Cloudflare, Inc., is an American technology company headquartered in San Francisco, California, that provides a range of internet services, including content delivery network (CDN) services, cloud cybersecurity, DDoS mitigation, and ICANN-accredited domain registration. The company's services act primarily as a reverse proxy between website visitors and a customer's hosting provider, improving performance and protecting against malicious traffic. Cloudflare was founded in 2009 by Matthew Prince, Lee Holloway, and Michelle Zatlyn. The company went public on the New York Stock Exchange in 2019 under the ticker symbol NET. Cloudflare has since expanded its offerings to include edge computing through its Workers platform, a public DNS resolver (1.1.1.1), and a VPN-like service known as WARP. In recent years, the company has integrated artificial intelligence into its infrastructure, acquiring companies such as Replicate and launching tools to manage AI bots and scrapers. According to W3Techs, Cloudflare is used by approximately 21.3% of all websites on the Internet as of January 2026. The company has been the subject of controversy regarding its policy of content neutrality. While Cloudflare executives have historically advocated for remaining a neutral infrastructure provider, the company has terminated services for specific high-profile websites associated with hate speech and violence, including The Daily Stormer, 8chan, and Kiwi Farms, following significant public pressure. Cloudflare has also faced criticism and litigation regarding copyright infringement by websites using its services, notably losing a lawsuit against Japanese publishers in 2025. The company experienced significant global outages in late 2025 which disrupted services for major platforms internationally. == History == Cloudflare was founded on July 26, 2009, by Matthew Prince, Lee Holloway, and Michelle Zatlyn. Prince and Holloway had previously collaborated on Project Honey Pot, a product of Unspam Technologies that partly inspired the basis of Cloudflare. In 2009, the company was venture-capital funded. On August 15, 2019, Cloudflare submitted its S-1 filing for an initial public offering on the New York Stock Exchange under the stock ticker NET. It opened for public trading on September 13, 2019, at $15 per share. According to the company, the name 'Cloudflare' was chosen, over the initial 'WebWall', because it best described what they were trying to do: build a "firewall in the cloud." In 2020, Cloudflare co-founder and COO Michelle Zatlyn was named president. Cloudflare has acquired web-services and security companies, including StopTheHacker (February 2014), CryptoSeal (June 2014), Eager Platform Co. (December 2016), Neumob (November 2017), S2 Systems (January 2020), Linc (December 2020), Zaraz (December 2021), Vectrix (February 2022), Area 1 Security (February 2022), Nefeli Networks (March 2024), BastionZero (May 2024), and Kivera (October 2024). Replicate (November 2025), and Human Native (January 2026). Since at least 2017, Cloudflare has used a wall of lava lamps at its San Francisco headquarters as a source of randomness for encryption keys, alongside double pendulums at its London offices and a Geiger counter at its Singapore offices. The lava lamp installation implements the Lavarand method, where a camera transforms the unpredictable shapes of the "lava" blobs into a digital image. In Q4 2022, Cloudflare provided paid services to 162,086 customers. In October 2024, Cloudflare won a lawsuit against patent troll Sable Networks. Sable paid Cloudflare $225,000, granted it a royalty-free license to its patent portfolio, and dedicated its patents to the public by abandoning its patent rights. In November 2025, it was announced Cloudflare had agreed to acquire Replicate, a San Francisco–based platform that enables software developers to run, fine-tune, and deploy open-source machine-learning models via an API without managing infrastructure. In January 2026, Cloudflare released an analysis regarding BGP routing leaks observed from the Venezuelan state-owned ISP CANTV (AS8048), which occurred on January 2 coincides with the arrest of Nicolás Maduro. While some security researchers had speculated that the outages were linked to U.S. cyber operations, Cloudflare's data indicated that the anomalies were consistent with a pattern of "insufficient routing export and import policies" by the ISP rather than malicious external interference. In January 2026, Cloudflare acquired Human Native, an AI data marketplace that brokers transactions between developers and content creators, for an undisclosed amount. On January 16, 2026, Cloudflare acquired The Astro Technology Company, the developers behind the open-source web framework Astro. In May 2026, Cloudflare announced the elimination of approximately 1,100 positions, around 20 percent of its workforce, in a restructuring the company attributed to the rapid adoption of artificial intelligence tools. The announcement coincided with the company's first-quarter 2026 earnings, which reported a record $639.8 million in quarterly revenue, a 34 percent year-over-year increase. CEO Matthew Prince stated the cuts were not driven by performance concerns but reflected roles made obsolete by AI, and that Cloudflare expected to employ more