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78-586: Data are collected using techniques such as measurement , observation , query , or analysis , and are typically represented as numbers or characters that may be further processed . Field data are data that are collected in an uncontrolled, in-situ environment. Experimental data are data that are generated in the course of a controlled scientific experiment. Data are analyzed using techniques such as calculation , reasoning , discussion, presentation , visualization , or other forms of post-analysis. Prior to analysis, raw data (or unprocessed data)

156-1169: A Bernoulli process . The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information. This is because H ( X ) = − ∑ i = 1 n p ( x i ) log b ⁡ p ( x i ) = − ∑ i = 1 2 1 2 log 2 ⁡ 1 2 = − ∑ i = 1 2 1 2 ⋅ ( − 1 ) = 1. {\displaystyle {\begin{aligned}\mathrm {H} (X)&=-\sum _{i=1}^{n}{p(x_{i})\log _{b}p(x_{i})}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\log _{2}{\frac {1}{2}}}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\cdot (-1)}=1.\end{aligned}}} However, if we know

234-488: A mass noun in singular form. This usage is common in everyday language and in technical and scientific fields such as software development and computer science . One example of this usage is the term " big data ". When used more specifically to refer to the processing and analysis of sets of data, the term retains its plural form. This usage is common in the natural sciences, life sciences, social sciences, software development and computer science, and grew in popularity in

312-420: A sigma-algebra on X {\displaystyle X} . The entropy of M {\displaystyle M} is H μ ( M ) = sup P ⊆ M H μ ( P ) . {\displaystyle \mathrm {H} _{\mu }(M)=\sup _{P\subseteq M}\mathrm {H} _{\mu }(P).} Finally, the entropy of the probability space

390-583: A climber's guidebook containing practical information on the best way to reach Mount Everest's peak may be considered "knowledge". "Information" bears a diversity of meanings that range from everyday usage to technical use. This view, however, has also been argued to reverse how data emerges from information, and information from knowledge. Generally speaking, the concept of information is closely related to notions of constraint, communication, control, data, form, instruction, knowledge, meaning, mental stimulus, pattern , perception, and representation. Beynon-Davies uses

468-490: A coin with probability p of landing on heads and probability 1 − p of landing on tails. The maximum surprise is when p = 1/2 , for which one outcome is not expected over the other. In this case a coin flip has an entropy of one bit . (Similarly, one trit with equiprobable values contains log 2 ⁡ 3 {\displaystyle \log _{2}3} (about 1.58496) bits of information because it can have one of three values.) The minimum surprise

546-403: A common view, data is collected and analyzed; data only becomes information suitable for making decisions once it has been analyzed in some fashion. One can say that the extent to which a set of data is informative to someone depends on the extent to which it is unexpected by that person. The amount of information contained in a data stream may be characterized by its Shannon entropy . Knowledge

624-498: A competing measure in structures dual to that of subsets of a universal set. Information is quantified as "dits" (distinctions), a measure on partitions. "Dits" can be converted into Shannon's bits , to get the formulas for conditional entropy, and so on. Another succinct axiomatic characterization of Shannon entropy was given by Aczél , Forte and Ng, via the following properties: It was shown that any function H {\displaystyle \mathrm {H} } satisfying

702-668: A description of other data. A similar yet earlier term for metadata is "ancillary data." The prototypical example of metadata is the library catalog, which is a description of the contents of books. Whenever data needs to be registered, data exists in the form of a data document . Kinds of data documents include: Some of these data documents (data repositories, data studies, data sets, and software) are indexed in Data Citation Indexes , while data papers are indexed in traditional bibliographic databases, e.g., Science Citation Index . Gathering data can be accomplished through

780-553: A few decades. Scientific publishers and libraries have been struggling with this problem for a few decades, and there is still no satisfactory solution for the long-term storage of data over centuries or even for eternity. Data accessibility . Another problem is that much scientific data is never published or deposited in data repositories such as databases . In a recent survey, data was requested from 516 studies that were published between 2 and 22 years earlier, but less than one out of five of these studies were able or willing to provide

