High-dimensional statistics
Encyclopedia
In statistical theory
, the field of high-dimensional statistics studies data whose dimension
is larger than dimensions considered in classical multivariate analysis
. High-dimensional statistics relies on the theory
of random vector
s. In many applications, the dimension of the data vectors may be larger than the sample size
.
considers a probability model for a population and considers data that arose as a sample from the population. For many problems, the estimates of the population characteristics ("parameters") can be substantially refined (in theory) as the sample size increases toward infinity. A traditional requirement of estimators is consistency
, that is, the convergence to the unknown true value of the parameter.
In 1968, A.N.Kolmogorov proposed another setting of statistical problems and another setting for the asymptotics, in which the dimension of variables p increases along with the sample size n so that the ratio p/n tends to a constant. It was called the “increasing dimension asymptotics” or “the Kolmogorov asymptotics” [1]. Kolmogorov's approach makes it possible to isolate many principal terms of error probabilities and of standard measures of the quality of estimators (quality functions) for large p and n.
references at http://hd-stat.narod.ru). A special parameter G that is a
function of the fourth moments of variables
was found having the property that a small value of G produces a number of specifically many-parametric phenomena. For increasing p and n so that p/n tends to a constant and G → 0, the principal terms of rotation invariant functionals occurring in statistics prove to be dependent on only the first two moments of variables. Under n and p tending to infinity, p/n → y > 0, and G → 0, these functionals have vanishing variance and converge to constants that represent the limit value of empirical means and
variances. As a consequence, some stable integral relations are produced between functions of parameters and functions of observable variables. They were called “stochastic canonical equations” or “dispersion equations” (see [3]). Using them one can express the principle parts of standard quality functions of regularized multivariate statistical procedures as functions of only observed variables. This provides
the possibility of choosing better procedures and finding asymptotically unimprovable solutions.
Statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that...
, the field of high-dimensional statistics studies data whose dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
is larger than dimensions considered in classical multivariate analysis
Multivariate analysis
Multivariate analysis is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical variable at a time...
. High-dimensional statistics relies on the theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
of random vector
Random element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line...
s. In many applications, the dimension of the data vectors may be larger than the sample size
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...
.
History
Traditionally, statistical inferenceStatistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...
considers a probability model for a population and considers data that arose as a sample from the population. For many problems, the estimates of the population characteristics ("parameters") can be substantially refined (in theory) as the sample size increases toward infinity. A traditional requirement of estimators is consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
, that is, the convergence to the unknown true value of the parameter.
In 1968, A.N.Kolmogorov proposed another setting of statistical problems and another setting for the asymptotics, in which the dimension of variables p increases along with the sample size n so that the ratio p/n tends to a constant. It was called the “increasing dimension asymptotics” or “the Kolmogorov asymptotics” [1]. Kolmogorov's approach makes it possible to isolate many principal terms of error probabilities and of standard measures of the quality of estimators (quality functions) for large p and n.
Mathematical theory
Extensive mathematical investigations were carried out that resulted in the creation of systematic theory for improved and asymptotically unimprovable versions of multivariate statistical procedures (seereferences at http://hd-stat.narod.ru). A special parameter G that is a
function of the fourth moments of variables
was found having the property that a small value of G produces a number of specifically many-parametric phenomena. For increasing p and n so that p/n tends to a constant and G → 0, the principal terms of rotation invariant functionals occurring in statistics prove to be dependent on only the first two moments of variables. Under n and p tending to infinity, p/n → y > 0, and G → 0, these functionals have vanishing variance and converge to constants that represent the limit value of empirical means and
variances. As a consequence, some stable integral relations are produced between functions of parameters and functions of observable variables. They were called “stochastic canonical equations” or “dispersion equations” (see [3]). Using them one can express the principle parts of standard quality functions of regularized multivariate statistical procedures as functions of only observed variables. This provides
the possibility of choosing better procedures and finding asymptotically unimprovable solutions.