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  Advanced automation functions

  In automation production technology the actions performed by an automated process are executed by a program of instructions which is run during a work cycle. To execute work cycle programs, an automated system should be available to execute these advanced functions. == Safety monitoring == If there is a need for workers in an automated system, a safety monitoring is required for the occupational safety and health of the workers. In a safety monitoring various steps can take place including a complete stop of the system, sounding an alarm or reducing the operating speed. Usually, limiting switches are sensors like temperature probes, heat and smoke detectors or pressure sensitive floor pads. == Maintenance and repair diagnostics == There are three modes of operations which are used in a cycle of maintenance and repair diagnostics: status monitoring, failure diagnostics and recommendation of the repair procedure. In the status monitoring mode, the current system status is displayed. The failure diagnostics mode takes place when a failure occurs. The system will then suggest an adequate repair procedure to a team of experts. == Error detection and recovery == The error detection mode is a step to determine if and when a failure occurs in automated system. The possible errors can be divided into three categories. random errors, systematic errors and aberrations. While in the error recovery mode, remedy actions take place for all detected errors.

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  Predictor–corrector method

  In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point. The next, "corrector" step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function's value at the same subsequent point. == Predictor–corrector methods for solving ODEs == When considering the numerical solution of ordinary differential equations (ODEs), a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step. === Example: Euler method with the trapezoidal rule === A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Consider the differential equation y ′ = f ( t , y ) , y ( t 0 ) = y 0 , {\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0},} and denote the step size by h {\displaystyle h} . First, the predictor step: starting from the current value y i {\displaystyle y_{i}} , calculate an initial guess value y ~ i + 1 {\displaystyle {\tilde {y}}_{i+1}} via the Euler method, y ~ i + 1 = y i + h f ( t i , y i ) . {\displaystyle {\tilde {y}}_{i+1}=y_{i}+hf(t_{i},y_{i}).} Next, the corrector step: improve the initial guess using trapezoidal rule, y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle y_{i+1}=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.} That value is used as the next step. === PEC mode and PECE mode === There are different variants of a predictor–corrector method, depending on how often the corrector method is applied. The Predict–Evaluate–Correct–Evaluate (PECE) mode refers to the variant in the above example: y ~ i + 1 = y i + h f ( t i , y i ) , y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}} It is also possible to evaluate the function f only once per step by using the method in Predict–Evaluate–Correct (PEC) mode: y ~ i + 1 = y i + h f ( t i , y ~ i ) , y i + 1 = y i + 1 2 h ( f ( t i , y ~ i ) + f ( t i + 1 , y ~ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},{\tilde {y}}_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},{\tilde {y}}_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}} Additionally, the corrector step can be repeated in the hope that this achieves an even better approximation to the true solution. If the corrector method is run twice, this yields the PECECE mode: y ~ i + 1 = y i + h f ( t i , y i ) , y ^ i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) , y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ^ i + 1 ) ) . {\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\{\hat {y}}_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )},\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\hat {y}}_{i+1}){\bigr )}.\end{aligned}}} The PECEC mode has one fewer function evaluation than PECECE mode. More generally, if the corrector is run k times, the method is in P(EC)k or P(EC)kE mode. If the corrector method is iterated until it converges, this could be called PE(CE)∞.

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  Wiki survey

  Wiki surveys or wikisurveys are a software-based survey method that crowdsource discussions and help participants to find areas of agreement. Other names include bridging systems and collective response systems. The approach, inspired by Wikipedia, is to open up surveys where participants can shape the questions, instead of traditional 'closed' surveys where participants can only respond to the questions asked. Wiki surveys have been used for purposes including facilitating deliberative democracy, crowdsourcing opinions from experts and figuring out common beliefs on a given topic. A notable usage of wiki surveys is in Taiwan's government system, where citizens can participate in crowdsourced lawmaking through Pol.is wiki surveys. == Implementations == === All Our Ideas === All Our Ideas founders coined the term "wiki survey," explaining how they took inspiration from the organic evolution of Wikipedia and hoped to create something similar for surveys. They hosted 5000 surveys between 2010 and 2014. A 2020 survey using the tool found 3 of its top 10 findings were user-generated. === Decidim === Decidim has been used by governments throughout Spain and Europe to help with participatory budgeting and other public policy decisions. === Polis === Polis (also known as Pol.is) was developed in 2012. The focus of Polis is to project participants into an 'opinion space' where they can see how their voting behavior compares to other participants. The opinion space clusters participants into groups of similar opinion and is designed in a way to avoid tyranny of the majority by being able to include groups that have small numbers of participants. The questions participants are presented with are agree/disagree/pass on a single 'comment' submitted by a participant. The code for Polis is free and open-source software under the GNU AGPL. === Remesh === Remesh was founded in 2013 and has partnered with the United Nations and Alliance for Middle East Peace efforts to bring peaceful resolutions to conflicts. Participants are anonymous and the algorithm can be fine-tuned to better understand local dialects in specific regions. == Examples == PlaNYC used All Our Ideas to gather ideas on how to establish New York City's sustainability plan vTaiwan, a citizen-lead government process in Taiwan, uses Polis for enabling large amounts of citizens to deliberate and consequently provide input on Taiwan's legislative decisions OECD used All Our Ideas to gather ideas from the public prior to meeting for a forum and meeting on which skills are most important to invest in for the 21st century March On, an offshoot of the Women's March Movement, used Polis to understand the opinions of people wanting to support the movement Residents of Harrogate use Polis to debate issues in their community, with the results being released publicly to everyone == Characteristics == Wiki surveys often have these three characteristics: === Collaborativeness === Wiki surveys allow participants to contribute questions, as well as answer questions created by its participants. === Adaptivity === Wiki surveys adapt to elicit the most useful information from its participants. One example involves changing the ordering of questions based on the voting behavior of previous participants so as to maximize consensus. The heuristic determining the ordering of questions highly values showing the comments that have been voted on the least. === 'Greediness' === In the context of wiki surveys, 'greediness' simply means making full use of information that participants are willing to provide. Wiki surveys do not require participants to answer a fixed amount of questions, so participants can answer as little or as much as they want. This is intended to be more efficient in capturing participants' preferences by allowing more organic sharing of their perspectives. == Traditional survey methods vs. wiki surveys == Questions in traditional survey methods fall into two categories: Open and closed questions. Open questions ask the person taking the survey to write an open response while closed questions give a fixed set of responses to select from. Wiki surveys are like a hybrid of the two, enabling insightful consensus in certain situations where traditional survey methods may lack. Closed questions are easy to analyze quantitively, but the limited options to select from for a given question may cause bias. Open questions are not as subject to bias, but are difficult to analyze quantitatively at scale. Wiki surveys allow for open responses by the users' contribution of survey questions (also called 'items'), and uses machine learning techniques to (at least partially) automate the quantitative analysis of the responses to those questions.

