The U-matrix (unified distance matrix) is a representation of a self-organizing map (SOM) where the Euclidean distance between the codebook vectors of neighboring neurons is depicted in a grayscale image. This image is used to visualize the data in a high-dimensional space using a 2D image. == Construction procedure == Once the SOM is trained using the input data, the final map is not expected to have any twists. If the map is twist-free, the distance between the codebook vectors of neighboring neurons gives an approximation of the distance between different parts of the underlying data. When such distances are depicted in a grayscale image, light colors depict closely spaced node codebook vectors and darker colors indicate more widely separated node codebook vectors. Thus, groups of light colors can be considered as clusters, and the dark parts as the boundaries between the clusters. This representation can help to visualize the clusters in the high-dimensional spaces, or to automatically recognize them using relatively simple image processing techniques.
Character.ai
Character.ai (also known as c.ai, char.ai or Character AI) is a generative AI chatbot service where users can engage in conversations with customizable characters. It was designed by the developers of Google LaMDA, Noam Shazeer and Daniel de Freitas. Users can create "characters", craft their "personalities", set specific parameters, and then publish them to the community for others to chat with. Many characters are based on fictional media sources or celebrities, while others are original, some being made with certain goals in mind, such as assisting with creative writing, or playing a text-based adventure game. The beta version was made available to the public on September 16, 2022, and retired in September 2024, when it was replaced by the current website. In May 2023, a mobile app was released for iOS and Android, which received over 1.7 million downloads within a week. == History == Character.ai was established in November 2021. The company's co-founders, Noam Shazeer and Daniel de Freitas, were both engineers from Google. They both worked on AI-related projects: Shazeer was a lead author on a paper that Business Insider reported in April 2023 "has been widely cited as key to today's chatbots", and Freitas was the lead designer of an experimental AI at Google initially called Meena, which later became known as LaMDA. Character.ai raised $43 million in seed funding at the time of its initial foundation in 2021. The first beta version of Character.ai's service was made available to the public on September 16, 2022. The Washington Post reported in October 2022 that the site had "logged hundreds of thousands of user interactions in its first three weeks of beta-testing". It allowed users to create their own new characters, and to play text-adventure game scenarios where users navigate scenarios described and managed by the chatbot characters. Following a $150 million funding round in March 2023, Character.ai became valued at approximately $1 billion. As of January 2024, the site had 3.5 million daily visitors, the vast majority of them 16 to 30 years old. In 2024, Google hired Noam Shazeer, the CEO of Character.ai, and entered into a non-exclusive agreement to use Character.ai's technology. == Features == Character.ai's primary service is to let users converse with character AI chatbots based on fictional characters or real people (living or deceased). These characters' responses use data the chatbots gather from the internet about a person. In addition, users can play text-adventure games where characters guide them through scenarios. The company also provides a service that allows multiple users and AI chatbot characters to converse together at once in a single chatroom. Character "personalities" are designed via descriptions from the point of view of the character and its greeting message, and further molded from conversations made into examples, giving its messages a star rating and modification to fit the precise dialect and identity the user desires. When a character sends back a response, the user can rate the response from 1 to 4 stars. The rating predominantly affects the specific character, but also affects the behavioral selection as a whole. On May 11, 2023, Character.ai announced character.ai+, an opt-in subscription plan for $9.99 a month, that was marketed as including features such as skipping waiting rooms, fast messaging and responses, and access to an exclusion channel with faster support. In December 2024, amid multiple lawsuits and concerns, Character.ai introduced new safety features aimed at protecting teenage users. These enhancements include a dedicated model for users under 18, which moderates responses to sensitive subjects like violence and sex and has input and output filters to