AI Grammar Tagalog

AI Grammar Tagalog — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Metadata repository

A metadata repository is a database created to store metadata. Metadata is information about the structures that contain the actual data. Metadata is often said to be "data about data", but this is misleading. Data profiles are an example of actual "data about data". Metadata adds one layer of abstraction to this definition– it is data about the structures that contain data. Metadata may describe the structure of any data, of any subject, stored in any format. A well-designed metadata repository typically contains data far beyond simple definitions of the various data structures. Typical repositories store dozens to hundreds of separate pieces of information about each data structure. Comparing the metadata of a couple data items - one digital and one physical - clarify what metadata is: First, digital: For data stored in a database one may have a table called "Patient" with many columns, each containing data which describes a different attribute of each patient. One of these columns may be named "Patient_Last_Name". What is some of the metadata about the column that contains the actual surnames of patients in the database? We have already used two items: the name of the column that contains the data (Patient_Last_Name) and the name of the table that contains the column (Patient). Other metadata might include the maximum length of last name that may be entered, whether or not last name is required (can we have a patient without Patient_Last_Name?), and whether the database converts any surnames entered in lower case to upper case. Metadata of a security nature may show the restrictions which limit who may view these names. Second, physical: For data stored in a brick and mortar library, one have many volumes and may have various media, including books. Metadata about books would include ISBN, Binding_Type, Page_Count, Author, etc. Within Binding_Type, metadata would include possible bindings, material, etc. This contextual information of business data include meaning and content, policies that govern, technical attributes, specifications that transform, and programs that manipulate. == Definition == The metadata repository is responsible for physically storing and cataloging metadata. Data in a metadata repository should be generic, integrated, current, and historical: Generic Meta model should store the metadata by generic terms instead of storing it by an applications-specific defined way, so that if your data base standard changes from one product to another the physical meta model of the metadata repository would not need to change. Integration of the metadata repository allows all business areas' metadata to be in an integrated fashion: Covering all domains and subject areas of the organization. current and historical The metadata repository should have accessible current and historical metadata. Metadata repositories used to be referred to as a data dictionary. With the transition of needs for the metadata usage for business intelligence has increased so is the scope of the metadata repository increased. Earlier data dictionaries are the closest place to interact technology with business. Data dictionaries are the universe of metadata repository in the initial stages but as the scope increased Business glossary and their tags to variety of status flags emerged in the business side while consumption of the technology metadata, their lineage and linkages made the repository, the source for valuable reports to bring business and technology together and helped data management decisions easier as well as assess the cost of the changes. Metadata repository explores the enterprise wide data governance, data quality and master data management (includes master data and reference data) and integrates this wealth of information with integrated metadata across the organization to provide decision support system for data structures, even though it only reflects the structures consumed from various systems. == Repository vs. registry == Repository has additional functionalities compared with registry. Metadata repository not only stores metadata like Metadata registry but also adds relationships with related metadata types. Metadata when related in a flow from its point of entry into organization up to the deliverables is considered as the lineage of that data point. Metadata when related across other related metadata types is called linkages. By providing the relationships to all the metadata points across the organization and maintaining its integrity with an architecture to handle the changes, metadata repository provides the basic material for understanding the complete data flow and their definitions and their impact. Also the important feature is to maintain the version control though this statement for contrasting is open for discussion. These definitions are still evolving, so the accuracy of the definitions needs refinement. The purpose of registry is to define the metadata element and maintained across the organization. And data models and other data management teams refer to the registry for any changes to follow. While Metadata repository sources metadata from various metadata systems in the organizations and reflects what is in the upstream. Repository never acts as an upstream while registry is used as an upstream for metadata changes. == Reason for use == Metadata repository enables all the structure of the organizations data containers to one integrated place. This opens plethora of resourceful information for making calculated business decisions. This tool uses one generic form of data model to integrate all the models thus brings all the applications and programs of the organization into one format. And on top of it applying the business definitions and business processes brings the business and technology closer that will help organizations make reliable roadmaps with definite goals. With one stop information, business will have more control on the changes, and can do impact analysis of the tool. Usually business spends much time and money to make decisions based on discovery and research on impacts to make changes or to add new data structures or remove structures in data management of the organization. With a structured and well maintained repository, moving the product from ideation to delivery takes the least amount of time (considering other variables are constant). To sum it up: Integration of the metadata across the organization Build relationship between various metadata types Build relationship between various disparate systems Define business golden copy of definitions Version control of the changes at structure level Interaction with Reference data Link view to master data Automatic synchronization with various authorized metadata source systems More control to business decisions Validate the structures by overlapping the models Discovering discrepancies, gaps, lineage, metrics at data structure level Each database management system (DBMS) and database tools have their own language for the metadata components within. Database applications already have their own repositories or registries that are expected to provide all of the necessary functionality to access the data stored within. Vendors do not want other companies to be capable of easily migrating data away from their products and into competitors products, so they are proprietary with the way they handle metadata. CASE tools, DBMS dictionaries, ETL tools, data cleansing tools, OLAP tools, and data mining tools all handle and store metadata differently. Only a metadata repository can be designed to store the metadata components from all of these tools. == Design == Metadata repositories should store metadata in four classifications: ownership, descriptive characteristics, rules and policies, and physical characteristics. Ownership, showing the data owner and the application owner. The descriptive characteristics, define the names, types and lengths, and definitions describing business data or business processes. Rules and policies, will define security, data cleanliness, timelines for data, and relationships. Physical characteristics define the origin or source, and physical location. Like building a logical data model for creating a database, a logical meta model can help identify the metadata requirements for business data. The metadata repository will be centralized, decentralized, or distributed. A centralized design means that there is one database for the metadata repository that stores metadata for all applications business wide. A centralized metadata repository has the same advantages and disadvantages of a centralized database. Easier to manage because all the data is in one database, but the disadvantage is that bottlenecks may occur. A decentralized metadata repository stores metadata in multiple databases, either separated by location and or departments of the business. This makes management of the repository more involved than a centraliz
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Friendica