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  Wiki survey

  Wiki surveys or wikisurveys are a software-based survey method that crowdsource discussions and help participants to find areas of agreement. Other names include bridging systems and collective response systems. The approach, inspired by Wikipedia, is to open up surveys where participants can shape the questions, instead of traditional 'closed' surveys where participants can only respond to the questions asked. Wiki surveys have been used for purposes including facilitating deliberative democracy, crowdsourcing opinions from experts and figuring out common beliefs on a given topic. A notable usage of wiki surveys is in Taiwan's government system, where citizens can participate in crowdsourced lawmaking through Pol.is wiki surveys. == Implementations == === All Our Ideas === All Our Ideas founders coined the term "wiki survey," explaining how they took inspiration from the organic evolution of Wikipedia and hoped to create something similar for surveys. They hosted 5000 surveys between 2010 and 2014. A 2020 survey using the tool found 3 of its top 10 findings were user-generated. === Decidim === Decidim has been used by governments throughout Spain and Europe to help with participatory budgeting and other public policy decisions. === Polis === Polis (also known as Pol.is) was developed in 2012. The focus of Polis is to project participants into an 'opinion space' where they can see how their voting behavior compares to other participants. The opinion space clusters participants into groups of similar opinion and is designed in a way to avoid tyranny of the majority by being able to include groups that have small numbers of participants. The questions participants are presented with are agree/disagree/pass on a single 'comment' submitted by a participant. The code for Polis is free and open-source software under the GNU AGPL. === Remesh === Remesh was founded in 2013 and has partnered with the United Nations and Alliance for Middle East Peace efforts to bring peaceful resolutions to conflicts. Participants are anonymous and the algorithm can be fine-tuned to better understand local dialects in specific regions. == Examples == PlaNYC used All Our Ideas to gather ideas on how to establish New York City's sustainability plan vTaiwan, a citizen-lead government process in Taiwan, uses Polis for enabling large amounts of citizens to deliberate and consequently provide input on Taiwan's legislative decisions OECD used All Our Ideas to gather ideas from the public prior to meeting for a forum and meeting on which skills are most important to invest in for the 21st century March On, an offshoot of the Women's March Movement, used Polis to understand the opinions of people wanting to support the movement Residents of Harrogate use Polis to debate issues in their community, with the results being released publicly to everyone == Characteristics == Wiki surveys often have these three characteristics: === Collaborativeness === Wiki surveys allow participants to contribute questions, as well as answer questions created by its participants. === Adaptivity === Wiki surveys adapt to elicit the most useful information from its participants. One example involves changing the ordering of questions based on the voting behavior of previous participants so as to maximize consensus. The heuristic determining the ordering of questions highly values showing the comments that have been voted on the least. === 'Greediness' === In the context of wiki surveys, 'greediness' simply means making full use of information that participants are willing to provide. Wiki surveys do not require participants to answer a fixed amount of questions, so participants can answer as little or as much as they want. This is intended to be more efficient in capturing participants' preferences by allowing more organic sharing of their perspectives. == Traditional survey methods vs. wiki surveys == Questions in traditional survey methods fall into two categories: Open and closed questions. Open questions ask the person taking the survey to write an open response while closed questions give a fixed set of responses to select from. Wiki surveys are like a hybrid of the two, enabling insightful consensus in certain situations where traditional survey methods may lack. Closed questions are easy to analyze quantitively, but the limited options to select from for a given question may cause bias. Open questions are not as subject to bias, but are difficult to analyze quantitatively at scale. Wiki surveys allow for open responses by the users' contribution of survey questions (also called 'items'), and uses machine learning techniques to (at least partially) automate the quantitative analysis of the responses to those questions.

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  List of Haskell software and tools

  This is a list of Haskell software and tools, including compilers, interpreters, build tools, package managers, integrated development environments, libraries, and other development utilities. == Compilers, interpreters and editors == Emacs — text editor Glasgow Haskell Compiler (GHC) Hugs — bytecode interpreter (discontinued) IntelliJ IDEA — IDE with Haskell support via plugins Vim — text editor Visual Studio Code — editor/IDE with Haskell support via extensions == Libraries and frameworks == Parsec — parser combinator library Servant — web framework Yesod — web framework == Build tools and package management == Cabal — build system and packaging infrastructure Haskell Platform — bundled distribution of Haskell tools and libraries (deprecated) Stack — build tool and dependency manager == Language tools and static analysis == Fourmolu — code formatter based on Ormolu Haskell Language Server — implementation of the Language Server Protocol for Haskell HLint — source code suggestion and linting tool Hoogle — Haskell API search engine Ormolu — code formatter Stan — static analysis tool Stylish Haskell — source code formatter == Interactive environments == GHCi — interactive REPL for the Glasgow Haskell Compiler IHaskell — Jupyter kernel for Haskell == Debugging and profiling tools == hp2ps — heap profiling visualization tool ThreadScope — parallel execution visualizer for Haskell programs == Documentation generators == Haddock — API documentation generator for Haskell == Parser and lexer generators == Alex — lexer generator for Haskell Happy — parser generator for Haskell == Testing frameworks == HUnit — unit testing framework QuickCheck — property-based testing library == Version control == Darcs — distributed version control system written in Haskell

