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Random polytopes can be constructed in many different ways. In this thesis two certain kinds are considered - random polytopes as the convex hull of random points and as the intersection of finitely many random half spaces. Concerning these two models different issues are treated.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第17837号 / 人博第658号 / 新制||人||158(附属図書館) / 25||人博||658(吉田南総合図書館) / 30652 / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 上木 直昌, 教授 森本 芳則, 教授 髙﨑 金久 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20887号 / 理博第4339号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 福島 竜輝, 教授 中島 啓, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We show that with the implementation presented in this paper, the Random Forest Classification model was able to predict whether or not a stock was going to increase in value during the coming day with an accuracy higher than 50\% for all stocks included in this study. Furthermore, we show that the active trading strategy presented in this paper generated higher returns and higher risk-adjusted returns than the passive investment in the stocks underlying the strategy. Therefore, we conclude \textit{(i)} that a Random Forest Classification model can be used to provide valuable insight on publicly traded stocks, and \textit{(ii)} that it is probably possible to create a profitable trading strategy based on a Random Forest Classifier, but that this requires a more sophisticated implementation than the one presented in this paper.

Testing the equality between two high-dimensional covariance matrices is challenging. As the most efficient way to measure evidential discrepancies in observed data, the likelihood ratio test is expected to be powerful when the null hypothesis is violated. However, when the data dimensionality becomes large and potentially exceeds the sample size by a substantial margin, likelihood ratio based approaches face practical and theoretical challenges. To solve this problem, this study proposes a method by which we first randomly project the original high-dimensional data into lower-dimensional space, and then apply the corrected likelihood ratio tests developed with random matrix theory. We show that testing with a single random projection is consistent under the null hypothesis. Through evaluating the power function, which is challenging in this context, we provide evidence that the test with a single random projection based on a random projection matrix with reasonable column sizes is more powerful when the two covariance matrices are unequal but component-wise discrepancy could be small -- a weak and dense signal setting. To more efficiently utilize this data information, we propose combined tests from multiple random projections from the class of meta-analyses. We establish the foundation of the combined tests from our theoretical analysis that the p-values from multiple random projections are asymptotically independent in the high-dimensional covariance matrices testing problem. Then, we show that combined tests from multiple random projections are consistent under the null hypothesis. In addition, our theory presents the merit of certain meta-analysis approaches over testing with a single random projection. Numerical evaluation of the power function of the combined tests from multiple random projections is also provided based on numerical evaluation of power function of testing with a single random projection. Extensive simulations and two real genetic data analyses confirm the merits and potential applications of our test. / Statistics

The role of topography on the spatial distribution of tornadoes was assessed through geospatial and statistical techniques. A 100-m digital elevation model was used to create slope, aspect, and surface roughness maps; and; tornado beginning and ending points and paths were used to extract terrain information. Tornado touchdowns, liftoffs, paths, and path-land angles were examined to determine whether tornado paths occur more frequently in or along certain terrain or slopes. Statistical analyses, such as bootstrapping, were used to analyze tornado touchdowns, liftoffs and paths and path-relative terrain angles. Results show that tornado paths are more common with downhill-movement. Tornadoes are not as likely to move uphill because the 73.6 percent northeast path bias represents the highest frequencies of path-angles. Tornado touchdowns and paths occur more often in smooth terrain, rather than rough terrain. Complex topographic variability seems to not have an effect on the spatial distribution of tornadoes.

Background: According to the South African National Health and Nutrition Examination Survey, approximately 69% of South African women and 29% of men are considered overweight/obese. In South Africa, overweight and obesity rates are on the rise and have reached epidemic proportions. Several conventional weight loss strategies have been employed and have been unsuccessful in addressing this issue. It is for this reason that this study strives to uncover if any improvement in weight and fitness status are associated with an improvement in health status. Methods: One of our main aims were to determine the success in weight loss and reduction in health risk factors in members who have completed 18-weeks of the Healthy weight programme. Therefore, this study is a retrospective, observational study of adults with a mean body mass index (BMI) of 32.63 ± 7.39 who completed an 18-week supervised exercise program. An online health questionnaire was completed followed by pre- intervention assessments which included: Anthropometric measures (height, weight, waist circumference, body fat percentage) followed by blood pressure, finger-prick random glucose and cholesterol measurements. A 12-min motion test to assess functional capacity were completed. All anthropometric, resting health measures and fitness tests were repeated at 12-weeks (post-beginner programme) and 18 weeks (post-intervention) after the start of the intervention. The intervention included 2 weekly classes with an optional gym class. The type of exercise intervention that was included in this study largely focused resistance training and cardiorespiratory fitness which targeted multiple large muscle groups. Each exercise session included 80 to 90 % exercise intensity as well as lumbopelvic core exercises. The data set was analysed as an intention to treat (ITT) protocol as some of the testing time points were missing. Results: Out of a possible 50 participants, 34 met the inclusion criteria of at least all 3 health measures taken on the 3 different occasions (baseline, post-beginner, postintervention). The 34 participants had a mean age of 47.26 ± 10.44 ranging between 24 years and 76 years old. 65 % of the participants were female. Anthropometric results showed significant changes in weight and waist circumference at 12- and at 18-weeks compared to baseline measures (p < 0.005). Additionally, body fat % significantly reduced at 12-weeks but increased slightly at 18 weeks diminishing the significant reduction compared to the baseline measure. In terms of fitness, participants succeeded in improving their average distance to complete a 12-min motion test by 14.5% (p < 0.001) at 12-weeks and 20% (p < 0.001) at 18-weeks compared to starting measures, respectively. After adjusting for multiple comparisons during post-hoc analysis, none of the health status outcomes (blood pressure, random glucose, random cholesterol) showed a significant improvement at any of the time points. In terms of associations, a change in body fat % could significantly explain some of the variance in the change (from baseline to 18-weeks) in cholesterol; and a change in weight could significantly explain some of the variance in the change in glucose over the same intervention time. Conclusion: The results of this study revealed the 18-week supervised exercise intervention led to a modest reduction (approximately 5%) in weight and a substantial improvement in fitness. However, no changes in health status were observed, suggesting that greater improvements in weight and fitness may be required to have a profound influence on health status.

Software testing is essential for quality assurance, with automated techniques such as random testing and adaptive random testing being cost-effective solutions compared to others. Adaptive random testing seeks to enhance random testing, and there is a conception that adaptive random testing always should replace random testing. Our research question investigates this conception by addressing a gap in the literature, where a comparison between the two techniques in terms of certain key metrics is missing, namely defect detection efficiency and test case generation time. Defect detection efficiency is the amount of defects detected divided by the number defects in the system multiplied by one hundred. Test case generation time is the time it takes to generate all of the test case inputs. These metrics where chosen as they can be seen as a measurement of the techniques effectiveness and efficiency respectively. In order to address this research question we employ a quantitative experiment where we compare the performance of random testing and adaptive random testing with a sole focus on these two metrics. The comparison is performed by implementing and testing both algorithms on eight error-seeded numerical programs and measuring the results. The results displayed that adaptive random testing had a defect detection efficiency total average of 21.59% and a test case generation time total average of 35.37 (ms), while random testing had a defect detection efficiency total average of 22.28% and a test case generation time total average of 0.26 (ms). These results might contribute to disproving the conception that adaptive random testing always should replace random testing, as random testing evidently performed better on both the measured metrics.

The random walk is a powerful model. Chemistry, Physics, and Finance are just a few of the disciplines that model with the random walk. It is clear from its varied uses that despite its simplicity, the simple random walk it very flexible. There is one major drawback, however, to the simple random walk and the geometric random walk. The limiting distribution is either normal, lognormal, or a levy process with infinite variance. This thesis introduces an new random walk aimed at overcoming this drawback. Because the simple random walk and the geometric random walk are special cases of the proposed walk, it is called a generalized random walk. Several properties of the generalized random walk are considered. First, the limiting distribution of the generalized random walk is shown to include a large class of distributions. Second and in conjunction with the first, the generalized random walk is compared to the geometric random walk. It is shown that when parametrized properly, the generalized random walk does converge to the lognormal distribution. Third, and perhaps most interesting, is one of the limiting properties of the generalized random walk. In the limit, generalized random walks are closely connected with a u function. The u function is the key link between generalized random walks and its difference equation. Last, we apply the generalized random walk to option pricing.