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52-499: (Redirected from S-curve ) [REDACTED] Look up S-curve  or S-shaped in Wiktionary, the free dictionary. S curve or S-curve may refer to: S-curve (art) , an S-shaped curve which serves a wide variety of compositional purposes S-curve (math), a characteristic S-shaped curve of a sigmoid function S-curve corset , an Edwardian corset style S-Curve Records ,

104-401: A < X < b  and  c < Y < d ) = ∫ a b ∫ c d f ( x , y ) d y d x ; {\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;} For two discrete random variables, it is beneficial to generate

156-451: A < b {\displaystyle a<b} , is therefore In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy 's inversion formula for

208-454: A ) = P ⁡ ( a < X ≤ b ) = ∫ a b f X ( x ) d x {\displaystyle F_{X}(b)-F_{X}(a)=\operatorname {P} (a<X\leq b)=\int _{a}^{b}f_{X}(x)\,dx} for all real numbers a {\displaystyle a} and b {\displaystyle b} . The function f X {\displaystyle f_{X}}

260-452: A continuous random variable X {\displaystyle X} can be expressed as the integral of its probability density function f X {\displaystyle f_{X}} as follows: F X ( x ) = ∫ − ∞ x f X ( t ) d t . {\displaystyle F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt.} In

312-476: A continuous random variable can be determined from the cumulative distribution function by differentiating using the Fundamental Theorem of Calculus ; i.e. given F ( x ) {\displaystyle F(x)} , f ( x ) = d F ( x ) d x {\displaystyle f(x)={\frac {dF(x)}{dx}}} as long as the derivative exists. The CDF of

364-627: A cumulative distribution function, in contrast to the lower-case f {\displaystyle f} used for probability density functions and probability mass functions . This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution uses Φ {\displaystyle \Phi } and ϕ {\displaystyle \phi } instead of F {\displaystyle F} and f {\displaystyle f} , respectively. The probability density function of

416-445: 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

468-422: A record company label Reverse curve , or "S" curve, in civil engineering See also [ edit ] All pages with titles beginning with S Curve Recurve (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title S Curve . If an internal link led you here, you may wish to change the link to point directly to

520-404: A shorter notation: F X ( x ) = P ⁡ ( X 1 ≤ x 1 , … , X N ≤ x N ) {\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})} Every multivariate CDF is: Not every function satisfying

572-604: A table of probabilities and address the cumulative probability for each potential range of X and Y , and here is the example: given the joint probability mass function in tabular form, determine the joint cumulative distribution function. Solution: using the given table of probabilities for each potential range of X and Y , the joint cumulative distribution function may be constructed in tabular form: For N {\displaystyle N} random variables X 1 , … , X N {\displaystyle X_{1},\ldots ,X_{N}} ,

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624-668: 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

676-594: Is binomial distributed . Then the CDF of X {\displaystyle X} is given by F ( k ; n , p ) = Pr ( X ≤ k ) = ∑ i = 0 ⌊ k ⌋ ( n i ) p i ( 1 − p ) n − i {\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i}} Here p {\displaystyle p}

728-1182: 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

780-463: Is exponential distributed . Then the CDF of X {\displaystyle X} is given by F X ( x ; λ ) = { 1 − e − λ x x ≥ 0 , 0 x < 0. {\displaystyle F_{X}(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}} Here λ > 0

832-527: Is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable. If X {\displaystyle X} is a purely discrete random variable , then it attains values x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } with probability p i = p ( x i ) {\displaystyle p_{i}=p(x_{i})} , and

884-411: Is called the survival function and denoted S ( x ) {\displaystyle S(x)} , while the term reliability function is common in engineering . While the plot of a cumulative distribution F {\displaystyle F} often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot , which folds the top half of

936-724: Is continuous at b {\displaystyle b} , this equals zero and there is no discrete component at b {\displaystyle b} . Every cumulative distribution function F X {\displaystyle F_{X}} is non-decreasing and right-continuous , which makes it a càdlàg function. Furthermore, lim x → − ∞ F X ( x ) = 0 , lim x → + ∞ F X ( x ) = 1. {\displaystyle \lim _{x\to -\infty }F_{X}(x)=0,\quad \lim _{x\to +\infty }F_{X}(x)=1.} Every function with these three properties

