Best AI Image Generators

Best AI Image Generators — hands-on reviews, top picks, pricing, pros and cons and a practical how-to guide on Aizhi.

Exercism

Exercism is an online, open-source, free coding platform that offers code practice and mentorship on 77 different programming languages. == History == Software developer Katrina Owen created Exercism while she was teaching programming at Jumpstart Labs. The platform was developed as an internal tool to solve the problem of her own students not receiving feedback on the coding problems they were practicing. Katrina put the site publicly online and found that people were sharing it with their friends, practicing together and giving each other feedback. Within 12 months, the site had organically grown to see over 6,000 users had submitted code or feedback, and hundreds of volunteers contribute to the languages or tooling on the platform. In 2016, Jeremy Walker joined as co-founder and CEO. In July 2018, the site was relaunched with a new design and centered around a formal mentoring mode, at which point Katrina stepped back from day-to-day involvement. == Product == In the past, the website differed from other coding platforms by requiring students to download exercises through a command line client, solve the code on their own computers then submit the solution for feedback, at which point they can also view other's solutions to the same problem. Since its second relaunch in 2021, solutions can be edited and submitted through a web editor, though the command line client remains available. Exercism has tracks for 74 programming languages. Among the notable languages taught: ABAP, C, C#, C++, CoffeeScript, Delphi, Elm, Erlang, F#, Gleam, Go, Java, JavaScript, Julia, Kotlin, Objective-C, PHP, Python, Raku, Red, Ruby, Rust, Scala, Swift, and V (Vlang). In 2023, the site launched a "12 in 23" challenge for users to learn the basics of 12 different languages - one per month in 2023. == Open source == The Exercism codebase is open source. In April 2016, it consisted of 50 repositories including website code, API code, command-line code and, most of all, over 40 stand-alone repositories for different language tracks. As of February 2024 Exercism has 14,344 contributors, maintains 366 repositories, and 19,603 mentors.
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Generalized blockmodeling

In generalized blockmodeling, the blockmodeling is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Generalized blockmodeling was introduced in 1994 by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj. == Definition == Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the blockmodel is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using criterion function. Using local optimization procedure, firstly the initial clustering (with specified number of clusters is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a vertex is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as blockmodels for the future exploration of the network. Different types of generalized blockmodeling are: generalized binary blockmodeling, generalized valued blockmodeling and generalized homogeneity blockmodeling. == Benefits == According to Patrick Doreian, the benefits of generalized blockmodeling, are as follows: usage of explicit criterion function, compatible with a given type of equivalence, results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; with generalized blockmodeling it's possible to specify new types of blockmodels; this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; researcher can speficy which (pair of) vertices must be (not) clustered together; this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block. == Problems == According to Doreian, the problems of generalized blockmodeling, are as follows: unknown sensitivity to particular data features, examination of boundary problems, computationally burdensome, which results in a constraint regarding practical network size (generalized blockmodeling is thus primarily used to analyse smaller networks (below 100 units)), identifying structure from incomplete network information, most of generalized blockmodeling is based on binary networks, but there is also development in the field of valued networks, criterion function is minimized for a specified blockmodel, with results in issues of evaluating statistically, based on the structural data alone, problems regarding three dimensional network data, problems regarding the evolution of fundamental network structure. == Book == The book with the same title, Generalized blockmodeling, written by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj, was in 2007 awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association.
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KXEN Inc.

KXEN was an American software company which existed from 1998 to 2013 when it was acquired by SAP AG. == History == KXEN was founded in June 1998 by Roger Haddad and Michel Bera. It was based in San Francisco, California with offices in Paris and London. On September 10, 2013, SAP AG announced plans to acquire KXEN. On October 1, 2013, a letter to KXEN customers announced the acquisition closed. KXEN primarily marketed predictive analytics software. == Predictive analytics == InfiniteInsight is a predictive modeling suite developed by KXEN that assists analytic professionals, and business executives to extract information from data. Among other functions, InfiniteInsight is used for variable importance, classification, regression, segmentation, time series, product recommendation, as described and expressed by the Java Data Mining interface, and for social network analysis. InfiniteInsight allows prediction of a behavior or a value, the forecast of a time series or the understanding of a group of individuals with similar behavior. Advanced functions include behavioral modeling, exporting the model code into different target environments or building predictive models on top of SAS or SPSS data files. Competitors are SAS Enterprise Miner, IBM SPSS Modeler, and Statistica. Open source predictive tools like the R package or Weka are also competitors, since they provide similar features free of charge.
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CN2 algorithm

