Talk:Complete spatial randomness

Expert tag

I have added this tag for two reasons. The context/assumptions are poorly defined ...is it being assumed that the distribution is spatially uniform as nothing here precludes non-uniform distributions (the actual contexts of the equations are poorly specified)? Secondly, if non-uniform distributions are excluded then various tests against a uniform or non-clustered non-hypothesis are available which could be linked or outlined. Melcombe (talk) 13:34, 1 June 2009 (UTC)

The tag was removed after improving the question of whether non-uniform distributions were included. I have reapplied it, as the question of useful information about testing the "complete spatial randomness" model is still open. Melcombe (talk) 09:58, 4 June 2009 (UTC)

I agree.

A poorly written article. LoneRubberDragon (talk) 09:49, 26 February 2010 (UTC)

There is no reference to random numbers articles in end-references, such as the simplest of:

http://en.wikipedia.org/wiki/Statistical_randomness

And "spatial" randomness need not be added (with "spatial"), per se, in this manner, as it is not clear what you are referring to, other than random numbers on a sample space. And CLEARLY a random distribution (statistical randomness) system will produce a random space on sampling, in zero, one, two, through N dimensions. And "complete" is a dangerous word to use because complete is a mathematical word used of systems, and can easily be confused. Only a subset of people in science math lingustics would ever use that word this way only, without cross references. So complete spatial randomness sounds like a specific concept, that you end up not actually describing, because the language of science and math is very specific. I understand what you mean (and see the redundancy and incompleteness causing ineffectiveness of the article word use), exactly, but the layman reader will often be misled by such articles. LoneRubberDragon (talk) 09:49, 26 February 2010 (UTC)

The dangers of your article dialect reference:

http://en.wikipedia.org/wiki/Completeness

In fact, the standard method of performing spatial randomness tests specifically, is to nyquist-grid the system, and convert it into a line, 2-D image, or N-D space, and then perform Wide Sense Stationary testing, on the partitioned signal. The resolution should be picked for the nyquist criterion of the clustering of data, and have enough spatial sampling space to reach a satisfactory answer stability of randomness (large data set). That grid resolution or larger will all be statistically stable, and grids smaller than the Nyquist Criterion will produce a more delta oriented image of Wide Sense Stationarity test. Numerical testing will illustrate these issues, easily. LoneRubberDragon (talk) 13:05, 26 February 2010 (UTC)

http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem

Additionally, for random textured surfaces, like an image of a cement wall or a uniform lawn or a sandy dune, they can be tested for complete spatial randomness by the same method, where the short order texture nonrandom information structure or character will be stored in the autocorrelation window of near field effect form and accompanying FFT for Power Spectral Density, and beyond that the autocorrelation will be virtually zero, indicating that the spatial pattern reaches statistical randomness, with no correlation present. Follow, for the methods. LoneRubberDragon (talk) 00:02, 28 February 2010 (UTC)

