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  Connected-component labeling

  Connected-component labeling (CCL), connected-component analysis (CCA), blob extraction, region labeling, blob discovery, or region extraction is an algorithmic application of graph theory, where subsets of connected components are uniquely labeled based on a given heuristic. Connected-component labeling is not to be confused with segmentation. Connected-component labeling is used in computer vision to detect connected regions in binary digital images, although color images and data with higher dimensionality can also be processed. When integrated into an image recognition system or human-computer interaction interface, connected component labeling can operate on a variety of information. Blob extraction is generally performed on the resulting binary image from a thresholding step, but it can be applicable to gray-scale and color images as well. Blobs may be counted, filtered, and tracked. Blob extraction is related to but distinct from blob detection. == Overview == A graph, containing vertices and connecting edges, is constructed from relevant input data. The vertices contain information required by the comparison heuristic, while the edges indicate connected 'neighbors'. An algorithm traverses the graph, labeling the vertices based on the connectivity and relative values of their neighbors. Connectivity is determined by the medium; image graphs, for example, can be 4-connected neighborhood or 8-connected neighborhood. Following the labeling stage, the graph may be partitioned into subsets, after which the original information can be recovered and processed . == Definition == The usage of the term connected-component labeling (CCL) and its definition is quite consistent in the academic literature, whereas connected-component analysis (CCA) varies both in terminology and in its definition of the problem. Rosenfeld et al. define connected components labeling as the “[c]reation of a labeled image in which the positions associated with the same connected component of the binary input image have a unique label.” Shapiro et al. define CCL as an operator whose “input is a binary image and [...] output is a symbolic image in which the label assigned to each pixel is an integer uniquely identifying the connected component to which that pixel belongs.” There is no consensus on the definition of CCA in the academic literature. It is often used interchangeably with CCL. A more extensive definition is given by Shapiro et al.: “Connected component analysis consists of connected component labeling of the black pixels followed by property measurement of the component regions and decision making.” The definition for connected-component analysis presented here is more general, taking the thoughts expressed in into account. == Algorithms == The algorithms discussed can be generalised to arbitrary dimensions, albeit with increased time and space complexity. === One component at a time === This is a fast and very simple method to implement and understand. It is based on graph traversal methods in graph theory. In short, once the first pixel of a connected component is found, all the connected pixels of that connected component are labelled before going onto the next pixel in the image. This algorithm is part of Vincent and Soille's watershed segmentation algorithm, other implementations also exist. In order to do that a linked list is formed that will keep the indexes of the pixels that are connected to each other, steps (2) and (3) below. The method of defining the linked list specifies the use of a depth or a breadth first search. For this particular application, there is no difference which strategy to use. The simplest kind of a last in first out queue implemented as a singly linked list will result in a depth first search strategy. It is assumed that the input image is a binary image, with pixels being either background or foreground and that the connected components in the foreground pixels are desired. The algorithm steps can be written as: Start from the first pixel in the image. Set current label to 1. Go to (2). If this pixel is a foreground pixel and it is not already labelled, give it the current label and add it as the first element in a queue, then go to (3). If it is a background pixel or it was already labelled, then repeat (2) for the next pixel in the image. Pop out an element from the queue, and look at its neighbours (based on any type of connectivity). If a neighbour is a foreground pixel and is not already labelled, give it the current label and add it to the queue. Repeat (3) until there are no more elements in the queue. Go to (2) for the next pixel in the image and increment current label by 1. Note that the pixels are labelled before being put into the queue. The queue will only keep a pixel to check its neighbours and add them to the queue if necessary. This algorithm only needs to check the neighbours of each foreground pixel once and doesn't check the neighbours of background pixels. The pseudocode is: algorithm OneComponentAtATime(data) input : imageData[xDim][yDim] initialization : label = 0, labelArray[xDim][yDim] = 0, statusArray[xDim][yDim] = false, queue1, queue2; for i = 0 to xDim do for j = 0 to yDim do if imageData[i][j] has not been processed do if imageData[i][j] is a foreground pixel do check its four neighbors(north, south, east, west) : if neighbor is not processed do if neighbor is a foreground pixel do add it to queue1 else update its status to processed end if labelArray[i][j] = label (give label) statusArray[i][j] = true (update status) while queue1 is not empty do For each pixel in the queue do : check its four neighbors if neighbor is not processed do if neighbor is a foreground pixel do add it to queue2 else update its status to processed end if give it the current label update its status to processed remove the current element from queue1 copy queue2 into queue1 end While increase the label end if else update its status to processed end if end if end if end for end for === Two-pass === Relatively simple to implement and understand, the two-pass algorithm, (also known as the Hoshen–Kopelman algorithm) iterates through 2-dimensional binary data. The algorithm makes two passes over the image: the first pass to assign temporary labels and record equivalences, and the second pass to replace each temporary label by the smallest label of its equivalence class. The input data can be modified in situ (which carries the risk of data corruption), or labeling information can be maintained in an additional data structure. Connectivity checks are carried out by checking neighbor pixels' labels (neighbor elements whose labels are not assigned yet are ignored), or say, the north-east, the north, the north-west and the west of the current pixel (assuming 8-connectivity). 4-connectivity uses only north and west neighbors of the current pixel. The following conditions are checked to determine the value of the label to be assigned to the current pixel (4-connectivity is assumed) Conditions to check: Does the pixel to the left (west) have the same value as the current pixel? Yes – We are in the same region. Assign the same label to the current pixel No – Check next condition Do both pixels to the north and west of the current pixel have the same value as the current pixel but not the same label? Yes – We know that the north and west pixels belong to the same region and must be merged. Assign the current pixel the minimum of the north and west labels, and record their equivalence relationship No – Check next condition Does the pixel to the left (west) have a different value and the one to the north the same value as the current pixel? Yes – Assign the label of the north pixel to the current pixel No – Check next condition Do the pixel's north and west neighbors have different pixel values than current pixel? Yes – Create a new label id and assign it to the current pixel The algorithm continues this way, and creates new region labels whenever necessary. The key to a fast algorithm, however, is how this merging is done. This algorithm uses the union-find data structure which provides excellent performance for keeping track of equivalence relationships. Union-find essentially stores labels which correspond to the same blob in a disjoint-set data structure, making it easy to remember the equivalence of two labels by the use of an interface method E.g.: findSet(l). findSet(l) returns the minimum label value that is equivalent to the function argument 'l'. Once the initial labeling and equivalence recording is completed, the second pass merely replaces each pixel label with its equivalent disjoint-set representative element. A faster-scanning algorithm for connected-region extraction is presented below. On the first pass: Iterate through each element of the data by column, then by row (Raster Scanning) If the element is not the background Get the neighboring elements of the current element If there are no neighbors, uniquely

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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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  Calibration (statistics)

  There are two main uses of the term calibration in statistics that denote special types of statistical inference problems. Calibration can mean a reverse process to regression, where instead of a future dependent variable being predicted from known explanatory variables, a known observation of the dependent variables is used to predict a corresponding explanatory variable; procedures in statistical classification to determine class membership probabilities which assess the uncertainty of a given new observation belonging to each of the already established classes. In addition, calibration is used in statistics with the usual general meaning of calibration. For example, model calibration can be also used to refer to Bayesian inference about the value of a model's parameters, given some data set, or more generally to any type of fitting of a statistical model. As Philip Dawid puts it, "a forecaster is well calibrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent." == In classification == Calibration in classification means transforming classifier scores into class membership probabilities. An overview of calibration methods for two-class and multi-class classification tasks is given by Gebel (2009). A classifier might separate the classes well, but be poorly calibrated, meaning that the estimated class probabilities are far from the true class probabilities. In this case, a calibration step may help improve the estimated probabilities. A variety of metrics exist that are aimed to measure the extent to which a classifier produces well-calibrated probabilities. Foundational work includes the Expected Calibration Error (ECE). Into the 2020s, variants include the Adaptive Calibration Error (ACE) and the Test-based Calibration Error (TCE), which address limitations of the ECE metric that may arise when classifier scores concentrate on narrow subset of the [0,1] range. A 2020s advancement in calibration assessment is the introduction of the Estimated Calibration Index (ECI). The ECI extends the concepts of the Expected Calibration Error (ECE) to provide a more nuanced measure of a model's calibration, particularly addressing overconfidence and underconfidence tendencies. Originally formulated for binary settings, the ECI has been adapted for multiclass settings, offering both local and global insights into model calibration. This framework aims to overcome some of the theoretical and interpretative limitations of existing calibration metrics. Through a series of experiments, Famiglini et al. demonstrate the framework's effectiveness in delivering a more accurate understanding of model calibration levels and discuss strategies for mitigating biases in calibration assessment. An online tool has been proposed to compute both ECE and ECI. The following univariate calibration methods exist for transforming classifier scores into class membership probabilities in the two-class case: Assignment value approach, see Garczarek (2002) Bayes approach, see Bennett (2002) Isotonic regression, see Zadrozny and Elkan (2002) Platt scaling (a form of logistic regression), see Lewis and Gale (1994) and Platt (1999) Bayesian Binning into Quantiles (BBQ) calibration, see Naeini, Cooper, Hauskrecht (2015) Beta calibration, see Kull, Filho, Flach (2017) === In probability prediction and forecasting === In prediction and forecasting, a Brier score is sometimes used to assess prediction accuracy of a set of predictions, specifically that the magnitude of the assigned probabilities track the relative frequency of the observed outcomes. Philip E. Tetlock employs the term "calibration" in this sense in his 2015 book Superforecasting. This differs from accuracy and precision. For example, as expressed by Daniel Kahneman, "if you give all events that happen a probability of .6 and all the events that don't happen a probability of .4, your discrimination is perfect but your calibration is miserable". In meteorology, in particular, as concerns weather forecasting, a related mode of assessment is known as forecast skill. == In regression == The calibration problem in regression is the use of known data on the observed relationship between a dependent variable and an independent variable to make estimates of other values of the independent variable from new observations of the dependent variable. This can be known as "inverse regression"; there is also sliced inverse regression. The following multivariate calibration methods exist for transforming classifier scores into class membership probabilities in the case with classes count greater than two: Reduction to binary tasks and subsequent pairwise coupling, see Hastie and Tibshirani (1998) Dirichlet calibration, see Gebel (2009) === Example === One example is that of dating objects, using observable evidence such as tree rings for dendrochronology or carbon-14 for radiometric dating. The observation is caused by the age of the object being dated, rather than the reverse, and the aim is to use the method for estimating dates based on new observations. The problem is whether the model used for relating known ages with observations should aim to minimise the error in the observation, or minimise the error in the date. The two approaches will produce different results, and the difference will increase if the model is then used for extrapolation at some distance from the known results.

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  Julia (programming language)

  Julia is a dynamic general-purpose programming language. As a high-level language, distinctive aspects of Julia's design include a type system with parametric polymorphism, the use of multiple dispatch as a core programming paradigm, just-in-time compilation and a parallel garbage collection implementation. Notably, Julia does not support classes with encapsulated methods but instead relies on the types of all of a function's arguments to determine which method will be called. By default, Julia is run similarly to scripting languages, using its runtime, and allows for interactions, but Julia programs can also be compiled to small binary standalone executables (or to small libraries for e.g. Python), with e.g. the JuliaC.jl compiler. Julia programs can reuse libraries from other languages, and vice versa. Julia has interoperability with C, C++, Fortran, Rust, Python, and R. Additionally, some Julia packages have bindings to be used from Python and R as libraries. Julia is supported by programmer tools like IDEs (see below) and by notebooks like Pluto.jl, Jupyter, and since 2025, Google Colab officially supports Julia natively. Julia is sometimes used in embedded systems (e.g. has been used in a satellite in space on a Raspberry Pi Compute Module 4; 64-bit Pis work best with Julia, and Julia is supported in Raspbian). == History == Work on Julia began in 2009, when Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman set out to create a free language that was both high-level and fast. On 14 February 2012, the team launched a website with a blog post explaining the language's mission. In an interview with InfoWorld in April 2012, Karpinski said about the name of the language, Julia: "There's no good reason, really. It just seemed like a pretty name." Bezanson said he chose the name on the recommendation of a friend, then years later wrote: Maybe julia stands for "Jeff's uncommon lisp is automated"? Julia's syntax is stable, since version 1.0 in 2018, and Julia has a backward compatibility guarantee for 1.x and also a stability promise for the documented (stable) API, while in the years before in the early development prior to 0.7 the syntax (and semantics) was changed in new versions. All of the (registered package) ecosystem uses the new and improved syntax, and in most cases relies on new APIs that have been added regularly, and in some cases minor additional syntax added in a forward compatible way e.g. in Julia 1.7. In the 10 years since the 2012 launch of pre-1.0 Julia, the community has grown. The Julia package ecosystem has over 11.8 million lines of code (including docs and tests). The JuliaCon academic conference for Julia users and developers has been held annually since 2014 with JuliaCon2020 welcoming over 28,900 unique viewers, and then JuliaCon2021 breaking all previous records (with more than 300 JuliaCon2021 presentations available for free on YouTube, up from 162 the year before), and 43,000 unique viewers during the conference. Three of the Julia co-creators are the recipients of the 2019 James H. Wilkinson Prize for Numerical Software (awarded every four years) "for the creation of Julia, an innovative environment for the creation of high-performance tools that enable the analysis and solution of computational science problems." Also, Alan Edelman, professor of applied mathematics at MIT, has been selected to receive the 2019 IEEE Computer Society Sidney Fernbach Award "for outstanding breakthroughs in high-performance computing, linear algebra, and computational science and for contributions to the Julia programming language." Version 0.3 was released in August 2014. Both Julia 0.7 and version 1.0 were released on 8 August 2018. Julia 1.4 added syntax for generic array indexing to handle e.g. 0-based arrays. The memory model was also changed. Julia 1.5 released in August 2020 added record and replay debugging support, for Mozilla's rr tool. The release changed the behavior in the REPL (to soft scope) to the one used in Jupyter, but keeps full compatible with non-REPL code (that retains hard scope). Julia 1.6 was the largest release since 1.0, and it was the long-term support (LTS) version for the longest time. Since Julia 1.7 development is back to time-based releases, and it was released in November 2021 with e.g. a new default random-number generator and Julia 1.7.3 fixed at least one security issue. Julia 1.8 added options for hiding source code when compiling Julia source code to executables. Julia 1.9 has added the ability to precompile packages to native machine code, done automatically; to improve precompilation of packages a new package PrecompileTools.jl was introduced, for use by package developers. Julia 1.10 was released on 25 December 2023 with new features such as parallel garbage collection. Julia 1.11 was released on 7 October 2024, and with it 1.10.5 became the next long-term support (LTS) version (i.e. those became the only two supported versions), since replaced by 1.10.10 released on 27 June, and 1.6 is no longer an LTS version. Julia 1.11 adds e.g. the new public keyword to signal safe public API (Julia users are advised to use such API, not internals, of Julia or packages, and package authors advised to use the keyword, generally indirectly, e.g. prefixed with the @compat macro, from Compat.jl, to also support older Julia versions, at least the LTS version). Julia 1.12 was released on 7 October 2025 (and 1.12.5 on 9 February 2026), and with it a JuliaC.jl package including the juliac compiler that works with it, for making rather small binary executables (much smaller than was possible before; through the use of new so-called trimming feature). Julia 1.10 LTS is an officially still-supported branch, but the 1.11 branch has also been maintained after 1.12 release, with 1.11.8 released and then 1.11.9 released on 8 February 2026. === JuliaCon === Since 2014, the Julia Community has hosted an annual Julia Conference focused on developers and users. The first JuliaCon took place in Chicago and kickstarted the annual occurrence of the conference. Since 2014, the conference has taken place across a number of locations including MIT and the University of Maryland, Baltimore. The event audience has grown from a few dozen people to over 28,900 unique attendees during JuliaCon 2020, which took place virtually. JuliaCon 2021 also took place virtually with keynote addresses from professors William Kahan, the primary architect of the IEEE 754 floating-point standard (which virtually all CPUs and languages, including Julia, use), Jan Vitek, Xiaoye Sherry Li, and Soumith Chintala, a co-creator of PyTorch. JuliaCon grew to 43,000 unique attendees and more than 300 presentations (still freely accessible, plus for older years). JuliaCon 2022 will also be virtual held between July 27 and July 29, 2022, for the first time in several languages, not just in English. === Sponsors === The Julia language became a NumFOCUS fiscally sponsored project in 2014 in an effort to ensure the project's long-term sustainability. Jeremy Kepner at MIT Lincoln Laboratory was the founding sponsor of the Julia project in its early days. In addition, funds from the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation, Intel, and agencies such as NSF, DARPA, NIH, NASA, and FAA have been essential to the development of Julia. Mozilla, the maker of Firefox web browser, with its research grants for H1 2019, sponsored "a member of the official Julia team" for the project "Bringing Julia to the Browser", meaning to Firefox and other web browsers. The Julia language is also supported by individual donors on GitHub. === The Julia company === JuliaHub, Inc. was founded in 2015 as Julia Computing, Inc. by Viral B. Shah, Deepak Vinchhi, Alan Edelman, Jeff Bezanson, Stefan Karpinski and Keno Fischer. In June 2017, Julia Computing raised US$4.6 million in seed funding from General Catalyst and Founder Collective, the same month was "granted $910,000 by the Alfred P. Sloan Foundation to support open-source Julia development, including $160,000 to promote diversity in the Julia community", and in December 2019 the company got $1.1 million funding from the US government to "develop a neural component machine learning tool to reduce the total energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings". In July 2021, Julia Computing announced they raised a $24 million Series A round led by Dorilton Ventures, which also owns Formula One team Williams Racing, that partnered with Julia Computing. Williams' Commercial Director said: "Investing in companies building best-in-class cloud technology is a strategic focus for Dorilton and Julia's versatile platform, with revolutionary capabilities in simulation and modelling, is hugely relevant to our business. We look forward to embedding Julia Computing in the world's most technologically advanced sport". In June 2023, JuliaHub received (again, now

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  Multisample anti-aliasing

  Multisample anti-aliasing (MSAA) is a type of spatial anti-aliasing, a technique used in computer graphics to remove jaggies. It is an optimization of supersampling, where only the necessary parts are sampled more. Jaggies are only noticed in a small area, so the area is quickly found, and only that is anti-aliased. == Definition == The term generally refers to a special case of supersampling. Initial implementations of full-scene anti-aliasing (FSAA) worked conceptually by simply rendering a scene at a higher resolution, and then downsampling to a lower-resolution output. Most modern GPUs are capable of this form of anti-aliasing, but it greatly taxes resources such as texture, bandwidth, and fillrate. (If a program is highly TCL-bound or CPU-bound, supersampling can be used without much performance hit.) According to the OpenGL GL_ARB_multisample specification, "multisampling" refers to a specific optimization of supersampling. The specification dictates that the renderer evaluate the fragment program once per pixel, and only "truly" supersample the depth and stencil values. (This is not the same as supersampling but, by the OpenGL 1.5 specification, the definition had been updated to include fully supersampling implementations as well.) In graphics literature in general, "multisampling" refers to any special case of supersampling where some components of the final image are not fully supersampled. The lists below refer specifically to the ARB_multisample definition. == Description == In supersample anti-aliasing, multiple locations are sampled within every pixel, and each of those samples is fully rendered and combined with the others to produce the pixel that is ultimately displayed. This is computationally expensive, because the entire rendering process must be repeated for each sample location. It is also inefficient, as aliasing is typically only noticed in some parts of the image, such as the edges, whereas supersampling is performed for every single pixel. In multisample anti-aliasing, if any of the multi sample locations in a pixel is covered by the triangle being rendered, a shading computation must be performed for that triangle. However this calculation only needs to be performed once for the whole pixel regardless of how many sample positions are covered; the result of the shading calculation is simply applied to all of the relevant multi sample locations. In the case where only one triangle covers every multi sample location within the pixel, only one shading computation is performed, and these pixels are little more expensive than (and the result is no different from) the non-anti-aliased image. This is true of the middle of triangles, where aliasing is not an issue. (Edge detection can reduce this further by explicitly limiting the MSAA calculation to pixels whose samples involve multiple triangles, or triangles at multiple depths.) In the extreme case where each of the multi sample locations is covered by a different triangle, a different shading computation will be performed for each location and the results then combined to give the final pixel, and the result and computational expense are the same as in the equivalent supersampled image. The shading calculation is not the only operation that must be performed on a given pixel; multisampling implementations may variously sample other operations such as visibility at different sampling levels. == Advantages == The pixel shader usually only needs to be evaluated once per pixel for every triangle covering at least one sample point. The edges of polygons (the most obvious source of aliasing in 3D graphics) are anti-aliased. Since multiple subpixels per pixel are sampled, polygonal details smaller than one pixel that might have been missed without MSAA can be captured and made a part of the final rendered image if enough samples are taken. == Disadvantages == === Alpha testing === Alpha testing is a technique common to older video games used to render translucent objects by rejecting pixels from being written to the framebuffer. If the alpha value of a translucent fragment (pixel) is below a specified threshold, it will be discarded. Because this is performed on a pixel by pixel basis, the image does not receive the benefits of multi-sampling (all of the multisamples in a pixel are discarded based on the alpha test) for these pixels. The resulting image may contain aliasing along the edges of transparent objects or edges within textures, although the image quality will be no worse than it would be without any anti-aliasing. Translucent objects that are modelled using alpha-test textures will also be aliased due to alpha testing. This effect can be minimized by rendering objects with transparent textures multiple times, although this would result in a high performance reduction for scenes containing many transparent objects. === Aliasing === Because multi-sampling calculates interior polygon fragments only once per pixel, aliasing and other artifacts will still be visible inside rendered polygons where fragment shader output contains high frequency components. === Performance === While less performance-intensive than SSAA (supersampling), it is possible in certain scenarios (scenes heavy in complex fragments) for MSAA to be multiple times more intensive for a given frame than post processing anti-aliasing techniques such as FXAA, SMAA and MLAA. Early techniques in this category tend towards a lower performance impact, but suffer from accuracy problems. More recent post-processing based anti-aliasing techniques such as temporal anti-aliasing (TAA), which reduces aliasing by combining data from previously rendered frames, have seen the reversal of this trend, as post-processing AA becomes both more versatile and more expensive than MSAA, which cannot antialias an entire frame alone. == Sampling methods == === Point sampling === In a point-sampled mask, the coverage bit for each multisample is only set if the multisample is located inside the rendered primitive. Samples are never taken from outside a rendered primitive, so images produced using point-sampling will be geometrically correct, but filtering quality may be low because the proportion of bits set in the pixel's coverage mask may not be equal to the proportion of the pixel that is actually covered by the fragment in question. === Area sampling === Filtering quality can be improved by using area sampled masks. In this method, the number of bits set in a coverage mask for a pixel should be proportionate to the actual area coverage of the fragment. This will result in some coverage bits being set for multisamples that are not actually located within the rendered primitive, and can cause aliasing and other artifacts. == Sample patterns == === Regular grid === A regular grid sample pattern, where multisample locations form an evenly spaced grid throughout the pixel, is easy to implement and simplifies attribute evaluation (i.e. setting subpixel masks, sampling color and depth). This method is computationally expensive due to the large number of samples. Edge optimization is poor for screen-aligned edges, but image quality is good when the number of multisamples is large. === Sparse regular grid === A sparse regular grid sample pattern is a subset of samples that are chosen from the regular grid sample pattern. As with the regular grid, attribute evaluation is simplified due to regular spacing. The method is less computationally expensive due to having a fewer samples. Edge optimization is good for screen aligned edges, and image quality is good for a moderate number of multisamples. === Stochastic sample patterns === A stochastic sample pattern is a random distribution of multisamples throughout the pixel. The irregular spacing of samples makes attribute evaluation complicated. The method is cost efficient due to low sample count (compared to regular grid patterns). Edge optimization with this method, although sub-optimal for screen aligned edges. Image quality is excellent for a moderate number of samples. == Quality == Compared to supersampling, multisample anti-aliasing can provide similar quality at higher performance, or better quality for the same performance. Further improved results can be achieved by using rotated grid subpixel masks. The additional bandwidth required by multi-sampling is reasonably low if Z and colour compression are available. Most modern GPUs support 2×, 4×, and 8× MSAA samples. Higher values result in better quality, but are slower.

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  Non-negative matrix factorization

  Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for o

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  IBM Watsonx

  Watsonx is a platform by IBM for building and managing artificial intelligence (AI) applications for business use. Released on May 9, 2023, the platform provides software tools and infrastructure for companies to work with both IBM's own AI models and models from third-party sources. The platform consists of three main components: watsonx.ai, a studio for training, validating, and deploying AI models; watsonx.data, a system for storing and managing data used by the models; and watsonx.governance, a toolkit to ensure AI applications are compliant with company policies and regulations. A key feature of the platform is that it can be trained on a company's private data to perform specialized tasks, a process known as fine-tuning. IBM states that this client-specific data is not used to train its own models. == History == Watsonx was introduced on May 9, 2023, at the annual IBM Think conference, as a platform that includes multiple services. Just like Watson AI computer with the similar name, Watsonx was named after Thomas J. Watson, IBM's founder and first CEO. On February 13, 2024, Anaconda partnered with IBM to embed its open-source Python packages into Watsonx. Watsonx is used at ESPN's Fantasy Football App for managing players' performance, and by Italian telecommunications company Wind Tre. It was employed to generate editorial content around nominees during the 66th Annual Grammy Awards. In 2025, Wimbledon integrated IBM watsonx generative AI into its app and website. Integrated with IBM Safer Payments, IBM watsonx has been used in banking sector fraud detection and anti-money laundering (AML) systems. == Services == === watsonx.ai === Watsonx.ai is a platform that allows AI developers to leverage a wide range of LLMs under IBM's own Granite series and others such as Facebook's LLaMA-2, free and open-source model Mistral, and many others present in the Hugging Face community. These models come pre-trained and optimized for various natural language processing (NLP) applications.The platform also allows fine-tuning with its Tuning Studio. === watsonx.data === Watsonx.data is a platform designed to assist clients in addressing issues related to data volume, complexity, cost, and governance.. The platform facilitates seamless data access, whether stored in the cloud or on-premises, through a single entry point. === watsonx.governance === Watsonx.governance is a platform that utilizes IBM's AI capabilities to implement AI lifecycle governance. This helps them manage risks and maintain compliance with evolving AI and industry regulations, while reducing AI bias through automated oversight.

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  Linear discriminant analysis

  Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis of variance (ANOVA) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable, whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is, as they also explain a categorical variable by the values of continuous independent variables. These other methods are preferable in applications where it is not reasonable to assume that the independent variables have a normal distribution, which is a fundamental assumption of the LDA method. LDA is also closely related to principal component analysis (PCA) and factor analysis in that they both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA, in contrast, does not take into account any difference in class, and factor analysis builds the feature combinations based on similarities rather than differences. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made. LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is discriminant correspondence analysis. Discriminant analysis is used when groups are known a priori (unlike in cluster analysis). Each case must have a score on one or more quantitative predictor measures, and a score on a group measure. In simple terms, discriminant function analysis is classification - the act of distributing things into groups, classes or categories of the same type. == History == The original dichotomous discriminant analysis was developed by Sir Ronald Fisher in 1936. It is different from an ANOVA or MANOVA, which is used to predict one (ANOVA) or multiple (MANOVA) continuous dependent variables by one or more independent categorical variables. Discriminant function analysis is useful in determining whether a set of variables is effective in predicting category membership. == LDA for two classes == Consider a set of observations x → {\displaystyle {\vec {x}}} (also called features, attributes, variables or measurements) for each sample of an object or event with known class y {\displaystyle y} . This set of samples is called the training set in a supervised learning context. The classification problem is then to find a good predictor for the class y {\displaystyle y} of any sample of the same distribution (not necessarily from the training set) given only an observation x → {\displaystyle {\vec {x}}} . LDA approaches the problem by assuming that the conditional probability density functions p ( x → | y = 0 ) {\displaystyle p({\vec {x}}|y=0)} and p ( x → | y = 1 ) {\displaystyle p({\vec {x}}|y=1)} are both the normal distribution with mean and covariance parameters ( μ → 0 , Σ 0 ) {\displaystyle \left({\vec {\mu }}_{0},\Sigma _{0}\right)} and ( μ → 1 , Σ 1 ) {\displaystyle \left({\vec {\mu }}_{1},\Sigma _{1}\right)} , respectively. Under this assumption, the Bayes-optimal solution is to predict points as being from the second class if the log of the likelihood ratios is bigger than some threshold T, so that: 1 2 ( x → − μ → 0 ) T Σ 0 − 1 ( x → − μ → 0 ) + 1 2 ln ⁡ | Σ 0 | − 1 2 ( x → − μ → 1 ) T Σ 1 − 1 ( x → − μ → 1 ) − 1 2 ln ⁡ | Σ 1 | > T {\displaystyle {\frac {1}{2}}({\vec {x}}-{\vec {\mu }}_{0})^{\mathrm {T} }\Sigma _{0}^{-1}({\vec {x}}-{\vec {\mu }}_{0})+{\frac {1}{2}}\ln |\Sigma _{0}|-{\frac {1}{2}}({\vec {x}}-{\vec {\mu }}_{1})^{\mathrm {T} }\Sigma _{1}^{-1}({\vec {x}}-{\vec {\mu }}_{1})-{\frac {1}{2}}\ln |\Sigma _{1}|\ >\ T} Without any further assumptions, the resulting classifier is referred to as quadratic discriminant analysis (QDA). LDA instead makes the additional simplifying homoscedasticity assumption (i.e. that the class covariances are identical, so Σ 0 = Σ 1 = Σ {\displaystyle \Sigma _{0}=\Sigma _{1}=\Sigma } ) and that the covariances have full rank. In this case, several terms cancel: x → T Σ 0 − 1 x → = x → T Σ 1 − 1 x → {\displaystyle {\vec {x}}^{\mathrm {T} }\Sigma _{0}^{-1}{\vec {x}}={\vec {x}}^{\mathrm {T} }\Sigma _{1}^{-1}{\vec {x}}} x → T Σ i − 1 μ → i = μ → i T Σ i − 1 x → {\displaystyle {\vec {x}}^{\mathrm {T} }{\Sigma _{i}}^{-1}{\vec {\mu }}_{i}={{\vec {\mu }}_{i}}^{\mathrm {T} }{\Sigma _{i}}^{-1}{\vec {x}}} because both sides are scalar and transpose to each other ( Σ i {\displaystyle \Sigma _{i}} is Hermitian) and the above decision criterion becomes a threshold on the dot product w → T x → > c {\displaystyle {\vec {w}}^{\mathrm {T} }{\vec {x}}>c} for some threshold constant c, where w → = Σ − 1 ( μ → 1 − μ → 0 ) {\displaystyle {\vec {w}}=\Sigma ^{-1}({\vec {\mu }}_{1}-{\vec {\mu }}_{0})} c = 1 2 w → T ( μ → 1 + μ → 0 ) {\displaystyle c={\frac {1}{2}}\,{\vec {w}}^{\mathrm {T} }({\vec {\mu }}_{1}+{\vec {\mu }}_{0})} This means that the criterion of an input x → {\displaystyle {\vec {x}}} being in a class y {\displaystyle y} is purely a function of this linear combination of the known observations. It is often useful to see this conclusion in geometrical terms: the criterion of an input x → {\displaystyle {\vec {x}}} being in a class y {\displaystyle y} is purely a function of projection of multidimensional-space point x → {\displaystyle {\vec {x}}} onto vector w → {\displaystyle {\vec {w}}} (thus, we only consider its direction). In other words, the observation belongs to y {\displaystyle y} if corresponding x → {\displaystyle {\vec {x}}} is located on a certain side of a hyperplane perpendicular to w → {\displaystyle {\vec {w}}} . The location of the plane is defined by the threshold c {\displaystyle c} . == Assumptions == The assumptions of discriminant analysis are the same as those for MANOVA. The analysis is quite sensitive to outliers and the size of the smallest group must be larger than the number of predictor variables. Multivariate normality: Independent variables are normal for each level of the grouping variable. Homogeneity of variance/covariance (homoscedasticity): Variances among group variables are the same across levels of predictors. Can be tested with Box's M statistic. It has been suggested, however, that linear discriminant analysis be used when covariances are equal, and that quadratic discriminant analysis may be used when covariances are not equal. Independence: Participants are assumed to be randomly sampled, and a participant's score on one variable is assumed to be independent of scores on that variable for all other participants. It has been suggested that discriminant analysis is relatively robust to slight violations of these assumptions, and it has also been shown that discriminant analysis may still be reliable when using dichotomous variables (where multivariate normality is often violated). == Discriminant functions == Discriminant analysis works by creating one or more linear combinations of predictors, creating a new latent variable for each function. These functions are called discriminant functions. The number of functions possible is either N g − 1 {\displaystyle N_{g}-1} where N g {\displaystyle N_{g}} = number of groups, or p {\displaystyle p} (the number of predictors), whichever is smaller. The first function created maximizes the differences between groups on that function. The second function maximizes differences on that function, but also must not be correlated with the previous function. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions. Given group j {\displaystyle j} , with R j {\displaystyle \mathbb {R} _{j}} sets of sample space, there is a discriminant rule such that if x ∈ R j {\displaystyle x\in \mathbb {R} _{j}} , then x ∈ j {\displaystyle x\in j} . Discriminant analysis then, finds “good” regions of R j {\displaystyle \mathbb {R} _{j}} to minimize classification error, therefore leading to a high percent correct classified in the classification table. Each function is given a discriminant score to determine how well it predicts group placement. Structure Corr

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  Toad (software)