people by the end of 2027 than at any point during 2026. == Products == Cloudflare provides network and security products for consumers and businesses, utilizing edge computing, reverse proxies for web traffic, data center interconnects, and a content distribution network to serve content across its network of servers. It supports transport layer protocols TCP, UDP, QUIC, and many application layer protocols such as DNS over HTTPS, SMTP, and HTTP/2 with support for HTTP/2 Server Push. As of 2023, Cloudflare handles an average of 45 million HTTP requests per second. As of 2024, Cloudflare servers are powered by AMD EPYC 9684X processors. Cloudflare also provides analysis and reports on large-scale outages, including Verizon's October 2024 outage. === Artificial intelligence === In 2023, Cloudflare launched "Workers AI", a framework allowing for use of Nvidia GPU's within Cloudflare's network. In 2024, Cloudflare launched a tool that prevents bots from scraping websites. To build automatic bot detector models, the company analyzed "AI" bots and crawler traffic. The company also launched an "AI" assistant to generate charts based on queries by leveraging "Workers AI". Cloudflare announced plans in September 2024 to launch a marketplace where website owners can sell "AI" model providers access to scrape their site's content. In March 2025, Cloudflare announced a new feature called "AI Labyrinth", which combats unauthorized "AI" data scraping by serving fake "AI"-generated content to LLM bots. In July, the company rolled out a permission-based setting to allow websites to automatically block online bots from scraping data and content. Cloudflare released AutoRAG into beta in 2025. AutoRAG (retrieval augmented generation) creates a vector database of a website's unstructured content to identify relationships between concepts. It is part of an initiative with Microsoft, alongside their NLWeb standard, to make websites easier for people and automated systems to query. Cloudflare and GoDaddy partnered in April 2026 to enable AI Crawl Control features on GoDaddy hosted websites. This would allow site owners to decide how AI bot crawlers interact with their content. === DDoS mitigation === Cloudflare provides free and paid DDoS mitigation services that protect customers from distributed denial of service (DDoS) attacks. Cloudflare received media attention in June 2011 for providing DDoS mitigation for the website of LulzSec, a black hat hacking group. In March 2013, The Spamhaus Project was targeted by a DDoS attack that Cloudflare reported exceeded 300 gigabits per second (Gbit/s). Patrick Gilmore, of Akamai, stated that at the time it was "the largest publicly announced DDoS attack in the history of the Internet". While trying to defend Spamhaus against the DDoS attacks, Cloudflare ended up being attacked as well; Google and other companies eventually came to Spamhaus' defense and helped it to absorb the unprecedented amount of attack traffic. In 2014, Cloudflare began providing free DDoS mitigation for artists, activists, journalists, and human rights groups under the name "Project Galileo". In 2017, it extended the service to electoral infrastructure and political campaigns under the name "Athenian Project". By 2025, more than 2,900 users and organizations were participating in Project Galileo, including 31 US states. In February 2014, Cloudflare claimed to have mitigated an NTP reflection attack against an unnamed European customer, which it stated peaked at 400 Gbit/s. In November 2014, it reported a 500 Gbit/s DDoS attack in Hong Kong. In July 2021, the company claimed to have absorbed a DDoS atta
AltStore
AltStore is an alternative app store for the iOS and iPadOS[1] mobile operating systems, which allows users to download applications that are not available on the App Store, most commonly tweaked apps, jailbreak apps, and apps including paid apps on the app store. It was publicly announced on September 25, 2019, and launched on September 28. == History == Riley Testut is an American developer who began to work on AltStore after Apple declined to allow his Nintendo emulator Delta on the App Store. Since Xcode allowed him to temporarily install his Delta app to his iOS device for 7 days of testing, he created AltStore in 2019 to replicate this functionality, which could be extended to other .ipa files. As of 2022, AltStore had been downloaded 1.5 million times. In the following years, AltStore expanded beyond its initial sideloading functionality. The platform was founded by Testut, with Shane Gill later joining as co-founder. AltStore was initially supported through Patreon contributions from its user community, and later saw increased adoption following regulatory developments in the European Union that enabled broader third-party app distribution. The project has also been involved in notable industry collaborations, including a partnership with Epic Games. == Features == AltStore exploits a loophole in the Xcode developer platform, which allows developers to sideload their own apps which they are working on without needing to jailbreak. Sideloaded apps are signed like a developer project for testing and will expire after 7 days with a free account or one year with a paid developer account, by which they will need to be refreshed or reinstalled.
