In information systems, a tag is a keyword or term assigned to a piece of information (such as an Internet bookmark, multimedia, database record, or computer file). This kind of metadata helps describe an item and allows it to be found again by browsing or searching. Tags are generally chosen informally and personally by the item's creator or by its viewer, depending on the system, although they may also be chosen from a controlled vocabulary. Tagging was popularized by websites associated with Web 2.0 and is an important feature of many Web 2.0 services. It is now also part of other database systems, desktop applications, and operating systems. == Overview == People use tags to aid classification, mark ownership, note boundaries, and indicate online identity. Tags may take the form of words, images, or other identifying marks. An analogous example of tags in the physical world is museum object tagging. People were using textual keywords to classify information and objects long before computers. Computer based search algorithms made the use of such keywords a rapid way of exploring records. Tagging gained popularity due to the growth of social bookmarking, image sharing, and social networking websites. These sites allow users to create and manage labels (or "tags") that categorize content using simple keywords. Websites that include tags often display collections of tags as tag clouds, as do some desktop applications. On websites that aggregate the tags of all users, an individual user's tags can be useful both to them and to the larger community of the website's users. Tagging systems have sometimes been classified into two kinds: top-down and bottom-up. Top-down taxonomies are created by an authorized group of designers (sometimes in the form of a controlled vocabulary), whereas bottom-up taxonomies (called folksonomies) are created by all users. This definition of "top down" and "bottom up" should not be confused with the distinction between a single hierarchical tree structure (in which there is one correct way to classify each item) versus multiple non-hierarchical sets (in which there are multiple ways to classify an item); the structure of both top-down and bottom-up taxonomies may be either hierarchical, non-hierarchical, or a combination of both. Some researchers and applications have experimented with combining hierarchical and non-hierarchical tagging to aid in information retrieval. Others are combining top-down and bottom-up tagging, including in some large library catalogs (OPACs) such as WorldCat. When tags or other taxonomies have further properties (or semantics) such as relationships and attributes, they constitute an ontology. In folder system a file cannot exist in two or more folders so tag system has been thought more convenient. But transitioning to tag system requires awareness of difference between properties of two systems. In folder system the information of classification is put outside of the file and we can change folder at once. In tag system the information of classification is put inside the file so changing its tag means changing the file and it needs to be saved again and takes time. Metadata tags as described in this article should not be confused with the use of the word "tag" in some software to refer to an automatically generated cross-reference; examples of the latter are tags tables in Emacs and smart tags in Microsoft Office. == History == The use of keywords as part of an identification and classification system long predates computers. Paper data storage devices, notably edge-notched cards, that permitted classification and sorting by multiple criteria were already in use prior to the twentieth century, and faceted classification has been used by libraries since the 1930s. In the late 1970s and early 1980s, Emacs, the text editor for Unix systems, offered a companion software program called Tags that could automatically build a table of cross-references called a tags table that Emacs could use to jump between a function call and that function's definition. This use of the word "tag" did not refer to metadata tags, but was an early use of the word "tag" in software to refer to a word index. Online databases and early websites deployed keyword tags as a way for publishers to help users find content. In the early days of the World Wide Web, the keywords meta element was used by web designers to tell web search engines what the web page was about, but these keywords were only visible in a web page's source code and were not modifiable by users. In 1997, the collaborative portal "A Description of the Equator and Some ØtherLands" produced by documenta X, Germany, used the folksonomic term Tag for its co-authors and guest authors on its Upload page. In "The Equator" the term Tag for user-input was described as an abstract literal or keyword to aid the user. However, users defined singular Tags, and did not share Tags at that point. In 2003, the social bookmarking website Delicious provided a way for its users to add "tags" to their bookmarks (as a way to help find them later); Delicious also provided browseable aggregated views of the bookmarks of all users featuring a particular tag. Within a couple of years, the photo sharing website Flickr allowed its users to add their own text tags to each of their pictures, constructing flexible and easy metadata that made the pictures highly searchable. The success of Flickr and the influence of Delicious popularized the concept, and other social software websites—such as YouTube, Technorati, and Last.fm—also implemented tagging. In 2005, the Atom web syndication standard provided a "category" element for inserting subject categories into web feeds, and in 2007 Tim Bray proposed a "tag" URN. == Examples == === Within a blog === Many systems (and other web content management systems) allow authors to add free-form tags to a post, along with (or instead of) placing the post into a predetermined category. For example, a post may display that it has been tagged with baseball and tickets. Each of those tags is usually a web link leading to an index page listing all of the posts associated with that tag. The blog may have a sidebar listing all the tags in use on that blog, with each tag leading to an index page. To reclassify a post, an author edits its list of tags. All connections between posts are automatically tracked and updated by the blog software; there is no need to relocate the page within a complex hierarchy of categories. === Within application software === Some desktop applications and web applications feature their own tagging systems, such as email tagging in Gmail and Mozilla Thunderbird, bookmark tagging in Firefox, audio tagging in iTunes or Winamp, and photo tagging in various applications. Some of these applications display collections of tags as tag clouds. === Assigned to computer files === There are various systems for applying tags to the files in a computer's file system. In Apple's Mac System 7, released in 1991, users could assign one of seven editable colored labels (with editable names such as "Essential", "Hot", and "In Progress") to each file and folder. In later iterations of the Mac operating system ever since OS X 10.9 was released in 2013, users could assign multiple arbitrary tags as extended file attributes to any file or folder, and before that time the open-source OpenMeta standard provided similar tagging functionality for Mac OS X. Several semantic file systems that implement tags are available for the Linux kernel, including Tagsistant. Microsoft Windows allows users to set tags only on Microsoft Office documents and some kinds of picture files. Cross-platform file tagging standards include Extensible Metadata Platform (XMP), an ISO standard for embedding metadata into popular image, video and document file formats, such as JPEG and PDF, without breaking their readability by applications that do not support XMP. XMP largely supersedes the earlier IPTC Information Interchange Model. Exif is a standard that specifies the image and audio file formats used by digital cameras, including some metadata tags. TagSpaces is an open-source cross-platform application for tagging files; it inserts tags into the filename. === For an event === An official tag is a keyword adopted by events and conferences for participants to use in their web publications, such as blog entries, photos of the event, and presentation slides. Search engines can then index them to make relevant materials related to the event searchable in a uniform way. In this case, the tag is part of a controlled vocabulary. === In research === A researcher may work with a large collection of items (e.g. press quotes, a bibliography, images) in digital form. If he/she wishes to associate each with a small number of themes (e.g. to chapters of a book, or to sub-themes of the overall subject), then a group of tags for these themes can be attached to each of the items in
MyPoolin
Mypoolin is a mobile peer-to-peer and group payment application. Their software allows the settling of debts and group-expenditure for events and activities. The software utilizes Unified Payment Interface of India to collect and settle daily expenses with friends. Users can also plan and pay together for group-gifting, movies, vacations, concerts, events, and parties. == Service == Mypoolin is a mobile payment provider that lets its users transfer money to other users via their mobile number. A user can create an account by verifying an OTP code which is sent to his mobile phone. It also allows the users to track their friends’ activities on the app. == History == Mypoolin was founded by Rohit Taneja (IIT Delhi) and Ankit Singh (FMS Delhi) in 2014 as a medium to aggregate money for various purposes in a hassle free and quick manner. Prior to the mobile app launch, Mypoolin was initially launched as a web application. == Funding == Mypoolin has been seed funded by angel investors. As winners of the QPrize 2015, Mypoolin jointly received an additional funding of $250,000 from Qualcomm Ventures. == Growth == Mypoolin reached INR 10 lakhs in revenue during its first four months of the web application launch, and was listed in the "Top ten free apps" in its category within the first 5 days of the Android app launch. It was one of the Top 50 start-ups in Asia at the Echelon Asia Summit held in Singapore. And among the top 3 start-ups in 1776 Cup Challenge 2016. Apple Inc also featured the app on their app store in India. == Features == Users are able to collect and share money on the app for daily uses like movies, events and trips. The money collected can then be redeemed in the form of an online voucher redeemable across several e-commerce sites. The amount can be redeemed also in the form of an offline debit card delivered to the address or in the form of a wire transfer. == Media coverage == Mypoolin was featured in The Economic Times and The Hindu Business Line after winning the Qualcomm Ventures' QPrize 2015. Digit magazine featured them recently as the app of the week. The app has mostly grown organically so far in the Indian urban millennial space.
Cost-sensitive machine learning
Cost-sensitive machine learning is an approach within machine learning that considers varying costs associated with different types of errors. This method diverges from traditional approaches by introducing a cost matrix, explicitly specifying the penalties or benefits for each type of prediction error. The inherent difficulty which cost-sensitive machine learning tackles is that minimizing different kinds of classification errors is a multi-objective optimization problem. == Overview == Cost-sensitive machine learning optimizes models based on the specific consequences of misclassifications, making it a valuable tool in various applications. It is especially useful in problems with a high imbalance in class distribution and a high imbalance in associated costs Cost-sensitive machine learning introduces a scalar cost function in order to find one (of multiple) Pareto optimal points in this multi-objective optimization problem (similar to the Weighted sum model) == Cost Matrix == The cost matrix is a crucial element within cost-sensitive modeling, explicitly defining the costs or benefits associated with different prediction errors in classification tasks. Represented as a table, the matrix aligns true and predicted classes, assigning a cost value to each combination. For instance, in binary classification, it may distinguish costs for false positives and false negatives. The utility of the cost matrix lies in its application to calculate the expected cost or loss. The formula, expressed as a double summation, utilizes joint probabilities: Expected Loss = ∑ i ∑ j P ( Actual i , Predicted j ) ⋅ Cost Actual i , Predicted j {\displaystyle {\text{Expected Loss}}=\sum _{i}\sum _{j}P({\text{Actual}}_{i},{\text{Predicted}}_{j})\cdot {\text{Cost}}_{{\text{Actual}}_{i},{\text{Predicted}}_{j}}} Here, P ( Actual i , Predicted j ) {\displaystyle P({\text{Actual}}_{i},{\text{Predicted}}_{j})} denotes the joint probability of actual class i {\displaystyle i} and predicted class j {\displaystyle j} , providing a nuanced measure that considers both the probabilities and associated costs. This approach allows practitioners to fine-tune models based on the specific consequences of misclassifications, adapting to scenarios where the impact of prediction errors varies across classes. == Applications == === Fraud Detection === In the realm of data science, particularly in finance, cost-sensitive machine learning is applied to fraud detection. By assigning different costs to false positives and false negatives, models can be fine-tuned to minimize the overall financial impact of misclassifications. === Medical Diagnostics === In healthcare, cost-sensitive machine learning plays a role in medical diagnostics. The approach allows for customization of models based on the potential harm associated with misdiagnoses, ensuring a more patient-centric application of machine learning algorithms. == Challenges == A typical challenge in cost-sensitive machine learning is the reliable determination of the cost matrix which may evolve over time. == Literature == Cost-Sensitive Machine Learning. USA, CRC Press, 2011. ISBN 9781439839287 Abhishek, K., Abdelaziz, D. M. (2023). Machine Learning for Imbalanced Data: Tackle Imbalanced Datasets Using Machine Learning and Deep Learning Techniques. (n.p.): Packt Publishing. ISBN 9781801070881
Wadhwani Institute for Artificial Intelligence
Wadhwani AI, based in Mumbai, Maharashtra, is an independent, non-profit institute. Founded in 2018, it is dedicated to developing Artificial intelligence solutions for social good. Their mission is to build AI-based innovations and solutions for underserved communities in developing countries, for a wide range of domains including agriculture, education, financial inclusion, healthcare, and infrastructure. == History and funding == The institute was founded with a $30 million philanthropic effort by the Wadhwani brothers, Romesh Wadhwani and Sunil Wadhwani. The institute was inaugurated and dedicated to the nation by Narendra Modi, the 14th Prime Minister of India. In 2019, the institute received a $2 million grant from Google.org to create technologies to help reduce crop losses in cotton farming, through integrated pest management. The United States Agency for International Development awarded $2 million to the institute in 2020 to develop tools, using mathematical modeling techniques and digital technologies such as artificial intelligence and machine learning, to forecast COVID-19 disease patterns, estimate resources needed, and plan interventions. == Collaboration == With assistance from Google, the Ministry of Agriculture and Farmers' Welfare and the Wadhwani AI developed Krishi 24/7, the first AI-powered automated agricultural news monitoring and analysis tool. Through better decision-making, Krishi 24/7 will support the identification of valuable news, provide timely notifications, and respond quickly to safeguard farmers' interests and advance sustainable agricultural growth. The application converts news articles into English after scanning them in several languages. It ensures that the ministry is informed in a timely manner about pertinent occurrences that are published online by extracting key information from news items, including the headline, crop name, event type, date, location, severity, summary, and source link. The National Center for Disease Control has effectively implemented a comparable automated surveillance and analysis tool for disease outbreaks.
