Determination of System Dimensionality from Observing Near-Normal Distributions
This paper identifies a previously undiscovered behavior of uniformly distributed data points or vectors in high dimensional ellipsoidal models. Such models give near normal distributions for each of its dimensions. Converse of this may also be true; that is, for a normal-like distribution of an obs...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/467195 |
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| author | Shahid Razzaq Shehzad Khalid |
| author_facet | Shahid Razzaq Shehzad Khalid |
| author_sort | Shahid Razzaq |
| collection | DOAJ |
| description | This paper identifies a previously undiscovered behavior of uniformly distributed data points or vectors in high dimensional ellipsoidal models. Such models give near normal distributions for each of its dimensions. Converse of this may also be true; that is, for a normal-like distribution of an observed variable, it is possible that the distribution is a result of uniform distribution of data points in a high dimensional ellipsoidal model, to which the observed variable belongs. Given the currently held notion of normal distributions, this new behavior raises many interesting questions. This paper also attempts to answer some of those questions. We cover both volume based (filled) and surface based (shell) ellipsoidal models. The phenomenon is demonstrated using statistical as well as mathematical approaches. We also show that the dimensionality of the latent model, that is, the number of hidden variables in a system, can be calculated from the observed distribution. We call the new distribution “Tanazur” and show through experiments that it is at least observed in one real world scenario, that of the motion of particles in an ideal gas. We show that the Maxwell-Boltzmann distribution of particle speeds can be explained on the basis of Tanazur distributions. |
| format | Article |
| id | doaj-art-681dbee178824937b64ea826800425c8 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-681dbee178824937b64ea826800425c82025-02-03T06:01:12ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/467195467195Determination of System Dimensionality from Observing Near-Normal DistributionsShahid Razzaq0Shehzad Khalid1Department of Computer Engineering, Bahria University, Islamabad 44000, PakistanDepartment of Computer Engineering, Bahria University, Islamabad 44000, PakistanThis paper identifies a previously undiscovered behavior of uniformly distributed data points or vectors in high dimensional ellipsoidal models. Such models give near normal distributions for each of its dimensions. Converse of this may also be true; that is, for a normal-like distribution of an observed variable, it is possible that the distribution is a result of uniform distribution of data points in a high dimensional ellipsoidal model, to which the observed variable belongs. Given the currently held notion of normal distributions, this new behavior raises many interesting questions. This paper also attempts to answer some of those questions. We cover both volume based (filled) and surface based (shell) ellipsoidal models. The phenomenon is demonstrated using statistical as well as mathematical approaches. We also show that the dimensionality of the latent model, that is, the number of hidden variables in a system, can be calculated from the observed distribution. We call the new distribution “Tanazur” and show through experiments that it is at least observed in one real world scenario, that of the motion of particles in an ideal gas. We show that the Maxwell-Boltzmann distribution of particle speeds can be explained on the basis of Tanazur distributions.http://dx.doi.org/10.1155/2015/467195 |
| spellingShingle | Shahid Razzaq Shehzad Khalid Determination of System Dimensionality from Observing Near-Normal Distributions Abstract and Applied Analysis |
| title | Determination of System Dimensionality from Observing Near-Normal Distributions |
| title_full | Determination of System Dimensionality from Observing Near-Normal Distributions |
| title_fullStr | Determination of System Dimensionality from Observing Near-Normal Distributions |
| title_full_unstemmed | Determination of System Dimensionality from Observing Near-Normal Distributions |
| title_short | Determination of System Dimensionality from Observing Near-Normal Distributions |
| title_sort | determination of system dimensionality from observing near normal distributions |
| url | http://dx.doi.org/10.1155/2015/467195 |
| work_keys_str_mv | AT shahidrazzaq determinationofsystemdimensionalityfromobservingnearnormaldistributions AT shehzadkhalid determinationofsystemdimensionalityfromobservingnearnormaldistributions |