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21-560: A sigmoid function is a function whose graph follows the logistic function . It is defined by the formula: In many fields, especially in the context of artificial neural networks , the term "sigmoid function" is correctly recognized as a synonym for the logistic function. While other S-shaped curves, such as the Gompertz curve or the ogee curve , may resemble sigmoid functions, they are distinct mathematical functions with different properties and applications. Sigmoid functions, particularly

42-460: A , b , c , d } {\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}} defined by f ( x ) = { a , if  x = 1 , d , if  x = 2 , c , if  x = 3 , {\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}

63-476: A , c , d } = { y : ∃ x ,  such that  ( x , y ) ∈ G ( f ) } {\displaystyle \{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}} . The codomain { a , b , c , d } {\displaystyle \{a,b,c,d\}} , however, cannot be determined from the graph alone. The graph of

84-415: A surface plot . In science , engineering , technology , finance , and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes ; see Plot (graphics) for details. A graph of a function is a special case of a relation . In the modern foundations of mathematics , and, typically, in set theory ,

105-399: A function is actually equal to its graph. However, it is often useful to see functions as mappings , which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain . For example, to say that a function is onto ( surjective ) or not the codomain should be taken into account. The graph of a function on its own does not determine

126-444: A progression from small beginnings that accelerates and approaches a climax over time. When a specific mathematical model is lacking, a sigmoid function is often used. The van Genuchten–Gupta model is based on an inverted S-curve and applied to the response of crop yield to soil salinity . Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth to water table in

147-668: A true sigmoid. This function is unusual because it actually attains the limiting values of -1 and 1 within a finite range, meaning that its value is constant at -1 for all x ≤ − 1 {\displaystyle x\leq -1} and at 1 for all x ≥ 1 {\displaystyle x\geq 1} . Nonetheless, it is smooth (infinitely differentiable, C ∞ {\displaystyle C^{\infty }} ) everywhere , including at x = ± 1 {\displaystyle x=\pm 1} . Many natural processes, such as those of complex system learning curves , exhibit

168-1182: Is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0. f ( x ) = { 2 1 + e − 2 m x 1 − x 2 − 1 , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 = { tanh ⁡ ( m x 1 − x 2 ) , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 {\displaystyle {\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}} using

189-480: Is a subset of the Cartesian product X × Y {\displaystyle X\times Y} . In the definition of a function in terms of set theory , it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph. The graph of the function f : { 1 , 2 , 3 } → {

210-406: Is recovered as the set of first component of each pair in the graph { 1 , 2 , 3 } = { x :   ∃ y ,  such that  ( x , y ) ∈ G ( f ) } {\displaystyle \{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}} . Similarly, the range can be recovered as {

231-480: Is the subset of the set { 1 , 2 , 3 } × { a , b , c , d } {\displaystyle \{1,2,3\}\times \{a,b,c,d\}} G ( f ) = { ( 1 , a ) , ( 2 , d ) , ( 3 , c ) } . {\displaystyle G(f)=\{(1,a),(2,d),(3,c)\}.} From the graph, the domain { 1 , 2 , 3 } {\displaystyle \{1,2,3\}}

SECTION 10

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252-550: The cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function , which is related to the cumulative distribution function of a normal distribution ; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution . A sigmoid function is constrained by a pair of horizontal asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function

273-489: The logit function . A sigmoid function is a bounded , differentiable , real function that is defined for all real input values and has a non-negative derivative at each point and exactly one inflection point . In general, a sigmoid function is monotonic , and has a first derivative which is bell shaped . Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus

294-604: The codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective. Given a function f : X → Y {\displaystyle f:X\to Y} from a set X (the domain ) to a set Y (the codomain ), the graph of the function is the set G ( f ) = { ( x , f ( x ) ) : x ∈ X } , {\displaystyle G(f)=\{(x,f(x)):x\in X\},} which

315-409: The cubic polynomial on the real line f ( x ) = x 3 − 9 x {\displaystyle f(x)=x^{3}-9x} is { ( x , x 3 − 9 x ) : x  is a real number } . {\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.} If this set is plotted on a Cartesian plane ,

336-399: The graph usually refers to the set of ordered triples ( x , y , z ) {\displaystyle (x,y,z)} where f ( x , y ) = z {\displaystyle f(x,y)=z} . This is a subset of three-dimensional space ; for a continuous real-valued function of two real variables, its graph forms a surface , which can be visualized as

357-404: The hyperbolic tangent mentioned above. Here, m {\displaystyle m} is a free parameter encoding the slope at x = 0 {\displaystyle x=0} , which must be greater than or equal to 3 {\displaystyle {\sqrt {3}}} because any smaller value will result in a function with multiple inflection points, which is therefore not

378-423: The logistic function, have a domain of all real numbers and typically produce output values in the range from 0 to 1, although some variations, like the hyperbolic tangent , produce output values between −1 and 1. These functions are commonly used as activation functions in artificial neurons and as cumulative distribution functions in statistics . The logistic sigmoid is also invertible, with its inverse being

399-619: The result is a curve (see figure). The graph of the trigonometric function f ( x , y ) = sin ⁡ ( x 2 ) cos ⁡ ( y 2 ) {\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})} is { ( x , y , sin ⁡ ( x 2 ) cos ⁡ ( y 2 ) ) : x  and  y  are real numbers } . {\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.} If this set

420-550: The sigmoid functions are used to blend colors or geometry between two values, smoothly and without visible seams or discontinuities. Titration curves between strong acids and strong bases have a sigmoid shape due to the logarithmic nature of the pH scale . The logistic function can be calculated efficiently by utilizing type III Unums . Graph of a function In the case of functions of two variables – that is, functions whose domain consists of pairs ( x , y ) {\displaystyle (x,y)} –,

441-554: The soil are shown in modeling crop response in agriculture . In artificial neural networks , sometimes non-smooth functions are used instead for efficiency; these are known as hard sigmoids . In audio signal processing , sigmoid functions are used as waveshaper transfer functions to emulate the sound of analog circuitry clipping . In biochemistry and pharmacology , the Hill and Hill–Langmuir equations are sigmoid functions. In computer graphics and real-time rendering, some of

SECTION 20

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