Hapax legomenon
In corpus linguistics, a hapax legomenon ( also or ; pl. hapax legomena; sometimes abbreviated to hapax, plural hapaxes) is a word or an expression that occurs only once within a context: either in the written record of an entire language, in the works of an author, or in a single text. The term is also sometimes used to describe a word that occurs in just one of an author's works but more than once in that particular work. Hapax legomenon is a transliteration of Greek ἅπαξ λεγόμενον, meaning "said once". The related terms dis legomenon, tris legomenon, and tetrakis legomenon respectively (, , ) refer to double, triple, or quadruple occurrences, but are far less commonly used. Hapax legomena are quite common, as predicted by Zipf's law, which states that the frequency of any word in a corpus is inversely proportional to its rank in the frequency table. For large corpora, about 40% to 60% of the words are hapax legomena, and another 10% to 15% are dis legomena. Thus, in the Brown Corpus of American English, about half of the 50,000 distinct words are hapax legomena within that corpus. Hapax legomenon refers to the appearance of a word or an expression in a body of text, not to either its origin or its prevalence in speech. It thus differs from a nonce word, which may never be recorded, may find currency and may be widely recorded, or may appear several times in the work which coins it, and so on. == Significance == Hapax legomena in ancient texts are usually difficult to decipher, since it is easier to infer meaning from multiple contexts than from just one. For example, many of the remaining undeciphered Mayan glyphs are hapax legomena, and Biblical (particularly Hebrew; see § Hebrew) hapax legomena sometimes pose problems in translation. Hapax legomena also pose challenges in natural language processing. Some scholars consider Hapax legomena useful in determining the authorship of written works. P. N. Harrison, in The Problem of the Pastoral Epistles (1921) made hapax legomena popular among Bible scholars, when he argued that there are considerably more of them in the three Pastoral Epistles than in other Pauline Epistles. He argued that the number of hapax legomena in a putative author's corpus indicates his or her vocabulary and is characteristic of the author as an individual. Harrison's theory has faded in significance due to a number of problems raised by other scholars. For example, in 1896, W. P. Workman found the following numbers of hapax legomena in each Pauline Epistle: At first glance, the last three totals (for the Pastoral Epistles) are not out of line with the others. To take account of the varying length of the epistles, Workman also calculated the average number of hapax legomena per page of the Greek text, which ranged from 3.6 to 13, as summarized in the diagram on the right. Although the Pastoral Epistles have more hapax legomena per page, Workman found the differences to be moderate in comparison to the variation among other Epistles. This was reinforced when Workman looked at several plays by Shakespeare, which showed similar variations (from 3.4 to 10.4 per page of Irving's one-volume edition), as summarized in the second diagram on the right. Apart from author identity, there are several other factors that can explain the number of hapax legomena in a work: text length: this directly affects the expected number and percentage of hapax legomena; the brevity of the Pastoral Epistles also makes any statistical analysis problematic. text topic: if the author writes on different subjects, of course many subject-specific words will occur only in limited contexts. text audience: if the author is writing to a peer rather than a student, or their spouse rather than their employer, again quite different vocabulary will appear. time: over the course of years, both the language and an author's knowledge and use of language will change. In the particular case of the Pastoral Epistles, all of these variables are quite different from those in the rest of the Pauline corpus, and hapax legomena are no longer widely accepted as strong indicators of authorship; those who reject Pauline authorship of the Pastorals rely on other arguments. There are also subjective questions over whether two forms amount to "the same word": dog vs. dogs, clue vs. clueless, sign vs. signature; many other gray cases also arise. The Jewish Encyclopedia points out that, although there are 1,500 hapaxes in the Hebrew Bible, only about 400 are not obviously related to other attested word forms. A final difficulty with the use of hapax legomena for authorship determination is that there is considerable variation among works known to be by a single author, and disparate authors often show similar values. In other words, hapax legomena are not a reliable