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  Knuth–Plass line-breaking algorithm

  The Knuth–Plass algorithm is a line-breaking algorithm designed for use in Donald Knuth's typesetting program TeX. It integrates the problems of text justification and hyphenation into a single algorithm by using a discrete dynamic programming method to minimize a loss function that attempts to quantify the aesthetic qualities desired in the finished output. The algorithm works by dividing the text into a stream of three kinds of objects: boxes, which are non-resizable chunks of content, glue, which are flexible, resizeable elements, and penalties, which represent places where breaking is undesirable (or, if negative, desirable). The loss function, known as "badness", is defined in terms of the deformation of the glue elements, and any extra penalties incurred through line breaking. Making hyphenation decisions follows naturally from the algorithm, but the choice of possible hyphenation points within words, and optionally their preference weighting, must be performed first, and that information inserted into the text stream in advance. Knuth and Plass' original algorithm does not include page breaking, but may be modified to interface with a pagination algorithm, such as the algorithm designed by Plass in his PhD thesis. Typically, the cost function for this technique should be modified so that it does not count the space left on the final line of a paragraph; this modification allows a paragraph to end in the middle of a line without penalty. The same technique can also be extended to take into account other factors such as the number of lines or costs for hyphenating long words. == Computational complexity == A naive brute-force exhaustive search for the minimum badness by trying every possible combination of breakpoints would take an impractical O ( 2 n ) {\displaystyle O(2^{n})} time. The classic Knuth-Plass dynamic programming approach to solving the minimization problem is a worst-case O ( n 2 ) {\displaystyle O(n^{2})} algorithm but usually runs much faster, in close to linear time. Solving for the Knuth-Plass optimum can be shown to be a special case of the convex least-weight subsequence problem, which can be solved in O ( n ) {\displaystyle O(n)} time. Methods to do this include the SMAWK algorithm. == Simple example of minimum raggedness metric == For the input text AAA BB CC DDDDD with line width 6, a greedy algorithm that puts as many words on a line as possible while preserving order before moving to the next line, would produce: ------ Line width: 6 AAA BB Remaining space: 0 CC Remaining space: 4 DDDDD Remaining space: 1 The sum of squared space left over by this method is 0 2 + 4 2 + 1 2 = 17 {\displaystyle 0^{2}+4^{2}+1^{2}=17} . However, the optimal solution achieves the smaller sum 3 2 + 1 2 + 1 2 = 11 {\displaystyle 3^{2}+1^{2}+1^{2}=11} : ------ Line width: 6 AAA Remaining space: 3 BB CC Remaining space: 1 DDDDD Remaining space: 1 The difference here is that the first line is broken before BB instead of after it, yielding a better right margin and a lower cost 11.

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  Iteration

  Iteration means repeating a process to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. == Mathematics == In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example. == Computing == In computing, iteration is a technique that marks out of a block of statements within a computer program for a defined number of repetitions. That block of statements is said to be iterated. A computer programmer might also refer to that block of statements as an iteration. === Implementations === Loops constitute the most common language constructs for performing iterations. The following pseudocode "iterates" three times the line of code between begin & end through a for loop, and uses the values of i as increments. It is permissible, and often necessary, to use values from other parts of the program outside the bracketed block of statements, to perform the desired function. Iterators constitute alternative language constructs to loops, which ensure consistent iterations over specific data structures. They can eventually save time and effort in later coding attempts. In particular, an iterator allows one to repeat the same kind of operation at each node of such a data structure, often in some pre-defined order. Iteratees are purely functional language constructs, which accept or reject data during the iterations. === Relation with recursion === Recursions and iterations have different algorithmic definitions, even though they can generate identical results. The primary difference is that recursion can be a solution without prior knowledge as to how many times the action must repeat, while a successful iteration requires that foreknowledge. Some types of programming languages, known as functional programming languages, are designed such that they do not set up a block of statements for explicit repetition, as with the for loop. Instead, those programming languages exclusively use recursion. Rather than call out a block of code to repeate a pre-defined number of times, the executing code block instead "divides" the work into a number of separate pieces, after which the code block executes itself on each individual piece. Each piece of work is divided repeatedly until the "amount" of work is as small as possible, at which point the algorithm does that work very quickly. The algorithm then "reverses" and reassembles the pieces into a complete whole. The classic example of recursion is in list-sorting algorithms, such as merge sort. The merge sort recursive algorithm first repeatedly divides the list into consecutive pairs. Each pair is then ordered, then each consecutive pair of pairs, and so forth until the elements of the list are in the desired order. The code below is an example of a recursive algorithm in the Scheme programming language that outputs the same result as the pseudocode under the previous heading. == Education == In some schools of pedagogy, iterations are used to describe the process of teaching or guiding students to repeat experiments, assessments, or projects, until more accurate results are found, or the student has mastered the technical skill. This idea is found in the old adage, "Practice makes perfect." In particular, "iterative" is defined as the "process of learning and development that involves cyclical inquiry, enabling multiple opportunities for people to revisit ideas and critically reflect on their implication." Unlike computing and math, educational iterations are not predetermined; instead, the task is repeated until success according to some external criteria (often a test) is achieved.