  Toad is a database management toolset from Quest Software for managing relational and non-relational databases using SQL aimed at database developers, database administrators, and data analysts. The Toad toolset runs against Oracle, SQL Server, IBM DB2 (LUW & z/OS), SAP and MySQL. A Toad product for data preparation supports many data platforms. == History == A practicing Oracle DBA, Jim McDaniel, designed Toad for his own use in the mid-1990s. He called it Tool for Oracle Application Developers, shortened to "TOAD". McDaniel initially distributed the tool as shareware and later online as freeware. Quest Software acquired TOAD in October 1998. Quest Software itself was acquired by Dell in 2012 to form Dell Software. In June 2016, Dell announced the sale of their software division, including the Quest business, to Francisco Partners and Elliott Management Corporation. On October 31, 2016, the sale was finalized. On November 1, 2016, the sale of Dell Software to Francisco Partners and Elliott Management was completed, and the company re-launched as Quest Software. == Features == Connection Manager - Allow users to connect natively to the vendor’s database whether on-premise or DBaaS. Browser - Allow users to browse all the different database/schema objects and their properties effective management. Editor - A way to create and maintain scripts and database code with debugging and integration with source control. Unit Testing (Oracle) - Ensures code is functionally tested before it is released into production. Static code review (Oracle) - Ensures code meets required quality level using a rules-based system. SQL Optimization - Provides developers with a way to tune and optimize SQL statements and database code without relying on a DBA. Advanced optimization enables DBAs to tune SQL effectively in production. Scalability testing and database workload replay - Ensures that database code and SQL will scale properly before it gets released into production. == Books == Toad Pocket Reference for Oracle plsql 1st Edition by Jim McDaniel and Patrick McGrath, O'Reilly, 2002 (ISBN 0596003374, ISBN 978-0-596-00337-1) Toad Pocket Reference for Oracle 2nd Edition by Jeff Smith, Bert Scalzo, and Patrick McGrath, O'Reilly, 2005 (ISBN 0596009712, ISBN 978-0-596-00971-7) TOAD Handbook by Bert Scalzo and Dan Hotka, Sams, 2003 (ISBN 0672324865, ISBN 978-0-672-32486-4) TOAD Handbook 2nd Edition by Bert Scalzo and Dan Hotka, Addison-Wesley Professional, 2009 (ISBN 0321649109, ISBN 978-0-321-64910-2). TOAD Handbook 2nd Edition by Bert Scalzo and Dan Hotka, Addison-Wesley Professional, 2009 (ISBN 0321649109, ISBN 978-0-321-64910-2).

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  Synaptic weight

  In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.

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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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  Occam learning

  In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set. Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability. == Introduction == Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al. that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power. == Definition of Occam learning == The succinctness of a concept c {\displaystyle c} in concept class C {\displaystyle {\mathcal {C}}} can be expressed by the length s i z e ( c ) {\displaystyle size(c)} of the shortest bit string that can represent c {\displaystyle c} in C {\displaystyle {\mathcal {C}}} . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} be concept classes containing target concepts and hypotheses respectively. Then, for constants α ≥ 0 {\displaystyle \alpha \geq 0} and 0 ≤ β < 1 {\displaystyle 0\leq \beta <1} , a learning algorithm L {\displaystyle L} is an ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} iff, given a set S = { x 1 , … , x m } {\displaystyle S=\{x_{1},\dots ,x_{m}\}} of m {\displaystyle m} samples labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} , L {\displaystyle L} outputs a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that h {\displaystyle h} is consistent with c {\displaystyle c} on S {\displaystyle S} (that is, h ( x ) = c ( x ) , ∀ x ∈ S {\displaystyle h(x)=c(x),\forall x\in S} ), and s i z e ( h ) ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle size(h)\leq (n\cdot size(c))^{\alpha }m^{\beta }} where n {\displaystyle n} is the maximum length of any sample x ∈ S {\displaystyle x\in S} . An Occam algorithm is called efficient if it runs in time polynomial in n {\displaystyle n} , m {\displaystyle m} , and s i z e ( c ) . {\displaystyle size(c).} We say a concept class C {\displaystyle {\mathcal {C}}} is Occam learnable with respect to a hypothesis class H {\displaystyle {\mathcal {H}}} if there exists an efficient Occam algorithm for C {\displaystyle {\mathcal {C}}} using H . {\displaystyle {\mathcal {H}}.} == The relation between Occam and PAC learning == Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows: === Theorem (Occam learning implies PAC learning) === Let L {\displaystyle L} be an efficient ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} . Then there exists a constant a > 0 {\displaystyle a>0} such that for any 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , for any distribution D {\displaystyle {\mathcal {D}}} , given m ≥ a ( 1 ϵ log ⁡ 1 δ + ( ( n ⋅ s i z e ( c ) ) α ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha }}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} samples drawn from D {\displaystyle {\mathcal {D}}} and labelled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, the algorithm L {\displaystyle L} will output a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } .Here, e r r o r ( h ) {\displaystyle error(h)} is with respect to the concept c {\displaystyle c} and distribution D {\displaystyle {\mathcal {D}}} . This implies that the algorithm L {\displaystyle L} is also a PAC learner for the concept class C {\displaystyle {\mathcal {C}}} using hypothesis class H {\displaystyle {\mathcal {H}}} . A slightly more general formulation is as follows: === Theorem (Occam learning implies PAC learning, cardinality version) === Let 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} . Let L {\displaystyle L} be an algorithm such that, given m {\displaystyle m} samples drawn from a fixed but unknown distribution D {\displaystyle {\mathcal {D}}} and labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, outputs a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} that is consistent with the labeled samples. Then, there exists a constant b {\displaystyle b} such that if log ⁡ | H n , m | ≤ b ϵ m − log ⁡ 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq b\epsilon m-\log {\frac {1}{\delta }}} , then L {\displaystyle L} is guaranteed to output a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } . While the above theorems show that Occam learning is sufficient for PAC learning, it doesn't say anything about necessity. Board and Pitt show that, for a wide variety of concept classes, Occam learning is in fact necessary for PAC learning. They proved that for any concept class that is polynomially closed under exception lists, PAC learnability implies the existence of an Occam algorithm for that concept class. Concept classes that are polynomially closed under exception lists include Boolean formulas, circuits, deterministic finite automata, decision-lists, decision-trees, and other geometrically defined concept classes. A concept class C {\displaystyle {\mathcal {C}}} is polynomially closed under exception lists if there exists a polynomial-time algorithm A {\displaystyle A} such that, when given the representation of a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} and a finite list E {\displaystyle E} of exceptions, outputs a representation of a concept c ′ ∈ C {\displaystyle c'\in {\mathcal {C}}} such that the concepts c {\displaystyle c} and c ′ {\displaystyle c'} agree except on the set E {\displaystyle E} . == Proof that Occam learning implies PAC learning == We first prove the Cardinality version. Call a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} bad if e r r o r ( h ) ≥ ϵ {\displaystyle error(h)\geq \epsilon } , where again e r r o r ( h ) {\displaystyle error(h)} is with respect to the true concept c {\displaystyle c} and the underlying distribution D {\displaystyle {\mathcal {D}}} . The probability that a set of samples S {\displaystyle S} is consistent with h {\displaystyle h} is at most ( 1 − ϵ ) m {\displaystyle (1-\epsilon )^{m}} , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in H n , m {\displaystyle {\mathcal {H}}_{n,m}} is at most | H n , m | ( 1 − ϵ ) m {\displaystyle |{\mathcal {H}}_{n,m}|(1-\epsilon )^{m}} , which is less than δ {\displaystyle \delta } if log ⁡ | H n , m | ≤ O ( ϵ m ) − log ⁡ 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq O(\epsilon m)-\log {\frac {1}{\delta }}} . This concludes the proof of the second theorem above. Using the second theorem, we can prove the first theorem. Since we have a ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm, this means that any hypothesis output by L {\displaystyle L} can be represented by at most ( n ⋅ s i z e ( c ) ) α m β {\displaystyle (n\cdot size(c))^{\alpha }m^{\beta }} bits, and thus log ⁡ | H n , m | ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq (n\cdot size(c))^{\alpha }m^{\beta }} . This is less than O ( ϵ m ) − log ⁡ 1 δ {\displaystyle O(\epsilon m)-\log {\frac {1}{\delta }}} if we set m ≥ a ( 1 ϵ log ⁡ 1 δ + ( ( n ⋅ s i z e ( c ) ) α ) ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha })}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} for some constant a > 0 {\displaystyle a>0} . Thus, by the Cardinality version Theorem, L {\displaystyle L} will output a consistent hypothesis h {\displaystyle h} with probability at least 1 − δ {\displaystyle 1-\delta } . This concludes the proof of the first theorem above. == Improving sample complexity for common problems == Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, co

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  Showcase Workshop

  Showcase Workshop, also referred to as Showcase, is a SaaS company that develops a presentation-building application for business use. Users upload files and images to a web platform which generates presentations viewable on a suite of mobile apps. Showcase was founded in 2011. The company’s headquarters are in Wellington, New Zealand. == History == Showcase Workshop was originally developed in response to dynamically changing content being presented on iPads at the 2012 Olympics. After market-testing a beta version of the core application, Showcase Workshop launched commercially in 2012. In 2014 Showcase partnered with Vodafone Global Enterprise. == Product == Users upload pre-existing PDFs, videos, images and Microsoft Office documents to a secure server, building presentations or ‘showcases’ which can then be downloaded via the mobile apps. The presentations are used for mobile sales enablement, training, or operational/health and safety purposes. == Reception == Reviewers have praised the ease of use of Showcase, calling it a “better alternative to developing a native app” and “intuitive”. Criticisms include the lack of differing templates and a lack of complex customisation controls. Showcase was nominated for a Tabby Award in 2014 and won a Tabby Award in 2015 for its Windows app.

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  Quadratic classifier

  In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}} Read more →

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  Probably approximately correct learning

  In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds). == Definitions and terminology == In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n {\displaystyle n} bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X {\displaystyle X} be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} . In the interval problem the instance space, X {\displaystyle X} , is the set of all bounded intervals in R {\displaystyle \mathbb {R} } , where R {\displaystyle \mathbb {R} } denotes the set of all real numbers. A concept is a subset c ⊂ X {\displaystyle c\subset X} . One concept is the set of all patterns of bits in X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive points. A concept class C {\displaystyle C} is a collection of concepts over X {\displaystyle X} . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} be a procedure that draws an example, x {\displaystyle x} , using a probability distribution D {\displaystyle D} and gives the correct label c ( x ) {\displaystyle c(x)} , that is 1 if x ∈ c {\displaystyle x\in c} and 0 otherwise. Now, given 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , assume there is an algorithm A {\displaystyle A} and a polynomial p {\displaystyle p} in 1 / ϵ , 1 / δ {\displaystyle 1/\epsilon ,1/\delta } (and other relevant parameters of the class C {\displaystyle C} ) such that, given a sample of size p {\displaystyle p} drawn according to EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} , then, with probability of at least 1 − δ {\displaystyle 1-\delta } , A {\displaystyle A} outputs a hypothesis h ∈ C {\displaystyle h\in C} that has an average error less than or equal to ϵ {\displaystyle \epsilon } on X {\displaystyle X} with the same distribution D {\displaystyle D} . Further if the above statement for algorithm A {\displaystyle A} is true for every concept c ∈ C {\displaystyle c\in C} and for every distribution D {\displaystyle D} over X {\displaystyle X} , and for all 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} then C {\displaystyle C} is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A {\displaystyle A} is a PAC learning algorithm for C {\displaystyle C} . == Equivalence == Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth

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