Random Walk: Will the Drunk Man Fall Off the Cliff?
By: Aakash Jhawar
Tags: Statistics, Gaussian Distribution, Random, Data Science, Random Walk

The Random Walk theory is based on the irregular motion of the individual pollen particles, studied by botanist, Mr. Robert Brown in 1828. In the process of researching on a random walk, scientists like Einstein and Smoluchowski studied similar subjects like random process, random noise, spectral analysis, and stochastic equations. The first simple model of Random Walk proposed was uncorrelated and unbiased.

Uncorrelated means the direction of movement is completely independent of the previous directions taken.

means the direction of movement is completely independent of the previous directions taken. Unbiased means there is no preferred direction, the direction moved at each step is completely random.

“A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.” (Source: Wikipedia)

It is a problem, which is closely related to Brownian motion.

Types of Random Walks

1. Correlated Random Walks (CRWs)

It involves a correlation between successive step orientations. This correlation is termed as Persistence. This produces a local directional bias, each step tends to point in the same direction as the previous one, although the influence of the initial direction of motion progressively diminishes over time and step, orientations are uniformly distributed in the long term. The nature of the motion of animals is similar, hence, CRWs have been constantly used to monitor motion paths of animals in various contexts.

2. Biased Random Walks (BRWs)

When the probability of traveling in a certain direction is greater, i.e. all directions are not equally likely, then such Random Walks are known as Biased Random walks.

Possible Factors of Bias :

Fixed external environmental factors such, as a movement under gyrotaxis.

Spatially varying factors, such as chemical gradients.

Mean-reversion mechanisms, such as movement within a home range.

Choice of direction by individuals at each step.

In BRWs, target direction and strength of bias vary with location and time.

3. General Random Walk & The Levy Walk

Generally, while considering the results from a random walk theory, we consider the step to be both fixed or variable. In variable lengths, those whose step lengths have fixed variance are considered.

Levy walk: Distribution of step lengths is heavy-tailed, i.e. has an infinite variance. In this case, the walk exhibits scale-invariant (i.e. fractal) characteristics. Many research papers suggest that the Levy model provides a suitable model for animal movements.

4. Simple Isotropic Random Walk

Let’s take a case where the walker is moving on an infinite one-dimensional uniform lattice. The motion is completely random. So the probability of moving either left or right is equal, i.e., ½. After one time step, the walker can either be at a distance 𝛿 to the left or to the right of the origin, with probability ½ each.

After the next time step, the walker will either be at a distance 2𝛿 to the left or right of the origin (with probability ¼ each) or will have returned to the origin (with probability ½). After an even (odd) number of steps, the walker can only be at an even (odd) distance away from the origin.

5. Analogy- Random Walker

A man starts from a point 0 and walks ‘x’ yards in a straight line. He then turns through any finite angle and walks another ‘x’ yards in a straight line. He repeats this process n times. We require the probability that after these n stretches he is at a distance between r and r + δr from his starting point 0.

The probability that a walker will be at a distance m to the right of the origin after n time steps (where m and n are even) is given by :

This is a form of the binomial distribution, with mean 0 and variance n. For large n, this converges to a normal (or Gaussian) distribution so, after a sufficiently large amount of time t = nτ, the location x = mδ of the walker is normally distributed with mean 0.