858-428: A logarithm mediates between these two operations. The conditional entropy and related quantities inherit simple relation, in turn. The measure theoretic definition in the previous section defined the entropy as a sum over expected surprisals μ ( A ) ⋅ ln ⁡ μ ( A ) {\displaystyle \mu (A)\cdot \ln \mu (A)} for an extremal partition. Here

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936-634: A message, as in data compression . For example, consider the transmission of sequences comprising the 4 characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are equally likely (25%), one cannot do better than using two bits to encode each letter. 'A' might code as '00', 'B' as '01', 'C' as '10', and 'D' as '11'. However, if the probabilities of each letter are unequal, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with 2% each, one could assign variable length codes. In this case, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and 'D' as '111'. With this representation, 70% of

1014-478: A particular number will win a lottery has high informational value because it communicates the occurrence of a very low probability event. The information content , also called the surprisal or self-information, of an event E {\displaystyle E} is a function which increases as the probability p ( E ) {\displaystyle p(E)} of an event decreases. When p ( E ) {\displaystyle p(E)}

1092-546: A perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his noisy-channel coding theorem . Entropy in information theory is directly analogous to the entropy in statistical thermodynamics . The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as combinatorics and machine learning . The definition can be derived from

1170-469: A primary source (the researcher is the first person to obtain the data) or a secondary source (the researcher obtains the data that has already been collected by other sources, such as data disseminated in a scientific journal). Data analysis methodologies vary and include data triangulation and data percolation. The latter offers an articulate method of collecting, classifying, and analyzing data using five possible angles of analysis (at least three) to maximize

1248-414: A set of axioms establishing that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable, differential entropy is analogous to entropy. The definition E [ − log ⁡ p ( X ) ] {\displaystyle \mathbb {E} [-\log p(X)]} generalizes the above. The core idea of information theory

1326-409: A variable. The concept of information entropy was introduced by Claude Shannon in his 1948 paper " A Mathematical Theory of Communication ", and is also referred to as Shannon entropy . Shannon's theory defines a data communication system composed of three elements: a source of data, a communication channel , and a receiver. The "fundamental problem of communication" – as expressed by Shannon –

1404-444: Is H μ ( Σ ) {\displaystyle \mathrm {H} _{\mu }(\Sigma )} , that is, the entropy with respect to μ {\displaystyle \mu } of the sigma-algebra of all measurable subsets of X {\displaystyle X} . Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as

1482-553: Is σ μ ( A ) = − ln ⁡ μ ( A ) . {\displaystyle \sigma _{\mu }(A)=-\ln \mu (A).} The expected surprisal of A {\displaystyle A} is h μ ( A ) = μ ( A ) σ μ ( A ) . {\displaystyle h_{\mu }(A)=\mu (A)\sigma _{\mu }(A).} A μ {\displaystyle \mu } -almost partition

1560-541: Is a set family P ⊆ P ( X ) {\displaystyle P\subseteq {\mathcal {P}}(X)} such that μ ( ∪ ⁡ P ) = 1 {\displaystyle \mu (\mathop {\cup } P)=1} and μ ( A ∩ B ) = 0 {\displaystyle \mu (A\cap B)=0} for all distinct A , B ∈ P {\displaystyle A,B\in P} . (This

1638-460: Is a relaxation of the usual conditions for a partition.) The entropy of P {\displaystyle P} is H μ ( P ) = ∑ A ∈ P h μ ( A ) . {\displaystyle \mathrm {H} _{\mu }(P)=\sum _{A\in P}h_{\mu }(A).} Let M {\displaystyle M} be

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1716-438: Is approximately 0.693 n nats or 0.301 n decimal digits. The meaning of the events observed (the meaning of messages ) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution , not the meaning of the events themselves. Another characterization of entropy uses

