AI Image Generators

Explore the best AI Image Generators — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

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  Digital supply chain security

  Digital supply chain security refers to efforts to enhance cyber security within the supply chain. It is a subset of supply chain security and is focused on the management of cyber security requirements for information technology systems, software and networks, which are driven by threats such as cyber-terrorism, malware, data theft and the advanced persistent threat (APT). Typical supply chain cyber security activities for minimizing risks include buying only from trusted vendors, disconnecting critical machines from outside networks, and educating users on the threats and protective measures they can take. The acting deputy undersecretary for the National Protection and Programs Directorate for the United States Department of Homeland Security, Greg Schaffer, stated at a hearing that he is aware that there are instances where malware has been found on imported electronic and computer devices sold within the United States. == Examples of supply chain cyber security threats == Network or computer hardware that is delivered with malware installed on it already. Malware that is inserted into software or hardware (by various means) Vulnerabilities in software applications and networks within the supply chain that are discovered by malicious hackers Counterfeit computer hardware == Related U.S. government efforts == Comprehensive National Cyber Initiative Defense Procurement Regulations: Noted in section 806 of the National Defense Authorization Act International Strategy for Cyberspace: White House lays out for the first time the U.S.’s vision for a secure and open Internet. The strategy outlines three main themes: diplomacy, development and defense. Diplomacy: The strategy sets out to “promote an open, interoperable, secure and reliable information and communication infrastructure” by establishing norms of acceptable state behavior built through consensus among nations. Development: Through this strategy the government seeks to “facilitate cybersecurity capacity-building abroad, bilaterally and through multilateral organizations.” The objective is to protect the global IT infrastructure and to build closer international partnerships to sustain open and secure networks. Defense: The strategy calls out that the government “will ensure that the risks associated with attacking or exploiting our networks vastly outweigh the potential benefits” and calls for all nations to investigate, apprehend and prosecute criminals and non-state actors who intrude and disrupt network systems. == Related government efforts around the world == Common Criteria offers with Evaluation Assurance Level(EAL) 4 an opportunity to evaluate all relevant aspects of the digital supply chain security like the product, the development environment, IT systems security, the processes in human resource, physical security and with the module ALC_FLR.3 (Systematic Flaw Remediation) also security update processes and methods even by physical site visits. EAL 4 is mutually recognized in countries that signed the SOGIS-MRA and up to ELA 2 in countries the signed the CCRA but including ALC_FRL.3. Russia: Russia has had non-disclosed functionality certification requirements for several years and has recently initiated the National Software Platform effort based on open-source software. This reflects the apparent desire for national autonomy, reducing dependence on foreign suppliers. India: Recognition of supply chain risk in its draft National Cybersecurity Strategy. Rather than targeting specific products for exclusion, it is considering Indigenous Innovation policies, giving preferences to domestic ITC suppliers in order to create a robust, globally competitive national presence in the sector. China: Deriving from goals in the 11th Five Year Plan (2006–2010), China introduced and pursued a mix of security-focused and aggressive Indigenous Innovation policies. China is requiring an indigenous innovation product catalog be used for its government procurement and implementing a Multi-level Protection Scheme (MLPS) which requires (among other things) product developers and manufacturers to be Chinese citizens or legal persons, and product core technology and key components must have independent Chinese or indigenous intellectual property rights. == Private sector efforts == SLSA (Supply-chain Levels for Software Artifacts) is an end-to-end framework for ensuring the integrity of software artifacts throughout the software supply chain. The requirements are inspired by Google’s internal "Binary Authorization for Borg" that has been in use for the past 8+ years and that is mandatory for all of Google's production workloads. The goal of SLSA is to improve the state of the industry, particularly open source, to defend against the most pressing integrity threats. With SLSA, consumers can make informed choices about the security posture of the software they consume. == Other references == Financial Sector Information Sharing and Analysis Center International Strategy for Cyberspace (from the White House) NSTIC SafeCode Whitepaper Archived 2013-10-21 at the Wayback Machine Trusted Technology Forum and the Open Trusted Technology Provider Standard (O-TTPS) Archived 2012-01-03 at the Wayback Machine Cyber Supply Chain Security Solution Malware Implants in Firmware Supply Chain in the Software Era INFORMATION AND COMMUNICATIONS TECHNOLOGY SUPPLY CHAIN RISK MANAGEMENT TASK FORCE: INTERIM REPORT

