Bayesovska regresijska analiza

master's thesis

• Author: Iva Bogojević
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-10, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu smo se bavili bayesovskom statistikom odnosno bayesovskim linearnim regresijskim modelima. Ukratko, u linearnoj regresiji pokusavamo objasniti varijabilnost zavisne varijable $Y$ pomoću jedne ili više nezavisnih varijabli $X$. Koristimo pretpostavku da je regresijski šum nezavisno normalno distribuiran, zavisna varijabla $Y$ također ima normalnu distribuciju, a $\textbf{X}$ je matrica dizajna, prvi stupac sadrži jedinice. Sad kad je model postavljen, bayesovska... More analiza traži aposteriornu distribuciju parametara i prediktivnu distribuciju za model. Analiza započinje apriornom distribucijom. Parametre ($\beta, \sigma^2$) označimo sa ($B, \Sigma^2$) kao slučajne veličine. U radu smo koristili pretpostavku da parametar ($B, \Sigma^2$) ima nepravilnu apriornu distribuciju $\pi (\beta, \sigma^2) \propto \frac{1}{\sigma^2}$, te dobili da je aposteriorna distribucija za $\beta$ uvjetno na $\Sigma^2=\sigma^2$ normalna, a aposteriorna distribucija za $\Sigma^2 Inv\chi^2$-distribucija. Drugi slučaj koji smo pokazali je slučaj apriorne distribucije. Pretpostavili smo da $B$ ima normalnu apriornu distribuciju (uvjetnu na $\sigma^2$), a $\sigma^2$ inverznu $\chi^2$ -apriornu distribuciju. Dobili smo da je i aposteriorna distribucija za $B$ dana sa $\pi ( \beta|$ y, X, $\sigma^2$) također normalna, a aposteriorna distribucija od $\Sigma^2$ ponovno inverzna $\chi^2$ distribucija. Pokazali smo i slučaj nejednakih varijanci, kad imamo dva modela, $Y_1=X_1 \beta + \epsilon_1$ i $Y_2=X_2 \beta + \epsilon_2$. Neinformativna apriorna distribucija se može utvrditi, pod pretpostavkom da su parametri nezavisni. Tada je apriorna distribucija zapisana kao: $\pi(\beta, \sigma_1, \sigma_2 ) \propto \frac{1}{\sigma_1 \sigma_2}$. Pokazuje se da je marginalna aposteriorna distribucija od $B$ produkt dvije multivarijatne Studentove t-gustoće. Također je pokazano da se marginalna aposteriorna distribucija od $B$ može aproksimirati sa normalnom distribucijom. Slično smo pokazali i za multivarijatni slučaj. Less

Abstract (english)

This thesis is concerned with a Bayesian statistics and Bayesian linear regression models. In short, in linear regression we are trying to explain the variability in dependent variable $Y$ with the help of one or more independent variables $X$. We assume that regression disturbances are independently and identically disturbed with normal distribution, the dependent variable $Y$ has also normal distribution and $\textbf{X}$ is design matrix, first column of $\textbf{X}$ is a column... More of ones. Once the model is specified, the Bayesian analysis seeks the posterior distribution of the parameters and a predictive distribution for the model's prediction. The analysis begins with a prior distribution. Parameters (($\beta, \sigma^2$) are denoted with ($B, \Sigma^2$) like random variables. We used assumption that parameter ($B, \Sigma^2$), has improper prior $\pi (\beta, \sigma^2) \propto \frac{1}{\sigma^2}$, it follows that posterior distribution of $\beta$ conditional on $\Sigma^2=\sigma^2$ is normal, and posterior distribution of $\Sigma^2$ is $Inv\chi^2$ distribution. Another case, which we have shown is case of informative prior distribution. We assume that $B$ has a normal prior distribution (conditional on $\sigma^2$), and $\Sigma^2$ inverted $\chi^2$ − prior distribution. It follows that posterior distribution for $B$ is given with: $\pi ( \beta|$ y, X, $\sigma^2$) is also normal and posterior distribution of $\Sigma^2$ is also inverted $\chi^2$ distribution. We also have shown the case of unequal variance, when we have two models, $Y_1=X_1 \beta + \epsilon_1$ and $Y_2=X_2 \beta + \epsilon_2$. A non-informative prior distribution can be asserted, by assuming that the parameters are independent. The prior is written, then, as: $\pi(\beta, \sigma_1, \sigma_2 ) \propto \frac{1}{\sigma_1 \sigma_2}$. It is shown that the marginal posterior distribution of $B$ is the product of two multivariate Student's t-densities. Also, it is shown that the marginal posterior of $B$ can be approximated with a normal distribution. Similarly, we have shown for the multivariate case. Less

• Keywords: bayesovska regresijska analiza; bayesovska statistika; bayesovski linearni regresijski model; apriorna distribucija; aposteriorna distribucija
• Keywords (english): Bayesian regression analisys; Bayesian statistics; Bayesian linear regression model; prior distribution; posterior distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:529822
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-06-06 12:06:06