Logistic functionS-shaped curve A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f(x)=L1+e−k(x−x0){\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}where L{\displaystyle L} is the carrying capacity, the supremum of the values of the function;k{\displaystyle k} is the logistic growth rate, the steepness of the curve; andx0{\displaystyle x_{0}} is the x{\displaystyle x} value of the function's midpoint.The logistic function has domain the real numbers, the limit as x{\displaystyle x} tends to −∞{\displaystyle -\infty } is 0, and the limit as x{\displaystyle x} tends to +∞{\displaystyle +\infty } is L{\displaystyle L}.
Generalised logistic functionThe generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F.J.Richards, who proposed the general form for the family of models in 1959.
Benjamin GompertzBenjamin GompertzFRS (5 March 1779 – 14 July 1865) was an English self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1825. He was the brother of the early animal rights activist and inventor Lewis Gompertz and the poet Isaac Gompertz.William MakehamWilliam Matthew Makeham (11 September 1826 – 17 November 1891) was an English actuary and mathematician. Makeham was responsible for proposing the age-independent Makeham term in the Gompertz–Makeham law of mortality that, together with the exponentially age-dependent Gompertz term, was one of the most effective theories to describe human mortality. Makeham was responsible for two important studies on human mortality: Makeham (1860).
Sigmoid functionA sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function".
E (mathematical constant)Base of natural logarithmsThe number e is a mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352e plays important and recurring roles across mathematics.
AsymptoteIn analytic geometry, an asymptote ( ) of a curve is a straight line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word "asymptote" derives from the Greek ἀσύμπτωτος (asumptōtos), which means "not falling together", from ἀ priv.
ActuaryAnalyst of business risk and uncertaintyAn actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require asset management, liability management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms.
Abraham de MoivreAbraham de Moivre (French pronunciation:; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau. He was a friend of Isaac Newton, Edmond Halley, and James Stirling.
Royal SocietyThe Royal Society of London for Improving Natural Knowledge, commonly referred to as the Royal Society, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, education and public engagement and fostering international and global co-operation.
Mathematical modelA mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations.
Time seriesIn mathematics, a time series is a sequence of data points indexed, listed, or graphed in chronological order. Most commonly, a time series consists of observations recorded at successive equally spaced points in time. Thus, it represents a form of discrete-time data. A time series may describe measurements collected over seconds, days, years, or even centuries.*As an Amazon Associate I earn from qualifying purchases.