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What is Measure

Handbook of Research on Novel Soft Computing Intelligent Algorithms: Theory and Practical Applications
Formally, a measure µ is a function defined on a s-algebra S over a set X and taking values in the extended interval [0, ?] such that the following properties are satisfied: (1)The empty set has measure zero . (2) A probability measure is a measure with total measure one i.e., p ( X )=1, a probability space is a measure space with a probability measure.
Published in Chapter:
Evaluation of Genetic Algorithm as Learning System in Rigid Space Interpretation
Bhupesh Kumar Singh (Govind Ballabh Pant University of Agriculture & Technology, India)
DOI: 10.4018/978-1-4666-4450-2.ch016
Genetic Algorithm (GA) (a structured framework of metaheauristics) has been used in various tasks such as search optimization and machine learning. Theoretically, there should be sound framework for genetic algorithms which can interpret/explain the various facts associated with it. There are various theories of the working of GA though all are subject to criticism. Hence an approach is being adopted that the legitimate theory of GA must be able to explain the learning process (a special case of the successive approximation) of GA. The analytical method of approximating some known function is expanding a complicated function an infinite series of terms containing some simpler (or otherwise useful) function. These infinite approximations facilitate the error to be made arbitrarily small by taking a progressive greater number of terms into consideration. The process of learning in an unknown environment, the form of function to be learned is known only by its form over the observation space. The problem of learning the possible form of the function is termed as experience problem. Various learning paradigms have ensured their legitimacy through the rigid space interpretation of the concentration of measure and Dvoretzky theorem. Hence it is being proposed that the same criterion should be applied to explain the learning capability of GA, various formalisms of explaining the working of GA should be evaluated by applying the criteria, and that learning capability can be used to demonstrate the probable capability of GA to perform beyond the limit cast by the No Free Lunch Theorem.
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Inconsistencies in the Disclosures of Discount Rates: The Case of Financial Reporting in Portugal
The result of applying a measurement basis to an asset or liability and related income and expenses.
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Statistical Dissemination Systems and the Web
The measure is a numeric value obtained by applying an aggregate function (such as count, sum, min, max, or average) to groups of data in a fact table.
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Social Network Analysis for Virtual Communities
Usually, is a function that assigns a number, for example a “size,” “volume,” or “probability,” to subsets of a given set. The concept is important in mathematical analysis and probability theory.
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