Talk:Lévy flight

The name Lévy

Is Lévy name of a person? Jay 04:19, 24 Mar 2004 (UTC)

Anyone would kindly illustrate the difference and relationship between Lévy flight and Power law? Thanks.

Seems clear to me: a "power law" is a probability distribution; a Lévy flight is a stochastic process, having a probability distribution at each point in time. Doesn't the article make that clear? Michael Hardy 22:10, 15 Mar 2005 (UTC)

There are certainly some inaccuracies in this article, unfortunately I dont know enough about Levy flights to correct them. For example take the paragraph:

"According to the central limit theorem, if the distribution were to have a finite variance, then after a large number of steps, the distance from the origin of the random walk would tend to a normal distribution. (This type of random walk is also known as Brownian motion)."

1. The distance (euclidian I suppose) from the origin of a stochastic process is always positive, which means it can not be normally distributed (except for the trivial/pathological case that the process stays at the origin forever). Maybe what is meant is that the projections of the process on its coordinates are normally distributed (or in other words the process is asymptotically Gaussian)

2. Normal distribution alone is not sufficient for a stochastic Process to make it a Brownian Motion.

I agree. In a Brownian motion, the increments even over short time intervals are normally distributed; the central limit theorem would apply only to sufficiently long intervals. Michael Hardy 22:00, 14 Jun 2005 (UTC)

How about this: "According to the central limit theorem, if the distribution were to have a finite variance, then after a large number of steps, the distance from the origin of the random walk would tend to a xxx distribution. (This type of random walk is also known as Brownian motion)."

I don't know what xxx should be. Its a bivariate normal distribution but with independent components. PAR 04:46, 15 Jun 2005 (UTC)

Yes, with the distance it is meant the projection to the y-axis. It must also be included that it has to be properly normalized. This convergence is Donskers Invariance Theorem. An advanced result in probability theory, which gives weak convergence in the Skorokhod-topology for a normalized sum to a Brownian-motion process. However, there does not exist any wikipedia-articles on any of these subjects. Billingsley's "Convergence of Probability Measures" is the classic in this field for those who want to check it out. --Steffen Grønneberg 19:17, 15 May 2006 (UTC)

"Distance" should probably read "Displacement" -- i.e. a vector (not Euclidean distance) which can indeed follow a Multivariate normal distribution, as I understand it. Equating that with Brownian motion should be more carefully worded. AFAIK Brownian motion results in normally distributed displacement, but Brownian motion means a specific process, whereas there could be other processes that would result in normal distribution, that aren't brownian. 188.221.54.25 (talk) 23:03, 1 June 2010 (UTC)

Hrm, actually, not sure if displacement or position, but either would be better than distance 188.221.54.25 (talk) 23:13, 1 June 2010 (UTC)

Too technical

Someone who can translate this article into layman's terms ought to have a crack at revising this article. —thames 18:42, 6 December 2005 (UTC)

To help someone achieve that, can you explain which parts of the article you found difficult to understand ? Have you read the related articles on random walks, probability distributions and the Lévy distribution ? Do these help you understand the Lévy flight article ? Gandalf61 09:17, 7 December 2005 (UTC)

No response to the above questions after waiting for a week, so I have removed the technical tag. Gandalf61 15:50, 14 December 2005 (UTC)

Hah! Too technical? Not even remotely technical enough. It's a stub at best. I'd recommend it for deletion if the plots were not marginally informative. The alpha and beta captions in the left-hand plot are so opaque as to be meaningless. Cauchy goes like 1/(1 + x^2). The overall low quality of this article is shown by the fact there's not even a link to the main article on the Levy distribution itself Daggilli (talk) 04:08, 30 June 2011 (UTC)

Biased toward axes?

