In this paper, a computational system is developed that estimates a survival curve and a point estimate when very few data are available and a high proportion of the data are censored. Standard statistical methods require a more complete data set. With any less data expert knowledge or heuristic methods are required. The system uses numerical methods to define fuzzy membership functions about each data point that quantify uncertainty due to censoring. The "fuzzy" data is then used to estimate a survival curve and the mean survival time is calculated from the curve. The new estimator converges to the Product-Limit estimator when a complete data set is available. In addition, this method allows for the incorporation of expert knowledge. Finally, simulation results are provided to demonstrate the performance of the new method and its improvement over the Product-Limit estimator.
A common and critical issue in survival data analysis is the way in which censored data are handled. The Kaplan-Meier (KM) estimator is a frequently used statistical method in survival analysis that works well with censored data. In small sample sizes with heavy censoring the estimates of KM are not reliable, because the assumptions of KM estimator are violated. In this study, fuzzy logic is used to obtain more reliable estimates when standard statistical methods cannot be used. Data analyzed in this study were the survival times of six AIDS patients under ten years old. One of the patients died after 197 days and the others were censored, giving a censor rate of 83%. The fuzzy-product-limit estimator (FPLE) and a modified FPLE were used to analyze the data. Mean survival time was calculated and associated confidence interval was calculated along with a measure of the amount of "fuzzy information" used to obtain the estimates. In addition, one to ten year survival rates estimated by the KM, FPLE and proposed methods are presented. The result of the simulations showed that the fuzzy methods with few and highly censored data provide more reasonable results than the standard statistical method.
The Fuzzy-Product-Limit Estimator (FPLE) is a method for estimating a survival curve and the mean survival time when very few data are available and a high proportion of the data are censored. Considering censored times as vague failure times, the censored values are represented by fuzzy membership functions that represent a belief of continued survival of the associated unit. Associated with any estimate is uncertainty. With the FPLE two distinct types of uncertainty exist in the estimate, the uncertainty due to the randomness in the recorded times and the vague uncertainty in the failure of the censored units. This paper addresses the problem of providing confidence bounds and estimates of uncertainty for the FPLE. Several methods for estimating the vague uncertainty in the estimator are suggested. Among them are the use of Efron's Bootstrap that obtains a confidence interval of the FPLE to quantify random uncertainty and produces an empirical distribution that is used to quantify properties of the vague uncertainty. Also, a method to obtain a graphical representation of the random and vague uncertainties is developed. The new methods provide confidence intervals that quantify statistical uncertainty as well-as the vague uncertainty in the estimates. Finally, results of simulations are provided to demonstrate the efficacy of the estimator and uncertainty in the estimates.Keywords: Bootstrap methods; fuzzy theory; product-limit estimator; confidence bounds. Int. J. Unc. Fuzz. Knowl. Based Syst. 2005.13:11-26. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 02/04/15. For personal use only. Uncertainty Estimates in the Fuzzy-Product-LimitEstimator 13The censored time t h the number of data points, the total operating time (TOT), the proportion of censored times, and the number of failures that occurred after time t t all provide information that is used to form the shape of the membership function for the censored time t t . In addition, the method allows the user to influence the membership functions through the parameters U and L. In general, the membership functions for censored and failed times are expressed as:
Students in all areas of computing require knowledge of the computing device including software implementation at the machine level. Several courses in computer science curricula address these low-level details such as computer architecture and assembly languages. For such courses, there are advantages to studying real architectures instead of simplified examples. However, real architectures and instruction sets introduce complexity that makes them difficult to grasp in a single semester course. Visualization techniques can help ease this burden, unfortunately existing tools are often difficult to use and consequently difficult to adopt in a course where time is already limited. To solve this problem, we present Frances. Frances graphically illustrates key differences between familiar high-level languages and unfamiliar low-level languages and also illustrates how familiar high-level programs behave on real architectures. Key to this tool is that we use a simple Web interface that requires no setup, easing course adoption hurdles. We also include several features that further enhance its usefulness in a classroom setting. These features include graphical relationships between high-level code and machine code, clearly illustrated step-by-step machine state transitions, color coding to make instruction behavior clear, and illustration of pointers. We have used Frances in courses and performed experimental evaluation. Our experiences with Frances in the classroom demonstrate its usability. Most notably, in our experimental setting, students with no computer architecture course experience were able to complete lessons using Frances with no guidance.
This paper presents experiences with creating a computer simulator as a student project in a CS1 course. Each student writes the simulator using C++ during the last ten weeks of the course. The project consists of a simulated memory, and simple CPU simulator including a machine language. Additionally, students implement an assembly language and a simple high-level language with associated compiler. The course has no programming prerequisite and can be taken to fulfill a general education requirement or as the first course for students majoring in computer science or information systems.Integrating such a project in an entry level course has a number of benefits as well as challenges. The project acts as a vehicle that engages students in a breadth of computer science topics, leading into discussions of theoretical considerations, languages, and computing devices. The project components provide an active learning environment. Students are introduced to numbering systems, number conversions, and numeric representations. The computer architecture components include introductions to main memory, CPU, and memory access techniques. The transition from and motivations for, the utilization of machine languages, assembly languages and high-level languages are demonstrated with the implementation of the project. Beginning students are given opportunity to practice programming and problem solving on a project of significant complexity. The biggest challenge is organization. Management of such a project requires a well-defined plan.
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