Mathematics and statistics

Order of operations

The order of operations, also called operator precedence, is the rule that mathematical expressions are evaluated parentheses first, then exponents, then multiplication and division, then addition and subtraction.

Order statistic

An order statistic is one of a statistical sample's values arranged in increasing order; the first is the sample minimum and the last is the sample maximum.

Order statistics

Order statistics are the values of a random sample arranged in increasing order, with the jth smallest called the jth order statistic; the theory underlies the sample median, quantiles, trimmed means, and extremes.

Order theory

Order theory is the branch of mathematics that studies partially ordered sets, or posets, together with structures built on them such as lattices and ordered algebraic systems.

Ordered abelian groups

An ordered abelian group is an abelian group with a total order compatible with addition, such as the integers, rationals, and reals; it is central to model theory.

Ordered logit

Ordered logit, also known as the proportional odds model, is a statistical regression method for ordinal outcomes with ordered categories, formulated by Peter McCullagh in 1980.

Ordered pair

An ordered pair (a, b) is a mathematical object with a first and second coordinate, so (a, b) equals (c, d) only when a = c and b = d.

Ordered probit model

The ordered probit model is a regression method for ordinal dependent variables, proposed by Aitchison and Silvey in 1957 and estimated by maximum likelihood, assuming normally distributed errors.

Ordinal analysis

Ordinal analysis, also known as proof theoretic ordinal, is a proof theory research program assigning each formal theory a characteristic ordinal, begun with Gentzen's 1936 consistency proof of Peano arithmetic.

Ordinal collapsing function

An ordinal collapsing function is a proof theory function that assigns large uncountable ordinals smaller countable ones, introduced by Heinz Bachmann and central to analyzing impredicative theories via the Bachmann–Howard ordinal.

Ordinal notation

An ordinal notation is a formal system of finite expressions with a computable ordering that names ordinals, central to ordinal analysis in proof theory and to proving program termination.

Ordinal number

An ordinal number (or ordinal) is the order type of a well-ordered set, introduced by Georg Cantor in 1883 to extend natural numbers to infinite positions.

Ordinary differential equation

An ordinary differential equation (ODE) is an equation involving an unknown function of one variable and its derivatives, used to model motion, circuits, chemical reactions, and populations.

Ordinary generating function

An ordinary generating function (OGF) encodes a sequence as a power series whose coefficients are the sequence's terms, distinguishing it from weighted variants such as exponential generating functions.

Ordinary least squares

Ordinary least squares (OLS) regression is the standard estimator of the linear regression model, choosing coefficients that minimize the sum of squared residuals, described as the workhorse of applied econometrics.

Orientability

Orientability is a property of a surface or manifold stating whether a consistent orientation, such as clockwise, can be assigned continuously over the entire space; spheres are orientable, Möbius bands are not.

Oriented matroid

An oriented matroid is a combinatorial structure recording the plus, minus, and zero sign patterns of finite point, vector, or hyperplane configurations, generalizing graphs, polytopes, and linear programming.

Ornstein–Uhlenbeck operator

The Ornstein–Uhlenbeck operator is the infinitesimal generator of the Ornstein–Uhlenbeck semigroup, acting as Δf − ⟨x, ∇f⟩; it is the Gaussian analog of the Laplacian, named for its 1930 paper.

Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process, introduced by Uhlenbeck and Ornstein in 1930, is a mean-reverting stochastic process that is Gaussian, Markovian, and stationary, used to model Brownian-particle velocity and interest rates.

Orthocenter

The orthocenter of a triangle is the point where its three altitudes meet, lying inside an acute triangle, outside an obtuse one, and at the right-angle vertex itself.

Orthogonal array

An orthogonal array, introduced by C. R. Rao in the 1940s, is a table of symbols in which every chosen set of columns contains all combinations equally often.

Orthogonal array design

Orthogonal array design is a design of experiments method using orthogonal arrays to estimate all main effects uncorrelatedly from relatively few test runs, widely used in industrial quality engineering.

Orthogonal collocation

Orthogonal collocation is a numerical method for solving differential equations by expanding the solution in orthogonal polynomials and forcing the equation to hold at their roots, introduced by Villadsen and Stewart in 1967.

Orthogonal complement

The orthogonal complement of a subset U of an inner product space is the set of all vectors whose inner product with every vector of U is zero.

Orthogonal coordinates

Orthogonal coordinates are a coordinate system in which coordinate lines or surfaces intersect at right angles, a common case of curvilinear coordinates including Cartesian, cylindrical, and spherical systems.

Orthogonal experimental design

Orthogonal experimental design is a design-of-experiments method that arranges factor levels in orthogonal arrays, estimating each factor's effects independently with far fewer runs than a full factorial experiment.

Orthogonal group

The orthogonal group, or general orthogonal group, is the classical group of linear transformations preserving a non-singular quadratic form; O(n) consists of the real orthogonal matrices preserving distances in n-dimensional Euclidean space.

Orthogonal iteration

Orthogonal iteration, also called subspace iteration or simultaneous iteration, is a numerical linear algebra algorithm that generalizes the power method to compute bases for eigenspaces.

Orthogonal matching pursuit

Orthogonal matching pursuit (OMP) is a greedy algorithm for sparse approximation, introduced by Mallat and Zhang in 1993, which recovers sparse signals from fewer measurements than the signal dimension.