Introduction
Large, high-dimensional data sets are common in the modern era
of computer-based instrumentation and electronic data storage. High-dimensional
data present many challenges for statistical visualization, analysis,
and modeling.
Data visualization, of course, is impossible beyond a few dimensions.
As a result, pattern recognition, data preprocessing, and model selection
must rely heavily on numerical methods.
A fundamental challenge in high-dimensional data analysis is
the so-called curse of dimensionality. Observations
in a high-dimensional space are necessarily sparser and less representative
than those in a low-dimensional space. In higher dimensions, data
over-represent the edges of a sampling distribution, because regions
of higher-dimensional space contain the majority of their volume near
the surface. (A d-dimensional spherical shell has
a volume, relative to the total volume of the sphere, that approaches
1 as d approaches infinity.) In high dimensions,
typical data points at the interior of a distribution are sampled
less frequently.
Often, many of the dimensions in a data set—the measured
features—are not useful in producing a model. Features may
be irrelevant or redundant. Regression and classification algorithms
may require large amounts of storage and computation time to process
raw data, and even if the algorithms are successful the resulting
models may contain an incomprehensible number of terms.
Because of these challenges, multivariate statistical methods
often begin with some type of dimension
reduction, in which data are approximated by points in
a lower-dimensional space. Dimension reduction is the goal of the
methods presented in this chapter. Dimension reduction often leads
to simpler models and fewer measured variables, with consequent benefits
when measurements are expensive and visualization is important.
 | Multivariate Methods | | Multidimensional Scaling |  |
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