988-475: 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

1040-396: Is equal to the derivative of F X {\displaystyle F_{X}} almost everywhere , and it is called the probability density function of the distribution of X {\displaystyle X} . If X {\displaystyle X} has finite L1-norm , that is, the expectation of | X | {\displaystyle |X|}

1092-1021: Is finite, then the expectation is given by the Riemann–Stieltjes integral E [ X ] = ∫ − ∞ ∞ t d F X ( t ) {\displaystyle \mathbb {E} [X]=\int _{-\infty }^{\infty }t\,dF_{X}(t)} and for any x ≥ 0 {\displaystyle x\geq 0} , x ( 1 − F X ( x ) ) ≤ ∫ x ∞ t d F X ( t ) {\displaystyle x(1-F_{X}(x))\leq \int _{x}^{\infty }t\,dF_{X}(t)} as well as x F X ( − x ) ≤ ∫ − ∞ − x ( − t ) d F X ( t ) {\displaystyle xF_{X}(-x)\leq \int _{-\infty }^{-x}(-t)\,dF_{X}(t)} as shown in

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1144-421: Is the function given by where the right-hand side represents the probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} . The probability that X {\displaystyle X} lies in the semi-closed interval ( a , b ] {\displaystyle (a,b]} , where

1196-746: Is the parameter of the distribution, often called the rate parameter. Suppose X {\displaystyle X} is normal distributed . Then the CDF of X {\displaystyle X} is given by F ( t ; μ , σ ) = 1 σ 2 π ∫ − ∞ t exp ⁡ ( − ( x − μ ) 2 2 σ 2 ) d x . {\displaystyle F(t;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{t}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,dx.} Here

1248-480: Is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n {\displaystyle n} independent experiments, and ⌊ k ⌋ {\displaystyle \lfloor k\rfloor } is the "floor" under k {\displaystyle k} , i.e. the greatest integer less than or equal to k {\displaystyle k} . Sometimes, it

1300-623: Is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function ( ccdf ) or simply the tail distribution or exceedance , and is defined as F ¯ X ( x ) = P ⁡ ( X > x ) = 1 − F X ( x ) . {\displaystyle {\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).} This has applications in statistical hypothesis testing , for example, because

1352-406: The characteristic function also rely on the "less than or equal" formulation. If treating several random variables X , Y , … {\displaystyle X,Y,\ldots } etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital F {\displaystyle F} for

1404-550: 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

1456-441: The generalized inverse distribution function , which is defined as Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are: The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions. The empirical distribution function is an estimate of the cumulative distribution function that generated

1508-423: The inverse distribution function or quantile function . Some distributions do not have a unique inverse (for example if f X ( x ) = 0 {\displaystyle f_{X}(x)=0} for all a < x < b {\displaystyle a<x<b} , causing F X {\displaystyle F_{X}} to be constant). In this case, one may use

1560-491: 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

1612-487: The (finite) expected value of the real-valued random variable X {\displaystyle X} can be defined on the graph of its cumulative distribution function as illustrated by the drawing in the definition of expected value for arbitrary real-valued random variables . As an example, suppose X {\displaystyle X} is uniformly distributed on the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Then

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1664-576: The CDF F X {\displaystyle F_{X}} of a real valued random variable X {\displaystyle X} is continuous , then X {\displaystyle X} is a continuous random variable ; if furthermore F X {\displaystyle F_{X}} is absolutely continuous , then there exists a Lebesgue-integrable function f X ( x ) {\displaystyle f_{X}(x)} such that F X ( b ) − F X (

1716-510: The CDF of X {\displaystyle X} is given by F X ( x ) = { 0 :   x < 0 x :   0 ≤ x ≤ 1 1 :   x > 1 {\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ x>1\end{cases}}} Suppose instead that X {\displaystyle X} takes only

1768-621: The CDF of X {\displaystyle X} will be discontinuous at the points x i {\displaystyle x_{i}} : F X ( x ) = P ⁡ ( X ≤ x ) = ∑ x i ≤ x P ⁡ ( X = x i ) = ∑ x i ≤ x p ( x i ) . {\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i}).} If