Friendica (formerly Friendika, originally Mistpark) is a free and open-source software distributed social network. It forms one part of the Fediverse, an interconnected and decentralized network of independently operated servers. == Features == Friendica users can connect with others via their own Friendica server, but may also fully integrate contacts from other platforms including Diaspora, Pump.io, GNU social, email, Discourse and more recently ActivityPub (including Mastodon, Pleroma and Pixelfed) and Bluesky into their 'newsfeed'. In addition to these two way connections, users can also use Friendica as a publishing platform to post content to WordPress, Tumblr, Insanejournal and Libertree. Posting to Google+ was also supported until that service was shut down. In addition, RSS feeds can be ingested. Because users are distributed across many servers, their "addresses" consist of a username, the "@" symbol, and the domain name of the Friendica instance in the same manner email addresses are formed. Twitter support was available but was deprecated due to API changes under Elon Musk's leadership rendering it unusable. Most of the functionality from major microblogging and social networking platforms are available in Friendica; for example, tagging users and groups via "@ mentions"; direct messages; hashtags; photo albums; "likes"; "dislikes"; comments; and re-shares of publicly visible posts. Published items can be edited and updated across the network. Comprehensive settings for privacy and the public visibility of posts allow users to regulate who can read which contributions, or see specific information about the user. Users can also create multiple profiles, allowing different groups of people (such as friends, or work mates) to see a different profile entirely when viewing the same page. User accounts can be downloaded or deleted, and can be imported to a different Friendica server if so required. Public forums can be created under different accounts, which can be switched between if the accounts are registered with the same email address. == Development == There is no corporation behind Friendica. The developers work on a voluntary basis and the project is run informally; the platform itself is used for the communication between the developers. There are different forums within Friendica, such as "Friendica Developers" and "Friendica Support". The source code of Friendica is hosted on GitHub. == Installation == The developers aim to make installation of the software as simple as possible for technical laymen. They argue that decentralization on small servers is a key condition for the freedom of users and their self-determination. The difficulty level is similar to an installation of WordPress. However, the installing on shared hosting is sometimes difficult because of missing PHP5 modules. Some volunteers also run public servers so that newcomers can also avoid the installation of their own software. == List of clients == Friendica implements multiple client-server API variants simultaneously. Along with endpoints needed to use enhanced Friendica features, it also implements the API used by GNU social, Twitter and since version 2021.06 also the one used by Mastodon. As a result, most GNU social and Mastodon clients can be used for Friendica. Examples of Friendica compatible clients include: Raccoon for Friendica, Friendiqa, Fedilab, AndStatus, Twidere and DiCa for Android, friendly for Sailfish OS, friclicli (CLI client), choqok and Friendiqa for Linux and Friendica Mobile for Windows 10. == Reception == Friendica was cited in January 2012 by Infoshop News as an "alternative to Google+ and Facebook" to be used on the Occupy Nigeria movement. In January 2012 Free Software Foundation Europe's blog cited Friendica as a reasonable alternative to centralized and controlled social networks such as Facebook or Google+. Biblical Notes writer J. Randal Matheny described Friendica in January 2012 as "One social networking option flying under the radar until recently deserves consideration as an already stable platform with a wide range of options, applications, plug-ins, and possibilities for opening up the Internet." In February 2012, the German computer magazine c't wrote: "Friendica demonstrates how decentralized social networks can become widely accepted." Another German publication, the professional magazine t3n listed Friendica as a Facebook rival in an online article in March 2012 about Facebook alternatives. It compared Friendica with similar social networks like Diaspora and identi.ca. MSN Tech & Gadgets contributor Emma Boyes wrote about Friendica in May 2012: "why you'll love it: you can use it to access all the other social networks and get recommendations of new friends and groups to join. Friendica is open source and decentralised. There's no corporation behind it and there are extensive privacy settings. You can choose from a variety of user interfaces and it boasts some cool features—for instance, being able to key in a list of your interests and use the 'profile match' feature to recommend other users who share them with you. A word of warning, though, the site is not as user-friendly as the others on this list, so it may be this one is one for the geeks." == Later reviews == Acquisition of Twitter by Elon Musk had revitalized public interest in Fediverse technologies in April 2022. Friendica received favorable reviews, with a PCMag article describing it as "mostly comparable to Facebook", drawing a parallel to Google+ and highlighting using it "for planning events, and its multiple profile feature means you can show a different face to your friends, coworkers, and family". The September 2022 issue of Linux Magazine contains a detailed comparison and walk-through of registering to and using basic functions of Diaspora, Friendica and Mastodon. They describe Friendica as "intuitive" and highlight the "huge choice of account settings" and that "Friendica does not require any specific hardware, so you can use an old computer system as a server." == Vulnerabilities == In September 2020, a hotfix was released to patch a security vulnerability that could leak sensitive information from the server environment since versions released in April 2019 (develop branch) and June 2019 (stable).
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The Outliner of Giants

The Outliner of Giants was commercial outlining software. Like other outliners, it allowed the user to create a document consisting of a series of nested lists. It was one of a number of browser-based outliners that are delivered as a web application, used through a web browser, rather than being installed as a stand-alone application. The Outliner of Giants was released in 2009. The service was shut down on December 31, 2017 and only exports are allowed at this time. == Feature set == Unlike most other browser-based outliners - which often focus on providing a minimum viable product - the Outliner of Giants had much of the functionality typically associated with a desktop outliner, such as the ability to use of columns to structure information. However, The Outliner of Giants did not support offline editing, requiring an active internet connection in order to make changes to an outline document. === Outlining === Like all outliners, The Outliner of Giants supported the creation of a hierarchy of items, with users modifying the parent-child relationship between items in order to structure a document. This included the ability to promote or demote items up or down the hierarchy, or move an item up or down a list of siblings on the same level. The Outliner of Giants did not support the true cloning of items (where an item can appear to be in multiple places within the hierarchy at the same time), although it did support the copying of single or multiple nodes. === Import === The Outliner of Giants could import both plain text and the OPML XML format, which is commonly used to transfer data between outlining applications. === Editing === Outline documents could be edited using a WYSIWYG editor, as well as the Markdown, and Textile markup languages. === Annotation === The Outliner of Giants supported functions to annotate an outline, such as the ability to add colored labels, highlights and text, as well as tags and hashtags. === Collaboration === The Outliner of Giants supported real-time collaboration, where multiple users could edit the same document, and can see the changes made by another user as they happened. === Publication === Outlines created through The Outliner of Giants could be published directly online through the service, either as outlines, pages or in a blog format. === Export === The Outliner of Giants can export outline data as plain text, HTML, as well as directly to the Google Docs word processor.
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MySocialCloud

MySocialCloud is a cloud-based bookmark vault and password website that allows users to log into all of their online accounts from a single, secure website. The company's investors include Sir Richard Branson, Insight Venture Partners’ Jerry Murdock, and PhotoBucket founder Alex Welch. The company and its founders have been featured in TechCrunch and The Huffington Post. == History == MySocialCloud was co-founded by Scott Ferreira, Stacey Ferreira, and Shiv Prakash in 2011. The idea for a one-stop password storage and login tool came when a computer crash left Scott without documents he used to store access information to his online data. In 2013, the siblings sold MySocialCloud to Reputation.com. == Services == MySocialCloud is cloud-based, and the platform lets users securely store passwords and automatically log into several social media websites simultaneously. The website auto-populates password fields, letting the user log into all of the sites at the push of a button. The service also provides users with security updates for the websites they have included in their profile, and informs users if a website has been hacked. Security played a major role during development of the platform. Passwords stored on the service are salted and hashed with a two-way encryption method known as AES.
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Azure Maps

Azure Maps is a suite of cloud-based, location-based services provided by Microsoft as part of the company's Azure platform. The platform provides geospatial and location-based services via REST APIs and software development kits (SDKs). The service is typically used to integrate maps or geospatial data into applications. Azure Maps differs from Microsoft's other enterprise mapping service, Bing Maps, in its pricing model, focus on privacy, and its level of integration into the broader Azure cloud ecosystem. == History == Azure Maps was first introduced in public preview mode under the name "Azure Location Based Services" in 2017, primarily as an enterprise solution. The services was intended to add mapping and location-based functionality onto the existing Azure cloud services suite, seen as a critical part of Microsoft's broader Internet-of-Things (IoT) strategy. The preview version included APIs which could be used to develop location aware apps for use cases such as logistics and mobility. In 2018, the software was renamed "Azure Maps," and became generally available to the public, and a number of new functions were added, including route calculation, travel time calculation, and incorporation of real-time traffic data and incident information. Azure Maps was integrated with Azure IoT Central in 2018, which added tracking, monitoring, and geofencing capabilities. A set of mobility APIs on were added in 2019, with applications such as use in public transport apps and shared bicycle fleet management. “Azure Maps Creator,” which converts private facility floor plans into indoor map data, was also introduced in 2019. Some commentators linked these services to Microsoft's broader development of augmented reality products. In 2020, Azure Maps Visual for Power BI was released, integrating location-based features and mapping capabilities into Microsoft's business intelligence software. An elevation API (which was later retired), geolocation services, and an iOS and Android software development kit were introduced in 2021. In 2022, support for historical weather, air quality, and tropical storm data was made generally available and custom styling for indoor maps was also introduced. In 2023, Azure Maps was certified as HIPAA compliant in a move to target healthcare and health insurance companies. == Functionality == === Geocoding === Geocoding is one of the core functionalities of Azure Maps, converting addresses or place names into geographic coordinates. Batch geocoding is used to process large amounts of address data, a function used for route optimization and spatial analysis. === Reverse geocoding === Reverse geocoding derives human-readable information from geographic coordinates like longitude and latitude, used in navigation and by geographic information systems. === Routing === Azure Maps uses map data and routing algorithms to calculate the shortest or fastest routes between locations based on factors like vehicle size and type, traffic conditions, and distance. Routing also supports multi-modal routing, which include multiple modes of transport in a single trip, including cycling, walking, and ferries. This functionality is used for location-based searches and route optimization in applications like fleet management, proximity marketing, and emergency services as well as logistics and delivery, urban planning, ride sharing apps, and outdoor activities. === Map visualization === The platform supports map visualizations that can be modified to reflect real-time data (including from IoT sensors) as well as historical data patterns. Visualizations include heat maps, street maps, satellite imagery and other custom data layers. Maps are rendered using raster or vector tiles which reduce the load of displaying large data sets or complex maps. This can be used in various applications in areas like transportation, smart cities, retail and marketing, public health, and environmental monitoring. For example, it can be used for tracking the spread of diseases or measuring the impact of changing climatic patterns. === Geofencing and spatial analytics === Azure Maps supports polygonal geofencing, which enables the definition of custom geographic boundaries. Geofenced areas can be monitored in real-time for events of interest. For example, an application could send an alert when equipment or persons enter or leave a defined area. Tools for analyzing historical geofencing data are also available via the APIs for optimization purposes. == Industry usage == Azure Maps' geofencing function has seen usage in the construction industry, designating hazardous areas for safety purposes and sending alerts if anyone enters the area. Private facility maps are used by construction companies for monitoring large construction sites to increase productivity and prevent accidents or damage. In emergency management, New Zealand based company Beca has used Azure Maps to provide analysis on the impact of earthquakes to users, including information on the severity and location of an earthquake and the impact on affected properties. Alaska's Department of Transportation uses Azure Maps as part of an information system providing weather-related warnings and analytics to road crews. Airmap, an airspace management platform for drones, uses Azure Maps. Azure Maps has also been used in conjunction with Azure Monitor for risk monitoring by an insurance company. Other companies that use or have used Azure Maps include BMW, Banco Santander, Jvion, MV Transportation, C.H. Robertson, Wise Skulls, Tata Consultancy Services, Providence Health and Services, Gas Brasiliano Distribuidora S.A., Shell plc, Persistent Systems, Phase 2 Dining and Entertainment, Symbio, HID, Globant, and Insight Enterprises. == Partnerships == Azure Maps and TomTom have been partners since 2016, and TomTom provides location data to Azure Maps and can process data from Azure Maps for mapping purposes. In 2021, Azure Maps partnered with AccuWeather to make climatic data available via its APIs, making weather data along all parts of calculated routes available for mobility and logistics purposes. Microsoft has partnered with Esri, the developer of ArcGIS, and there is cross-compatibility between Azure and ArcGIS so that data from Azure Maps can be integrated into ArcGIS and vice versa. Azure Maps partnered with Moovit in 2019, a startup providing software that interfaces with public transport data. Moovit's database on global public transit networks, including information on which stations and facilities are wheelchair accessible, was linked to Azure Maps. This service was noted for its use increasing accessibility to public transport for the visually impaired by means of voice activated route planning assistance. NORAD has used some Azure Maps functions for their NORAD Tracks Santa website during Christmas holidays. == Components == === REST APIs === Various APIs cover the major functionalities across Azure Maps: Data registry API Geolocation API Render API Route API Search API Spatial API Time zone API Traffic API Weather API === SDKs === Azure Maps SDKs uses MapLibre-style specifications and open source MapLibre GL-based libraries as a rendering engine. The Web SDK is used for developing web apps with maps and location-based data and functionality. It includes a map control module as well as modules with drawing tools. It also supports Azure Maps Creator and various spatial data formats. The platform also includes a set of REST SDKs for developers integrating Azure Maps REST APIs into Python, C#, Java or JavaScript applications. Azure Maps also includes Android and iOS SDKs used for developing applications for Android and Apple devices. === Azure Maps Creator === Azure Maps Creator is a tool for generating custom maps for locations like large office complexes, construction sites, or university campuses. These maps can then be integrated into applications and used with other Azure Maps functions for purposes such as wayfinding and maintenance and security in building automation contexts. === Azure Maps Visual for Power BI === Azure Maps is integrated with Microsoft Power BI, a graphical tool for producing data visualizations. Since July 2020, Power BI can be used in conjunction with Azure Maps for developing map-based data visualizations. This functionality entered general availability in May 2023.
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Closest point method

The closest point method (CPM) is an embedding method for solving partial differential equations on surfaces. The closest point method uses standard numerical approaches such as finite differences, finite element or spectral methods in order to solve the embedding partial differential equation (PDE) which is equal to the original PDE on the surface. The solution is computed in a band surrounding the surface in order to be computationally efficient. In order to extend the data off the surface, the closest point method uses a closest point representation. This representation extends function values to be constant along directions normal to the surface. == Definitions == Closest Point function: Given a surface S , c p ( x ) {\displaystyle {\mathcal {S}},cp(\mathbf {x} )} refers to a (possibly non-unique) point belonging to S {\displaystyle {\mathcal {S}}} , which is closest to x {\displaystyle \mathbf {x} } [SE]. Closest point extension: Let S {\displaystyle {\mathcal {S}}} , be a smooth surface in R d {\displaystyle \mathbb {R} ^{d}} . The closest point extension of a function u : S → R {\displaystyle u:{\mathcal {S}}\rightarrow \mathbb {R} } , to a neighborhood Ω {\displaystyle \Omega } of S {\displaystyle {\mathcal {S}}} , is the function v : Ω → R {\displaystyle v:\Omega \rightarrow \mathbb {R} } , defined by v ( x ) = u ( c p ( x ) ) {\displaystyle v(\mathbf {x} )=u(cp(\mathbf {x} ))} . == Closest point method == Initialization consists of these steps [EW]: If it is not already given, a closest point representation of the surface is constructed. A computational domain is chosen. Typically this is a band around the surface. Replace surface gradients by standard gradients in R 3 {\displaystyle \mathbb {R} ^{3}} . Solution is initialized by extending the initial surface data on to the computational domain using the closest point function. After initialization, alternate between the following two steps: Using the closest point function, extend the solution off the surface to the computational domain. Compute the solution to the embedding PDE on a Cartesian mesh in the computational domain for one time step. == Banding == The surface PDE is extended into R 3 {\displaystyle \mathbb {R} ^{3}} however it is only necessary to solve this new PDE near the surface. Hence, we solve the PDE in a band surrounding the surface for efficient computational purposes. Ω c x : ‖ x − c p ( x ) ‖ 2 ≤ λ {\displaystyle \Omega _{c}{x:\|x-cp(x)\|_{2}\leq \lambda }} where λ {\displaystyle \lambda } is the bandwidth. == Example: Heat equation on a circle == Using initial profile u S ( θ , t ) = sin ⁡ ( θ ) {\displaystyle u_{S}(\theta ,t)=\sin(\theta )} leads to the solution u S ( θ , t ) = exp ⁡ ( − t ) sin ⁡ ( θ ) {\displaystyle u_{S}(\theta ,t)=\exp(-t)\sin(\theta )} for the heat equation. Forward Euler time-stepping is used with relation Δ t = 0.1 Δ x 2 {\displaystyle \Delta t=0.1\Delta x^{2}} and degree-four interpolation polynomials for the interpolations. Second-order centered differences are used for the spatial discretization. The CPM results in the expected second order error in the solution u {\displaystyle u} . == Applications == The closest point method can be applied to various PDEs on surfaces. Reaction–diffusion problems on point clouds [RD], eigenvalue problems [EV], and level set equations [LS] are a few examples.
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List of Fortran software and tools

This is a list of Fortran software and tools, including IDEs, compilers, libraries, debugging tools, numerical and scientific computing tools, and related projects. == Fortran compilers == Absoft Pro Fortran — Absoft Pro Fortran is discontinued and ran on Linux and macOS AOCC — from AMD Classic Flang — part of the LLVM Project LLVM Flang — part of the LLVM Project Fortran 77 — Fortran 77 was developed by Digital Equipment Corporation, it is discontinued. G95 – portable open-source Fortran 95 compiler GCC (GNU Fortran) PGI compilers – NVIDIA developed compilers after acquiring The Portland Group IBM XL Fortran — IBM XL Fortran is current and runs on Linux (Power/AIX) and integrates with Eclipse Intel Fortran Compiler – part of Intel OneAPI HPC toolkit LFortran — LFortran is current, cross-platform, and has IDE support. MinGW – cross compiler and forked into Mingw-w64 nAG Fortran Compiler - from nAG Open64 — Open64 is an open-source compiler that has been terminated and ran on Linux Open Watcom — Open Watcom is current, runs on MS-DOS and OS/2, and has IDE support. Oracle Fortran — Oracle Fortran is discontinued, ran on Linux and Solaris. ROSE — source-to-source compiler framework developed at Lawrence Livermore National Laboratory Silverfrost FTN95 — FTN95 from Silverfrost is current, runs on Windows, and has IDE support. == Integrated development environments (IDEs) and editors == Code::Blocks — supports Fortran with plugins Eclipse IDE — with Fortran support via Photran Emacs — extensible text editor with built-in Fortran modes and support for modern tooling via language servers Geany — lightweight cross-platform IDE based on GTK IntelliJ IDEA — cross-platform IDE by JetBrains with Fortran pluggin KDevelop — KDE-based IDE NetBeans — Apache software foundation IDE with Fortran configuration OpenWatcom — IDE and compiler suite for C, C++, and Fortran Simply Fortran — standalone Fortran IDE for Windows, Linux, and macOS Vim — modal text editor with native Fortran syntax support and extensive plugin-based development features Visual Studio — with Intel Fortran integration Visual Studio Code — supports Fortran via extensions == Mathematical libraries == == Scientific libraries == ABINIT — software suite to calculate optical, mechanical, vibrational, and other observable properties of materials Cantera — chemical kinetics, thermodynamics, and transport tool suite CERN Program Library — collection of Fortran libraries for physics applications from CERN CP2K — quantum chemistry and solid-state physics software package for atomistic simulations Dalton — molecular electronic structure program FFTPACK — subroutines for the fast Fourier transform Kinetic PreProcessor – open-source software tool used in atmospheric chemistry MESA — Modules for Experiments in Stellar Astrophysics Nek5000 — MPI parallel higher-order spectral element CFD solver NWChem — open-source high-performance computational chemistry software Octopus — real-space Time-Dependent Density Functional Theory code MODTRAN – model atmospheric propagation of electromagnetic radiation MOLCAS — quantum chemistry software package for multiconfigurational electronic structure calculations NOVAS – software library for astrometry-related numerical computations Physics Analysis Workstation – data analysis and graphical presentation in high-energy physics Quantum ESPRESSO — integrated suite for electronic-structure calculations and materials modeling SIESTA — first-principles materials simulation code using density functional theory Tinker — software tools for molecular design == Debugging and performance tools == GDB — GNU Debugger with Fortran support Valgrind — memory debugging and profiling tool VTune Profiler — performance analysis tool Allinea Forge — debugger and profiler for HPC applications == Build and package management == Autotools — build system supporting Fortran projects CMake — cross-platform build system supporting Fortran Make — build automation tool Spack — package manager for HPC software including Fortran libraries == Machine learning and AI libraries == Athena Fiats (Functional Inference And Training for Surrogates) FNN (Fortran Neural Network) FortNN Fortran-TF-lib (Fortran interface to TensorFlow) FTorch (Fortran interface to PyTorch) MlFortran RoseNNa == Parallel and high-performance computing tools == MPI Fortran bindings — standard interface for distributed-memory parallelism OpenMP — shared-memory parallel programming support through compiler directives Coarray Fortran — parallel programming model introduced in Fortran 2008 ScaLAPACK — parallel linear algebra package built on top of LAPACK == Testing frameworks == FUnit — open-source unit testing framework developed at NASA’s Langley Research Center, for Fortran 90, 95, and 2003. pFUnit — unit testing framework for Fortran, modeled after JUnit == Documentation and code analysis tools == FORD — automatic documentation generator for modern Fortran projects SQuORE — software quality and management platform with code analysis support Understand — static analysis and code comprehension tool for large Fortran projects
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Comparison gallery of image scaling algorithms

This gallery shows the results of numerous image scaling algorithms. == Scaling methods == An image size can be changed in several ways. Consider resizing a 160x160 pixel photo to the following 40x40 pixel thumbnail and then scaling the thumbnail to a 160x160 pixel image. Also consider doubling the size of the following image containing text. == Examples of enlarged images == Below are examples of various images enlarged 4x using each scaling algorithm.
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Standard test image

A standard test image is a digital image file used across different institutions to test image processing and image compression algorithms. By using the same standard test images, different labs are able to compare results, both visually and quantitatively. The images are in many cases chosen to represent natural or typical images that a class of processing techniques would need to deal with. Other test images are chosen because they present a range of challenges to image reconstruction algorithms, such as the reproduction of fine detail and textures, sharp transitions and edges, and uniform regions. == Historical origins == Test images as transmission system calibration material probably date back to the original Paris to Lyon pantelegraph link. Analogue fax equipment (and photographic equipment for the printing trade) were the largest user groups of the standardized image for calibration technology until the coming of television and digital image transmission systems. == Common test image resolutions == The standard resolution of the images is usually 512×512 or 720×576. Most of these images are available as TIFF files from the University of Southern California's Signal and Image Processing Institute. Kodak has released 768×512 images, available as PNGs, that was originally on Photo CD with higher resolution, that are widely used for comparing image compression techniques.
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Pandas (software)

Pandas (styled as pandas) is a software library written for the Python programming language for data manipulation and analysis. In particular, it offers data structures and operations for manipulating numerical tables and time series. It is free software released under the three-clause BSD license. The name is derived from the term "panel data", an econometrics term for data sets that include observations over multiple time periods for the same individuals, as well as a play on the phrase "Python data analysis". Wes McKinney started building what would become Pandas at AQR Capital while he was a researcher there from 2007 to 2010. The development of Pandas introduced into Python many comparable features of working with DataFrames that were established in the R programming language. The library is built upon another library, NumPy. == History == Developer Wes McKinney started working on Pandas in 2008 while at AQR Capital Management out of the need for a high performance, flexible tool to perform quantitative analysis on financial data. Before leaving AQR, he was able to convince management to allow him to open source the library in 2009. Another AQR employee, Chang She, joined the effort in 2012 as the second major contributor to the library. In 2015, Pandas signed on as a fiscally sponsored project of NumFOCUS, a 501(c)(3) nonprofit charity in the United States. == Data model == Pandas is built around data structures called Series and DataFrames. Data for these collections can be imported from various file formats such as comma-separated values, JSON, Parquet, SQL database tables or queries, and Microsoft Excel. === Series === A Series is a one-dimensional array-like object that stores a sequence of values together with an associated set of labels, called an index. It is built on top of NumPy's array and affords many similar functionalities, but instead of using implicit integer positions, a Series allows explicit index labels of many data types. A Series can be created from Python lists, dictionaries, or NumPy arrays. If no index is provided, pandas automatically assigns a default integer index ranging from 0 to n-1, where n is the number of items in the Series. A simple example with customized labels is: To access a value or list of values from a Series, use its index or list of indices: Series can be used arithmetically, as in the statement series_3 = series_1 + series_2. This will align data points with corresponding index values in series_1 and series_2 (similar to a join in relational algebra), then add them together to produce new values in series_3. A Series has various attributes, such as name (Series name), dtype (data type of values), shape (number of rows), values, and index. They can be used in many of the same operations as NumPy arrays, with additional methods for reindexing, label-based selection, and handling missing data. === DataFrame === A DataFrame is a two-dimensional, tabular data structure with labeled rows and columns. Each column is stored internally as a Series and may hold a different data type (numeric, string, boolean, etc.). DataFrames can be created by a variety of means, including dictionaries of lists, NumPy arrays, and external files such as CSV or Excel spreadsheets: To retrieve a DataFrame column as a Series, use either 1) the index (dict-like notation) or 2) the name of column if the name is a valid Python identifier (attribute-like access). DataFrames support operations such as column assignment, row and column deletion, label-based indexing with loc, position-based indexing with iloc, reshaping, grouping, and joining. Merge operations implement a subset of relational algebra and allow one-to-one, many-to-one, and many-to-many joins. Some common attributes of a DataFrame include dtypes (data type of each column), shape (dimensions of the DataFrame returned as a tuple with form (number of rows, number of columns)), index/columns (labels of the DataFrame's rows/columns, respectively, returned as an Index object), values (data in the DataFrame returned as a 2D array), and empty (returns True if the DataFrame is empty). === Index === Index objects hold metadata for Series and Dataframe objects, such as axis labels and names, and are automatically created from input data. By default, a pandas index is a series of integers ascending from 0, similar to the indices of Python arrays. However, indices can also use any NumPy data type, including floating point, timestamps, or strings. Indices are also immutable, which allows them to be safely shared across multiple objects. pandas' syntax for mapping index values to relevant data is the same syntax Python uses to map dictionary keys to values. For example, if s is a Series, s['a'] will return the data point at index a. Unlike dictionary keys, index values are not guaranteed to be unique. If a Series uses the index value a for multiple data points, then s['a'] will instead return a new Series containing all matching values. A DataFrame's column names are stored and implemented identically to an index. As such, a DataFrame can be thought of as having two indices: one column-based and one row-based. Because column names are stored as an index, these are not required to be unique. If data is a Series, then data['a'] returns all values with the index value of a. However, if data is a DataFrame, then data['a'] returns all values in the column(s) named a. To avoid this ambiguity, Pandas supports the syntax data.loc['a'] as an alternative way to filter using the index. Pandas also supports the syntax data.iloc[n], which always takes an integer n and returns the nth value, counting from 0. This allows a user to act as though the index is an array-like sequence of integers, regardless of how it is actually defined. pandas also supports hierarchical indices with multiple values per data point through the "MultiIndex" class. MultiIndex objects allow a single DataFrame to represent multiple dimensions, similar to a pivot table in Microsoft Excel, where each level can optionally carry its own unique name. In practice, data with more than 2 dimensions is often represented using DataFrames with hierarchical indices, instead of the higher-dimension Panel and Panel4D data structures. == Functionality == pandas supports a variety of indexing and subsetting techniques, allowing data to be selected by label, index, or Boolean conditions. For example, df[df['col1'] > 5] will return all rows in the DataFrame df for which the value of the column col1 exceeds 5. The library also implements grouping operations based on the split-apply-combine approach, enabling users to aggregate, transform, or restructure data according to column values or functions applied to index labels. For example, df['col1'].groupby(df['col2']) groups the data in 'col1' by their values in 'col2', while df.groupby(lambda i: i % 2) groups all data in the whole DataFrame by whether their index is even. The library also provides extensive tools for transforming, filtering and summarizing data. Users may apply arbitrary functions to Series and DataFrames, and because the library is built on top of Numpy, most NumPy functions can be applied directly to pandas objects as well. The library also includes built-in operations for arithmetic operations, string processing, and descriptive statistics such as mean, median, and standard deviation. These built-in functions are designed to handle missing data, usually represented by the floating-point value NaN. In addition, pandas includes tools for reorganizing data into different structural formats, with methods that can reshape tabular data between "wide" and "long" formats and pivot values based on column labels. pandas also implements a flexible set of relational operations for combining datasets. For instance, merge() links row in DataFrames based on one or more shared keys or indices, supporting one-to-one, one-to-many, and many-to-many relationships in a manner analogous to join operations in relational databases like SQL. DataFrames can also be concatenated or stacked together along an axis through the concat() method, and overlapping data can be further spliced together using combine_first() to fill in missing values. Furthermore, the library includes specialized support for working with time-series data. Features include the ability to interpolate values and filter using a range of timestamps, such as data['1/1/2023':'2/2/2023'] , which will return all dates between January 1 and February 2. Missing values in time-series data are represented by a dedicated NaT (Not a Timestamp) object, instead of the NaN value it uses elsewhere. == Criticisms == Pandas has been criticized for its inefficiency. The entire dataset must be loaded in RAM, and the library does not optimize query plans or support parallel computing across multiple cores. Wes McKinney, the creator of Pandas, has recommended Apache Arrow as an alternative to address these performance concerns and ot
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Clara.io

Clara.io is web-based freemium 3D computer graphics software developed by Exocortex, a Canadian software company. The free or "Basic" component of their freemium offering, however, places severe restrictions, such as on saving models and importing texture maps, which are undisclosed in the company's own descriptions of their plans.vf TMN == History == Clara.io was announced in July 2013, and first presented as part of the official SIGGRAPH 2013 program later that month. By November 2013, when the open beta period started, Clara.io had 14,000 registered users. Clara.io claimed to have 26,000 registered users in January 2014, which grew to 85,000 by December 2014. Clara.io was permanently shut down on December 31, 2022, but the site is currently still partially functional to logged-in users. == Features == Polygonal modeling Constructive solid geometry Key frame animation Skeletal animation Hierarchical scene graph Texture mapping Photorealistic rendering (streaming cloud rendering using V-Ray Cloud) Scene publishing via HTML iframe embedding FBX, Collada, OBJ, STL and Three.js import/export Collaborative real-time editing Revision control (versioning & history) Scripting, Plugins & REST APIs 3D model library Unlisted and Private scenes (paid subscriptions only). == Technology == Clara.io is developed using HTML5, JavaScript, WebGL and Three.js. Clara.io does not rely on any browser plugins and thus runs on any platform that has a modern standards compliant browser. == Screenshots ==
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Princh

Princh is a Danish software company, which is headquartered in Aarhus, Denmark. Founded in 2015, Princh develops cloud printing and electronic payment products. The company is headquartered in the city of Aarhus. While utilizing a smartphone or web app, users can locate a nearby printer to their current location, get directions to access said printer, and/or authorize a print and pay for the print job in question. The product is available as a native mobile apps for Android and iOS, as well as on web and desktop products for businesses and libraries. The app connects a network of printer owners and users around the world. Princh supports an array of printable files. == History == The company was founded in 2015. The company is currently based in the southern part of Aarhus. The Princh printing service was officially launched on June 23, 2015. Currently, Princh is available as a service in a multitude of locations such as print shops, libraries, hotels, or universities. Princh is a popular printing and payment product among libraries and can among other places be found in Denmark, Sweden, Norway, Germany, United Kingdom, United States, and Canada. == How it works == With the Princh app, users will be able to locate their nearest printer. Once the user is at the printer, the user chooses the document to be printed out and shares it with the Princh app. The user then selects the desired nearby printer entering the printer ID number or scanning the QR-code located on top of the printer, pays electronically and the print job is processed by the printer. Printer owners get access to a personal control panel where they can set printing prices and monitor all Princh activity for their business. == Notes and references ==
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Biohybrid system

Biohybrid systems refer to the integration of biological materials, such as cells or tissues, with artificial components, including electronics or mechanical structure. This combination incorporates the capabilities of living organisms with the precision of man-made technology. As a result, these systems perform tasks that neither biology nor machines could achieve independently. Biohybrid systems might use lab-cultured muscle cells to power small robots or combine sensors with living tissue for better health sensing. The intent behind these systems is to combine the benefits of biological and technological components to introduce new solutions for complex medical challenges. Biohybrid systems may have transformative potential across sectors, such as robotics to create actuators and sensors that mimic natural muscle and nerve function, medicine in developing smart implants and drug delivery systems, in prosthetics for enhancing user control through neural or muscular interfaces and environmental sustainability for deploying biohybrid solutions for pollution sensing or remediation. == Origin == The term "biohybrid" is a compound of "bio" from biology (meaning life) and "hybrid" (referring to a combination of distinct elements), denoting a field of study. Its use helps distinguish such systems from purely biological constructs or entirely synthetic machines. Early academic mentions may include bio actuated robotics papers and foundational tissue-robot integration studies published in journals like Nature Biotechnology or Science Robotics. The emergence of the term reflects a growing recognition of the need to describe systems that do not fit cleanly into traditional categories. == Design principles == One of the most significant biohybrid challenges is to engineer interfaces between living tissue and artificial materials that are efficient. This means having precise control over adhesion at the surface, diffusion of nutrients, and signal conduction. Actuation mechanisms within the heart of these systems generate movement or mechanical response. These may be in the form of living muscle cells such as skeletal myocytes or cardiomyocytes, soft pneumatic actuators, or electrical stimulation-responsive tissues. Materials selection is equally critical. Hydrogels, elastomers like PDMS (polydimethylsiloxane), and biopolymers are commonly used due to their softness and biocompatibility. These materials must support cell viability, resist immune attack, and allow the integration of mechanical or electrical components. == Key components == At their core, biohybrid systems work by bridging living biological parts with technology. Through this integration, functionality that neither system could accomplish singularly is possible. Biological parts may be cells, tissues, or even organs—occasionally cultured in a laboratory setting. These biological parts carry out biologically inspired behaviors, such as muscle contraction or chemical sensing in the body. Technological components may constitute devices like sensors, electronic components, and mechanical structure. These manipulate the system, supply power, or transfer data. An example is a sensor that is implantable within a body and detects glucose levels as it sends information to a smart phone. By integrating these artificial and biological parts, biohybrid systems can perform advanced functions, such as tissue regeneration, real-time health monitoring, or the recovery of motor function in paralysis patients. Biohybrid systems generally consist of two major components: the biological and the mechanical. Biological components may include muscle cells for contraction, endothelial cells for vascularization, and stem cells for regenerative capabilities. Mechanical components comprise soft actuators that mimic organic motion, synthetic scaffolds that provide support and structure, and microfluidic systems that facilitate the delivery of nutrients and removal of waste. These components are combined in a manner that allows for dynamic, lifelike behavior—such as the contraction of tissue or the propagation of mechanical waves—while maintaining biocompatibility and durability. == Applications == The range of applications for biohybrid systems is broad and continuously expanding. In robotics, biohybrid structures have been used to engineer microscopic, muscle-driven machines, such as Harvard University's biohybrid stingray robot. In medical applications, they offer new alternatives for organ repair and augmentation, including biohybrid heart valves and esophageal scaffolds. Biohybrids are also promising in neural interfaces, where the goal is to create long-lasting, stable interaction between mechanical devices and brain tissue. Muscle-actuated drug response platforms are under exploration in pharmacology for modelling and real-time screening. == Examples == Several high-profile research projects have demonstrated the potential of biohybrid systems: Harvard researchers developed a biohybrid swimming ray powered by rat cardiac cells layered onto a gold skeleton, mimicking the motion of a real stingray. At the Massachusetts Institute of Technology, a cardiac pump actuated entirely by living heart muscle cells was engineered to simulate the behavior of a beating heart. Bio actuated soft robots have been built to simulate gut peristalsis, using muscle contractions to replicate natural wave-like movement in the digestive tract. == Challenges and limitations == As with many technologies that involve living systems, biohybrid systems raise important ethical and biomedical questions. Cell sourcing remains a key issue, particularly when embryonic or animal-derived cells are used. Long-term viability is another concern—living tissues must be kept alive with nutrients and oxygen, and they often degrade or elicit immune responses when implanted. Powering these biological parts presents logistical and ethical hurdles as well. Systems must either include internal mechanisms for nutrient delivery or be supported externally, which can limit portability and independence. == Future directions == Researchers are exploring self-directed, self-regulated organ substitutes and regenerative implants that can respond to their surroundings in real-time. These systems may be integrated with artificial intelligence to make them adjust to stimuli and coordinate complex behaviors. Future potential applications are wearable biohybrid systems for rehabilitation, space medicine devices for long-duration missions, and implantable devices that integrate into human physiology.
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Co-occurrence matrix

