Stochastic

It is called stochastic (from the Latin stochasticus, which in turn comes from the Greek στοχαστικός stochastikós "skillful in conjecturing") to the system whose intrinsic behavior is non-deterministic. A stochastic process is one whose behavior is non-deterministic, insofar as the subsequent state of the system is determined both by the predictable actions of the process and by random elements. However, according to Mark Kac and E. Nelson, any temporal development (whether deterministic or essentially probabilistic) that can be analyzed in terms of probability deserves the description of a stochastic process. Specifically: the term stochastic applies to processes, algorithms, and models in which there is a changing sequence of probabilistically analyzable events as time passes.

Origin

The use of the term stochastic to refer to something based on probability theory can be traced back in time to Ladislaus Bortkiewicz, who gave it the meaning of "make conjectures", and which carries the Greek term from the ancient philosophers, from the title Ars Conjectandi that Jakob Bernoulli gave to his work on probability theory.

In mathematics, specifically in probability theory, the field of stochastic processes has been an important area of research. Although in many probabilistic problems fixed random variables are used, in stochastic processes it is necessary to specify the random variables for each instant of time.

A stochastic matrix is a matrix that has nonnegative real values that sum to one in each column.

Applications

Natural Sciences

An example of a stochastic process in the natural world is the pressure in a gas as modeled in a Wiener process. Even though (in classical terms) each molecule moves following a deterministic pattern, the movement of a set of them is informatics and practically unpredictable. A sufficiently large set of molecules will show stochastic characteristics, such as filling their container, exerting the same pressure, diffusing down concentration gradients, etc. These are emergent properties of systems.

Physics

The name "Monte Carlo" for the stochastic Monte Carlo method was popularized by physics researchers Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, among others. The name refers to the Monte Carlo Casino in Monaco, where Ulam's uncle would borrow money to gamble on. The use of randomness and the repetitive nature of the process are analogous to the activities that take place in a casino.

Biology

In biological systems, it has been found that the introduction of "noise" Stochastic contributes to enhancing the signal strength of internal feedback loops used for balance and other forms of vestibular communication. Multitudes of biochemical events also lead by themselves to stochastic analysis. Gene expression, for example, is a stochastic process resulting from the inherent unpredictability of molecular collisions (eg, RNA polymerase attaching and detaching from a promoter) resulting from Brownian motion.

Geomorphology

Other disciplines

Medicine

The stochastic effect or the "chance effect" is one of the classifications of radiation effects that refers to the random and statistical nature of the damage. Unlike the deterministic effect, the severity is independent of the dose. Only the probability of an effect increases with dose. Cancer is a stochastic effect.

Stochastic differentiation is where a population of stem cells is maintained by the balance between stem cell divisions that generate either two self-renewing stem cells or two that differentiate.

Artificial intelligence

In artificial intelligence, a stochastic program operates using probabilistic methods to solve problems, such as the simulated annealing algorithm, stochastic neural networks, stochastic optimization, genetic algorithms, and genetic programming. A problem may itself be stochastic, such as when planning under uncertainty.

Music

In music, stochastic elements can be generated by mathematical processes. It is feasible to use stochastic processes to compose fixed pieces, which can be produced in one performance.

One of the precursors of stochastic music is Iannis Xenakis, who used probability, game theory, group theory, set theory and Boolean algebra to compose. He also frequently used computers to produce his work. Previously, John Cage and others had composed random music, or indeterminate music, which is created by random processes but lacks a strict mathematical basis.

Colour reproduction

When performing color reproduction, the image is separated into its shaping colors by taking multiple photographs filtered for each color. The resulting movie represents the different data for cyan, magenta, yellow, and black. Color printing is a binary system, where ink may or may not be present, so all color separations to be printed must be dot-transferable at some stage in the work process. Traditional screen lines, which are amplitude modulated, suffer from problems with the Moiré pattern, but they continued to be used until the stochastic process known as stochastic screening appeared. A stochastic (or frequency modulated) dot process creates a higher sharpness image.

Language and linguistics

Non-deterministic approaches in language studies are mainly inspired by the work of Ferdinand de Saussure. In usage-based linguistic theories, for example, where it is argued that competence, or speaking, is based on performance, in the sense of linguistic knowledge being based on the frequency of experience, it is often said that the Grammar is probabilistic and variable rather than fixed and absolute. This is so, because a person's competence can change according to their own experience in linguistic terms. In this way, the frequency of use events determines the knowledge that an individual has of the language in question.

Social sciences and economics

Stochastic social science theory is similar to systems theory in that events are interactions of systems, albeit with a strong emphasis on unconscious processes. The event creates its own conditions of possibility, making it unpredictable for the variables that participate in it. Stochastic social science theory can be seen as an elaboration of a kind of "third axis" in which human behavior can be placed in line with the traditional opposition between "nature vs care".

Industry

It is assumed that industrial processes are stochastic processes. This assumption is generally valid for both continuous and batch industrial processes. Process testing and monitoring is recorded using a process control map that traces a control process parameter over time. Typically a dozen or more parameters will be monitored simultaneously. Statistical models are used to define boundary lines that establish when corrective actions must be taken to bring the process towards its intentional operating window.

Finances

Financial markets use stochastic models to represent the seemingly random behavior of assets such as securities, commodities, and interest rates. These models are used, then, by quantitative analysts to assess the price of securities, fixed income titles or interest rates. These techniques are equally used in the insurance industry.