Deterministic vs Stochastic Machine Learning

In machine learning, both deterministic and stochastic methods are used in different sectors depending on their usefulness. A deterministic process considers that known average rates without random deviations are applied to huge populations. A stochastic process, on the other hand, defines a collection of time-ordered random variables that reflect potential sampling paths. In this article, we will discuss the main differences between how they work and their applications. The main points discussed in this article are described below.

Contents

  1. Deterministic and stochastic process modeling
  2. When could they both be used?
  3. How do these approaches work?
  4. Different forms of stochastic and deterministic algorithms
  5. Advantages and disadvantages of deterministic and stochastic
  6. Applications of deterministic and stochastic algorithms

Let’s start with a high-level overview of deterministic and stochastic processes.

Deterministic and stochastic process modeling

Deterministic modeling produces consistent results for a given set of inputs, no matter how many times the model is recalculated. The mathematical characteristics are known in this case. None of them are random and each problem has only one set of specified values ​​along with an answer or solution. The unknown components of a deterministic model are external to the model. It deals with definitive results as opposed to random results and does not take error into account.

In contrast, stochastic modeling is inherently unpredictable and unknown components are built into the model. The model generates a large number of answers, guesses, and results, much like adding variables to a difficult math problem to see how they affect the solution. The identical procedure is then repeated several times in different contexts.

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When could they both be used?

A deterministic model is applied where outcomes are precisely determined through a known relationship between states and events where there is no randomness or uncertainty.

For example, if we know that consuming a fixed amount of “y” sugar will increase the fat in one’s body by “2 times”. So ‘y’ can always be determined exactly when the value of ‘x’ is known.

Similarly, when the relationship between variables is unknown or uncertain, stochastic modeling can be used because it relies on estimating the likelihood of the likelihood of events.

For example, the insurance industry depends primarily on stochastic modeling to predict how corporate balance sheets will look in the future.

How do these approaches work?

As deterministic models show the relationship between the results and the factors affecting the results. For this type of model, the relationship between the variables must be known or determined.

Consider building machine learning that can help an athlete in a 100 meter sprint, the most important factor in the 100 meter sprint is time. The goal of the model would be to minimize the athlete’s time. The two most important factors affecting time are speed and distance.

The distance covered by each athlete is the same, it is constant for everyone, the only thing that varies is the speed. But the variable speed could be controlled because the factors affecting the speed are known like body position, flight time, etc. Since we know that time depends on speed and distance, this makes this problem deterministic.

The stochastic aspect of machine learning algorithms is most evident in the complicated, nonlinear approaches used to solve classification and predictive regression modeling problems. These methods use randomization in the process of building a model from the training data, which results in a different model fit each time the same algorithm is run on the same data.

Therefore, when tested on an exclusion test dataset, slightly modified models perform differently. Because of this stochastic behavior, model performance should be described using summary statistics that indicate average or predicted model performance rather than model performance from a single training session.

Consider a dice rolling problem. You roll a dice in a casino. If you roll a six or a one, you win the prize money. Initially, a sample space that includes all possible die roll results will be generated. The probability for any number to be rolled is calculated, i.e. ‘0.17’. But we are only interested in two numbers, ‘6’ and ‘1’. The final probability would therefore be 0.33. This is how a stochastic model would work.

Let’s see how a linear regression model can work as both a deterministic and a stochastic model in different scenarios.

Deterministic models define a precise link between variables. In the deterministic scenario, the linear regression has three components. The dependent variable ‘y’, the independent variable ‘x’ and the y-intercept ‘c’. There is no room for errors in the prediction of y for a given x. Here is an example equation to replicate the above explanation.

F=95C+32

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The equation above would have a graph something like this with all the data points in a straight line.

A stochastic model that takes into account random error. There is a deterministic component as well as a random error component. A probabilistic link between y and x is assumed in this paradigm. Here is an example equation to replicate the above explanation.

y= 1.5x+error

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In the graph above, it can be observed that due to the error component in the linear regression equation, there is randomness in the data.

Different forms of stochastic and deterministic algorithms

Principal component analysis (PCA)

PCA is a deterministic approach because there are no parameters to initialize. PCA finds the line through the centroid with the smallest sum of squared distances between points given a set of points in n-dimensional space. Identifying the line for which the projections of the points on that line are as large as possible is the same thing (measured by the sum of the squared lengths).

Then, subject to being orthogonal to the first line, it finds the line through the centroid with the smallest sum of the squared distances to the points. The third main component, the fourth, and so on. Since all of these procedures are simply geometric, the principal components are deterministic data functions.

Weighted nearest neighbors

A weighted nearest neighbor method could also be called a basic KNN is a deterministic method. This technique uses a statistic known as a “weighing function”. The weight is determined by taking the reciprocal of the distance. Since the distance between each data point and the query point would be the same on each iteration, the weights would be a deterministic term.

Poisson process

The Poisson method is a stochastic process that displays a random number of points or occurrences over time. The number of points in a process that falls between zero and a specific period is characterized as a time-dependent Poisson random variable. The index set of this process is composed of non-negative integers, while the state space is composed of natural numbers. This approach is known as the Poisson counting process because it can be thought of as a counting operation.

Bernoulli process

Bernoulli’s process is a randomly distributed set of random variables, each with a chance of one or zero. This procedure is analogous to continuously tossing a coin, the probability of winning being p and the value being one, and the probability of getting a tail being zero. As the result is probabilistic, this is the reason why this method is a stochastic process.

Random Walk

The simple random walk is a discrete-time stochastic process using integers as its state space based on a Bernoulli process, with each Bernoulli variable taking a positive or negative value.

Advantages and disadvantages of deterministic and stochastic

Let’s take a look at the advantages and disadvantages of these two methods.

Advantages

  • Deterministic models have the advantage of being simple.
  • The deterministic is easier to grasp and may therefore be more suitable for certain cases.
  • Stochastic models provide a variety of possible outcomes and the relative probability of each.
  • The stochastic model uses the most common approach to get the results.

Disadvantages

  • In the deterministic approach, there are no cumulative probabilities due to which the low reserve cases are too optimistic.
  • In stochastic approach, the model is more complex, also called black box approach.
  • Biases can be hidden in the stochastic model and it focuses on the extremes.

Applications of deterministic and stochastic algorithms

  • Deterministic models are used in flood risk analysis.
  • The deterministic model used in the Turing machine is a machine (automaton) capable of enumerating any arbitrary subset of acceptable alphabetic strings; these strings are part of a recursively enumerable set. A Turing machine has an infinitely long tape on which to perform read and write operations.
  • Stochastic investment models aim to estimate changes in prices, returns on assets (ROA), and asset classes (such as bonds and stocks) over time. It uses Monte Carlo simulation, which can simulate the performance of a portfolio based on the probability distributions of individual stock returns.
  • Stochastic modeling influences marketing and the evolution of audience tastes and preferences, as well as the solicitation and scientific appeal of specific cinematic cameos (i.e. name recognition and other broadcast and advertising elements on social networks).

Conclusion

A deterministic approach has a simple and understandable structure that can only be applied when the relationship between variables is determined; on the other hand, a stochastic approach has a complex and incomprehensible structure that works on the likelihood of probabilities. With this article, we understood the difference between deterministic and stochastic approaches in machine learning.

References

Sherry J. Basler