*Summary*

*statistics*. Weight function. Weighted average cost of capital. Weighted geometric mean. Weighted harmonic mean. Weighted least squares. Weighted median. Weighting.

Statistical theory provides a basis for good data collection and the structuring of investigations in the topics of: The task of summarising statistical data in conventional forms (also known as *descriptive* *statistics*) is considered in theoretical statistics as a problem of defining what aspects of statistical samples need to be described and how well they can be described from a typically limited sample of data.

In minimizing *description* length (or *descriptive* *complexity*), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling. The MDL principle has been applied in communication-coding theory in information theory, in linear regression, and in data mining. The evaluation of MDL-based inferential procedures often uses techniques or criteria from computational complexity theory.

Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see *descriptive* *complexity*. The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems (an example is FP), counting problems (e.g. #P), optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, boolean circuits, quantum Turing machines, monotone circuits, etc.

Thus the main application areas of FMT are: *descriptive* *complexity* *theory*, database theory and formal language theory. FMT is mainly about discrimination of structures. The usual motivating question is whether a given class of structures can be described (up to isomorphism) in a given language. For instance, can all cyclic graphs be discriminated (from the non-cyclic ones) by a sentence of the first-order logic of graphs? This can also be phrased as: is the property "cyclic" FO expressible?.

Nonparametric statistics includes both *descriptive* *statistics* and statistical inference. The term "nonparametric statistics" has been imprecisely defined in the following two ways, among others. Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in ordinal data. As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods.

The semantics of these are captured as *description* *logic* concepts, roles, and individuals, and typically implemented as classes, properties, and individuals in the Web Ontology Language. The most general ontologies are called upper ontologies, which attempt to provide a foundation for all other knowledge by acting as mediators between domain ontologies that cover specific knowledge about a particular knowledge domain (field of interest or area of concern).

It involves publishing in languages specifically designed for data: Resource *Description* Framework (RDF), Web Ontology Language (OWL), and Extensible Markup Language (XML). HTML describes documents and the links between them. RDF, OWL, and XML, by contrast, can describe arbitrary things such as people, meetings, or airplane parts. These technologies are combined in order to provide *descriptions* that supplement or replace the content of Web documents.

In *descriptive* *complexity*, a query is a mapping from structures of one signature to structures of another vocabulary. Neil Immerman, in his book *Descriptive* *Complexity*, "use[s] the concept of query as the fundamental paradigm of computation" (p. 17). Given signatures \sigma and \tau, we define the set of structures on each language, and. A query is then any mapping Computational complexity theory can then be phrased in terms of the power of the mathematical logic necessary to express a given query. A query is order-independent if the ordering of objects in the structure does not affect the results of the query.

Fagin's theorem is a result in *descriptive* *complexity* *theory* that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It is remarkable since it is a characterization of the class NP that does not invoke a model of computation such as a Turing machine. It was proven by Ronald Fagin in 1973 in his doctoral thesis. The arity required by the second-order formula was improved (in one direction) in Lynch's theorem, and several results of Grandjean have provided tighter bounds on nondeterministic random-access machines. Immerman 1999 provides a detailed proof of the theorem.

OIL is based on concepts developed in *Description* *Logic* (DL) and frame-based systems and is compatible with RDFS. OIL was developed by Dieter Fensel, Frank van Harmelen (Vrije Universiteit, Amsterdam) and Ian Horrocks (University of Manchester) as part of the IST OntoKnowledge project. Much of the work in OIL was subsequently incorporated into DAML+OIL and the Web Ontology Language (OWL). de:Ontology Inference Layer DARPA Agent Markup Language (DAML). DAML+OIL. Ontology.

The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in *descriptive* *statistics*. These can all be seen as minimizing variation by some measure; see. The most frequently occurring number in a list is called the mode. For example, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode. Some authors say they are all modes and some say there is no mode. The median is the middle number of the group when they are ranked in order.

Neil Immerman's *descriptive* *complexity* page, including a diagram. Complexity zoo about FO, see also the following classes.

In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy:

He is one of the key developers of *descriptive* *complexity*, an approach he is currently applying to research in model checking, database theory, and computational complexity theory. Professor Immerman is an editor of the SIAM Journal on Computing and of Logical Methods in Computer Science. He received B.S. and M.S. degrees from Yale University in 1974 and his Ph.D. from Cornell University in 1980 under the supervision of Juris Hartmanis, a Turing award winner at Cornell. His book "*Descriptive* *Complexity*" appeared in 1999.

The interpretation of these triplets is (implicitly) based on the semantics of a simple *Description* *logic* (DL). E.g., the triplet Common Cold – causative agent – Virus, corresponds to the first-order expression or the more intuitive DL expression In the Common cold example the concept *description* is "primitive", which means that necessary criteria are given that must be met for each instance, without being sufficient for classifying a disorder as an instance of Common Cold .

In natural sciences and social sciences, quantitative research is the systematic empirical investigation of observable phenomena via statistical, mathematical, or computational techniques. The objective of quantitative research is to develop and employ mathematical models, theories, and hypotheses pertaining to phenomena. The process of measurement is central to quantitative research because it provides the fundamental connection between empirical observation and mathematical expression of quantitative relationships.

The unique name assumption is a simplifying assumption made in some ontology languages and *description* *logics*. In logics with the unique name assumption, different names always refer to different entities in the world. It was included in Ray_Reiter's discussion of the Closed-world_assumption often tacitly included in Database Management Systems (e.g. SQL) in his 1984 article "Towards a logical reconstruction of relational database theory (in M.L. Brodie, J. Mylopoulos, J. W. Schmidt (Hrsg.), Data modelling in Artificial Intelligence, Database and Programming Languages, Springer 1984, S. 191–233).

Gruber introduced the term as a specification of a conceptualization: An ontology is a *description* (like a formal specification of a program) of the concepts and relationships that can formally exist for an agent or a community of agents. This definition is consistent with the usage of ontology as set of concept definitions, but more general. And it is a different sense of the word than its use in philosophy.

He is considered by some to be the godfather of *Description* *Logic*, the logic-based knowledge representation formalism underlying the Web Ontology Language OWL.] He is the co-author with Hector Levesque of a popular book on knowledge representation and reasoning and many scientific papers. * External biography