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  Pascale Fung

  Pascale Fung (馮雁) (born in Shanghai, China) is a co-founder and Chief Research and Innovation Officer of AMI Labs, an artificial intelligence research company focused on world models. She is a professor in the Department of Electronic & Computer Engineering and the Department of Computer Science & Engineering at the Hong Kong University of Science & Technology(HKUST). She is the director of the Centre for AI Research (CAiRE) at HKUST. She is an elected Fellow of the Institute of Electrical and Electronics Engineers (IEEE) for her “contributions to human-machine interactions”, an elected Fellow of the International Speech Communication Association for “fundamental contributions to the interdisciplinary area of spoken language human-machine interactions” and an elected Fellow of the Association for Computational Linguistics (ACL) for her “significant contributions toward statistical NLP, comparable corpora, and building intelligent systems that can understand and empathize with humans”. She is a member of the Global Future Council on Artificial Intelligence and Robotics, a think tank of the World Economic Forum, and blogs for the Forum's online publication Agenda. She is a member of the Partnership on AI. She has been invited as an AI expert to different government initiatives in China, Japan, the UAE, India, the European Union and the United Nations. Fung's publication topics include spoken language systems, natural language processing, and empathetic human-robot interaction. She co-founded the Human Language Technology Center (HLTC) and is an affiliated faculty with the Robotics Institute and the Big Data Institute, both at HKUST. Additionally, she is the founding chair of the Women Faculty Association at HKUST. She is actively involved in encouraging young women into careers in engineering and science. == Career and research interests == Fung's work is focused on building systems that try to understand and empathize with humans. She has authored and co-authored hundreds of publications, along with many journal listings and book chapters. Fung is often found in the media, among others as a writer for Scientific American, the World Economic Forum, and the London School of Economics, and the Design Society. She was a pioneer in using statistical models for natural language understanding. Her PhD thesis proposed unsupervised methods for aligning texts and mining dictionary translations in different languages by distributional properties. She is an expert in spoken language understanding and computer emotional intelligence, and is a strong proponent of technology transfer. Fung has applied many of her research group's results in the fields of, among others, robotics, IoT, and financial analytics. Her efforts led to the launch of the world's first Chinese natural language search engine in 2001, the first Chinese virtual assistant for smartphones in 2010, and the first emotional intelligent speaker in 2017. == Honors == Elected Fellow, Association for the Advancement of Artificial Intelligence (AAAI), for “significant contributions to the field of Conversational AI and to the development of ethical AI principles and algorithms” Elected Fellow, Association for Computational Linguistics (ACL), for “significant contributions toward statistical NLP, comparable corpora, and building intelligent systems that can understand and empathize with humans” Nominee, the VentureBeat AI Innovation Awards at Transform 2020, for "AI for Good" Awardee, 2017 Outstanding Women Professionals & Entrepreneurs Award, Hong Kong Women Professionals & Entrepreneurs Association Elected Fellow, Institute of Electrical and Electronics Engineers (IEEE), for “contributions to human-machine interactions” Elected Fellow, International Speech Communication Association (ISCA), for “fundamental contributions to the interdisciplinary area of spoken language human-machine interactions" Member, Global Future Council on AI and Robotics, World Economic Forum (2016–) One of the Top 50 Women of Hope, selected by List Magazine in 2014 Selected as “My Favorite Teacher” by top engineering students in 2007 and in 2009 == Affiliations == Fung is affiliated with the following institutions and organizations: Hong Kong University of Science and Technology World Economic Forum Institute of Electrical and Electronics Engineers Association for Computational Linguistics International Speech Communication Association Association for Computing Machinery Association for the Advancement of Artificial Intelligence

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  The Best Free AI Website Builder for Beginners

  In search of the best AI website builder? An AI website builder is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI website builder slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.

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  State complexity

  State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a deterministic finite automaton requires exactly 2 n {\displaystyle 2^{n}} states in the worst case. == Transformation between variants of finite automata == Finite automata can be deterministic and nondeterministic, one-way (DFA, NFA) and two-way (2DFA, 2NFA). Other related classes are unambiguous (UFA), self-verifying (SVFA) and alternating (AFA) finite automata. These automata can also be two-way (2UFA, 2SVFA, 2AFA). All these machines can accept exactly the regular languages. However, the size of different types of automata necessary to accept the same language (measured in the number of their states) may be different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle f(n)} is the least number of states in automata of the second type sufficient to recognize every language recognized by an n {\displaystyle n} -state automaton of the first type. The following results are known. NFA to DFA: 2 n {\displaystyle 2^{n}} states. This is the subset construction by Rabin and Scott, proved optimal by Lupanov. UFA to DFA: 2 n {\displaystyle 2^{n}} states, see Leung, An earlier lower bound by Schmidt was smaller. NFA to UFA: 2 n − 1 {\displaystyle 2^{n}-1} states, see Leung. There was an earlier smaller lower bound by Schmidt. SVFA to DFA: Θ ( 3 n / 3 ) {\displaystyle \Theta (3^{n/3})} states, see Jirásková and Pighizzini 2DFA to DFA: n ( n n − ( n − 1 ) n ) {\displaystyle n(n^{n}-(n-1)^{n})} states, see Kapoutsis. Earlier construction by Shepherdson used more states, and an earlier lower bound by Moore was smaller. 2DFA to NFA: ( 2 n n + 1 ) = O ( 4 n n ) {\displaystyle {\binom {2n}{n+1}}=O({\frac {4^{n}}{\sqrt {n}}})} , see Kapoutsis. Earlier construction by Birget used more states. 2NFA to NFA: ( 2 n n + 1 ) {\displaystyle {\binom {2n}{n+1}}} , see Kapoutsis. 2NFA to NFA accepting the complement: O ( 4 n ) {\displaystyle O(4^{n})} states, see Vardi. AFA to DFA: 2 2 n {\displaystyle 2^{2^{n}}} states, see Chandra, Kozen and Stockmeyer. AFA to NFA: 2 n {\displaystyle 2^{n}} states, see Fellah, Jürgensen and Yu. 2AFA to DFA: 2 n 2 n {\displaystyle 2^{n2^{n}}} , see Ladner, Lipton and Stockmeyer. 2AFA to NFA: 2 Θ ( n log ⁡ n ) {\displaystyle 2^{\Theta (n\log n)}} , see Geffert and Okhotin. === The 2DFA vs. 2NFA problem and logarithmic space === It is an open problem whether all 2NFAs can be converted to 2DFAs with polynomially many states, i.e. whether there is a polynomial p ( n ) {\displaystyle p(n)} such that for every n {\displaystyle n} -state 2NFA there exists a p ( n ) {\displaystyle p(n)} -state 2DFA. The problem was raised by Sakoda and Sipser, who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas discovered a formal relation between this problem and the L vs. NL open problem. This relation was further elaborated by Kapoutsis. == State complexity of operations for finite automata == Given a binary regularity-preserving operation on languages ∘ {\displaystyle \circ } and a family of automata X (DFA, NFA, etc.), the state complexity of ∘ {\displaystyle \circ } is an integer function f ( m , n ) {\displaystyle f(m,n)} such that for each m-state X-automaton A and n-state X-automaton B there is an f ( m , n ) {\displaystyle f(m,n)} -state X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} , and for all integers m, n there is an m-state X-automaton A and an n-state X-automaton B such that every X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} must have at least f ( m , n ) {\displaystyle f(m,n)} states. Analogous definition applies for operations with any number of arguments. The first results on state complexity of operations for DFAs were published by Maslov and by Yu, Zhuang and Salomaa. Holzer and Kutrib pioneered the state complexity of operations on NFA. The known results for basic operations are listed below. === Union === If language L 1 {\displaystyle L_{1}} requires m states and language L 2 {\displaystyle L_{2}} requires n states, how many states does L 1 ∪ L 2 {\displaystyle L_{1}\cup L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n + 1 {\displaystyle m+n+1} states, see Holzer and Kutrib. UFA: at least min ( n , m ) Ω ( log ⁡ ( min ( n , m ) ) ) {\displaystyle \min(n,m)^{\Omega (\log(\min(n,m)))}} ; between m n + m + n {\displaystyle mn+m+n} and m + n m 2 0.79 m {\displaystyle m+nm2^{0.79m}} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and 4 m + n + 4 {\displaystyle 4m+n+4} states, see Kunc and Okhotin. 2NFA: m + n {\displaystyle m+n} states, see Kunc and Okhotin. === Intersection === How many states does L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m n {\displaystyle mn} states, see Holzer and Kutrib. UFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. 2NFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. === Complementation === If language L requires n states then how many states does its complement require? DFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. NFA: 2 n {\displaystyle 2^{n}} states, see Birget. or Jirásková UFA: at least n Ω ~ ( log ⁡ n ) {\displaystyle n^{{\tilde {\Omega }}(\log n)}} states, see Göös, Kiefer and Yuan, (this follows an earlier bound by Raskin); and at most n + 1 ⋅ 2 0.5 n {\displaystyle {\sqrt {n+1}}\cdot 2^{0.5n}} states, see Indzhev and Kiefer. SVFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. 2DFA: at least n {\displaystyle n} and at most 4 n {\displaystyle 4n} states, see Geffert, Mereghetti and Pighizzini. === Concatenation === How many states does L 1 L 2 = { w 1 w 2 ∣ w 1 ∈ L 1 , w 2 ∈ L 2 } {\displaystyle L_{1}L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}} require? DFA: m ⋅ 2 n − 2 n − 1 {\displaystyle m\cdot 2^{n}-2^{n-1}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n {\displaystyle m+n} states, see Holzer and Kutrib. UFA: 3 4 2 m + n − 1 {\displaystyle {\frac {3}{4}}2^{m+n}-1} states, see Jirásek, Jirásková and Šebej. SVFA: Θ ( 3 n / 3 2 m ) {\displaystyle \Theta (3^{n/3}2^{m})} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 2 Ω ( n ) log ⁡ m {\displaystyle {\frac {2^{\Omega (n)}}{\log m}}} and at most 2 m m + 1 ⋅ 2 n n + 1 {\displaystyle 2m^{m+1}\cdot 2^{n^{n+1}}} states, see Jirásková and Okhotin. === Kleene star === DFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Šebej. SVFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 1 n 2 n 2 − 1 {\displaystyle {\frac {1}{n}}2^{{\frac {n}{2}}-1}} and at most 2 O ( n n + 1 ) {\displaystyle 2^{O(n^{n+1})}} states, see Jirásková and Okhotin. === Reversal === DFA: 2 n {\displaystyle 2^{n}} states, see Mirkin, Leiss, and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: n {\displaystyle n} states. SVFA: 2 n + 1 {\displaystyle 2n+1} states, see Jirásek, Jirásková and Szabari. 2DFA: between n + 1 {\displaystyle n+1} and n + 2 {\displaystyle n+2} states, see Jirásková and Okhotin. == Finite automata over a unary alphabet == State complexity of finite automata with a one-letter (unary) alphabet, pioneered by Chrobak, is different from the multi-letter case. Let g ( n ) = e Θ ( n ln ⁡ n ) {\displaystyle g(n)=e^{\Theta ({\sqrt {n\ln n}})}} be Landau's function. === Transformation between models === For a one-letter alphabet, transformations between different types of finite automata are sometimes more efficient than in the general case. NFA to DFA: g ( n ) + O ( n 2 ) {\displaystyle g(n)+O(n^{2})} states, see Chrobak. 2DFA to DFA: g ( n ) + O ( n ) {\displaystyle g(n)+O(n)} states, see Chrobak and Kunc and Okhotin. 2NFA to DFA: O ( g ( n ) ) {\displaystyle O(g(n))} states, see Mereghetti and Pighizzini. and Geffert, Mereghetti and Pighizzini. NFA to 2DFA: at most O ( n 2 ) {\displaystyle O(n^{2})} states, see Chrobak. 2NFA to 2DFA: at most n O ( log ⁡ n ) {\displaystyle n^{O(\log n)}} states, proved by implementing the method of Savitch's theorem, see

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  Auto-defrost

  Auto-defrost, automatic defrost or self-defrosting is a technique which regularly defrosts the evaporator in a refrigerator or freezer. Appliances using this technique are often called frost free, frostless, or no-frost. == Mechanism == The defrost mechanism in a refrigerator heats the cooling element (evaporator coil) for a short period of time and melts the frost that has formed on it. The resulting water drains through a duct at the back of the unit. Defrosting is controlled by an electric or electronic timer. For every 6, 8, 10, 12 or 24 hours of compressor operation, it turns on a defrost heater for 15 minutes to half an hour. The defrost heater, having a typical power rating of 350W to 600W, is often mounted just below the evaporator in top and bottom-freezer models. It can also be located below and in the middle of the evaporator in side-by-side models. It may be protected from short circuits by means of fusible links. In older refrigerators, the timer runs continuously. In newer designs, the timer only runs while the compressor runs, so the longer the refrigerator door is closed, the less time the heater will run for and the more energy is saved. A defrost thermostat opens the heater circuit when the evaporator temperature rises above a preset temperature, 40°F (5°C) or more, thereby preventing excessive heating of the freezer compartment. The defrost timer is such that either the compressor or the defrost heater is on, but not both at the same time. Inside the freezer, air is circulated by means of one or more fans. In a typical design cold air from the freezer compartment is ducted to the fresh food compartment and circulated back into the freezer compartment. Air circulation helps sublimate any ice or frost that may form on frozen items in the freezer compartment. While defrosting, this fan is stopped to prevent heated-up air from reaching the food compartment. Instead of the normal cooling elements being embedded in the freezer liner, auto-defrost elements are behind or beneath the liner. This allows them to be heated for short periods of time to dispose of frost, without heating the contents of the freezer. Alternatively, some systems use the hot gas in the condenser to defrost the evaporator. This is done by means of a circuit that is cross-linked by a three-way valve. The hot gas quickly heats up the evaporator and defrosts it. This system is primarily used in commercial applications such as ice-cream displays. == Application == While this technique was originally applied to the refrigerator compartment, it was later used for freezer compartment as well. A combined refrigerator/freezer which applies self-defrosting to the refrigerator compartment only is usually called "partial frost free" or semi-automatic defrost (some brands call these "Auto Defrost" while Frigidaire referred to their semi-automatic models as "Cycla-Matic," Kelvinator often named these models as "Cyclic Defrost" ). These refrigerators usually have a pan underneath where water from the melted frost in the refrigerator section evaporates. Freezers with automatic defrosting and combined refrigerator/freezer units which also apply self defrosting to their freezer compartment are called "frost free". The latter usually feature an air connection between the two compartments with the air passage to the refrigerator compartment regulated by a damper. By this means, a controlled portion of the air coming from the freezer reaches the refrigerator. Some older models have no air circulation between their freezer and refrigerator sections. Instead, they use an independent cooling system (for example: an evaporator coil with a defrost heater and a circulating fan in the freezer and a cold-plate or open-coil evaporator in the refrigerator. "Frost-Free" refrigerator/freezer units usually use a heating element to defrost their evaporators, a pan to collect and evaporate water from the frost that melts from the cold plate and/or evaporator coil, a timer which turns off the compressor and turns on the defrost element usually from once to 4 times a day for periods usually ranging from 15 to 30 minutes, a defrost limiter thermostat that turns off the heating element before the temperature rises too much while the timer is still in its defrost phase. Some models also feature a drain heater to prevent ice from blocking the drain. Other early types of refrigerators also use hot gas defrost instead of electric heaters. These reverse the evaporator and condenser sides for the defrost cycle. Some newer refrigerator/freezer models have a computer that monitors how many times each door is opened and uses this data to control defrost scheduling thereby reducing power use. == Advantages == No need to manually defrost the frost buildup, therefore power consumption will not increase with time. Food packaging is easier to see. Most frozen food will not stick together. Smells are limited, especially in total frost-free appliances because the air always circulates. Better temperature management. == Disadvantages == The system can be more expensive to run when usage is high and if the fan continues or starts to run when the door is opened. A thermal cutout safety device is required to prevent overheating of the heating element. Increased electrical and mechanical complexity compared to a basic upright freezer or chest freezer, making it more prone to component failure. The temperature of the freezer contents rises during the defrosting cycles, especially if there is a light load in the freezer. This can cause "freezer burn" on articles placed in the freezer, from partially defrosting, then re-freezing On hot, humid days condensation will sometimes form around the refrigerator doors. Defrosting may not be completed by the time the defrost timer cycles back to normal operation (especially in hot, humid conditions with frequent door openings), leaving ice/frost on the evaporator coils. This condition can lead to "icing" which will interfere with the operation of the refrigerator. In laboratories, self-defrosting freezers must not be used to store certain delicate reagents such as enzymes, because the temperature cycling can degrade them. In addition, water can evaporate out of containers that do not have a very tight seal, altering the concentration of the reagents. Self-defrosting freezers should never be used to store flammable chemicals.

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  Stephen Wolfram

  Stephen Wolfram ( WUUL-frəm; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society. As a businessman, Wolfram is the founder and CEO of the software company Wolfram Research, where he works as chief designer of Mathematica and the Wolfram Alpha answer engine. == Early life == === Family === Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander. Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. Wolfram's mother, Sybil Wolfram, was a Fellow and Tutor in Philosophy at Lady Margaret Hall at University of Oxford from 1964 to 1993. Wolfram is married to a mathematician. They have four children together. === Education === Wolfram was educated at Eton College, but left prematurely in 1976. As a young child, Wolfram had difficulties learning arithmetic. He entered St. John's College, Oxford, at age 17 and left in 1978 without graduating to attend the California Institute of Technology the following year, where he received a PhD in particle physics in 1980. Wolfram's thesis committee was composed of Richard Feynman, Peter Goldreich, Frank J. Sciulli, and Steven Frautschi, and chaired by Richard D. Field. == Early career == Wolfram, at the age of 15, began research in applied quantum field theory and particle physics and published scientific papers in peer-reviewed scientific journals; by the time he left Oxford, he had published ten such papers. Following his PhD, Wolfram joined the faculty at Caltech and became the youngest recipient of a MacArthur Fellowship in 1981, at age 21. == Later career == === Complex systems and cellular automata === In 1983, Wolfram left for the School of Natural Sciences of the Institute for Advanced Study in Princeton. By that time, he was no longer interested in particle physics. Instead, he began pursuing investigations into cellular automata, mainly with computer simulations. He produced a series of papers investigating the class of elementary cellular automata, conceiving the Wolfram code, a naming system for one-dimensional cellular automata, and a classification scheme for the complexity of their behaviour. He conjectured that the Rule 110 cellular automaton might be Turing complete, which a research assistant to Wolfram, Matthew Cook, later proved correct. Wolfram sued Cook and temporarily blocked publication of the work on Rule 110 for allegedly violating a non-disclosure agreement until Wolfram could publish the work in his controversial book A New Kind of Science. Wolfram's cellular-automata work came to be cited in more than 10,000 papers. In the mid-1980s, Wolfram worked on simulations of physical processes (such as turbulent fluid flow) with cellular automata on the Connection Machine alongside Richard Feynman and helped initiate the field of complex systems. In 1984, he was a participant in the Founding Workshops of the Santa Fe Institute, along with Nobel laureates Murray Gell-Mann, Manfred Eigen, and Philip Warren Anderson, and future laureate Frank Wilczek. In 1986, he founded the Center for Complex Systems Research (CCSR) at the University of Illinois Urbana–Champaign. In 1987, he founded the journal Complex Systems. === Symbolic Manipulation Program === Wolfram led the development of the computer algebra system SMP (Symbolic Manipulation Program) in the Caltech physics department during 1979–1981. A dispute with the administration over the intellectual property rights regarding SMP—patents, copyright, and faculty involvement in commercial ventures—eventually led him to resign from Caltech. SMP was further developed and marketed commercially by Inference Corp. of Los Angeles during 1983–1988. === Mathematica === In 1986, Wolfram left the Institute for Advanced Study for the University of Illinois Urbana–Champaign, where he had founded their Center for Complex Systems Research, and started to develop the computer algebra system Mathematica, which was released on 23 June 1988, when he left academia. In 1987, he founded Wolfram Research, which continues to develop and market the program. === A New Kind of Science === From 1992 to 2002, Wolfram worked on his controversial book A New Kind of Science, which presents an empirical study of simple computational systems. Additionally, it argues that for fundamental reasons these types of systems, rather than traditional mathematics, are needed to model and understand complexity in nature. Wolfram's conclusion is that the universe is discrete in its nature, and runs on fundamental laws that can be described as simple programs. He predicts that a realization of this within scientific communities will have a revolutionary influence on physics, chemistry, biology, and most other scientific areas, hence the book's title. The book was met with skepticism and criticism that Wolfram took credit for the work of others and made conclusions without evidence to support them. === Wolfram Alpha computational knowledge engine === In March 2009, Wolfram announced Wolfram Alpha, an answer engine. Wolfram Alpha launched in May 2009, and a paid-for version with extra features launched in February 2012 that was met with criticism for its high price, which later dropped from $50 to $2. The engine is based on natural language processing and a large library of rules-based algorithms. The application programming interface allows other applications to extend and enhance Wolfram Alpha. === Touchpress === In 2010, Wolfram co-founded Touchpress with Theodore Gray, Max Whitby, and John Cromie. The company specialised in creating in-depth premium apps and games covering a wide range of educational subjects designed for children, parents, students, and educators. Touchpress published more than 100 apps. The company is no longer active. === Wolfram Language === In March 2014, at the annual South by Southwest (SXSW) event, Wolfram officially announced the Wolfram Language as a new general multi-paradigm programming language, though it was previously available through Mathematica and not an entirely new programming language. The documentation for the language was pre-released in October 2013 to coincide with the bundling of Mathematica and the Wolfram Language on every Raspberry Pi computer with some controversy because of the proprietary nature of the Wolfram Language. While the Wolfram Language has existed for over 30 years as the primary programming language used in Mathematica, it was not officially named until 2014, and is not widely used. === Wolfram Physics Project === In April 2020, Wolfram announced the "Wolfram Physics Project" as an effort to reduce and explain all the laws of physics within a paradigm of a hypergraph that is transformed by minimal rewriting rules that obey the Church–Rosser property. The effort is a continuation of the ideas he originally described in A New Kind of Science. Wolfram claims that "From an extremely simple model, we're able to reproduce special relativity, general relativity and the core results of quantum mechanics." Physicists are generally unimpressed with Wolfram's claim, and say his results are non-quantitative and arbitrary. == Personal interests and activities == Wolfram has a log of personal analytics, including emails received and sent, keystrokes made, meetings and events attended, recordings of phone calls, and even physical movement dating back to the 1980s. In the preface of A New Kind of Science, he noted that he recorded over 100 million keystrokes and 100 mouse miles. He has said that personal analytics "can give us a whole new dimension to experiencing our lives." Wolfram was a scientific consultant for the 2016 film Arrival. He and his son Christopher Wolfram wrote some of the code featured on screen, such as the code in graphics depicting an analysis of the alien logograms, for which they used the Wolfram Language.

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  Language and Computers

  Language and Computers: Studies in Practical Linguistics (ISSN 0921-5034) is a book series on corpus linguistics and related areas. As studies in linguistics, volumes in the series have, by definition, their foundations in linguistic theory; however, they are not concerned with theory for theory's sake, but always with a definite direct or indirect interest in the possibilities of practical application in the dynamic area where language and computers meet. The book series was founded in 1988, and is published by Brill|Rodopi. == Editors == Christian Mair Charles F. Meyer == Volumes == Volumes include: # 77. English Corpus Linguistics: Variation in Time, Space and Genre. Selected papers from ICAME 32., Edited by Gisle Andersen and Kristin Bech. ISBN 978-90-420-3679-6 E-ISBN 978-94-012-0940-3 # 76. English Corpus Linguistics: Crossing Paths., Edited by Merja Kytö. ISBN 978-90-420-3518-8 E-ISBN 978-94-012-0793-5 # 75. Corpus Linguistics and Variation in English.Theory and Description., Edited by Joybrato Mukherjee and Magnus Huber. ISBN 978-90-420-3495-2 E-ISBN 978-94-012-0771-3 # 74. English Corpus Linguistics: Looking back, Moving forward. Papers from the 30th International Conference on English Language Research on Computerized Corpora (ICAME 30), Lancaster, UK, 27–31 May 2009., Edited by Sebastian Hoffmann, Paul Rayson and Geoffrey Leech. ISBN 978-90-420-3466-2 E-ISBN 978-94-012-0747-8 #73. Corpus-based Studies in Language Use, Language Learning, and Language Documentation., Edited by John Newman, Harald Baayen and Sally Rice. ISBN 978-90-420-3401-3 E-ISBN 978-94-012-0688-4 #72. The Progressive in Modern English. A Corpus-Based Study of Grammaticalization and Related Changes., by Svenja Kranich. ISBN 978-90-420-3143-2 E-ISBN 978-90-420-3144-9 #71. Corpus-linguistic applications. Current studies, new directions, Edited by Stefan Th. Gries, Stefanie Wulff, and Mark Davies.. ISBN 978-90-420-2800-5 #70. A resource-light approach to morpho-syntactic tagging., by Anna Feldman and Jirka Hana. ISBN 978-90-420-2768-8 #69. Corpus Linguistics. Refinements and Reassessments., Edited by Antoinette Renouf and Andrew Kehoe. ISBN 978-90-420-2597-4 #68. Corpora: Pragmatics and Discourse. Papers from the 29th International Conference on English Language Research on Computerized Corpora (ICAME 29). Ascona, Switzerland, 14–18 May 2008., Edited by Andreas H. Jucker, Daniel Schreier and Marianne Hundt. ISBN 978-90-420-2592-9 #67. Modals and Quasi-modals in English., by Peter Collins. ISBN 978-90-420-2532-5 #66. Linking up contrastive and learner corpus research., Edited by Gaëtanelle Gilquin, Szilvia Papp and María Belén Díez-Bedmar. ISBN 978-90-420-2446-5 #64. Language, People, Numbers. Corpus Linguistics and Society., Edited by Andrea Gerbig and Oliver Mason. ISBN 978-90-420-2350-5 #63. Variation and change in the lexicon. A corpus-based analysis of adjectives in English ending in –ic and –ical. , by Mark Kaunisto. ISBN 978-90-420-2233-1 #62. Corpus Linguistics 25 Years on., Edited by Roberta Facchinetti. ISBN 978-90-420-2195-2 #61. Corpora in the Foreign Language Classroom. Selected papers from the Sixth International Conference on Teaching and Language Corpora (TaLC 6), Edited by Encarnación Hidalgo, Luis Quereda and Juan Santana. ISBN 978-90-420-2142-6 #60. Corpus Linguistics Beyond the Word. Corpus Research from Phrase to Discourse, Edited by Eileen Fitzpatrick. ISBN 978-90-420-2135-8 #59. Corpus Linguistics and the Web., Edited by Marianne Hundt, Nadja Nesselhauf and Carolin Biewer. ISBN 978-90-420-2128-0 #58. English mediopassive constructions. A cognitive, corpus-based study of their origin, spread, and current status, by Marianne Hundt. ISBN 978-90-420-2127-3 / ISBN 90-420-2127-6

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  Self-verifying finite automaton

  In automata theory, a self-verifying finite automaton (SVFA) is a special kind of a nondeterministic finite automaton (NFA) with a symmetric kind of nondeterminism introduced by Hromkovič and Schnitger. Generally, in self-verifying nondeterminism, each computation path is concluded with any of the three possible answers: yes, no, and I do not know. For each input string, no two paths may give contradictory answers, namely both answers yes and no on the same input are not possible. At least one path must give answer yes or no, and if it is yes then the string is considered accepted. SVFA accept the same class of languages as deterministic finite automata (DFA) and NFA but have different state complexity. == Formal definition == An SVFA is represented formally by a 6-tuple, A=(Q, Σ, Δ, q0, Fa, Fr) such that (Q, Σ, Δ, q0, Fa) is an NFA, and Fa, Fr are disjoint subsets of Q. For each word w = a1a2 … an, a computation is a sequence of states r0,r1, …, rn, in Q with the following conditions: r0 = q0 ri+1 ∈ Δ(ri, ai+1), for i = 0, …, n−1. If rn ∈ Fa then the computation is accepting, and if rn ∈ Fr then the computation is rejecting. There is a requirement that for each w there is at least one accepting computation or at least one rejecting computation but not both. == Results == Each DFA is a SVFA, but not vice versa. Jirásková and Pighizzini proved that for every SVFA of n states, there exists an equivalent DFA of g ( n ) = Θ ( 3 n / 3 ) {\displaystyle g(n)=\Theta (3^{n/3})} states. Furthermore, for each positive integer n, there exists an n-state SVFA such that the minimal equivalent DFA has exactly g ( n ) {\displaystyle g(n)} states. Other results on the state complexity of SVFA were obtained by Jirásková and her colleagues.

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  AARON

  AARON is the collective name for a series of computer programs written by artist Harold Cohen that create original artistic images autonomously, which set it apart from previous programs. Proceeding from Cohen's initial question "What are the minimum conditions under which a set of marks functions as an image?", AARON was in development between 1972 and the 2010s. As the software is not open source, its development effectively ended with Cohen's death in 2016. The name "AARON" does not seem to be an acronym; rather, it was a name chosen to start with the letter "A" so that the names of successive programs could follow it alphabetically. However, Cohen did not create any other major programs. Initial versions of AARON created abstract drawings that grew more complex through the 1970s. More representational imagery was added in the 1980s; first rocks, then plants, then people. In the 1990s more representational figures set in interior scenes were added, along with color. AARON returned to more abstract imagery, this time in color, in the early 2000s. Cohen used machines that allowed AARON to produce physical artwork. The first machines drew in black and white using a succession of custom-built "turtle" and flatbed plotter devices. Cohen would sometimes color these images by hand in fabric dye (Procion), or scale them up to make larger paintings and murals. In the 1990s Cohen built a series of digital painting machines to output AARON's images in ink and fabric dye. His later work used a large-scale inkjet printer on canvas. Development of AARON began in the C programming language then switched to Lisp in the early 1990s. Cohen credits Lisp with helping him solve the challenges he faced in adding color capabilities to AARON. An article about Cohen appeared in Computer Answers that describes AARON and shows two line drawings that were exhibited at the Tate gallery. The article goes on to describe the workings of AARON, then running on a DEC VAX 750 minicomputer. Raymond Kurzweil's company has produced a downloadable screensaver of AARON for Microsoft Windows PCs. This version of AARON can also produce printable images. AARON's source code is not publicly available, but Cohen has described AARON's operations in various essays and it is discussed in abstract in Pamela McCorduck's book. AARON cannot learn new styles or imagery on its own; each new capability must be hand-coded by Cohen. It is capable of producing a practically infinite supply of distinct images in its own style. Examples of these images have been exhibited in galleries worldwide. AARON's artwork has been used as an artistic equivalent of the Turing test. It does seem however that AARON's output follows a noticeable formula (figures standing next to a potted plant, framed within a colored square is a common theme). Cohen is very careful not to claim that AARON is creative. But he does ask "If what AARON is making is not art, what is it exactly, and in what ways, other than its origin, does it differ from the 'real thing?' If it is not thinking, what exactly is it doing?" — The further exploits of AARON, Painter. The Whitney Museum featured AARON in 2024, showcasing the evolution of AARON as the earliest artificial intelligence (AI) program for artmaking.