1794-401: Is central to the definition of information entropy. The connection between thermodynamics and what is now known as information theory was first made by Ludwig Boltzmann and expressed by his equation : where S {\displaystyle S} is the thermodynamic entropy of a particular macrostate (defined by thermodynamic parameters such as temperature, volume, energy, etc.), W

1872-419: Is close to 1, the surprisal of the event is low, but if p ( E ) {\displaystyle p(E)} is close to 0, the surprisal of the event is high. This relationship is described by the function log ⁡ ( 1 p ( E ) ) , {\displaystyle \log \left({\frac {1}{p(E)}}\right),} where log {\displaystyle \log }

1950-425: Is different from data analysis which transforms data and information into insights. Data reporting is the previous step that translates raw data into information. When data is not reported, the problem is known as underreporting ; the opposite problem leads to false positives . Data reporting can be difficult. Census bureaus may hire perhaps hundreds of thousands of workers to achieve the task of counting all of

2028-515: Is distributed according to p : X → [ 0 , 1 ] {\displaystyle p\colon {\mathcal {X}}\to [0,1]} , the entropy is H ( X ) := − ∑ x ∈ X p ( x ) log ⁡ p ( x ) , {\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),} where Σ {\displaystyle \Sigma } denotes

2106-415: Is fairly predictable. We can be fairly certain that, for example, 'e' will be far more common than 'z', that the combination 'qu' will be much more common than any other combination with a 'q' in it, and that the combination 'th' will be more common than 'z', 'q', or 'qu'. After the first few letters one can often guess the rest of the word. English text has between 0.6 and 1.3 bits of entropy per character of

2184-401: Is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his source coding theorem that the entropy represents an absolute mathematical limit on how well data from the source can be losslessly compressed onto

2262-474: Is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the Boltzmann constant . Adding heat to a system increases its thermodynamic entropy because it increases

2340-530: Is that the "informational value" of a communicated message depends on the degree to which the content of the message is surprising. If a highly likely event occurs, the message carries very little information. On the other hand, if a highly unlikely event occurs, the message is much more informative. For instance, the knowledge that some particular number will not be the winning number of a lottery provides very little information, because any particular chosen number will almost certainly not win. However, knowledge that

2418-414: Is the base of the logarithm used. Common values of b are 2, Euler's number e , and 10, and the corresponding units of entropy are the bits for b = 2 , nats for b = e , and bans for b = 10 . In the case of p ( x ) = 0 {\displaystyle p(x)=0} for some x ∈ X {\displaystyle x\in {\mathcal {X}}} ,

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2496-533: Is the expected value operator , and I is the information content of X . I ⁡ ( X ) {\displaystyle \operatorname {I} (X)} is itself a random variable. The entropy can explicitly be written as: H ( X ) = − ∑ x ∈ X p ( x ) log b ⁡ p ( x ) , {\displaystyle \mathrm {H} (X)=-\sum _{x\in {\mathcal {X}}}p(x)\log _{b}p(x),} where b

2574-727: Is the logarithm , which gives 0 surprise when the probability of the event is 1. In fact, log is the only function that satisfies а specific set of conditions defined in section § Characterization . Hence, we can define the information, or surprisal, of an event E {\displaystyle E} by I ( E ) = − log 2 ⁡ ( p ( E ) ) , {\displaystyle I(E)=-\log _{2}(p(E)),} or equivalently, I ( E ) = log 2 ⁡ ( 1 p ( E ) ) . {\displaystyle I(E)=\log _{2}\left({\frac {1}{p(E)}}\right).} Entropy measures

2652-405: Is the trace . At an everyday practical level, the links between information entropy and thermodynamic entropy are not evident. Physicists and chemists are apt to be more interested in changes in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the second law of thermodynamics , rather than an unchanging probability distribution. As the minuteness of

2730-405: Is the awareness of its environment that some entity possesses, whereas data merely communicates that knowledge. For example, the entry in a database specifying the height of Mount Everest is a datum that communicates a precisely-measured value. This measurement may be included in a book along with other data on Mount Everest to describe the mountain in a manner useful for those who wish to decide on

