AI Writing Tools

Explore the best AI Writing Tools — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

Shaded Picture System

The Shaded Picture System was a 3D raster computer display processor introduced by Evans & Sutherland in October 1973. The Shaded Picture System was the first general-purpose, commercially available raster computer graphics display processor capable of real-time, shaded 3D graphics. It could only display black and white graphics at a resolution of 256 by 256. It was extremely expensive, and very few units were ever sold. == History == The principles of shaded, hidden-line true 3D graphics were pioneered at the University of Utah in 1967. However, this algorithm was slow and would take several minutes to produce an image. In 1970, Gary Watkins developed a FORTRAN simulator of a faster algorithm that would theoretically generate shaded 3D images in real-time, "if implemented in suitable hardware". The simulator itself was still not capable of real-time shaded 3D image rendering. Evans & Sutherland developed a functional prototype of this "suitable hardware", which was later sold as the Shaded Picture System in 1973. About a year earlier in 1972, Evans & Sutherland sold the first and only CT1 to Case Western Reserve University. The CT1, or Continuous Tone 1, was a specialized image generator, not meant as a marketable or mass-produced product. At the time, the CT1, along with G.E./NASA's upgraded Electronic Scene Generator from 1971, would have been the only real-time raster graphics systems sold to customers comparable to the Shaded Picture System, although both the CT1 and Electronic Scene Generator were intentionally produced as one-off products and specialized for the needs of their customers. The Shaded Picture System, in contrast, was intentionally marketed.In early 1975, Evans & Sutherland demonstrated a random-access video frame buffer using relatively low-cost semiconductor memory, which was much more capable than the Shaded Picture System. When interfaced with a (non-shaded) E&S Picture System, the frame buffer had a resolution of 512 by 512 in grayscale and partial color capabilities. By the end of 1975, this frame buffer was commercially available.
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Relation network

A relation network (RN) is an artificial neural network component with a structure that can reason about relations among objects. An example category of such relations is spatial relations (above, below, left, right, in front of, behind). RNs can infer relations, they are data efficient, and they operate on a set of objects without regard to the objects' order. == History == In June 2017, DeepMind announced the first relation network. It claimed that the technology had achieved "superhuman" performance on multiple question-answering problem sets. == Design == RNs constrain the functional form of a neural network to capture the common properties of relational reasoning. These properties are explicitly added to the system, rather than established by learning just as the capacity to reason about spatial, translation-invariant properties is explicitly part of convolutional neural networks (CNN). The data to be considered can be presented as a simple list or as a directed graph whose nodes are objects and whose edges are the pairs of objects whose relationships are to be considered. The RN is a composite function: R N ( O ) = f ϕ ( ∑ i , j g θ ( o i , o j , q ) ) , {\displaystyle RN\left(O\right)=f_{\phi }\left(\sum _{i,j}g_{\theta }\left(o_{i},o_{j},q\right)\right),} where the input is a set of "objects" O = { o 1 , o 2 , . . . , o n } , o i ∈ R m {\displaystyle O=\left\lbrace o_{1},o_{2},...,o_{n}\right\rbrace ,o_{i}\in \mathbb {R} ^{m}} is the ith object, and fφ and gθ are functions with parameters φ and θ, respectively and q is the question. fφ and gθ are multilayer perceptrons, while the 2 parameters are learnable synaptic weights. RNs are differentiable. The output of gθ is a "relation"; therefore, the role of gθ is to infer any ways in which two objects are related. Image (128x128 pixel) processing is done with a 4-layer CNN. Outputs from the CNN are treated as the objects for relation analysis, without regard for what those "objects" explicitly represent. Questions were processed with a long short-term memory network.
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Probit model

In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model. A probit model is a popular specification for a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. It is most often estimated using the maximum likelihood procedure, such an estimation being called a probit regression. == Conceptual framework == Suppose a response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example, Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes the form P ( Y = 1 ∣ X ) = Φ ( X T β ) , {\displaystyle P(Y=1\mid X)=\Phi (X^{\operatorname {T} }\beta ),} where P is the probability and Φ {\displaystyle \Phi } is the cumulative distribution function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood. It is possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable Y ∗ = X T β + ε , {\displaystyle Y^{\ast }=X^{T}\beta +\varepsilon ,} where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive: Y = { 1 Y ∗ > 0 0 otherwise } = { 1 X T β + ε > 0 0 otherwise } {\displaystyle Y=\left.{\begin{cases}1&Y^{}>0\\0&{\text{otherwise}}\end{cases}}\right\}=\left.{\begin{cases}1&X^{\operatorname {T} }\beta +\varepsilon >0\\0&{\text{otherwise}}\end{cases}}\right\}} The use of the standard normal distribution causes no loss of generality compared with the use of a normal distribution with an arbitrary mean and standard deviation, because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount. To see that the two models are equivalent, note that P ( Y = 1 ∣ X ) = P ( Y ∗ > 0 ) = P ( X T β + ε > 0 ) = P ( ε > − X T β ) = P ( ε < X T β ) by symmetry of the normal distribution = Φ ( X T β ) {\displaystyle {\begin{aligned}P(Y=1\mid X)&=P(Y^{\ast }>0)\\&=P(X^{\operatorname {T} }\beta +\varepsilon >0)\\&=P(\varepsilon >-X^{\operatorname {T} }\beta )\\&=P(\varepsilon 0 {\displaystyle t,\lim _{n\rightarrow \infty }n_{t}/n=c_{t}>0} . Denote p ^ t = r t / n t {\displaystyle {\hat {p}}_{t}=r_{t}/n_{t}} σ ^ t 2 = 1 n t p ^ t ( 1 − p ^ t ) φ 2 ( Φ − 1 ( p ^ t ) ) {\displaystyle {\hat {\sigma }}_{t}^{2}={\frac {1}{n_{t}}}{\frac {{\hat {p}}_{t}(1-{\hat {p}}_{t})}{\varphi ^{2}{\big (}\Phi ^{-1}({\hat {p}}_{t}){\big )}}}} Then Berkson's minimum chi-square estimator is a generalized least squares estimator in a regression of Φ − 1 ( p ^ t ) {\displaystyle \Phi ^{-1}({\hat {p}}_{t})} on x ( t ) {\displaystyle x_{(t)}} with weights σ ^ t − 2 {\displaystyle {\hat {\sigma }}_{t}^{-2}} : β ^ = ( ∑ t = 1 T σ ^ t − 2 x ( t ) x ( t ) T ) − 1 ∑ t = 1 T σ ^ t − 2 x ( t ) Φ − 1 ( p ^ t ) {\displaystyle {\hat {\beta }}={\Bigg (}\sum _{t=1}^{T}{\hat {\sigma }}_{t}^{-2}x_{(t)}x_{(t)}^{\operatorname {T} }{\Bigg )}^{-1}\sum _{t=1}^{T}{\hat {\sigma }}_{t}^{-2}x_{(t)}\Phi ^{-1}({\hat {p}}_{t})} It can be shown that this estimator is consistent (as n→∞ and T fixed), asymptotically normal and efficient. Its advantage is the presence of a closed-form formula for the estimator. However, it is only meaningful to carry out this analysis when individual observations are not available, only their aggregated counts r t {\displaystyle r_{t}} , n t {\disp
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ID3 algorithm

In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.
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Collabora Online

Collabora Online (often abbreviated as COOL) is an open-source online office suite developed by Collabora, based on LibreOffice Online, the web-based edition of the LibreOffice office suite. It enables real-time collaborative editing of documents, spreadsheets, presentations, and vector graphics in a web browser. Optional applications are available for offline use on Android, ChromeOS, iOS, iPadOS, Linux distributions, macOS, and Windows. It supports the OpenDocument format and is compatible with other major formats, including those used by Microsoft Office. The Document Foundation (TDF), the nonprofit organization behind LibreOffice, states that a majority of the LibreOffice software development is done by its partners like Collabora. Collabora Online is an open-source alternative to proprietary cloud office platforms such as Google Workspace and Microsoft 365. Unlike these services, it can be self-hosted or hosted by third-party providers. The platform is marketed particularly toward enterprises and public institutions seeking greater digital sovereignty and independence from U.S.-based "big tech" companies. Collabora also develops Collabora Office, a standalone desktop and mobile app suite based on LibreOffice. Although Collabora Online has increasingly taken on a central role, both products may be used in parallel, similar to Microsoft Office and Microsoft 365. In November 2025, Collabora released Collabora Office Desktop and renamed the previous product Collabora Office Classic. The new product shares code with Collabora Online and brings the same user interface to the desktop on Linux, Windows and MacOS. A separate version, the Collabora Online Development Edition (CODE), is offered free of charge and is recommended for individuals, small teams, and developers. CODE provides early access to new features and serves as a testing and development platform for open-source community contributors. As TDF does not offer a free version of LibreOffice Online, CODE represents the primary freely available option for organizations and individuals interested in deploying LibreOffice in a web-based, collaborative setting. == Applications == Collabora Online includes several applications for document editing, available through the web-based interface and optional desktop and mobile apps: Collabora Writer – A word processor based on LibreOffice Writer, comparable to Microsoft Word and Google Docs. It supports WYSIWYG editing, styles, formatting tools, comment threads, and change tracking. Collabora Calc – A spreadsheet editor based on LibreOffice Calc, similar to Microsoft Excel and Google Sheets. Features include pivot tables, formulas, data validation, conditional formatting, advanced sorting and filtering, charts, and support for up to 16,000 columns. Compatible with some macros written in VBA. Collabora Impress – A presentation program based on LibreOffice Impress, comparable to Microsoft PowerPoint and Google Slides. It supports master slides, transitions, speaker notes, and multimedia elements. Collabora Draw is not a separate application, most of the functionality of the Draw application is now integrated in Writer and Impress – vector graphics editor based on LibreOffice Draw, comparable to Microsoft Visio and Google Drawings. == Features == Collabora Online can be accessed from modern web browsers without the need for plug-ins or add-ons. It supports real-time collaborative editing of word processing documents, spreadsheets, presentations, and vector graphics. Collaboration features include commenting, version tracking with document comparison and restoration, and integration with communication tools such as chat or video calls. These functions are often enabled through integration with enterprise open-source cloud platforms like Nextcloud, ownCloud, Seafile, EGroupware, GroupOffice and others. Collabora Online can also be embedded or integrated into a variety of third-party applications. Although client apps are not required to use the web-based suite, optional applications are available for offline use on Android, ChromeOS, iOS, iPadOS, Linux distributions, macOS, and Windows. These apps share the same LibreOffice-based core as the server version, ensuring document compatibility across platforms. Development of the LibreOffice core benefits both the online server and the client applications simultaneously. The mobile apps offer touch-optimized interfaces that adapt to different screen sizes and can be used offline, with optional integration into cloud storage services. Collabora Online supports OpenDocument formats (ODF; .odt, .odp, .ods, .odg) in accordance with ISO/IEC 26300. It is also compatible with Microsoft Office formats, including Office Open XML (.docx, .pptx, .xlsx) and legacy binary formats (.doc, .ppt, .xls). Additional supported formats include PDF, PNG, CSV, TSV, RTF, EPUB, and others. The suite can import a range of formats supported by LibreOffice, including Microsoft Visio and Publisher files, Apple Keynote, Numbers, and Pages files, as well as legacy formats used by Lotus 1-2-3, Microsoft Works, and Quattro Pro. The core of Collabora Online is written in C++ and utilizes LibreOfficeKit, a programming interface that enables reuse of much of LibreOffice's existing code for document saving, loading, and rendering. Collabora Online operates on the principle that documents remain on the server, with users viewing tile-rendered images of the document and sending their edits back to the server. The user interface is implemented in JavaScript. For file access and authentication with file hosting services, Collabora Online uses Microsoft's WOPI protocol, allowing compatibility with any service supporting Microsoft 365 integration. == Server == The server component can be self-hosted or deployed through third-party enterprise open-source cloud platforms, allowing organizations to maintain control over data and infrastructure. It is available for various Linux distributions and as a Docker image. The server enables features such as in-browser document editing, file synchronization, and real-time communication. These third-party cloud platforms typically offer additional functionality comparable to services such as Dropbox, Google Workspace, Microsoft 365, or Zoom, including file sharing, calendars, email, contacts, chat, and video conferencing. Collabora Online can be integrated into these applications, as well as with other services such as learning management systems and enterprise content platforms, through open APIs and an SDK. == Reception == Various online and print publications have discussed Collabora Online. In December 2016 the technology website Softpedia mentioned the availability of collaborative editing in version 2.0 and the integration with ownCloud, Nextcloud, and other file synchronization and sharing solutions. In June 2020, ZDNET reported that Collabora Online would be included as the standard office suite in Nextcloud version 19, noting that direct document editing was added to the native video conferencing software Talk. The technology blog OMG! Ubuntu! covered the release of Collabora's Android and iOS apps, emphasizing their offline functionality. In September 2020, Linux Magazine compared Collabora Online with OnlyOffice, noting the flexibility and platform independence of both tools and highlighting Collabora's extensive feature set derived from LibreOffice. === Digital sovereignty === Collabora Online's open-source design and support for self-hosting have made it notable in discussions about digital sovereignty—the ability of users and organizations to control their own data. This is particularly relevant in Europe, where concerns about dependence on U.S.-based "big tech" companies and data privacy have grown in recent years. On 10th June 2025, Microsoft executives under oath in the French Senate admitted that they cannot guarantee data sovereignty and would be compelled to pass French (and by implication the wider European Union) information to the US administration if requested via a warrant or subpoena. The Cloud Act is a law that gives the US government authority to obtain digital data held by US-based tech corporations, irrespective of whether that data is stored on servers at home or on foreign soil. A 2020 briefing by the European Parliament highlighted risks associated with reliance on major technology companies that collect and exploit user data. Legal decisions such as the Schrems II ruling have further underscored these concerns. Several European government agencies have adopted private cloud solutions using Collabora Online and related platforms to enhance data security and maintain control over sensitive information. == History == The former LibreOffice development team from SUSE joined Collabora in September 2013, forming the subsidiary Collabora Productivity. In 2015 Collabora and IceWarp announced the development of an enterprise-ready version of LibreOffice Online to compete wi
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Weighted majority algorithm (machine learning)