indicator. Authorship studies now usually use a wide range of measures to look for patterns rather than relying upon single measurements. == Computer science == In the fields of computational linguistics and natural language processing (NLP), esp. corpus linguistics and machine-learned NLP, it is common to disregard hapax legomena (and sometimes other infrequent words), as they are likely to have little value for computational techniques. This disregard has the added benefit of significantly reducing the memory use of an application, since, by Zipf's law, many words are hapax legomena. == Examples == The following are some examples of hapax legomena in languages or corpora. === Arabic === In the Qurʾān: The proper nouns Iram (Q 89:7, Iram of the Pillars), Bābil (Q 2:102, Babylon), Bakka(t) (Q 3:96, Bakkah), Jibt (Q 4:51), Ramaḍān (Q 2:185, Ramadan), ar-Rūm (Q 30:2, Byzantine Empire), Tasnīm (Q 83:27), Qurayš (Q 106:1, Quraysh), Majūs (Q 22:17, Magian/Zoroastrian), Mārūt (Q 2:102, Harut and Marut), Makka(t) (Q 48:24, Mecca), Nasr (Q 71:23), (Ḏū) an-Nūn (Q 21:87) and Hārūt (Q 2:102, Harut and Marut) occur only once. zanjabīl (زَنْجَبِيل – ginger) is a Qurʾānic hapax (Q 76:17). zamharīr (زَمْهَرِيرًۭ) is a Qurʾānic hapax (Q 76:13), usually glossed as referring to extreme cold. The epitheton ornans aṣ-ṣamad (الصَّمَد – the One besought) is a Qurʾānic hapax (Q 112:2). ṭūd (طُودْ - mountain) is a Qurʾānic hapax (Q 26:63). === Chinese and Japanese === Classical Chinese and Japanese literature contains many Chinese characters that feature only once in the corpus, and their meaning and pronunciation has often been lost. Known in Japanese as kogo (孤語), literally "lonely characters", these can be considered a type of hapax legomenon. For example, the Classic of Poetry (c. 1000 BC) uses the character 篪 exactly once in the verse 「伯氏吹塤, 仲氏吹篪」, and it was only through the discovery of a description by Guo Pu (276–324 AD) that the character could be associated with a specific type of ancient flute. === English === It is fairly common for authors to "coin" new words to convey a particular meaning or for the sake of entertainment, without any suggestion that they are "proper" words. For example, P.G. Wodehouse and Lewis Carroll frequently coined novel words. Indexy, below, appears to be an example of this. Flother, as a synonym for snowflake, is a hapax legomenon of written English found in a manuscript entitled The XI Pains of Hell (c. 1275). Honorificabilitudinitatibus is a hapax legomenon of Shakespeare's works, coming from Erasmus' Adagia Indexy, in Bram Stoker's Dracula, used as an adjective to describe a situational state with no other further use in the language: "If that man had been an ordinary lunatic I would have taken my chance of trusting him; but he seems so mixed up with the Count in an indexy kind of way that I am afraid of doing anything wrong by helping his fads." Manticratic, meaning "of the rule by the Prophet's family or clan", was apparently invented by T. E. Lawrence and appears once in Seven Pillars of Wisdom. Nortelrye, a word for "education", occurs only once in Chaucer's The Reeve's Tale. Sassigassity, perhaps with the meaning of "audacity", occurs only once in Dickens's short story "A Christmas Tree". Slæpwerigne, "sleep-weary", occurs exactly once in the Old English corpus, in the Exeter Book. There is debate over whether it means "weary with sleep" or "weary for sleep". === German === The name of the 9th-century poem Muspilli is a back-formation from "muspille", Old High German hapax legomenon of unclear meaning only found in this text (see Muspilli § Etymology for discussion). === Ancient Greek === According to classical scholar Clyde Pharr, "the Iliad has 1,097 hapax legomena, while the Odyssey has 868". Others have defined the term differently, however, and count as few as 303 in the Iliad and 191 in the Odyssey. panaōrios (παναώριος), ancient Greek for "very untimely", is one of many words that occur only once in the Iliad. The Greek New Testament contains 686 local hapax legomena, which are sometimes called "New Testament hapaxes". 62 of these occur in 1 Peter and 54 occur in 2 Peter
Best AI Essay Writers in 2026
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Glottochronology
