Probability and Measure
01 Jan 1996-Vol. 53, pp 97-98
About: The article was published on 1996-01-01 and is currently open access. It has received 1282 citations till now. The article focuses on the topics: Probability measure & Law of the unconscious statistician.
Citations
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Book•
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11 Jan 2013
TL;DR: Outlier Analysis is a comprehensive exposition, as understood by data mining experts, statisticians and computer scientists, and emphasis was placed on simplifying the content, so that students and practitioners can also benefit.
Abstract: With the increasing advances in hardware technology for data collection, and advances in software technology (databases) for data organization, computer scientists have increasingly participated in the latest advancements of the outlier analysis field. Computer scientists, specifically, approach this field based on their practical experiences in managing large amounts of data, and with far fewer assumptions the data can be of any type, structured or unstructured, and may be extremely large. Outlier Analysisis a comprehensive exposition, as understood by data mining experts, statisticians and computer scientists. The book has been organized carefully, and emphasis was placed on simplifying the content, so that students and practitioners can also benefit. Chapters will typically cover one of three areas: methods and techniques commonly used in outlier analysis, such as linear methods, proximity-based methods, subspace methods, and supervised methods; data domains, such as, text, categorical, mixed-attribute, time-series, streaming, discrete sequence, spatial and network data; and key applications of these methods as applied to diverse domains such as credit card fraud detection, intrusion detection, medical diagnosis, earth science, web log analytics, and social network analysis are covered.
1,278 citations
Cites background or methods from "Probability and Measure"
...Subsequently, it was extended by Aleksandr Lyapunov to cases where the variables are not necessarily identically distributed [70], but they do need to be independent....
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...Interested readers are referred to [70]....
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...The central limit theorem has been studied extensively in probability and statistics [70]....
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TL;DR: In this article, the authors propose an online aggregation interface that allows users to both observe the progress of their aggregation queries and control execution on the fly, and present a suite of techniques that extend a database system to meet these requirements.
Abstract: Aggregation in traditional database systems is performed in batch mode: a query is submitted, the system processes a large volume of data over a long period of time, and, eventually, the final answer is returned. This archaic approach is frustrating to users and has been abandoned in most other areas of computing. In this paper we propose a new online aggregation interface that permits users to both observe the progress of their aggregation queries and control execution on the fly. After outlining usability and performance requirements for a system supporting online aggregation, we present a suite of techniques that extend a database system to meet these requirements. These include methods for returning the output in random order, for providing control over the relative rate at which different aggregates are computed, and for computing running confidence intervals. Finally, we report on an initial implementation of online aggregation in POSTGRES.
1,109 citations
05 Mar 2013
TL;DR: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. as discussed by the authors introduces the basic concepts in the design and analysis of randomized algorithms and provides a comprehensive and representative selection of the algorithms that might be used in each of these areas.
Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.
785 citations
01 Feb 2005
TL;DR: This paper clarifies the mathematical structure of this measure space and its relationship to the underlying spaces associated with each of the basic random variables.
Abstract: The basic random variables on which random uncertainties can in a given model depend can be viewed as defining a measure space with respect to which the solution to the mathematical problem can be defined. This measure space is defined on a product measure associated with the collection of basic random variables. This paper clarifies the mathematical structure of this space and its relationship to the underlying spaces associated with each of the random variables. Cases of both dependent and independent basic random variables are addressed. Bases on the product space are developed that can be viewed as generalizations of the standard polynomial chaos approximation. Moreover, two numerical constructions of approximations in this space are presented along with the associated convergence analysis.
577 citations
Cites background from "Probability and Measure"
...These are typically mixtures of probability measures, each tagging the probabilistic content of some related function [3]....
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TL;DR: In this paper, the authors prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of auto-gressive moving-average models with noise sequence driven by a GARCH model.
Abstract: We prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of autoregressive moving-average models with noise sequence driven by a GARCH model. Results are obtained under mild conditions.
