  Kernel methods are powerful machine learning techniques which implement
generic non-linear functions to solve complex tasks in a simple way. They Have
a solid mathematical background and exhibit excellent performance in practice.
However, kernel machines are still considered black-box models as the feature
mapping is not directly accessible and difficult to interpret.The aim of this
work is to show that it is indeed possible to interpret the functions learned
by various kernel methods is intuitive despite their complexity. Specifically,
we show that derivatives of these functions have a simple mathematical
formulation, are easy to compute, and can be applied to many different
problems. We note that model function derivatives in kernel machines is
proportional to the kernel function derivative. We provide the explicit
analytic form of the first and second derivatives of the most common kernel
functions with regard to the inputs as well as generic formulas to compute
higher order derivatives. We use them to analyze the most used supervised and
unsupervised kernel learning methods: Gaussian Processes for regression,
Support Vector Machines for classification, Kernel Entropy Component Analysis
for density estimation, and the Hilbert-Schmidt Independence Criterion for
estimating the dependency between random variables. For all cases we expressed
the derivative of the learned function as a linear combination of the kernel
function derivative. Moreover we provide intuitive explanations through
illustrative toy examples and show how to improve the interpretation of real
applications in the context of spatiotemporal Earth system data cubes. This
work reflects on the observation that function derivatives may play a crucial
role in kernel methods analysis and understanding.
