Affine function in the context of Multiple linear regression

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

The point where the tangent line and the curve meet or intersect is called the point of tangency. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.The tangent line to a point on a differentiable curve can also be thought of as a tangent line approximation, the graph of the affine function that best approximates the original function at the given point.

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists.

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable is a simple linear regression; a model with two or more explanatory variables is a multiple linear regression. This term is distinct from multivariate linear regression, which predicts multiple correlated dependent variables rather than a single dependent variable.

In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a kind of function between vector spaces.