540 citations
Cites background or result from "Probability and Measure"
...We refer to Bougerol and Picard (1992) for the proof of the first result, and to Nelson (1990) and Berkes et al....
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...1 (see Billingsley 1995, pp. 284 and 495)....
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...The central limit theorem of Billingsley (1961) and the Wold–Cramér device allow us to derive the asymptotic normality result (v)....
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References
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01 Jan 2006
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.
1,371 citations
Book•
[...]
11 Jan 2013
TL;DR: Outlier Analysis is a comprehensive exposition, as understood by data mining experts, statisticians and computer scientists, and emphasis was placed on simplifying the content, so that students and practitioners can also benefit.
Abstract: With the increasing advances in hardware technology for data collection, and advances in software technology (databases) for data organization, computer scientists have increasingly participated in the latest advancements of the outlier analysis field. Computer scientists, specifically, approach this field based on their practical experiences in managing large amounts of data, and with far fewer assumptions the data can be of any type, structured or unstructured, and may be extremely large. Outlier Analysisis a comprehensive exposition, as understood by data mining experts, statisticians and computer scientists. The book has been organized carefully, and emphasis was placed on simplifying the content, so that students and practitioners can also benefit. Chapters will typically cover one of three areas: methods and techniques commonly used in outlier analysis, such as linear methods, proximity-based methods, subspace methods, and supervised methods; data domains, such as, text, categorical, mixed-attribute, time-series, streaming, discrete sequence, spatial and network data; and key applications of these methods as applied to diverse domains such as credit card fraud detection, intrusion detection, medical diagnosis, earth science, web log analytics, and social network analysis are covered.
1,278 citations
[...]
TL;DR: In this article, the authors propose an online aggregation interface that allows users to both observe the progress of their aggregation queries and control execution on the fly, and present a suite of techniques that extend a database system to meet these requirements.
Abstract: Aggregation in traditional database systems is performed in batch mode: a query is submitted, the system processes a large volume of data over a long period of time, and, eventually, the final answer is returned. This archaic approach is frustrating to users and has been abandoned in most other areas of computing. In this paper we propose a new online aggregation interface that permits users to both observe the progress of their aggregation queries and control execution on the fly. After outlining usability and performance requirements for a system supporting online aggregation, we present a suite of techniques that extend a database system to meet these requirements. These include methods for returning the output in random order, for providing control over the relative rate at which different aggregates are computed, and for computing running confidence intervals. Finally, we report on an initial implementation of online aggregation in POSTGRES.
1,109 citations
05 Mar 2013
TL;DR: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. as discussed by the authors introduces the basic concepts in the design and analysis of randomized algorithms and provides a comprehensive and representative selection of the algorithms that might be used in each of these areas.
Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.
785 citations
TL;DR: A central limit theorem is proved for MIMO channels with a large number of antennas, which is described as a "channel-hardening" result for data and voice services, scheduling, and rate feedback.
Abstract: Wireless data traffic is expected to grow over the next few years and the technologies that will provide data services are still being debated. One possibility is to use multiple antennas at base stations and terminals to get very high spectral efficiencies in rich scattering environments. Such multiple-input/multiple-output (MIMO) channels can then be used in conjunction with scheduling and rate-feedback algorithms to further increase channel throughput. This paper provides an analysis of the expected gains due to scheduling and bits needed for rate feedback. Our analysis requires an accurate approximation of the distribution of the MIMO channel mutual information. Because the exact distribution of the mutual information in a Rayleigh-fading environment is difficult to analyze, we prove a central limit theorem for MIMO channels with a large number of antennas. While the growth in average mutual information (capacity) of a MIMO channel with the number of antennas is well understood, it turns out that the variance of the mutual information can grow very slowly or even shrink as the number of antennas grows. We discuss implications of this "channel-hardening" result for data and voice services, scheduling, and rate feedback. We also briefly discuss the implications when shadow fading effects are included.
725 citations