2808-443: Is the longevity of data. Scientific research generates huge amounts of data, especially in genomics and astronomy , but also in the medical sciences , e.g. in medical imaging . In the past, scientific data has been published in papers and books, stored in libraries, but more recently practically all data is stored on hard drives or optical discs . However, in contrast to paper, these storage devices may become unreadable after

2886-471: Is the number of microstates (various combinations of particles in various energy states) that can yield the given macrostate, and k B is the Boltzmann constant . It is assumed that each microstate is equally likely, so that the probability of a given microstate is p i = 1/ W . When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently k B times

2964-402: Is the plural of datum , "(thing) given," and the neuter past participle of dare , "to give". The first English use of the word "data" is from the 1640s. The word "data" was first used to mean "transmissible and storable computer information" in 1946. The expression "data processing" was first used in 1954. When "data" is used more generally as a synonym for "information", it is treated as

3042-625: Is typically cleaned: Outliers are removed, and obvious instrument or data entry errors are corrected. Data can be seen as the smallest units of factual information that can be used as a basis for calculation, reasoning, or discussion. Data can range from abstract ideas to concrete measurements, including, but not limited to, statistics . Thematically connected data presented in some relevant context can be viewed as information . Contextually connected pieces of information can then be described as data insights or intelligence . The stock of insights and intelligence that accumulate over time resulting from

3120-419: Is when p = 0 or p = 1 , when the event outcome is known ahead of time, and the entropy is zero bits. When the entropy is zero bits, this is sometimes referred to as unity, where there is no uncertainty at all – no freedom of choice – no information . Other values of p give entropies between zero and one bits. Information theory is useful to calculate the smallest amount of information required to convey

3198-542: Is worth noting that if we drop the "small for small probabilities" property, then H {\displaystyle \mathrm {H} } must be a non-negative linear combination of the Shannon entropy and the Hartley entropy . The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the expected amount of information learned (or uncertainty eliminated) by revealing

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3276-417: The Boltzmann constant k B indicates, the changes in S / k B for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in data compression or signal processing . In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which

3354-424: The Boltzmann constant , and p i is the probability of a microstate . The Gibbs entropy was defined by J. Willard Gibbs in 1878 after earlier work by Boltzmann (1872). The Gibbs entropy translates over almost unchanged into the world of quantum physics to give the von Neumann entropy introduced by John von Neumann in 1927: where ρ is the density matrix of the quantum mechanical system and Tr

3432-847: The base for the logarithm . Thus, entropy is characterized by the above four properties. This differential equation leads to the solution I ⁡ ( u ) = k log ⁡ u + c {\displaystyle \operatorname {I} (u)=k\log u+c} for some k , c ∈ R {\displaystyle k,c\in \mathbb {R} } . Property 2 gives c = 0 {\displaystyle c=0} . Property 1 and 2 give that I ⁡ ( p ) ≥ 0 {\displaystyle \operatorname {I} (p)\geq 0} for all p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} , so that k < 0 {\displaystyle k<0} . The different units of information ( bits for

3510-415: The binary logarithm log 2 , nats for the natural logarithm ln , bans for the decimal logarithm log 10 and so on) are constant multiples of each other. For instance, in case of a fair coin toss, heads provides log 2 (2) = 1 bit of information, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity, n tosses provide n bits of information, which

3588-442: The 20th and 21st centuries. Some style guides do not recognize the different meanings of the term and simply recommend the form that best suits the target audience of the guide. For example, APA style as of the 7th edition requires "data" to be treated as a plural form. Data, information , knowledge , and wisdom are closely related concepts, but each has its role concerning the other, and each term has its meaning. According to

3666-407: The Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate. In the view of Jaynes (1957), thermodynamic entropy, as explained by statistical mechanics , should be seen as an application of Shannon's information theory: the thermodynamic entropy

