Redundancy (information theory)In information theory, redundancy measures the fractional difference between the entropy H(X) of an ensemble X, and its maximum possible value . Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while forward error correction is a way of adding desired redundancy for purposes of error detection and correction when communicating over a noisy channel of limited capacity.
Binary logarithmIn mathematics, the binary logarithm (log2n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5. The binary logarithm is the logarithm to the base 2 and is the inverse function of the power of two function. As well as log2, an alternative notation for the binary logarithm is lb (the notation preferred by ISO 31-11 and ISO 80000-2).
Nat (unit)The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or information entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the shannon. This unit is also known by its unit symbol, the nat. One nat is the information content of an event when the probability of that event occurring is 1/e. One nat is equal to 1/ln 2 shannons ≈ 1.44 Sh or, equivalently, 1/ln 10 hartleys ≈ 0.434 Hart.
Joint entropyIn information theory, joint entropy is a measure of the uncertainty associated with a set of variables. The joint Shannon entropy (in bits) of two discrete random variables and with images and is defined as where and are particular values of and , respectively, is the joint probability of these values occurring together, and is defined to be 0 if . For more than two random variables this expands to where are particular values of , respectively, is the probability of these values occurring together, and is defined to be 0 if .
Cross-entropyIn information theory, the cross-entropy between two probability distributions and over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution , rather than the true distribution . The cross-entropy of the distribution relative to a distribution over a given set is defined as follows: where is the expected value operator with respect to the distribution .
Units of informationIn computing and telecommunications, a unit of information is the capacity of some standard data storage system or communication channel, used to measure the capacities of other systems and channels. In information theory, units of information are also used to measure information contained in messages and the entropy of random variables. The most commonly used units of data storage capacity are the bit, the capacity of a system that has only two states, and the byte (or octet), which is equivalent to eight bits.
Diversity indexA diversity index is a quantitative measure that reflects how many different types (such as species) there are in a dataset (a community), and that can simultaneously take into account the phylogenetic relations among the individuals distributed among those types, such as richness, divergence or evenness. These indices are statistical representations of biodiversity in different aspects (richness, evenness, and dominance).
Entropy rateIn the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process. For stochastic processes with a countable index, the entropy rate is the limit of the joint entropy of members of the process divided by , as tends to infinity: when the limit exists. An alternative, related quantity is: For strongly stationary stochastic processes, .
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.
Boltzmann's entropy formulaIn statistical mechanics, Boltzmann's equation (also known as the Boltzmann–Planck equation) is a probability equation relating the entropy , also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the number of real microstates corresponding to the gas's macrostate: where is the Boltzmann constant (also written as simply ) and equal to 1.380649 × 10−23 J/K, and is the natural logarithm function (also written as , as in the image above).
