Collected data may simultaneously be of low sample size and high dimension. Such data exhibit some geometric regularities consisting of a single observation being a rotation on a sphere, and a pair of observations being orthogonal. This thesis investigates these geometric properties in some detail. Background is provided and various approaches to the result are discussed. An approach based on the mean value theorem is eventually chosen, being the only candidate investigated that gives explicit convergence bounds. The bounds are tested employing Monte Carlo simulation and found to be adequate. / Data som insamlas kan samtidigt ha en liten stickprovsstorlek men vara högdimensionell. Sådan data uppvisar vissa geometriska mönster som består av att en enskild observation är en rotation på en sfär, och att ett par av observationer är rätvinkliga. Den här uppsatsen undersöker dessa geometriska egenskaper mer detaljerat. En bakgrund ges och olika typer av angreppssätt diskuteras. Till slut väljs en metod som baseras på medelvärdessatsen eftersom detta är den enda av de undersökta metoderna som ger explicita konvergensgränser. Gränserna testas sedermera med Monte Carlo-simulering och visar sig stämma.

<div>Classification is an important supervised learning technique with numerous applications. This dissertation addresses two research problems in this area. The first is multicategory classification methods for high dimensional data. To handle high dimension low sample size (HDLSS) data with uneven group sizes (i.e., imbalanced data), we develop a new classification method called angle-based multicategory distance-weighted support vector machine (MDWSVM). It is motivated from its binary counterpart and has the merits of both the support vector machine (SVM) and distance-weighted discrimination (DWD) methods while alleviating both the data piling issue of SVM and the imbalanced data issue of DWD. Theoretical results and numerical studies are used to demonstrate the advantages of our MDWSVM method over existing methods.</div><div><br></div><div>The second part of the dissertation is on the application of classification methods to precision medicine problems. Because one-stage precision medicine problems can be reformulated as weighted classification problems, the subtle differences between classification methods may lead to different application performances under this setting. Among the margin-based classification methods, we propose to use the distance weighted discrimination outcome weighted learning (DWD-OWL) method. We also extend the model to handle negative rewards for better generality and apply the angle-based idea to handle multiple treatments. The proofs of Fisher consistency for DWD-OWL in both the binary and multicategory cases are provided. Under mild conditions, the insensitivity of DWD-OWL for imbalanced setting is also demonstrated.</div>