High Dimensional Random Walks Can Appear Low Dimensional: Application to Influenza H3N2 Evolution

Abstract

One important feature of the mammalian immune system is the highly specific binding of antigens to antibodies. Antibodies generated in response to one infection may also provide some level of cross immunity to other infections. One model to describe this cross immunity is the notion of antigenic space, which assigns each antibody and each virus a point in $\mathbb{R}^n$. Past studies have suggested the dimensionality of antigenic space, $n$, may be small. In this study we show that data from hemagglutination assays suggest a high dimensional random walk (or self avoiding random random walk). The discrepancy between our result and prior studies is due to the fact that random walks can appear low dimensional according to a variety of analyses. including principal component analysis (PCA) and multidimensional scaling (MDS).