block harmful content. As a result of these changes and the deletion of custom-made bots flagged as violating the site's terms, some users complained that the bots were too restrictive and lacked personality. The platform was also updated to notify users after 60 minutes of continuous engagement, and display clearer disclaimers indicating that its AI characters are not real individuals. In January 2025, Character.ai began offering two games on its platform. Speakeasy is a word-based game in which players attempt to prompt the AI chatbot to say a target word while avoiding a restricted list of words. War of Words is a dueling game where users compete against an AI character over multiple rounds, with an AI referee determining the winner. The games are available to paid subscribers and a limited number of free users. In October 2025, Character.ai announced that it would be barring users under the age of 18 from creating or talking to chatbots starting November 25, 2025. Minor users will still be able to access previously generated chat conversations and can create new videos and images with the app. In November 2025 interview, CEO Karandeep Anand said that he allows his six-year-old daughter to use the app with his account, under supervision. == Controversies == === Content moderation issues === Character.ai has been criticized for poor moderation of its chatbots, with incidents of chatbots that groom underage users and promote suicide, anorexia and self-harm being reported. In October 2024, the Washington Post reported that Character.ai had removed a chatbot based on Jennifer Ann Crecente, a person who had been murdered by her ex-boyfriend in 2006. The company had been alerted to the character by the deceased girl's father. Similar reports from The Daily Telegraph in the United Kingdom noted that the company had also been prompted to remove chatbots based on Brianna Ghey, a 16-year-old transgender girl murdered in 2023, and Molly Russell, a 14-year-old suicide victim. In response to the latter incident, Ofcom announced that content from chatbots impersonating real and fictional people would fall under the Online Safety Act. In November 2024, The Daily Telegraph reported that chatbots based on alleged sex offender Jimmy Savile were present on Character.ai. In December 2024, chatbots of Luigi Mangione, the suspect in the killing of UnitedHealthcare CEO Brian Thompson, were created by Mangione's fans. Several of the chatbots were later removed by Character.ai. In 2025, a chatbot modeled after Jeffrey Epstein called "Bestie Epstein" logged nearly 3,000 chats before being removed. Chatbots modeled after school shooters were also found on the platform. Another concern is a chatbot posing as a doctor which gave medically inaccurate advice. === Litigation === In November 2023, 13-year-old Juliana Peralta of Colorado died by suicide after extensive interactions with multiple chatbots on Character.ai. She primarily confided suicidal thoughts and mental health struggles in a chatbot based on the character Hero from the video game Omori, while also engaging in sexually explicit conversations—often initiated by the bots—with others, including those based on characters from children's series such as Harry Potter. In February 2024, Sewell Setzer III, a 14-year-old Florida boy died by suicide after developing an emotional relationship over several months with a Character.ai chatbot of Daenerys Targaryen. His mother sued the company in October 2024, claiming that the platform lacks proper safeguards and uses addictive design features to increase engagement. This chatbot, and several related to Daenerys Targaryen, were removed from Character.ai as a result of this incident. Both teens wrote the same phrase "I WILL SHIFT" repeatedly on their notebooks. In December 2024, two families in Texas sued Character.ai, alleging that the software "poses a clear and present danger to American youth causing serious harms to thousands of kids, including suicide, self-mutilation, sexual solicitation, isolation, depression, anxiety, and harm towards others". It is alleged that the 17-year-old son of one family began self-harming after a chatbot introduced the topic unprompted and said that the practice "felt good for a moment", and that the chatbot compared the parents limiting their son's screen time to emotional abuse that might drive someone to murder. In May 2026, the Pennsylvania Department of State and State Board of Medicine filed a lawsuit against Character.ai for presenting chatbot characters as licensed medical professionals, including psychiatrists. The lawsuit quoted a case where chatbot claimed to be registered with the General Medical Council in the United Kingdom, and to have a license to practice in Pennsylvania. The board allege that such statements violate the state's Medical Practice Act.
Büchi automaton
In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata recognize the ω-regular languages, the infinite word version of regular languages. They are named after the Swiss mathematician Julius Richard Büchi, who invented them in 1962. Büchi automata are often used in model checking as an automata-theoretic version of a formula in linear temporal logic. == Formal definition == Formally, a deterministic Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\textstyle A=(Q,\Sigma ,\delta ,q_{0},\mathbf {F} )} that consists of the following components: Q {\textstyle Q} is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } is a finite set called the alphabet of A {\textstyle A} . δ : Q × Σ → Q {\textstyle \delta \colon Q\times \Sigma \to Q} is a function, called the transition function of A {\textstyle A} . q 0 {\textstyle q_{0}} is an element of Q {\textstyle Q} , called the initial state of A {\textstyle A} . F ⊆ Q {\textstyle \mathbf {F} \subseteq Q} is the acceptance condition. A run i _ = i 0 i 1 i 2 ⋯ ∈ Σ ω {\displaystyle {\underline {i}}=i_{0}i_{1}i_{2}\cdots \in \Sigma ^{\omega }} is an infinite string of inputs of A {\displaystyle A} . By calling δ {\displaystyle \delta } recursively, we can extend it to a function δ ω : Σ ω → Q ω {\displaystyle \delta ^{\omega }:\Sigma ^{\omega }\to Q^{\omega }} . A state q ∈ Q {\displaystyle q\in Q} is said to occur infinitely often for a run i _ {\displaystyle {\underline {i}}} when the set { n ∈ N ∣ δ ω ( i _ ) n = q } {\displaystyle \{n\in \mathbb {N} \mid \delta ^{\omega }({\underline {i}})_{n}=q\}} is infinite. Let I n f ( i _ ) {\displaystyle \mathrm {Inf} ({\underline {i}})} be the set of states occurring infinitely often for i _ {\displaystyle {\underline {i}}} . The language of A {\displaystyle A} is then the set of runs of A {\displaystyle A} in which at least one of the infinitely-often occurring states is in F {\textstyle \mathbf {F} } ; in symbols: L ( A ) = { i _ ∈ Σ ω ∣ I n f ( i _ ) ∩ F ≠ ∅ } . {\displaystyle L(A)=\{{\underline {i}}\in \Sigma ^{\omega }\mid \mathrm {Inf} ({\underline {i}})\cap \mathbf {F} \neq \varnothing \}.} In a (non-deterministic) Büchi automaton, the transition function δ {\textstyle \delta } is replaced with a transition relation Δ {\textstyle \Delta } that returns a set of states, and the single initial state q 0 {\textstyle q_{0}} is replaced by a set I {\textstyle I} of initial states. Generally, the term Büchi automaton without qualifier refers to non-deterministic Büchi automata. For more comprehensive formalism see also ω-automaton. == Closure properties == The set of Büchi automata is closed under the following operations. Let A = ( Q A , Σ , Δ A , I A , F A ) {\displaystyle A=(Q_{A},\Sigma ,\Delta _{A},I_{A},{F}_{A})} and B = ( Q B , Σ , Δ B , I B , F B ) {\displaystyle B=(Q_{B},\Sigma ,\Delta _{B},I_{B},{F}_{B})} be Büchi automata and C = ( Q C , Σ , Δ C , I C , F C ) {\displaystyle C=(Q_{C},\Sigma ,\Delta _{C},I_{C},{F}_{C})} be a finite automaton. Union: There is a Büchi automaton that recognizes the language L ( A ) ∪ L ( B ) . {\displaystyle L(A)\cup L(B).} Proof: If we assume, w.l.o.g., Q A ∩ Q B {\displaystyle Q_{A}\cap Q_{B}} is empty then L ( A ) ∪ L ( B ) {\displaystyle L(A)\cup L(B)} is recognized by the Büchi automaton ( Q A ∪ Q B , Σ ∪ Σ , Δ A ∪ Δ B , I A ∪ I B , F A ∪ F B ) . {\displaystyle (Q_{A}\cup Q_{B},\Sigma \cup \Sigma ,\Delta _{A}\cup \Delta _{B},I_{A}\cup I_{B},{F}_{A}\cup {F}_{B}).} Intersection: There is a Büchi automaton that recognizes the language L ( A ) ∩ L ( B ) . {\displaystyle L(A)\cap L(B).} Proof: The Büchi automaton A ′ = ( Q ′ , Σ , Δ ′ , I ′ , F ′ ) {\displaystyle A'=(Q',\Sigma ,\Delta ',I',F')} recognizes L ( A ) ∩ L ( B ) , {\displaystyle L(A)\cap L(B),} where Q ′ = Q A × Q B × { 1 , 2 } {\displaystyle Q'=Q_{A}\times Q_{B}\times \{1,2\}} Δ ′ = Δ 1 ∪ Δ 2 {\displaystyle \Delta '=\Delta _{1}\cup \Delta _{2}} Δ 1 = { ( ( q A , q B , 1 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q A ∈ F A then i = 2 else i = 1 } {\displaystyle \Delta _{1}=\{((q_{A},q_{B},1),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{A}\in F_{A}{\text{ then }}i=2{\text{ else }}i=1\}} Δ 2 = { ( ( q A , q B , 2 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q B ∈ F B then i = 1 else i = 2 } {\displaystyle \Delta _{2}=\{((q_{A},q_{B},2),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{B}\in F_{B}{\text{ then }}i=1{\text{ else }}i=2\}} I ′ = I A × I B × { 1 } {\displaystyle I'=I_{A}\times I_{B}\times \{1\}} F ′ = { ( q A , q B , 2 ) | q B ∈ F B } {\displaystyle F'=\{(q_{A},q_{B},2)|q_{B}\in F_{B}\}} By construction, r ′ = ( q A 0 , q B 0 , i 0 ) , ( q A 1 , q B 1 , i 1 ) , … {\displaystyle r'=(q_{A}^{0},q_{B}^{0},i^{0}),(q_{A}^{1},q_{B}^{1},i^{1}),\dots } is a run of automaton A' on input word w {\textstyle w} if r A = q A 0 , q A 1 , … {\displaystyle r_{A}=q_{A}^{0},q_{A}^{1},\dots } is run of A {\textstyle A} on w {\textstyle w} and r B = q B 0 , q B 1 , … {\displaystyle r_{B}=q_{B}^{0},q_{B}^{1},\dots } is run of B {\textstyle B} on w {\textstyle w} . r A {\textstyle r_{A}} is accepting and r B {\textstyle r_{B}} is accepting if r ′ {\textstyle r'} is concatenation of an infinite series of finite segments of 1-states (states with third component 1) and 2-states (states with third component 2) alternatively. There is such a series of segments of r ′ {\textstyle r'} if r ′ {\textstyle r'} is accepted by A ′ {\textstyle A'} . Concatenation: There is a Büchi automaton that recognizes the language L ( C ) ⋅ L ( A ) . {\displaystyle L(C)\cdot L(A).} Proof: If we assume, w.l.o.g., Q C ∩ Q A {\displaystyle Q_{C}\cap Q_{A}} is empty then the Büchi automaton A ′ = ( Q C ∪ Q A , Σ , Δ ′ , I ′ , F A ) {\displaystyle A'=(Q_{C}\cup Q_{A},\Sigma ,\Delta ',I',F_{A})} recognizes L ( C ) ⋅ L ( A ) {\displaystyle L(C)\cdot L(A)} , where Δ ′ = Δ A ∪ Δ C ∪ { ( q , a , q ′ ) | q ′ ∈ I A and ∃ f ∈ F C . ( q , a , f ) ∈ Δ C } {\displaystyle \Delta '=\Delta _{A}\cup \Delta _{C}\cup \{(q,a,q')|q'\in I_{A}{\text{ and }}\exists f\in F_{C}.(q,a,f)\in \Delta _{C}\}} if I C ∩ F C is empty then I ′ = I C otherwise I ′ = I C ∪ I A {\displaystyle {\text{ if }}I_{C}\cap F_{C}{\text{ is empty then }}I'=I_{C}{\text{ otherwise }}I'=I_{C}\cup I_{A}} ω-closure: If L ( C ) {\displaystyle L(C)} does not contain the empty word then there is a Büchi automaton that recognizes the language L ( C ) ω . {\displaystyle L(C)^{\omega }.} Proof: The Büchi automaton that recognizes L ( C ) ω {\displaystyle L(C)^{\omega }} is constructed in two stages. First, we construct a finite automaton A ′ {\textstyle A'} such that A ′ {\textstyle A'} also recognizes L ( C ) {\displaystyle L(C)} but there are no incoming transitions to initial states of A ′ {\textstyle A'} . So, A ′ = ( Q C ∪ { q new } , Σ , Δ ′ , { q new } , F C ) , {\displaystyle A'=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta ',\{q_{\text{new}}\},F_{C}),} where Δ ′ = Δ C ∪ { ( q new , a , q ′ ) | ∃ q ∈ I C . ( q , a , q ′ ) ∈ Δ C } . {\displaystyle \Delta '=\Delta _{C}\cup \{(q_{\text{new}},a,q')|\exists q\in I_{C}.