Stochastic point processes and their practical value

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Poisson binomial point processes are gaining significant momentum as their real-world applications are numerous.

Point processes map a collection of data points, sometimes called events, that occur over a period of time. When collections of random variables model events that show the evolution of a given system over time, they are referred to as stochastic point processes.

Author Vincent Granville, a data scientist and machine learning expert who co-founded Data Science Central (acquired by TechTarget in 2020), wrote Stochastic processes and simulations serve as an introductory course in stochastic processes. It covers more areas than traditional college courses or textbooks would normally do. His approach is to introduce a new but intuitive type of random structure for modeling mathematical points, called the Poisson binomial process.

Looking at the distinction between binomial and Poisson process examples will allow readers to better understand Poisson binomial, which he uses as a gateway to understanding all stochastic point processes. For example, a binomial distribution is used to model the probability of the number of successes we can expect from a given number of independent and identical success/failure trials (known as Bernoulli trials). A Poisson distribution, in this context, is attached to the number of points found in any region – say a square – assuming independence between non-overlapping regions.

In the case of Poisson binomial processes, however, the scores are location-dependent and are therefore neither identically distributed nor independent. So they are distinct from binomial and Poisson, despite the other similarities discussed in this book. The name Poisson binomial has historical connotations and generalizes the classic Poisson process: the score distribution is not Poisson but Poisson binomial.

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They are particularly useful for cellular networks, chemistry, physics, engineering applications, sensor analysis, and sensor network modeling and optimization. Granville explains that they are also known as perturbed lattice point processes, which he believes is a more modern name that emphasizes these real-world applications as well as their topological features.

The first section of the book introduces the key concepts a data scientist needs to understand the formulas involved in Poisson binomial processes. The second section explains how to use them. For example, subsection 2.1.2 presents them with illustrations, and section 2.4 teaches readers how to create data videos and illustrates several topics within the book.

Later sections delve further into the methods and principles with numerous resources including: theorems, source code for developers, original exercises, videos, references and a glossary. Anyone interested in reading more about stochastic point processes can do so Click here to download a sample chapter.

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