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  Global serializability

  In concurrency control of databases, transaction processing (transaction management), and other transactional distributed applications, global serializability (or modular serializability) is a property of a global schedule of transactions. A global schedule is the unified schedule of all the individual database (and other transactional object) schedules in a multidatabase environment (e.g., federated database). Complying with global serializability means that the global schedule is serializable, has the serializability property, while each component database (module) has a serializable schedule as well. In other words, a collection of serializable components provides overall system serializability, which is usually incorrect. A need in correctness across databases in multidatabase systems makes global serializability a major goal for global concurrency control (or modular concurrency control). With the proliferation of the Internet, Cloud computing, Grid computing, and small, portable, powerful computing devices (e.g., smartphones), as well as increase in systems management sophistication, the need for atomic distributed transactions and thus effective global serializability techniques, to ensure correctness in and among distributed transactional applications, seems to increase. In a federated database system or any other more loosely defined multidatabase system, which are typically distributed in a communication network, transactions span multiple (and possibly distributed) databases. Enforcing global serializability in such system, where different databases may use different types of concurrency control, is problematic. Even if every local schedule of a single database is serializable, the global schedule of a whole system is not necessarily serializable. The massive communication exchanges of conflict information needed between databases to reach conflict serializability globally would lead to unacceptable performance, primarily due to computer and communication latency. Achieving global serializability effectively over different types of concurrency control has been open for several years. == The global serializability problem == === Problem statement === The difficulties described above translate into the following problem: Find an efficient (high-performance and fault tolerant) method to enforce Global serializability (global conflict serializability) in a heterogeneous distributed environment of multiple autonomous database systems. The database systems may employ different concurrency control methods. No limitation should be imposed on the operations of either local transactions (confined to a single database system) or global transactions (span two or more database systems). === Quotations === Lack of an appropriate solution for the global serializability problem has driven researchers to look for alternatives to serializability as a correctness criterion in a multidatabase environment (e.g., see Relaxing global serializability below), and the problem has been characterized as difficult and open. The following two quotations demonstrate the mindset about it by the end of the year 1991, with similar quotations in numerous other articles: "Without knowledge about local as well as global transactions, it is highly unlikely that efficient global concurrency control can be provided... Additional complications occur when different component DBMSs [Database Management Systems] and the FDBMSs [Federated Database Management Systems] support different concurrency mechanisms... It is unlikely that a theoretically elegant solution that provides conflict serializability without sacrificing performance (i.e., concurrency and/or response time) and availability exists." === Proposed solutions === Several solutions, some partial, have been proposed for the global serializability problem. Among them: Global conflict graph (serializability graph, precedence graph) checking Distributed Two-phase locking (Distributed 2PL) Distributed Timestamp ordering Tickets (local logical timestamps which define local total orders, and are propagated to determine global partial order of transactions) == Relaxing global serializability == Some techniques have been developed for relaxed global serializability (i.e., they do not guarantee global serializability; see also Relaxing serializability). Among them (with several publications each): Quasi serializability Two-level serializability Another common reason nowadays for Global serializability relaxation is the requirement of availability of internet products and services. This requirement is typically answered by large scale data replication. The straightforward solution for synchronizing replicas' updates of a same database object is including all these updates in a single atomic distributed transaction. However, with many replicas such a transaction is very large, and may span several computers and networks that some of them are likely to be unavailable. Thus such a transaction is likely to end with abort and miss its purpose. Consequently, Optimistic replication (Lazy replication) is often utilized (e.g., in many products and services by Google, Amazon, Yahoo, and alike), while global serializability is relaxed and compromised for eventual consistency. In this case relaxation is done only for applications that are not expected to be harmed by it. Classes of schedules defined by relaxed global serializability properties either contain the global serializability class, or are incomparable with it. What differentiates techniques for relaxed global conflict serializability (RGCSR) properties from those of relaxed conflict serializability (RCSR) properties that are not RGCSR is typically the different way global cycles (span two or more databases) in the global conflict graph are handled. No distinction between global and local cycles exists for RCSR properties that are not RGCSR. RCSR contains RGCSR. Typically RGCSR techniques eliminate local cycles, i.e., provide local serializability (which can be achieved effectively by regular, known concurrency control methods); however, obviously they do not eliminate all global cycles (which would achieve global serializability).

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  Exploratory blockmodeling

  Exploratory blockmodeling is an (inductive) approach (or a group of approaches) in blockmodeling regarding the specification of an ideal blockmodel. This approach, also known as hypotheses-generating, is the simplest approach, as it "merely involves the definition of the block types permitted as well as of the number of clusters." With this approach, researcher usually defines the best possible blockmodel, which then represent the base for the analysis of the whole network. This approach is usually based on: previous analyses and theoretical considerations, using stricker blockmodel and block types, where the structural equivalence is stricker than the regular equivalence and using smaller number of classes. The opposite approach is called a confirmatory blockmodeling.

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  Population model (evolutionary algorithm)

  The population model of an evolutionary algorithm (EA) describes the structural properties of its population to which its members are subject. A population is the set of all proposed solutions of an EA considered in one iteration, which are also called individuals according to the biological role model. The individuals of a population can generate further individuals as offspring with the help of the genetic operators of the procedure. The simplest and widely used population model in EAs is the global or panmictic model, which corresponds to an unstructured population. It allows each individual to choose any other individual of the population as a partner for the production of offspring by crossover, whereby the details of the selection are irrelevant as long as the fitness of the individuals plays a significant role. Due to global mate selection, the genetic information of even slightly better individuals can prevail in a population after a few generations (iteration of an EA), provided that no better other offspring have emerged in this phase. If the solution found in this way is not the optimum sought, that is called premature convergence. This effect can be observed more often in panmictic populations. In nature global mating pools are rarely found. What prevails is a certain and limited isolation due to spatial distance. The resulting local neighbourhoods initially evolve independently and mutants have a higher chance of persisting over several generations. As a result, genotypic diversity in the gene pool is preserved longer than in a panmictic population. It is therefore obvious to divide the previously global population by substructures. Two basic models were introduced for this purpose, the island models, which are based on a division of the population into fixed subpopulations that exchange individuals from time to time, and the neighbourhood models, which assign individuals to overlapping neighbourhoods, also known as cellular genetic or evolutionary algorithms (cGA or cEA). The associated division of the population also suggests a corresponding parallelization of the procedure. For this reason, the topic of population models is also frequently discussed in the literature in connection with the parallelization of EAs. == Island models == In the island model, also called the migration model or coarse grained model, evolution takes place in strictly divided subpopulations. These can be organised panmictically, but do not have to be. From time to time an exchange of individuals takes place, which is called migration. The time between an exchange is called an epoch and its end can be triggered by various criteria: E.g. after a given time or given number of completed generations, or after the occurrence of stagnation. Stagnation can be detected, for example, by the fact that no fitness improvement has occurred in the island for a given number of generations. Island models introduce a variety of new strategy parameters: Number of subpopulations Size of the subpopulations Neighbourhood relations between islands: they determine which islands are considered neighbouring and can thus exchange individuals, see picture of a simple unidirectional ring (black arrows) and its extension by additional bidirectional neighbourhood relations (additional green arrows) Criteria for the termination of an epoch, synchronous or asynchronous migration Migration rate: number or proportion of individuals involved in migration. Migrant selection: There are many alternatives for this. E.g. the best individuals can replace the worst or randomly selected ones. Depending on the migration rate, this can affect one or more individuals at a time. With these parameters, the selection pressure can be influenced to a considerable extent. For example, it increases with the interconnectedness of the islands and decreases with the number of subpopulations or the epoch length. == Neighbourhood models or cellular evolutionary algorithms == The neighbourhood model, also called diffusion model or fine grained model, defines a topological neighbouhood relation between the individuals of a population that is independent of their phenotypic properties. The fundamental idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual that communicates with its nearest neighbours. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as isolation by distance. The set of potential mates of an individual is called its neighbourhood or deme. The adjacent figure illustrates that by showing two slightly overlapping neighbourhoods of two individuals marked yellow, through which genetic information can spread between the two demes. It is known that in this kind of algorithm, similar individuals tend to cluster and create niches that are independent of the deme boundaries and, in particular, can be larger than a deme. There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive ones and maybe merge solution contents during this process. Simultaneously, farther niches can be affected more slowly. EAs with this type of population are also well known as cellular EAs (cEA) or cellular genetic algorithms (cGA). A commonly used structure for arranging the individuals of a population is a 2D toroidal grid, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring, see the figure on the right). The neighbourhood of a particular individual in the grid is defined in terms of the Manhattan distance from it to others in the population. In the basic algorithm, all the neighbourhoods have the same size and identical shapes. The two most commonly used neighbourhoods for two-dimensional cEAs are L5 and C9, see the figure on the left. Here, L stands for Linear while C stands for Compact. Each deme represents a panmictic subpopulation within which mate selection and the acceptance of offspring takes place by replacing the parent. The rules for the acceptance of offspring are local in nature and based on the neighbourhood: for example, it can be specified that the best offspring must be better than the parent being replaced or, less strictly, only better than the worst individual in the deme. The first rule is elitist and creates a higher selective pressure than the second non-elitist rule. In elitist EAs, the best individual of a population always survives. In this respect, they deviate from the biological model. The overlap of the neighbourhoods causes a mostly slow spread of genetic information across the neighbourhood boundaries, hence the name diffusion model. A better offspring now needs more generations than in panmixy to spread in the population. This promotes the emergence of local niches and their local evolution, thus preserving genotypic diversity over a longer period of time. The result is a better and dynamic balance between breadth and depth search adapted to the search space during a run. Depth search takes place in the niches and breadth search in the niche boundaries and through the evolution of the different niches of the whole population. For the same neighbourhood size, the spread of genetic information is larger for elongated figures like L9 than for a block like C9, and again significantly larger than for a ring. This means that ring neighbourhoods are well suited for achieving high quality results, even if this requires comparatively long run times. On the other hand, if one is primarily interested in fast and good, but possibly suboptimal results, 2D topologies are more suitable. == Comparison == When applying both population models to genetic algorithms, evolutionary strategy and other EAs, the splitting of a total population into subpopulations usually reduces the risk of premature convergence and leads to better results overall more reliably and faster than would be expected with panmictic EAs. Island models have the disadvantage compared to neighbourhood models that they introduce a large number of new strategy parameters. Despite the existing studies on this topic in the literature, a certain risk of unfavourable settings remains for the user. With neighbourhood models, on the other hand, only the size of the neighbourhood has to be specified and, in the case of the two-dimensional model, the choice of the neighbourhood figure is added. == Parallelism == Since both population models imply population partitioning, they are well suited as a basis for parallelizing an EA. This applies even more to cellular EAs, since they rely only on locally available information about the members of their respective demes. Thus, in the extreme case, an independent execution thread can be assigned to each individual, so that the entire cEA can run on a parallel hardware platform. The island model also supports p