3744-493: The above properties must be a constant multiple of Shannon entropy, with a non-negative constant. Compared to the previously mentioned characterizations of entropy, this characterization focuses on the properties of entropy as a function of random variables (subadditivity and additivity), rather than the properties of entropy as a function of the probability vector p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} . It

3822-617: The act of observation as constitutive, is offered as an alternative to data for visual representations in the humanities. The term data-driven is a neologism applied to an activity which is primarily compelled by data over all other factors. Data-driven applications include data-driven programming and data-driven journalism . Data reporting Data reporting is the process of collecting and submitting data . The effective management of any organization relies on accurate data. Inaccurate data reporting can lead to poor decision-making based on erroneous evidence. Data reporting

3900-432: The best method to climb it. Awareness of the characteristics represented by this data is knowledge. Data are often assumed to be the least abstract concept, information the next least, and knowledge the most abstract. In this view, data becomes information by interpretation; e.g., the height of Mount Everest is generally considered "data", a book on Mount Everest geological characteristics may be considered "information", and

3978-434: The binary alphabet. Some special forms of data are distinguished. A computer program is a collection of data, that can be interpreted as instructions. Most computer languages make a distinction between programs and the other data on which programs operate, but in some languages, notably Lisp and similar languages, programs are essentially indistinguishable from other data. It is also useful to distinguish metadata , that is,

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4056-1280: The coin is not fair, but comes up heads or tails with probabilities p and q , where p ≠ q , then there is less uncertainty. Every time it is tossed, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than one full bit of information. For example, if p = 0.7, then H ( X ) = − p log 2 ⁡ ( p ) − q log 2 ⁡ ( q ) = − 0.7 log 2 ⁡ ( 0.7 ) − 0.3 log 2 ⁡ ( 0.3 ) ≈ − 0.7 ⋅ ( − 0.515 ) − 0.3 ⋅ ( − 1.737 ) = 0.8816 < 1. {\displaystyle {\begin{aligned}\mathrm {H} (X)&=-p\log _{2}(p)-q\log _{2}(q)\\&=-0.7\log _{2}(0.7)-0.3\log _{2}(0.3)\\&\approx -0.7\cdot (-0.515)-0.3\cdot (-1.737)\\&=0.8816<1.\end{aligned}}} Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, then, can only decrease from

4134-615: The concept of a sign to differentiate between data and information; data is a series of symbols, while information occurs when the symbols are used to refer to something. Before the development of computing devices and machines, people had to manually collect data and impose patterns on it. With the development of computing devices and machines, these devices can also collect data. In the 2010s, computers were widely used in many fields to collect data and sort or process it, in disciplines ranging from marketing , analysis of social service usage by citizens to scientific research. These patterns in

4212-408: The data are seen as information that can be used to enhance knowledge. These patterns may be interpreted as " truth " (though "truth" can be a subjective concept) and may be authorized as aesthetic and ethical criteria in some disciplines or cultures. Events that leave behind perceivable physical or virtual remains can be traced back through data. Marks are no longer considered data once the link between

4290-543: The efficiency of a source set with n symbols can be defined simply as being equal to its n -ary entropy. See also Redundancy (information theory) . The characterization here imposes an additive property with respect to a partition of a set . Meanwhile, the conditional probability is defined in terms of a multiplicative property, P ( A ∣ B ) ⋅ P ( B ) = P ( A ∩ B ) {\displaystyle P(A\mid B)\cdot P(B)=P(A\cap B)} . Observe that

4368-573: The ethos of data as "given". Peter Checkland introduced the term capta (from the Latin capere , "to take") to distinguish between an immense number of possible data and a sub-set of them, to which attention is oriented. Johanna Drucker has argued that since the humanities affirm knowledge production as "situated, partial, and constitutive," using data may introduce assumptions that are counterproductive, for example that phenomena are discrete or are observer-independent. The term capta , which emphasizes

