Gaussian adaptation

Gaussian adaptation (GA), also called normal or natural adaptation (NA) is an evolutionary algorithm designed for the maximization of manufacturing yield due to statistical deviation of component values of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n-dimensional vector x[xT = (x1, x2, ..., xn)] are taken from a multivariate Gaussian distribution, N(m, M), having mean m and moment matrix M. The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass samples are m and M. The outcome of x as a pass sample is determined by a function s(x), 0 < s(x) < q ≤ 1, such that s(x) is the probability that x will be selected as a pass sample. The average probability of finding pass samples (yield) is P ( m ) = ∫ s ( x ) N ( x − m ) d x {\displaystyle P(m)=\int s(x)N(x-m)\,dx} Then the theorem of GA states: For any s(x) and for any value of P < q, there always exist a Gaussian p. d. f. [ probability density function ] that is adapted for maximum dispersion. The necessary conditions for a local optimum are m = m and M proportional to M. The dual problem is also solved: P is maximized while keeping the dispersion constant (Kjellström, 1991). Proofs of the theorem may be found in the papers by Kjellström, 1970, and Kjellström & Taxén, 1981. Since dispersion is defined as the exponential of entropy/disorder/average information it immediately follows that the theorem is valid also for those concepts. Altogether, this means that Gaussian adaptation may carry out a simultaneous maximisation of yield and average information (without any need for the yield or the average information to be defined as criterion functions). The theorem is valid for all regions of acceptability and all Gaussian distributions. It may be used by cyclic repetition of random variation and selection (like the natural evolution). In every cycle a sufficiently large number of Gaussian distributed points are sampled and tested for membership in the region of acceptability. The centre of gravity of the Gaussian, m, is then moved to the centre of gravity of the approved (selected) points, m. Thus, the process converges to a state of equilibrium fulfilling the theorem. A solution is always approximate because the centre of gravity is always determined for a limited number of points. It was used for the first time in 1969 as a pure optimization algorithm making the regions of acceptability smaller and smaller (in analogy to simulated annealing, Kirkpatrick 1983). Since 1970 it has been used for both ordinary optimization and yield maximization. == Natural evolution and Gaussian adaptation == It has also been compared to the natural evolution of populations of living organisms. In this case s(x) is the probability that the individual having an array x of phenotypes will survive by giving offspring to the next generation; a definition of individual fitness given by Hartl 1981. The yield, P, is replaced by the mean fitness determined as a mean over the set of individuals in a large population. Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it may push the centre of gravity of the Gaussian to the centre of gravity of the selected individuals. This may be accomplished by the Hardy–Weinberg law. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of the structure (Kjellström, 1996). In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense Gaussian adaptation may be seen as a genetic algorithm. == How to climb a mountain == Mean fitness may be calculated provided that the distribution of parameters and the structure of the landscape is known. The real landscape is not known, but figure below shows a fictitious profile (blue) of a landscape along a line (x) in a room spanned by such parameters. The red curve is the mean based on the red bell curve at the bottom of figure. It is obtained by letting the bell curve slide along the x-axis, calculating the mean at every location. As can be seen, small peaks and pits are smoothed out. Thus, if evolution is started at A with a relatively small variance (the red bell curve), then climbing will take place on the red curve. The process may get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small. If the mutation rate is sufficiently high, the disorder or variance may increase and the parameter(s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue up to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides — dependent on the landscape — the process may become very jerky, and if the ratio between the time spent by the process at a local peak and the time of transition to the next peak is very high, it may as well look like a punctuated equilibrium as suggested by Gould (see Ridley). == Computer simulation of Gaussian adaptation == Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically. The implementation of normal adaptation on a computer is a fairly simple task. The adaptation of m may be done by one sample (individual) at a time, for example m(i + 1) = (1 – a) m(i) + ax where x is a pass sample, and a < 1 a suitable constant so that the inverse of a represents the number of individuals in the population. M may in principle be updated after every step y leading to a feasible point x = m + y according to: M(i + 1) = (1 – 2b) M(i) + 2byyT, where yT is the transpose of y and b << 1 is another suitable constant. In order to guarantee a suitable increase of average information, y should be normally distributed with moment matrix μ2M, where the scalar μ > 1 is used to increase average information (information entropy, disorder, diversity) at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WWT = M. Thus, we have y = Wg, where g is normally distributed with the moment matrix μU, and U is the unit matrix. W and WT may be updated by the formulas W = (1 – b)W + bygT and WT = (1 – b)WT + bgyT because multiplication gives M = (1 – 2b)M + 2byyT, where terms including b2 have been neglected. Thus, M will be indirectly adapted with good approximation. In practice it will suffice to update W only W(i + 1) = (1 – b)W(i) + bygT. This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999). The figure below illustrates the effect of increased average information in a Gaussian p.d.f. used to climb a mountain Crest (the two lines represent the contour line). Both the red and green cluster have equal mean fitness, about 65%, but the green cluster has a much higher average information making the green process much more efficient. The effect of this adaptation is not very salient in a 2-dimensional case, but in a high-dimensional case, the efficiency of the search process may be increased by many orders of magnitude. == The evolution in the brain == In the brain the evolution of DNA-messages is supposed to be replaced by an evolution of signal patterns and the phenotypic landscape is replaced by a mental landscape, the complexity of which will hardly be second to the former. The metaphor with the mental landscape is based on the assumption that certain signal patterns give rise to a better well-being or performance. For instance, the control of a group of muscles leads to a better pronunciation of a word or performance of a piece of music. In this simple model it is assumed that the brain consists of interconnected components that may add, multiply and delay signal values. A nerve cell kernel may add signal values, a synapse may multiply with a constant and An axon may delay values. This is a basis of the theory of digital filters and neural networks consisting of components that may add, multiply and delay signalvalues and also of many brain models, Levine 1991. In the figure below the brain stem is supposed to deliver Gaussian distributed signal patterns. This may be possible since certai

