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- Title
- Independence and Graphical Models for Fitting Real Data
- Creator
- Cho, Jason Y.
- Date
- 2023
- Description
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Given some real life dataset where the attributes of the dataset take on categorical values, with corresponding r(1) × r(2) × … × r(m)...
Show moreGiven some real life dataset where the attributes of the dataset take on categorical values, with corresponding r(1) × r(2) × … × r(m) contingency table with nonzero rows or nonzero columns, we will be testing the goodness-of-fit of various independence models to the dataset using a variation of Metropolis-Hastings that uses Markov bases as a tool to get a Monte Carlo estimate of the p-value. This variation of Metropolis-Hastings can be found in Algorithm 3.1.1. Next we will consider the problem: ``out of all possible undirected graphical models each associated to some graph with m vertices that we test to fit on our dataset, which one best fits the dataset?" Here, the m attributes are labeled as vertices for the graph. We would have to conduct 2^(mC2) goodness-of-fit tests since there are 2^(mC2) possible undirected graphs on m vertices. Instead, we consider a backwards selection method likelihood-ratio test algorithm. We first start with the complete graph G = K(m), and call the corresponding undirected graphical model ℳ(G) as the parent model. Then for each edge e in E(G), we repeatedly apply the likelihood-ratio test to test the relative fit of the model ℳ(G-e), the child model, vs. ℳ(G), the parent model, where ℳ(G-e) ⊆ℳ(G). More details on this iterative process can be found in Algorithm 4.1.3. For our dataset, we will be using the alcohol dataset found in https://www.kaggle.com/datasets/sooyoungher/smoking-drinking-dataset, where the four attributes of the dataset we will use are ``Gender" (male, female), ``Age", ``Total cholesterol (mg/dL)", and ``Drinks alcohol or not?". After testing the goodness-of-fit of three independence models corresponding to the independence statements ``Gender vs Drink or not?", ``Age vs Drink or not?", and "Total cholesterol vs Drink or not?", we found that the data came from a distribution from the two independence models corresponding to``Age vs Drink or not?" and "Total cholesterol vs Drink or not?" And after applying the backwards selection likelihood-ratio method on the alcohol dataset, we found that the data came from a distribution from the undirected graphical model associated to the complete graph minus the edge {``Total cholesterol”, ``Drink or not?”}.
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- Title
- Dynamic Risk and Dynamic Performance Measures Generated by Distortion Functions and Diversification Benefits Optimization
- Creator
- Liu, Hao
- Date
- 2023
- Description
-
This thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk...
Show moreThis thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk management is made via developing dynamic risk measures and dynamic acceptability indices that can be characterized by distortion functions. In particular, we proved a representation theorem illustrating that the class of dynamic coherent risk measures generated by distortion functions coincides with a specific type of dynamic risk measures, the dynamic WV@R. We also investigate thoroughly various types of time consistencies for dynamic risk measures and dynamic acceptability indices in terms of distortion functions. Another contribution to risk management is proving strong consistency and asymptotic normality of two estimators of dynamic WV@R. In contrast to the exist- ing literature, our results do not rely on the assumptions of distribution of random variables. Instead, we investigate the asymptotic normality of estimators in terms of the generating distortion functions. Last but not least, we give counterexample to show that a sufficient condition of asymptotic normality is not necessary. The contribution to optimization is twofold. On the one hand, we formulate the (scalar) diversification optimization problem as a vector optimization problem (VOP), and show that a set-valued Bellman principle is satisfied by this VOP. On the other hand, we derive explicit policy gradient formula and implement the deep neural network to solve diversification optimization problem numerically. This deep learning technique allows to overcome computation difficulty caused by the non-convexity of VOP.
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