The CN2 induction algorithm is a learning algorithm for rule induction. It is designed to work even when the training data is imperfect. It is based on ideas from the AQ algorithm and the ID3 algorithm. As a consequence it creates a rule set like that created by AQ but is able to handle noisy data like ID3. == Description of algorithm == The algorithm must be given a set of examples, TrainingSet, which have already been classified in order to generate a list of classification rules. A set of conditions, SimpleConditionSet, which can be applied, alone or in combination, to any set of examples is predefined to be used for the classification. routine CN2(TrainingSet) let the ClassificationRuleList be empty repeat let the BestConditionExpression be Find_BestConditionExpression(TrainingSet) if the BestConditionExpression is not nil then let the TrainingSubset be the examples covered by the BestConditionExpression remove from the TrainingSet the examples in the TrainingSubset let the MostCommonClass be the most common class of examples in the TrainingSubset append to the ClassificationRuleList the rule 'if ' the BestConditionExpression ' then the class is ' the MostCommonClass until the TrainingSet is empty or the BestConditionExpression is nil return the ClassificationRuleList routine Find_BestConditionExpression(TrainingSet) let the ConditionalExpressionSet be empty let the BestConditionExpression be nil repeat let the TrialConditionalExpressionSet be the set of conditional expressions, {x and y where x belongs to the ConditionalExpressionSet and y belongs to the SimpleConditionSet}. remove all formulae in the TrialConditionalExpressionSet that are either in the ConditionalExpressionSet (i.e., the unspecialized ones) or null (e.g., big = y and big = n) for every expression, F, in the TrialConditionalExpressionSet if F is statistically significant and F is better than the BestConditionExpression by user-defined criteria when tested on the TrainingSet then replace the current value of the BestConditionExpression by F while the number of expressions in the TrialConditionalExpressionSet > user-defined maximum remove the worst expression from the TrialConditionalExpressionSet let the ConditionalExpressionSet be the TrialConditionalExpressionSet until the ConditionalExpressionSet is empty return the BestConditionExpression
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Computer Graphics: Principles and Practice

Computer Graphics: Principles and Practice is a textbook written by James D. Foley, Andries van Dam, Steven K. Feiner, John Hughes, Morgan McGuire, David F. Sklar, and Kurt Akeley and published by Addison–Wesley. First published in 1982 as Fundamentals of Interactive Computer Graphics, it is widely considered a classic standard reference book on the topic of computer graphics. It is sometimes known as the bible of computer graphics (due to its size). == Editions == === First Edition === The first edition, published in 1982 and titled Fundamentals of Interactive Computer Graphics, discussed the SGP library, which was based on ACM's SIGGRAPH CORE 1979 graphics standard, and focused on 2D vector graphics. === Second Edition === The second edition, published 1990, was completely rewritten and covered 2D and 3D raster and vector graphics, user interfaces, geometric modeling, anti-aliasing, advanced rendering algorithms and an introduction to animation. The SGP library was replaced by SRGP (Simple Raster Graphics Package), a library for 2D raster primitives and interaction handling, and SPHIGS (Simple PHIGS), a library for 3D primitives, which were specifically written for the book. === Second Edition in C === In the second edition in C, published in 1995, all examples were converted from Pascal to C. New implementations for the SRGP and SPHIGS graphics packages in C were also provided. === Third Edition === A third edition covering modern GPU architecture was released in July 2013. Examples in the third edition are written in C++, C#, WPF, GLSL, OpenGL, G3D, or pseudocode. == Awards == The book has won a Front Line Award (Hall of Fame) in 1998.
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Santa Fe Trail problem

The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.
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Correspondence analysis

Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ⁡ ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ⁡ ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for
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Evolutionary algorithm