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  Vinberg's algorithm

  In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. == Description of the algorithm == Let Γ < I s o m ( H n ) {\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})} be a hyperbolic reflection group. Choose any point v 0 ∈ H n {\displaystyle v_{0}\in \mathbb {H} ^{n}} ; we shall call it the basic (or initial) point. The fundamental domain P 0 {\displaystyle P_{0}} of its stabilizer Γ v 0 {\displaystyle \Gamma _{v_{0}}} is a polyhedral cone in H n {\displaystyle \mathbb {H} ^{n}} . Let H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} be the faces of this cone, and let a 1 , . . . , a m {\displaystyle a_{1},...,a_{m}} be outer normal vectors to it. Consider the half-spaces H k − = { x ∈ R n , 1 | ( x , a k ) ≤ 0 } . {\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.} There exists a unique fundamental polyhedron P {\displaystyle P} of Γ {\displaystyle \Gamma } contained in P 0 {\displaystyle P_{0}} and containing the point v 0 {\displaystyle v_{0}} . Its faces containing v 0 {\displaystyle v_{0}} are formed by faces H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} of the cone P 0 {\displaystyle P_{0}} . The other faces H m + 1 , . . . {\displaystyle H_{m+1},...} and the corresponding outward normals a m + 1 , . . . {\displaystyle a_{m+1},...} are constructed by induction. Namely, for H j {\displaystyle H_{j}} we take a mirror such that the root a j {\displaystyle a_{j}} orthogonal to it satisfies the conditions (1) ( v 0 , a j ) < 0 {\displaystyle (v_{0},a_{j})<0} ; (2) ( a i , a j ) ≤ 0 {\displaystyle (a_{i},a_{j})\leq 0} for all i < j {\displaystyle i Read more →

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  LRE Map

  The LRE Map (Language Resources and Evaluation) is a freely accessible large database on resources dedicated to Natural language processing. The original feature of LRE Map is that the records are collected during the submission of different major Natural language processing conferences. The records are then cleaned and gathered into a global database called "LRE Map". The LRE Map is intended to be an instrument for collecting information about language resources and to become, at the same time, a community for users, a place to share and discover resources, discuss opinions, provide feedback, discover new trends, etc. It is an instrument for discovering, searching and documenting language resources, here intended in a broad sense, as both data and tools. The large amount of information contained in the Map can be analyzed in many different ways. For instance, the LRE Map can provide information about the most frequent type of resource, the most represented language, the applications for which resources are used or are being developed, the proportion of new resources vs. already existing ones, or the way in which resources are distributed to the community. == Context == Several institutions worldwide maintain catalogues of language resources (ELRA, LDC, NICT Universal Catalogue, ACL Data and Code Repository, OLAC, LT World, etc.) However, it has been estimated that only 10% of existing resources are known, either through distribution catalogues or via direct publicity by providers (web sites and the like). The rest remains hidden, the only occasions where it briefly emerges being when a resource is presented in the context of a research paper or report at some conference. Even in this case, nevertheless, it might be that a resource remains in the background simply because the focus of the research is not on the resource per se. == History == The LRE Map originated under the name "LREC Map" during the preparation of LREC 2010 conference. More specifically, the idea was discussed within the FlaReNet project, and in collaboration with ELRA and the Institute of Computational Linguistics of CNR in Pisa, the Map was put in place at LREC 2010. The LREC organizers asked the authors to provide some basic information about all the resources (in a broad sense, i.e. including tools, standards and evaluation packages), either used or created, described in their papers. All these descriptors were then gathered in a global matrix called the LREC Map. The same methodology and requirements from the authors has been then applied and extended to other conferences, namely COLING-2010, EMNLP-2010, RANLP-2011, LREC 2012, LREC 2014 and LREC 2016. After this generalization to other conferences, the LREC Map has been renamed as the LRE Map. == Size and content == The size of the database increases over time. The data collected amount to 4776 entries. Each resource is described according to the following attributes: Resource type, e.g. lexicon, annotation tool, tagger/parser. Resource production status, e.g. newly created finished, existing-updated. Resource availability, e.g. freely available, from data center. Resource modality, e.g. speech, written, sign language. Resource use, e.g. named entity recognition, language identification, machine translation. Resource language, e.g. English, 23 European Union languages, official languages of India. == Uses == The LRE map is a very important tool to chart the NLP field. Compared to other studied based on subjective scorings, the LRE map is made of real facts. The map has a great potential for many uses, in addition to being an information gathering tool: It is a great instrument for monitoring the evolution of the field (useful for funders), if applied in different contexts and times. It can be seen as a huge joint effort, the beginning of an even larger cooperative action not just among few leaders but among all the researchers. It is also an "educational" means towards the broad acknowledgment of the need of meta-research activities with the active involvement of many. It is also instrumental in introducing the new notion of "citation of resources" that could provide an award and a means of scholarly recognition for researchers engaged in resource creation. It is used to help the organization of the conferences of the field like LREC. == Derived matrices == The data were then cleaned and sorted by Joseph Mariani (CNRS-LIMSI IMMI) and Gil Francopoulo (CNRS-LIMSI IMMI + Tagmatica) in order to compute the various matrices of the final FLaReNet reports. One of them, the matrix for written data at LREC 2010 is as follows: English is the most studied language. Secondly, come French and German languages and then Italian and Spanish. == Future == The LRE Map has been extended to Language Resources and Evaluation Journal and other conferences.

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  Apple Intelligence

  Apple Intelligence is a generative artificial intelligence system developed by Apple Inc. Relying on a combination of on-device and server processing, it was announced on June 10, 2024, at the 2024 Worldwide Developers Conference, as a built-in feature of Apple's iOS 18, iPadOS 18, and macOS Sequoia, which were announced alongside Apple Intelligence. Apple Intelligence is free for all users with supported devices. On macOS, Apple Intelligence is available only on Apple silicon Mac computers; Intel-based Mac computers are not supported. Features include writing tools that assist users with grammar and proofreading, image generation, summaries of system notifications, AI-assisted image retouching in the Photos app, and integration with ChatGPT, the popular chatbot by OpenAI. As of March 2026, Apple Intelligence is not available yet on devices purchased in mainland China or on any device using an Apple Account set to mainland China, even if the device was bought elsewhere. == History == === Background === Apple first implemented artificial intelligence features in its products with the release of Siri in the iPhone 4S in 2011. In the years after its release, Apple engaged in efforts to ensure its artificial intelligence operations remained covert; according to University of California, Berkeley professor Trevor Darrell, the company's secrecy deterred graduate students. The company started expanding its artificial intelligence team in 2015, opening up its operations by publishing more scientific papers and joining AI industry research groups. Apple reportedly acquired more AI companies from 2016 to 2020. In 2017, Apple released the iPhone 8 and the iPhone X with the A11 Bionic processor, which featured its first dedicated Neural Engine for accelerating common machine learning tasks. Despite its investments in artificial intelligence, Siri was criticized both by reviewers and internally at Apple for lagging behind other AI assistants. The rapid development of generative artificial intelligence and the release of ChatGPT in late 2022 reportedly blindsided Apple executives and forced the company to refocus its efforts on AI. In an interview with Good Morning America, Apple CEO Tim Cook stated that generative AI had "great promise" but had some potential dangers, and that it was "looking closely" at ChatGPT. It was first reported in July 2023 that Apple was creating its own internal large language model, codenamed "Ajax". In October 2023, Apple was reportedly on track to release new generative AI features into its operating systems by 2024, including a significantly redeveloped Siri. In an earnings call in February 2024, Cook stated that the company was spending a "tremendous amount of time and effort" into AI features that would be shared "later that year". === Google deal === In January 2026, Apple and Google announced a multi-year partnership under which Apple’s next-generation foundation models are expected to incorporate Google’s Gemini models and cloud infrastructure. According to the companies, the collaboration is intended to support future Apple Intelligence features, including enhancements to Siri, while Apple Intelligence will continue to operate on Apple devices and through Apple’s Private Cloud Compute system, which Apple states is designed to preserve user privacy. On an earnings call, Apple reported to investors that they were integrating an on-device model of the Google Gemini AI to Siri, as the development of their model was beset with setbacks. Apple has previously tested and used other third-party AI models like ChatGPT, but according to a Bloomberg article by Mark Gurman, Apple pushed forward the proposed Google deal; by using Google's Gemini model possessing 1.2 trillion parameters, Apple would integrate a much larger and more complex model than those it previously developed and used. Of note, comparable AI models from other major companies (including OpenAI and Meta) have also been reported to operate at a similar “trillion-parameter” scale and to compete against Gemini-class systems on benchmarks. == Models == Apple Intelligence consists of an on-device model as well as a cloud model running on servers primarily using Apple silicon. Both models consist of a generic foundation model, as well as multiple adapter models that are more specialized to particular tasks like text summarization and tone adjustment. It was launched for developers and testers on July 29, 2024, in U.S. English, with the developer betas of iOS 18.1, macOS 15.1, and iPadOS 18.1, released partially on October 28, 2024, and will fully launch by 2026. According to a human evaluation done by Apple's machine learning division, the on-device foundation model beat or tied equivalent small models by Mistral AI, Microsoft, and Google, while the server foundation models beat the performance of OpenAI's GPT-3, while roughly matching the performance of GPT-4. Apple's cloud models are built on a Private Cloud Compute platform which is allegedly designed with user privacy and end-to-end encryption in mind. Unlike other generative AI services like ChatGPT which use servers from third parties, Apple Intelligence's cloud models are run entirely on Apple servers with custom Apple silicon hardware built for end-to-end encryption. It was also designed to make sure that the software running on said servers matches the independently verifiable software accessible to researchers. In case of a software mismatch, Apple devices will refuse to connect to the servers. On June 10, 2025, Apple announced that Apple's on-device foundation models will be available to third-party applications as part of the Foundation Models API, with support for structured data response and tool calling. == Features == === Writing tools === Apple Intelligence features writing tools that are powered by LLMs. Selected text can be proofread, rewritten, made more friendly, concise or professional, similar to the AI writing features of the popular online English-language writing assistant tool Grammarly. It can also be used to generate summaries, key points, tables, and lists from an article or piece of writing. In iOS 18.2 and macOS 15.2, a ChatGPT integration was added to Writing Tools through "Compose" and "Describe your change" features. === Real-time Translation === Apple Intelligence enables the real-time translation of messages, photos and videos, and phone calls, through Apple's hardware. For communicating with foreigners, using the Translate app on iPhone to show subtitles in their language or to play back the translated audio naturally in their language, and also by wearing AirPods with Live Translation can now help to understand what someone is saying in users' preferred language in conversation. If both have headphones, simultaneous interpretation can be achieved. === Image Playground === Apple Intelligence can be used to generate images on-device with the Image Playground app. Similarly to OpenAI's DALL-E, it can be used to generate images using AI, using phrases and descriptions to output an image with customizable styles such as Animation and Sketch. In Notes, users can access Image Playground on iPad through the Image Wand tool in the Apple Pencil palette without having to open the Image Playground app. Rough sketches made with Apple Pencil can be transformed into images. As part of iOS, iPadOS, and macOS 26, Image Playground now integrates with the image generation models built into ChatGPT. === Genmoji === Using Apple Intelligence text-to-image models, users can generate unique "Genmoji" images by typing descriptions (prompting). Users can pick people in photos to have Genmoji generate images that resemble them. Similarly to emoji, Genmoji can be added inline to text messages, tapbacks, stickers and can be shared in Messages as well in third-party applications as inline messages or as stickers. === Siri overhaul === Siri, which used to be Apple's virtual assistant, has been updated to be an LLM chatbot, with enhanced capabilities made possible by Apple Intelligence. The latest iteration features an updated user interface, improved natural language processing, and the option to interact via text by double tapping the home bar without enabling the feature in the Accessibility menu, or double-clicking the command key on macOS. In a later update, Apple Intelligence will add the ability for Siri to use personal context from device activities to answer queries. === Mail === Apple Intelligence adds a feature called Priority Messages to the Mail app, which shows urgent emails such as same-day invitations or boarding passes, with AI generated summaries of the email. The Mail app also gains the ability to categorize incoming mail into Primary, Transactions, Updates, and Promotions based on what the email contains, which Apple claims is done all on-device. === Photos === Apple's Photos app includes a feature to create custom memory movies and enhanced search capabilities. Users can describe

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  Sequential algorithm

  In computer science, a sequential algorithm or serial algorithm is an algorithm that is executed sequentially – once through, from start to finish, without other processing executing – as opposed to concurrently or in parallel. The term is primarily used to contrast with concurrent algorithm or parallel algorithm; most standard computer algorithms are sequential algorithms, and not specifically identified as such, as sequentialness is a background assumption. Concurrency and parallelism are in general distinct concepts, but they often overlap – many distributed algorithms are both concurrent and parallel – and thus "sequential" is used to contrast with both, without distinguishing which one. If these need to be distinguished, the opposing pairs sequential/concurrent and serial/parallel may be used. "Sequential algorithm" may also refer specifically to an algorithm for decoding a convolutional code.

