Alexander Gammerman (born 2 November 1944) is a British computer scientist, and professor at Royal Holloway University of London. He is the co-inventor of conformal prediction. He is the founding director of the Centre for Machine Learning at Royal Holloway, University of London, and a Fellow of the Royal Statistical Society. == Career == Gammerman's academic career has been pursued in the Soviet Union and the United Kingdom. He started working as a Research Fellow in the Agrophysical Research Institute, St. Petersburg. In 1983, he emigrated to the United Kingdom and was appointed as a lecturer in the Computer Science Department at Heriot-Watt University, Edinburgh. Together with Roger Thatcher, Gammerman published several articles on Bayesian inference. In 1993, he was appointed to the established chair in Computer Science at University of London tenable at Royal Holloway and Bedford New College, where he served as the Head of Computer Science department from 1995 to 2005. In 1998, the Centre for Reliable Machine Learning was established, and Gammerman became the first director of the centre. Gammerman has written 7 books. == Honours and awards == In 1996, Gammerman received the P.W. Allen Award from the Forensic Science Society. In 2006, he became an Honorary Professor, at University College London. In 2009, he became a Distinguished Professor at Complutense University of Madrid, Spain. In 2019, he received a research grant funded by the energy company Centrica about predicting the time to the next failure of equipment. In 2020, he received the Amazon Research Award for the project titled Conformal Martingales for Change-Point Detection == Selected books == Measures of Complexity (2016), Springer, ISBN 3319357786. Algorithmic Learning in a Random World (2005), Springer, ISBN 0387001522. Causal Models and Intelligent Data Management (1999), Springer, ISBN 978-3-642-58648-4. Probabilistic Reasoning and Bayesian Belief Networks (1998), Nelson Thornes Ltd, ISBN 1872474268. Computational Learning and Probabilistic Reasoning (1996), Wiley, ISBN 0471962791.
Control engineering
Control engineering, also known as control systems engineering and, in some European countries, automation engineering, is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls overlaps and is usually taught along with electrical engineering, chemical engineering and mechanical engineering at many institutions around the world. The practice uses sensors and detectors to measure the output performance of the process being controlled; these measurements are used to provide corrective feedback helping to achieve the desired performance. Systems designed to perform without requiring human input are called automatic control systems (such as cruise control for regulating the speed of a car). Multi-disciplinary in nature, control systems engineering activities focus on implementation of control systems mainly derived by mathematical modeling of a diverse range of systems. == Overview == Modern day control engineering is a relatively new field of study that gained significant attention during the 20th century with the advancement of technology. It can be broadly defined or classified as practical application of control theory. Control engineering plays an essential role in a wide range of control systems, from simple household washing machines to high-performance fighter aircraft. It seeks to understand physical systems, using mathematical modelling, in terms of inputs, outputs and various components with different behaviors; to use control system design tools to develop controllers for those systems; and to implement controllers in physical systems employing available technology. A system can be mechanical, electrical, fluid, chemical, financial or biological, and its mathematical modelling, analysis and controller design uses control theory in one or many of the time, frequency and complex-s domains, depending on the nature of the design problem. Control engineering is the engineering discipline that focuses on the modeling of a diverse range of dynamic systems (e.g. mechanical systems) and the design of controllers that will cause these systems to behave in the desired manner. Although such controllers need not be electrical, many are and hence control engineering is often viewed as a subfield of electrical engineering. Electrical circuits, digital signal processors and microcontrollers can all be used to implement control systems. Control engineering has a wide range of applications from the flight and propulsion systems of commercial airliners to the cruise control present in many modern automobiles. In most cases, control engineers utilize feedback when designing control systems. This is often accomplished using a proportional–integral–derivative controller (PID controller) system. For example, in an automobile with cruise control the vehicle's speed