Evolutionary algorithms (EA) reproduce essential elements of biological evolution in a computer algorithm in order to solve "difficult" problems, at least approximately, for which no exact or satisfactory solution methods are known. They are metaheuristics and population-based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, recombination and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolution (microevolutionary processes) and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity. == Generic definition == The following is an example of a generic evolutionary algorithm: Randomly generate the initial population of individuals, the first generation. Evaluate the fitness of each individual in the population. Check, if the goal is reached and the algorithm can be terminated. Select individuals as parents, preferably of higher fitness. Produce offspring with optional crossover (mimicking reproduction). Apply mutation operations on the offspring. Select individuals preferably of lower fitness for replacement with new individuals (mimicking natural selection). Return to 2 == Types == Similar techniques differ in genetic representation and other implementation details, and the nature of the particular applied problem. Genetic algorithm – This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved), by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in optimization problems. Genetic programming – Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming: Cartesian genetic programming Gene expression programming Grammatical evolution Linear genetic programming Multi expression programming Evolutionary programming – Similar to evolution strategy, but with a deterministic selection of all parents. Evolution strategy (ES) – Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks. CMA-ES Natural evolution strategy Differential evolution – Based on vector differences and is therefore primarily suited for numerical optimization problems. Coevolutionary algorithm – Similar to genetic algorithms and evolution strategies, but the created solutions are compared on the basis of their outcomes from interactions with other solutions. Solutions can either compete or cooperate during the search process. Coevolutionary algorithms are often used in scenarios where the fitness landscape is dynamic, complex, or involves competitive interactions. Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect. Learning classifier system – Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression types. Fitness is typically determined with either a strength or accuracy based reinforcement learning or supervised learning approach. Quality–Diversity algorithms – QD algorithms simultaneously aim for high-quality and diverse solutions. Unlike traditional optimization algorithms that solely focus on finding the best solution to a problem, QD algorithms explore a wide variety of solutions across a problem space and keep those that are not just high performing, but also diverse and unique. == Theoretical background == The following theoretical principles apply to all or almost all EAs. === No free lunch theorem === The no free lunch theorem of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust. === Convergence === For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence under the condition that an optimum exists. Without loss of generality, a maximum search is assumed for the proof: From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation k {\displaystyle k} an improvement of the fitness F {\displaystyle F} of the respective best individual x ′ {\displaystyle x'} will occur with a probability P > 0 {\displaystyle P>0} . Thus: F ( x 1 ′ ) ≤ F ( x 2 ′ ) ≤ F ( x 3 ′ ) ≤ ⋯ ≤ F ( x k ′ ) ≤ ⋯ {\displaystyle F(x'_{1})\leq F(x'_{2})\leq F(x'_{3})\leq \cdots \leq F(x'_{k})\leq \cdots } I.e., the fitness values represent a monotonically non-decreasing sequence, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum. Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (see step 4 of the generic definition) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection. === Virtual alphabets === With the theory of virtual alphabets, David E. Goldberg showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion. == Comparison to other concepts == === Biological processes === A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis to become a mature p
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EXAPT

EXAPT (a portmanteau of "Extended Subset of APT") is a production-oriented programming language that allows users to generate NC programs with control information for machining tools and facilitates decision-making for production-related issues that may arise during various machining processes. EXAPT was first developed to address industrial requirements. Through the years, the company created additional software for the manufacturing industry. Today, EXAPT offers a suite of SAAS products and services for the manufacturing industry. The trade name, EXAPT, is most commonly associated with the CAD/CAM-System, production data, and tool management software of the German company EXAPT Systemtechnik GmbH based in Aachen, DE. == General == EXAPT is a modularly built programming system for all NC machining operations as Drilling Turning Milling Turn-Milling Nibbling Flame-, laser-, plasma- and water jet cutting Wire eroding Operations with industrial robots Due to the modular structure, the main product groups, EXAPTcam and EXAPTpdo, are gradually expandable and permit individual software for the manufacturing industry used individually and also in a compound with an existing IT environment. == Functionality == EXAPTcam meets the requirements for NC planning, especially for the cutting operations such as turning, drilling, and milling up to 5-axis simultaneous machining. Thereby new process technologies, tool, and machine concepts are constantly involved. In the NC programming data from different sources such as 3D CAD models, drawings or tables can flow in. The possibilities of NC programming reaches from language-oriented to feature-oriented NC programming. The integrated EXAPT knowledge database and intelligent and scalable automatisms support the user. The EXAPT NC planning also covers the generation of production information as clamping and tool plans, presetting data or time calculations. The realistic simulation possibilities of NC planning and NC control data provide with production reliability. EXAPTpdo (EXAPT ProductionsDataOrganization) provides a neutrally applicable technology platform for the information compound of the NC planning - to the shop floor. This applies to all NC production data that are necessary for the set-up of NC machines, for the provision, presetting, and stocking of manufacturing resources and provided by EXAPTpdo in a central database. Besides classical functions of the tool management system (TMS) as the management of cutting tools, measuring, testing and clamping devices the technology data management and tool lifecycle management (TLM) is also included. System-supported "where-used lists" helps to handle the manufacturing resource cycle by secured requirement determination and requirement fulfillment. Unnecessary transports and unplanned dispositive adjustments are dropped, stocks are reduced, set-up times reduced and the throughput is increased. EXAPTpdo synchronizes involved systems within the value chain. Stock systems, MES systems or ERP systems (e.g. from the purchasing or production areas) do not work in isolation from each other but they interact with each other. EXAPTpdo provides the base to Smart Factory, for more flexibility in production and faster communication. == History == With the foundation of the EXAPT-Verein in 1967 as spin-off of the universities Aachen, Berlin and Stuttgart the further development "EXAPT (EXtended Subset of APT)" of the programming language "APT (Automatically Programmed Tool)" was focused and so the first milestone for the EXAPT history was set. In the same year the system EXAPT 1 for drilling and simple milling tasks became available. 1969 The industrial application of EXAPT 2 for the programming of NC machines with 2-axis linear and path control begins. In the following year, the development of the EXAPT modular system starts. 1972 BASIC-EXAPT is provided for the universal, homogeneous programming of all NC tasks. The support is made by the EXAPT applications consultancy. 1973 EXAPT 1.1 is provided for the programming of straight-cut and continuous-path controlled drilling and milling machines and machining centers. At the Hanover Fair (IHA 73) the interactive access to a mainframe via a time-sharing terminal for the part program entry and correction is presented and starts the replacement of the punch card. 1974 The possibilities for the use of process computers for the NC data transfer are leveled out. EXAPT offers the possibility of the result simulation when using plotters with display of tool paths and tools in assignment to the workpiece. In April 1975, the EXAPT NC Systemtechnik GmbH was founded with the aim, of enabling entry into the NC technique for small and medium-sized companies by a complete product and service program. In the following year, the system portfolio is extended with further system modules and service programs and the provision of postprocessors. 1978 The development activities on the EXAPT module system started in 1970 are completed. Using modern software techniques, the different system parts BASIC-EXAPT, EXAPT 1, EXAPT 1.1, and EXAPT 2 are composed of a total system. System support and applications consultancy become a new working focus. From the beginning to the middle of the 1980s Beside new portable software modules for CAD/CAM applications (e. g. CAPEX, NESTEX, CADEX, CADCPL), the first version of the EXAPT DNC system and extensions of the EXAPT NC programming system for the machining of sculptured surfaces are presented. 1988 EXAPT expands the software product range by systems for tool data management (BMO) and production data management (FDO). EXAPT trains more than 1,300 course participants including company-specific courses. 1992 The first version of the completely new product generation EXAPTplus is presented and the agency in Dresden is opened. 1993 The company name "EXAPT NC Systemtechnik GmbH" is changed to "EXAPT Systemtechnik GmbH." EXAPTplus is presented on PC under Windows NT at the EMO '93. The decentralization of the use of EXAPT systems expands the range of applications. In the following year, EXAPT-DNC is executable under Windows on a customary PC. Special hardware is not needed and so it can be used in compound with the database-supported EXAPT production data management system (FDO). 1995 EXAPTplus is also ready for complex application cases such as machining of tubes at extrusion tools. EXAPT-CADI provides the transfer of 2D CAD data to EXAPTplus. With the new office Gießen the marketing is strengthened. In the following year the EXAPT NC editor is developed for the direct processing of NC control data with tool path display and visualization of the tools. In the course of the market entry of more comfortable 3D CAD systems for the solid modelling of components a detailed evaluation of current systems is made in 1997. It is decided to use SolidWorks as a reference system for the solid-oriented NC planning with EXAPT. 1998 The first solution for the transfer of geometry data between SolidWorks and EXAPTplus is generated. The EXAPT organization systems are (beside SQL) also executable under Oracle now. The use of client server solutions supports the data flow in the production. 1999 AFR functions are provided in connection with EXAPTsolid to support a workpiece modelling for NC. The millennium capability is ensured for all EXAPT systems. AFR is a ground-breaking for the integration of third-party products. 2002 EXAPT-BMG is developed for the generation and visualization of tools with additional functions for the assembly from components. The acquisition of tools with their geometric and technological presentation offers extensive support of the NC planning with EXAPT systems. 2003 EXAPTpdo is available to optimize the process chains in production planning and production execution optimally regarding the increasing requirements of changing production conditions. 2004 Diverse system extensions are made in EXAPTplus, EXAPTsolid, EXAPT NC editor, EXAPTpdo for the complete machining on turning/milling centres with result reliability because of more extensive simulation based on realNC (Tecnomatix), for the use of new complex tool systems and the compound use between ERP systems as SAP and intelligent CNC systems. In the following year, EXAPTpdo is extended for the cross-order set-up optimization and provision of manufacturing re-sources especially for single and small series production with connection to purchase and physical portfolio management. 2006 The EXAPT systems are available for extended use as an information platform for production, the time management, and similar requirements. EXAPTsolid is extended for the feature-oriented milling operation and machine simulation. The NC programming of complex machine tools, e.g. three-turret-turning/milling centers is supported by EXAPT systems, as well as the use of multi-functional tools. 2007 A module for 3-5-axis simultaneous milling machining is presented.
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Abess

abess (Adaptive Best Subset Selection, also ABESS) is a machine learning method designed to address the problem of best subset selection. It aims to determine which features or variables are crucial for optimal model performance when provided with a dataset and a prediction task. abess was introduced by Zhu in 2020 and it dynamically selects the appropriate model size adaptively, eliminating the need for selecting regularization parameters. abess is applicable in various statistical and machine learning tasks, including linear regression, the Single-index model, and other common predictive models. abess can also be applied in biostatistics. == Basic Form == The basic form of abess is employed to address the optimal subset selection problem in general linear regression. abess is an l 0 {\displaystyle l_{0}} method, it is characterized by its polynomial time complexity and the property of providing both unbiased and consistent estimates. In the context of linear regression, assuming we have knowledge of n {\displaystyle n} independent samples ( x i , y i ) , i = 1 , … , n {\displaystyle (x_{i},y_{i}),i=1,\ldots ,n} , where x i ∈ R p × 1 {\displaystyle x_{i}\in \mathbb {R} ^{p\times 1}} and y i ∈ R {\displaystyle y_{i}\in \mathbb {R} } , we define X = ( x 1 , … , x n ) ⊤ {\displaystyle X=(x_{1},\ldots ,x_{n})^{\top }} and y = ( y 1 , … , y n ) ⊤ {\displaystyle y=(y_{1},\ldots ,y_{n})^{\top }} . The following equation represents the general linear regression model: y = X β + ε . {\displaystyle y=X\beta +\varepsilon .} To obtain appropriate parameters β {\displaystyle \beta } , one can consider the loss function for linear regression: L n LR ( β ; X , y ) = 1 2 n ‖ y − X β ‖ 2 2 . {\displaystyle {\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y)={\frac {1}{2n}}\|y-X\beta \|_{2}^{2}.} In abess, the initial focus is on optimizing the loss function under the l 0 {\displaystyle l_{0}} constraint. That is, we consider the following problem: min β ∈ R p × 1 L n LR ( β ; X , y ) , subject to ‖ β ‖ 0 ≤ s , {\displaystyle \min _{\beta \in \mathbb {R} ^{p\times 1}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y),{\text{ subject to }}\|\beta \|_{0}\leq s,} where s {\displaystyle s} represents the desired size of the support set, and ‖ β ‖ 0 = ∑ i = 1 p I ( β i ≠ 0 ) {\displaystyle \|\beta \|_{0}=\sum _{i=1}^{p}{\mathcal {I}}_{(\beta _{i}\neq 0)}} is the l 0 {\displaystyle l_{0}} norm of the vector. To address the optimization problem described above, abess iteratively exchanges an equal number of variables between the active set and the inactive set. In each iteration, the concept of sacrifice is introduced as follows: For j in the active set ( j ∈ A ^ {\displaystyle j\in {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ∖ { j } ) − L n LR ( β ^ A ) = X j ⊤ X j 2 n ( β ^ j ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\hat {\beta }}_{j}\right)^{2}} For j in the inactive set ( j ∉ A ^ {\displaystyle j\notin {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ) − L n LR ( β ^ A + t ^ { j } ) = X j ⊤ X j 2 n ( d ^ j X j ⊤ X j / n ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\hat {\boldsymbol {t}}}^{\{j\}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\frac {{\hat {\mathrm {d} }}_{j}}{{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}/n}}\right)^{2}} Here are the key elements in the above equations: β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} : This represents the estimate of β {\displaystyle \beta } obtained in the previous iteration. A ^ {\displaystyle {\hat {\mathcal {A}}}} : It denotes the estimated active set from the previous iteration. β ^ A ∖ { j } {\displaystyle {\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}} : This is a vector where the j-th element is set to 0, while the other elements are the same as β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} . t ^ { j } = arg ⁡ min t L n LR ( β ^ A + t { j } ) {\displaystyle {\hat {\boldsymbol {t}}}^{\{j\}}=\arg \min _{t}{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\boldsymbol {t}}^{\{j\}}\right)} : Here, t { j } {\displaystyle t^{\{j\}}} represents a vector where all elements are 0 except the j-th element. d ^ j = X j ⊤ ( y − X β ^ ) / n {\displaystyle {\hat {d}}_{j}={\boldsymbol {X}}_{j}^{\top }({\boldsymbol {y}}-{\boldsymbol {X}}{\hat {\boldsymbol {\beta }}})/n} : This