Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.
Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics.
In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes.
For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs.
Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics.
Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory.
Partially ordered vector spaces and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem.[1] Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues.
Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam.
In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties.
Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities
pi=Pr(X=i), i=0,1,2
and these numbers satisfy
| 2 | |
| \sum | |
| i=0 |
pi=1 and 0\leqpi\leq1.
Now suppose X is a binomial random variable with parameter q and n = 2, i.e. X represents the number of successes when repeating a certain experiment two times, where each experiment has an individual success probability of q. Then
pi=Pr(X=i)={2\choosei}qi(1-q)2-i
4p0p2-p
| 2=0. | |
| 1 |
| 2 | |
| \sum | |
| i=0 |
pi=1 and 0\leqpi\leq1,
Algebraic geometry has also recently found applications to statistical learning theory, including a generalization of the Akaike information criterion to singular statistical models.[2]