In machine learning, weighted majority algorithm (WMA) is a meta learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well. Assume that the problem is a binary decision problem. To construct the compound algorithm, a positive weight is given to each of the algorithms in the pool. The compound algorithm then collects weighted votes from all the algorithms in the pool, and gives the prediction that has a higher vote. If the compound algorithm makes a mistake, the algorithms in the pool that contributed to the wrong predicting will be discounted by a certain ratio β where 0<β<1. It can be shown that the upper bounds on the number of mistakes made in a given sequence of predictions from a pool of algorithms A {\displaystyle \mathbf {A} } is O ( l o g | A | + m ) {\displaystyle \mathbf {O(log|A|+m)} } if one algorithm in x i {\displaystyle \mathbf {x} _{i}} makes at most m {\displaystyle \mathbf {m} } mistakes. There are many variations of the weighted majority algorithm to handle different situations, like shifting targets, infinite pools, or randomized predictions. The core mechanism remains similar, with the final performances of the compound algorithm bounded by a function of the performance of the specialist (best performing algorithm) in the pool.
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World Programming System

The World Programming System, also known as WPS Analytics or WPS, is a software product developed by a company called World Programming (acquired by Altair Engineering). WPS Analytics supports users of mixed ability to access and process data and to perform data science tasks. It has interactive visual programming tools using data workflows, and it has coding tools supporting the use of the SAS language mixed with Python, R and SQL. == About == WPS can use programs written in the language of SAS without the need for translating them into any other language. In this regard WPS is compatible with the SAS system. WPS has a built-in language interpreter able to process the language of SAS and produce similar results. WPS is available to run on z/OS, Windows, macOS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX. On all supported platforms, programs written in the language of SAS can be executed from a WPS command line interface, often referred to as running in batch mode. WPS can also be used from a graphical user interface known as the WPS Workbench for managing, editing and running programs written in the language of SAS. The WPS Workbench user interface is based on Eclipse. WPS version 4 (released in March 2018) introduced a drag-and-drop workflow canvas providing interactive blocks for data retrieval, blending and preparation, data discovery and profiling, predictive modelling powered by machine learning algorithms, model performance validation and scorecards. WPS version 3 (released in February 2012) provided a new client/server architecture that allows the WPS Workbench GUI to execute SAS programs on remote server installations of WPS in a network or cloud. The resulting output, data sets, logs, etc., can then all be viewed and manipulated from inside the Workbench as if the workloads had been executed locally. SAS programs do not require any special language statements to use this feature. == Summary of main features == Runs on Windows, macOS, z/OS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX An integrated development environment based on Eclipse for Linux, macOS and Windows. Support for language of SAS elements. Support for the language of SAS Macros. Matrix Programming support using PROC IML. Support for generating band plots, bar charts, box plots, bubble plots, contour plots, dendrogram plots, ellipse plots, fringe plots, heat maps, high-low plots, histograms, loess plots, needle plots, pie charts, penalised b-spline, radar charts, reference lines, scatter plots, series plots, step plots, regression plots and vector plots. Support for statistical procedures ACECLUS, ASSOCRULES, ANOVA, BIN, BOXPLOT, CANCORR, CANDISC, CLUSTER, CORRESP, DISCRIM, DISTANCE, FACTOR, FASTCLUS, FREQ, GAM, GANNO, GENMOD, GLIMMIX, GLM, GLMMOD, GLMSELECT, ICLIFETEST, KDE, LIFEREG, LIFETEST, LOESS, LOGISTIC, MDS, MEANS, MI, MIANALYSE, MIXED, MODECLUS, NESTED, NLIN, NPAR1WAY, PHREG, PLAN, PLS, POWER, PRINCOMP, PROBIT, QUANTREG, RBF, REG, ROBUSTREG, RSREG, SCORE, SEGMENT, SIMNORMAL, STANDARD, STDSIZE, STDRATE, STEPDISC, SUMMARY, SURVEYMEANS, SURVEYSELECT, TPSPLINE, TRANSREG, TREE, TTEST, UNIVARIATE, VARCLUS, VARCOMP Support for time series procedures ARIMA, AUTOREG, ESM, EXPAND, FORECAST, LOAN, SEVERITY, SPECTRA, TIMESERIES, X12 Support for machine learning procedures DECISIONFOREST, DECISIONTREE, GMM, MLP, OPTIMALBIN, SEGMENT, SVM Support for ODS. Reads and writes SAS datasets (compressed or uncompressed). Access: Actian Matrix (previously known as ParAccel), DASD, DB2, Excel, Greenplum, Hadoop, Informix, Kognitio Archived 2012-08-24 at the Wayback Machine, MariaDB, MySQL, Netezza, ODBC, OLEDB, Oracle, PostgreSQL, SAND, Snowflake, SPSS/PSPP, SQL Server, Sybase, Sybase IQ, Teradata, VSAM, Vertica and XML. Support for SAS Tape Format. Direct output of reports to CSV, PDF and HTML. Support to connect WPS systems programmatically, remote submit parts of a program to execute on connected remote servers, upload and download data between the connected systems. Support for Hadoop Support for R Support for Python == Industry recognition == Gartner recognized World Programming in their Cool Vendors in Data Science, 2014 Report. == Lawsuit == In 2010 World Programming defended its use of the language of SAS in the High Court of England and Wales in SAS Institute Inc. v World Programming Ltd. The software was the subject of a lawsuit by SAS Institute. The EU Court of Justice ruled in favor of World Programming, stating that the copyright protection does not extend to the software functionality, the programming language used and the format of the data files used by the program. It stated that there is no copyright infringement when a company which does not have access to the source code of a program studies, observes and tests that program to create another program with the same functionality.
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Probabilistic latent semantic analysis

Probabilistic latent semantic analysis (PLSA), also known as probabilistic latent semantic indexing (PLSI, especially in information retrieval circles) is a statistical technique for the analysis of two-mode and co-occurrence data. In effect, one can derive a low-dimensional representation of the observed variables in terms of their affinity to certain hidden variables, just as in latent semantic analysis, from which PLSA evolved. Compared to standard latent semantic analysis which stems from linear algebra and downsizes the occurrence tables (usually via a singular value decomposition), probabilistic latent semantic analysis is based on a mixture decomposition derived from a latent class model. == Model == Considering observations in the form of co-occurrences ( w , d ) {\displaystyle (w,d)} of words and documents, PLSA models the probability of each co-occurrence as a mixture of conditionally independent multinomial distributions: P ( w , d ) = ∑ c P ( d ) P ( c | d ) P ( w | c ) = P ( d ) ∑ c P ( c | d ) P ( w | c ) {\displaystyle P(w,d)=\sum _{c}P(d)P(c|d)P(w|c)=P(d)\sum _{c}P(c|d)P(w|c)} with c {\displaystyle c} being the words' topic. Note that the number of topics is a hyperparameter that must be chosen in advance and is not estimated from the data. The first formulation is the symmetric formulation, where w {\displaystyle w} and d {\displaystyle d} are both generated from the latent class c {\displaystyle c} in similar ways (using the conditional probabilities P ( d | c ) {\displaystyle P(d|c)} and P ( w | c ) {\displaystyle P(w|c)} ), whereas the second formulation is the asymmetric formulation, where, for each document d {\displaystyle d} , a latent class is chosen conditionally to the document according to P ( c | d ) {\displaystyle P(c|d)} , and a word is then generated from that class according to P ( w | c ) {\displaystyle P(w|c)} . Although we have used words and documents in this example, the co-occurrence of any couple of discrete variables may be modelled in exactly the same way. So, the number of parameters is equal to c d + w c {\displaystyle cd+wc} . The number of parameters grows linearly with the number of documents. In addition, although PLSA is a generative model of the documents in the collection it is estimated on, it is not a generative model of new documents. Their parameters are learned using the EM algorithm. == Application == PLSA may be used in a discriminative setting, via Fisher kernels. PLSA has applications in information retrieval and filtering, natural language processing, machine learning from text, bioinformatics, and related areas. It is reported that the aspect model used in the probabilistic latent semantic analysis has severe overfitting problems. == Extensions == Hierarchical extensions: Asymmetric: MASHA ("Multinomial ASymmetric Hierarchical Analysis") Symmetric: HPLSA ("Hierarchical Probabilistic Latent Semantic Analysis") Generative models: The following models have been developed to address an often-criticized shortcoming of PLSA, namely that it is not a proper generative model for new documents. Latent Dirichlet allocation – adds a Dirichlet prior on the per-document topic distribution Higher-order data: Although this is rarely discussed in the scientific literature, PLSA extends naturally to higher order data (three modes and higher), i.e. it can model co-occurrences over three or more variables. In the symmetric formulation above, this is done simply by adding conditional probability distributions for these additional variables. This is the probabilistic analogue to non-negative tensor factorisation. == History == This is an example of a latent class model (see references therein), and it is related to non-negative matrix factorization. The present terminology was coined in 1999 by Thomas Hofmann.
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Uniphore