Glottochronology (from Attic Greek γλῶττα 'tongue, language' and χρόνος 'time') is the part of lexicostatistics which involves comparative linguistics and deals with the chronological relationship between languages. The idea was developed by Morris Swadesh in the 1950s in his article on Salish internal relationships. He developed the idea under two assumptions: there indeed exists a relatively stable basic vocabulary (referred to as Swadesh lists) in all languages of the world; and, any replacements happen in a way analogous to radioactive decay in a constant percentage per time elapsed. Using mathematics and statistics, Swadesh developed an equation to determine when languages separated and give an approximate time of when the separation occurred. His methods aimed to aid linguistic anthropologists by giving them a definitive way to determine a separation date between two languages. The formula provides an approximate number of centuries since two languages were supposed to have separated from a singular common ancestor. His methods also purported to provide information on when ancient languages may have existed. Despite multiple studies and literature containing the information of glottochronology, it is not widely used today and is surrounded with controversy. Glottochronology tracks language separation from thousands of years ago but many linguists are skeptical of the concept because it is more of a 'probability' rather than a 'certainty.' On the other hand, some linguists may say that glottochronology is gaining traction because of its relatedness to archaeological dates. Glottochronology is not as accurate as archaeological data, but some linguists still believe that it can provide a solid estimate. Over time many different extensions of the Swadesh method evolved; however, Swadesh's original method is so well known that 'glottochronology' is usually associated with him. == Methodology == The original method of glottochronology presumed that the core vocabulary of a language is replaced at a constant (or constant average) rate across all languages and cultures and so can be used to measure the passage of time. The process makes use of a list of lexical terms and morphemes which are similar to multiple languages. Lists were compiled by Morris Swadesh and assumed to be resistant against borrowing (originally designed in 1952 as a list of 200 items, but the refined 100-word list in Swadesh (1955) is much more common among modern day linguists). The core vocabulary was designed to encompass concepts common to every human language such as personal pronouns, body parts, heavenly bodies and living beings, verbs of basic actions, numerals, basic adjectives, kin terms, and natural occurrences and events. Through a basic word list, one eliminates concepts that are specific to a particular culture or time period. It has been found through differentiating word lists that the ideal is really impossible and that the meaning set may need to be tailored to the languages being compared. Word lists are not homogenous throughout studies and they are often changed and designed to suit both languages being studied. Linguists find that it is difficult to find a word list where all words used are culturally unbiased. Many alternative word lists have been compiled by other linguists and often use fewer meaning slots. The percentage of cognates (words with a common origin) in the word lists is then measured. The larger the percentage of cognates, the more recently the two languages being compared are presumed to have separated. === Glottochronologic constant === Determining word lists rely on morpheme decay or change in vocabulary. Morpheme decay must stay at a constant rate for glottochronology to be applied to a language. This leads to a critique of the glottochronologic formula because some linguists argue that the morpheme decay rate is not guaranteed to stay the same throughout history. American Linguist Robert Lees obtained a value for the "glottochronological constant" (r) of words by considering the known changes in 13 pairs of languages using the 200 word list. He obtained a value of 0.8048 ± 0.0176 with 90% confidence. For his 100-word list Swadesh obtained a value of 0.86, the higher value reflecting the elimination of semantically unstable words. === Divergence time === The basic formula of glottochronology proposed by Morris Swadesh is: t = − ln ( c ) 2 ln ( r ) {\displaystyle t=-{\frac {\ln(c)}{2\ln(r)}}} t = a given period of time from one stage of the language to another (measured in millennia), c = proportion of wordlist items retained at the end of that period and r = rate of replacement for that word list. By testing historically verifiable cases in which t is known by nonlinguistic data (such as the approximate distance from Classical Latin to modern Romance languages), Swadesh arrived at the empirical value of approximately 0.14 for L, (c?) which means that the rate of replacement constitutes around 14 words from the 100-wordlist per millennium. This is represented in the table below. === Results === Glottochronology was applied to a range of language families, including Salishan, Indo-European, Japonic, Afro-Asiatic, Chinese and Mayan and other American languages. For Amerind, correlations have been obtained with radiocarbon dating and blood groups as well as archaeology. === Example Wordlist === Below is an example of a basic word list composed of basic Turkish words and their English translations. == Discussion == The concept of language change is old, and its history is reviewed in Hymes (1973) and Wells (1973). In some sense, glottochronology is a reconstruction of history and can often be closely related to archaeology. Many linguistic studies find the success of glottochronology to be found alongside archaeological data. Glottochronology itself dates back to the mid-20th century. An introduction to the subject is given in Embleton (1986) and in McMahon and McMahon (2005). Glottochronology has been controversial ever since, partly because of issues of accuracy but also because of the question of whether its basis is sound (for example, Bergsland 1958; Bergsland and Vogt 1962; Fodor 1961; Chrétien 1962; Guy 1980). The concerns have been addressed by Dobson et al. (1972), Dyen (1973) and Kruskal, Dyen and Black (1973). The assumption of a single-word replacement rate can distort the divergence-time estimate when borrowed words are included (Thomason and Kaufman 1988). The presentations vary from "Why linguists don't do dates" to the one by Starostin discussed below. Since its original inception, glottochronology has been rejected by many linguists, mostly Indo-Europeanists of the school of the traditional comparative method. Criticisms have been answered in particular around three points of discussion: Criticism levelled against the higher stability of lexemes in Swadesh lists alone (Haarmann 1990) misses the point because a certain amount of losses only enables the computations (Sankoff 1970). The non-homogeneity of word lists often leads to lack of understanding between linguists. Linguists also have difficulties finding a completely unbiased list of basic cultural words. it can take a long time for linguists to find a viable word list which can take several test lists to find a usable list. Traditional glottochronology presumes that language changes at a stable rate. Thus, in Bergsland & Vogt (1962), the authors make an impressive demonstration, on the basis of actual language data verifiable by extralinguistic sources, that the "rate of change" for Icelandic constituted around 4% per millennium, but for closely connected Riksmal (Literary Norwegian), it would amount to as much as 20% (Swadesh's proposed "constant rate" was supposed to be around 14% per millennium). That and several other similar examples effectively proved that Swadesh's formula would not work on all available material, which is a serious accusation since evidence that can be used to "calibrate" the meaning of L (language history recorded during prolonged periods of time) is not overwhelmingly large in the first place. It is highly likely that the chance of replacement is different for every word or feature ("each word has its own history", among hundreds of other sources:). That global assumption has been modified and downgraded to single words, even in single languages, in many newer attempts (see below). There is a lack of understanding of Swadesh's mathematical/statistical methods. Some linguists reject the methods in full because the statistics lead to 'probabilities' when linguists trust 'certainties' more. A serious argument is that language change arises from socio-historical events that are, of course, unforeseeable and, therefore, uncomputable. == Modifications == Somewhere in between the original concept of Swadesh and the rejection of glottochronology in its entirety lies the idea that glottochronology as a formal method of linguistic
Evolvability (computer science)
The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ( f , r ) = Prob ( f ( x ) = r ( x ) ) − Prob ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ( f , r ′ ) − Perf s ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ( f , r ′ ) − Perf s ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)
P4-metric
The P4 metric (also known as FS or Symmetric F ) enables performance evaluation of a binary classifier. The P4 metric is calculated from precision, recall, specificity, and NPV (negative predictive value). The definition of the P4 metric is similar to that of the F1 metric, however the P4 metric definition addresses criticisms leveled against the definition of the F1 metric. The definition of the P4 metric may, therefore, be understood as an extension of the F1 metric. Like the other known metrics, the P4 metric is a function of: TP (true positives), TN (true negatives), FP (false positives), FN (false negatives). == Justification == The key concept of the P4 metric is to leverage the four key conditional probabilities: P ( + ∣ C + ) {\displaystyle P(+\mid C{+})} — the probability that the sample is positive, provided the classifier result was positive. P ( C + ∣ + ) {\displaystyle