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  Gremlin (query language)

  Gremlin is a graph traversal language and virtual machine developed by Apache TinkerPop of the Apache Software Foundation. Gremlin works for both OLTP-based graph databases as well as OLAP-based graph processors. Gremlin's automata and functional language foundation enable Gremlin to naturally support imperative and declarative querying, host language agnosticism, user-defined domain specific languages, an extensible compiler/optimizer, single- and multi-machine execution models, and hybrid depth- and breadth-first evaluation with Turing completeness. As an explanatory analogy, Apache TinkerPop and Gremlin are to graph databases what the JDBC and SQL are to relational databases. Likewise, the Gremlin traversal machine is to graph computing as what the Java virtual machine is to general purpose computing. == History == 2009-10-30 the project is born, and immediately named "TinkerPop" 2009-12-25 v0.1 is the first release 2011-05-21 v1.0 is released 2012-05-24 v2.0 is released 2015-01-16 TinkerPop becomes an Apache Incubator project 2015-07-09 v3.0.0-incubating is released 2016-05-23 Apache TinkerPop becomes a top-level project 2016-07-18 v3.1.3 and v3.2.1 are first releases as Apache TinkerPop 2017-12-17 v3.3.1 is released 2018-05-08 v3.3.3 is released 2019-08-05 v3.4.3 is released 2020-02-20 v3.4.6 is released 2021-05-01 v3.5.0 is released 2022-04-04 v3.6.0 is released 2023-07-31 v3.7.0 is released 2025-11-12 v3.8.0 is released == Vendor integration == Gremlin is an Apache2-licensed graph traversal language that can be used by graph system vendors. There are typically two types of graph system vendors: OLTP graph databases and OLAP graph processors. The table below outlines those graph vendors that support Gremlin. == Traversal examples == The following examples of Gremlin queries and responses in a Gremlin-Groovy environment are relative to a graph representation of the MovieLens dataset. The dataset includes users who rate movies. Users each have one occupation, and each movie has one or more categories associated with it. The MovieLens graph schema is detailed below. === Simple traversals === For each vertex in the graph, emit its label, then group and count each distinct label. What year was the oldest movie made? What is Die Hard's average rating? === Projection traversals === For each category, emit a map of its name and the number of movies it represents. For each movie with at least 11 ratings, emit a map of its name and average rating. Sort the maps in decreasing order by their average rating. Emit the first 10 maps (i.e. top 10). === Declarative pattern matching traversals === Gremlin supports declarative graph pattern matching similar to SPARQL. For instance, the following query below uses Gremlin's match()-step. What 80's action movies do 30-something programmers like? Group count the movies by their name and sort the group count map in decreasing order by value. Clip the map to the top 10 and emit the map entries. === OLAP traversal === Which movies are most central in the implicit 5-stars graph? == Gremlin graph traversal machine == Gremlin is a virtual machine composed of an instruction set as well as an execution engine. An analogy is drawn between Gremlin and Java. === Gremlin steps (instruction set) === The following traversal is a Gremlin traversal in the Gremlin-Java8 dialect. The Gremlin language (i.e. the fluent-style of expressing a graph traversal) can be represented in any host language that supports function composition and function nesting. Due to this simple requirement, there exists various Gremlin dialects including Gremlin-Groovy, Gremlin-Scala, Gremlin-Clojure, etc. The above Gremlin-Java8 traversal is ultimately compiled down to a step sequence called a traversal. A string representation of the traversal above provided below. The steps are the primitives of the Gremlin graph traversal machine. They are the parameterized instructions that the machine ultimately executes. The Gremlin instruction set is approximately 30 steps. These steps are sufficient to provide general purpose computing and what is typically required to express the common motifs of any graph traversal query. Given that Gremlin is a language, an instruction set, and a virtual machine, it is possible to design another traversal language that compiles to the Gremlin traversal machine (analogous to how Scala compiles to the JVM). For instance, the popular SPARQL graph pattern match language can be compiled to execute on the Gremlin machine. The following SPARQL query would compile to In Gremlin-Java8, the SPARQL query above would be represented as below and compile to the identical Gremlin step sequence (i.e. traversal). === Gremlin Machine (virtual machine) === The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).

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  Cloud manufacturing

  Cloud manufacturing (CMfg) is a new manufacturing paradigm developed from existing advanced manufacturing models (e.g., ASP, AM, NM, MGrid) and enterprise information technologies under the support of cloud computing, Internet of Things (IoT), virtualization and service-oriented technologies, and advanced computing technologies. It transforms manufacturing resources and manufacturing capabilities into manufacturing services, which can be managed and operated in an intelligent and unified way to enable the full sharing and circulating of manufacturing resources and manufacturing capabilities. CMfg can provide safe and reliable, high quality, cheap and on-demand manufacturing services for the whole lifecycle of manufacturing. The concept of manufacturing here refers to big manufacturing that includes the whole lifecycle of a product (e.g. design, simulation, production, test, maintenance). The concept of Cloud manufacturing was initially proposed by the research group led by Prof. Bo Hu Li and Prof. Lin Zhang in China in 2010. Related discussions and research were conducted hereafter, and some similar definitions (e.g. Cloud-Based Design and Manufacturing (CBDM). ) to cloud manufacturing were introduced. Cloud manufacturing is a type of parallel, networked, and distributed system consisting of an integrated and inter-connected virtualized service pool (manufacturing cloud) of manufacturing resources and capabilities as well as capabilities of intelligent management and on-demand use of services to provide solutions for all kinds of users involved in the whole lifecycle of manufacturing. == Types == Cloud Manufacturing can be divided into two categories. The first category concerns deploying manufacturing software on the Cloud, i.e. a “manufacturing version” of Computing. CAx software can be supplied as a service on the Manufacturing Cloud (MCloud). The second category has a broader scope, cutting across production, management, design and engineering abilities in a manufacturing business. Unlike with computing and data storage, manufacturing involves physical equipment, monitors, materials and so on. In this kind of Cloud Manufacturing system, both material and non-material facilities are implemented on the Manufacturing Cloud to support the whole supply chain. Costly resources are shared on the network. This means that the utilisation rate of rarely used equipment rises and the cost of expensive equipment is reduced. According to the concept of Cloud technology, there will not be direct interaction between Cloud Users and Service Providers. The Cloud User should neither manage nor control the infrastructure and manufacturing applications. As a matter of fact, the former can be considered part of the latter. In CMfg system, various manufacturing resources and abilities can be intelligently sensed and connected into wider Internet, and automatically managed and controlled using IoT technologies (e.g., RFID, wired and wireless sensor network, embedded system). Then the manufacturing resources and abilities are virtualized and encapsulated into different manufacturing cloud services (MCSs), that can be accessed, invoked, and deployed based on knowledge by using virtualization technologies, service-oriented technologies, and cloud computing technologies. The MCSs are classified and aggregated according to specific rules and algorithms, and different kinds of manufacturing clouds are constructed. Different users can search and invoke the qualified MCSs from related manufacturing cloud according to their needs, and assemble them to be a virtual manufacturing environment or solution to complete their manufacturing task involved in the whole life cycle of manufacturing processes under the support of cloud computing, service-oriented technologies, and advanced computing technologies. Four types of cloud deployment modes (public, private, community and hybrid clouds) are ubiquitous as a single point of access. Private cloud refers to a centralized management effort in which manufacturing services are shared within one company or its subsidiaries. Enterprises' mission-critical and core-business applications are often kept in a private cloud. Community cloud is a collaborative effort in which manufacturing services are shared between several organizations from a specific community with common concerns. Public cloud realizes the key concept of sharing services with the general public in a multi-tenant environment. Hybrid cloud is a composition of two or more clouds (private, community or public) that remain distinct entities but are also bound together, offering the benefits of multiple deployment modes. == Resources == From the resource’s perspective, each kind of manufacturing capability requires support from the related manufacturing resource. For each type of manufacturing capability, its related manufacturing resource comes in two forms, soft resources and hard resources. === Soft resources === Software: software applications throughout the product lifecycle including design, analysis, simulation, process planning, and are only beginning to be embraced by the electronics manufacturing industry. Knowledge: experience and know-how needed to complete a production task, i.e. engineering knowledge, product models, standards, evaluation procedures and results, customer feedback, and manufacturing in the cloud provides just as many solutions as the number of questions it also raises for manufacturing executives wanting to make the best possible decision. Skill: expertise in performing a specific manufacturing task. Personnel: human resource engaged in the manufacturing process, i.e. designers, operators, managers, technicians, project teams, customer service, etc. Experience: performance, quality, client evaluation, etc. Business Network: business relationships and business opportunity networks that exist in an enterprise. === Hard resources === Manufacturing Equipment: facilities needed for completing a manufacturing task, e.g. machine tools, cutters, test and monitoring equipment and other fabrication tools. Monitoring/Control Resource: devices used to identify and control other manufacturing resource, for instance, RFID (Radio-Frequency IDentification), WSN (Wireless Sensor Network), virtual managers and remote controllers. Computational Resource: computing devices to support production process, e.g. servers, computers, storage media, control devices, etc. Materials: inputs and outputs in a production system, e.g. raw material, product-in-progress, finished product, power, water, lubricants, etc. Storage: automated storage and retrieval systems, logic controllers, location of warehouses, volume capacity and schedule/optimization methods. Transportation: movement of manufacturing inputs/outputs from one location to another. It includes the modes of transport, e.g. air, rail, road, water, cable, pipeline and space, and the related price, and time taken.

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  Softmax function

  The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes. == Definition == The softmax function takes as input a tuple z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. Formally, the standard (unit) softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where ⁠ K > 1 {\displaystyle K>1} ⁠, takes a tuple z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}} with σ ( z ) i = e z i ∑ j = 1 K e z j . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{z_{i}}}{\sum _{j=1}^{K}e^{z_{j}}}}\,.} In words, the softmax applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input tuple z {\displaystyle \mathbf {z} } (consisting of K {\displaystyle K} real numbers), and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input tuple. For example, the standard softmax of ( 1 , 2 , 8 ) {\displaystyle (1,2,8)} is approximately ( 0.001 , 0.002 , 0.997 ) {\displaystyle (0.001,0.002,0.997)} , which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8). In general, instead of e a different base b > 0 can be used. As above, if b > 1 then larger input components will result in larger output probabilities, and increasing the value of b will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if 0 < b < 1 then smaller input components will result in larger output probabilities, and decreasing the value of b will create probability distributions that are more concentrated around the positions of the smallest input values. Writing b = e β {\displaystyle b=e^{\beta }} or b = e − β {\displaystyle b=e^{-\beta }} (for real β) yields the expressions: σ ( z ) i = e β z i ∑ j = 1 K e β z j or σ ( z ) i = e − β z i ∑ j = 1 K e − β z j for i = 1 , … , K . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{\beta z_{i}}}{\sum _{j=1}^{K}e^{\beta z_{j}}}}{\text{ or }}\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K.} A value proportional to the reciprocal of β is sometimes referred to as the temperature: β = 1 / k T {\textstyle \beta =1/kT} , where k is typically 1 or the Boltzmann constant and T is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, with one value dominating. In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter β (or T) is varied. The softmax function is a multiple-variable generalization of the logistic function. == Interpretations == === Smooth arg max === The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity. Formally, instead of considering the arg max as a function with categorical output 1 , … , n {\displaystyle 1,\dots ,n} (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): a r g m a x ⁡ ( z 1 , … , z n ) = ( y 1 , … , y n ) = ( 0 , … , 0 , 1 , 0 , … , 0 ) , {\displaystyle \operatorname {arg\,max} (z_{1},\,\dots ,\,z_{n})=(y_{1},\,\dots ,\,y_{n})=(0,\,\dots ,\,0,\,1,\,0,\,\dots ,\,0),} where the output coordinate y i = 1 {\displaystyle y_{i}=1} if and only if i {\displaystyle i} is the arg max of ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} , meaning z i {\displaystyle z_{i}} is the unique maximum value of ( z 1 , … , z n ) {\displaystyle (z_{1},\,\dots ,\,z_{n})} . For example, in this encoding a r g m a x ⁡ ( 1 , 5 , 10 ) = ( 0 , 0 , 1 ) , {\displaystyle \operatorname {arg\,max} (1,5,10)=(0,0,1),} since the third argument is the maximum. This can be generalized to multiple arg max values (multiple equal z i {\displaystyle z_{i}} being the maximum) by dividing the 1 between all max args; formally 1/k where k is the number of arguments assuming the maximum. For example, a r g m a x ⁡ ( 1 , 5 , 5 ) = ( 0 , 1 / 2 , 1 / 2 ) , {\displaystyle \operatorname {arg\,max} (1,\,5,\,5)=(0,\,1/2,\,1/2),} since the second and third argument are both the maximum. In case all arguments are equal, this is simply a r g m a x ⁡ ( z , … , z ) = ( 1 / n , … , 1 / n ) . {\displaystyle \operatorname {arg\,max} (z,\dots ,z)=(1/n,\dots ,1/n).} Points z with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points. With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input z as ⁠ β → ∞ {\displaystyle \beta \to \infty } ⁠, σ β ( z ) → a r g m a x ⁡ ( z ) . {\displaystyle \sigma _{\beta }(\mathbf {z} )\to \operatorname {arg\,max} (\mathbf {z} ).} However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, σ β ( 1 , 1.0001 ) → ( 0 , 1 ) , {\displaystyle \sigma _{\beta }(1,\,1.0001)\to (0,1),} but σ β ( 1 , 0.9999 ) → ( 1 , 0 ) , {\displaystyle \sigma _{\beta }(1,\,0.9999)\to (1,\,0),} and σ β ( 1 , 1 ) = 1 / 2 {\displaystyle \sigma _{\beta }(1,\,1)=1/2} for all inputs: the closer the points are to the singular set ( x , x ) {\displaystyle (x,x)} , the slower they converge. However, softargmax does converge compactly on the non-singular set. Conversely, as ⁠ β → − ∞ {\displaystyle \beta \to -\infty } ⁠, softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization". It is also the case that, for any fixed β, if one input ⁠ z i {\displaystyle z_{i}} ⁠ is much larger than the others relative to the temperature, T = 1 / β {\displaystyle T=1/\beta } , the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: σ ( 0 , 10 ) := σ 1 ( 0 , 10 ) = ( 1 / ( 1 + e 10 ) , e 10 / ( 1 + e 10 ) ) ≈ ( 0.00005 , 0.99995 ) {\displaystyle \sigma (0,\,10):=\sigma _{1}(0,\,10)=\left(1/\left(1+e^{10}\right),\,e^{10}/\left(1+e^{10}\right)\right)\approx (0.00005

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  Handwriting recognition

  Handwriting recognition (HWR), also known as handwritten text recognition (HTR), is the ability of a computer to receive and interpret intelligible handwritten input from sources such as paper documents, photographs, touch-screens and other devices. The image of the written text may be sensed "off line" from a piece of paper by optical scanning (optical character recognition) or intelligent word recognition. Alternatively, the movements of the pen tip may be sensed "on line", for example by a pen-based computer screen surface, a generally easier task as there are more clues available. A handwriting recognition system handles formatting, performs correct segmentation into characters, and finds the most possible words. == Offline recognition == Offline handwriting recognition involves the automatic conversion of text in an image into letter codes that are usable within computer and text-processing applications. The data obtained by this form is regarded as a static representation of handwriting. Offline handwriting recognition is comparatively difficult, as different people have different handwriting styles. And, as of today, OCR engines are primarily focused on machine printed text and ICR for hand "printed" (written in capital letters) text. === Traditional techniques === ==== Character extraction ==== Offline character recognition often involves scanning a form or document. This means the individual characters contained in the scanned image will need to be extracted. Tools exist that are capable of performing this step. However, there are several common imperfections in this step. The most common is when characters that are connected are returned as a single sub-image containing both characters. This causes a major problem in the recognition stage. Yet many algorithms are available that reduce the risk of connected characters. ==== Character recognition ==== After individual characters have been extracted, a recognition engine is used to identify the corresponding computer character. Several different recognition techniques are currently available. ===== Feature extraction ===== Feature extraction works in a similar fashion to neural network recognizers. However, programmers must manually determine the properties they feel are important. This approach gives the recognizer more control over the properties used in identification. Yet any system using this approach requires substantially more development time than a neural network because the properties are not learned automatically. === Modern techniques === Where traditional techniques focus on segmenting individual characters for recognition, modern techniques focus on recognizing all the characters in a segmented line of text. Particularly they focus on machine learning techniques that are able to learn visual features, avoiding the limiting feature engineering previously used. State-of-the-art methods use convolutional networks to extract visual features over several overlapping windows of a text line image which a recurrent neural network uses to produce character probabilities. == Online recognition == Online handwriting recognition involves the automatic conversion of text as it is written on a special digitizer or PDA, where a sensor picks up the pen-tip movements as well as pen-up/pen-down switching. This kind of data is known as digital ink and can be regarded as a digital representation of handwriting. The obtained signal is converted into letter codes that are usable within computer and text-processing applications. The elements of an online handwriting recognition interface typically include: a pen or stylus for the user to write with a touch sensitive surface, which may be integrated with, or adjacent to, an output display. a software application which interprets the movements of the stylus across the writing surface, translating the resulting strokes into digital text. The process of online handwriting recognition can be broken down into a few general steps: preprocessing, feature extraction and classification The purpose of preprocessing is to discard irrelevant information in the input data, that can negatively affect the recognition. This concerns speed and accuracy. Preprocessing usually consists of binarization, normalization, sampling, smoothing and denoising. The second step is feature extraction. Out of the two- or higher-dimensional vector field received from the preprocessing algorithms, higher-dimensional data is extracted. The purpose of this step is to highlight important information for the recognition model. This data may include information like pen pressure, velocity or the changes of writing direction. The last big step is classification. In this step, various models are used to map the extracted features to different classes and thus identifying the characters or words the features represent. === Hardware === Commercial products incorporating handwriting recognition as a replacement for keyboard input were introduced in the early 1980s. Examples include handwriting terminals such as the Pencept Penpad and the Inforite point-of-sale terminal. With the advent of the large consumer market for personal computers, several commercial products were introduced to replace the keyboard and mouse on a personal computer with a single pointing/handwriting system, such as those from Pencept, CIC and others. The first commercially available tablet-type portable computer was the Write-Top from Linus Technologies, released in July 1988. Its operating system was based on MS-DOS. In the early 1990s, hardware makers including NCR, IBM and EO released tablet computers running the PenPoint operating system developed by GO Corp. PenPoint used handwriting recognition and gestures throughout and provided the facilities to third-party software. IBM's tablet computer was the first to use the ThinkPad name and used IBM's handwriting recognition. This recognition system was later ported to Microsoft Windows for Pen Computing, and IBM's Pen for OS/2. None of these were commercially successful. Advancements in electronics allowed the computing power necessary for handwriting recognition to fit into a smaller form factor than tablet computers, and handwriting recognition is often used as an input method for hand-held PDAs. The first PDA to provide written input was the Apple Newton, which exposed the public to the advantage of a streamlined user interface. However, the device was not a commercial success, owing to the unreliability of the software, which tried to learn a user's writing patterns. By the time of the release of the Newton OS 2.0, wherein the handwriting recognition was greatly improved, including unique features still not found in current recognition systems such as modeless error correction, the largely negative first impression had been made. After discontinuation of Apple Newton, the feature was incorporated in Mac OS X 10.2 and later as Inkwell. Palm later launched a successful series of PDAs based on the Graffiti recognition system. Graffiti improved usability by defining a set of "unistrokes", or one-stroke forms, for each character. This narrowed the possibility for erroneous input, although memorization of the stroke patterns did increase the learning curve for the user. The Graffiti handwriting recognition was found to infringe on a patent held by Xerox, and Palm replaced Graffiti with a licensed version of the CIC handwriting recognition which, while also supporting unistroke forms, pre-dated the Xerox patent. The court finding of infringement was reversed on appeal, and then reversed again on a later appeal. The parties involved subsequently negotiated a settlement concerning this and other patents. A Tablet PC is a notebook computer with a digitizer tablet and a stylus, which allows a user to handwrite text on the unit's screen. The operating system recognizes the handwriting and converts it into text. Windows Vista and Windows 7 include personalization features that learn a user's writing patterns or vocabulary for English, Japanese, Chinese Traditional, Chinese Simplified and Korean. The features include a "personalization wizard" that prompts for samples of a user's handwriting and uses them to retrain the system for higher accuracy recognition. This system is distinct from the less advanced handwriting recognition system employed in its Windows Mobile OS for PDAs. Although handwriting recognition is an input form that the public has become accustomed to, it has not achieved widespread use in either desktop computers or laptops. It is still generally accepted that keyboard input is both faster and more reliable. As of 2006, many PDAs offer handwriting input, sometimes even accepting natural cursive handwriting, but accuracy is still a problem, and some people still find even a simple on-screen keyboard more efficient. === Software === Early software could understand print handwriting where the characters were separated; however, cursive handwriting

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  FERET (facial recognition technology)

  The Facial Recognition Technology (FERET) program was a government-sponsored project that aimed to create a large, automatic face-recognition system for intelligence, security, and law enforcement purposes. The program began in 1993 under the combined leadership of Dr. Harry Wechsler at George Mason University (GMU) and Dr. Jonathon Phillips at the Army Research Laboratory (ARL) in Adelphi, Maryland and resulted in the development of the Facial Recognition Technology (FERET) database. The goal of the FERET program was to advance the field of face recognition technology by establishing a common database of facial imagery for researchers to use and setting a performance baseline for face-recognition algorithms. Potential areas where this face-recognition technology could be used include: Automated searching of mug books using surveillance photos Controlling access to restricted facilities or equipment Checking the credentials of personnel for background and security clearances Monitoring airports, border crossings, and secure manufacturing facilities for particular individuals Finding and logging multiple appearances of individuals over time in surveillance videos Verifying identities at ATM machines Searching photo ID records for fraud detection The FERET database has been used by more than 460 research groups and is currently managed by the National Institute of Standards and Technology (NIST). By 2017, the FERET database has been used to train artificial intelligence programs and computer vision algorithms to identify and sort faces. == History == The origin of facial recognition technology is largely attributed to Woodrow Wilson Bledsoe and his work in the 1960s, when he developed a system to identify faces from a database of thousands of photographs. The FERET program first began as a way to unify a large body of face-recognition technology research under a standard database. Before the program's inception, most researchers created their own facial imagery database that was attuned to their own specific area of study. These personal databases were small and usually consisted of images from less than 50 individuals. The only notable exceptions were the following: Alex Pentland’s database of around 7500 facial images at the Massachusetts Institute of Technology (MIT) Joseph Wilder's database of around 250 individuals at Rutgers University Christoph von der Malsburg’s database of around 100 facial images at the University of Southern California (USC) The lack of a common database made it difficult to compare the results of face recognition studies in the scientific literature because each report involved different assumptions, scoring methods, and images. Most of the papers that were published did not use images from a common database nor follow a standard testing protocol. As a result, researchers were unable to make informed comparisons between the performances of different face-recognition algorithms. In September 1993, the FERET program was spearheaded by Dr. Harry Wechsler and Dr. Jonathon Phillips under the sponsorship of the U.S. Department of Defense Counterdrug Technology Development Program through DARPA with ARL serving as technical agent. === Phase I === The first facial images for the FERET database were collected from August 1993 to December 1994, a time period known as Phase I. The pictures were initially taken with a 35-mm camera at both GMU and ARL facilities, and the same physical setup was used in each photography session to keep the images consistent. For each individual, the pictures were taken in sets, including two frontal views, a right and left profile, a right and left quarter profile, a right and left half profile, and sometimes at five extra locations. Therefore, a set of images consisted of 5 to 11 images per person. At the end of Phase I, the FERET database had collected 673 sets of images, resulting in over 5000 total images. At the end of Phase I, five organizations were given the opportunity to test their face-recognition algorithm on the newly created FERET database in order to compare how they performed against each other. There five principal investigators were: MIT, led by Alex Pentland Rutgers University, led by Joseph Wilder The Analytic Science Company (TASC), led by Gale Gordon The University of Illinois at Chicago (UIC) and the University of Illinois at Urbana-Champaign, led by Lewis Sadler and Thomas Huang USC, led by Christoph von der Malsburg During this evaluation, three different automatic tests were given to the principal investigators without human intervention: The large gallery test, which served to baseline how algorithms performed against a database when it has not been properly tuned. The false-alarm test, which tested how well the algorithm monitored an airport for suspected terrorists. The rotation test, which measured how well the algorithm performed when the images of an individual in the gallery had different poses compared to those in the probe set. For most of the test trials, the algorithms developed by USC and MIT managed to outperform the other three algorithms for the Phase I evaluation. === Phase II === Phase II began after Phase I, and during this time, the FERET database acquired more sets of facial images. By the start of the Phase II evaluation in March 1995, the database contained 1109 sets of images for a total of 8525 images of 884 individuals. During the second evaluation, the same algorithms from the Phase I evaluation were given a single test. However, the database now contained significantly more duplicate images (463, compared to the previous 60), making the test more challenging. === Phase III === Afterwards, the FERET program entered Phase III where another 456 sets of facial images were added to the database. The Phase III evaluation, which took place in September 1996, aimed to not only gauge the progress of the algorithms since the Phase I assessment but also identify the strengths and weaknesses of each algorithm and determine future objectives for research. By the end of 1996, the FERET database had accumulated a total of 14,126 facial images pertaining to 1199 different individuals as well as 365 duplicate sets of images. As a result of the FERET program, researchers were able to establish a common baseline for comparing different face-recognition algorithms and create a large standard database of facial images that is open for research. In 2003, DARPA released a high-resolution, 24-bit color version of the images in the FERET database (existing reference).

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  PCVC Speech Dataset

  The PCVC (Persian Consonant Vowel Combination) Speech Dataset is a Modern Persian speech corpus for speech recognition and also speaker recognition. The dataset contains sound samples of Modern Persian combination of vowel and consonant phonemes from different speakers. Every sound sample contains just one consonant and one vowel So it is somehow labeled in phoneme level. This dataset consists of 23 Persian consonants and 6 vowels. The sound samples are all possible combinations of vowels and consonants (138 samples for each speaker). The sample rate of all speech samples is 48000 which means there are 48000 sound samples in every 1 second. Every sound sample starts with consonant then continues with vowel. In each sample, in average, 0.5 second of each sample is speech and the rest is silence. Each sound sample ends with silence. All of sound samples are denoised with "Adaptive noise reduction" algorithm. Compared to Farsdat speech dataset and Persian speech corpus it is more easy to use because it is prepared in .mat data files. Also it is more based on phoneme based separation and all samples are denoised. == Contents == The corpus is downloadable from its Kaggle web page, and contains the following: .mat data files of sound samples in a 23630000 matrix, in which 23 is number of consonants, 6 is the number of vowels and 30000 is the length of sound sample.

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  NETtalk (artificial neural network)

  NETtalk is an artificial neural network that learns to pronounce written English text by supervised learning. It takes English text as input, and produces a matching phonetic transcriptions as output. It is the result of research carried out in the mid-1980s by Terrence Sejnowski and Charles Rosenberg. The intent behind NETtalk was to construct simplified models that might shed light on the complexity of learning human level cognitive tasks, and their implementation as a connectionist model that could also learn to perform a comparable task. The authors trained it by backpropagation. The network was trained on a large amount of English words and their corresponding pronunciations, and is able to generate pronunciations for unseen words with a high level of accuracy. The output of the network was a stream of phonemes, which fed into DECtalk to produce audible speech. It achieved popular success, appearing on the Today show. From the point of view of modeling human cognition, NETtalk does not specifically model the image processing stages and letter recognition of the visual cortex. Rather, it assumes that the letters have been pre-classified and recognized. It is NETtalk's task to learn proper associations between the correct pronunciation with a given sequence of letters based on the context in which the letters appear. A similar architecture was subsequently used for the opposite task, that of converting continuous speech signal to a phoneme sequence. == Training == The training dataset was a 20,008-word subset of the Brown Corpus, with manually annotated phoneme and stress for each letter. The development process was described in a 1993 interview. It took three months -- 250 person-hours -- to create the training dataset, but only a few days to train the network. After it was run successfully on this, the authors tried it on a phonological transcription of an interview with a young Latino boy from a barrio in Los Angeles. This resulted in a network that reproduced his Spanish accent. The original NETtalk was implemented on a Ridge 32, which took 0.275 seconds per learning step (one forward and one backward pass). Training NETtalk became a benchmark to test for the efficiency of backpropagation programs. For example, an implementation on Connection Machine-1 (with 16384 processors) ran at 52x speedup. An implementation on a 10-cell Warp ran at 340x speedup. The following table compiles the benchmark scores as of 1988. Speed is measured in "millions of connections per second" (MCPS). For example, the original NETtalk on Ridge 32 took 0.275 seconds per forward-backward pass, giving 18629 / 10 6 0.275 = 0.068 {\displaystyle {\frac {18629/10^{6}}{0.275}}=0.068} MCPS. Relative times are normalized to the MicroVax. == Architecture == The network had three layers and 18,629 adjustable weights, large by the standards of 1986. There were worries that it would overfit the dataset, but it was trained successfully. The input of the network has 203 units, divided into 7 groups of 29 units each. Each group is a one-hot encoding of one character. There are 29 possible characters: 26 letters, comma, period, and word boundary (whitespace). To produce the pronunciation of a single character, the network takes the character itself, as well as 3 characters before and 3 characters after it. The hidden layer has 80 units. The output has 26 units. 21 units encode for articulatory features (point of articulation, voicing, vowel height, etc.) of phonemes, and 5 units encode for stress and syllable boundaries. Sejnowski studied the learned representation in the network, and found that phonemes that sound similar are clustered together in representation space. The output of the network degrades, but remains understandable, when some hidden neurons are removed.

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  Cross-entropy

  In information theory, the cross-entropy between two probability distributions p {\displaystyle p} and q {\displaystyle q} , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribution q {\displaystyle q} , rather than the true distribution p {\displaystyle p} . == Definition == The cross-entropy of the distribution q {\displaystyle q} relative to a distribution p {\displaystyle p} over a given set is defined as follows: H ( p , q ) = − E p ⁡ [ log ⁡ q ] , {\displaystyle H(p,q)=-\operatorname {E} _{p}[\log q],} where E p ⁡ [ ⋅ ] {\displaystyle \operatorname {E} _{p}[\cdot ]} is the expected value operator with respect to the distribution p {\displaystyle p} . The definition may be formulated using the Kullback–Leibler divergence D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , divergence of p {\displaystyle p} from q {\displaystyle q} (also known as the relative entropy of p {\displaystyle p} with respect to q {\displaystyle q} ). H ( p , q ) = H ( p ) + D K L ( p ∥ q ) , {\displaystyle H(p,q)=H(p)+D_{\mathrm {KL} }(p\parallel q),} where H ( p ) {\displaystyle H(p)} is the entropy of p {\displaystyle p} . For discrete probability distributions p {\displaystyle p} and q {\displaystyle q} with the same support X {\displaystyle {\mathcal {X}}} , this means The situation for continuous distributions is analogous. We have to assume that p {\displaystyle p} and q {\displaystyle q} are absolutely continuous with respect to some reference measure r {\displaystyle r} (usually r {\displaystyle r} is a Lebesgue measure on a Borel σ-algebra). Let P {\displaystyle P} and Q {\displaystyle Q} be probability density functions of p {\displaystyle p} and q {\displaystyle q} with respect to r {\displaystyle r} . Then − ∫ X P ( x ) log ⁡ Q ( x ) d x = E p ⁡ [ − log ⁡ Q ] , {\displaystyle -\int _{\mathcal {X}}P(x)\,\log Q(x)\,\mathrm {d} x=\operatorname {E} _{p}[-\log Q],} and therefore NB: The notation H ( p , q ) {\displaystyle H(p,q)} is also used for a different concept, the joint entropy of p {\displaystyle p} and q {\displaystyle q} . == Motivation == In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} can be seen as representing an implicit probability distribution q ( x i ) = ( 1 2 ) ℓ i {\displaystyle q(x_{i})=\left({\frac {1}{2}}\right)^{\ell _{i}}} over { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} , where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, cross-entropy can be interpreted as the expected message-length per datum when a wrong distribution q {\displaystyle q} is assumed while the data actually follows a distribution p {\displaystyle p} . That is why the expectation is taken over the true probability distribution p {\displaystyle p} and not q . {\displaystyle q.} Indeed the expected message-length under the true distribution p {\displaystyle p} is E p ⁡ [ ℓ ] = − E p ⁡ [ ln ⁡ q ( x ) ln ⁡ ( 2 ) ] = − E p ⁡ [ log 2 ⁡ q ( x ) ] = − ∑ x i p ( x i ) log 2 ⁡ q ( x i ) = − ∑ x p ( x ) log 2 ⁡ q ( x ) = H ( p , q ) . {\displaystyle {\begin{aligned}\operatorname {E} _{p}[\ell ]&=-\operatorname {E} _{p}\left[{\frac {\ln {q(x)}}{\ln(2)}}\right]\\[1ex]&=-\operatorname {E} _{p}\left[\log _{2}{q(x)}\right]\\[1ex]&=-\sum _{x_{i}}p(x_{i})\,\log _{2}q(x_{i})\\[1ex]&=-\sum _{x}p(x)\,\log _{2}q(x)=H(p,q).\end{aligned}}} == Estimation == There are many situations where cross-entropy needs to be measured but the distribution of p {\displaystyle p} is unknown. An example is language modeling, where a model is created based on a training set T {\displaystyle T} , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, p {\displaystyle p} is the true distribution of words in any corpus, and q {\displaystyle q} is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula: H ( T , q ) = − ∑ i = 1 N 1 N log 2 ⁡ q ( x i ) {\displaystyle H(T,q)=-\sum _{i=1}^{N}{\frac {1}{N}}\log _{2}q(x_{i})} where N {\displaystyle N} is the size of the test set, and q ( x ) {\displaystyle q(x)} is the probability of event x {\displaystyle x} estimated from the training set. In other words, q ( x i ) {\displaystyle q(x_{i})} is the probability estimate of the model that the i-th word of the text is x i {\displaystyle x_{i}} . The sum is averaged over the N {\displaystyle N} words of the test. This is a Monte Carlo estimate of the true cross-entropy, where the test set is treated as samples from p ( x ) {\displaystyle p(x)} . == Relation to maximum likelihood == The cross entropy arises in classification problems when introducing a logarithm in the guise of the log-likelihood function. This section concerns the estimation of the probabilities of different discrete outcomes. To this end, denote a parametrized family of distributions by q θ {\displaystyle q_{\theta }} , with θ {\displaystyle \theta } subject to the optimization effort. Consider a given finite sequence of N {\displaystyle N} values x i {\displaystyle x_{i}} from a training set, obtained from conditionally independent sampling. The likelihood assigned to any considered parameter θ {\displaystyle \theta } of the model is then given by the product over all probabilities q θ ( X = x i ) {\displaystyle q_{\theta }(X=x_{i})} . Repeated occurrences are possible, leading to equal factors in the product. If the count of occurrences of the value equal to x {\displaystyle x} is denoted by # x {\displaystyle \#x} , then the frequency of that value equals # x / N {\displaystyle \#x/N} . If p ( X = x ) {\displaystyle p(X=x)} is the underlying probability distribution, for large N {\displaystyle N} we expect p ( X = x ) ≈ # x / N {\displaystyle p(X=x)\approx \#x/N} , by the law of large numbers. Writing our likelihood function as the product of observations from the distribution q θ {\displaystyle q_{\theta }} : L ( θ ; x ) = ∏ i q θ ( X = x i ) = ∏ x q θ ( X = x ) # x ≈ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) = exp ⁡ log ⁡ [ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) ] = exp ⁡ ( ∑ x N ⋅ p ( X = x ) log ⁡ q θ ( X = x ) ) , {\displaystyle {\begin{aligned}{\mathcal {L}}(\theta ;{\mathbf {x} })&=\prod _{i}q_{\theta }(X=x_{i})=\prod _{x}q_{\theta }(X=x)^{\#x}\\&\approx \prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}=\exp \log \left[\prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}\right]\\&=\exp \left(\sum _{x}N\cdot p(X=x)\log q_{\theta }(X=x)^{}\right),\end{aligned}}} where we have used the calculation rules for the logarithm in the final line. Notice how the exponent contains a − H ( p , q θ ) {\displaystyle -H(p,q_{\theta })} term. Taking the logarithm of both sides gives: log ⁡ L ( θ ; x ) = − N ⋅ H ( p , q θ ) . {\displaystyle \log {\mathcal {L}}(\theta ;{\mathbf {x} })=-N\cdot H(p,q_{\theta }).} Since the logarithm is a monotonically increasing function, the maximizing value of θ {\displaystyle \theta } is unaffected by this final step. Similarly, the maximizing value of θ {\displaystyle \theta } is unaffected by the factor of N {\displaystyle N} . So we observe that the likelihood maximization amounts to minimization of the cross-entropy. == Cross-entropy minimization == Cross-entropy minimization is frequently used in optimization and rare-event probability estimation. When comparing a distribution q {\displaystyle q} against a fixed reference distribution p {\displaystyle p} , cross-entropy and KL divergence are identical up to an additive constant (since p {\displaystyle p} is fixed): According to the Gibbs' inequality, both take on their minimal values when p = q {\displaystyle p=q} , which is 0 {\displaystyle 0} for KL divergence, and H ( p ) {\displaystyle \mathrm {H} (p)} for cross-entropy. In the engineering literature, the principle of minimizing KL divergence (Kullback's "Principle of Minimum Discrimination Information") is often called the Principle of Minimum Cross-Entropy (MCE), or Minxent. However, as discussed in the article Kullback–Leibler divergence, sometimes the distribution q {\displaystyle q} is the fixed prior reference distribution, and the distribution p {\displaystyle p} is optimized to be as close to q {\displaystyle q} as possible, subject to some constraint. In this case the two minimizations are not equivalent. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by restating cross-entropy to be D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , rather than H (

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  Modes of variation

  In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA). == Formulation == Modes of variation are a natural extension of PCA and FPCA. === Modes of variation in PCA === If a random vector X = ( X 1 , X 2 , ⋯ , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}} has the mean vector μ p {\displaystyle {\boldsymbol {\mu }}_{p}} , and the covariance matrix Σ p × p {\displaystyle \mathbf {\Sigma } _{p\times p}} with eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0} and corresponding orthonormal eigenvectors e 1 , e 2 , ⋯ , e p {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}} , by eigendecomposition of a real symmetric matrix, the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } can be decomposed as Σ = Q Λ Q T , {\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},} where Q {\displaystyle \mathbf {Q} } is an orthogonal matrix whose columns are the eigenvectors of Σ {\displaystyle \mathbf {\Sigma } } , and Λ {\displaystyle \mathbf {\Lambda } } is a diagonal matrix whose entries are the eigenvalues of Σ {\displaystyle \mathbf {\Sigma } } . By the Karhunen–Loève expansion for random vectors, one can express the centered random vector in the eigenbasis X − μ = ∑ k = 1 p ξ k e k , {\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},} where ξ k = e k T ( X − μ ) {\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})} is the principal component associated with the k {\displaystyle k} -th eigenvector e k {\displaystyle \mathbf {e} _{k}} , with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.} Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } is the set of vectors, indexed by α {\displaystyle \alpha } , m k , α = μ ± α λ k e k , α ∈ [ − A , A ] , {\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in [-A,A],} where A {\displaystyle A} is typically selected as 2 or 3 {\displaystyle 2\ {\text{or}}\ 3} . === Modes of variation in FPCA === For a square-integrable random function X ( t ) , t ∈ T ⊂ R p {\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}} , where typically p = 1 {\displaystyle p=1} and T {\displaystyle {\mathcal {T}}} is an interval, denote the mean function by μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),} where λ 1 ≥ λ 2 ≥ ⋯ ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0} are the eigenvalues and { φ 1 , φ 2 , ⋯ } {\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}} are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.} By the Karhunen–Loève theorem, one can express the centered function in the eigenbasis, X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),} where ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt} is the k {\displaystyle k} -th principal component with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.} Then the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} is the set of functions, indexed by α {\displaystyle \alpha } , m k , α ( t ) = μ ( t ) ± α λ k φ k ( t ) , t ∈ T , α ∈ [ − A , A ] {\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in [-A,A]} that are viewed simultaneously over the range of α {\displaystyle \alpha } , usually for A = 2 or 3 {\displaystyle A=2\ {\text{or}}\ 3} . == Estimation == The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. === Modes of variation in PCA === Suppose the data x 1 , x 2 , ⋯ , x n {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}} represent n {\displaystyle n} independent drawings from some p {\displaystyle p} -dimensional population X {\displaystyle \mathbf {X} } with mean vector μ {\displaystyle {\boldsymbol {\mu }}} and covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . These data yield the sample mean vector x ¯ {\displaystyle {\overline {\mathbf {x} }}} , and the sample covariance matrix S {\displaystyle \mathbf {S} } with eigenvalue-eigenvector pairs ( λ ^ 1 , e ^ 1 ) , ( λ ^ 2 , e ^ 2 ) , ⋯ , ( λ ^ p , e ^ p ) {\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})} . Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } can be estimated by m ^ k , α = x ¯ ± α λ ^ k e ^ k , α ∈ [ − A , A ] . {\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in [-A,A].} === Modes of variation in FPCA === Consider n {\displaystyle n} realizations X 1 ( t ) , X 2 ( t ) , ⋯ , X n ( t ) {\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)} of a square-integrable random function X ( t ) , t ∈ T {\displaystyle X(t),t\in {\mathcal {T}}} with the mean function μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} and the covariance function G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))} . Functional principal component analysis provides methods for the estimation of μ ( t ) {\displaystyle \mu (t)} and G ( s , t ) {\displaystyle G(s,t)} in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} can be estimated by m ^ k , α ( t ) = μ ^ ( t ) ± α λ ^ k φ ^ k ( t ) , t ∈ T , α ∈ [ − A , A ] . {\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in [-A,A].} == Applications == Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , {\displaystyle \alpha =\pm 1,\pm 2,} and ± 3 {\displaystyle \pm 3} , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality

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  Neural network Gaussian process

