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In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ] [ 2 ] [ 3 ] This concept first arose in calculus , and was later generalized to the more abstract setting of order theory .
A perfectly monotonic increasing relationship implies that for any two pairs of data values X i, Y i and X j, Y j, that X i − X j and Y i − Y j always have the same sign. A perfectly monotonic decreasing relationship implies that these differences always have opposite signs. The Spearman correlation coefficient is often described as being ...
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.
Once it had been realized that INT models may be perceived as special cases of a much broader general approach for modeling non-linear monotone convex relationships, the new Response Modeling Methodology had been initiated and developed (Shore, 2005a, [2] 2011 [3] and references therein).
[4] [5] Another related concept is that of a completely/absolutely monotonic sequence. This notion was introduced by Hausdorff in 1921. This notion was introduced by Hausdorff in 1921. The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics.
In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation .
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. [1] It is closely allied to the operator concave and operator concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality.