4 مرفق
Extensions to the Dirichlet distribution
بسم الله الرحمن الرحيم
تحية طيبة وبعد
الملفات المرفقة عبارة عن البحث الذي أجريته لرسالة الدكتوراة عام 1999م
وهي مرتبة رقميا
الملفات المرفقة عددها 30 ملفا (ب. د. ف. , PDF )
بعضها مفقود و سيتم إلحاقها مستقبلا إن شاء الله, والمفقود لا يؤثر على مجمل البحث و فكرته.
عنوان البحث
Extensions to the Dirichlet distribution
for data on the simplex
and
extensions to the Liouville distribution
for data on the positive orthant
عناوين الأبواب المكونة للبحث
Chapter 1
Introduction, background, and overview
Chapter 2
Compositional data
Chapter 3
Known generalizations of the Dirichlet distribution
Chapter 4
Liouville distributions and some known generalizations
Chapter 5
Adaptive Dirichlet distributions with dependent ratios
Chapter 6
Extended Liouville distributions
Chapter 7
Perfect aggregation in reliability analysis
Chapter 8
Snowmelt runoff problem
Chapter 9
Simulation results
Chapter 10
Conclusion and suggestions for future research
Abstract
The primary contributions of this dissertation are:
1)
the development of new distributions on the unit simplex and the positive orthant;
2)
the application of some of our new distributions to both actual snowmelt runoff data and hypothetical simulated data,
to compare the performance of these new distributions to that of other distributions that have been used in the past;
and
3)
an application of some of these new distributions to the study of perfect aggregation in reliability analysis.
We begin by reviewing what is known about distributions for analyzing compositional data; i.e., non-negative multivariate data that satisfy a unit-sum constraint.
Unfortunately, the unit-sum constraint is sometimes just ignored in practice. In other cases, the Dirichlet distribution is used to model compositional data,
even when the conditional independence properties of the Dirichlet are not appropriate to the data-generating process.
Distributions other than the Dirichlet also have limitations, including either unknown or restrictive correlation sign structures.
Similar problems arise for non-normal multivariate data on the positive orthant.
Due to the above concerns, we propose new families of distributions on both the unit simplex and the positive orthant. In particular,
we develop a generalization to the Dirichlet distribution on the unit simplex, and a generalization to the Liouville distribution on the positive orthant.
Researchers working in this area have pointed out the importance of knowing the correlation sign structures achievable by a given model,
but this information is not available for many existing distributions. We are able to provide closed-form expressions for the moments
and correlation coefficients of most of the distributions developed in this dissertation, and also show that they have more flexible correlation sign structures
than many existing distributions on the unit simplex or the positive orthant.
Next, we apply one of our new unit simplex distributions to both actual snowmelt runoff data and hypothetical simulated data.
Our analysis shows that this distribution is in some respects an improvement over the distribution that had been used to analyze such data in the past by other researchers.
In particular, the distribution we developed is able to more closely match the observed (empirical or simulated) correlation matrices.
We also explore the issue of perfect aggregation in reliability analysis (i.e., the conditions under which a Bayesian reliability analysis using only system-level data
will yield the same results as an analysis performed using component-level data). Previous work for Bernoulli systems with dependent components was able to
demonstrate only that perfect aggregation holds if the system state probabilities have a joint Dirichlet distribution (i.e., sufficient condition for perfect aggregation).
We give necessary and sufficient conditions that the joint prior distribution of the system state probabilities must satisfy in order for perfect aggregation to hold for such systems,
and also for parallel Poisson systems. We then provide a number of examples of joint probability distributions for the system state probabilities that satisfy perfect aggregation,
including both existing probability distributions from the literature, and also some new distributions original to this dissertation.