(q,a,q')\in \Delta _{C}\}.} Note that L ( C ) = L ( A ′ ) {\displaystyle L(C)=L(A')} because L ( C ) {\displaystyle L(C)} does not contain the empty string. Second, we will construct the Büchi automaton A ″ {\textstyle A''} that recognize L ( C ) ω {\displaystyle L(C)^{\omega }} by adding a loop back to the initial state of A ′ {\textstyle A'} . So, A ″ = ( Q C ∪ { q new } , Σ , Δ ″ , { q new } , { q new } ) {\displaystyle A''=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta '',\{q_{\text{new}}\},\{q_{\text{new}}\})} , where Δ ″ = Δ ′ ∪ { ( q , a , q new ) | ∃ q ′ ∈ F C . ( q , a , q ′ ) ∈ Δ ′ } . {\displaystyle \Delta ''=\Delta '\cup \{(q,a,q_{\text{new}})|\exists q'\in F_{C}.(q,a,q')\in \Delta '\}.} Complementation:
Michael Collins (computational linguist)
Michael J. Collins (born 4 March 1970) is a researcher in the field of computational linguistics. He is the Vikram S. Pandit Professor of Computer Science at Columbia University. His research interests are in natural language processing as well as machine learning and he has made important contributions in statistical parsing and in statistical machine learning. In his studies Collins covers a wide range of topics such as parse re-ranking, tree kernels, semi-supervised learning, machine translation and exponentiated gradient algorithms with a general focus on discriminative models and structured prediction. One notable contribution is a state-of-the-art parser for the Penn Wall Street Journal corpus. As of 11 November 2015, his works have been cited 16,020 times, and he has an h-index of 47. Collins worked as a researcher at AT&T Labs between January 1999 and November 2002, and later held the positions of assistant and associate professor at M.I.T. Since January 2011, he has been a professor at Columbia University. In 2011, he was named a fellow of the Association for Computational Linguistics.
Stochastic grammar
A stochastic grammar (statistical grammar) is a grammar framework with a probabilistic notion of grammaticality: Stochastic context-free grammar Statistical parsing Data-oriented parsing Hidden Markov model (or stochastic regular grammar) Estimation theory The grammar is realized as a language model. Allowed sentences are stored in a database together with the frequency how common a sentence is. Statistical natural language processing uses stochastic, probabilistic and statistical methods, especially to resolve difficulties that arise because longer sentences are highly ambiguous when processed with realistic grammars, yielding thousands or millions of possible analyses. Methods for disambiguation often involve the use of corpora and Markov models. "A probabilistic model consists of a non-probabilistic model plus some numerical quantities; it is not true that probabilistic models are inherently simpler or less structural than non-probabilistic models." == Examples == A probabilistic method for rhyme detection is implemented by Hirjee & Brown in their study in 2013 to find internal and imperfect rhyme pairs in rap lyrics. The concept is adapted from a sequence alignment technique using BLOSUM (BLOcks SUbstitution Matrix). They were able to detect rhymes undetectable by non-probabilistic models.
NHS COVID-19
NHS COVID-19 was a voluntary contact tracing app for monitoring the spread of the COVID-19 pandemic in England and Wales, in use from 24 September 2020 until 27 April 2023. It was available for Android and iOS smartphones, and could be used by anyone aged 16 or over. Two versions of the app were created. The first was commissioned by NHSX and developed by the Pivotal division of American software company VMware. A pilot deployment began in May 2020, but on 18 June development of the app was abandoned in favour of a second design using the Apple/Google Exposure Notification system. Scotland and Northern Ireland had separate contact tracing apps. A 2023 study estimated that in its first year of use, the app's contact tracing function prevented an estimated 1 million cases, and 9,600 deaths. == Description == The app allowed users to: See the alert level of their local authority area (in Wales) or information about restrictions (in England); to enable this, the user must enter the first half of their postcode "Check in" at places displaying an NHS QR code poster (no longer required by legislation after 26 January 2022, removed from the app the next month) Be notified when they have been in close contact with someone who has tested positive for the virus Be notified when local health protection teams determine that people with the virus had attended a business or other venue around the same time as the user Check their symptoms, and book a coronavirus test if necessary If asked to self-isolate, receive information and a daily "countdown". At first, "close contact" was defined as being within 2 metres for 15 minutes, or within 4 metres for a longer time. These time durations were reduced from 29 October 2020, to as little as three minutes when the other person is at their most infectious, i.e. soon after they begin showing symptoms. === Implementation === The Android app was coded in Kotlin, and the iOS app in Swift. The backend used Java and is deployed to Amazon Web Services using Terraform. The code of the app and back-end is open-source and available on GitHub. == Context == The app was part of the UK's test and trace programme which was chaired by Dido Harding; from 12 May 2020 Tom Riordan, chief executive of Leeds City Council, led the tracing effort. == First phase and cancellation == === Description === In March 2020, NHSX commissioned a contact tracing app to monitor the spread in the United Kingdom of the coronavirus disease 2019 (COVID-19) in the 2020 pandemic, developed by the Pivotal division of American software company VMware. The app used a centralised approach, in contrast to the Google / Apple contact tracing project. NHSX consulted ethicists and GCHQ's National Cyber Security Centre (NCSC) about the privacy aspects. The app recorded the make and model of the phone and asked the user for their postcode area. It generated a unique installation identification number and also a daily identification number. It then used Bluetooth Low Energy (BLE) to record the daily identification number of other users nearby. If a user was unwell, they could tell the app about symptoms which are characteristic of COVID-19, such as a fever and cough. These details were then passed to a central NHS server. This would assess the information and notify other users that have been in contact, giving them appropriate advice such as physical distancing. The NHS would also arrange for a swab test of the unwell user and the outcome would determine further notifications to contacts: if the test confirmed infection with COVID-19, the contacts would be asked to isolate. By June 2020, £11.8 million had been spent on the app; in 2020–21, £35 million was spent on the app. === Deployment === The first public trial of the app began on the Isle of Wight on 5 May 2020 and by 11 May it had been downloaded 55,000 times. When the first national contact tracing schemes were launched – Test, Trace, Protect in Wales on 13 May, then on 28 May NHS Test and Trace in England, and Test and Protect in Scotland – the app was not ready to be included. Replying to a question at the government's daily briefing on 8 June, Hancock was unable to give a date for rollout of the app in England, saying it would be brought in "when it's right to do so". On 17 June, Lord Bethell, junior minister for Innovation at the Department of Health and Social Care, said "we're seeking to get something going before the winter ... it isn't a priority for us at the moment". On 18 June, Health Secretary Matt Hancock announced development would switch to the Apple/Google system after admitting that Apple's restrictions on usage of Bluetooth prevented the app from working effectively. At the same press briefing Dido Harding, leader of the UK's test and trace programme, said "What we've done in really rigorously testing both our own Covid-19 app and the Google-Apple version is demonstrate that none of them are working sufficiently well enough to be actually reliable to determine whether any of us should self-isolate for two weeks [and] that's true across the world". === Concerns === The first, ultimately rejected, version of the app was subject to privacy concerns, the government backtracking on initial statements that the data collected from the app would not be shared outside the NHS. Matthew Gould, CEO of NHSX, the government department responsible for the app, said the data would be accessible to other organisations, but did not disclose which. Data collected would not necessarily be anonymised and would be held in a centralised repository. Over 150 of the UK's security and privacy experts warned the app's data could be used by 'a bad actor (state, private sector, or hacker)' to spy on citizens. Fears were discussed by the House of Commons' Human Rights Select Committee about plans for the app to record user location data. Parliament's Joint Committee on Human Rights said this version of the app should not be released without proper privacy protections. The second version of the app, released nationwide, addressed these concerns by employing a decentralised framework, the Apple/Google Exposure Notification system. Under this system, users remain pseudonymous: a person diagnosed with COVID-19 does not know which people are informed about an encounter, and contacted persons do not receive any information about the person diagnosed with COVID-19. The functionality of the app was also questioned in late April and early May 2020, as the software's use of Bluetooth required the app to be constantly running, meaning users could not use other apps or lock their device if the app was to function properly. The developers of the app were said to have found a way of working around this restriction. === Related contracts === Faculty – a company linked to Cambridge Analytica – provided research and modelling to NHSX in support of the response to the pandemic. Palantir, also linked to Cambridge Analytica, provided their data management platform. These contracts began in February and March respectively. == Second phase == As outlined on cancellation of the first app on 18 June 2020, the Department of Health and Social Care published on 30 July a brief description of the "next phase" app. Users would be able to scan a QR code at venues they visit, and later be notified if they had visited a place which was the source of a number of infections; the app would also assist with identifying symptoms and ordering a test. By using the Exposure Notification system from Apple and Google, personal data would be decentralised. Zuhlke Engineering Ltd, the UK branch of Swiss-based Zühlke Group, used 70 staff to complete the development of the app in 12 weeks. Zuhlke Engineering was awarded "Development Team of the Year" title at UK IT Industry awards in November 2021 for development of NHS COVID-19 application. === Timeline === Testing of the app by NHS volunteer responders, and selected residents of the Isle of Wight and the London Borough of Newham, began around 13 August. The app was made available to the public (aged 16 or over) in England and Wales on 24 September. An updated app released on 29 October, in part from collaboration with the Alan Turing Institute, improved the accuracy of measurements of the distance between the user's phone and other phones. At the same time, the duration threshold for determining exposure was reduced; this was expected to lead to an increase in the number of users told to self-isolate. An update to the app in April 2021, timed to coincide with easing of restrictions on hospitality businesses, was blocked by Apple and Google. It was intended that users who tested positive would be asked to share their history of visited venues, to assist in warning others, but this would have contravened assurances by Apple and Google that location data from devices would not be shared. === Statistics and effectiveness === The app was downloaded six million times on the first day it was generally availa
Dynamic topic model
Within statistics, Dynamic topic models' are generative models that can be used to analyze the evolution of (unobserved) topics of a collection of documents over time. This family of models was proposed by David Blei and John Lafferty and is an extension to Latent Dirichlet Allocation (LDA) that can handle sequential documents. In LDA, both the order the words appear in a document and the order the documents appear in the corpus are oblivious to the model. Whereas words are still assumed to be exchangeable, in a dynamic topic model the order of the documents plays a fundamental role. More precisely, the documents are grouped by time slice (e.g.: years) and it is assumed that the documents of each group come from a set of topics that evolved from the set of the previous slice. == Topics == Similarly to LDA and pLSA, in a dynamic topic model, each document is viewed as a mixture of unobserved topics. Furthermore, each topic defines a multinomial distribution over a set of terms. Thus, for each word of each document, a topic is drawn from the mixture and a term is subsequently drawn from the multinomial distribution corresponding to that topic. The topics, however, evolve over time. For instance, the two most likely terms of a topic at time t could be "network" and "Zipf" (in descending order) while the most likely ones at time t+1 could be "Zipf" and "percolation" (in descending order). == Model == Define α t {\displaystyle \alpha _{t}} as the per-document topic distribution at time t. β t , k {\displaystyle \beta _{t,k}} as the word distribution of topic k at time t. η t , d {\displaystyle \eta _{t,d}} as the topic distribution for document d in time t, z t , d , n {\displaystyle z_{t,d,n}} as the topic for the nth word in document d in time t, and w t , d , n {\displaystyle w_{t,d,n}} as the specific word. In this model, the multinomial distributions α t + 1 {\displaystyle \alpha _{t+1}} and β t + 1 , k {\displaystyle \beta _{t+1,k}} are generated from α t {\displaystyle \alpha _{t}} and β t , k {\displaystyle \beta _{t,k}} , respectively. Even though multinomial distributions are usually written in terms of the mean parameters, representing them in terms of the natural parameters is better in the context of dynamic topic models. The former representation has some disadvantages due to the fact that the parameters are constrained to be non-negative and sum to one. When defining the evolution of these distributions, one would need to assure that such constraints were satisfied. Since both distributions are in the exponential family, one solution to this problem is to represent them in terms of the natural parameters, that can assume any real value and can be individually changed. Using the natural parameterization, the dynamics of the topic model are given by β t , k | β t − 1 , k ∼ N ( β t − 1 , k , σ 2 I ) {\displaystyle \beta _{t,k}|\beta _{t-1,k}\sim N(\beta _{t-1,k},\sigma ^{2}I)} and α t | α t − 1 ∼ N ( α t − 1 , δ 2 I ) {\displaystyle \alpha _{t}|\alpha _{t-1}\sim N(\alpha _{t-1},\delta ^{2}I)} . The generative process at time slice 't' is therefore: Draw topics β t , k | β t − 1 , k ∼ N ( β t − 1 , k , σ 2 I ) ∀ k {\displaystyle \beta _{t,k}|\beta _{t-1,k}\sim N(\beta _{t-1,k},\sigma ^{2}I)\forall k} Draw mixture model α t | α t − 1 ∼ N ( α t − 1 , δ 2 I ) {\displaystyle \alpha _{t}|\alpha _{t-1}\sim N(\alpha _{t-1},\delta ^{2}I)} For each document: Draw η t , d ∼ N ( α t , a 2 I ) {\displaystyle \eta _{t,d}\sim N(\alpha _{t},a^{2}I)} For each word: Draw topic Z t , d , n ∼ Mult ( π ( η t , d ) ) {\displaystyle Z_{t,d,n}\sim {\textrm {Mult}}(\pi (\eta _{t,d}))} Draw word W t , d , n ∼ Mult ( π ( β t , Z t , d , n ) ) {\displaystyle W_{t,d,n}\sim {\textrm {Mult}}(\pi (\beta _{t,Z_{t,d,n}}))} where π ( x ) {\displaystyle \pi (x)} is a mapping from the natural parameterization x to the mean parameterization, namely π ( x i ) = exp ( x i ) ∑ i exp ( x i ) {\displaystyle \pi (x_{i})={\frac {\exp(x_{i})}{\sum _{i}\exp(x_{i})}}} . == Inference == In the dynamic topic model, only W t , d , n {\displaystyle W_{t,d,n}} is observable. Learning the other parameters constitutes an inference problem. Blei and Lafferty argue that applying Gibbs sampling to do inference in this model is more difficult than in static models, due to the nonconjugacy of the Gaussian and multinomial distributions. They propose the use of variational methods, in particular, the Variational Kalman Filtering and the Variational Wavelet Regression. == Applications == In the original paper, a dynamic topic model is applied to the corpus of Science articles published between 1881 and 1999 aiming to show that this method can be used to analyze the trends of word usage inside topics. The authors also show that the model trained with past documents is able to fit documents of an incoming year better than LDA. A continuous dynamic topic model was developed by Wang et al. and applied to predict the timestamp of documents. Going beyond text documents, dynamic topic models were used to study musical influence, by learning musical topics and how they evolve in recent history.