Stochastic is a term used synonymously with the term random. In statistics, stochastic models are applied when random variations in data are expected, observed, or unable to be ruled out. Stochastic models are the counterpart to deterministic models where randomness is not involved.

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The canonical example of stochastic is rolling a dice: there are a known number of possible outcomes but for any single outcome there is a degree of randomness. Other examples include the growth rates of bacteria, random walk simulations, molecular movements in liquids, and many other techniques of modeling natural phenomena (Spanos, 1999).

## Examples of Stochastic Processes

Stochastic processes are those that involve a degree of randomness. Many statistical modeling techniques account for randomness by either anticipating the observation of random natural phenomena or by attempting to simulate them. Common examples of Stochastic processes are as follows:

- Rolling Dice
- Monte Carlo Simulations
- Brownian fluids
- Random Walk algorithms
- Price action in stock markets

These are but a few of the thousands of applications, domains, and observed cases where stochastic are known to be present.

## Stochastic vs. Random

The terms stochastic and random are often used synonymously and ultimately mean the same thing. Stochastic tends to be the more formal term one expects to find cited among more scientific or formal texts.

For example, stochastic modeling is highly utilized within the field of machine learning. Traditional statistics will also use the term stochastic but will also offer more informal nomenclature such as *random* sampling.

The term *stochastic* is often used to indicate **natural randomness** that is being accounted for whereas the term *random* is more generalized for **artificial randomness** that is being introduced to account for natural phenomena.

In either case, stochastic or random are commonly used to describe the lack of dependence among model features and predictor variables (independent variables) over a period of time. Modeling techniques such as correlation analysis and autocorrelation can help identify cases where variable independence is present.

## Applications

Stochastic modeling is applied in many fields where randomness is expected or, in some cases, even preferred. Weather modeling is one case where the purposeful incorporation of randomness in data is deemed to increase model accuracy. Other fields of study incorporate stochastic modeling to account for the randomness that may, or may not, be anticipated. Below are some common fields where stochastic modeling is commonly applied.

### Earth Science

Stochastic models can be applied to generate random weather simulation data to create advanced forecasts, early warning systems, and more accurate predictive measures. Earth science modeling often relies on time-series data where the incorporation of randomness helps better model probabilistic fluctuations and uncertainty in actual weather conditions (Breinl, 2017).

### Medicine

Stochastic modeling is used in medical and clinical scenarios to account for the random variability in a range of observational studies. For example, stochastic modeling can describe random fluctuations in outcome measures in response to alterations in dose within pharmacokinetic observations (Leenhouts, 1989).

### Epidemiology

Stochastic models are used to model the infection, transmission, and spread of pathogens on epidemic levels. These models are particularly useful when populations are small or variability in transmission, environmental conditions, recovery rates, age-related factors, and other demographic factors are likely to be highly variable (Allen, 2017).

## References

- Leenhouts, H P, and K H Chadwick. “The molecular basis of stochastic and nonstochastic effects.”
*Health physics*vol. 57 Suppl 1 (1989): 343-8. doi:10.1097/00004032-198907001-00048 - Breinl, Korbinian et al. “Can weather generation capture precipitation patterns across different climates, spatial scales and under data scarcity?.”
*Scientific reports*vol. 7,1 5449. 14 Jul. 2017, doi:10.1038/s41598-017-05822-y - Allen, Linda J S. “A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis.”
*Infectious Disease Modelling*vol. 2,2 128-142. 11 Mar. 2017, doi:10.1016/j.idm.2017.03.001 - Spanos, Aris.
*Probability Theory and Statistical Inference: Econometric Modeling with Observational Data*. Cambridge University Press, 1999.