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Generalized Logistic Function
The 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. Definition Richards's curve has the following form: :Y(t) = A + where Y = weight, height, size etc., and t = time. It has five parameters: *A: the lower (left) asymptote; *K: the upper (right) asymptote when C=1. If A=0 and C=1 then K is called the carrying capacity; *B: the growth rate; *\nu > 0 : affects near which asymptote maximum growth occurs. *Q: is related to the value Y(0) *C: typically takes a value of 1. Otherwise, the upper asymptote is A + The equation can also be written: :Y(t) = A + where M can be thought of as a starting time, at which Y(M) = A + . Including both Q and M can be convenient: :Y(t) = A + this representation simplifies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Logistic Function A0 K1 B1
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +\infty and approaching zero as x approaches -\infty. The logistic function finds applications in a range of fields, including biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, statistics, and artificial neural networks. A generalization of the logistic function is the hyperbolastic function of type I. The standard logistic function, where L=1,k=1,x_0=0, is sometimes simply called ''the sigmoid''. It is also sometimes called the ''expit'', being the inverse of the logit. History The logistic function was introduced in a series of three papers by Pierre François Verhulst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigmoid Function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \frac=1-S(-x). Other standard 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 an alias for the logistic function. Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis John Richards
Francis John Richards (1 October 1901 – 2 January 1965) was an English plant physiologist who specialized in quantitative studies on the mineral nutrition requirements of crops. He first described the general form of the generalised logistic function in 1959. Early life and education Richards was born at Burton-on-Trent, the third child of Robert with two brothers and a sister. His eldest brother died in World War I and the other brother took over the family business as a butcher. Richards took an interest in natural history and astronomy at young age. He attended the Burton-on-Trent Grammar School and took an interest in biology, encouraged by his schoolmaster Storer. He earned higher certificates in mathematics with subsidiary physics and chemistry in 1919 and in biology with distinction in 1920. In 1921, Richards began his studies at Birmingham University. Under the direction of R. H. Yapp, Richards studied salt marsh ecology and led surveys of Dovey estuary. He gradua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carrying Capacity
The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the environment's maximal load, which in population ecology corresponds to the population equilibrium, when the number of deaths in a population equals the number of births (as well as immigration and emigration). The effect of carrying capacity on population dynamics is modelled with a logistic function. Carrying capacity is applied to the maximum population an environment can support in ecology, agriculture and fisheries. The term carrying capacity has been applied to a few different processes in the past before finally being applied to population limits in the 1950s. The notion of carrying capacity for humans is covered by the notion of sustainable population. At the global scale, scientific data indicates that humans are living beyon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gompertz Curve
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the Logistic function, simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations. History Benjamin Gompertz (1779–1865) was an actuary in London who was privately educated. He was elected a fellow of the Royal Society in 1819. The function was first presented in his June 16, 1825 paper at the bottom of page 518. The G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