What you refer to specifically, as a possible synonymous industiral term, is thus already covered in Stationary Processes, and covers these Wide Sense Stationary processes (WSS) in cross references, covering 1-D, 2-D, N-D processes. Wide Sense Stationary is implied by stationarity, where there's a mean, and the covariance of WSS is constant for a stationary sequence (Poisson PDF or otherwise), that is, the covariance or standard of deviation measure is also: shift invariant, that is linear space invairant, linear time invairant, or isotropic. You can show a distribution is stationary by a cross-correlation showing shift-invairance, by showing that the cross-correlation, that is specifically the auto-correlatiion function, is shift-invariant. This shift inviariance is shown by an autocorrelation function that drops to zero as displacement increases beyond one sample, showing time invariance in no correlation over time (thus randomness), and a perfect correlation only at time = 0, where the two correlate perfecty on only itself, aligned, for an infinite bandwidth random signal. However, it should be noted the converse of a demonsrated shift-invariant autocorrelation does not Always Imply WSS in certain structured signals, but WSS are always shift invairiant autocorrelations. The autocorrelation not implying randomness can be seen when you consider, for example, a degenerate function of a delta function case where it has autocorrelation at t = 0, and zero autocorrelation everywhere else, and yet it isn't random per-se. And a long random sequence that repeats shows autocorrelations at that period of repeition and zero everywhere else, showing randomness, but on a period, where the autocorrelation function shows the character of randomness bandwidths and periodic structures. A special case of randomness consideration is a set of randomly placed delta functions has a strong correlation at 0, virtually zero correlation at random autocorrelation dispalcements, and zero everywhere else, and yet it isn't very random "looking", having numerous zeros in a row, and randomly distributed ones at the deltas, but remembering that any random probability density function can be used, and knowing that it is a nonuniform density function with, say 95% zeros and 5% ones randomly, one knows that it IS random because the autocorrelation for this nondegenerate case goes to zero correlation. Also, for random signals without infinite frequency bandwidth, that is for band limited random signals, the autocorrelation of a random signal produces a sinc function related to the square window placed on the random signal spectrum, in fourier relationship. And a frequency band limited window autocorrelation is 1 at t=0, and rapidly oscilates exponentially toward zero, as time shifts increase in the autocorrelation, and thus is random proven outside of the bandwidth. They are technically not random signals, even if of white noise, because the neighboring samples have some band limiting relationship to its nearest neighbors with a rapid decline in correlation with distance due to the randomness it contains. The subject also leads to sudies of the autocorrelation Power Spectral Density, closely related through fourier analysis. Anyway, this is (VERY BRIEFLY) a part of what makes a random system describable by mean and standard deviation, irrespective of the time or space of sampling. And in the Buddhist sense, all signals are perfectly spatially random, because on large enough scales, all signals correlate at t = 0 on only themselves, and do not correlate on the infinity of cosmic scales, where no correlation exists except zero correlation, where all samples of data are but a point, except maybe in a cyclic universe model on the repeating cycle of countless trillions of years, random, but with periodic correlation blips on cosmic scales. LoneRubberDragon (talk) 10:12, 26 February 2010 (UTC)

http://en.wikipedia.org/wiki/Wide_sense_stationary

http://en.wikipedia.org/wiki/Stationary_process

http://en.wikipedia.org/wiki/Covariance

http://en.wikipedia.org/wiki/Standard_deviation

http://en.wikipedia.org/wiki/Standard_error_(statistics)

http://en.wikipedia.org/wiki/Time_invariant_system

http://en.wikipedia.org/wiki/LTI_system_theory

http://en.wikipedia.org/wiki/Convergence_of_random_variables

http://en.wikipedia.org/wiki/Proofs_of_convergence_of_random_variables

http://en.wikipedia.org/wiki/Random_variable#Convergence

http://en.wikipedia.org/wiki/Correlation_function

http://en.wikipedia.org/wiki/Autocorrelation

http://en.wikipedia.org/wiki/Ergodic

http://en.wikipedia.org/wiki/Ensemble_average

http://en.wikipedia.org/wiki/Power_spectral_density

http://en.wikipedia.org/wiki/Fourier_transform#Uniform_continuity_and_the_Riemann.E2.80.93Lebesgue_lemma (Sinc and BW window)

http://en.wikipedia.org/wiki/Kronecker_delta

For exmple of applications of analytical methods for detecting WSS spatial randomness, can be found in sub-pixel image processing validation procedures, of your other article, User AI, IF the autocorrelation of pixel intensity, of a perfect flat gray image, is not zero for displacements other than 0,0, as of a perfectly infinite-spectral-bandwidth stationary-random, THEN it indicates either electronic pixel-crosstalk, band limiting, or improper focussing. Such a system would show poor behavior in superresolution / microscan algorithms, due to the attenuation of sub-pixel information, decreasing the SNR, and ruining the superresolution algorithm. Now why does this work? Because photons are randomly distributed points, and are designed by camera-engineers to be binned by the pixel, in a manner that is nyquist satisfied, for performing WSS testing of randomness. LoneRubberDragon (talk) 12:45, 26 February 2010 (UTC)

http://en.wikipedia.org/wiki/Crosstalk

http://en.wikipedia.org/wiki/Band_limited

http://en.wikipedia.org/wiki/Focus_(optics)

But lacking cross references to standard terminology, to link your specific synonymous term, makes it difficult for layman to understand, for synonymous and standard terminology. This is pedagogically suspect. I understand exactly what you mean, but that is just myself, who is a well read "Dragon". Referenceless articles like this actually hurt my Dragon Eyes, because it so closely resembles Stationary Processes, and WSS testing, for its obvious intimate relationship. But for the lack of direct and general linkage, with terms more specific than necessary, and other terms missing! That is why I refer to Microscan and Superresolution, interchangeably, in my comments on your User A1 article, so that the layman can see the university universe of related terms that are in quite-intimate common use, and less common use. LoneRubberDragon (talk) 10:12, 26 February 2010 (UTC)