P(C{+}\mid +)} — the probability that the classifier result will be positive, provided the sample is positive. P ( C − ∣ − ) {\displaystyle P(C{-}\mid -)} — the probability that the classifier result will be negative, provided the sample is negative. P ( − ∣ C − ) {\displaystyle P(-\mid C{-})} — the probability the sample is negative, provided the classifier result was negative. The main assumption behind this metric is that all the probabilities mentioned above are close to 1 for a properly designed binary classifier. Indeed, P 4 = 1 {\displaystyle \mathrm {P} _{4}=1} if, and only if, all of the probabilities above are equal to 1. Another important feature is that P 4 {\displaystyle \mathrm {P} _{4}} tends to zero any of the above probabilities tend to zero. == Definition == P4 is defined as a harmonic mean of four key conditional probabilities: P 4 = 4 1 P ( + ∣ C + ) + 1 P ( C + ∣ + ) + 1 P ( C − ∣ − ) + 1 P ( − ∣ C − ) = 4 1 p r e c i s i o n + 1 r e c a l l + 1 s p e c i f i c i t y + 1 N P V . {\displaystyle \mathrm {P} _{4}={\frac {4}{{\frac {1}{P(+\mid C{+})}}+{\frac {1}{P(C{+}\mid +)}}+{\frac {1}{P(C{-}\mid -)}}+{\frac {1}{P(-\mid C{-})}}}}={\frac {4}{{\frac {1}{\mathit {precision}}}+{\frac {1}{\mathit {recall}}}+{\frac {1}{\mathit {specificity}}}+{\frac {1}{\mathit {NPV}}}}}.} In terms of TP,TN,FP,FN it can be calculated as follows: P 4 = 4 ⋅ T P ⋅ T N 4 ⋅ T P ⋅ T N + ( T P + T N ) ⋅ ( F P + F N ) . {\displaystyle \mathrm {P} _{4}={\frac {4\cdot \mathrm {TP} \cdot \mathrm {TN} }{4\cdot \mathrm {TP} \cdot \mathrm {TN} +(\mathrm {TP} +\mathrm {TN} )\cdot (\mathrm {FP} +\mathrm {FN} )}}.} == Evaluation of the binary classifier performance == Evaluating the performance of binary classifiers is a multidisciplinary concept. It spans from the evaluation of medical tests, psychiatric tests to machine learning classifiers from a variety of fields. Thus, many of the metrics in use exist under several names, some defined independently. == Properties of P4 metric == Symmetry — contrasting to the F1 metric, P4 is symmetrical. It means - it does not change its value when dataset labeling is changed - positives named negatives and negatives named positives. Range: P 4 ∈ [ 0 , 1 ] {\displaystyle \mathrm {P} _{4}\in [0,1]} . Achieving P 4 ≈ 1 {\displaystyle \mathrm {P} _{4}\approx 1} requires all the key four conditional probabilities being close to 1. For P 4 ≈ 0 {\displaystyle \mathrm {P} _{4}\approx 0} it is sufficient that one of the key four conditional probabilities is close to 0. == Examples, comparing with the other metrics == Dependency table for selected metrics ("true" means depends, "false" - does not depend): Metrics that do not depend on a given probability are prone to misrepresentation when the probability approaches 0. === Example 1: Rare disease detection test === Let us consider a medical test used to detect a rare disease. Suppose a population size of 100000 and 0.05% of the population is infected. Further suppose the following test performance: 95% of all positive individuals are classified correctly (TPR=0.95) and 95% of all negative individuals are classified correctly (TNR=0.95). In such a case, due to high population imbalance and in spite of having high test accuracy (0.95), the probability that an individual who has been classified as positive is in fact positive is very low: P ( + ∣ C + ) = 0.0095. {\displaystyle P(+\mid C{+})=0.0095.} We can observe how this low probability is reflected in some of the metrics: P 4 = 0.0370 {\displaystyle \mathrm {P} _{4}=0.0370} , F 1 = 0.0188 {\displaystyle \mathrm {F} _{1}=0.0188} , J = 0.9100 {\displaystyle \mathrm {J} =\mathbf {0.9100} } (Informedness / Youden index), M K = 0.0095 {\displaystyle \mathrm {MK} =0.0095} (Markedness). === Example 2: Image recognition — cats vs dogs === Consider the problem of training a neural network based image classifier with only two types of images: those containing dogs (labeled as 0) and those containing cats (labeled as 1). Thus, the goal is to distinguish between the cats and dogs. Suppose that the classifier overpredicts in favour of cats ("positive" samples): 99.99% of cats are classified correctly and only 1% of dogs are classified correctly. Further, suppose that the image dataset consists of 100000 images, 90% of which are pictures of cats and 10% are pictures of dogs. In this situation, the probability that the picture containing dog will be classified correctly is pretty low: P ( C − | − ) = 0.01. {\displaystyle P(C-|-)=0.01.} Not all metrics are notice this low probability: P 4 = 0.0388 {\displaystyle \mathrm {P} _{4}=0.0388} , F 1 = 0.9478 {\displaystyle \mathrm {F} _{1}=\mathbf {0.9478} } , J = 0.0099 {\displaystyle \mathrm {J} =0.0099} (Informedness / Youden index), M K = 0.8183 {\displaystyle \mathrm {MK} =\mathbf {0.8183} } (Markedness).