlling the expected range of the functions represented by the GP. The polynomial kernel has a hyperparameter d that controls the order of the polynomial in the features, and the RBF kernel contains a length-scale hyperparameter γ that controls how close x and x must be in the feature space to be considered "similar" and the expected roughness of the functions represented by the GP. Implicitly, each nonlinear kernel maps the two vectors x and x into a new, higher-dimensional feature space through a mapping τ , then takes the inner product of the vectors in the new feature space: