The LENA Foundation is an American nonprofit organisation which provides tools for measuring children's language acquisition and exposure. Specifically, the LENA system consists of a digital language processor which is worn by a child and records and analyses their auditory environment, using propriety software. It then presents a summary of child-adult conversation, such as conversation turns and word counts. The purpose of the LENA system is to encourage interactive talk between children (between the age of two to forty-eight months) and their caretakers. The LENA system is also used for research; while useful for researchers who wish to save transcription costs or observe the child in its natural state, the accuracy of this system, while often quite high, varies between contexts, for example notably in the case of hard of hearing children. Because of this, several researchers recommend caution in using only the LENA system on its own for the purposes of scientific research. == History == The LENA Foundation was established in 2009 by Terrance and Judith Paul, founders of Renaissance Learning, Inc., with the purpose of aiding children with disabilities and assisting with early learning. They were inspired by the book "Meaningful Differences in the Everyday Experience of American Children" by Dr. Betty Hart and Dr. Todd Risley. A pilot version of the LENA system was launched in February 2006. The LENA Research Foundation was registered as a tax-exempt 501(c)(3) nonprofit in September 2010. The organisation was renamed simply LENA in 2018 and adopted the tagline "Building brains through early talk." LENA has been used for parental feedback, linguistics or paediatrics research, and for specific clinical cases. == Scientific background == In 2018, research using the LENA system showed that there was a link between children's conversational turns and activation of Broca's area (a part of the brain responsible, although not necessarily essential, for language processing). The LENA foundation cites research by its own employees as evidence for the scientific basis of its technology. Said research claims that verbal interaction with young children has an effect on language acquisition, including verbal comprehension skills during adolescence. == LENA System == The LENA software analyses a child's natural language environment, such as verbal exposure, and provides several metrics, such as adult and child speech time, television/recorded audio time, word count, or conversation turn count. The LENA hardware is a recorder that is usually placed into a child's specially-designed vest. The software was trained on over 65,000 hours of manually annotated American English audio recordings. It splits the audio into segments which are categorised as "key child", "other child", "male adult", "noise", etc. The advantages of LENA as opposed to manual transcription are its speed and ease of use; the disadvantages are its potential inaccuracies and lack of transcription capability (which LENA does not profess to attempt). The LENA system has also been criticised for prioritising quantity of speaking over quality (i.e., mastery of the language, as opposed to babble). == Product lines == === LENA Start === LENA Start is a program for parents that utilises feedback from the LENA System in conjunction with weekly group sessions in order to address the home language environment. It was introduced in 2015 and implemented across several U.S. states. In October 2020, during the restrictions of the COVID-19 pandemic, Read Aloud Delaware began a virtual LENA Start program with families statewide, where parents received feedback and participated in one-hour Zoom workshops each week during the 10-week program. === LENA Grow === LENA Grow is a professional development program for teachers in early childhood classrooms. Before launching at sites around the country, the program was first piloted in Escambia County, Florida. === LENA Home === LENA Home is a supplement to existing parent coaching curricula. Typically, home visitors facilitate the use of the LENA System to help parents track their progress towards increasing interactive talk in their homes. === Developmental Snapshot === The LENA Developmental Snapshot, based on a 52-question parent survey, assesses both expressive and receptive language skills and provides an estimate of a child's developmental age from 2 months to 36 months.
Spanner (database)
Spanner is a distributed SQL database management and storage service developed by Google. It provides features such as global transactions, strongly consistent reads, and automatic multi-site replication and failover. Spanner is used in Google F1, the database for its advertising business Google Ads, as well as Gmail and Google Photos. == Features == Spanner stores large amounts of mutable structured data. Spanner allows users to perform arbitrary queries using SQL with relational data while maintaining strong consistency and high availability for that data with synchronous replication. Key features of Spanner: Transactions can be applied across rows, columns, tables, and databases within a Spanner universe. Clients can control the replication and placement of data using automatic multi-site replication and failover. Replication is synchronous and strongly consistent. Reads are strongly consistent and data is versioned to allow for stale reads: clients can read previous versions of data, subject to garbage collection windows. Supports a native SQL interface for reading and writing data. Support for Graph Query Language == History == Spanner was first described in 2012 for internal Google data centers. Spanner's SQL capability was added in 2017 and documented in a SIGMOD 2017 paper. It became available as part of Google Cloud Platform in 2017, under the name "Cloud Spanner". == Architecture == Spanner uses the Paxos algorithm as part of its operation to shard (partition) data across up to hundreds of servers. It makes heavy use of hardware-assisted clock synchronization using GPS clocks and atomic clocks to ensure global consistency. TrueTime is the brand name for Google's distributed cloud infrastructure, which provides Spanner with the ability to generate monotonically increasing timestamps in data centers around the world. Google's F1 SQL database management system (DBMS) is built on top of Spanner, replacing Google's custom MySQL variant.
Vowpal Wabbit
Vowpal Wabbit (VW) is an open-source fast online interactive machine learning system library and program developed originally at Yahoo! Research, and currently at Microsoft Research. It was started and is led by John Langford. Vowpal Wabbit's interactive learning support is particularly notable including Contextual Bandits, Active Learning, and forms of guided Reinforcement Learning. Vowpal Wabbit provides an efficient scalable out-of-core implementation with support for a number of machine learning reductions, importance weighting, and a selection of different loss functions and optimization algorithms. == Notable features == The VW program supports: Multiple supervised (and semi-supervised) learning problems: Classification (both binary and multi-class) Regression Active learning (partially labeled data) for both regression and classification Multiple learning algorithms (model-types / representations) OLS regression Matrix factorization (sparse matrix SVD) Single layer neural net (with user specified hidden layer node count) Searn (Search and Learn) Latent Dirichlet Allocation (LDA) Stagewise polynomial approximation Recommend top-K out of N One-against-all (OAA) and cost-sensitive OAA reduction for multi-class Weighted all pairs Contextual-bandit (with multiple exploration/exploitation strategies) Multiple loss functions: squared error quantile hinge logistic poisson Multiple optimization algorithms Stochastic gradient descent (SGD) BFGS Conjugate gradient Regularization (L1 norm, L2 norm, & elastic net regularization) Flexible input - input features may be: Binary Numerical Categorical (via flexible feature-naming and the hash trick) Can deal with missing values/sparse-features Other features On the fly generation of feature interactions (quadratic and cubic) On the fly generation of N-grams with optional skips (useful for word/language data-sets) Automatic test-set holdout and early termination on multiple passes bootstrapping User settable online learning progress report + auditing of the model Hyperparameter optimization == Scalability == Vowpal wabbit has been used to learn a tera-feature (1012) data-set on 1000 nodes in one hour. Its scalability is aided by several factors: Out-of-core online learning: no need to load all data into memory The hashing trick: feature identities are converted to a weight index via a hash (uses 32-bit MurmurHash3) Exploiting multi-core CPUs: parsing of input and learning are done in separate threads. Compiled C++ code
Evolutionary multimodal optimization
In applied mathematics, multimodal optimization deals with optimization tasks that involve finding all or most of the multiple (at least locally optimal) solutions of a problem, as opposed to a single best solution. Evolutionary multimodal optimization is a branch of evolutionary computation, which is closely related to machine learning. Wong provides a short survey, wherein the chapter of Shir and the book of Preuss cover the topic in more detail. == Motivation == Knowledge of multiple solutions to an optimization task is especially helpful in engineering, when due to physical (and/or cost) constraints, the best results may not always be realizable. In such a scenario, if multiple solutions (locally and/or globally optimal) are known, the implementation can be quickly switched to another solution and still obtain the best possible system performance. Multiple solutions could also be analyzed to discover hidden properties (or relationships) of the underlying optimization problem, which makes them important for obtaining domain knowledge. In addition, the algorithms for multimodal optimization usually not only locate multiple optima in a single run, but also preserve their population diversity, resulting in their global optimization ability on multimodal functions. Moreover, the techniques for multimodal optimization are usually borrowed as diversity maintenance techniques to other problems. == Background == Classical techniques of optimization would need multiple restart points and multiple runs in the hope that a different solution may be discovered every run, with no guarantee however. Evolutionary algorithms (EAs) due to their population based approach, provide a natural advantage over classical optimization techniques. They maintain a population of possible solutions, which are processed every generation, and if the multiple solutions can be preserved over all these generations, then at termination of the algorithm we will have multiple good solutions, rather than only the best solution. Note that this is against the natural tendency of classical optimization techniques, which will always converge to the best solution, or a sub-optimal solution (in a rugged, “badly behaving” function). Finding and maintenance of multiple solutions is wherein lies the challenge of using EAs for multi-modal optimization. Niching is a generic term referred to as the technique of finding and preserving multiple stable niches, or favorable parts of the solution space possibly around multiple solutions, so as to prevent convergence to a single solution. The field of Evolutionary algorithms encompasses genetic algorithms (GAs), evolution strategy (ES), differential evolution (DE), particle swarm optimization (PSO), and other methods. Attempts have been made to solve multi-modal