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Sigmoid function: What is it? What’s it for?

An essential concept in mathematics, the Sigmoid function applies perfectly to neural networks. Find out more about this function, its different phases and applications.

What is the Sigmoid function?

Also known as the S-curve, the Sigmoid function is expressed as follows:

$f\left(x\right)=\frac{1}{1+{e}^{-\lambda x}}$

The visual representation of the curve is characterized by an exponential increase in one variable, which then becomes linear, then asymptotic. The whole thing then forms a kind of S. Hence the name of the sigmoid function.

What are the phases of the Sigmoid curve?

The S-curve is visually characterized by 4 very distinct phases. These are shown below.

The offset phase

Initially, the curve grows relatively slowly. This is the latency period.

This can be seen, for example, in market research for the launch of a new product. At first, few buyers dare to take the plunge and acquire a product that is still unknown.

The exponential phase

Gradually, the curve begins to increase until it reaches exponential growth.

Again, this type of curve can be found in market research. It’s only when the first buyers have tested the product or service that the majority of consumers are ready to take the plunge. The snowball effect is set in motion. Customer satisfaction and word-of-mouth help organizations acquire more and more new customers. The result is exponential growth.

The transitional phase

After exponential growth, the Sigmoid function is characterized by a transitional phase in which the curve slows down, rises or falls slightly. Small variations are then observed.

In the context of our market research, this refers to the period when the majority of potential customers have already been targeted. Here, there aren’t many new leads/prospects left. At the same time, the first signs of customer dissatisfaction can appear. This can lead to a slowdown or even a drop in sales. But this doesn’t happen automatically.

And finally, the Sigmoid curve stabilizes until it resembles a plateau. All values remain more or less identical.

In the commercial world, this applies to well-established companies that have acquired a stable customer base.

Good to know: beyond market research, the Sigmoid function can be observed in a wide variety of cases, such as contagions that turn into epidemics (covid crisis), an individual’s height or weight over the course of a lifetime, not to mention neural networks.

What's the link between the Sigmoid function and machine learning?

The Sigmoid function is often used in neural networks. It acts as an activation function. More precisely, it enables the network to produce a result based on the available data.

So, when input data enters the neural network, each neuron applies a transformation to it. This enables them to produce a new result.

But with the Sigmoid function, the result is non-linear. The spatial representation of the data is therefore modified. In other words, the neural network is able to explore the data from different angles. These different perspectives enable us to better understand the data, its meaning and to arrive at the most relevant result.

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Like many mathematical and statistical formulas, the Sigmoid function is indispensable for creating machine learning models. But mastering mathematics is not enough to design and use neural networks. You also need to know programming languages, machine learning, deep learning, data engineering and more. Datascientest offers training in all these disciplines.

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