Shepp–Logan phantom

The Shepp–Logan phantom is a standard test image created by Larry Shepp and Benjamin F. Logan for their 1974 paper "The Fourier Reconstruction of a Head Section". It serves as the model of a human head in the development and testing of image reconstruction algorithms. == Definition == The function describing the phantom is defined as the sum of 10 ellipses inside a 2×2 square:

Perry Rhodan

Perry Rhodan is a German space opera franchise, named after its hero. It commenced in 1961 and has been ongoing for decades, written by an ever-changing team of authors. Having sold approximately two billion copies (in novella format) worldwide (including over one billion in Germany alone), it is the most successful science fiction book series ever written. The first billion of worldwide sales was celebrated in 1986. The series has spun off into comic books, audio dramas, video games and the like. A reboot, Perry Rhodan NEO, was launched in 2011 and began publication in English in April 2021. == Print publication == The series has spun off into many different forms of media, but originated as a serial novella published weekly since 8 September 1961 in the Romanheft (Meaning "Magazine novel") format. These are digest-sized booklets, usually containing 66 pages, the German equivalent of the now-defunct (and generally longer) American pulp magazine. They are published by Pabel-Moewig Verlag, a subsidiary of Bauer Media Group headquartered in Hamburg. As of February 2019, 3000 booklet novels of the original series, 850 spinoff novels of the sister series Atlan and over 400 paperbacks and 200 hardcover editions have been published, totalling over 300,000 pages. == English translation == The first 126 novels (plus five novels of the spinoff series Atlan) were translated into English and published by Ace Books between 1969 and 1978, with the same translations used for the British edition published by Futura Publications which issued only 39 novels. When Ace cancelled its translation of the series, translator Wendayne Ackerman self-published the following 19 novels (under the business name 'Master Publications') and made them available by subscription only. Financial disputes with the German publishers led to the cancellation of the American translation in 1979. An attempt to revive the series in English was made in 1997–1998 by Vector Publications of the US, which published translations of four issues (1800–1803) from the current storyline being published in Germany at the time. The series and its spin-offs have captured a substantial fraction of the original German science fiction output and exert influence on many German writers in the field. == Structure == The series is told in an arc storyline structure. An arc—called a "cycle"—would have anywhere from 25 to 100 issues devoted to it. Similar subsequent cycles are referred to as a "grand-cycle". == History == ‘Perry Rhodan, der Erbe des Universums’ (Eng: ‘The Heir to the Universe’, though the American/British editions instead used the subtitle 'Peacelord of the Universe') was created by German science fiction authors K. H. Scheer and Walter Ernsting and launched in 1961 by German publishing house Arthur Moewig Verlag (now Pabel-Moewig Verlag). Originally planned as a 30 to 50 volume series, it has been published continuously every week since, celebrating the 3000th issue in 2019. Written by an ever-changing team of authors, many of whom, however, remained with the series for decades or life, Perry Rhodan is issued in weekly novella-size installments in the traditional German Heftroman (pulp booklet) format. Unlike most German Heftromane, Perry Rhodan consists not of unconnected novels but is a series with a continuous, increasingly complex plotline, with frequent back references to events. In addition to its original Heftroman form, the series now also appears in hardcovers, paperbacks, e-books, comics and audiobooks. Over the decades there have also been comic strips, numerous collectibles, several encyclopedias, audio plays, inspired music, etc. The series has seen partial translations into several languages. It also spawned the German-Italian-Spanish 1967 movie Mission Stardust, which is widely considered so terrible that many fans of the series pretend it never existed. Coinciding with the 50th-anniversary World Con, on 30 September 2011, a new series named Perry Rhodan Neo began publication, attracting new readers with a reboot of the story, starting in the year 2036 instead of 1971, and a related but independent story-line. On 2 April 2021, light novel and manga publisher J-Novel Club announced Perry Rhodan NEO as a launch title for its new J-Novel Pulp imprint, making this the first ongoing English release of new Perry Rhodan serials in over 20 years. It has become the most popular science fiction book series of all time. == Overview == === Fictional history === The story begins in 1971. During the first human Moon landing by US Space Force Major Perry Rhodan and his crew, they discover a marooned extraterrestrial space ship from the fictional planet Arkon, located in the (real) M13 cluster. Appropriating the Arkonide technology, they proceed to unify Terra and carve out a place for humanity in the galaxy and the cosmos. Two of the accomplishments that enable them to do so are positronic brains and starship drives for near-instantaneous hyperspatial translation. These were directly borrowed from Isaac Asimov's science fiction. As the series progresses, major characters, including the title character, are granted relative immortality. They are immune to age and disease, but not to violent death. The story continues over the course of millennia and includes flashbacks thousands and even millions of years into the past. The scope widens to encompass other galaxies, even more remote regions of space, parallel universes and cosmic structures, time travel, paranormal powers, a variety of aliens ranging from threatening to endearing, and bodiless entities, some of which have godlike powers. === Multiverse === The universe in which the main plot generally takes place is called the Einstein Universe (or "Meekorah"). Its laws are for the most part identical to those of the real universe, as known by late 20th century science. Newer theories about dark matter and dark energy are currently not used in the series. The laws of nature follow old theories that have been disproven, in order to protect series continuity. There are many other universes, each to a greater or lesser extent different from the familiar one, in which, for example one in which time runs slower, an anti-matter universe, a shrinking universe, etc. Each universe possesses its owntimelines, which are for the most part unreachable from each other but may be accessed by special means, thereby itself creating many more parallel timelines. The Einstein Universe is embedded in a high-dimensional manifold, called Hyperspace. This hyperspace consists of several subspaces use for faster-than-light travel by technological means. The exact traits of those higher dimensions are got yhr mode pity unexplained. The border of the universe is a dimension called the deep, once used for construction of the gigantic disc-shaped world Deepland. === Psionic Web and Moral Code === The Psionic Web crosses the whole universe, constantly emitting "vital energy" and "psionic energy", guaranteeing normal (organic among others) life and the wellbeing of higher entities. The Moral Code crosses through all universes, and is linked to the Psionic Web. It is subdivided into the Cosmogenes, which are again subdivided into the Cosmonucleotids. The Cosmonucleotids determine reality and fate for their respective parts of a given universe, via messengers. Higher beings are trying to gain control of this Code to rule reality. The Moral Code itself was not installed by the higher beings, the higher powers by themselves have no clue why or by whom the Code was made. Once the Cosmocrats ordered Perry Rhodan to find the answer to the third ultimate question: "Who initiated the LAW and what does it accomplish?" Perry Rhodan had the chance to receive the answer at the mountain of creation, but refused, as he knew that the answer would destroy his mind. The negative Superintelligence Koltoroc had received the answer to the last ultimate question, 69 million years BC at Negane Mountain, but it is not known if it made any use of the information. === Onion-shell model === An evolutionary schema, similar to the Great Chain of Being, called the "onion-shell model" is employed in relationship to all life. Here, continuous evolution is from lower to higher lifeforms, culminating in bodiless entities. Later in the series, further lifeforms, representing stages between the known shells, were introduced. The main shells are: Lifeless matter Bacteria Higher animals Intelligent species Intelligent species that have contacted other species Superintelligences (SI) Matter sources/ Matter sinks Cosmocrats / Chaotarchs (High Powers) Powers close to the "Horizon of the LAW", the essence of the Multiverse The Superintelligences are the next step above normal minds. They can be born, for example, when a species collectively gives up its bodies and unites their spirits. Such Superintelligences may claim as their domain areas consisting of up to several galaxies (the entity known as "E

The Raimones

The Raimones (stylized as THE RAiMONES) is a 2017 generative music project that utilized artificial intelligence to compose music in the style of the American punk rock band The Ramones. Developed by Matthias Frey, a researcher at Sony CSL Tokyo, the project was an early experiment in applying deep learning to high-energy, minimalist musical genres. == Technical Development == The project utilized Long short-term memory (LSTM) recurrent neural networks to generate musical structures and lyrics. The model was trained on a dataset consisting of 130 Ramones songs in MIDI format and the band's complete lyrical catalog. The technical framework was built using Python and Jupyter Notebook, drawing influence from the character-level RNN text generation models popularized by Andrej Karpathy. Unlike contemporary AI music projects that focused on the harmonic complexities of classical or pop music, THE RAiMONES sought to determine if neural networks could replicate the "1-2-3-4" rhythmic consistency and formulaic nature of early punk. == "I'm Alive" == The primary output of the project was the song "I'm Alive," released in 2017. The work is described as a form of "augmented intelligence," a hybrid approach where the AI provides the compositional foundation and human musicians handle the arrangement and performance. The song was recorded by the musician Mr. Ratboy (Gilbert Avondet). Avondet's involvement provided a stylistic link to the subject material, as he had previously served as a touring guitarist for Marky Ramone and the Intruders in 1996. The project's discography has since been made available on major streaming platforms, including Apple Music. == Reception and Significance == The project has been cited as a "proof of concept" for AI's ability to tackle "noisy" and aggressive aesthetics. In 2019, the Belgian magazine Knack Focus profiled the project alongside other AI pioneers such as Holly Herndon, noting the project's attempt to recreate the sound of "deceased legends" while maintaining a distinct, machine-like quality. It has also been featured in academic settings, such as at UC Santa Cruz, as a case study for AI-driven genre mimicry.

Evolving intelligent system

In computer science, an evolving intelligent system is a fuzzy logic system which improves the own performance by evolving rules. The technique is known from machine learning, in which external patterns are learned by an algorithm. Fuzzy logic based machine learning works with neuro-fuzzy systems. Intelligent systems have to be able to evolve, self-develop, and self-learn continuously in order to reflect a dynamically evolving environment. The concept of Evolving Intelligent Systems (EISs) was conceived around the turn of the century with the phrase EIS itself coined for the first time by Angelov and Kasabov in a 2006 IEEE newsletter and expanded in a 2010 text. EISs develop their structure, functionality and internal knowledge representation through autonomous learning from data streams generated by the possibly unknown environment and from the system self-monitoring. EISs consider a gradual development of the underlying (fuzzy or neuro-fuzzy) system structure and differ from evolutionary and genetic algorithms which consider such phenomena as chromosomes crossover, mutation, selection and reproduction, parents and off-springs. The evolutionary fuzzy and neuro systems are sometimes also called "evolving" which leads to some confusion. This was more typical for the first works on this topic in the late 1990s. == Implementations == EISs can be implemented, for example, using neural networks or fuzzy rule-based models. The first neural networks which consider an evolving structure were published in. These were later expanded by N. Kasabov and P. Angelov for the neuro-fuzzy models. P. Angelov introduced the evolving fuzzy rule-based systems (EFSs) as the first mathematical self-learning model that can dynamically evolve its internal structure and is human interpretable and coined the phrase EFS. Contemporarily, the offline incremental approach for learning an EIS, namely, EFuNN, was proposed by N. Kasabov. P. Angelov, D. Filev, N. Kasabov and O. Cordon organised the first IEEE Symposium on EFSs in 2006 (the proceedings of the conference can be found in). EFSs include a formal (and mathematically sound) learning mechanism to extract it from streaming data. One of the earliest and the most widely cited comprehensive survey on EFSs was done in 2008. Later comprehensive surveys on EFS methods with real applications were done in 2011 and 2016 by E. Lughofer. Other works that contributed further to this area in the following years expanded it to evolving participatory learning, evolving grammar, evolving decision trees, evolving human behaviour modelling, self-calibrating (evolving) sensors (eSensors), evolving fuzzy rule-based classifiers, evolving fuzzy controllers, autonomous fault detectors. More recently, the stability of the evolving fuzzy rule-based systems that consist of the structure learning and the fuzzily weighted recursive least square parameter update method has been proven by Rong. Generalized EFS, which allow rules to be arbitrarily rotated in the feature space and thus to improve their data representability, have been proposed in with significant extensions in towards 'smartness' of the rule bases (thus, termed as "Generalized Smart EFS"), allowing more interpretability and reducing curse of dimensionality. The generalized rule structure was also successfully used in the context of evolving neuro-fuzzy systems. Several facets and challenges for achieving more transparent and understandable rule bases in EFS have been discussed by E. Lughofer in. EISs form the theoretical and methodological basis for the Autonomous Learning Machines (ALMA) and autonomous multi-model systems (ALMMo) as well as of the Autonomous Learning Systems. Evolving Fuzzy Rule-based classifiers, in particular, is a very powerful new concept that offers much more than simply incremental or online classifiers – it can cope with new classes being added or existing classes being merged. This is much more than just adapting to new data samples being added or classification surfaces being evolved. Fuzzy rule-based classifiers are the methodological basis of a new approach to deep learning that was until now considered as a form of multi-layered neural networks. Deep Learning offers high precision levels surpassing the level of human ability and grabbed the imagination of the researchers, industry and the wider public. However, it has a number of intrinsic constraints and limitations. These include: The "black box", opaque internal structure which has millions of parameters and involves ad hoc decisions on the number of layers and algorithm parameters. The requirement for a huge amount of training data samples, computational resources (usually requiring GPUs and/or HPC) and time (usually requiring many hours of training). Iterative search. Requires retraining for new situations (is not evolving). Does not have proven convergence and stability. Most, if not all, of the above limitations can be avoided with the use of the Deep (Fuzzy) Rule-based Classifiers, which were recently introduced based on ALMMo, while achieving similar or even better performance. The resulting prototype-based IF...THEN...models are fully interpretable and dynamically evolving (they can adapt quickly and automatically to new data patterns or even new classes). They are non-parametric and, therefore, their training is non-iterative and fast (it can take few milliseconds per data sample/image on a normal laptop which contrasts with the multiple hours the current deep learning methods require for training even when they use GPUs and HPC). Moreover, they can be trained incrementally, online, or in real-time. Another aspect of Evolving Fuzzy Rule-based classifiers has been proposed in, which, in case of multi-class classification problems, achieves the reduction of class imbalance by cascadability into class sub-spaces and an increased flexibility and performance for adding new classes on the fly from streaming samples.

Artipic

Artipic is a graphics editor developed for Microsoft Windows. An older version for macOS is still available but unsupported. Artipic features drawing, editing, retouching, transforming and composing images including color corrections, effects and layer-based operations. It converts all common image formats and imports camera raw formats. In the global image editing ecosystem Artipic can be positioned somewhere in the middle. It differs from simple free photo editors by more advanced capabilities, however it does not cover the complete professional-level functionality pack provided by industry leaders like Adobe Photoshop. == History == Artipic developed by Swedish company Artipic AB. Artipic 1.0 was released in March 2014 as a free version. The first commercial version on Microsoft Windows was released in November 2014, on macOS – in October 2015. == Features == Supports Microsoft Windows and macOS Standard tools: select, crop, move, rotate, transform, stamp, color picking, text Advanced tools: custom brushes, gradients, shapes, paths, layers and masks Special tools: healing brush, red-eye effect reduction, dodge and burn brushes Adjustments: Brightness & Contrast, Hue & Saturation, Curves, Levels, Color Balance, Gamma Correction, Exposure, Color Temperature, Tint, Color Enhancer, Photo Filter Simulation, Posterization, Thresholding Filters: Smoothen, Sharpen, Vignetting, High-pass, Diffuse Glow, Shadow, Gaussian Blur Reversible (non-destructive) stylization presets Batch processing White balance RAW-converter including Gray Card Adobe Photoshop images supported == Version history ==