optimization in all these realms and most, if not all the various methods implement niching in some form or the other. == Multimodal optimization using genetic algorithms/evolution strategies == De Jong's crowding method, Goldberg's sharing function approach, Petrowski's clearing method, restricted mating, maintaining multiple subpopulations are some of the popular approaches that have been proposed by the community. The first two methods are especially well studied, however, they do not perform explicit separation into solutions belonging to different basins of attraction. The application of multimodal optimization within ES was not explicit for many years, and has been explored only recently. A niching framework utilizing derandomized ES was introduced by Shir, proposing the CMA-ES as a niching optimizer for the first time. The underpinning of that framework was the selection of a peak individual per subpopulation in each generation, followed by its sampling to produce the consecutive dispersion of search-points. The biological analogy of this machinery is an alpha-male winning all the imposed competitions and dominating thereafter its ecological niche, which then obtains all the sexual resources therein to generate its offspring. Recently, an evolutionary multiobjective optimization (EMO) approach was proposed, in which a suitable second objective is added to the originally single objective multimodal optimization problem, so that the multiple solutions form a weak pareto-optimal front. Hence, the multimodal optimization problem can be solved for its multiple solutions using an EMO algorithm. Improving upon their work, the same authors have made their algorithm self-adaptive, thus eliminating the need for pre-specifying the parameters. An approach that does not use any radius for separating the population into subpopulations (or species) but employs the space topology instead is proposed in.
(1+ε)-approximate nearest neighbor search
(1+ε)-approximate nearest neighbor search is a variant of the nearest neighbor search problem. A solution to the (1+ε)-approximate nearest neighbor search is a point or multiple points within distance (1+ε) R from a query point, where R is the distance between the query point and its true nearest neighbor. Reasons to approximate nearest neighbor search include the space and time costs of exact solutions in high-dimensional spaces (see curse of dimensionality) and that in some domains, finding an approximate nearest neighbor is an acceptable solution. Approaches for solving (1+ε)-approximate nearest neighbor search include k-d trees, locality-sensitive hashing and brute-force search.
Cloud computing
Cloud computing is defined by the International Organization for Standardization (ISO) as "a paradigm for enabling network access to a scalable and elastic pool of shareable physical or virtual resources with self-service provisioning and administration on demand". It is commonly referred to as "the cloud". == Characteristics == In 2011, the National Institute of Standards and Technology (NIST) identified five "essential characteristics" for cloud systems. Below are the exact definitions according to NIST: On-demand self-service: "A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automatically without requiring human interaction with each service provider." Broad network access: "Capabilities are available over the network and accessed through standard mechanisms that promote use by heterogeneous thin or thick client platforms (e.g., mobile phones, tablets, laptops, and workstations)." Resource pooling: " The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer demand." Rapid elasticity: "Capabilities can be elastically provisioned and released, in some cases automatically, to scale rapidly outward and inward commensurate with demand. To the consumer, the capabilities available for provisioning often appear unlimited and can be appropriated in any quantity at any time." Measured service: "Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency for both the provider and consumer of the utilized service. By 2023, the International Organization for Standardization (ISO) had expanded and refined the list. == History == The history of cloud computing extends to the 1960s, with the initial concepts of time-sharing becoming popularized via remote job entry (RJE). The "data center" model, where users submitted jobs to operators to run on mainframes, was predominantly used during this era. This period saw broad experimentation with making large-scale computing power more accessible through time-sharing, while optimizing infrastructure, platforms, and applications to improve efficiency for end users. The "cloud" metaphor for virtualized services dates to 1994, when it was used by General Magic for the universe of "places" that mobile agents in the Telescript environment could "go". The metaphor is credited to David Hoffman, a General Magic communications specialist, based on its long-standing use in networking and telecom. The expression cloud computing became more widely known in 1996 when Compaq Computer Corporation drew up a business plan for future computing and the Internet. The company's ambition was to supercharge sales with "cloud computing-enabled applications". The business plan foresaw that online consumer file storage would likely be commercially successful. As a result, Compaq decided to sell server hardware to internet service providers. In the 2000s, the application of cloud computing began to take shape with the establishment of Amazon Web Services (AWS) in 2002, which allowed developers to build applications independently. In 2006 Amazon Simple Storage Service, known as Amazon S3, and the Amazon Elastic Compute Cloud (EC2) were released. In 2008 NASA's development of the first open-source software for deploying private and hybrid clouds. The following decade saw the launch of various cloud services. In 2010, Microsoft launched Microsoft Azure, and Rackspace Hosting and NASA initiated an open-source cloud-software project, OpenStack. IBM introduced the IBM SmartCloud framework in 2011, and Oracle announced the Oracle Cloud in 2012. In December 2019, Amazon launched AWS Outposts, a service that extends AWS infrastructure, services, APIs, and tools to customer data centers, co-location spaces, or on-premises facilities. == Value proposition == Cloud computing can shorten time to market by offering pre-configured tools, scalable resources, and managed services, allowing users to focus on core business value rather than maintaining infrastructure. Cloud platforms can enable organizations and individuals to reduce upfront capital expenditures on physical infrastructure by shifting to an operational expenditure model, where costs scale with usage. Cloud platforms also offer managed services and tools, such as artificial intelligence, data analytics, and machine learning, which might otherwise require significant in-house expertise and infrastructure investment. While cloud computing can offer cost advantages through effective resource optimization, organizations often face challenges such as unused resources, inefficient configurations, and hidden costs without proper oversight and governance. Many cloud platforms provide cost management tools, such as AWS Cost Explorer and Azure Cost Management, and frameworks like FinOps have emerged to standardize financial operations in the cloud. Cloud computing also facilitates collaboration, remote work, and global service delivery by enabling secure access to data and applications from any location with an internet connection. Cloud providers offer various redundancy options for core services, such as managed storage and managed databases, though redundancy configurations often vary by service tier. Advanced redundancy strategies, such as cross-region replication or failover systems, typically require explicit configuration and may incur additional costs or licensing fees. Cloud environments operate under a shared responsibility model, where providers are typically responsible for infrastructure security, physical hardware, and software updates, while customers are accountable for data encryption, identity and access management (IAM), and application-level security. These responsibilities vary depending on the cloud service model—Infrastructure as a Service (IaaS), Platform as a Service (PaaS), or Software as a Service (SaaS)—with customers typically having more control and responsibility in IaaS environments and progressively less in PaaS and SaaS models, often trading control for convenience and managed services. == Adoption and suitability == The decision to adopt cloud computing or maintain on-premises infrastructure depends on factors such as scalability, cost structure, latency requirements, regulatory constraints, and infrastructure customization. Organizations with variable or unpredictable workloads, limited capital for upfront investments, or a focus on rapid scalability benefit from cloud adoption. Startups, SaaS companies, and e-commerce platforms often prefer the pay-as-you-go operational expenditure (OpEx) model of cloud infrastructure. Additionally, companies prioritizing global accessibility, remote workforce enablement, disaster recovery, and leveraging advanced services such as AI/ML and analytics are well-suited for the cloud. In recent years, some cloud providers have started offering specialized services for high-performance computing and low-latency applications, addressing some use cases previously exclusive to on-premises setups. On the other hand, organizations with strict regulatory requirements, highly predictable workloads, or reliance on deeply integrated legacy systems may find cloud infrastructure less suitable. Businesses in industries like defense, government, or those handling highly sensitive data often favor on-premises setups for greater control and data sovereignty. Additionally, companies with ultra-low latency requirements, such as high-frequency trading (HFT) firms, rely on custom hardware (e.g., FPGAs) and physical proximity to exchanges, which most cloud providers cannot fully replicate despite recent advancements. Similarly, tech giants like Google, Meta, and Amazon build their own data centers due to economies of scale, predictable workloads, and the ability to customize hardware and network infrastructure for optimal efficiency. However, these companies also use cloud services selectively for certain workloads and applications where it aligns with their operational needs. In practice, many organizations are increasingly adopting hybrid cloud architectures, combining on-premises infrastructure with cloud services. This approach allows businesses to balance scalability, cost-effectiveness, and control, offering the benefits of both deployment models while mitigating their respective limitations. == Challenges and limitations == One of the primary challenges of cloud computing, compared with traditional on-premises systems, is maintaining data security and privacy. Cloud users entrust their sensitive data to third-party providers, who may not have adequate measures to protect it from unau
