37-610: [REDACTED] Look up growing in Wiktionary, the free dictionary. Growing may refer to: Growth (disambiguation) Growing (band) , an American noise band Growing (Rina Chinen album) , 1998 Growing (Sleeping People album) , 2007 Growing , a 1961 autobiographical book by Leonard Woolf See also [ edit ] All pages with titles beginning with Growing All pages with titles containing Growing Grow (disambiguation) Topics referred to by
74-488: A 2010 American horror film Izaugsme ( Growth ), a Latvian political party Grown (album) , by 2PM See also [ edit ] [REDACTED] Wikiquote has quotations related to Growth . Grow (disambiguation) Growth curve (disambiguation) Growth impairment (disambiguation) Growth industry (disambiguation) Growth model (disambiguation) Growth rate (disambiguation) Growth regulator (disambiguation) Topics referred to by
111-889: A peptide hormone that stimulates growth Human development (biology) Plant growth Secondary growth , growth that thickens woody plants A tumor or other such neoplasm Economics [ edit ] Economic growth , the increase in the inflation-adjusted market value of the goods and services Growth investing , a style of investment strategy focused on capital appreciation Mathematics [ edit ] Exponential growth , also called geometric growth Hyperbolic growth Linear growth , refers to two distinct but related notions Logistic growth , characterized as an S curve Social science [ edit ] Developmental psychology Erikson's stages of psychosocial development Human development (humanity) Personal development Population growth Other uses [ edit ] Growth (film) ,
148-511: A physical model). This yields an unstable equilibrium at 0 and a stable equilibrium at 1, and thus for any function value greater than 0 and less than 1, it grows to 1. The logistic equation is a special case of the Bernoulli differential equation and has the following solution: f ( x ) = e x e x + C . {\displaystyle f(x)={\frac {e^{x}}{e^{x}+C}}.} Choosing
185-452: A probability p . In more detail, p can be interpreted as the probability of one of two alternatives (the parameter of a Bernoulli distribution ); the two alternatives are complementary, so the probability of the other alternative is q = 1 − p {\displaystyle q=1-p} and p + q = 1 {\displaystyle p+q=1} . The two alternatives are coded as 1 and 0, corresponding to
222-452: A traditional division of Greek mathematics . As a word derived from ancient Greek mathematical terms, the name of this function is unrelated to the military and management term logistics , which is instead from French : logis "lodgings", though some believe the Greek term also influenced logistics ; see Logistics § Origin for details. The standard logistic function
259-415: Is 0, and the limit as x → + ∞ {\displaystyle x\to +\infty } is L {\displaystyle L} . The standard logistic function , depicted at right, where L = 1 , k = 1 , x 0 = 0 {\displaystyle L=1,k=1,x_{0}=0} , has the equation and is sometimes simply called the sigmoid . It
296-431: Is a location–scale family , which corresponds to parameters of the logistic function. If L = 1 {\displaystyle L=1} is fixed, then the midpoint x 0 {\displaystyle x_{0}} is the location and the slope k {\displaystyle k} is the scale. Conversely, its antiderivative can be computed by
333-453: Is also sometimes called the expit , being the inverse function of the logit . 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 . There are various generalizations , depending on
370-611: Is an odd function . The sum of the logistic function and its reflection about the vertical axis, f ( − x ) {\displaystyle f(-x)} , is 1 1 + e − x + 1 1 + e − ( − x ) = e x e x + 1 + 1 e x + 1 = 1. {\displaystyle {\frac {1}{1+e^{-x}}}+{\frac {1}{1+e^{-(-x)}}}={\frac {e^{x}}{e^{x}+1}}+{\frac {1}{e^{x}+1}}=1.} The logistic function
407-478: Is different from Wikidata All article disambiguation pages All disambiguation pages Growth (disambiguation) (Redirected from Growth (disambiguation) ) [REDACTED] Look up growth in Wiktionary, the free dictionary. Growth may refer to: Biology [ edit ] Auxology , the study of all aspects of human physical growth Bacterial growth Cell growth Growth hormone ,
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#1732787779485444-396: Is different from Wikidata All article disambiguation pages All disambiguation pages Logistic function A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve ) with the equation where The logistic function has domain the real numbers , the limit as x → − ∞ {\displaystyle x\to -\infty }
