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Author:Nummenpalo, Jerri
Title:Polytopal Big Data Statistics
Polytoopit tilastotieteessä ja Big Datassa
Publication type:Master's thesis
Publication year:2014
Pages:v + 41      Language:   eng
Department/School:Perustieteiden korkeakoulu
Main subject:Matematiikka   (Mat-1)
Supervisor:Engström, Alexander
Instructor:
Electronic version URL: http://urn.fi/URN:NBN:fi:aalto-201507013710
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
Instructions

Reading digital theses in the closed network of the Aalto University Harald Herlin Learning Centre

In the closed network of Learning Centre you can read digital and digitized theses not available in the open network.

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Location:P1 Ark Aalto  1163   | Archive
Keywords:extension complexity
polytopes
statistics
Big Data
polytoopit
tilastotiede
Abstract (eng): Polytopes are geometric objects which arise in combinatorial problems and problems in optimization.
Polytope theory can be used to construct new statistical techniques that help us analyze modern massive data sets.
Big Data refers to methodologies that take into consideration the computational limitations of algorithms when dealing with large data sets.

In the first part of this thesis the topic of extension complexities of polytopes is considered.
There has been a lot of recent research on the matter and it is closely related to Big Data.
A combinatorial proof on the extension complexity of the correlation polytope from a new paper is presented and a minor error is corrected.
As a new result, the strength of the proof is displayed and a possible stronger result is discussed.

In the second part of this thesis a result from a recent article on computational statistics is generalized.
The original research displays the trade-offs between statistical and computational aspects of recovering a high-dimensional vector corrupted by Gaussian noise.
The contribution of this thesis is a result that includes the possibility of correlation in the noise.
Results on error bounds similar to those in the article are reported.
Abstract (fin): Polytoopit ovat geometrisia kappaleita matematiikkassa ja niitä esiintyy erityisesti kombinatoriikkaan liittyvissä ongelmissa.
Tässä diplomityössä tutustutaan kahteen matemaattiseen ongelmaan, joissa molemmissa esiintyy polytooppeja.
Kumpikin ongelma on läheisesti yhteydessä tilastotieteeseen ja isojen tietomäärien - Big Datan - analysointiin.

Ensimmäinen ongelma liittyy polytooppien tehokkaaseen esittämistapaan.
Tarkastelemme tuloksia viimeaikaisista tutkimuksista, jotka liittyvät polytooppien esittämiseen korkeampiulotteisissa Euklidisissa avaruuksissa.
Tutkimme lähemmin korrelaatiopolytoopin esittämistä ja raportoimme tuloksia uudesta artikkelista.
Lisäksi näytämme artikkelissa esitetyn menetelmän rajat, korjaamme siellä esiintyneen virheen ja keskustelemme mahdollisesta vahvemmasta tuloksesta.

Toisessa ongelmassa keskitymme yleistämään tuloksen artikkelista vuodelta 2013.
Yleistämme artikkelissa käytetyn tilastollisen mallin kattamaan korrelaation ja todistamme virherajoja.
Näytämme, että artikkelissa esitetyt väitteet pitävät paikkansa myös yleisemmälle mallille.
ED:2014-06-27
INSSI record number: 49344
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