Probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns each event in the event space a probability, which is a number between 0 and 1.
Transparency (data compression)In data compression and psychoacoustics, transparency is the result of lossy data compression accurate enough that the compressed result is perceptually indistinguishable from the uncompressed input, i.e. perceptually lossless. A transparency threshold is a given value at which transparency is reached. It is commonly used to describe compressed data bitrates. For example, the transparency threshold for MP3 to linear PCM audio is said to be between 175 and 245 kbit/s, at 44.
Weak orderingIn mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders.
Apple Lossless Audio CodecThe Apple Lossless Audio Codec (ALAC), also known as Apple Lossless, or Apple Lossless Encoder (ALE), is an audio coding format, and its reference audio codec implementation, developed by Apple Inc. for lossless data compression of digital music. After initially keeping it proprietary from its inception in 2004, in late 2011 Apple made the codec available open source and royalty-free. Traditionally, Apple has referred to the codec as Apple Lossless, though more recently it has begun to use the abbreviated term ALAC when referring to the codec.
Information theoryInformation theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy.
Product orderIn mathematics, given a partial order and on a set and , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the Cartesian product Given two pairs and in declare that if and Another possible ordering on is the lexicographical order, which is a total ordering. However the product order of two total orders is not in general total; for example, the pairs and are incomparable in the product order of the ordering with itself.
Frequentist probabilityFrequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation.
Algorithmic information theoryAlgorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as strings or any other data structure. In other words, it is shown within algorithmic information theory that computational incompressibility "mimics" (except for a constant that only depends on the chosen universal programming language) the relations or inequalities found in information theory.
Audio coding formatAn audio coding format (or sometimes audio compression format) is a content representation format for storage or transmission of digital audio (such as in digital television, digital radio and in audio and video files). Examples of audio coding formats include MP3, AAC, Vorbis, FLAC, and Opus. A specific software or hardware implementation capable of audio compression and decompression to/from a specific audio coding format is called an audio codec; an example of an audio codec is LAME, which is one of several different codecs which implements encoding and decoding audio in the MP3 audio coding format in software.
Order topologyIn mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals together with the above rays form a base for the order topology.
Cyclic orderIn mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation [a, b, c], meaning "after a, one reaches b before c". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected.
Rényi entropyIn information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as index of diversity.