481-753: Is known as the softplus function and (with scaling) is a smooth approximation of the ramp function , just as the logistic function (with scaling) is a smooth approximation of the Heaviside step function . The unique standard logistic function is the solution of the simple first-order non-linear ordinary differential equation d d x f ( x ) = f ( x ) ( 1 − f ( x ) ) {\displaystyle {\frac {d}{dx}}f(x)=f(x){\big (}1-f(x){\big )}} with boundary condition f ( 0 ) = 1 / 2 {\displaystyle f(0)=1/2} . This equation
518-1787: Is known as the density of the logistic distribution : f ( x ) = 1 1 + e − x = e x 1 + e x , {\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}},} d d x f ( x ) = e x ⋅ ( 1 + e x ) − e x ⋅ e x ( 1 + e x ) 2 = e x ( 1 + e x ) 2 = ( e x 1 + e x ) ( 1 1 + e x ) = ( e x 1 + e x ) ( 1 − e x 1 + e x ) = f ( x ) ( 1 − f ( x ) ) {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)&={\frac {e^{x}\cdot (1+e^{x})-e^{x}\cdot e^{x}}{(1+e^{x})^{2}}}\\&={\frac {e^{x}}{(1+e^{x})^{2}}}\\&=\left({\frac {e^{x}}{1+e^{x}}}\right)\left({\frac {1}{1+e^{x}}}\right)\\&=\left({\frac {e^{x}}{1+e^{x}}}\right)\left(1-{\frac {e^{x}}{1+e^{x}}}\right)\\&=f(x)\left(1-f(x)\right)\end{aligned}}} from which all higher derivatives can be derived algebraically. For example, f ″ = ( 1 − 2 f ) ( 1 − f ) f {\displaystyle f''=(1-2f)(1-f)f} . The logistic distribution
555-635: Is the hyperbolic angle on the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1} , which factors as ( x + y ) ( x − y ) = 1 {\displaystyle (x+y)(x-y)=1} , and thus has asymptotes the lines through the origin with slope − 1 {\displaystyle -1} and with slope 1 {\displaystyle 1} , and vertex at ( 1 , 0 ) {\displaystyle (1,0)} corresponding to
592-533: Is the continuous version of the logistic map . Note that the reciprocal logistic function is solution to a simple first-order linear ordinary differential equation. The qualitative behavior is easily understood in terms of the phase line : the derivative is 0 when the function is 1; and the derivative is positive for f {\displaystyle f} between 0 and 1, and negative for f {\displaystyle f} above 1 or less than 0 (though negative populations do not generally accord with
629-681: Is the logistic function with parameters k = 1 {\displaystyle k=1} , x 0 = 0 {\displaystyle x_{0}=0} , L = 1 {\displaystyle L=1} , which yields f ( x ) = 1 1 + e − x = e x e x + 1 = e x / 2 e x / 2 + e − x / 2 . {\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}={\frac {e^{x/2}}{e^{x/2}+e^{-x/2}}}.} In practice, due to
666-414: Is the shifted and scaled sigmoid a S ( k ( x − r ) ) {\displaystyle aS{\big (}k(x-r){\big )}} . When the capacity L = 1 {\displaystyle L=1} , the value of the logistic function is in the range ( 0 , 1 ) {\displaystyle (0,1)} and can be interpreted as
703-1931: Is thus rotationally symmetrical about the point (0, 1/2). The logistic function is the inverse of the natural logit function and so converts the logarithm of odds into a probability . The conversion from the log-likelihood ratio of two alternatives also takes the form of a logistic curve. The logistic function is an offset and scaled hyperbolic tangent function: f ( x ) = 1 2 + 1 2 tanh ( x 2 ) , {\displaystyle f(x)={\frac {1}{2}}+{\frac {1}{2}}\tanh \left({\frac {x}{2}}\right),} or tanh ( x ) = 2 f ( 2 x ) − 1. {\displaystyle \tanh(x)=2f(2x)-1.} This follows from tanh ( x ) = e x − e − x e x + e − x = e x ⋅ ( 1 − e − 2 x ) e x ⋅ ( 1 + e − 2 x ) = f ( 2 x ) − e − 2 x 1 + e − 2 x = f ( 2 x ) − e − 2 x + 1 − 1 1 + e − 2 x = 2 f ( 2 x ) − 1. {\displaystyle {\begin{aligned}\tanh(x)&={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{x}\cdot \left(1-e^{-2x}\right)}{e^{x}\cdot \left(1+e^{-2x}\right)}}\\&=f(2x)-{\frac {e^{-2x}}{1+e^{-2x}}}=f(2x)-{\frac {e^{-2x}+1-1}{1+e^{-2x}}}=2f(2x)-1.\end{aligned}}} The hyperbolic-tangent relationship leads to another form for
740-759: The substitution u = 1 + e x {\displaystyle u=1+e^{x}} , since f ( x ) = e x 1 + e x = u ′ u , {\displaystyle f(x)={\frac {e^{x}}{1+e^{x}}}={\frac {u'}{u}},} so (dropping the constant of integration ) ∫ e x 1 + e x d x = ∫ 1 u d u = ln u = ln ( 1 + e x ) . {\displaystyle \int {\frac {e^{x}}{1+e^{x}}}\,dx=\int {\frac {1}{u}}\,du=\ln u=\ln(1+e^{x}).} In artificial neural networks , this
777-454: The analytical solution, the logistic curve shows early exponential growth for negative argument, which reaches to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap. The differential equation derived above is a special case of a general differential equation that only models the sigmoid function for x > 0 {\displaystyle x>0} . In many modeling applications,