4446-435: The expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that rolling a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability ( p = 1 / 6 {\displaystyle p=1/6} ) than each outcome of a coin toss ( p = 1 / 2 {\displaystyle p=1/2} ). Consider

4524-413: The following properties. We denote p i = Pr( X = x i ) and Η n ( p 1 , ..., p n ) = Η( X ) . The rule of additivity has the following consequences: for positive integers b i where b 1 + ... + b k = n , Choosing k = n , b 1 = ... = b n = 1 this implies that the entropy of a certain outcome is zero: Η 1 (1) = 0 . This implies that

4602-407: The logarithm is ad hoc and the entropy is not a measure in itself. At least in the information theory of a binary string, log 2 {\displaystyle \log _{2}} lends itself to practical interpretations. Motivated by such relations, a plethora of related and competing quantities have been defined. For example, David Ellerman 's analysis of a "logic of partitions" defines

4680-537: The mark and observation is broken. Mechanical computing devices are classified according to how they represent data. An analog computer represents a datum as a voltage, distance, position, or other physical quantity. A digital computer represents a piece of data as a sequence of symbols drawn from a fixed alphabet . The most common digital computers use a binary alphabet, that is, an alphabet of two characters typically denoted "0" and "1". More familiar representations, such as numbers or letters, are then constructed from

4758-926: The message. Named after Boltzmann's Η-theorem , Shannon defined the entropy Η (Greek capital letter eta ) of a discrete random variable X {\textstyle X} , which takes values in the set X {\displaystyle {\mathcal {X}}} and is distributed according to p : X → [ 0 , 1 ] {\displaystyle p:{\mathcal {X}}\to [0,1]} such that p ( x ) := P [ X = x ] {\displaystyle p(x):=\mathbb {P} [X=x]} : H ( X ) = E [ I ⁡ ( X ) ] = E [ − log ⁡ p ( X ) ] . {\displaystyle \mathrm {H} (X)=\mathbb {E} [\operatorname {I} (X)]=\mathbb {E} [-\log p(X)].} Here E {\displaystyle \mathbb {E} }

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4836-447: The number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer. (See article: maximum entropy thermodynamics ). Maxwell's demon can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as Landauer (from 1961) and co-workers have shown, to function

4914-431: The observation of event i follows from Shannon's solution of the fundamental properties of information : Given two independent events, if the first event can yield one of n equiprobable outcomes and another has one of m equiprobable outcomes then there are mn equiprobable outcomes of the joint event. This means that if log 2 ( n ) bits are needed to encode the first value and log 2 ( m ) to encode

4992-586: The only possible values of I {\displaystyle \operatorname {I} } are I ⁡ ( u ) = k log ⁡ u {\displaystyle \operatorname {I} (u)=k\log u} for k < 0 {\displaystyle k<0} . Additionally, choosing a value for k is equivalent to choosing a value x > 1 {\displaystyle x>1} for k = − 1 / log ⁡ x {\displaystyle k=-1/\log x} , so that x corresponds to

5070-522: The petabyte scale. Using traditional data analysis methods and computing, working with such large (and growing) datasets is difficult, even impossible. (Theoretically speaking, infinite data would yield infinite information, which would render extracting insights or intelligence impossible.) In response, the relatively new field of data science uses machine learning (and other artificial intelligence (AI)) methods that allow for efficient applications of analytic methods to big data. The Latin word data

5148-404: The problem of reproducibility is the attempt to require FAIR data , that is, data that is Findable, Accessible, Interoperable, and Reusable. Data that fulfills these requirements can be used in subsequent research and thus advances science and technology. Although data is also increasingly used in other fields, it has been suggested that the highly interpretive nature of them might be at odds with

5226-545: The remaining randomness in the random variable X {\displaystyle X} given the random variable Y {\displaystyle Y} . Entropy can be formally defined in the language of measure theory as follows: Let ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} be a probability space . Let A ∈ Σ {\displaystyle A\in \Sigma } be an event . The surprisal of A {\displaystyle A}

