A smartphone kill switch is a software-based security feature that allows a smartphone's owner to remotely render it inoperable if it is lost or stolen, thereby deterring theft. There have been a number of initiatives to legally require kill switches on smartphones. Smartphones have high resale value, and are therefore often the target of theft, with thieves selling them to cartels for resale. A kill switch can deter theft by making devices worthless. == Legal requirements == In the United States, Minnesota was the first state to pass a bill requiring smartphones to have such a feature, and California was the first to require that the feature be turned on by default. The California law requires the kill switch to be resistant to reinstallation of the phone's operating system. The CTIA initially resisted the legislation, fearing that it would make phones easier to hack, but later supported kill switches. There is evidence that this legislation has been effective, with smartphone theft declining by 50% between 2013 and 2017 in San Francisco. Secure Our Smartphones (S.O.S.), a New York State and San Francisco initiative started by New York State Attorney General Eric Schneiderman and San Francisco District Attorney George Gascón. The initiative is co-chaired by Schneiderman, Gascón and Boris Johnson, and has 105 members. == Examples == An Android phone signed into a Google account can be remotely locked and erased via Google's Find My Device service, as long as it is connected to the Internet. To prevent this, a thief must sign the device out of Google before the owner locks or erases it. iPhones have a similar service.
Robot learning
Robot learning is a research field at the intersection of machine learning and robotics. It studies techniques allowing a robot to acquire novel skills or adapt to its environment through learning algorithms. The embodiment of the robot, situated in a physical embedding, provides at the same time specific difficulties (e.g. high-dimensionality, real time constraints for collecting data and learning) and opportunities for guiding the learning process (e.g. sensorimotor synergies, motor primitives). Example of skills that are targeted by learning algorithms include sensorimotor skills such as locomotion, grasping, active object categorization, as well as interactive skills such as joint manipulation of an object with a human peer, and linguistic skills such as the grounded and situated meaning of human language. Learning can happen either through autonomous self-exploration or through guidance from a human teacher, like for example in robot learning by imitation. Robot learning can be closely related to adaptive control, reinforcement learning as well as developmental robotics which considers the problem of autonomous lifelong acquisition of repertoires of skills. While machine learning is frequently used by computer vision algorithms employed in the context of robotics, these applications are usually not referred to as "robot learning". == Imitation learning == Many research groups are developing techniques where robots learn by imitating. This includes various techniques for learning from demonstration (sometimes also referred to as "programming by demonstration") and observational learning. == Sharing learned skills and knowledge == In Tellex's "Million Object Challenge", the goal is robots that learn how to spot and handle simple items and upload their data to the cloud to allow other robots to analyze and use the information. RoboBrain is a knowledge engine for robots which can be freely accessed by any device wishing to carry out a task. The database gathers new information about tasks as robots perform them, by searching the Internet, interpreting natural language text, images, and videos, object recognition as well as interaction. The project is led by Ashutosh Saxena at Stanford University. RoboEarth is a project that has been described as a "World Wide Web for robots" − it is a network and database repository where robots can share information and learn from each other and a cloud for outsourcing heavy computation tasks. The project brings together researchers from five major universities in Germany, the Netherlands and Spain and is backed by the European Union. Google Research, DeepMind, and Google X have decided to allow their robots share their experiences. == Vision-language-action model == Research groups and companies are developing vision-language-action models, foundation models that allow robotic control through the combination of vision and language. Google DeepMind, Figure AI and Hugging Face are actively working on that.
