Algebraic statistics

Algebraic statistics is the use of

Traditionally, algebraic statistics has been associated with the design of experiments and

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.

Design of experiments

For example,

Algebraic analysis and abstract statistical inference]

stationary stochastic processes, which is important in time series

statistics.

Encompassing previous results on probability theory on algebraic structures,

lattice theory

.

Partially ordered sets and lattices

infinite dimensions) by Jonathan Borwein

and colleagues.

statistical decision theory by Lucien Le Cam

.

Recent work using commutative algebra and algebraic geometry

In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of

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

.

Introductory example

Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities

${\displaystyle p_{i}=\mathrm {Pr} (X=i),\quad i=0,1,2}$

and these numbers satisfy

${\displaystyle \sum _{i=0}^{2}p_{i}=1\quad {\mbox{and}}\quad 0\leq p_{i}\leq 1.}$

Conversely, any three such numbers unambiguously specify a random variable, so we can identify the random variable X with the tuple (p₀,p₁,p₂)∈R³.

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

${\displaystyle p_{i}=\mathrm {Pr} (X=i)={2 \choose i}q^{i}(1-q)^{2-i}}$

and it is not hard to show that the tuples (p₀,p₁,p₂) which arise in this way are precisely the ones satisfying

${\displaystyle 4p_{0}p_{2}-p_{1}^{2}=0.\ }$

The latter is a polynomial equation defining an algebraic variety (or surface) in R³, and this variety, when intersected with the simplex given by

${\displaystyle \sum _{i=0}^{2}p_{i}=1\quad {\mbox{and}}\quad 0\leq p_{i}\leq 1,}$

yields a piece of an algebraic curve which may be identified with the set of all 3-state Bernoulli variables. Determining the parameter q amounts to locating one point on this curve; testing the hypothesis that a given variable X is Bernoulli amounts to testing whether a certain point lies on that curve or not.

Application of algebraic geometry to statistical learning theory

Algebraic geometry has also recently found applications to

• ISBN 0-521-82446-X
  . (Chapters from preliminary draft are available on-line)
• Caliński, Tadeusz; Kageyama, Sanpei (2003). Block designs: A Randomization approach, Volume II: Design. Lecture Notes in Statistics. Vol. 170. New York: Springer-Verlag.
  ISBN 0-387-95470-8
  .
• Hinkelmann, Klaus;
  ISBN 978-0-471-55177-5
  .
• H. B. Mann. 1949. Analysis and Design of Experiments: Analysis of Variance and Analysis-of-Variance Designs. Dover.
• Raghavarao, Damaraju
  (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover.
• Raghavarao, Damaraju
  ; Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.
• ISBN 0-19-853256-3
  .
• L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology. Cambridge University Press 2005.
• G. Pistone, E. Riccomango, H. P. Wynn. Algebraic Statistics. CRC Press, 2001.
• Drton, Mathias, Sturmfels, Bernd, Sullivant, Seth. Lectures on Algebraic Statistics, Springer 2009.
• Watanabe, Sumio. Algebraic Geometry and Statistical Learning Theory, Cambridge University Press 2009.
• Paolo Gibilisco, Eva Riccomagno, Maria-Piera Rogantin, Henry P. Wynn. Algebraic and Geometric Methods in Statistics, Cambridge 2009.

External links

• Algebraic Statistics

• Journal of Algebraic Statistics
• Archives of Journal of Algebraic Statistics