is calculated based on the equation mentioned. The iterative process involves exchanging variables, with the aim of minimizing the sacrifices in the active set while maximizing the sacrifices in the inactive set during each iteration. This approach allows abess to efficiently search for the optimal feature subset. In abess, select an appropriate s max {\displaystyle s_{\max }} and optimize the above problem for active sets size s = 1 , … , s max {\displaystyle s=1,\ldots ,s_{\max }} using the information criterion GIC = n log ⁡ L n LR + s log ⁡ p log ⁡ log ⁡ n , {\displaystyle {\text{GIC}}=n\log {\mathcal {L}}_{n}^{\text{LR}}+s\log p\log \log n,} to adaptively choose the appropriate active set size s {\displaystyle s} and obtain its corresponding abess estimator. == Generalizations == The splicing algorithm in abess can be employed for subset selection in other models. === Distribution-Free Location-Scale Regression === In 2023, Siegfried extends abess to the case of Distribution-Free and Location-Scale. Specifically, it considers the optimization problem max ϑ ∈ R P , β ∈ R J , γ ∈ R J ∑ i = 1 N ℓ i ( ϑ , x i ⊤ β , exp ⁡ ( x i ⊤ γ ) − 1 ) , {\displaystyle \max _{{\boldsymbol {\vartheta }}\in \mathbb {R} ^{P},{\boldsymbol {\beta }}\in \mathbb {R} ^{J},{\boldsymbol {\gamma }}\in \mathbb {R} ^{J}}\sum _{i=1}^{N}\ell _{i}\left({\boldsymbol {\vartheta }},{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }},{\sqrt {\exp \left({\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\gamma }}\right)}}^{-1}\right),} subject to ‖ ( β ⊤ , γ ⊤ ) ⊤ ‖ 0 ≤ s , {\displaystyle \left\|\left({\boldsymbol {\beta }}^{\top },{\boldsymbol {\gamma }}^{\top }\right)^{\top }\right\|_{0}\leq s,} where ℓ i {\displaystyle \ell _{i}} is a loss function, ϑ {\displaystyle {\boldsymbol {\vartheta }}} is a parameter vector, β {\displaystyle {\boldsymbol {\beta }}} and γ {\displaystyle {\boldsymbol {\gamma }}} are vectors, and x i {\displaystyle {\boldsymbol {x}}_{i}} is a data vector. This approach, demonstrated across various applications, enables parsimonious regression modeling for arbitrary outcomes while maintaining interpretability through innovative subset selection procedures. === Groups Selection === In 2023, Zhang applied the splicing algorithm to group selection, optimizing the following model: min β ∈ R p L n LR ( β ; X , y ) subject to ∑ j = 1 J I ( ‖ β G j ‖ 2 ≠ 0 ) ≤ s {\displaystyle \min _{{\boldsymbol {\beta }}\in \mathbb {R} ^{p}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y){\text{ subject to }}\sum _{j=1}^{J}I\left(\|{\boldsymbol {\beta }}_{G_{j}}\|_{2}\neq 0\right)\leq s} Here are the symbols involved: J {\displaystyle J} : Total number of feature groups, representing the existence of J {\displaystyle J} non-overlapping feature groups in the dataset. G j {\displaystyle G_{j}} : Index set for the j {\displaystyle j} -th feature group, where j {\displaystyle j} ranges from 1 to J {\displaystyle J} , representing the feature grouping structure in the data. s {\displaystyle s} : Model size, a positive integer determined from the data, limiting the number of selected feature groups. === Regression with Corrupted Data === Zhang applied the splicing algorithm to handle corrupted data. Corrupted data refers to information that has been disrupted or contains errors during the data collection or recording process. This interference may include sensor inaccuracies, recording errors, communication issues, or other external disturbances, leading to inaccurate or distorted observations within the dataset. === Single Index Models === In 2023, Tang applied the splicing algorithm to optimal subset selection in the Single-index model. The form of the Single Index Model (SIM) is given by y i = g ( b ⊤ x i , e i ) , i = 1 , … , n , {\displaystyle y_{i}=g({\boldsymbol {b}}^{\top }{\boldsymbol {x}}_{i},e_{i}),\quad i=1,\ldots ,n,} where b {\displaystyle {\boldsymbol {b}}} is the parameter vector, e i {\displaystyle e_{i}} is the error term. The corresponding loss function is defined as l n ( β ) = ∑ i = 1 n ( r i n − 1 2 − x i ⊤ β ) 2 , {\displaystyle l_{n}({\boldsymbol {\beta }})=\sum _{i=1}^{n}\left({\frac {r_{i}}{n}}-{\frac {1}{2}}-{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }}\right)^{2},} where r {\disp
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Canonical correspondence analysis

