mathematics

Sigmoid function

Sigmoid function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sigmoid function rather than just read about it. In short: A 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.

Key takeaways

Sigmoid function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sigmoid function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sigmoid function from memory before moving on to harder problems.

Reference excerpt

A 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". Special cases of sigmoid functions 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. There is also the Heaviside step function, which instantaneously transitions between 0 and 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function.

Theory In mathematics, a unitary sigmoid function is a bounded sigmoid-type function normalized to the unit range, typically with lower and upper asymptotes at 0 and 1. The theory proposed by Grebenc distinguishes three kinds of unitary sigmoid functions according to their asymptotic behavior and the presence or absence of oscillation near the asymptotes. A general form of a unitary sigmoid function is

y = A S ( f ( x ) ) + B , {\displaystyle y=A\,S(f(x))+B,}

where S {\displaystyle S} is an increasing sigmoid function, f ( x ) {\displaystyle f(x)} is a transformation of the independent variable, and A {\displaystyle A} and B {\displaystyle B} are constants controlling scaling and translation.

Classification

1st kind A unitary sigmoid function of the first kind is a bounded increasing function that approaches its lower and upper asymptotes monotonically, without oscillation. This class includes many of the standard sigmoid functions used in statistics, biomathematics, and engineering, such as the logistic function and related generalizations.

2nd kind A unitary sigmoid function of the second kind is a bounded increasing function that oscillates near the upper asymptote while preserving an overall sigmoid transition.

3rd kind A unitary sigmoid function of the third kind is a bounded increasing function that oscillates near both the lower and upper asymptotes. These functions retain the global shape of a sigmoid curve but exhibit oscillatory behavior in the vicinity of both limiting states.

Taxonomy The tables below show the taxonomy of unitary sigmoid functions of all three kinds. Table 1. Taxonomy matrix with examples of sigmoid functions of the 1st kind

Table 2. Taxonomy matrix with examples of sigmoid functions of the 2nd kind on the unbounded interval

Table 3. Taxonomy matrix with examples of sigmoid functions of the 3rd kind

Construction methods The same theory presents a list of 30 methods for constructing sigmoid functions.. These include algebraic transformations, integration and convolution methods, constructions from bell-shaped functions, solutions of ordinary and partial differential equations, recursive schemes, stochastic differential equations, feedback systems, and chaotic systems.

M0: Construction method for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a sigmoid function to the power M4: Exponentiating a sigmoid function M5: Symmetric sigmoid functions derived from asymmetric ones M6: Sigmoid functions of the reciprocal independent variable M7: Embedding a sigmoid function into other function M8: Sum of sigmoid functions M9: Multiplication of sigmoid functions M10: Integral of the product of an increasing and a decreasing function M11: Derivation from lambda (bell-shaped) functions M12: Integration of lambda (bell-shaped) function M13: Integration of the sum of lambda (bell-shaped) functions M14: Integration of the product of two lambda (bell-shaped) functions M15: Integration of the difference of two shifted sigmoid functions M16: Integration of the product of two shifted sigmoid functions M17: Convolution of sigmoid functions M18: Integration of the product of lambda and sigmoid function M19: Solutions of ordinary differential equations M20: Solutions of partial differential equation (PDE) M21: Solutions of functional differential equation (FDE) M22: Sum of a sigmoid function and some derivatives M23: Combination of sigmoid functions, its derivative and integral M24: Filtering sigmoid functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward loops M29: Solutions of stochastic differential equation M30: Chaotic sigmoid functions Consult reference for more details.

Definition A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point.

… excerpt ends here. Continue reading the full article.

Illustrations

Sigmoid function: Fig. 1. Graphical representation of the three kinds of sigmoid functions on the infinite domain: the 1st kind (red solid line), the 2nd kind (violet dashed line), the 3rd kind (blue dotted line). — Fig. 1. Graphical representation of the three kinds of sigmoid functions on the infinite domain: the 1st kind (red solid line), the 2nd kind (violet dashed line), the 3rd kind (blue dotted line).

Worked examples

Example 1 — a first encounter with Sigmoid function

Start with the simplest possible case. Write down what Sigmoid function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sigmoid function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sigmoid function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sigmoid function

In research: Sigmoid function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sigmoid function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sigmoid function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Artificial neural networks, Sigmoid functions, so understanding it makes those chapters shorter.
In everyday life: Look for Sigmoid function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sigmoid function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sigmoid function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sigmoid function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sigmoid function in simple terms?

A 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.

Why does Sigmoid function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sigmoid function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sigmoid function.