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Information Dimension
In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rényi in 1959. Simply speaking, it is a measure of the fractal dimension of a probability distribution. It characterizes the growth rate of the Shannon entropy given by successively finer discretizations of the space. In 2010, Wu and Verdú gave an operational characterization of Rényi information dimension as the fundamental limit of almost lossless data compression for analog sources under various regularity constraints of the encoder/decoder. Definition and Properties The entropy of a discrete random variable Z is :\mathbb_0(Z)=\sum_P_Z(z)\log_2\frac where P_Z(z) is the probability measure of Z when Z=z, and the supp(P_Z) denotes a set \. Let X be an arbitrary real-valued random variable. Given a positive integer m, we create a new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a Fair coin, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rate Distortion
Rate or rates may refer to: Finance * Rate (company), an American residential mortgage company formerly known as Guaranteed Rate * Rates (tax), a type of taxation system in the United Kingdom used to fund local government * Exchange rate, rate at which one currency will be exchanged for another Mathematics and science * Rate (mathematics), a specific kind of ratio, in which two measurements are related to each other (often with respect to time) * Rate function, a function used to quantify the probabilities of a rare event * Reaction rate, in chemistry the speed at which reactants are converted into products Military * Naval rate, a junior enlisted member of a navy * Rating system of the Royal Navy, a former method of indicating a British warship's firepower People * Ed Rate (1899–1990), American football player * José Carlos Rates (1879–1945), General Secretary of the Portuguese Communist Party * Peter of Rates (died 60 AD), traditionally considered to be the first bisho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mean Value Theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval (mathematics), interval starting from local hypotheses about derivatives at points of the interval. History A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
A Simple Continuous Function Which Are Used To Be Quantized
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rectified Gaussian Distribution
In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval (0,\infty)) as a result of censoring (statistics), censoring. Density function The probability density function of a rectified Gaussian distribution, for which random variables ''X'' having this distribution, derived from the normal distribution \mathcal(\mu,\sigma^2), are displayed as X \sim \mathcal^(\mu,\sigma^2) , is given by f(x;\mu,\sigma^2) =\Phi\delta(x)+ \frac\; e^\textrm(x). Here, \Phi(x) is the cumulative distribution function (cdf) of the standard normal distribution: \Phi(x) = \frac \int_^x e^ \, dt \quad x\in\mathbb, \delta(x) is the Dirac delta function \delta(x) = \begin +\infty, & x = 0 \\ 0, & x \ne 0 \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rectifier
A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. The process is known as ''rectification'', since it "straightens" the direction of current. Physically, rectifiers take a number of forms, including Vacuum tube#Diodes, vacuum tube diodes, wet chemical cells, mercury-arc valves, stacks of copper and selenium rectifier, selenium oxide plates, Diode#Semiconductor diodes, semiconductor diodes, silicon-controlled rectifiers and other silicon-based semiconductor switches. Historically, even synchronous electromechanical switches and motor-generator sets have been used. Early radio receivers, called crystal radios, used a "Crystal detector#Cat whisker detector, cat's whisker" of fine wire pressing on a crystal of galena (lead sulfide) to serve as a point-contact rectifier or "crystal detector". Rectifiers have many uses, but are often found serving as component ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Probability Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
A Standard Gaussian Distribution For Illustration An Example
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lebesgue's Decomposition Theorem
In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure. Definition The theorem states that if (\Omega,\Sigma) is a measurable space and \mu and \nu are σ-finite signed measures on \Sigma, then there exist two uniquely determined σ-finite signed measures \nu_0 and \nu_1 such that: * \nu=\nu_0+\nu_1\, * \nu_0\ll\mu (that is, \nu_0 is absolutely continuous with respect to \mu) * \nu_1\perp\mu (that is, \nu_1 and \mu are singular). Refinement Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem. That is, let (\Omega,\Sigma) be a measure space, \mu a σ-finite positive measure on \Sigma and \lambda a complex measure on \Sigma. * There is a unique pair of complex measures on \Sigma such that \lambda = \lambda_a + \lambda_s, \quad \lambda_a \ll \mu, \quad \lambda_s \perp \mu. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Shannon's Entropy
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X, which may be any member x within the set \mathcal and is distributed according to p\colon \mathcal\to, 1/math>, the entropy is \Eta(X) := -\sum_ p(x) \log p(x), where \Sigma denotes the sum over the variable's possible values. The choice of base for \log, the logarithm, varies for different applications. Base 2 gives the unit of bits (or " shannons"), while base ''e'' gives "natural units" nat, and base 10 gives units of "dits", "bans", or " hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable. The concept of information entropy was int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