Algorithm
In mathematics and computer science, an algorithm ( ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a heuristic is an approach to solving problems without well-defined correct or optimal results. For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and input, a computation occurs at each step, eventually producing output and terminating. The transition between states can be non-deterministic; randomized algorithms incorporate random input. == Etymology == Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of these texts involving the Hindu–Arabic numeral system and arithmetic appeared, for example Liber Alghoarismi de practica arismetrice, attributed to John of Seville, and Liber Algoritmi de numero Indorum, attributed to Adelard of Bath. Here, alghoarismi or algoritmi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algoritmi, or "Thus spoke Al-Khwarizmi". The word algorism in English came to mean the use of place-value notation in calculations; it occurs in the Ancrene Wisse from circa 1225. By the time Geoffrey Chaucer wrote The Canterbury Tales in the late 14th century, he used a variant of the same word in describing augrym stones, stones used for place-value calculation. In the 15th century, under the influence of the Greek word ἀριθμός (arithmos, "number"; cf. "arithmetic"), the Latin word was altered to algorithmus. By 1596, this form of the word was used in English, as algorithm, by Thomas Hood. == Definition == One informal definition is "a set of rules that precisely defines a sequence of operations", which would include all computer programs, and any bureaucratic procedure or cook-book recipe. In general, a program is an algorithm only if it stops eventually. Formally, algorithm is an explicit set of instructions to produce an output, that can be followed by a computer or a human performing specific operations on symbols.. == History == === Ancient algorithms === Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in Babylonian mathematics (around 2500 BC), Egyptian mathematics (around 1550 BC), Indian mathematics (around 800 BC and later), the Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC), Chinese mathematics (around 200 BC and later), and Arabic mathematics (around 800 AD). The earliest evidence of algorithms is found in ancient Mesopotamian mathematics. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to c. 2500 BC describes the earliest division algorithm. During the Hammurabi dynasty c. 1800 – c. 1600 BC, Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus c. 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).Examples of ancient Indian mathematics included the Shulba Sutras, the Kerala School, and the Brāhmasphuṭasiddhānta. In the 9th century, Muḥammad ibn Mūsā al-Khwārizmī revolutionized the field by establishing the algorithm as a systematic, finite sequence of logical steps to solve mathematical problems. In his influential work, The Compendious Book on Calculation by Completion and Balancing, he moved beyond specific numerical solutions to introduce general procedures for algebraic reduction and balancing. This transformed mathematics into a 'mechanical' process of well-defined rules—a fundamental shift that laid the groundwork for modern algorithmic theory. The Latin translation of his arithmetic treatise, titled Algoritmi de numero Indorum, led to the term algorithm being derived from the Latinization of his name, Algoritmi, specifically to describe this new rule-based approach to mathematics. The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm. === Computers === ==== Weight-driven clocks ==== Weight-driven clocks were a key European invention in Middle Ages, specifically the verge escapement mechanism producing the tick of mechanical clocks. Accurate automatic machines led to mechanical automata in the 13th century and computational machines—the difference and analytical engines of Charles Babbage and Ada Lovelace in the mid-19th century. Lovelace designed the first algorithm intended for a computer, Babbage's analytical engine, the first real Turing-complete computer, more than the mechanical calculators of the time. Although the full implementation of Babbage's second device was only built decades after her lifetime, Lovelace has been called "history's first programmer". ==== Electromechanical relay ==== The Jacquard loom, a precursor to punch cards, and telephone switching machines led to the development of the first computers. By the mid-19th century, the telegraph, was in use throughout the world. By the late 19th century, ticker tape (c. 1870s) and punch cards (c. 1890) were developed. Then came the teleprinter (c. 1910) with its punched-paper use of Baudot code on tape. Telephone-switching networks of electromechanical relays were invented in 1835. These led to the invention of the digital adding device by George Stibitz in 1937. While working in Bell Laboratories, he observed the "burdensome" use of mechanical calculators with gears, prompting him to experiment create an experimental digital adder at home. === Formalization === In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve David Hilbert's Entscheidungsproblem (decision problem). Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939. === Modern Algorithms === For decades, it was assumed that algorithm evolution progresses from heuristics to formal algorithms. A Symbolic integration provides a classic illustration. In 1961, James Slagle’s program SAINT used heuristics to solve 52 of 54 freshman calculus exercises from an MIT textbook (≈96%). In 1967, Larry Moses’s SIN refined the heuristics and achieved 100% success, though it remained heuristic. Finally, in 1969, Robert Risch introduced the Risch Algorithm with formal guarantees. This trajectory defined the traditional path: heuristics evolving until a definitive, guaranteed algorithm emerged. However, the rise of transformer-based AI has inverted this sequence — classical algorithms are now being displaced by heuristics once again. Algorithms have evolved and improved in many ways as time goes on. Common uses of algorithms today include social media apps like Instagram and YouTube. Algorithms are used as a way to analyze what people like and push more of those things to the people who interact with them. Quantum computing uses quantum algorithm procedures to solve problems faster. More recently, in 2024, NIST updated their post-quantum encryption standards, which includes new encryption algorithms to enhance defenses against attacks using quantum computing. == Representations == Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables. Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algor