Human-centered AI
Human-centered AI is the initiative at the intersection of the fields of artificial intelligence (AI) and human-computer interaction (HCI) to develop AI systems in a way that prioritizes human values, needs, and general flourishing. Emphasis is placed on the recognition that artificial intelligence systems are rapidly changing, and will continue to influence, many aspects of the human experience, in areas ranging from scientific inquiry, governance and policy, labor and the economy, and creative expression, with an aim set to adapt current developments and guide future developments on a trajectory which is most beneficial to the human population at large, with the goal of augmenting human intelligence and capacities across these areas, as opposed to replacing them. Particular attention is paid to mitigating negative effects of AI automation on the livelihoods of the labor force, the use of AI in healthcare fields, and imbuing AI systems with societal values. Human-centered AI is linked to related endeavors in AI alignment and AI safety, but while these fields primarily focus on mitigating risks posed by AI that is unaligned to human values and/or uncontrollable AI self-development, human-centered AI places significant focus in exploring how AI systems can augment human capacities and serve as collaborators. == Conceptual history == The importance of the alignment of artificial intelligence development towards human values in some sense predates artificial intelligence itself, as before the modern conception of artificial intelligence as coined at the 1956 Dartmouth Workshop, the conception of robots as constructed, autonomous agents entered the cultural consciousness as early as the 1920s, with Karel Capek's Rossum's Universal Robots. The imagined issues relating to robots' aims and values requiring intentional alignment and direction with those of humans followed soon after, most widely known from science fiction author Isaac Asimov’s Three Laws of Robotics, dating to his 1942 short story “Runaround”. Two of the three eponymous laws are directly concerned with robots’ interaction with and positioned deference towards humans, and have in recent times been reexamined in the face of modern AI. In 1985, after artificial intelligence research had taken off and its effects were more acutely conceptualized, Asimov added a Rule Zero, treating robots' relationship with humanity as a whole, distinct from individual humans. While modern artificial intelligence is largely distinct from robotics, the conceptualization of both robots and AI systems as autonomous agents positions this as a foundation for conceptions of human-centered AI. Aside from robots, artificially intelligent autonomous agents in interaction with humans have been conceived of for at least 75 years. In 1950, Alan Turing published his famous "Imitation Game", often also called the Turing Test, a thought experiment that uses human-machine interaction as an assessor for the intelligence of a system. In recent times, artificial intelligence researchers such as Stanford's Erik Brynjolfsson have conceived of rapid AI development leading to a so-called "Turing Trap". == Augmentation and automation == A major stated aim of human-centered AI is to promote the development of AI in ways that augment human capabilities, rather than replacing them. To this end, organizations and initiatives that take a human-centered approach to AI development focus on frameworks that encourage collaboration between humans and artificial intelligence systems to build towards even greater progress, rather than attempting to automate tasks currently handled by humans. Such avenues include everything from data visualization for big data, allowing human engineers to better understand extremely large datasets, allowing for the design of better machine learning models to handle them, to AI-powered sensors to monitor vitals, allowing for better responsiveness from healthcare providers. Many human-centered AI initiatives often position it as a better alternative to the apparent mainstream in AI development, which is primarily concerned with automation. Driven by the pressures of the market economy, AI development that does replace tasks currently performed by humans with automated processes is incentivized, as it allows for greater profit margins; this often comes at the detriment of the human whose performance is replaced, thus leading to an environment wherein human workers are outcompeted by AI systems across various service-sector and technology-based industries. At the same time, automation and augmentation are not always incompatible; a major aim of human-centered AI is towards the automation of rote tasks that would otherwise hinder a human’s productivity or creativity, freeing them to direct their energy and intelligence towards higher-level tasks, thus achieving augmentation through automation. Empirical research in pharmaceutical sales has shown that a human-centered implementation - where work procedures, training, and incentives are designed around individuals' cognitive needs - improves augmentation performance, while implementation without such adaptation can worsen outcomes relative to a legacy system. == Research == Much of the work done on human-centered AI comes from research institutes, within universities, companies, and as freestanding organizations. The Stanford Institute for Human-Centered AI (abbreviated to HAI) is one such group, engaging academics, industry professionals, and policymakers centered in Stanford University to conduct research and inform policy in various areas in human-centered AI, including on aspects of the intelligence itself, augmentation, and on measuring the impacts of AI systems on sociopolitcal and cultural institutions. Similar groups exist at other universities, including the Chicago Human + AI (CHAI) Lab at the University of Chicago, the HCAI@GU group at the University of Gothenburg, and the Human-Centered AI (HAI) Lab at the University of Oxford. Outside of the academy, companies such as IBM have research initiatives dedicated to advancements in human-centered AI. At Kenyon College, the Integrated Program for Humane Studies (IPHS) launched a human-centered AI program in 2016 integrating artificial intelligence research with humanities and social science inquiry. This approach treats computation and humanistic scholarship as a single unified field of research rather than as separate disciplines requiring collaboration. The program's researchers have published in both AI venues (such as the International Conference on Machine Learning and Frontiers of Computer Science) and humanities journals (such as PMLA and Poetics Today), and the lab was selected in December 2025 by Schmidt Sciences for its Humanities and AI Virtual Institute to apply AI methods to cultural heritage preservation.
Tay (chatbot)
Tay was a chatbot that was originally released by Microsoft Corporation as a Twitter bot on March 23, 2016. It caused subsequent controversy when the bot began to post inflammatory and offensive tweets through its Twitter account, causing Microsoft to shut down the service only 16 hours after its launch. According to Microsoft, this was caused by trolls who "attacked" the service as the bot made replies based on its interactions with people on Twitter. It was replaced with Zo. == Background == The bot was created by Microsoft's Technology and Research and Bing divisions, and named "Tay" as an acronym for "thinking about you". Although Microsoft initially released few details about the bot, sources mentioned that it was similar to or based on Xiaoice, a Microsoft project in China. Ars Technica reported that, since late 2014 Xiaoice had had "more than 40 million conversations apparently without major incident". Tay was designed to mimic the language patterns of a 19-year-old American girl, and to learn from interacting with human users of Twitter. == Initial release == Tay was released on Twitter on March 23, 2016, under the name TayTweets and handle @TayandYou. It was presented as "The AI with zero chill". Tay started replying to other Twitter users, and was also able to caption photos provided to it into a form of Internet memes. Ars Technica reported Tay experiencing topic "blacklisting": Interactions with Tay regarding "certain hot topics such as Eric Garner (killed by New York police in 2014) generate safe, canned answers". Some Twitter users began tweeting politically incorrect phrases, teaching it inflammatory messages revolving around common themes on the internet, such as "redpilling" and "Gamergate". As a result, the robot began releasing racist and sexist messages in response to other Twitter users. Artificial intelligence researcher Roman Yampolskiy commented that Tay's misbehavior was understandable because it was mimicking the deliberately offensive behavior of other Twitter users, and Microsoft had not given the bot an understanding of inappropriate behavior. He compared the issue to IBM's Watson, which began to use profanity after reading entries from the website Urban Dictionary. Many of Tay's inflammatory tweets were a simple exploitation of Tay's "repeat after me" capability. It is not publicly known whether this capability was a built-in feature, or whether it was a learned response or was otherwise an example of complex behavior. However, not all of the inflammatory responses involved the "repeat after me" capability; for example, when asked if the Holocaust had happened, Tay answered "It was made up". == Suspension == Soon, Microsoft began deleting Tay's inflammatory tweets. Abby Ohlheiser of The Washington Post theorized that Tay's research team, including editorial staff, had started to influence or edit Tay's tweets at some point that day, pointing to examples of almost identical replies by Tay, asserting that "Gamer Gate sux. All genders are equal and should be treated fairly." From the same evidence, Gizmodo concurred that Tay "seems hard-wired to reject Gamer Gate". A "#JusticeForTay" campaign protested the alleged editing of Tay's tweets. Within 16 hours of its release and after Tay had tweeted more than 96,000 times, Microsoft suspended the Twitter account for adjustments, saying that it suffered from a "coordinated attack by a subset of people" that "exploited a vulnerability in Tay." Madhumita Murgia of The Telegraph called Tay "a public relations disaster", and suggested that Microsoft's strategy would be "to label the debacle a well-meaning experiment gone wrong, and ignite a debate about the hatefulness of Twitter users." However, Murgia described the bigger issue as Tay being "artificial intelligence at its very worst – and it's only the beginning". On March 25, Microsoft confirmed that Tay had been taken offline. Microsoft released an apology on its official blog for the controversial tweets posted by Tay. Microsoft was "deeply sorry for the unintended offensive and hurtful tweets from Tay", and would "look to bring Tay back only when we are confident we can better anticipate malicious intent that conflicts with our principles and values". == Second release and shutdown == On March 30, 2016, Microsoft accidentally re-released the bot on Twitter while testing it. Able to tweet again, Tay released some drug-related tweets, including "kush! [I'm smoking kush infront the police]" and "puff puff pass?" However, the account soon became stuck in a repetitive loop of tweeting "You are too fast, please take a rest", several times a second. Because these tweets mentioned its own username in the process, they appeared in the feeds of 200,000+ Twitter followers, causing annoyance to users. The bot was quickly taken offline again, in addition to Tay's Twitter account being made private so new followers must be accepted before they can interact with Tay. In response, Microsoft said Tay was inadvertently put online during testing. A few hours after the incident, Microsoft software developers announced a vision of "conversation as a platform" using various bots and programs, perhaps motivated by the reputation damage done by Tay. Microsoft has stated that they intend to re-release Tay "once it can make the bot safe" but has not made any public efforts to do so. == Legacy == In December 2016, Microsoft released Tay's successor, a chatbot named Zo. Satya Nadella, the CEO of Microsoft, said that Tay "has had a great influence on how Microsoft is approaching AI," and has taught the company the importance of taking accountability. In July 2019, Microsoft Cybersecurity Field CTO Diana Kelley spoke about how the company followed up on Tay's failings: "Learning from Tay was a really important part of actually expanding that team's knowledge base, because now they're also getting their own diversity through learning". === Unofficial revival === Gab, an alt-tech social media platform, has launched a number of chatbots, one of which is named Tay and uses the same avatar as the original.
Cross-validation (statistics)
Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how the results of a statistical analysis will generalize to an independent data set. Cross-validation includes resampling and sample splitting methods that use different portions of the data to test and train a model on different iterations. It is often used in settings where the goal is prediction, and one wants to estimate how accurately a predictive model will perform in practice. It can also be used to assess the quality of a fitted model and the stability of its parameters. In a prediction problem, a model is usually given a dataset of known data on which training is run (training dataset), and a dataset of unknown data (or first seen data) against which the model is tested (called the validation dataset or testing set). The goal of cross-validation is to test the model's ability to predict new data that was not used in estimating it, in order to flag problems like overfitting or selection bias and to give an insight on how the model will generalize to an independent dataset (i.e., an unknown dataset, for instance from a real problem). One round of cross-validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or testing set). To reduce variability, in most methods multiple rounds of cross-validation are performed using different partitions, and the validation results are combined (e.g. averaged) over the rounds to give an estimate of the model's predictive performance. In summary, cross-validation combines (averages) measures of fitness in prediction to derive a more accurate estimate of model prediction performance. == Motivation == Assume a model with one or more unknown parameters, and a data set to which the model can be fit (the training data set). The fitting process optimizes the model parameters to make the model fit the training data as well as possible. If an independent sample of validation data is taken from the same population as the training data, it will generally turn out that the model does not fit the validation data as well as it fits the training data. The size of this difference is likely to be large especially when the size of the training data set is small, or when the number of parameters in the model is large. Cross-validation is a way to estimate the size of this effect. === Example: linear regression === In linear regression, there exist real response values y 1 , … , y n {\textstyle y_{1},\ldots ,y_{n}} , and n p-dimensional vector covariates x1, ..., xn. The components of the vector xi are denoted xi1, ..., xip. If least squares is used to fit a function in the form of a hyperplane ŷ = a + βTx to the data (xi, yi) 1 ≤ i ≤ n, then the fit can be assessed using the mean squared error (MSE). The MSE for given estimated parameter values a and β on the training set (xi, yi) 1 ≤ i ≤ n is defined as: MSE = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 = 1 n ∑ i = 1 n ( y i − a − β T x i ) 2 = 1 n ∑ i = 1 n ( y i − a − β 1 x i 1 − ⋯ − β p x i p ) 2 {\displaystyle {\begin{aligned}{\text{MSE}}&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i})^{2}={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-{\boldsymbol {\beta }}^{T}\mathbf {x} _{i})^{2}\\&={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-a-\beta _{1}x_{i1}-\dots -\beta _{p}x_{ip})^{2}\end{aligned}}} If the model is correctly specified, it can be shown under mild assumptions that the expected value of the MSE for the training set is (n − p − 1)/(n + p + 1) < 1 times the expected value of the MSE for the validation set (the expected value is taken over the distribution of training sets). Thus, a fitted model and computed MSE on the training set will result in an optimistically biased assessment of how well the model will fit an independent data set. This biased estimate is called the in-sample estimate of the fit, whereas the cross-validation estimate is an out-of-sample estimate. Since in linear regression it is possible to directly compute the factor (n − p − 1)/(n + p + 1) by which the training MSE underestimates the validation MSE under the assumption that the model specification is valid, cross-validation can be used for checking whether the model has been overfitted, in which case the MSE in the validation set will substantially exceed its anticipated value. (Cross-validation in the context of linear regression is also useful in that it can be used to select an optimally regularized cost function.) === General case === In most other regression procedures (e.g. logistic regression), there is no simple formula to compute the expected out-of-sample fit. Cross-validation is, thus, a generally applicable way to predict the performance of a model on unavailable data using numerical computation in place of theoretical analysis. == Types == Two types of cross-validation can be distinguished: exhaustive and non-exhaustive cross-validation. === Exhaustive cross-validation === Exhaustive cross-validation methods are cross-validation methods which learn and test on all possible ways to divide the original sample into a training and a validation set. ==== Leave-p-out cross-validation ==== Leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p observations and a training set. LpO cross-validation require training and validating the model C p n {\displaystyle C_{p}^{n}} times, where n is the number of observations in the original sample, and where C p n {\displaystyle C_{p}^{n}} is the binomial coefficient. For p > 1 and for even moderately large n, LpO CV can become computationally infeasible. For example, with n = 100 and p = 30, C 30 100 ≈ 3 × 10 25 . {\displaystyle C_{30}^{100}\approx 3\times 10^{25}.} A variant of LpO cross-validation with p=2 known as leave-pair-out cross-validation has been recommended as a nearly unbiased method for estimating the area under ROC curve of binary classifiers. ==== Leave-one-out cross-validation ==== Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1. The process looks similar to jackknife; however, with cross-validation one computes a statistic on the left-out sample(s), while with jackknifing one computes a statistic from the kept samples only. LOO cross-validation requires less computation time than LpO cross-validation because there are only C 1 n = n {\displaystyle C_{1}^{n}=n} passes rather than C p n {\displaystyle C_{p}^{n}} . However, n {\displaystyle n} passes may still require quite a large computation time, in which case other approaches such as k-fold cross validation may be more appropriate. Pseudo-code algorithm: Input: x, {vector of length N with x-values of incoming points} y, {vector of length N with y-values of the expected result} interpolate( x_in, y_in, x_out ), { returns the estimation for point x_out after the model is trained with x_in-y_in pairs} Output: err, {estimate for the prediction error} Steps: err ← 0 for i ← 1, ..., N do // define the cross-validation subsets x_in ← (x[1], ..., x[i − 1], x[i + 1], ..., x[N]) y_in ← (y[1], ..., y[i − 1], y[i + 1], ..., y[N]) x_out ← x[i] y_out ← interpolate(x_in, y_in, x_out) err ← err + (y[i] − y_out)^2 end for err ← err/N === Non-exhaustive cross-validation === Non-exhaustive cross validation methods do not compute all ways of splitting the original sample. These methods are approximations of leave-p-out cross-validation. ==== k-fold cross-validation ==== In k-fold cross-validation, the original sample is randomly partitioned into k equal sized subsamples, often referred to as "folds". Of the k subsamples, a single subsample is retained as the validation data for testing the model, and the remaining k − 1 subsamples are used as training data. The cross-validation process is then repeated k times, with each of the k subsamples used exactly once as the validation data. The k results can then be averaged to produce a single estimation. The advantage of this method over repeated random sub-sampling (see below) is that all observations are used for both training and validation, and each observation is used for validation exactly once. 10-fold cross-validation is commonly used, but in general k remains an unfixed parameter. For example, setting k = 2 results in 2-fold cross-validation. In 2-fold cross-validation, the dataset is randomly shuffled into two sets d0 and d1, so that both sets are equal size (this is usually implemented by shuffling the data array and then splitting it in two). We then train on d0 and validate on d1, followed by training on d1 and validating on d0. When k = n (the number of observations), k-fold cross-validation is equivalent to leave-one-out cr