Uniphore is an American software company that develops artificial intelligence platforms for business use. The company is headquartered in Palo Alto, California, with offices in the United States, United Kingdom, Spain, Israel, United Arab Emirates, and India. Uniphore is known for its "Business AI Cloud," an enterprise AI platform that combines data, knowledge, models, and software agents for use in sales, marketing, and service. The company has also acquired firms in video emotion AI, AI agents, low-code automation, knowledge automation, voice and screen capture, customer data platforms, and data engineering. == History == Uniphore Software Systems was founded by Umesh Sachdev and Ravi Saraogi in 2008 and was incubated at IIT Madras. The company received an initial grant of $100,000 from the National Research Development Corporation. Early work focused on speech technologies for emerging markets. Uniphore partnered with companies that specialized in English and European languages, and adapting the technology for Indian languages and dialects. In 2014, Uniphore released its first flagship products, auMina, along with two other products, Akeira and amVoice. Uniphore raised series A funding, led by Kris Gopalakrishnan (cofounder of Infosys), in April 2015. The next month, Uniphore received additional investment from IDG Ventures. With input from its investors, Uniphore changed its business model from license fee-based income to a software as a service-based subscription fee model in 2015. By June 2016, it had added more than 70 global languages and expanded its services to Southeast Asia, the Middle East, and the United States. The company opened operations in Singapore in October 2016. The company raised Series B funding in October 2017, led by John Chambers and existing investors. Series C funding of $51 million was announced in August 2019 and led by March Capital. Uniphore acquired an exclusive third-party license for robotic process automation technology from NTT DATA in October 2020. In January 2021, Uniphore acquired Emotion Research Lab, a startup based in Spain that uses artificial intelligence and machine learning to analyze video and interpret emotions. The company received $140 million in Series D funding, led by Sorenson Capital Partners, in March 2021, bringing total funding to $210 million. In January 2021, Uniphore acquired Emotion Research Lab. In July 2021, it agreed to acquire Jacada, a provider of low-code/no-code automation; the transaction closed in October 2021. On February 16, 2022, Uniphore announced a $400 million Series E financing led by NEA, which valued the company at $2.5 billion. Hilarie Koplow-McAdams, an NEA venture partner and former Salesforce/New Relic executive, joined Uniphore's board in 2022. Uniphore's board has also included former Cisco CEO John Chambers, former Convergys CEO Andrea J. Ayers, and CrowdStrike CFO Burt Podbere (appointed January 2021). In February 2023, Uniphore acquired UK-based Red Box, a platform for capturing voice and screen recordings used in regulated and large-scale environments. It also acquired France-based Hexagone, a behavioral analytics firm combining computer vision and natural-language techniques. On December 5, 2024, Uniphore announced agreements to acquire ActionIQ, a customer data platform (CDP) vendor, and Infoworks, an enterprise data engineering platform. Uniphore launched the Business AI Cloud on June 9, 2025. The Business AI Cloud consists of a single, unified platform that includes data, knowledge, AI models, and AI agents. Uniphore announced in August 2025 that it had acquired Orby AI and intended to acquire Autonom8 to extend multi-agent and workflow automation capabilities. As of September 2025, Uniphore's customers included the United States Coast Guard, Singapore Police Force, London Underground, DirecTV, JPMorgan Chase, LG, DHL, UPS, Vodafone, Verizon, NTT Data, and as of May 2021, Firstsource. In October 2025, Uniphore raised $260 million in a Series F round at a reported valuation of $2.5 billion. Investors included March Capital, NEA, Nvidia, AMD, Snowflake, and Databricks. In January 2026, KPMG and Uniphore announced a collaboration focused on deploying AI agents powered by specialized small language models. The announcement was made at the World Economic Forum held in Davos. Cognizant and Uniphore announced a partnership in February 2026 to develop industry-specific AI tools for regulated sectors, which would initially focus on life sciences and finance. Uniphore and Rackspace also announced a partnership in March 2026. This partnership was announced in order to create an "Infrastructure-to-Agents" architecture, focusing on Business AI as a private cloud service. == Products == As of 2025, Uniphore's core offering is the Business AI Cloud and Business AI Suite of agentic AI applications. === Business AI Cloud === Uniphore’s Business AI Cloud is a full-stack platform that organizes enterprise data and knowledge for agentic AI applications. The platform enables deployment across clouds and existing data sources. Key layers and capabilities include the following. Agentic layer: Includes prebuilt agents, a natural-language agent builder, and orchestration based on Business Process Model and Notation (BPMN) to run AI workflows across business units. Model layer: Supports an open, interoperable mix of closed and open-source large language models (LLMs). Models can be orchestrated, governed, and replaced as needed. Knowledge layer: Organizes raw data into structured knowledge used for retrieval, explainability, and fine-tuning of small language models (SLMs). Data layer: Connects to data across multiple platforms and clouds through a zero-copy, composable fabric, enabling in-place preparation and supporting data residency and sovereignty requirements. === Business AI Suite === The Uniphore Business AI Suite has various prebuilt AI agents that can be used in customer service, sales, marketing, and human resources. The Uniphore Business AI Suite includes several LOBs (Lines of Business) for business functions with intelligent agents that are prebuilt, but composable. Built on the Uniphore Business AI Cloud, each application combines agentic automation and fine-tuned models. Marketing AI, Customer Service AI, Sales AI, and People AI (for human resources) are included. Competitors include Palantir, Microsoft Azure, Amazon Bedrock, Google's Vertex AI, Databricks, and Snowflake. == Recognition == Deloitte Technology Fast 50 India identified Uniphore as the 17th fastest-growing technology company in India in 2012 and one of the top 500 fastest growing companies in the Asia-Pacific region in 2014. In 2016, Time included Sachdev on its list of "10 millennials who are changing the world" for “building a phone that can understand almost any language”. NASSCOM named Uniphore to its "League of 10" emerging Indian technology companies in 2017. In 2020, the San Francisco Business Times ranked Uniphore as No. 7 among small companies in its list of the best places to work in the San Francisco Bay Area. In 2022, the company was featured on the Forbes AI 50 list. Uniphore was mentioned in the Deloitte Technology Fast 500 list in 2023, 2024, and 2025. In 2025, Inc. included Uniphore in its Best in Business program.
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Multi-label classification