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  Timo Honkela

  Timo Untamo Honkela (August 4, 1962 – May 9, 2020) was a computer scientist at the University of Helsinki, Aalto University School of Science and Aalto University School of Art, Design and Architecture. He holds a PhD from Helsinki University of Technology. From 2014 until 2018 he held a fixed-term professorship at the University of Helsinki. Before joining the University of Helsinki he worked as a non-tenured professor in two Schools of the Aalto University, The School of Art, Design and Architecture and the School of Science. He has presented his thoughts on his studies and work in the joint blog 375 Humanists. Timo Honkela conducted research on several areas related to knowledge engineering, cognitive modeling and natural language processing. Honkela was born in Kalajoki. From 1998 to 2000 he worked as a professor in the Aalto Media Lab. To the media Lab Honkela brought his expertise in Kohonen self-organising map (SOM) and worked closely with artist and designers around the topic. In 2001 Honkela collaborated with George Legrady to produce an interactive museum installation, Pockets Full of Memories to the Centre Georges Pompidou, National Museum of Modern Art in Paris. The concept, created by Legrady, provided for visitors a possibility to scan their own objects to a database and then organise them by Kohonen Self-Organizing Map algorithm. In 2017 Honkela published a book in Finnish. The book Rauhankone (English: Peace Machine) presents his idea of designing artificial intelligence and machine learning to serve humanity, in practice to help people to live in peace with each other. He died in Helsinki. == Publications == Timo Honkela, Wlodzislaw Duch, Mark Girolami and Samuel Kaski (editors): Artificial Neural Networks and Machine Learning, Springer, 2011. Jorma Laaksonen and Timo Honkela (editors): Advances in Self-Organizing Maps, Springer, 2011. Timo Honkela: Rauhankone. Gaudeamus, 2017.

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  Simon Godsill

  Simon John Godsill (born 2 December 1965) is professor of statistical signal processing at the University of Cambridge, and a professorial fellow at Corpus Christi College. He is also a member of the Centre for Science and Policy. His main area of research is Bayesian statistics and stochastic sampling methodologies, particularly particle filtering. == Education == Godsill obtained both undergraduate and Ph.D. degrees from the Department of Engineering at Cambridge University, whilst a member of Selwyn College. He obtained a first class degree in the Electrical and Information Sciences Tripos. The title of his 1993 Ph.D. thesis was "The Restoration of Degraded Audio Signals" and his Ph.D. supervisor was Peter Rayner, whom he shared with Michael Richard Lynch. == Career == Godsill has published over 250 articles in peer reviewed journals, along with the books Digital audio restoration: a statistical model based approach and Compressed sensing & sparse filtering. == Business interests == Godsill is currently a director of CEDAR Audio Ltd, a Cambridge-based company that applies Bayesian mathematics for purposes of noise reduction in audio data. In February 2005, the company received a Sci-Tech Academy Award (a 'Technical Oscar') for its services to the movie industry, and a stream of innovations appeared over the following years with corresponding recognition including induction into the Audio Technology Hall of Fame (2008), a Cinema Audio Society Award (2009). Godsill is also a director at Input Dynamics Ltd, a Cambridge-based company that applies Bayesian techniques to touch screen technology. Godsill is involved with the research effort at BMLL Technologies, a Cambridge spin-off working in the field of machine learning application in the financial sector.

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  Claire Cardie

  Claire Cardie is an American computer scientist specializing in natural language processing. Since 2006, she has been a professor of computer science and information science at Cornell University, and from 2010 to 2011 she was the first Charles and Barbara Weiss Chair of Information Science at Cornell. Her research interests include coreference resolution and sentiment analysis. == Education and career == Cardie is a 1982 graduate of Yale University, majoring in computer science. After working for several companies as a computer programmer, she returned to graduate study in the late 1980s and completed her Ph.D. at the University of Massachusetts Amherst in 1994. Her dissertation, Domain-Specific Knowledge Acquisition for Conceptual Sentence Analysis, was supervised by Wendy Lehnert. She has been on the Cornell University faculty since 1994, initially in computer science and since 2005 also in information science. She was an assistant professor (1994–2000) and associate professor (2000–06), before being promoted to a full professorship in 2006. In 2007 she founded a start-up company, Appinions, and she was its chief scientist until 2015. Her doctoral students at Cornell have included Amit Singhal and Kiri Wagstaff. == Recognition == Cardie became a Fellow of the Association for Computational Linguistics in 2016. She was elected as an ACM Fellow in 2019 "for contributions to natural language processing, including coreference resolution, information and opinion extraction". She was named to the 2021 class of Fellows of the American Association for the Advancement of Science.

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  Vicuna LLM

  Vicuna LLM is an omnibus large language model used in AI research. Its methodology is to enable the public at large to contrast and compare the accuracy of LLMs "in the wild" (an example of citizen science) and to vote on their output; a question-and-answer chat format is used. At the beginning of each round two LLM chatbots from a diverse pool of nine are presented randomly and anonymously, their identities only being revealed upon voting on their answers. The user has the option of either replaying ("regenerating") a round, or beginning an entirely fresh one with new LLMs. (The user also has the option of choosing which LLMs to do battle.) Based on Llama 2, it is an open source project, and it itself has become the subject of academic research in the burgeoning field. A non-commercial, public demo of the Vicuna-13b model is available to access using LMSYS.

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  Best AI Code-review Tools in 2026

  Looking for the best AI code-review tool? An AI code-review tool is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI code-review tool slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Markov switching multifractal

  In financial econometrics (the application of statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations. MSM captures the outliers, log-memory-like volatility persistence and power variation of financial returns. In currency and equity series, MSM compares favorably with standard volatility models such as GARCH(1,1) and FIGARCH both in- and out-of-sample. MSM is used by practitioners in the financial industry for different types of forecasts. == MSM specification == The MSM model can be specified in both discrete time and continuous time. === Discrete time === Let P t {\displaystyle P_{t}} denote the price of a financial asset, and let r t = ln ⁡ ( P t / P t − 1 ) {\displaystyle r_{t}=\ln(P_{t}/P_{t-1})} denote the return over two consecutive periods. In MSM, returns are specified as r t = μ + σ ¯ ( M 1 , t M 2 , t . . . M k ¯ , t ) 1 / 2 ϵ t , {\displaystyle r_{t}=\mu +{\bar {\sigma }}(M_{1,t}M_{2,t}...M_{{\bar {k}},t})^{1/2}\epsilon _{t},} where μ {\displaystyle \mu } and σ {\displaystyle \sigma } are constants and { ϵ t {\displaystyle \epsilon _{t}} } are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector: M t = ( M 1 , t M 2 , t … M k ¯ , t ) ∈ R + k ¯ . {\displaystyle M_{t}=(M_{1,t}M_{2,t}\dots M_{{\bar {k}},t})\in R_{+}^{\bar {k}}.} Given the volatility state M t {\displaystyle M_{t}} , the next-period multiplier M k , t + 1 {\displaystyle M_{k,t+1}} is drawn from a fixed distribution M with probability γ k {\displaystyle \gamma _{k}} , and is otherwise left unchanged. The transition probabilities are specified by γ k = 1 − ( 1 − γ 1 ) ( b k − 1 ) {\displaystyle \gamma _{k}=1-(1-\gamma _{1})^{(b^{k-1})}} . The sequence γ k {\displaystyle \gamma _{k}} is approximately geometric γ k ≈ γ 1 b k − 1 {\displaystyle \gamma _{k}\approx \gamma _{1}b^{k-1}} at low frequency. The marginal distribution M has a unit mean, has a positive support, and is independent of k. ==== Binomial MSM ==== In empirical applications, the distribution M is often a discrete distribution that can take the values m 0 {\displaystyle m_{0}} or 2 − m 0 {\displaystyle 2-m_{0}} with equal probability. The return process r t {\displaystyle r_{t}} is then specified by the parameters θ = ( m 0 , μ , σ ¯ , b , γ 1 ) {\displaystyle \theta =(m_{0},\mu ,{\bar {\sigma }},b,\gamma _{1})} . Note that the number of parameters is the same for all k ¯ > 1 {\displaystyle {\bar {k}}>1} . === Continuous time === MSM is similarly defined in continuous time. The price process follows the diffusion: d P t P t = μ d t + σ ( M t ) d W t , {\displaystyle {\frac {dP_{t}}{P_{t}}}=\mu dt+\sigma (M_{t})\,dW_{t},} where σ ( M t ) = σ ¯ ( M 1 , t … M k ¯ , t ) 1 / 2 {\displaystyle \sigma (M_{t})={\bar {\sigma }}(M_{1,t}\dots M_{{\bar {k}},t})^{1/2}} , W t {\displaystyle W_{t}} is a standard Brownian motion, and μ {\displaystyle \mu } and σ ¯ {\displaystyle {\bar {\sigma }}} are constants. Each component follows the dynamics: The intensities vary geometrically with k: γ k = γ 1 b k − 1 . {\displaystyle \gamma _{k}=\gamma _{1}b^{k-1}.} When the number of components k ¯ {\displaystyle {\bar {k}}} goes to infinity, continuous-time MSM converges to a multifractal diffusion, whose sample paths take a continuum of local Hölder exponents on any finite time interval. == Inference and closed-form likelihood == When M {\displaystyle M} has a discrete distribution, the Markov state vector M t {\displaystyle M_{t}} takes finitely many values m 1 , . . . , m d ∈ R + k ¯ {\displaystyle m^{1},...,m^{d}\in R_{+}^{\bar {k}}} . For instance, there are d = 2 k ¯ {\displaystyle d=2^{\bar {k}}} possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix A = ( a i , j ) 1 ≤ i , j ≤ d {\displaystyle A=(a_{i,j})_{1\leq i,j\leq d}} with components a i , j = P ( M t + 1 = m j | M t = m i ) {\displaystyle a_{i,j}=P\left(M_{t+1}=m^{j}|M_{t}=m^{i}\right)} . Conditional on the volatility state, the return r t {\displaystyle r_{t}} has Gaussian density f ( r t | M t = m i ) = 1 2 π σ 2 ( m i ) exp ⁡ [ − ( r t − μ ) 2 2 σ 2 ( m i ) ] . {\displaystyle f(r_{t}|M_{t}=m^{i})={\frac {1}{\sqrt {2\pi \sigma ^{2}(m^{i})}}}\exp \left[-{\frac {(r_{t}-\mu )^{2}}{2\sigma ^{2}(m^{i})}}\right].} === Conditional distribution === === Closed-form Likelihood === The log likelihood function has the following analytical expression: ln ⁡ L ( r 1 , … , r T ; θ ) = ∑ t = 1 T ln ⁡ [ ω ( r t ) . ( Π t − 1 A ) ] . {\displaystyle \ln L(r_{1},\dots ,r_{T};\theta )=\sum _{t=1}^{T}\ln[\omega (r_{t}).(\Pi _{t-1}A)].} Maximum likelihood provides reasonably precise estimates in finite samples. === Other estimation methods === When M {\displaystyle M} has a continuous distribution, estimation can proceed by simulated method of moments, or simulated likelihood via a particle filter. == Forecasting == Given r 1 , … , r t {\displaystyle r_{1},\dots ,r_{t}} , the conditional distribution of the latent state vector at date t + n {\displaystyle t+n} is given by: Π ^ t , n = Π t A n . {\displaystyle {\hat {\Pi }}_{t,n}=\Pi _{t}A^{n}.\,} MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH, and Fractionally Integrated GARCH. Lux obtains similar results using linear predictions. == Applications == === Multiple assets and value-at-risk === Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities. === Asset pricing === In financial economics, MSM has been used to analyze the pricing implications of multifrequency risk. The models have had some success in explaining the excess volatility of stock returns compared to fundamentals and the negative skewness of equity returns. They have also been used to generate multifractal jump-diffusions. == Related approaches == MSM is a stochastic volatility model with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton. MSM is closely related to the Multifractal Model of Asset Returns. MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process. MSM provides a pure regime-switching formulation of multifractal measures, which were pioneered by Benoit Mandelbrot.

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