5304-450: The requested data. Overall, the likelihood of retrieving data dropped by 17% each year after publication. Similarly, a survey of 100 datasets in Dryad found that more than half lacked the details to reproduce the research results from these studies. This shows the dire situation of access to scientific data that is not published or does not have enough details to be reproduced. A solution to

5382-457: The research's objectivity and permit an understanding of the phenomena under investigation as complete as possible: qualitative and quantitative methods, literature reviews (including scholarly articles), interviews with experts, and computer simulation. The data is thereafter "percolated" using a series of pre-determined steps so as to extract the most relevant information. An important field in computer science , technology , and library science

5460-416: The residents of a country. Teachers use data from student assessments to determine grades; manufacturers rely on sales data from retailers to indicate which products should have increased production, and which should be curtailed or discontinued. Shannon entropy In information theory , the entropy of a random variable quantifies the average level of uncertainty or information associated with

5538-477: The second, one needs log 2 ( mn ) = log 2 ( m ) + log 2 ( n ) to encode both. Shannon discovered that a suitable choice of I {\displaystyle \operatorname {I} } is given by: I ⁡ ( p ) = log ⁡ ( 1 p ) = − log ⁡ ( p ) . {\displaystyle \operatorname {I} (p)=\log \left({\tfrac {1}{p}}\right)=-\log(p).} In fact,

5616-419: The sum over the variable's possible values. The choice of base for log {\displaystyle \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

5694-472: The synthesis of data into information, can then be described as knowledge . Data has been described as "the new oil of the digital economy ". Data, as a general concept , refers to the fact that some existing information or knowledge is represented or coded in some form suitable for better usage or processing . Advances in computing technologies have led to the advent of big data , which usually refers to very large quantities of data, usually at

5772-439: The time only one bit needs to be sent, 26% of the time two bits, and only 4% of the time 3 bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the high prevalence of 'A' followed by 'B' – together 96% of characters). The calculation of the sum of probability-weighted log probabilities measures and captures this effect. English text, treated as a string of characters, has fairly low entropy; i.e. it

5850-536: The value associated with uniform probability. The extreme case is that of a double-headed coin that never comes up tails, or a double-tailed coin that never results in a head. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain. To understand the meaning of −Σ p i log( p i ) , first define an information function I in terms of an event i with probability p i . The amount of information acquired due to

5928-450: The value of a random variable X : The inspiration for adopting the word entropy in information theory came from the close resemblance between Shannon's formula and very similar known formulae from statistical mechanics . In statistical thermodynamics the most general formula for the thermodynamic entropy S of a thermodynamic system is the Gibbs entropy where k B is

6006-1486: The value of the corresponding summand 0 log b (0) is taken to be 0 , which is consistent with the limit : lim p → 0 + p log ⁡ ( p ) = 0. {\displaystyle \lim _{p\to 0^{+}}p\log(p)=0.} One may also define the conditional entropy of two variables X {\displaystyle X} and Y {\displaystyle Y} taking values from sets X {\displaystyle {\mathcal {X}}} and Y {\displaystyle {\mathcal {Y}}} respectively, as: H ( X | Y ) = − ∑ x , y ∈ X × Y p X , Y ( x , y ) log ⁡ p X , Y ( x , y ) p Y ( y ) , {\displaystyle \mathrm {H} (X|Y)=-\sum _{x,y\in {\mathcal {X}}\times {\mathcal {Y}}}p_{X,Y}(x,y)\log {\frac {p_{X,Y}(x,y)}{p_{Y}(y)}},} where p X , Y ( x , y ) := P [ X = x , Y = y ] {\displaystyle p_{X,Y}(x,y):=\mathbb {P} [X=x,Y=y]} and p Y ( y ) = P [ Y = y ] {\displaystyle p_{Y}(y)=\mathbb {P} [Y=y]} . This quantity should be understood as

6084-404: 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 {\displaystyle X} , which takes values in the set X {\displaystyle {\mathcal {X}}} and

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