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  External memory algorithm

  In computing, external memory algorithms or out-of-core algorithms are algorithms that are designed to process data that are too large to fit into a computer's main memory at once. Such algorithms must be optimized to efficiently fetch and access data stored in slow bulk memory (auxiliary memory) such as hard drives or tape drives, or when memory is on a computer network. External memory algorithms are analyzed in the external memory model. == Model == External memory algorithms are analyzed in an idealized model of computation called the external memory model (or I/O model, or disk access model). The external memory model is an abstract machine similar to the RAM machine model, but with a cache in addition to main memory. The model captures the fact that read and write operations are much faster in a cache than in main memory, and that reading long contiguous blocks is faster than reading randomly using a disk read-and-write head. The running time of an algorithm in the external memory model is defined by the number of reads and writes to memory required. The model was introduced by Alok Aggarwal and Jeffrey Vitter in 1988. The external memory model is related to the cache-oblivious model, but algorithms in the external memory model may know both the block size and the cache size. For this reason, the model is sometimes referred to as the cache-aware model. The model consists of a processor with an internal memory or cache of size M, connected to an unbounded external memory. Both the internal and external memory are divided into blocks of size B. One input/output or memory transfer operation consists of moving a block of B contiguous elements from external to internal memory, and the running time of an algorithm is determined by the number of these input/output operations. == Algorithms == Algorithms in the external memory model take advantage of the fact that retrieving one object from external memory retrieves an entire block of size B. This property is sometimes referred to as locality. Searching for an element among N objects is possible in the external memory model using a B-tree with branching factor B. Using a B-tree, searching, insertion, and deletion can be achieved in O ( log B ⁡ N ) {\displaystyle O(\log _{B}N)} time (in Big O notation). Information theoretically, this is the minimum running time possible for these operations, so using a B-tree is asymptotically optimal. External sorting is sorting in an external memory setting. External sorting can be done via distribution sort, which is similar to quicksort, or via a M B {\displaystyle {\tfrac {M}{B}}} -way merge sort. Both variants achieve the asymptotically optimal runtime of O ( N B log M B ⁡ N B ) {\displaystyle O\left({\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)} to sort N objects. This bound also applies to the fast Fourier transform in the external memory model. The permutation problem is to rearrange N elements into a specific permutation. This can either be done either by sorting, which requires the above sorting runtime, or inserting each element in order and ignoring the benefit of locality. Thus, permutation can be done in O ( min ( N , N B log M B ⁡ N B ) ) {\displaystyle O\left(\min \left(N,{\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)\right)} time. == Applications == The external memory model captures the memory hierarchy, which is not modeled in other common models used in analyzing data structures, such as the random-access machine, and is useful for proving lower bounds for data structures. The model is also useful for analyzing algorithms that work on datasets too big to fit in internal memory. A typical example is geographic information systems, especially digital elevation models, where the full data set easily exceeds several gigabytes or even terabytes of data. This methodology extends beyond general purpose CPUs and also includes GPU computing as well as classical digital signal processing. In general-purpose computing on graphics processing units (GPGPU), powerful graphics cards (GPUs) with little memory (compared with the more familiar system memory, which is most often referred to simply as RAM) are utilized with relatively slow CPU-to-GPU memory transfer (when compared with computation bandwidth). == History == An early use of the term "out-of-core" as an adjective is in 1962 in reference to devices that are other than the core memory of an IBM 360. An early use of the term "out-of-core" with respect to algorithms appears in 1971.

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  Photo-consistency

  In computer vision, photo-consistency determines whether a given voxel is occupied. A voxel is considered to be photo consistent when its color appears to be similar to all the cameras that can see it. Most voxel coloring or space carving techniques require using photo consistency as a check condition in Image-based modeling and rendering applications. == Usage == 3D Volumetric Reconstruction. Image registration. Multi-view reconstruction.