External memory algorithm
In computing, external memory algorithms or out-of-core algorithms are algorithms that are designed to process data that are too large to fit into a computer's main memory at once. Such algorithms must be optimized to efficiently fetch and access data stored in slow bulk memory (auxiliary memory) such as hard drives or tape drives, or when memory is on a computer network. External memory algorithms are analyzed in the external memory model. == Model == External memory algorithms are analyzed in an idealized model of computation called the external memory model (or I/O model, or disk access model). The external memory model is an abstract machine similar to the RAM machine model, but with a cache in addition to main memory. The model captures the fact that read and write operations are much faster in a cache than in main memory, and that reading long contiguous blocks is faster than reading randomly using a disk read-and-write head. The running time of an algorithm in the external memory model is defined by the number of reads and writes to memory required. The model was introduced by Alok Aggarwal and Jeffrey Vitter in 1988. The external memory model is related to the cache-oblivious model, but algorithms in the external memory model may know both the block size and the cache size. For this reason, the model is sometimes referred to as the cache-aware model. The model consists of a processor with an internal memory or cache of size M, connected to an unbounded external memory. Both the internal and external memory are divided into blocks of size B. One input/output or memory transfer operation consists of moving a block of B contiguous elements from external to internal memory, and the running time of an algorithm is determined by the number of these input/output operations. == Algorithms == Algorithms in the external memory model take advantage of the fact that retrieving one object from external memory retrieves an entire block of size B. This property is sometimes referred to as locality. Searching for an element among N objects is possible in the external memory model using a B-tree with branching factor B. Using a B-tree, searching, insertion, and deletion can be achieved in O ( log B N ) {\displaystyle O(\log _{B}N)} time (in Big O notation). Information theoretically, this is the minimum running time possible for these operations, so using a B-tree is asymptotically optimal. External sorting is sorting in an external memory setting. External sorting can be done via distribution sort, which is similar to quicksort, or via a M B {\displaystyle {\tfrac {M}{B}}} -way merge sort. Both variants achieve the asymptotically optimal runtime of O ( N B log M B N B ) {\displaystyle O\left({\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)} to sort N objects. This bound also applies to the fast Fourier transform in the external memory model. The permutation problem is to rearrange N elements into a specific permutation. This can either be done either by sorting, which requires the above sorting runtime, or inserting each element in order and ignoring the benefit of locality. Thus, permutation can be done in O ( min ( N , N B log M B N B ) ) {\displaystyle O\left(\min \left(N,{\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)\right)} time. == Applications == The external memory model captures the memory hierarchy, which is not modeled in other common models used in analyzing data structures, such as the random-access machine, and is useful for proving lower bounds for data structures. The model is also useful for analyzing algorithms that work on datasets too big to fit in internal memory. A typical example is geographic information systems, especially digital elevation models, where the full data set easily exceeds several gigabytes or even terabytes of data. This methodology extends beyond general purpose CPUs and also includes GPU computing as well as classical digital signal processing. In general-purpose computing on graphics processing units (GPGPU), powerful graphics cards (GPUs) with little memory (compared with the more familiar system memory, which is most often referred to simply as RAM) are utilized with relatively slow CPU-to-GPU memory transfer (when compared with computation bandwidth). == History == An early use of the term "out-of-core" as an adjective is in 1962 in reference to devices that are other than the core memory of an IBM 360. An early use of the term "out-of-core" with respect to algorithms appears in 1971.
List of artificial intelligence projects
The following is a list of current and past, non-classified notable artificial intelligence projects. == Specialized projects == === Brain-inspired === Blue Brain Project, an attempt to create a synthetic brain by reverse-engineering the mammalian brain down to the molecular level. Google Brain, a deep learning project part of Google X attempting to have intelligence similar or equal to human-level. Human Brain Project, ten-year scientific research project, based on exascale supercomputers. === Cognitive architectures === 4CAPS, developed at Carnegie Mellon University under Marcel A. Just ACT-R, developed at Carnegie Mellon University under John R. Anderson. AIXI, Universal Artificial Intelligence developed by Marcus Hutter at IDSIA and ANU. CALO, a DARPA-funded, 25-institution effort to integrate many artificial intelligence approaches (natural language processing, speech recognition, machine vision, probabilistic logic, planning, reasoning, many forms of machine learning) into an AI assistant that learns to help manage your office environment. CHREST, developed under Fernand Gobet at Brunel University and Peter C. Lane at the University