In machine learning, multi-label classification or multi-output classification is a variant of the classification problem where multiple nonexclusive labels may be assigned to each instance. Multi-label classification is a generalization of multiclass classification, which is the single-label problem of categorizing instances into precisely one of several (greater than or equal to two) classes. In the multi-label problem the labels are nonexclusive and there is no constraint on how many of the classes the instance can be assigned to. The formulation of multi-label learning was first introduced by Shen et al. in the context of Semantic Scene Classification, and later gained popularity across various areas of machine learning. Formally, multi-label classification is the problem of finding a model that maps inputs x to binary vectors y; that is, it assigns a value of 0 or 1 for each element (label) in y. == Problem transformation methods == Several problem transformation methods exist for multi-label classification, and can be roughly broken down into: === Transformation into binary classification problems === The baseline approach, called the binary relevance method, amounts to independently training one binary classifier for each label. Given an unseen sample, the combined model then predicts all labels for this sample for which the respective classifiers predict a positive result. Although this method of dividing the task into multiple binary tasks may resemble superficially the one-vs.-all (OvA) and one-vs.-rest (OvR) methods for multiclass classification, it is essentially different from both, because a single classifier under binary relevance deals with a single label, without any regard to other labels whatsoever. A classifier chain is an alternative method for transforming a multi-label classification problem into several binary classification problems. It differs from binary relevance in that labels are predicted sequentially, and the output of all previous classifiers (i.e. positive or negative for a particular label) are input as features to subsequent classifiers. Classifier chains have been applied, for instance, in HIV drug resistance prediction. Bayesian network has also been applied to optimally order classifiers in Classifier chains. In case of transforming the problem to multiple binary classifications, the likelihood function reads L = ∏ i = 1 n ( ∏ k ( ∏ j k ( p k , j k ( x i ) δ y i , k , j k ) ) ) {\displaystyle L=\prod _{i=1}^{n}(\prod _{k}(\prod _{j_{k}}(p_{k,j_{k}}(x_{i})^{\delta _{y_{i,k},j_{k}}})))} where index i {\displaystyle i} runs over the samples, index k {\displaystyle k} runs over the labels, j k {\displaystyle j_{k}} indicates the binary outcomes 0 or 1, δ a , b {\displaystyle \delta _{a,b}} indicates the Kronecker delta, y i , k ∈ 0 , 1 {\displaystyle y_{i,k}\in {0,1}} indicates the multiple hot encoded labels of sample i {\displaystyle i} . === Transformation into multi-class classification problem === The label powerset (LP) transformation creates one binary classifier for every label combination present in the training set. For example, if possible labels for an example were A, B, and C, the label powerset representation of this problem is a multi-class classification problem with the classes [0 0 0], [1 0 0], [0 1 0], [0 0 1], [1 1 0], [1 0 1], [0 1 1], and [1 1 1] where for example [1 0 1] denotes an example where labels A and C are present and label B is absent. === Ensemble methods === A set of multi-class classifiers can be used to create a multi-label ensemble classifier. For a given example, each classifier outputs a single class (corresponding to a single label in the multi-label problem). These predictions are then combined by an ensemble method, usually a voting scheme where every class that receives a requisite percentage of votes from individual classifiers (often referred to as the discrimination threshold) is predicted as a present label in the multi-label output. However, more complex ensemble methods exist, such as committee machines. Another variation is the random k-labelsets (RAKEL) algorithm, which uses multiple LP classifiers, each trained on a random subset of the actual labels; label prediction is then carried out by a voting scheme. A set of multi-label classifiers can be used in a similar way to create a multi-label ensemble classifier. In this case, each classifier votes once for each label it predicts rather than for a single label. == Adapted algorithms == Some classification algorithms/models have been adapted to the multi-label task, without requiring problem transformations. Examples of these including for multi-label data are k-nearest neighbors: the ML-kNN algorithm extends the k-NN classifier to multi-label data. decision trees: "Clare" is an adapted C4.5 algorithm for multi-label classification; the modification involves the entropy calculations. MMC, MMDT, and SSC refined MMDT, can classify multi-labeled data based on multi-valued attributes without transforming the attributes into single-values. They are also named multi-valued and multi-labeled decision tree classification methods. kernel methods for vector output neural networks: BP-MLL is an adaptation of the popular back-propagation algorithm for multi-label learning. == Learning paradigms == Based on learning paradigms, the existing multi-label classification techniques can be classified into batch learning and online machine learning. Batch learning algorithms require all the data samples to be available beforehand. It trains the model using the entire training data and then predicts the test sample using the found relationship. The online learning algorithms, on the other hand, incrementally build their models in sequential iterations. In iteration t, an online algorithm receives a sample, xt and predicts its label(s) ŷt using the current model; the algorithm then receives yt, the true label(s) of xt and updates its model based on the sample-label pair: (xt, yt). == Multi-label stream classification == Data streams are possibly infinite sequences of data that continuously and rapidly grow over time. Multi-label stream classification (MLSC) is the version of multi-label classification task that takes place in data streams. It is sometimes also called online multi-label classification. The difficulties of multi-label classification (exponential number of possible label sets, capturing dependencies between labels) are combined with difficulties of data streams (time and memory constraints, addressing infinite stream with finite means, concept drifts). Many MLSC methods resort to ensemble methods in order to increase their predictive performance and deal with concept drifts. Below are the most widely used ensemble methods in the literature: Online Bagging (OzaBagging)-based methods: Observing the probability of having K many of a certain data point in a bootstrap sample is approximately Poisson(1) for big datasets, each incoming data instance in a data stream can be weighted proportional to Poisson(1) distribution to mimic bootstrapping in an online setting. This is called Online Bagging (OzaBagging). Many multi-label methods that use Online Bagging are proposed in the literature, each of which utilizes different problem transformation methods. EBR, ECC, EPS, EBRT, EBMT, ML-Random Rules are examples of such methods. ADWIN Bagging-based methods: Online Bagging methods for MLSC are sometimes combined with explicit concept drift detection mechanisms such as ADWIN (Adaptive Window). ADWIN keeps a variable-sized window to detect changes in the distribution of the data, and improves the ensemble by resetting the components that perform poorly when there is a drift in the incoming data. Generally, the letter 'a' is used as a subscript in the name of such ensembles to indicate the usage of ADWIN change detector. EaBR, EaCC, EaHTPS are examples of such multi-label ensembles. GOOWE-ML-based methods: Interpreting the relevance scores of each component of the ensemble as vectors in the label space and solving a least squares problem at the end of each batch, Geometrically-Optimum Online-Weighted Ensemble for Multi-label Classification (GOOWE-ML) is proposed. The ensemble tries to minimize the distance between the weighted prediction of its components and the ground truth vector for each instance over a batch. Unlike Online Bagging and ADWIN Bagging, GOOWE-ML utilizes a weighted voting scheme where better performing components of the ensemble are given more weight. The GOOWE-ML ensemble grows over time, and the lowest weight component is replaced by a new component when it is full at the end of a batch. GOBR, GOCC, GOPS, GORT are the proposed GOOWE-ML-based multi-label ensembles. Multiple Windows : Here, BR models that use a sliding window are replaced with two windows for each label, one for relevant and one for non-relevant examples. Instances are oversampled or undersampled according to a load factor that is kept
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Concept class

