(such as ADMIXTURE) have been developed using other estimation techniques. Estimated proportions can be visualized using bar plots — each bar represents an individual, and is subdivided to represent the proportion of an individual's genetic ancestry from one of the K populations. Varying K can illustrate different scales of population structure; using a small K for the entire human population will subdivide people roughly by continent, while using large K will partition populations into finer subgroups. Though clustering methods are popular, they are open to misinterpretation: for non-simulated data, there is never a "true" value of K, but rather an approximation considered useful for a given question. They are sensitive to sampling strategies, sample size, and close relatives in data sets; there may be no discrete populations at all; and there may be hierarchical structure where subpopulations are nested. Clusters may be admixed themselves, and may not have a useful interpretation as source populations. Dimensionality reduction Genetic data are high dimensional and dimensionality reduction techniques can capture population structure. Principal component analysis (PCA) was first applied in population genetics in 1978 by Cavalli-Sforza and colleagues and resurged with high-throughput sequencing. Initially PCA was used on allele frequencies at known genetic markers for populations, though later it was found that by coding SNPs as integers (for example, as the number of non-reference alleles) and normalizing the values, PCA could be applied at the level of individuals. One formulation considers individuals and bi-allelic SNPs. For each individual , the value at locus is is the number of non-reference alleles (one of ). If the allele frequency at is , then the resulting matrix of normalized genotypes has entries: PCA transforms data to maximize variance; given enough data, when each individual is visualized as point on a plot, discrete clusters can form. Individuals with admixed ancestries will tend to fall between clusters, and when there is homogenous isolation by distance in the data, the top PC vectors will reflect geographic variation. The eigenvectors generated by PCA can be explicitly written in terms of the mean coalescent times for pairs of individuals, making PCA useful for inference about the population histories of groups in a given sample. PCA cannot, however, distinguish between different processes that lead to the same mean coalescent times. Multidimensional scaling and discriminant analysis have been used to study differentiation, population assignment, and to analyze genetic distances. Neighborhood graph approaches like