In multivariate analysis, canonical correspondence analysis (CCA) is an ordination technique that determines axes from the response data as a unimodal combination of measured predictors. CCA is commonly used in ecology in order to extract gradients that drive the composition of ecological communities. CCA extends correspondence analysis (CA) with regression, in order to incorporate predictor variables. == History == CCA was developed in 1986 by Cajo ter Braak and implemented in the program CANOCO, an extension of DECORANA. To date, CCA is one of the most popular multivariate methods in ecology, despite the availability of contemporary alternatives. CCA was originally derived and implemented using an algorithm of weighted averaging, though Legendre & Legendre (1998) derived an alternative algorithm. == Assumptions == The requirements of a CCA are that the samples are random and independent. Also, the data are categorical and that the independent variables are consistent within the sample site and error-free. The original publication states the need for equal species tolerances, equal species maxima, and equispaced or uniformly distributed species optima and site scores.
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Random indexing

Random indexing is a dimensionality reduction method and computational framework for distributional semantics, based on the insight that very-high-dimensional vector space model implementations are impractical, that models need not grow in dimensionality when new items (e.g. new terminology) are encountered, and that a high-dimensional model can be projected into a space of lower dimensionality without compromising L2 distance metrics if the resulting dimensions are chosen appropriately. This is the original point of the random projection approach to dimension reduction first formulated as the Johnson–Lindenstrauss lemma, and locality-sensitive hashing has some of the same starting points. Random indexing, as used in representation of language, originates from the work of Pentti Kanerva on sparse distributed memory, and can be described as an incremental formulation of a random projection. It can be also verified that random indexing is a random projection technique for the construction of Euclidean spaces—i.e. L2 normed vector spaces. In Euclidean spaces, random projections are elucidated using the Johnson–Lindenstrauss lemma. The TopSig technique extends the random indexing model to produce bit vectors for comparison with the Hamming distance similarity function. It is used for improving the performance of information retrieval and document clustering. In a similar line of research, Random Manhattan Integer Indexing (RMII) is proposed for improving the performance of the methods that employ the Manhattan distance between text units. Many random indexing methods primarily generate similarity from co-occurrence of items in a corpus. Reflexive Random Indexing (RRI) generates similarity from co-occurrence and from shared occurrence with other items.
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Salience (neuroscience)