In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning. In this setting, if one takes a set Y as a set of (classifier output) labels, and X is a set of examples, the map c : X → Y {\displaystyle c:X\to Y} , i.e. from examples to classifier labels (where Y = { 0 , 1 } {\displaystyle Y=\{0,1\}} and where c is a subset of X), c is then said to be a concept. A concept class C {\displaystyle C} is then a collection of such concepts. Given a class of concepts C, a subclass D is reachable if there exists a sample s such that D contains exactly those concepts in C that are extensions to s. Not every subclass is reachable. == Background == A sample s {\displaystyle s} is a partial function from X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} . Identifying a concept with its characteristic function mapping X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} , it is a special case of a sample. Two samples are consistent if they agree on the intersection of their domains. A sample s ′ {\displaystyle s'} extends another sample s {\displaystyle s} if the two are consistent and the domain of s {\displaystyle s} is contained in the domain of s ′ {\displaystyle s'} . == Examples == Suppose that C = S + ( X ) {\displaystyle C=S^{+}(X)} . Then: the subclass { { x } } {\displaystyle \{\{x\}\}} is reachable with the sample s = { ( x , 1 ) } {\displaystyle s=\{(x,1)\}} ; the subclass S + ( Y ) {\displaystyle S^{+}(Y)} for Y ⊆ X {\displaystyle Y\subseteq X} are reachable with a sample that maps the elements of X − Y {\displaystyle X-Y} to zero; the subclass S ( X ) {\displaystyle S(X)} , which consists of the singleton sets, is not reachable. == Applications == Let C {\displaystyle C} be some concept class. For any concept c ∈ C {\displaystyle c\in C} , we call this concept 1 / d {\displaystyle 1/d} -good for a positive integer d {\displaystyle d} if, for all x ∈ X {\displaystyle x\in X} , at least 1 / d {\displaystyle 1/d} of the concepts in C {\displaystyle C} agree with c {\displaystyle c} on the classification of x {\displaystyle x} . The fingerprint dimension F D ( C ) {\displaystyle FD(C)} of the entire concept class C {\displaystyle C} is the least positive integer d {\displaystyle d} such that every reachable subclass C ′ ⊆ C {\displaystyle C'\subseteq C} contains a concept that is 1 / d {\displaystyle 1/d} -good for it. This quantity can be used to bound the minimum number of equivalence queries needed to learn a class of concepts according to the following inequality: F D ( C ) − 1 ≤ # E Q ( C ) ≤ ⌈ F D ( C ) ln ⁡ ( | C | ) ⌉ {\textstyle FD(C)-1\leq \#EQ(C)\leq \lceil FD(C)\ln(|C|)\rceil } .
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Plate notation

In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The assumptions are that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition. == Example == In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate; α is the parameter of the uniform Dirichlet prior on the per-document topic distributions, and β is the parameter of the uniform Dirichlet prior on the per-topic word distribution. The outermost plate represents all the variables related to a specific document, including θ i {\displaystyle \theta _{i}} , the topic distribution for document i. The M in the corner of the plate indicates that the variables inside are repeated M times, once for each document. The inner plate represents the variables associated with each of the N i {\displaystyle N_{i}} words in document i: z i j {\displaystyle z_{ij}} is the topic distribution for the jth word in document i, and w i j {\displaystyle w_{ij}} is the actual word used. The N in the corner represents the repetition of the variables in the inner plate N j {\displaystyle N_{j}} times, once for each word in document i. The circle representing the individual words is shaded, indicating that each w i j {\displaystyle w_{ij}} is observable, and the other circles are empty, indicating that the other variables are latent variables. The directed edges between variables indicate dependencies between the variables: for example, each w i j {\displaystyle w_{ij}} depends on z i j {\displaystyle z_{ij}} and β. == Extensions == A number of extensions have been created by various authors to express more information than simply the conditional relationships. However, few of these have become standard. Perhaps the most commonly used extension is to use rectangles in place of circles to indicate non-random variables—either parameters to be computed, hyperparameters given a fixed value (or computed through empirical Bayes), or variables whose values are computed deterministically from a random variable. The diagram on the right shows a few more non-standard conventions used in some articles in Wikipedia (e.g. variational Bayes): Variables that are actually random vectors are indicated by putting the vector size in brackets in the middle of the node. Variables that are actually random matrices are similarly indicated by putting the matrix size in brackets in the middle of the node, with commas separating row size from column size. Categorical variables are indicated by placing their size (without a bracket) in the middle of the node. Categorical variables that act as "switches", and which pick one or more other random variables to condition on from a large set of such variables (e.g. mixture components), are indicated with a special type of arrow containing a squiggly line and ending in a T junction. Boldface is consistently used for vector or matrix nodes (but not categorical nodes). == Software implementation == Plate notation has been implemented in various TeX/LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab and PyMC.
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Crucible (software)

Crucible is a collaborative code review application by Australian software company Atlassian. Like other Atlassian products, Crucible is a Web-based application primarily aimed at enterprise, and certain features that enable peer review of a codebase may be considered enterprise social software. Crucible is particularly tailored to remote workers, and facilitates asynchronous review and commenting on code. Crucible also integrates with popular source control tools, such as Git and Subversion. Crucible is not open source, but customers are allowed to view and modify the code for their own use.
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Ho–Kashyap algorithm