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#1732787779485814-504: The choice of the term "logistic" (French: logistique ), but it is presumably in contrast to the logarithmic curve, and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of arithmetic growth and geometric growth (whose curve he calls a logarithmic curve , instead of the modern term exponential curve ), and thus "logistic growth" is presumably named by analogy, logistic being from Ancient Greek : λογῐστῐκός , romanized : logistikós ,
851-446: The constant of integration C = 1 {\displaystyle C=1} gives the other well known form of the definition of the logistic curve: f ( x ) = e x e x + 1 = 1 1 + e − x . {\displaystyle f(x)={\frac {e^{x}}{e^{x}+1}}={\frac {1}{1+e^{-x}}}.} More quantitatively, as can be seen from
888-401: The field. The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet . Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838, then presented an expanded analysis and named
925-413: The function in 1844 (published 1845); the third paper adjusted the correction term in his model of Belgian population growth. The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth approaches the limit with an exponentially decaying gap, like the initial stage in reverse. Verhulst did not explain
962-405: The hyperbolic tangent. Similarly, ( e t / 2 + e − t / 2 , e t / 2 ) {\displaystyle {\bigl (}e^{t/2}+e^{-t/2},e^{t/2}{\bigr )}} parametrizes the hyperbola x y − y 2 = 1 {\displaystyle xy-y^{2}=1} , with quotient
999-671: The linear transformation ( 1 1 0 1 ) {\displaystyle {\bigl (}{\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}{\bigr )}} , while the parametrization of the unit hyperbola (for the hyperbolic tangent) corresponds to the linear transformation 1 2 ( 1 1 − 1 1 ) {\displaystyle {\tfrac {1}{2}}{\bigl (}{\begin{smallmatrix}1&1\\-1&1\end{smallmatrix}}{\bigr )}} . The standard logistic function has an easily calculated derivative . The derivative
1036-403: The lines through the origin with slope 0 {\displaystyle 0} and with slope 1 {\displaystyle 1} , and vertex at ( 2 , 1 ) {\displaystyle (2,1)} , corresponding to the range and midpoint ( 1 / 2 {\displaystyle 1/2} ) of
1073-423: The logistic function's derivative: d d x f ( x ) = 1 4 sech 2 ( x 2 ) , {\displaystyle {\frac {d}{dx}}f(x)={\frac {1}{4}}\operatorname {sech} ^{2}\left({\frac {x}{2}}\right),} which ties the logistic function into the logistic distribution . Geometrically, the hyperbolic tangent function
1110-411: The logistic function. Parametrically, hyperbolic cosine and hyperbolic sine give coordinates on the unit hyperbola: ( ( e t + e − t ) / 2 , ( e t − e − t ) / 2 ) {\displaystyle \left((e^{t}+e^{-t})/2,(e^{t}-e^{-t})/2\right)} , with quotient
1147-471: The logistic function. These correspond to linear transformations (and rescaling the parametrization) of the hyperbola x y = 1 {\displaystyle xy=1} , with parametrization ( e − t , e t ) {\displaystyle (e^{-t},e^{t})} : the parametrization of the hyperbola for the logistic function corresponds to t / 2 {\displaystyle t/2} and
Growing - Misplaced Pages Continue
1184-430: The more general form d f ( x ) d x = k a f ( x ) ( a − f ( x ) ) , f ( 0 ) = a 1 + e k r {\displaystyle {\frac {df(x)}{dx}}={\frac {k}{a}}f(x){\big (}a-f(x){\big )},\quad f(0)={\frac {a}{1+e^{kr}}}} can be desirable. Its solution
1221-413: The nature of the exponential function e − x {\displaystyle e^{-x}} , it is often sufficient to compute the standard logistic function for x {\displaystyle x} over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1. The logistic function has
1258-431: The range and midpoint ( 1 {\displaystyle {1}} ) of tanh. Analogously, the logistic function can be viewed as the hyperbolic angle on the hyperbola x y − y 2 = 1 {\displaystyle xy-y^{2}=1} , which factors as y ( x − y ) = 1 {\displaystyle y(x-y)=1} , and thus has asymptotes
1295-411: The same term [REDACTED] This disambiguation page lists articles associated with the title Growing . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Growing&oldid=1141732075 " Category : Disambiguation pages Hidden categories: Short description
1332-409: The same term [REDACTED] This disambiguation page lists articles associated with the title Growth . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Growth&oldid=1259648613 " Category : Disambiguation pages Hidden categories: Short description
1369-508: The symmetry property that 1 − f ( x ) = f ( − x ) . {\displaystyle 1-f(x)=f(-x).} This reflects that the growth from 0 when x {\displaystyle x} is small is symmetric with the decay of the gap to the limit (1) when x {\displaystyle x} is large. Further, x ↦ f ( x ) − 1 / 2 {\displaystyle x\mapsto f(x)-1/2}
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