1820-514: The above four properties is a multivariate CDF, unlike in the single dimension case. For example, let F ( x , y ) = 0 {\displaystyle F(x,y)=0} for x < 0 {\displaystyle x<0} or x + y < 1 {\displaystyle x+y<1} or y < 0 {\displaystyle y<0} and let F ( x , y ) = 1 {\displaystyle F(x,y)=1} otherwise. It

1872-534: The case of a random variable X {\displaystyle X} which has distribution having a discrete component at a value b {\displaystyle b} , P ⁡ ( X = b ) = F X ( b ) − lim x → b − F X ( x ) . {\displaystyle \operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x).} If F X {\displaystyle F_{X}}

1924-403: The case of a scalar continuous distribution , it gives the area under the probability density function from negative infinity to x {\displaystyle x} . Cumulative distribution functions are also used to specify the distribution of multivariate random variables . The cumulative distribution function of a real-valued random variable X {\displaystyle X}

1976-577: The diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of F X {\displaystyle F_{X}} ). In particular, we have lim x → − ∞ x F X ( x ) = 0 , lim x → + ∞ x ( 1 − F X ( x ) ) = 0. {\displaystyle \lim _{x\to -\infty }xF_{X}(x)=0,\quad \lim _{x\to +\infty }x(1-F_{X}(x))=0.} In addition,

2028-554: The discrete values 0 and 1, with equal probability. Then the CDF of X {\displaystyle X} is given by F X ( x ) = { 0 :   x < 0 1 / 2 :   0 ≤ x < 1 1 :   x ≥ 1 {\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\1/2&:\ 0\leq x<1\\1&:\ x\geq 1\end{cases}}} Suppose X {\displaystyle X}

2080-437: The distribution or of the empirical results. If the CDF F is strictly increasing and continuous then F − 1 ( p ) , p ∈ [ 0 , 1 ] , {\displaystyle F^{-1}(p),p\in [0,1],} is the unique real number x {\displaystyle x} such that F ( x ) = p {\displaystyle F(x)=p} . This defines

2132-413: The graph over, that is where 1 { A } {\displaystyle 1_{\{A\}}} denotes the indicator function and the second summand is the survivor function , thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median , dispersion (specifically, the mean absolute deviation from the median ) and skewness of

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2184-404: 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

2236-423: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=S_Curve&oldid=1193841392 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Sigmoid function A sigmoid function is a function whose graph follows the logistic function . It

2288-444: The joint CDF F X 1 , … , X N {\displaystyle F_{X_{1},\ldots ,X_{N}}} is given by Interpreting the N {\displaystyle N} random variables as a random vector X = ( X 1 , … , X N ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}} yields

2340-568: The joint CDF F X Y {\displaystyle F_{XY}} is given by where the right-hand side represents the probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} and that Y {\displaystyle Y} takes on a value less than or equal to y {\displaystyle y} . Example of joint cumulative distribution function: For two continuous variables X and Y : Pr (

2392-423: 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

2444-749: The one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic , T , has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value t {\displaystyle t} of the test statistic p = P ⁡ ( T ≥ t ) = P ⁡ ( T > t ) = 1 − F T ( t ) . {\displaystyle p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).} In survival analysis , F ¯ X ( x ) {\displaystyle {\bar {F}}_{X}(x)}

2496-449: The parameter μ {\displaystyle \mu } is the mean or expectation of the distribution; and σ {\displaystyle \sigma } is its standard deviation. A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the standard normal table , the unit normal table , or the Z table . Suppose X {\displaystyle X}

2548-478: The points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function. When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X , Y {\displaystyle X,Y} ,

2600-639: The real numbers, discrete or "mixed" as well as continuous , is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) F : R → [ 0 , 1 ] {\displaystyle F\colon \mathbb {R} \rightarrow [0,1]} satisfying lim x → − ∞ F ( x ) = 0 {\displaystyle \lim _{x\rightarrow -\infty }F(x)=0} and lim x → ∞ F ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }F(x)=1} . In

2652-440: 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 . Cumulative distribution function Every probability distribution supported on

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2704-555: 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

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