  A Neural Network Gaussian Process (NNGP) is a Gaussian process (GP) obtained as the limit of a certain type of sequence of neural networks. Specifically, a wide variety of network architectures converges to a GP in the infinitely wide limit, in the sense of distribution. The concept constitutes an intensional definition, i.e., a NNGP is just a GP, but distinguished by how it is obtained. == Motivation == Bayesian networks are a modeling tool for assigning probabilities to events, and thereby characterizing the uncertainty in a model's predictions. Deep learning and artificial neural networks are approaches used in machine learning to build computational models which learn from training examples. Bayesian neural networks merge these fields. They are a type of neural network whose parameters and predictions are both probabilistic. While standard neural networks often assign high confidence even to incorrect predictions, Bayesian neural networks can more accurately evaluate how likely their predictions are to be correct. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width. When we consider a sequence of Bayesian neural networks with increasingly wide layers (see figure), they converge in distribution to a NNGP. This large width limit is of practical interest, since the networks often improve as layers get wider. And the process may give a closed form way to evaluate networks. NNGPs also appears in several other contexts: It describes the distribution over predictions made by wide non-Bayesian artificial neural networks after random initialization of their parameters, but before training; it appears as a term in neural tangent kernel prediction equations; it is used in deep information propagation to characterize whether hyperparameters and architectures will be trainable. It is related to other large width limits of neural networks. === Scope === The first correspondence result had been established in the 1995 PhD thesis of Radford M. Neal, then supervised by Geoffrey Hinton at University of Toronto. Neal cites David J. C. MacKay as inspiration, who worked in Bayesian learning. Today the correspondence is proven for: Single hidden layer Bayesian neural networks; deep fully connected networks as the number of units per layer is taken to infinity; convolutional neural networks as the number of channels is taken to infinity; transformer networks as the number of attention heads is taken to infinity; recurrent networks as the number of units is taken to infinity. In fact, this NNGP correspondence holds for almost any architecture: Generally, if an architecture can be expressed solely via matrix multiplication and coordinatewise nonlinearities (i.e., a tensor program), then it has an infinite-width GP. This in particular includes all feedforward or recurrent neural networks composed of multilayer perceptron, recurrent neural networks (e.g., LSTMs, GRUs), (nD or graph) convolution, pooling, skip connection, attention, batch normalization, and/or layer normalization. === Illustration === Every setting of a neural network's parameters θ {\displaystyle \theta } corresponds to a specific function computed by the neural network. A prior distribution p ( θ ) {\displaystyle p(\theta )} over neural network parameters therefore corresponds to a prior distribution over functions computed by the network. As neural networks are made infinitely wide, this distribution over functions converges to a Gaussian process for many architectures. The notation used in this section is the same as the notation used below to derive the correspondence between NNGPs and fully connected networks, and more details can be found there. The figure to the right plots the one-dimensional outputs z L ( ⋅ ; θ ) {\displaystyle z^{L}(\cdot ;\theta )} of a neural network for two inputs x {\displaystyle x} and x ∗ {\displaystyle x^{}} against each other. The black dots show the function computed by the neural network on these inputs for random draws of the parameters from p ( θ ) {\displaystyle p(\theta )} . The red lines are iso-probability contours for the joint distribution over network outputs z L ( x ; θ ) {\displaystyle z^{L}(x;\theta )} and z L ( x ∗ ; θ ) {\displaystyle z^{L}(x^{};\theta )} induced by p ( θ ) {\displaystyle p(\theta )} . This is the distribution in function space corresponding to the distribution p ( θ ) {\displaystyle p(\theta )} in parameter space, and the black dots are samples from this distribution. For infinitely wide neural networks, since the distribution over functions computed by the neural network is a Gaussian process, the joint distribution over network outputs is a multivariate Gaussian for any finite set of network inputs. == Discussion == === Infinitely wide fully connected network === This section expands on the correspondence between infinitely wide neural networks and Gaussian processes for the specific case of a fully connected architecture. It provides a proof sketch outlining why the correspondence holds, and introduces the specific functional form of the NNGP for fully connected networks. The proof sketch closely follows the approach by Novak and coauthors. ==== Network architecture specification ==== Consider a fully connected artificial neural network with inputs x {\displaystyle x} , parameters θ {\displaystyle \theta } consisting of weights W l {\displaystyle W^{l}} and biases b l {\displaystyle b^{l}} for each layer l {\displaystyle l} in the network, pre-activations (pre-nonlinearity) z l {\displaystyle z^{l}} , activations (post-nonlinearity) y l {\displaystyle y^{l}} , pointwise nonlinearity ϕ ( ⋅ ) {\displaystyle \phi (\cdot )} , and layer widths n l {\displaystyle n^{l}} . For simplicity, the width n L + 1 {\displaystyle n^{L+1}} of the readout vector z L {\displaystyle z^{L}} is taken to be 1. The parameters of this network have a prior distribution p ( θ ) {\displaystyle p(\theta )} , which consists of an isotropic Gaussian for each weight and bias, with the variance of the weights scaled inversely with layer width. This network is illustrated in the figure to the right, and described by the following set of equations: x ≡ input y l ( x ) = { x l = 0 ϕ ( z l − 1 ( x ) ) l > 0 z i l ( x ) = ∑ j W i j l y j l ( x ) + b i l W i j l ∼ N ( 0 , σ w 2 n l ) b i l ∼ N ( 0 , σ b 2 ) ϕ ( ⋅ ) ≡ nonlinearity y l ( x ) , z l − 1 ( x ) ∈ R n l × 1 n L + 1 = 1 θ = { W 0 , b 0 , … , W L , b L } {\displaystyle {\begin{aligned}x&\equiv {\text{input}}\\y^{l}(x)&=\left\{{\begin{array}{lcl}x&&l=0\\\phi \left(z^{l-1}(x)\right)&&l>0\end{array}}\right.\\z_{i}^{l}(x)&=\sum _{j}W_{ij}^{l}y_{j}^{l}(x)+b_{i}^{l}\\W_{ij}^{l}&\sim {\mathcal {N}}\left(0,{\frac {\sigma _{w}^{2}}{n^{l}}}\right)\\b_{i}^{l}&\sim {\mathcal {N}}\left(0,\sigma _{b}^{2}\right)\\\phi (\cdot )&\equiv {\text{nonlinearity}}\\y^{l}(x),z^{l-1}(x)&\in \mathbb {R} ^{n^{l}\times 1}\\n^{L+1}&=1\\\theta &=\left\{W^{0},b^{0},\dots ,W^{L},b^{L}\right\}\end{aligned}}} ==== ==== z l | y l {\displaystyle z^{l}|y^{l}} is a Gaussian process We first observe that the pre-activations z l {\displaystyle z^{l}} are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . This result holds even at finite width. Each pre-activation z i l {\displaystyle z_{i}^{l}} is a weighted sum of Gaussian random variables, corresponding to the weights W i j l {\displaystyle W_{ij}^{l}} and biases b i l {\displaystyle b_{i}^{l}} , where the coefficients for each of those Gaussian variables are the preceding activations y j l {\displaystyle y_{j}^{l}} . Because they are a weighted sum of zero-mean Gaussians, the z i l {\displaystyle z_{i}^{l}} are themselves zero-mean Gaussians (conditioned on the coefficients y j l {\displaystyle y_{j}^{l}} ). Since the z l {\displaystyle z^{l}} are jointly Gaussian for any set of y l {\displaystyle y^{l}} , they are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . The covariance or kernel of this Gaussian process depends on the weight and bias variances σ w 2 {\displaystyle \sigma _{w}^{2}} and σ b 2 {\displaystyle \sigma _{b}^{2}} , as well as the second moment matrix K l {\displaystyle K^{l}} of the preceding activations y l {\displaystyle y^{l}} , z i l ∣ y l ∼ G P ( 0 , σ w 2 K l + σ b 2 ) K l ( x , x ′ ) = 1 n l ∑ i y i l ( x ) y i l ( x ′ ) {\displaystyle {\begin{aligned}z_{i}^{l}\mid y^{l}&\sim {\mathcal {GP}}\left(0,\sigma _{w}^{2}K^{l}+\sigma _{b}^{2}\right)\\K^{l}(x,x')&={\frac {1}{n^{l}}}\sum _{i}y_{i}^{l}(x)y_{i}^{l}(x')\end{aligned}}} The effect of the weight scale σ w 2 {\displaystyle \sigma _{w}^{2}} is to rescale the contribution to the covariance matrix from K l {\displaystyle K^{l}} , while the bias is shared for all inputs, and so σ b 2 {\displaystyle \sigma _{b}^{2}} makes the z i l {\displaystyle z_{i}^{l}} for different datapoints more similar and

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  Least-squares support vector machine

  Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods. == From support-vector machine to least-squares support-vector machine == Given a training set { x i , y i } i = 1 N {\displaystyle \{x_{i},y_{i}\}_{i=1}^{N}} with input data x i ∈ R n {\displaystyle x_{i}\in \mathbb {R} ^{n}} and corresponding binary class labels y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} , the SVM classifier, according to Vapnik's original formulation, satisfies the following conditions: { w T ϕ ( x i ) + b ≥ 1 , if y i = + 1 , w T ϕ ( x i ) + b ≤ − 1 , if y i = − 1 , {\displaystyle {\begin{cases}w^{T}\phi (x_{i})+b\geq 1,&{\text{if }}\quad y_{i}=+1,\\w^{T}\phi (x_{i})+b\leq -1,&{\text{if }}\quad y_{i}=-1,\end{cases}}} which is equivalent to y i [ w T ϕ ( x i ) + b ] ≥ 1 , i = 1 , … , N , {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1,\quad i=1,\ldots ,N,} where ϕ ( x ) {\displaystyle \phi (x)} is the nonlinear map from original space to the high- or infinite-dimensional space. === Inseparable data === In case such a separating hyperplane does not exist, we introduce so-called slack variables ξ i {\displaystyle \xi _{i}} such that { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N . {\displaystyle {\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N.\end{cases}}} According to the structural risk minimization principle, the risk bound is minimized by the following minimization problem: min J 1 ( w , ξ ) = 1 2 w T w + c ∑ i = 1 N ξ i , {\displaystyle \min J_{1}(w,\xi )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}\xi _{i},} Subject to { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N , {\displaystyle {\text{Subject to }}{\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N,\end{cases}}} To solve this problem, we could construct the Lagrangian function: L 1 ( w , b , ξ , α , β ) = 1 2 w T w + c ∑ i = 1 N ξ i − ∑ i = 1 N α i { y i [ w T ϕ ( x i ) + b ] − 1 + ξ i } − ∑ i = 1 N β i ξ i , {\displaystyle L_{1}(w,b,\xi ,\alpha ,\beta )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}{\xi _{i}}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{y_{i}\left[{w^{T}\phi (x_{i})+b}\right]-1+\xi _{i}\right\}-\sum \limits _{i=1}^{N}\beta _{i}\xi _{i},} where α i ≥ 0 , β i ≥ 0 ( i = 1 , … , N ) {\displaystyle \alpha _{i}\geq 0,\ \beta _{i}\geq 0\ (i=1,\ldots ,N)} are the Lagrangian multipliers. The optimal point will be in the saddle point of the Lagrangian function, and then we obtain By substituting w {\displaystyle w} by its expression in the Lagrangian formed from the appropriate objective and constraints, we will get the following quadratic programming problem: max Q 1 ( α ) = − 1 2 ∑ i , j = 1 N α i α j y i y j K ( x i , x j ) + ∑ i = 1 N α i , {\displaystyle \max Q_{1}(\alpha )=-{\frac {1}{2}}\sum \limits _{i,j=1}^{N}{\alpha _{i}\alpha _{j}y_{i}y_{j}K(x_{i},x_{j})}+\sum \limits _{i=1}^{N}\alpha _{i},} where K ( x i , x j ) = ⟨ ϕ ( x i ) , ϕ ( x j ) ⟩ {\displaystyle K(x_{i},x_{j})=\left\langle \phi (x_{i}),\phi (x_{j})\right\rangle } is called the kernel function. Solving this QP problem subject to constraints in (1), we will get the hyperplane in the high-dimensional space and hence the classifier in the original space. === Least-squares SVM formulation === The least-squares version of the SVM classifier is obtained by reformulating the minimization problem as min J 2 ( w , b , e ) = μ 2 w T w + ζ 2 ∑ i = 1 N e i 2 , {\displaystyle \min J_{2}(w,b,e)={\frac {\mu }{2}}w^{T}w+{\frac {\zeta }{2}}\sum \limits _{i=1}^{N}e_{i}^{2},} subject to the equality constraints y i [ w T ϕ ( x i ) + b ] = 1 − e i , i = 1 , … , N . {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]=1-e_{i},\quad i=1,\ldots ,N.} The least-squares SVM (LS-SVM) classifier formulation above implicitly corresponds to a regression interpretation with binary targets y i = ± 1 {\displaystyle y_{i}=\pm 1} . Using y i 2 = 1 {\displaystyle y_{i}^{2}=1} , we have ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i e i ) 2 = ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 , {\displaystyle \sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}(y_{i}e_{i})^{2}=\sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2},} with e i = y i − ( w T ϕ ( x i ) + b ) . {\displaystyle e_{i}=y_{i}-(w^{T}\phi (x_{i})+b).} Notice, that this error would also make sense for least-squares data fitting, so that the same end results holds for the regression case. Hence the LS-SVM classifier formulation is equivalent to J 2 ( w , b , e ) = μ E W + ζ E D {\displaystyle J_{2}(w,b,e)=\mu E_{W}+\zeta E_{D}} with E W = 1 2 w T w {\displaystyle E_{W}={\frac {1}{2}}w^{T}w} and E D = 1 2 ∑ i = 1 N e i 2 = 1 2 ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 . {\displaystyle E_{D}={\frac {1}{2}}\sum \limits _{i=1}^{N}e_{i}^{2}={\frac {1}{2}}\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2}.} Both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } should be considered as hyperparameters to tune the amount of regularization versus the sum squared error. The solution does only depend on the ratio γ = ζ / μ {\displaystyle \gamma =\zeta /\mu } , therefore the original formulation uses only γ {\displaystyle \gamma } as tuning parameter. We use both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } as parameters in order to provide a Bayesian interpretation to LS-SVM. The solution of LS-SVM regressor will be obtained after we construct the Lagrangian function: { L 2 ( w , b , e , α ) = J 2 ( w , e ) − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , = 1 2 w T w + γ 2 ∑ i = 1 N e i 2 − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , {\displaystyle {\begin{cases}L_{2}(w,b,e,\alpha )\;=J_{2}(w,e)-\sum \limits _{i=1}^{N}\alpha _{i}\left\{{\left[{w^{T}\phi (x_{i})+b}\right]+e_{i}-y_{i}}\right\},\\\quad \quad \quad \quad \quad \;={\frac {1}{2}}w^{T}w+{\frac {\gamma }{2}}\sum \limits _{i=1}^{N}e_{i}^{2}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{\left[w^{T}\phi (x_{i})+b\right]+e_{i}-y_{i}\right\},\end{cases}}} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } are the Lagrange multipliers. The conditions for optimality are { ∂ L 2 ∂ w = 0 → w = ∑ i = 1 N α i ϕ ( x i ) , ∂ L 2 ∂ b = 0 → ∑ i = 1 N α i = 0 , ∂ L 2 ∂ e i = 0 → α i = γ e i , i = 1 , … , N , ∂ L 2 ∂ α i = 0 → y i = w T ϕ ( x i ) + b + e i , i = 1 , … , N . {\displaystyle {\begin{cases}{\frac {\partial L_{2}}{\partial w}}=0\quad \to \quad w=\sum \limits _{i=1}^{N}\alpha _{i}\phi (x_{i}),\\{\frac {\partial L_{2}}{\partial b}}=0\quad \to \quad \sum \limits _{i=1}^{N}\alpha _{i}=0,\\{\frac {\partial L_{2}}{\partial e_{i}}}=0\quad \to \quad \alpha _{i}=\gamma e_{i},\;i=1,\ldots ,N,\\{\frac {\partial L_{2}}{\partial \alpha _{i}}}=0\quad \to \quad y_{i}=w^{T}\phi (x_{i})+b+e_{i},\,i=1,\ldots ,N.\end{cases}}} Elimination of w {\displaystyle w} and e {\displaystyle e} will yield a linear system instead of a quadratic programming problem: [ 0 1 N T 1 N Ω + γ − 1 I N ] [ b α ] = [ 0 Y ] , {\displaystyle \left[{\begin{matrix}0&1_{N}^{T}\\1_{N}&\Omega +\gamma ^{-1}I_{N}\end{matrix}}\right]\left[{\begin{matrix}b\\\alpha \end{matrix}}\right]=\left[{\begin{matrix}0\\Y\end{matrix}}\right],} with Y = [ y 1 , … , y N ] T {\displaystyle Y=[y_{1},\ldots ,y_{N}]^{T}} , 1 N = [ 1 , … , 1 ] T {\displaystyle 1_{N}=[1,\ldots ,1]^{T}} and α = [ α 1 , … , α N ] T {\displaystyle \alpha =[\alpha _{1},\ldots ,\alpha _{N}]^{T}} . Here, I N {\displaystyle I_{N}} is an N × N {\displaystyle N\times N} identity matrix, and Ω ∈ R N × N {\displaystyle \Omega \in \mathbb {R} ^{N\times N}} is the kernel matrix defined by Ω i j = ϕ ( x i ) T ϕ ( x j ) = K ( x i , x j ) {\displaystyle \Omega _{ij}=\phi (x_{i})^{T}\phi (x_{j})=K(x_{i},x_{j})} . === Kernel function K === For the kernel function K(•, •) one typically has the following choices: Linear kernel : K ( x , x i ) = x i T x , {\displaystyle K(x,x_{i})=x_{i}^{T}x,} Polynomial kernel of degree d {\displaystyle d} : K ( x , x i ) = ( 1 + x i T x / c ) d , {\displaystyle K(x,x_{i})=\left({1+x_{i}^{T}x/c}\right)^{d},} Radial basis function RBF kernel : K ( x , x i ) = exp ⁡ ( − ‖ x − x i ‖ 2 / σ 2 ) , {\displaystyle K(x,x_{i})=\exp \left({-\left\|{x-x_{i}}\right\|^{2}/\sigma ^{2}}\right),} MLP kernel : K ( x , x i ) = tanh ⁡ ( k x i T x + θ ) , {\displaystyle K(x,x_{i})=\tanh \left({k

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  World Programming System

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

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