Also, your bibliography article now doesn't download, when I tested it. LoneRubberDragon (talk) 10:21, 26 February 2010 (UTC)

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA291151&Location=U2&doc=GetTRDoc.pdf

And you should not mention Poisson distributions. Poisson distributions only happen is say quantized 2-D sampling of scalar fields, like in image procssing, as well as other physical quantized processes. However WSS and stationary processes can refer to uniform random distributions, Poisson random distributions, Gaussian random distributions, and any form of random distribution, that is in some sense stationary on 1 or more dimensions of sampling. Poisson distributions, I reiterate, apply specifically to systems like image processing, where pixel sampling of spatially randomly distributed photons on a grid system of light intensity follows a Poisson distribution, which degenerates into a gaussian distribution, if the lambda increases beyond 5, and simplifies some math models by switching to Gaussian Approximations. But stationary processes can be with ANY Random Probability Density Function / random (PDF). And if a spatial distribution is random but clustered and being tested for its properties, they are contained in the Probability Density Function derived from the autocorrelation function of a grid systematized translation of the point distribution, like random placed photon's in grid binned WSS SP autocorrelation tests. LoneRubberDragon (talk) 10:40, 26 February 2010 (UTC)

$f(k;\lambda )={\frac {\lambda ^{k}e^{-\lambda }}{k!}},\,\!$

http://en.wikipedia.org/wiki/Probability_density_function

http://en.wikipedia.org/wiki/Poisson_process

Remember, a uniform, gaussian, or any distribution PDF can be a stationary process if it satisfies randomness convergence, as a WSS system, with average and deviation / covariance measure sufficing for the sampling dimensions, and it's random cluster analyis in probability density function contained PDF data. BUT if you are also using this term verging on relating to the measuring of spatial distributions, any coarser resolution than a nyquist grid on the scale that is ONLY of few points on a plane, then this leads to Cluster Analysis of Small Samples of spatial data, that are too small for PURE spatial randomness testing in the statistical sense of nyquist grid binned spatial points of your article, specifically. Nyquist relates to this at the saddle point between statistics and clusters of below-nyquist-finite data. LoneRubberDragon (talk) 10:50, 26 February 2010 (UTC)

http://en.wikipedia.org/wiki/Cluster_analysis

http://en.wikipedia.org/wiki/Kernel_principal_component_analysis (intimately related sub-nyquist point spatial analysis)

This passage from Zen and the Art of Motorcycle Maintenance, Pirsig, relates my feelings here well. LoneRubberDragon (talk) 11:32, 26 February 2010 (UTC)

ZatAoMM, Prisig, 1974 (1982), pg. 264, "At present we're snowed under with an irrational expansion of blind data-gathering in the sciences because there's no rational format for any understanding of scientific creativity. At present we are snowed under with a lot of stylishness in the arts - thin art - because there's very little assimilation or extension into underlying form. We have artists with no scientific knowledge, and scientists with no artistic knowledge, and both are with not any spiritual sense of gravity at all. And the result is not just bad, it is ghastly. The time for real unification of technology and art is really long overdue."

http://en.wikipedia.org/wiki/Zen_and_the_Art_of_Motorcycle_Maintenance

This leads to a Tower of Babel. A tower of confusion, without any cross referencing, details, or art, as the metaphorical confusion arising from compartmentalization of cliques instead of cooperation of corporations (bodies of individuals as one body). This is the formation of numerous "private" languages of personal interpretation, and opens the door for pointless argumentation, not aligned with truth, for the mere fact of making terms withour relating them to the existant body of known proven forms. It happened in Peleg's time about Babylon, it can happen today. I think you understand this now, User A1. And you don't need to take This Dragon's word for it, for the sources exist, in The Word, if you research around or study. For example, I checked your term, having the PDF fail, and could not easily find CSR with random. Complete Spatial Randomness (14,000) does get hits as a synonym, but Wide Sense Stationary (38,000) and Stationary Process (200,000) get more viable hits. But lacking cross references, makes this a stan alone tool to divert terminology, instead of encompassing everything in a university style of full understanding. But not bad for five hours of writing, watching TV, snacking, and breaks, huh? LoneRubberDragon (talk) 11:32, 26 February 2010 (UTC)

http://en.wikipedia.org/wiki/Tower_of_babel

Sorry, again, if the comments section is longer than the article, as in Superresolution. These are hard subjects of sticky technicalities. LoneRubberDragon (talk)