Radial basis function network
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. == Network architecture == Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . The output of the network is then a scalar function of the input vector, φ : R n → R {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by φ ( x ) = ∑ i = 1 N a i ρ ( | | x − c i | | ) {\displaystyle \varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)} where N {\displaystyle N} is the number of neurons in the hidden layer, c i {\displaystyle \mathbf {c} _{i}} is the center vector for neuron i {\displaystyle i} , and a i {\displaystyle a_{i}} is the weight of neuron i {\displaystyle i} in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian ρ ( ‖ x − c i ‖ ) = exp [ − β i ‖ x − c i ‖ 2 ] {\displaystyle \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]} . The Gaussian basis functions are local to the center vector in the sense that lim | | x | | → ∞ ρ ( ‖ x − c i ‖ ) = 0 {\displaystyle \lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0} i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. The parameters a i {\displaystyle a_{i}} , c i {\displaystyle \mathbf {c} _{i}} , and β i {\displaystyle \beta _{i}} are determined in a manner that optimizes the fit between φ {\displaystyle \varphi } and the data. === Normalization === ==== Normalized architecture ==== In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is φ ( x ) = d e f ∑ i = 1 N a i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N a i u ( ‖ x − c i ‖ ) {\displaystyle \varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} where u ( ‖ x − c i ‖ ) = d e f ρ ( ‖ x − c i ‖ ) ∑ j = 1 N ρ ( ‖ x − c j ‖ ) {\displaystyle u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}} is known as a normalized radial basis function. ==== Theoretical motivation for normalization ==== There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density P ( x ∧ y ) = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) σ ( | y − e i | ) {\displaystyle P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}} where the weights c i {\displaystyle \mathbf {c} _{i}} and e i {\displaystyle e_{i}} are exemplars from the data and we require the kernels to be normalized ∫ ρ ( ‖ x − c i ‖ ) d n x = 1 {\displaystyle \int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1} and ∫ σ ( | y − e i | ) d y = 1 {\displaystyle \int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1} . The probability densities in the input and output spaces are P ( x ) = ∫ P ( x ∧ y ) d y = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) {\displaystyle P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and The expectation of y given an input x {\displaystyle \mathbf {x} } is φ ( x ) = d e f E ( y ∣ x ) = ∫ y P ( y ∣ x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy} where P ( y ∣ x ) {\displaystyle P\left(y\mid \mathbf {x} \right)} is the conditional probability of y given x {\displaystyle \mathbf {x} } . The conditional probability is related to the joint probability through Bayes' theorem P ( y ∣ x ) = P ( x ∧ y ) P ( x ) {\displaystyle P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}} which yields φ ( x ) = ∫ y P ( x ∧ y ) P ( x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy} . This becomes φ ( x ) = ∑ i = 1 N e i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N e i u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} when the integrations are performed. === Local linear models === It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order, φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) ρ ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} in the unnormalized and normalized cases, respectively. Here b i {\displaystyle \mathbf {b} _{i}} are weights to be determined. Higher order linear terms are also possible. This result can be written φ ( x ) = ∑ i = 1 2 N ∑ j = 1 n e i j v i j ( x − c i ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}} where e i j = { a i , if i ∈ [ 1 , N ] b i j , if i ∈ [ N + 1 , 2 N ] {\displaystyle e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} and v i j ( x − c i ) = d e f { δ i j ρ ( ‖ x − c i ‖ ) , if i ∈ [ 1 , N ] ( x i j − c i j ) ρ ( ‖ x − c i ‖ ) , if i ∈ [ N + 1 , 2 N ] {\displaystyle v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} in the unnormalized case and in the normalized case. Here δ i j {\displaystyle \delta _{ij}} is a Kronecker delta function defined as δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}} . == Training == RBF networks are typically trained from pairs of input and target values x ( t ) , y ( t ) {\displaystyle \mathbf {x} (t),y(t)} , t = 1 , … , T {\displaystyle t=1,\dots ,T} by a two-step algorithm. In the first step, the center vectors c i {\displaystyle \mathbf {c} _{i}} of the RBF functions in the hidden layer