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  Virtual directory

  In computing, the term virtual directory has a couple of meanings. It may simply designate (for example in IIS) a folder which appears in a path but which is not actually a subfolder of the preceding folder in the path. However, this article will discuss the term in the context of directory services and identity management. A virtual directory or virtual directory server (VDS) in this context is a software layer that delivers a single access point for identity management applications and service platforms. A virtual directory operates as a high-performance, lightweight abstraction layer that resides between client applications and disparate types of identity-data repositories, such as proprietary and standard directories, databases, web services, and applications. A virtual directory receives queries and directs them to the appropriate data sources by abstracting and virtualizing data. The virtual directory integrates identity data from multiple heterogeneous data stores and presents it as though it were coming from one source. This ability to reach into disparate repositories makes virtual directory technology ideal for consolidating data stored in a distributed environment. As of 2011, virtual directory servers most commonly use the LDAP protocol, but more sophisticated virtual directories can also support SQL as well as DSML and SPML. Industry experts have heralded the importance of the virtual directory in modernizing the identity infrastructure. According to Dave Kearns of Network World, "Virtualization is hot and a virtual directory is the building block, or foundation, you should be looking at for your next identity management project." In addition, Gartner analyst, Bob Blakley said that virtual directories are playing an increasingly vital role. In his report, “The Emerging Architecture of Identity Management,” Blakley wrote: “In the first phase, production of identities will be separated from consumption of identities through the introduction of a virtual directory interface.” == Capabilities == Virtual directories can have some or all of the following capabilities: Aggregate identity data across sources to create a single point of access. Create high-availability for authoritative data stores. Act as identity firewall by preventing denial-of-service attacks on the primary data stores through an additional virtual layer. Support a common searchable namespace for centralized authentication. Present a unified virtual view of user information stored across multiple systems. Delegate authentication to backend sources through source-specific security means. Virtualize data sources to support migration from legacy data stores without modifying the applications that rely on them. Enrich identities with attributes pulled from multiple data stores, based on a link between user entries. Some advanced identity virtualization platforms can also: Enable application-specific, customized views of identity data without violating internal or external regulations governing identity data. Reveal contextual relationships between objects through hierarchical directory structures. Develop advanced correlation across diverse sources using correlation rules. Build a global user identity by correlating unique user accounts across various data stores, and enrich identities with attributes pulled from multiple data stores, based on a link between user entries. Enable constant data refresh for real-time updates through a persistent cache. == Advantages == Virtual directories: Enable faster deployment because users do not need to add and sync additional application-specific data sources Leverage existing identity infrastructure and security investments to deploy new services Deliver high availability of data sources Provide application-specific views of identity data which can help avoid the need to develop a master enterprise schema Allow a single view of identity data without violating internal or external regulations governing identity data Act as identity firewalls by preventing denial-of-service attacks on the primary data-stores and providing further security on access to sensitive data Can reflect changes made to authoritative sources in real-time Leverages existing update processes of authoritative sources, so no separate (sometimes manual) process to update a central directory is needed Present a unified virtual view of user information from multiple systems so that it appears to reside in a single system Can secure all backend storage locations with a single security policy == Disadvantages == An original disadvantage is public perception of "push & pull technologies" which is the general classification of "virtual directories" depending on the nature of their deployment. Virtual directories were initially designed and later deployed with "push technologies" in mind, which also contravened with privacy laws of the United States. This is no longer the case. There are, however, other disadvantages in the current technologies. The classical virtual directory based on proxy cannot modify underlying data structures or create new views based on the relationships of data from across multiple systems. So if an application requires a different structure, such as a flattened list of identities, or a deeper hierarchy for delegated administration, a virtual directory is limited. Many virtual directories cannot correlate same-users across multiple diverse sources in the case of duplicate users Virtual directories without advanced caching technologies cannot scale to heterogeneous, high-volume environments. == Sample terminology == Unify metadata: Extract schemas from the local data source, map them to a common format, and link the same identities from different data silos based on a unique identifier. Namespace joining: Create a single large directory by bringing multiple directories together at the namespace level. For instance, if one directory has the namespace "ou=internal,dc=domain,dc=com" and a second directory has the namespace "ou=external,dc=domain,dc=com," then creating a virtual directory with both namespaces is an example of namespace joining. Identity joining: Enrich identities with attributes pulled from multiple data stores, based on a link between user entries. For instance if the user joeuser exists in a directory as "cn=joeuser,ou=users" and in a database with a username of "joeuser" then the "joeuser" identity can be constructed from both the directory and the database. Data remapping: The translation of data inside of the virtual directory. For instance, mapping “uid” to “samaccountname,” so a client application that only supports a standard LDAP-compliant data source is able to search an Active Directory namespace, as well. Query routing: Route requests based on certain criteria, such as “write operations going to a master, while read operations are forwarded to replicas.” Identity routing: Virtual directories may support the routing of requests based on certain criteria (such as write operations going to a master while read operations being forwarded to replicas). Authoritative source: A "virtualized" data repository, such as a directory or database, that the virtual directory can trust for user data. Server groups: Group one or more servers containing the same data and functionality. A typical implementation is the multi-master, multi-replica environment in which replicas process "read" requests and are in one server group, while masters process "write" requests and are in another, so that servers are grouped by their response to external stimuli, even though all share the same data. == Use cases == The following are sample use cases of virtual directories: Integrating multiple directory namespaces to create a central enterprise directory. Supporting infrastructure integrations after mergers and acquisitions. Centralizing identity storage across the infrastructure, making identity information available to applications through various protocols (including LDAP, JDBC, and web services). Creating a single access point for web access management (WAM) tools. Enabling web single sign-on (SSO) across varied sources or domains. Supporting role-based, fine-grained authorization policies Enabling authentication across different security domains using each domain’s specific credential checking method. Improving secure access to information both inside and outside of the firewall.

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  Least-squares spectral analysis