is continuously monitored and fed back to the system, which adjusts the motor's torque accordingly. Where there is regular feedback, control theory can be used to determine how the system responds to such feedback. In practically all such systems stability is important and control theory can help ensure stability is achieved. Although feedback is an important aspect of control engineering, control engineers may also work on the control of systems without feedback. This is known as open loop control. A classic example of open loop control is a washing machine that runs through a pre-determined cycle without the use of sensors. == History == Automatic control systems were first developed over two thousand years ago. The first feedback control device on record is thought to be the ancient Ktesibios's water clock in Alexandria, Egypt, around the third century BCE. It kept time by regulating the water level in a vessel and, therefore, the water flow from that vessel. This certainly was a successful device as water clocks of similar design were still being made in Baghdad when the Mongols captured the city in 1258 CE. A variety of automatic devices have been used over the centuries to accomplish useful tasks or simply just to entertain. The latter includes the automata, popular in Europe in the 17th and 18th centuries, featuring dancing figures that would repeat the same task over and over again; these automata are examples of open-loop control. Milestones among feedback, or "closed-loop" automatic control devices, include the temperature regulator of a furnace attributed to Drebbel, circa 1620, and the centrifugal flyball governor used for regulating the speed of steam engines by James Watt in 1788. In his 1868 paper "On Governors", James Clerk Maxwell was able to explain instabilities exhibited by the flyball governor using differential equations to describe the control system. This demonstrated the importance and usefulness of mathematical models and methods in understanding complex phenomena, and it signaled the beginning of mathematical control and systems theory. Elements of control theory had appeared earlier but not as dramatically and convincingly as in Maxwell's analysis. Control theory made significant strides over the next century. New mathematical techniques, as well as advances in electronic and computer technologies, made it possible to control significantly more complex dynamical systems than the original flyball governor could stabilize. New mathematical techniques included developments in optimal control in the 1950s and 1960s followed by progress in stochastic, robust, adaptive, nonlinear control methods in the 1970s and 1980s. Applications of control methodology have helped to make possible space travel and communication satellites, safer and more efficient aircraft, cleaner automobile engines, and cleaner and more efficient chemical processes. Before it emerged as a unique discipline, control engineering was practiced as a part of mechanical engineering and control theory was studied as a part of electrical engineering since electrical circuits can often be easily described using control theory techniques. In the first control relationships, a current output was represented by a voltage control input. However, not having adequate technology to implement electrical control systems, designers were left with the option of less efficient and slow responding mechanical systems. A very effective mechanical controller that is still widely used in some hydro plants is the governor. Later on, previous to modern power electronics, process control systems for industrial applications were devised by mechanical engineers using pneumatic and hydraulic control devices, many of which are still in use today. === Mathematical modelling === David Quinn Mayne, (1930–2024) was among the early developers of a rigorous mathematical method for analysing Model predictive control algorithms (MPC). It is currently used in tens of thousands of applications and is a core part of the advanced control technology by hundreds of process control producers. MPC's major strength is its capacity to deal with nonlinearities and hard constraints in a simple and intuitive fashion. His work underpins a class of algorithms that are probably correct, heuristically explainable, and yield control system designs which meet practically important objectives. == Control systems == == Control theory == == Education == At many universities around the world, control engineering courses are taught primarily in electrical engineering and mechanical engineering, but some courses can be instructed in mechatronics engineering, and aerospace engineering. In others, control engineering is connected to computer science, as most control techniques today are implemented through computers, often as embedded systems (as in the automotive field). The field of control within chemical engineering is often known as process control. It deals primarily with the control of variables in a chemical process in a plant. It is taught as part of the undergraduate curriculum of any chemical engineering program and employs many of the same principles in control engineering. Other engineering disciplines also overlap with control engineering as it can be applied to any system for which a suitable model can be derived. However, specialised control engineering departments do exist, for example, in Italy there are several master in Automation & Robotics that are fully specialised in Control engineering or the Department of Automatic Control and Systems Engineering at the University of Sheffield or the Department of Robotics and Control Engineering at the United States Naval Academy and the Department of Control and Automation Engineering at the Istanbul Technical University. Control engineering has diversified applications that include science, finance management, and even human behavior. Students of control engineering may start with a linear control system course dealing with the time and complex-s domain, which req
Kaggle
Kaggle is a data science competition platform and online community for data scientists and machine learning practitioners under Google LLC. Kaggle enables users to find and publish datasets, explore and build models in a web-based data science environment, work with other data scientists and machine learning engineers, and enter competitions to solve data science challenges. Kaggle has also facilitated the use of unethical and unreliable data in medical research. == History == Kaggle was founded by Anthony Goldbloom in April 2010. Jeremy Howard, one of the first Kaggle users, joined in November 2010 and served as the President and Chief Scientist. Also on the team was Nicholas Gruen serving as the founding chair. In 2011, the company raised $12.5 million and Max Levchin became the chairman. On March 8, 2017, Fei-Fei Li, Chief Scientist at Google, announced that Google was acquiring Kaggle. In June 2017, Kaggle surpassed 1 million registered users, and as of October 2023, it has over 15 million users in 194 countries. In 2022, founders Goldbloom and Hamner stepped down from their positions and D. Sculley became the CEO. In February 2023, Kaggle introduced Models, allowing users to discover and use pre-trained models through deep integrations with the rest of Kaggle’s platform. In April 2025, Kaggle partnered with Wikimedia Foundation. == Site overview == === Competitions === Many machine-learning competitions have been run on Kaggle since the company was founded. Notable competitions include gesture recognition for Microsoft Kinect, making a association football AI for Manchester City, coding a trading algorithm for Two Sigma Investments, and improving the search for the Higgs boson at CERN. The competition host prepares the data and a description of the problem; the host may choose whether it's going to be rewarded with money or be unpaid. Participants experiment with different techniques and compete against each other to produce the best models. Work is shared publicly through Kaggle Kernels to achieve a better benchmark and to inspire new ideas. Submissions can be made through Kaggle Kernels, via manual upload or using the Kaggle API. For most competitions, submissions are scored immediately (based on their predictive accuracy relative to a hidden solution file) and summarized on a live leaderboard. After the deadline passes, the competition host pays the prize money in exchange for "a worldwide, perpetual, irrevocable and royalty-free license [...] to use the winning Entry", i.e. the algorithm, software and related intellectual property developed, which is "non-exclusive unless otherwise specified". Alongside its public competitions, Kaggle also offers private competitions, which are limited to Kaggle's top participants. Kaggle offers a free tool for data science teachers to run academic machine-learning competitions. Kaggle also hosts recruiting competitions in which data scientists compete for a chance to interview at leading data science companies like Facebook, Winton Capital, and Walmart. Kaggle's competitions have resulted in successful projects such as furthering HIV research, chess ratings and traffic forecasting. Geoffrey Hinton and George Dahl used deep neural networks to win a competition hosted by Merck. Vlad Mnih (one of Hinton's students) used deep neural networks to win a competition hosted by Adzuna. This resulted in the technique being taken up by others in the Kaggle community. Tianqi Chen from the University of Washington also used Kaggle to show the power of XGBoost, which has since replaced Random Forest as one of the main methods used to win Kaggle competitions. Several academic papers have been published based on findings from Kaggle competitions. A contributor to this is the live leaderboard, which encourages participants to continue innovating beyond existing best practices. The winning methods are frequently written on the Kaggle Winner's Blog. === Progression system === Kaggle has implemented a progression system to recognize and reward users based on their contributions and achievements within the platform. This system consists of five tiers: Novice, Contributor, Expert, Master, and Grandmaster. Each tier is achieved by meeting specific criteria in competitions, datasets, kernels (code-sharing), and discussions. The highest tier, Kaggle Grandmaster, is awarded to users who have ranked at the top of multiple competitions including high ranking in a solo team. As of April 2, 2025, out of 23.29 million Kaggle accounts, 2,973 have achieved Kaggle Master status and 612 have achieved Kaggle Grandmaster status. === Kaggle Notebooks === Kaggle includes a free, browser-based online integrated development environment, called Kaggle Notebooks, designed for data science and machine learning. Users can write and execute code in Python or R, import datasets, use popular libraries, and train models on CPUs, GPUs, or TPUs directly in the cloud. This environment is often used for competition submissions, tutorials, education, and exploratory data analysis. == Medical Research Problems == In December 2025, an article was published in The Transmitter titled "Exclusive: Springer Nature retracts, removes nearly 40 publications that trained neural networks on ‘bonkers’ dataset". The dataset in question was uploaded to Kaggle containing photographs of autistic and non-autistic children's faces. This dataset contained more than 2,900 images and it is unlikely that these children or their families gave consent for the photos for use in medical research or the images were ethically approved for research. The articles using the dataset in Springer Nature were retracted from the scientific literature. At least 90 other publications cite a version of the dataset. In April 2026, another two datasets were identified on Kaggle with no data provenance having been published in Nature titled: "Dozens of AI disease-prediction models were trained on dubious data". These datasets were used in 124 clinical prediction models, at least two of which have been used in hospitals in Indonesia and Spain, while one article using the dataset was referenced in a medical device patent. As of April 17, 2026, three of the articles using these datasets have been retracted from the scientific literature. In May 2026, an additional research publication using two image datasets from Kaggle is under investigation in Scientific Reports. An article in Retraction Watch "‘Comically bad’ datasets used to train clinical models for stroke and diabetes" highlighted the images included famous actors such as Sylvester Stallone as Rambo, George Clooney, Angelina Jolie and Daniel Craig as well as children. It would be unethical for the use of these child images in medical research without consent. Reverse searching images saw some of the images were not for stroke but for bell's palsy. One of the datasets is no longer available on Kaggle while the other one still remains and mentions the images may be subject to copyright. Kaggle relies on the community self-reporting metadata and provenance and mentions the stroke and diabetes dataset identified in "Evidence of unreliable data and poor data provenance in clinical prediction model research and clinical practice" does not violate their terms of service and they would have been removed if they had.
Knowledge processing for robots
KnowRob (Knowledge processing for robots) is a system which combines knowledge representation and reasoning methods to acquire and ground knowledge. This system is the backbone of openEASE. both under developing at the Institute for Artificial Intelligence at the University of Bremen, Germany. == The framework == KnowRob can serve as a common sense framework for the integration of knowledge. This knowledge can be static encyclopedic knowledge, common sense knowledge, task descriptions, environment models, object information, observed actions, etc., which can come from different sources, like manually axiomatized, derived from observations, or imported from the web. KnowRob has been used by different research groups, as the Rice University using the ontological knowledge base in a robotic platform. As well by the Eindhoven University of Technology research group competing in the RoboCup league, in the "at Home" category, with the RoboEarth project. As well, KnowRob is mentioned in the work of some research groups from the Lucian Blaga University of Sibiu, Middle East Technical University in their combination of different knowledge bases, Keio University as related work because of the ontology service, University of Texas at Austin as related work as well because of the relation with the work presented, Hanyang University as related work as an OWL based knowledge processing framework. == Representations == To represent the knowledge, KnowRob uses the OWL ontology language and an extended first-order logic knowledge representation with computable predicates. To give the order of subactions, KnowRob includes a pair-wise ordering constrain, which gives a partial ordering. KnowRob adopts the closed-world assumption Prolog, and an open-world assumption by the use of computables. To include reasoning rules into Prolog, KnowRob uses an inference procedure beyond the capabilities of OWL to extract information about tasks executions. In its second version, KnowRob provides a logic interface to the hybrid reasoning kernel as a logic based language. This language presents the hybrid reasoning kernel as if everything were entities retrievable by providing partial descriptions for them. This entities descriptions include objects, their parts, and articulation models, environments composed of objects, software components, actions, and events. === Episodic memories === Episodic memory is related to the experience information, which is organized temporally and spatially, alongside combined with context information. In KnowRob, an episodic memory is understood as a recording that the agent makes of the ongoing activity, which includes very detailed information about the actions, motions, their purposes, effects and the behavior they generate, it also includes the images captured during execution, etc. == Usage == The knowledge is computed by external methods using Prolog queries. In the second version of the KnowRob system, is included a better structure of the packages and documentations. Which includes some extensions from the previous version, as well as a logic based language. For example, a cup description from perception can be represented in this language as: entity(Cup,[an, object, [type, cup], [shape, cylinder], [color, orange]]) As well, a controller could represent the same object as: entity(Cup, [an, object, [type, cup], [proper_physical_parts, [an, object, [type, handle], [grasp−pose, G−pose]]]]) The interface language is comparable to other query languages for symbolic knowledge bases. KnowRob's query language integrates reasoning methods, such as the simulation-based reasoning. == Goals == The goal of the KnowRob framework is to make semantic knowledge available for service robots. It is able to answer queries about missing information in vague instructions for tasks. This is possible with the actions hierarchical representation and information about objects which can be included in certain action.
Demis Hassabis
Sir Demis Hassabis (/ˈdɛ.mɪs/ DE-mis /hɑːˈsɑː.bis/ hah-SAH-bees; born Dimitrios Hassapis, Greek: Δημήτριος Χασάπης, 27 July 1976) is a British artificial intelligence (AI) researcher and entrepreneur. He is the chief executive officer and co-founder of Google DeepMind and Isomorphic Labs, and a UK Government AI Adviser. In 2024, Hassabis and John M. Jumper were jointly awarded the Nobel Prize in Chemistry for their AI research contributions to protein structure prediction. Hassabis is a Fellow of the Royal Society and has won awards for his research efforts, including the Breakthrough Prize, the Canada Gairdner International Award and the Lasker Award. He was appointed a CBE in 2017, and knighted in 2024 for his work on AI. He was also listed among the Time 100 most influential people in the world in 2017 and 2025, and was one of the "Architects of AI" collectively chosen as Time's 2025 Person of the Year. == Early life and education == Hassabis was born to Costas and Angela Hassapis. His father is a Greek Cypriot and his mother is a Chinese Singaporean. Demis grew up in North London. His original surname was "Hassapis" (Greek: Χασάπης), meaning "butcher" in Greek, but he later, according to Ingo Althöfer, "executed a point mutation by changing ‘p’ to ‘b’". One of his younger brothers still carries the original surname. In his early career, he was a video game AI programmer and designer, and an expert board games player. A child prodigy in chess from the age of four, when he first learnt chess by watching his father playing against his uncle, Hassabis reached master standard at the age of 13 with an Elo rating of 2300 and captained many of the England junior chess teams. He represented the University of Cambridge in the Oxford–Cambridge varsity chess matches of 1995, 1996 and 1997, winning a half blue. He first got interested in technology after buying his first computer in 1984, a ZX Spectrum 48K, funded from chess winnings. He taught himself how to program from books. He subsequently wrote his first AI program on a Commodore Amiga to play the reversi board game. Between 1988 and 1990, Hassabis was educated at Queen Elizabeth's School, Barnet, a boys' grammar school in North London. He was subsequently home-schooled by his parents for a year, before studying at the comprehensive school of Christ's College in East Finchley. He completed his A-level exams two years early at 16. === Bullfrog Productions === Asked by Cambridge University to take a gap year owing to his young age, Hassabis began his computer games career at Bullfrog Productions after entering an Amiga Power "Win-a-job-at-Bullfrog" competition. He began by playtesting on Syndicate and then at 17 co-designing and lead-programming on the 1994 game Theme Park, with the game's designer Peter Molyneux. Theme Park, a simulation video game, sold several million copies and inspired a whole genre of simulation sandbox games. Despite being offered a seven-figure sum to remain in the games industry, he turned it down. He earned enough from his gap year to pay his own way through university. === University of Cambridge === Hassabis left Bullfrog to study at Queens' College of the University of Cambridge, where he completed the Computer Science Tripos and graduated in 1997 with a double first. == Career and research == === Lionhead === After graduating from Cambridge, Hassabis worked at Lionhead Studios. Games designer Peter Molyneux, with whom Hassabis had worked at Bullfrog Productions, had recently founded the company. At Lionhead, Hassabis worked as lead AI programmer on the 2001 god game Black & White. === Elixir Studios === Hassabis left Lionhead in 1998 to found Elixir Studios, a London-based independent games developer, signing publishing deals with Eidos Interactive, Vivendi Universal and Microsoft. In addition to managing the company, Hassabis served as executive designer of the games Republic: The Revolution and Evil Genius. Each received BAFTA nominations for their interactive music scores, created by James Hannigan. The release of Elixir's first game, Republic: The Revolution, a highly ambitious and unusual political simulation game, was delayed due to its huge scope, which involved an AI simulation of the workings of an entire fictional country. The final game was reduced from its original vision and greeted with lukewarm reviews, receiving a Metacritic score of 62/100. Evil Genius, a tongue-in-cheek Austin Powers parody, fared much better with a score of 75/100. In April 2005 the intellectual property and technology rights were sold to various publishers and the studio was closed. === Neuroscience research === Following Elixir Studios, Hassabis returned to academia to obtain his PhD in cognitive neuroscience from UCL Queen Square Institute of Neurology in 2009 supervised by Eleanor Maguire. He sought to find inspiration in the human brain for new AI algorithms. He continued his neuroscience and artificial intelligence research as a visiting scientist jointly at Massachusetts Institute of Technology (MIT), in the lab of Tomaso Poggio, and Harvard University, before earning a Henry Wellcome postdoctoral research fellowship to the Gatsby Computational Neuroscience Unit at UCL in 2009 working with Peter Dayan. Working in the field of imagination, memory, and amnesia, he co-authored several influential papers published in Nature, Science, Neuron, and PNAS. His very first academic work, published in PNAS, was a landmark paper that showed systematically for the first time that patients with damage to their hippocampus, known to cause amnesia, were also unable to imagine themselves in new experiences. The finding established a link between the constructive process of imagination and the reconstructive process of episodic memory recall. Based on this work and a follow-up functional magnetic resonance imaging (fMRI) study, Hassabis developed a new theoretical account of the episodic memory system identifying scene construction, the generation and online maintenance of a complex and coherent scene, as a key process underlying both memory recall and imagination. This work received widespread coverage in the mainstream media and was listed in the top 10 scientific breakthroughs of the year by the journal Science. He later generalised these ideas to advance the notion of a 'simulation engine of the mind' whose role it was to imagine events and scenarios to aid with better planning. === DeepMind === Hassabis is the CEO and co-founder of DeepMind, a machine learning AI startup, founded in London in 2010 with Shane Legg and Mustafa Suleyman. Hassabis met Legg when both were postdocs at the Gatsby Computational Neuroscience Unit, and he and Suleyman had been friends through family. Hassabis also recruited his university friend and Elixir partner David Silver. DeepMind's mission is to "solve intelligence" and then use intelligence "to solve everything else". More concretely, DeepMind aims to combine insights from systems neuroscience with new developments in machine learning and computing hardware to unlock increasingly powerful general-purpose learning algorithms that will work towards the creation of an artificial general intelligence (AGI). The company has focused on training learning algorithms to master games, and in December 2013 it announced that it had made a pioneering breakthrough by training an algorithm called a Deep Q-Network (DQN) to play Atari games at a superhuman level by using only the raw pixels on the screen as inputs. DeepMind's early investors included several high-profile tech entrepreneurs. In 2014, Google purchased DeepMind for £400 million. Although most of the company has remained an independent entity based in London, DeepMind Health has since been directly incorporated into Google Health. Since the Google acquisition, the company has notched up a number of significant achievements, perhaps the most notable being the creation of AlphaGo, a program that defeated world champion Lee Sedol at the complex game of Go. Go had been considered a holy grail of AI, for its high number of possible board positions and resistance to existing programming techniques. However, AlphaGo beat European champion Fan Hui 5–0 in October 2015 before winning 4–1 against former world champion Lee Sedol in March 2016 and winning 3–0 against the world's top-ranked player Ke Jie in 2017. Additional DeepMind accomplishments include creating a neural Turing machine, reducing the energy used by the cooling systems in Google's data centres by 40%, and advancing research on AI safety. DeepMind has also been responsible for technical advances in machine learning, having produced a number of award-winning papers. In particular, the company has made significant advances in deep learning and reinforcement learning, and pioneered the field of deep reinforcement learning which combines these two methods. Hassabis has predicted that artificial intelligence will be "one of the most beneficial techn
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = 1/h K(x/h). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ( h ) = AMISE ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the
Linagora
Linagora is a French open source software editor, founded in June 2000 by Alexandre Zapolsky and Michel-Marie Maudet. Located in France, as well as in Belgium, Canada, Vietnam, the United States and Tunisia, the company employs around 200 people. In 2023, Linagora created the OpenLLM France community, alongside other French Artificial Intelligence companies and organizations. In 2025, the company launched Lucie, an opensource Large Language Model. == History == Linagora was founded on June 28, 2000. Its name is a contraction of the words "Linux" and "Agora". The company was founded by Alexandre Zapolsky and Michel-Marie Maudet. Soon after, the two entrepreneurs were joined by Alexandre Zapolsky's wife and brother, who took on the roles of commercial director and administrative and financial director of the SME. In 2007, the company was selected by the French National Assembly to provide the software for Linux computers, replacing Microsoft Windows. Linagora then claimed the position of the leading French open source software company by revenue. In 2015, French Prime Minister Manuel Valls allocated €10.7 million from the "Investments for the Future" fund for a research program aimed at developing a new generation of open source software platforms based on Linagora's offerings. In September 2016, Linagora launched the social network "La Cerise" for the newspaper L'Humanité. This app offered a service and tool for readers and citizens mobilizing for causes. It aimed to share engagement through petitions, discussions, agendas, and contacts. In October 2016, the company won two public contracts for supporting open source software in forty-two French ministries and other administrative entities. In May 2019, Linagora organized a fundraising event in the presence of the French Secretary of State for Digital Affairs, Cédric O, to celebrate its 19th anniversary. The funds were intended for: Supporting parents of hospitalized Polynesian children in France. Equipping primary school students with digital devices (tablets or PCs). Establishing a digital academy "OpenHackademy" in French Polynesia to train unemployed youth in digital skills and help them find jobs. In December 2022, Linagora acquired a property known as "Maison Rocher" and later "Maison Chocolat," located on the Île Saint-Germain in Issy-les-Moulineaux. Renamed "Villa Good Tech" by Linagora, this award-winning architectural work by Éric Daniel-Lacombe became the company's new headquarters, aiming to provide a space for associative actors and companies to develop technologies that contribute to a better world. In July 2023, Linagora launched OpenLLM France, a community initially comprising around twenty actors focused on generative AI. The goal was to develop a sovereign and open source large language model. This initiative, led by co-founder and CEO Michel-Marie Maudet, had more than four hundred French members by early 2024. and announced its expansion to the European sphere during Fosdem 2024. In February 2024, the CNRS and Linagora signed a framework agreement to strengthen their research collaboration. In January 2025, Linagora released Lucie, an open source and sovereign AI that faced ridicule due to tests on an unfinished, uncensored version designed for scientific and experimental use. The platform divided opinions between those who saw it as a technological achievement and those who criticized it as "French bashing" compared to American and Chinese AIs. == Acquisitions == The company acquired: In July 2007, the SME AliaSource, based in Ramonville-Saint-Agne and led by its founder, Pierre Baudracco. In 2008, the open source web hosting company Netaktiv, a member of the GIE Gitoyen, announced during the 2008 Solutions Linux trade show. In 2012, the Toulouse-based company EBM Websourcing, the publisher of the open-source software Petals Link, and took over its development. In 2016, the digital agency Neoma Interactive, specializing in UX design and digital communication strategy. == Locations == In 2017, the company's headquarters was located in Issy-les-Moulineaux, with branches in Lyon, Toulouse, Marseille, and internationally in Brussels, San Francisco, Montreal, Vietnam, and Tunisia. In 2005, the company attempted to establish a presence in Nantes. In 2024, the headquarters was moved to Issy-les-Moulineaux. == Activity == === Software === Twake Workplace One of Linagora's flagship products is Twake Workplace, which stands out as a 100% open-source solution compared with those of the GAFAMs. Twake Workplace is available as a complete platform or module by module. It includes : Twake Mail, a powerful modern messaging solution based on the JMAP protocol and the James email server from the Apache Foundation, for which Linagora provides technical management; Twake Chat, an instant communications solution for businesses developed using the Matrix protocol and compatible with the French government's chat solution, Tchap; Twake Drive, an easy-to-use collaborative platform for group work using OnlyOffice. ==== OpenPaaS ==== In 2018, the search engine Qwant announced that its email service Qwantmail would be based on the OpenPaaS product. In 2022, Qwant announced the abandonment of its Qwantmail project due to Linagora's collection of personal email addresses and serious security breaches. The site Next (formerly PC INpact) published an article in January 2020 criticizing the "failures and delays" of the Qwantmail project led by Linagora, which led to the CNIL's intervention regarding Qwant and Linagora. ==== LinTO ==== In 2017, Linagora launched its open source voice assistant project named LinTO. This enterprise voice assistant, described as "GAFAM Free," was presented at CES 2018 in Las Vegas. The LinTO voice framework was developed as part of the eponymous research project funded by Bpifrance (Grands Défis du Numérique instrument). === Services === ==== OSSA (Open Source Software Assurance) ==== One of the company's main activities is OSSA. Through OSSA, Linagora provided support for open source software for 42 ministries and other administrative entities in 2012. == Legal issues == === Dispute with BlueMind === In 2012, a legal dispute arose between BlueMind and Linagora. Linagora accused BlueMind of copyright infringement, unfair competition, and breach of a non-compete clause, leading to several legal actions. Linagora sued BlueMind for copyright infringement and unfair competition in the Bordeaux court, which ruled in Linagora's favor for unfair competition and parasitism but rejected the copyright claim. BlueMind was ordered to pay nearly €170,000 to Linagora. Linagora sued former associates Pierre Baudracco and Pierre Carlier in the Paris Commercial Court for breach of a non-compete clause and violation of a warranty of eviction. The court dismissed Linagora's claims and ordered it to pay €20,000 each to Baudracco and Carlier. Linagora appealed, and the Paris Court of Appeal partially overturned the decision, awarding Linagora €480,000. BlueMind sued Linagora for defamation and public insult in the Toulouse Criminal Court. The court ruled against Linagora, but the decision was overturned by the Court of Cassation in January 2024, and the case was remanded for retrial. === Conviction for wrongful termination and harassment === On June 14, 2017, France 3 reported on a decision by the Versailles Court of Appeal, which ruled that Linagora had wrongfully terminated an employee and subjected them to moral harassment. The court ordered Linagora to pay the employee €22,000 for wrongful termination, €11,000 for notice pay, €6,600 for legal severance pay, €3,200 for conservative suspension, and €3,000 for moral harassment.