Statistics and machine learning algorithms increasingly have to be able to handle complex data, involving higher-level descriptors (features) of the characteristics of an object or system. Such descriptors are usually inspired by knowledge of the intrinsic structure of the object, e.g. a covariance matrix modeling variability and correlation between pixels in an image, or by physical understanding of the system, such as a probability distribution of a flow characteristic in a turbulent flow. In such cases, each data point does not simply represent a set of numbers (coordinates in a vector space), but has substructure of its own, representing more complex notions like a matrix, a probability distribution, a function, a shape, etc. In this talk, I will discuss several applications from the field of nuclear fusion plasma physics, wherein we characterize fluctuation and measurement uncertainty by probability distributions. We employ a metric on the Riemannian space of Gaussian probability distributions to discriminate between various types of plasma instabilities and classify them. Furthermore, we describe a new, very robust regression technique, called geodesic least squares regression, for estimating relations between plasma quantities that are affected by a considerable amount of fluctuation or measurement uncertainty.