  Least-squares spectral analysis (LSSA) is a class of methods for estimating a frequency spectrum by fitting sinusoids to data using a least-squares fit. Unlike Fourier analysis, the most widely used spectral method in science, data need not be equally spaced to use LSSA. Furthermore, while Fourier analysis generally amplifies long-period noise in long or gapped records, LSSA mitigates such problems. The first strictly least-squares LSSA method was developed in 1969 and 1971, and is known as the Vaníček method or the Gauss–Vaniček method, after its inventor Petr Vaníček and Carl Friedrich Gauss, the inventor of the least-squares method for error minimization. A widely known LSSA variant is the Lomb method or the Lomb–Scargle periodogram, based on dated computational simplifications of the Vaníček method introduced in the 1970s and 1980s, first by Nicholas R. Lomb and later by Jeffrey D. Scargle. Other LSSA variants have been subsequently developed. == Historical background == The close connections between Fourier analysis, the periodogram, and the least-squares fitting of sinusoids have been known for a long time. However, most developments are restricted to complete data sets of equally spaced samples. In 1963, Freek J. M. Barning of Mathematisch Centrum, Amsterdam, handled unequally spaced data by similar techniques, including both a periodogram analysis equivalent to what nowadays is called the Lomb method and least-squares fitting of selected frequencies of sinusoids determined from such periodograms — and connected by a procedure known today as the matching pursuit with post-back fitting or the orthogonal matching pursuit. Petr Vaníček, a Canadian geophysicist and geodesist of the University of New Brunswick, proposed in 1969 also the matching-pursuit approach for equally and unequally spaced data, which he called "successive spectral analysis" and the result a "least-squares periodogram". He generalized this method to account for any systematic components beyond a simple mean, such as a "predicted linear (quadratic, exponential, ...) secular trend of unknown magnitude", and applied it to a variety of samples, in 1971. Vaníček's strictly least-squares method was then simplified in 1976 by Nicholas R. Lomb of the University of Sydney, who pointed out its close connection to periodogram analysis. Subsequently, the definition of a periodogram of unequally spaced data was modified and analyzed by Jeffrey D. Scargle of NASA Ames Research Center, who showed that, with minor changes, it becomes identical to Lomb's least-squares formula for fitting individual sinusoid frequencies. Scargle states that his paper "does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced," and further points out regarding least-squares fitting of sinusoids compared to periodogram analysis, that his paper "establishes, apparently for the first time, that (with the proposed modifications) these two methods are exactly equivalent." Press summarizes the development this way: A completely different method of spectral analysis for unevenly sampled data, one that mitigates these difficulties and has some other very desirable properties, was developed by Lomb, based in part on earlier work by Barning and Vanicek, and additionally elaborated by Scargle. In 1989, Michael J. Korenberg of Queen's University in Kingston, Ontario, developed the "fast orthogonal search" method of more quickly finding a near-optimal decomposition of spectra or other problems, similar to the technique that later became known as the orthogonal matching pursuit. == Development of LSSA and variants == === The Vaníček method === In the Vaníček method, a discrete data set is approximated by a weighted sum of sinusoids of progressively determined frequencies using a standard linear regression or least-squares fit. The frequencies are chosen using a method similar to Barning's, but going further in optimizing the choice of each successive new frequency by picking the frequency that minimizes the residual after least-squares fitting (equivalent to the fitting technique now known as matching pursuit with pre-backfitting). The number of sinusoids must be less than or equal to the number of data samples (counting sines and cosines of the same frequency as separate sinusoids). The relationship between the DFT and the approximation of trigonometric functions using the least-squares method is well explained in (Strutz, 2017). A data vector Φ is represented as a weighted sum of sinusoidal basis functions, tabulated in a matrix A by evaluating each function at the sample times, with weight vector x: ϕ ≈ A x , {\displaystyle \phi \approx {\textbf {A}}x,} where the weights vector x is chosen to minimize the sum of squared errors in approximating Φ. The solution for x is closed-form, using standard linear regression: x = ( A T A ) − 1 A T ϕ . {\displaystyle x=({\textbf {A}}^{\mathrm {T} }{\textbf {A}})^{-1}{\textbf {A}}^{\mathrm {T} }\phi .} Here the matrix A can be based on any set of functions mutually independent (not necessarily orthogonal) when evaluated at the sample times; functions used for spectral analysis are typically sines and cosines evenly distributed over the frequency range of interest. If we choose too many frequencies in a too-narrow frequency range, the functions will be insufficiently independent, the matrix ill-conditioned, and the resulting spectrum meaningless. When the basis functions in A are orthogonal (that is, not correlated, meaning the columns have zero pair-wise dot products), the matrix ATA is diagonal; when the columns all have the same power (sum of squares of elements), then that matrix is an identity matrix times a constant, so the inversion is trivial. The latter is the case when the sample times are equally spaced and sinusoids chosen as sines and cosines equally spaced in pairs on the frequency interval 0 to a half cycle per sample (spaced by 1/N cycles per sample, omitting the sine phases at 0 and maximum frequency where they are identically zero). This case is known as the discrete Fourier transform, slightly rewritten in terms of measurements and coefficients. x = A T ϕ {\displaystyle x={\textbf {A}}^{\mathrm {T} }\phi } — DFT case for N equally spaced samples and frequencies, within a scalar factor. === The Lomb method === Trying to lower the computational burden of the Vaníček method in 1976 (no longer an issue), Lomb proposed using the above simplification in general, except for pair-wise correlations between sine and cosine bases of the same frequency, since the correlations between pairs of sinusoids are often small, at least when they are not tightly spaced. This formulation is essentially that of the traditional periodogram but adapted for use with unevenly spaced samples. The vector x is a reasonably good estimate of an underlying spectrum, but since we ignore any correlations, Ax is no longer a good approximation to the signal, and the method is no longer a least-squares method — yet in the literature continues to be referred to as such. Rather than just taking dot products of the data with sine and cosine waveforms directly, Scargle modified the standard periodogram formula so to find a time delay τ {\displaystyle \tau } first, such that this pair of sinusoids would be mutually orthogonal at sample times t j {\displaystyle t_{j}} and also adjusted for the potentially unequal powers of these two basis functions, to obtain a better estimate of the power at a frequency. This procedure made his modified periodogram method exactly equivalent to Lomb's method. Time delay τ {\displaystyle \tau } by definition equals to tan ⁡ 2 ω τ = ∑ j sin ⁡ 2 ω t j ∑ j cos ⁡ 2 ω t j . {\displaystyle \tan {2\omega \tau }={\frac {\sum _{j}\sin 2\omega t_{j}}{\sum _{j}\cos 2\omega t_{j}}}.} Then the periodogram at frequency ω {\displaystyle \omega } is estimated as: P x ( ω ) = 1 2 [ [ ∑ j X j cos ⁡ ω ( t j − τ ) ] 2 ∑ j cos 2 ⁡ ω ( t j − τ ) + [ ∑ j X j sin ⁡ ω ( t j − τ ) ] 2 ∑ j sin 2 ⁡ ω ( t j − τ ) ] , {\displaystyle P_{x}(\omega )={\frac {1}{2}}\left[{\frac {\left[\sum _{j}X_{j}\cos \omega (t_{j}-\tau )\right]^{2}}{\sum _{j}\cos ^{2}\omega (t_{j}-\tau )}}+{\frac {\left[\sum _{j}X_{j}\sin \omega (t_{j}-\tau )\right]^{2}}{\sum _{j}\sin ^{2}\omega (t_{j}-\tau )}}\right],} which, as Scargle reports, has the same statistical distribution as the periodogram in the evenly sampled case. At any individual frequency ω {\displaystyle \omega } , this method gives the same power as does a least-squares fit to sinusoids of that frequency and of the form: ϕ ( t ) = A sin ⁡ ω t + B cos ⁡ ω t . {\displaystyle \phi (t)=A\sin \omega t+B\cos \omega t.} In practice, it is always difficult to judge if a given Lomb peak is significant or not, especially when the nature of the noise is unknown, so for example a false-alarm spectr

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