It seems that the Lévy flight given in the example is biased toward going large distances along either the x-axis or the y-axis. Would it not be more accurate to have each step of the Lévy flight going in a randomly selected direction? --Zemyla^t 16:55, 25 May 2006 (UTC)

I agree. The biasing is caused by choosing the x and y increments independently, as stated in the figure caption. Large steps are rare, and it is even rarer that they occur simultaneously both along x and y. Thus polar coordinates should be used instead, with a random uniform distributed angle, and a random Lévy distributed modulus. GuidoGer 11:55, 19 September 2006 (UTC)

Lévy Dust and the Interval Distribution

Does the probability distribution of the interval $\left|x\right|^{-\alpha }$ have a well-known name? I haven't been able to find one, but would think this would be a "standard" definition. If so, we should name it and link to it.

My understanding is that the stopping points on the Lévy flight, in composite, form a Lévy dust. The comment has been made that a 2-D flight created from two orthogonal and independent identically distributed random variables is not visually satisfying, and instead should be expressed in polar coordinates as a uniform RV in angle and a Lévy distribution in relative range. Probably the flight is statistically different, but is the dust? I think that comparing these two techniques for producing a Lévy flight is quite illuminating, as it expresses the Lévy-distributed character (and therefore explains the name) at the same time showing that in dimensions greater than 1 the decomposition of the intervals is not also Lévy. --Phays 01:38, 28 June 2007 (UTC)

Question re search methods

Is this topic related to search methods in AI/ CS, particularly hill climbing, and particularly to the problem of avoiding local maxima? Or simulated annealing? Mcswell 21:27, 24 October 2007 (UTC)

Yes, since by following a Levy flight there is more probability that you will search part of the parameter space that is very far away. If your search followed Brownian motion, the chance of looking far away from your current position on the next step is exponentially small. —Preceding unsigned comment added by 131.215.100.230 (talk) 23:47, 13 March 2009 (UTC)

This article doesn't make sense.

Early in the article it says:

Specifically, the distribution used is a power law of the form y = x^−α where 1 < α < 3 and therefore has an infinite variance.

Then later it says:

Lévy (i.e. stable) distribution with α = 1 and β = 0 (i.e. a Cauchy distribution):

and then:

a Lévy (i.e. stable) distribution with α = 2 and β = 0 (i.e., a normal distribution).

There's no beta in the power law mentioned originally, and how in Hell either the Cauchy distribution or the normal distribution can be described as a "power law of the form y = x^−α" is not explained.

A precise mathematical definition is needed and is not there.

Are people who write such absurdities sober when they edit Wikipedia? Michael Hardy (talk) 00:58, 1 July 2011 (UTC)

Who knows. Improved now I hope. Melcombe (talk) 13:33, 3 August 2011 (UTC)

Proposed merge with Lévy walk

Levy flight and Levy walk mean the same. Per [1]. Levy walk is a plausible redirect for Levy flight. Harsh (talk) 12:09, 2 December 2014 (UTC)

I am proceeding with a blank-and-redirect of Lévy walk to here per WP:BLAR, since the content isn't worthy to be merged and the title of the two pages are closely related words. Harsh (talk) 17:10, 6 December 2014 (UTC)

I have a question: Are Levy walks and Levy walks exactly the same? If so, this should be stated in the first paragraph to help the reader. Some authors do make a distinction, e.g. here: http://link.springer.com/chapter/10.1007/978-94-009-5165-5_29. How relevant is this? --141.5.11.5 (talk) 11:44, 10 February 2015 (UTC)

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This article reads poorly.

Lévy Flight : Random walk trajectories which are composed of self-similar jumps. They are described by the Lévy distribution.

Random Walk : A random process consisting of a sequence of discrete steps of fixed length. The random thermal perturbations in a liquid are responsible for a random walk phenomenon known as Brownian motion, and the collisions of molecules in a gas are a random walk responsible for diffusion. Random walks have interesting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice.

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. All Lévy processes are additive processes.

This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). 49.183.158.187 (talk) 02:34, 13 May 2022 (UTC)