A co-occurrence matrix or co-occurrence distribution (also referred to as : gray-level co-occurrence matrices GLCMs) is a matrix that is defined over an image to be the distribution of co-occurring pixel values (grayscale values, or colors) at a given offset. It is used as an approach to texture analysis with various applications especially in medical image analysis. == Method == Given a grey-level image I {\displaystyle I} , co-occurrence matrix computes how often pairs of pixels with a specific value and offset occur in the image. The offset, ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} , is a position operator that can be applied to any pixel in the image (ignoring edge effects): for instance, ( 1 , 2 ) {\displaystyle (1,2)} could indicate "one down, two right". An image with p {\displaystyle p} different pixel values will produce a p × p {\displaystyle p\times p} co-occurrence matrix, for the given offset. The ( i , j ) th {\displaystyle (i,j)^{\text{th}}} value of the co-occurrence matrix gives the number of times in the image that the i th {\displaystyle i^{\text{th}}} and j th {\displaystyle j^{\text{th}}} pixel values occur in the relation given by the offset. For an image with p {\displaystyle p} different pixel values, the p × p {\displaystyle p\times p} co-occurrence matrix C is defined over an n × m {\displaystyle n\times m} image I {\displaystyle I} , parameterized by an offset ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} , as: C Δ x , Δ y ( i , j ) = ∑ x = 1 n ∑ y = 1 m { 1 , if I ( x , y ) = i and I ( x + Δ x , y + Δ y ) = j 0 , otherwise {\displaystyle C_{\Delta x,\Delta y}(i,j)=\sum _{x=1}^{n}\sum _{y=1}^{m}{\begin{cases}1,&{\text{if }}I(x,y)=i{\text{ and }}I(x+\Delta x,y+\Delta y)=j\\0,&{\text{otherwise}}\end{cases}}} where: i {\displaystyle i} and j {\displaystyle j} are the pixel values; x {\displaystyle x} and y {\displaystyle y} are the spatial positions in the image I; the offsets ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} define the spatial relation for which this matrix is calculated; and I ( x , y ) {\displaystyle I(x,y)} indicates the pixel value at pixel ( x , y ) {\displaystyle (x,y)} . The 'value' of the image originally referred to the grayscale value of the specified pixel, but could be anything, from a binary on/off value to 32-bit color and beyond. (Note that 32-bit color will yield a 232 × 232 co-occurrence matrix!) Co-occurrence matrices can also be parameterized in terms of a distance, d {\displaystyle d} , and an angle, θ {\displaystyle \theta } , instead of an offset ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} . Any matrix or pair of matrices can be used to generate a co-occurrence matrix, though their most common application has been in measuring texture in images, so the typical definition, as above, assumes that the matrix is an image. It is also possible to define the matrix across two different images. Such a matrix can then be used for color mapping. == Aliases == Co-occurrence matrices are also referred to as: GLCMs (gray-level co-occurrence matrices) GLCHs (gray-level co-occurrence histograms) spatial dependence matrices == Application to image analysis == Whether considering the intensity or grayscale values of the image or various dimensions of color, the co-occurrence matrix can measure the texture of the image. Because co-occurrence matrices are typically large and sparse, various metrics of the matrix are often taken to get a more useful set of features. Features generated using this technique are usually called Haralick features, after Robert Haralick. Texture analysis is often concerned with detecting aspects of an image that are rotationally invariant. To approximate this, the co-occurrence matrices corresponding to the same relation, but rotated at various regular angles (e.g. 0, 45, 90, and 135 degrees), are often calculated and summed. Texture measures like the co-occurrence matrix, wavelet transforms, and model fitting have found application in medical image analysis in particular. == Other applications == Co-occurrence matrices are also used for words processing in natural language processing (NLP).
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Model-based clustering