The Ho–Kashyap algorithm is an iterative method in machine learning for finding a linear decision boundary that separates two linearly separable classes. It was developed by Yu-Chi Ho and Rangasami L. Kashyap in 1965, and usually presented as a problem in linear programming. == Setup == Given a training set consisting of samples from two classes, the Ho–Kashyap algorithm seeks to find a weight vector w {\displaystyle \mathbf {w} } and a margin vector b {\displaystyle \mathbf {b} } such that: Y w = b {\displaystyle \mathbf {Yw} =\mathbf {b} } where Y {\displaystyle \mathbf {Y} } is the augmented data matrix with samples from both classes (with appropriate sign conventions, e.g., samples from class 2 are negated), w {\displaystyle \mathbf {w} } is the weight vector to be determined, and b {\displaystyle \mathbf {b} } is a positive margin vector. The algorithm minimizes the criterion function: J ( w , b ) = | | Y w − b | | 2 {\displaystyle J(\mathbf {w} ,\mathbf {b} )=||\mathbf {Yw} -\mathbf {b} ||^{2}} subject to the constraint that b > 0 {\displaystyle \mathbf {b} >\mathbf {0} } (element-wise). Given a problem of linearly separating two classes, we consider a dataset of elements { ( x i , y i ) } i ∈ 1 : N {\displaystyle \{(\mathbf {x_{i}} ,y_{i})\}_{i\in 1:N}} where y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} . Linearly separating them by a perceptron is equivalent to finding weight and bias w , b {\displaystyle \mathbf {w} ,b} for a perceptron, such that: [ y 1 x 1 1 ⋮ ⋮ y N x N 1 ] [ w b ] > 0 {\displaystyle {\begin{bmatrix}y_{1}\mathbf {x} _{1}&1\\\vdots &\vdots \\y_{N}\mathbf {x} _{N}&1\\\end{bmatrix}}{\begin{bmatrix}\mathbf {w} \\b\end{bmatrix}}>0} == Algorithm == The idea of the Ho–Kashyap algorithm is as follows: Given any b {\displaystyle \mathbf {b} } , the corresponding w {\displaystyle \mathbf {w} } is known: It is simply w = Y + b {\displaystyle \mathbf {w} =\mathbf {Y} ^{+}\mathbf {b} } , where Y + {\displaystyle \mathbf {Y} ^{+}} denotes the Moore–Penrose pseudoinverse of Y {\displaystyle \mathbf {Y} } . Therefore, it only remains to find b {\displaystyle \mathbf {b} } by gradient descent. However, the gradient descent may sometimes decrease some of the coordinates of b {\displaystyle \mathbf {b} } , which may cause some coordinates of b {\displaystyle \mathbf {b} } to become negative, which is undesirable. Therefore, whenever some coordinates of b {\displaystyle \mathbf {b} } would have decreased, those coordinates are unchanged instead. As for the coordinates of b {\displaystyle \mathbf {b} } that would increase, those would increase without issue. Formally, the algorithm is as follows: Initialization: Set b ( 0 ) {\displaystyle \mathbf {b} (0)} to an arbitrary positive vector, typically b ( 0 ) = 1 {\displaystyle \mathbf {b} (0)=\mathbf {1} } (a vector of ones). Set the iteration counter k = 0 {\displaystyle k=0} . Set w ( 0 ) = Y + b ( 0 ) {\displaystyle \mathbf {w} (0)=\mathbf {Y} ^{+}\mathbf {b} (0)} Loop until convergence, or until iteration counter exceeds some k m a x {\displaystyle k_{max}} . Error calculation: Compute the error vector: e ( k ) = Y w ( k ) − b ( k ) {\displaystyle \mathbf {e} (k)=\mathbf {Yw} (k)-\mathbf {b} (k)} . Margin update: Update the margin vector: b ( k + 1 ) = b ( k ) + 2 η k ( e ( k ) + | e ( k ) | ) {\displaystyle \mathbf {b} (k+1)=\mathbf {b} (k)+2\eta _{k}(\mathbf {e} (k)+|\mathbf {e} (k)|)} where η k {\displaystyle \eta _{k}} is a positive learning rate parameter, and | e ( k ) | {\displaystyle |\mathbf {e} (k)|} denotes the element-wise absolute value. Weight calculation: Compute the weight vector using the pseudoinverse: w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} . Convergence check: If | | e ( k ) | | ≤ θ {\displaystyle ||\mathbf {e} (k)||\leq \theta } for some predetermined threshold θ {\displaystyle \theta } (close to zero), then return b ( k + 1 ) , w ( k + 1 ) {\displaystyle \mathbf {b} (k+1),\mathbf {w} (k+1)} . if e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } (all components non-positive), return "Samples not separable.". Return "Algorithm failed to converge in time.". == Properties == If the training data is linearly separable, the algorithm converges to a solution (where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } ) in a finite number of iterations. If the data is not linearly separable, the algorithm may or may not ever reach the point where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } . However, if it does happen that e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } at some iteration, this proves non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability of the data. == Relationship to other algorithms == Perceptron algorithm: Both seek linear separators. The perceptron updates weights incrementally based on individual misclassified samples, while Ho–Kashyap is a batch method that processes all samples to compute the pseudoinverse and updates based on an overall error vector. Linear discriminant analysis (LDA): LDA assumes underlying Gaussian distributions with equal covariances for the classes and derives the decision boundary from these statistical assumptions. Ho–Kashyap makes no explicit distributional assumptions and instead tries to solve a system of linear inequalities directly. Support vector machines (SVM): For linearly separable data, SVMs aim to find the maximum-margin hyperplane. The Ho–Kashyap algorithm finds a separating hyperplane but not necessarily the one with the maximum margin. If the data is not separable, soft-margin SVMs allow for some misclassifications by optimizing a trade-off between margin size and misclassification penalty, while Ho–Kashyap provides a least-squares solution. == Variants == Modified Ho–Kashyap algorithm changes weight calculation step w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} to w ( k + 1 ) = w ( k ) + η k Y + | e ( k ) | {\displaystyle \mathbf {w} (k+1)=\mathbf {w} (k)+\eta _{k}\mathbf {Y} ^{+}|\mathbf {e} (k)|} . Kernel Ho–Kashyap algorithm: Applies kernel methods (the "kernel trick") to the Ho–Kashyap framework to enable non-linear classification by implicitly mapping data to a higher-dimensional feature space.
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Conference on Computer Vision and Pattern Recognition

The Conference on Computer Vision and Pattern Recognition is an annual conference on computer vision and pattern recognition. == Affiliations == The conference was first held in 1983 in Washington, DC, organized by Takeo Kanade and Dana H. Ballard. From 1985 to 2010 it was sponsored by the IEEE Computer Society. In 2011 it was also co-sponsored by University of Colorado Colorado Springs. Since 2012 it has been co-sponsored by the IEEE Computer Society and the Computer Vision Foundation, which provides open access to the conference papers. == Scope == The conference considers a wide range of topics related to computer vision and pattern recognition—basically any topic that is extracting structures or answers from images or video or applying mathematical methods to data to extract or recognize patterns. Common topics include object recognition, image segmentation, motion estimation, 3D reconstruction, and deep learning. The conference generally has less than 30% acceptance rates for all papers and less than 5% for oral presentations. It is managed by a rotating group of volunteers who are chosen in a public election at the Pattern Analysis and Machine Intelligence-Technical Community (PAMI-TC) meeting four years before the meeting. The conference uses a multi-tier double-blind peer review process. The program chairs, who cannot submit papers, select area chairs who manage the reviewers for their subset of submissions. == Location and time == The conference is usually held in June in North America. == Awards == === Best Paper Award === These awards are picked by committees delegated by the program chairs of the conference. === Longuet-Higgins Prize === The Longuet-Higgins Prize recognizes papers from ten years ago that have made a significant impact on computer vision research. === PAMI Young Researcher Award === The Pattern Analysis and Machine Intelligence Young Researcher Award is an award given by the Technical Committee on Pattern Analysis and Machine Intelligence of the IEEE Computer Society to a researcher within 7 years of completing their Ph.D. for outstanding early career research contributions. Candidates are nominated by the computer vision community, with winners selected by a committee of senior researchers in the field. This award was originally instituted in 2012 by the journal Image and Vision Computing, also presented at the conference, and the journal continues to sponsor the award. === PAMI Thomas S. Huang Memorial Prize === The Thomas Huang Memorial Prize was established at the 2020 conference and is awarded annually starting from 2021 to honor researchers who are recognized as examples in research, teaching/mentoring, and service to the computer vision community.
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