Type-1 OWA operators

Type-1 OWA operators are a set of aggregation operators that generalise the Yager's OWA (ordered weighted averaging) operators in the interest of aggregating fuzzy sets rather than crisp values in soft decision making and data mining. These operators provide a mathematical technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets. The two definitions for type-1 OWA operators are based on Zadeh's Extension Principle and α {\displaystyle \alpha } -cuts of fuzzy sets. The two definitions lead to equivalent results. == Definitions == === Definition 1 === Let F ( X ) {\displaystyle F(X)} be the set of fuzzy sets with domain of discourse X {\displaystyle X} , a type-1 OWA operator is defined as follows: Given n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,1]} , a type-1 OWA operator is a mapping, Φ {\displaystyle \Phi } , Φ : F ( X ) × ⋯ × F ( X ) ⟶ F ( X ) {\displaystyle \Phi \colon F(X)\times \cdots \times F(X)\longrightarrow F(X)} ( A 1 , ⋯ , A n ) ↦ Y {\displaystyle (A^{1},\cdots ,A^{n})\mapsto Y} such that μ Y ( y ) = sup ∑ k = 1 n w ¯ i a σ ( i ) = y ( μ W 1 ( w 1 ) ∧ ⋯ ∧ μ W n ( w n ) ∧ μ A 1 ( a 1 ) ∧ ⋯ ∧ μ A n ( a n ) ) {\displaystyle \mu _{Y}(y)=\displaystyle \sup _{\displaystyle \sum _{k=1}^{n}{\bar {w}}_{i}a_{\sigma (i)}=y}\left({\begin{array}{{1}l}\mu _{W^{1}}(w_{1})\wedge \cdots \wedge \mu _{W^{n}}(w_{n})\wedge \mu _{A^{1}}(a_{1})\wedge \cdots \wedge \mu _{A^{n}}(a_{n})\end{array}}\right)} where w ¯ i = w i ∑ i = 1 n w i {\displaystyle {\bar {w}}_{i}={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}} , and σ : { 1 , ⋯ , n } ⟶ { 1 , ⋯ , n } {\displaystyle \sigma \colon \{1,\cdots ,n\}\longrightarrow \{1,\cdots ,n\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\ \forall i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th highest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . === Definition 2 === Using the alpha-cuts of fuzzy sets: Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , then for each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,\;1]} , an α {\displaystyle \alpha } -level type-1 OWA operator with α {\displaystyle \alpha } -level sets { W α i } i = 1 n {\displaystyle \left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}} to aggregate the α {\displaystyle \alpha } -cuts of fuzzy sets { A i } i = 1 n {\displaystyle \left\{{A^{i}}\right\}_{i=1}^{n}} is: Φ α ( A α 1 , … , A α n ) = { ∑ i = 1 n w i a σ ( i ) ∑ i = 1 n w i | w i ∈ W α i , a i ∈ A α i , i = 1 , … , n } {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}} where W α i = { w | μ W i ( w ) ≥ α } , A α i = { x | μ A i ( x ) ≥ α } {\displaystyle W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}} , and σ : { 1 , ⋯ , n } → { 1 , ⋯ , n } {\displaystyle \sigma :\{\;1,\cdots ,n\;\}\to \{\;1,\cdots ,n\;\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th largest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . == Representation theorem of Type-1 OWA operators == Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , and the fuzzy sets A 1 , ⋯ , A n {\displaystyle A^{1},\cdots ,A^{n}} , then we have that Y = G {\displaystyle Y=G} where Y {\displaystyle Y} is the aggregation result obtained by Definition 1, and G {\displaystyle G} is the result obtained by in Definition 2. == Programming problems for Type-1 OWA operators == According to the Representation Theorem of Type-1 OWA Operators, a general type-1 OWA operator can be decomposed into a series of α {\displaystyle \alpha } -level type-1 OWA operators. In practice, this series of α {\displaystyle \alpha } -level type-1 OWA operators is used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals Φ α ( A α 1 , ⋯ , A α n ) {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)} . Then, the resulting aggregation fuzzy set is constructed with the membership function as follows: μ G ( x ) = ⋁ α : x ∈ Φ α ( A α 1 , ⋯ , A α n ) α ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha } For the left end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) − = min W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\operatorname {\min } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} while for the right end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) + = max W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\operatorname {\max } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper. == Alpha-level approach to Type-1 OWA operation == Three-step process: Step 1—To set up the α {\displaystyle \alpha } - level resolution in [0, 1]. Step 2—For each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,1]} , Step 2.1—To calculate ρ α + i 0 ∗ {\displaystyle \rho _{\alpha +}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α + i 0 ≥ A α + σ ( i 0 ) {\displaystyle \rho _{\alpha +}^{i_{0}}\geq A_{\alpha +}^{\sigma (i_{0})}} , stop, ρ α + i 0 {\displaystyle \rho _{\alpha +}^{i_{0}}} is the solution; otherwise go to Step 2.1-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to Step 2.1-2. Step 2.2 To calculate ρ α − i 0 ∗ {\displaystyle \rho _{\alpha -}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α − i 0 ≥ A α − σ ( i 0 ) {\displaystyle \rho _{\alpha -}^{i_{0}}\geq A_{\alpha -}^{\sigma (i_{0})}} , stop, ρ α − i 0 {\displaystyle \rho _{\alpha -}^{i_{0}}} is the solution; otherwise go to Step 2.2-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to step Step 2.2-2. Step 3—To construct the aggregation resulting fuzzy set G {\displaystyle G} based on all the available intervals [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] {\displaystyle \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]} : μ G ( x ) = ⋁ α : x ∈ [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]}\alpha } == Some Examples == The type-1 OWA operator with the weights shown in the top figure is used to aggregate the fuzzy sets (solide lines) in the bottom figure, and the dashed line is the aggregation result. == Special cases == Any OWA operators, like maximum, minimum, mean operators; Join operators of (type-1) fuzzy sets, i.e., fuzzy maximum operators; Meet operators of (type-1) fuzzy sets, i.e., fuzzy minimum operators; Join-like operators of (type-1) fuzzy sets; Meet-like operators of (type-1) fuzzy sets. == Generalizations == Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making. == Applications == Type-1 OWA operators have been applied to different domains for soft decision making. Improved efficiency of computing approach ; Type reduction of type-2 fuzzy sets ; Group decision making ; Credit risk evaluation ; Information fusion ; Linguistic expressions and symbolic translation ; Sentiment analysis ; Ro