In statistics, cluster analysis is the algorithmic grouping of objects into homogeneous groups based on numerical measurements. Model-based clustering based on a statistical model for the data, usually a mixture model. This has several advantages, including a principled statistical basis for clustering, and ways to choose the number of clusters, to choose the best clustering model, to assess the uncertainty of the clustering, and to identify outliers that do not belong to any group. == Model-based clustering == Suppose that for each of n {\displaystyle n} observations we have data on d {\displaystyle d} variables, denoted by y i = ( y i , 1 , … , y i , d ) {\displaystyle y_{i}=(y_{i,1},\ldots ,y_{i,d})} for observation i {\displaystyle i} . Then model-based clustering expresses the probability density function of y i {\displaystyle y_{i}} as a finite mixture, or weighted average of G {\displaystyle G} component probability density functions: p ( y i ) = ∑ g = 1 G τ g f g ( y i ∣ θ g ) , {\displaystyle p(y_{i})=\sum _{g=1}^{G}\tau _{g}f_{g}(y_{i}\mid \theta _{g}),} where f g {\displaystyle f_{g}} is a probability density function with parameter θ g {\displaystyle \theta _{g}} , τ g {\displaystyle \tau _{g}} is the corresponding mixture probability where ∑ g = 1 G τ g = 1 {\displaystyle \sum _{g=1}^{G}\tau _{g}=1} . Then in its simplest form, model-based clustering views each component of the mixture model as a cluster, estimates the model parameters, and assigns each observation to cluster corresponding to its most likely mixture component. === Gaussian mixture model === The most common model for continuous data is that f g {\displaystyle f_{g}} is a multivariate normal distribution with mean vector μ g {\displaystyle \mu _{g}} and covariance matrix Σ g {\displaystyle \Sigma _{g}} , so that θ g = ( μ g , Σ g ) {\displaystyle \theta _{g}=(\mu _{g},\Sigma _{g})} . This defines a Gaussian mixture model. The parameters of the model, τ g {\displaystyle \tau _{g}} and θ g {\displaystyle \theta _{g}} for g = 1 , … , G {\displaystyle g=1,\ldots ,G} , are typically estimated by maximum likelihood estimation using the expectation-maximization algorithm (EM); see also EM algorithm and GMM model. Bayesian inference is also often used for inference about finite mixture models. The Bayesian approach also allows for the case where the number of components, G {\displaystyle G} , is infinite, using a Dirichlet process prior, yielding a Dirichlet process mixture model for clustering. === Choosing the number of clusters === An advantage of model-based clustering is that it provides statistically principled ways to choose the number of clusters. Each different choice of the number of groups G {\displaystyle G} corresponds to a different mixture model. Then standard statistical model selection criteria such as the Bayesian information criterion (BIC) can be used to choose G {\displaystyle G} . The integrated completed likelihood (ICL) is a different criterion designed to choose the number of clusters rather than the number of mixture components in the model; these will often be different if highly non-Gaussian clusters are present. === Parsimonious Gaussian mixture model === For data with high dimension, d {\displaystyle d} , using a full covariance matrix for each mixture component requires estimation of many parameters, which can result in a loss of precision, generalizabity and interpretability. Thus it is common to use more parsimonious component covariance matrices exploiting their geometric interpretation. Gaussian clusters are ellipsoidal, with their volume, shape and orientation determined by the covariance matrix. Consider the eigendecomposition of a matrix Σ g = λ g D g A g D g T , {\displaystyle \Sigma _{g}=\lambda _{g}D_{g}A_{g}D_{g}^{T},} where D g {\displaystyle D_{g}} is the matrix of eigenvectors of Σ g {\displaystyle \Sigma _{g}} , A g = diag { A 1 , g , … , A d , g } {\displaystyle A_{g}={\mbox{diag}}\{A_{1,g},\ldots ,A_{d,g}\}} is a diagonal matrix whose elements are proportional to the eigenvalues of Σ g {\displaystyle \Sigma _{g}} in descending order, and λ g {\displaystyle \lambda _{g}} is the associated constant of proportionality. Then λ g {\displaystyle \lambda _{g}} controls the volume of the ellipsoid, A g {\displaystyle A_{g}} its shape, and D g {\displaystyle D_{g}} its orientation. Each of the volume, shape and orientation of the clusters can be constrained to be equal (E) or allowed to vary (V); the orientation can also be spherical, with identical eigenvalues (I). This yields 14 possible clustering models, shown in this table: It can be seen that many of these models are more parsimonious, with far fewer parameters than the unconstrained model that has 90 parameters when G = 4 {\displaystyle G=4} and d = 9 {\displaystyle d=9} . Several of these models correspond to well-known heuristic clustering methods. For example, k-means clustering is equivalent to estimation of the EII clustering model using the classification EM algorithm. The Bayesian information criterion (BIC) can be used to choose the best clustering model as well as the number of clusters. It can also be used as the basis for a method to choose the variables in the clustering model, eliminating variables that are not useful for clustering. Different Gaussian model-based clustering methods have been developed with an eye to handling high-dimensional data. These include the pgmm method, which is based on the mixture of factor analyzers model, and the HDclassif method, based on the idea of subspace clustering. The mixture-of-experts framework extends model-based clustering to include covariates. == Example == We illustrate the method with a dateset consisting of three measurements (glucose, insulin, sspg) on 145 subjects for the purpose of diagnosing diabetes and the type of diabetes present. The subjects were clinically classified into three groups: normal, chemical diabetes and overt diabetes, but we use this information only for evaluating clustering methods, not for classifying subjects. The BIC plot shows the BIC values for each combination of the number of clusters, G {\displaystyle G} , and the clustering model from the Table. Each curve corresponds to a different clustering model. The BIC favors 3 groups, which corresponds to the clinical assessment. It also favors the unconstrained covariance model, VVV. This fits the data well, because the normal patients have low values of both sspg and insulin, while the distributions of the chemical and overt diabetes groups are elongated, but in different directions. Thus the volumes, shapes and orientations of the three groups are clearly different, and so the unconstrained model is appropriate, as selected by the model-based clustering method. The classification plot shows the classification of the subjects by model-based clustering. The classification was quite accurate, with a 12% error rate as defined by the clinical classification. Other well-known clustering methods performed worse with higher error rates, such as single-linkage clustering with 46%, average link clustering with 30%, complete-linkage clustering also with 30%, and k-means clustering with 28%. == Outliers in clustering == An outlier in clustering is a data point that does not belong to any of the clusters. One way of modeling outliers in model-based clustering is to include an additional mixture component that is very dispersed, with for example a uniform distribution. Another approach is to replace the multivariate normal densities by t {\displaystyle t} -distributions, with the idea that the long tails of the t {\displaystyle t} -distribution would ensure robustness to outliers. However, this is not breakdown-robust. A third approach is the "tclust" or data trimming approach which excludes observations identified as outliers when estimating the model parameters. == Non-Gaussian clusters and merging == Sometimes one or more clusters deviate strongly from the Gaussian assumption. If a Gaussian mixture is fitted to such data, a strongly non-Gaussian cluster will often be represented by several mixture components rather than a single one. In that case, cluster merging can be used to find a better clustering. A different approach is to use mixtures of complex component densities to represent non-Gaussian clusters. == Non-continuous data == === Categorical data === Clustering multivariate categorical data is most often done using the latent class model. This assumes that the data arise from a finite mixture model, where within each cluster the variables are independent. === Mixed data === These arise when variables are of different types, such as continuous, categorical or ordinal data. A latent class model for mixed data assumes local independence between the variable. The location model relaxes the local independence assumption. The clustMD approach assumes that the observed variables are manifestations of underlying continuous Gaussian latent
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