Salience (also called saliency, from Latin saliō meaning "leap, spring") is the property by which some thing stands out. Salient events are an attentional mechanism by which organisms learn and survive; those organisms can focus their limited perceptual and cognitive resources on the pertinent (that is, salient) subset of the sensory data available to them. Saliency typically arises from contrasts between items and their neighborhood. They might be represented, for example, by a red dot surrounded by white dots, or by a flickering message indicator of an answering machine, or a loud noise in an otherwise quiet environment. Saliency detection is often studied in the context of the visual system, but similar mechanisms operate in other sensory systems. Just what is salient can be influenced by training: for example, for human subjects particular letters can become salient by training. There can be a sequence of necessary events, each of which has to be salient, in turn, in order for successful training in the sequence; the alternative is a failure, as in an illustrated sequence when tying a bowline; in the list of illustrations, even the first illustration is a salient: the rope in the list must cross over, and not under the bitter end of the rope (which can remain fixed, and not free to move); failure to notice that the first salient has not been satisfied means the knot will fail to hold, even when the remaining salient events have been satisfied. When attention deployment is driven by salient stimuli, it is considered to be bottom-up, memory-free, and reactive. Conversely, attention can also be guided by top-down, memory-dependent, or anticipatory mechanisms, such as when looking ahead of moving objects or sideways before crossing streets. Humans and other animals have difficulty paying attention to more than one item simultaneously, so they are faced with the challenge of continuously integrating and prioritizing different bottom-up and top-down influences. == Neuroanatomy == The brain component named the hippocampus helps with the assessment of salience and context by using past memories to filter new incoming stimuli, and placing those that are most important into long term memory. The entorhinal cortex is the pathway into and out of the hippocampus, and is an important part of the brain's memory network; research shows that it is a brain region that suffers damage early on in Alzheimer's disease, one of the effects of which is altered (diminished) salience. The pulvinar nuclei (in the thalamus) modulate physical/perceptual salience in attentional selection. One group of neurons (i.e., D1-type medium spiny neurons) within the nucleus accumbens shell (NAcc shell) assigns appetitive motivational salience ("want" and "desire", which includes a motivational component), aka incentive salience, to rewarding stimuli, while another group of neurons (i.e., D2-type medium spiny neurons) within the NAcc shell assigns aversive motivational salience to aversive stimuli. The primary visual cortex (V1) generates a bottom-up saliency map from visual inputs to guide reflexive attentional shifts or gaze shifts. According to V1 Saliency Hypothesis, the saliency of a location is higher when V1 neurons give higher responses to that location relative to V1 neurons' responses to other visual locations. For example, a unique red item among green items, or a unique vertical bar among horizontal bars, is salient since it evokes higher V1 responses and attracts attention or gaze. The V1 neural responses are sent to the superior colliculus to guide gaze shifts to the salient locations. A fingerprint of the saliency map in V1 is that attention or gaze can be captured by the location of an eye-of-origin singleton in visual inputs, e.g., a bar uniquely shown to the left eye in a background of many other bars shown to the right eye, even when observers cannot tell the difference between the singleton and the background bars. == In psychology == The term is widely used in the study of perception and cognition to refer to any aspect of a stimulus that, for any of many reasons, stands out from the rest. Salience may be the result of emotional, motivational or cognitive factors and is not necessarily associated with physical factors such as intensity, clarity or size. Although salience is thought to determine attentional selection, salience associated with physical factors does not necessarily influence selection of a stimulus. === Salience bias === Salience bias (also referred to as perceptual salience) is a cognitive bias that predisposes individuals to focus on or attend to items, information, or stimuli that are more prominent, visible, or emotionally striking. This is as opposed to stimuli that are unremarkable, or less salient, even though this difference is often irrelevant by objective standards. The American Psychological Association (APA) defines the salience hypothesis as a theory regarding perception where "motivationally significant" information is more readily perceived than information with little or less significant motivational importance. Perceptual salience (salience bias) is linked to the vividness effect, whereby a more pronounced response is produced by a more vivid perception of a stimulus than the mere knowledge of the stimulus. Salience bias assumes that more dynamic, conspicuous, or distinctive stimuli engage attention more than less prominent stimuli, disproportionately impacting decision making, it is a bias which favors more salient information. ==== Application ==== ===== Cognitive Psychology ===== Salience bias, like all other cognitive biases, is an applicable concept to various disciplines. For example, cognitive psychology investigates cognitive functions and processes, such as perception, attention, memory, problem solving, and decision making, all of which could be influenced by salience bias. Salience bias acts to combat cognitive overload by focusing attention on prominent stimuli, which affects how individuals perceive the world as other, less vivid stimuli that could add to or change this perception, are ignored. Human attention gravitates towards novel and relevant stimuli and unconsciously filters out less prominent information, demonstrating salience bias, which influences behavior as human behavior is affected by what is attended to. Behavioral economists Tversky and Kahneman also suggest that the retrieval of instances is influenced by their salience, such as how witnessing or experiencing an event first-hand has a greater impact than when it is less salient, like if it were read about, implying that memory is affected by salience. ===== Language ===== It is also relevant in language understanding and acquisition. Focusing on more salient phenomena allows people to detect language patterns and dialect variations more easily, making dialect categorization more efficient. ===== Social Behavior ===== Furthermore, social behaviors and interactions can also be influenced by perceptual salience. Changes in the perceptual salience of an individual heavily influences their social behavior and subjective experience of their social interactions, confirming a "social salience effect". Social salience relates to how individuals perceive and respond to other people. ===== Behavioral Science ===== The connection between salience bias and other heuristics, like availability and representativeness, links it to the fields of behavioral science and behavioral economics. Salience bias is closely related to the availability heuristic in behavioral economics, based on the influence of information vividness and visibility, such as recency or frequency, on judgements, for example:Accessibility and salience are closely related to availability, and they are important as well. If you have personally experienced a serious earthquake, you're more likely to believe that an earthquake is likely than if you read about it in a weekly magazine. Thus, vivid and easily imagined causes of death (for example, tornadoes) often receive inflated estimates of probability, and less-vivid causes (for example, asthma attacks) receive low estimates, even if they occur with a far greater frequency (here, by a factor of twenty). Timing counts too: more recent events have a greater impact on our behavior, and on our fears, than earlier ones.Humans have bounded rationality, which refers to their limited ability to be rational in decision making, due to a limited capacity to process information and cognitive ability. Heuristics, such as availability, are employed to reduce the complexity of cognitive and social tasks or judgements, in order to decrease the cognitive load that result from bounded rationality. Despite the effectiveness of heuristics in doing so, they are limited by systematic errors that occur, often the result of influencing biases, such as salience. This can lead to misdirected or misinformed judgements, based on an overemphasis or overweighting of
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Semantic mapping (statistics)

Semantic mapping (SM) is a statistical method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used in a set of multidimensional vectors of features to extract a few new features that preserves the main data characteristics. SM performs dimensionality reduction by clustering the original features in semantic clusters and combining features mapped in the same cluster to generate an extracted feature. Given a data set, this method constructs a projection matrix that can be used to map a data element from a high-dimensional space into a reduced dimensional space. SM can be applied in construction of text mining and information retrieval systems, as well as systems managing vectors of high dimensionality. SM is an alternative to random mapping, principal components analysis and latent semantic indexing methods.
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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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