of Hertfordshire. CLARION, developed under Ron Sun at Rensselaer Polytechnic Institute and University of Missouri. CoJACK, an ACT-R inspired extension to the JACK multi-agent system that adds a cognitive architecture to the agents for eliciting more realistic (human-like) behaviors in virtual environments. Copycat, by Douglas Hofstadter and Melanie Mitchell at the Indiana University. DUAL, developed at the New Bulgarian University under Boicho Kokinov. FORR developed by Susan L. Epstein at The City University of New York. IDA and LIDA, implementing Global Workspace Theory, developed under Stan Franklin at the University of Memphis. OpenCog Prime, developed using the OpenCog Framework. Procedural Reasoning System (PRS), developed by Michael Georgeff and Amy L. Lansky at SRI International. Psi-Theory developed under Dietrich Dörner at the Otto-Friedrich University in Bamberg, Germany. Soar, developed under Allen Newell and John Laird at Carnegie Mellon University and the University of Michigan. Society of Mind and its successor The Emotion Machine proposed by Marvin Minsky. Subsumption architectures, developed e.g. by Rodney Brooks (though it could be argued whether they are cognitive). === Games === AlphaGo, software developed by Google that plays the Chinese board game Go. Chinook, a computer program that plays English draughts; the first to win the world champion title in the competition against humans. Deep Blue, a chess-playing computer developed by IBM which beat Garry Kasparov in 1997. Halite, an artificial intelligence programming competition created by Two Sigma in 2016. Libratus, a poker AI that beat world-class poker players in 2017, intended to be generalisable to other applications. The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Michie in 1961. Quick, Draw!, an online game developed by Google that challenges players to draw a picture of an object or idea and then uses a neural network to guess what the drawing is. The Samuel Checkers-playing Program (1959) was among the world's first successful self-learning programs, and as such a very early demonstration of the fundamental concept of artificial intelligence (AI). Stockfish AI, an open source chess engine currently ranked the highest in many computer chess rankings. TD-Gammon, a program that learned to play world-class backgammon partly by playing against itself (temporal difference learning with neural networks). === Internet activism === Serenata de Amor, project for the analysis of public expenditures and detect discrepancies. === Knowledge and reasoning === Alice (Microsoft), a project from Microsoft Research Lab aimed at improving decision-making in Economics Braina, an intelligent personal assistant application with a voice interface for Windows OS. Cyc, an attempt to assemble an ontology and database of everyday knowledge, enabling human-like reasoning. Eurisko, a language by Douglas Lenat for solving problems which consists of heuristics, including some for how to use and change its heuristics. Google Now, an intelligent personal assistant with a voice interface in Google's Android and Apple Inc.'s iOS, as well as Google Chrome web browser on personal computers. Holmes a new AI created by Wipro. Microsoft Cortana, an intelligent personal assistant with a voice interface in Microsoft's various Windows 10 editions. MindsDB, is an AI automation platform for building AI/ML powered features and applications. Mycin, an early medical expert system. Open Mind Common Sense, a project based at the MIT Media Lab to build a large common sense knowledge base from online contributions. Siri, an intelligent personal assistant and knowledge navigator with a voice-interface in Apple Inc.'s iOS and macOS. SNePS, simultaneously a logic-based, frame-based, and network-based knowledge representation, reasoning, and acting system. Viv (software), a new AI by the creators of Siri. Wolfram Alpha, an online service that answers queries by computing the answer from structured data. === Motion and manipulation === AIBO, the robot pet for the home, grew out of Sony's Computer Science Laboratory (CSL). Cog, a robot developed by MIT to study theories of cognitive science and artificial intelligence, now discontinued. === Music === Melomics, a bioinspired technology for music composition and synthesization of music, where computers develop their own style, rather than mimic musicians. === Natural language processing === AIML, an XML dialect for creating natural language software agents. Apache Lucene, a high-performance, full-featured text search engine library written entirely in Java. Apache OpenNLP, a machine learning based toolkit for the processing of natural language text. It supports the most common NLP tasks, such as tokenization, sentence segmentation, part-of-speech tagging, named entity extraction, chunking and parsing. Artificial Linguistic Internet Computer Entity (A.L.I.C.E.), a natural language processing chatterbot. ChatGPT, a chatbot built on top of OpenAI's GPT-3.5 and GPT-4 family of large language models. Claude, a family of large language models developed by Anthropic and launched in 2023. Claude LLMs achieved high coding scores in several recognized LLM benchmarks. Cleverbot, successor to Jabberwacky, now with 170m lines of conversation, Deep Context, fuzziness and parallel processing. Cleverbot learns from around 2 million user interactions per month. DeepSeek: Chinese chatbot funded by hedge fund High-Flyer. DBRX, 136 billion parameter open sourced large language model developed by Mosaic ML and Databricks. ELIZA, a famous 1966 computer program by Joseph Weizenbaum, which parodied person-centered therapy. FreeHAL, a self-learning conversation simulator (chatterbot) which uses semantic nets to organize its knowledge to imitate a very close human behavior within conversations. Gemini, a family of multimodal large language model developed by Google's DeepMind. Drives the Gemini chatbot, formerly known as Bard. GigaChat, a chatbot by Russian Sberbank. GPT-3, a 2020 language model developed by OpenAI that can produce text difficult to distinguish from that written by a human. Jabberwacky, a chatbot by Rollo Carpenter, aiming to simulate natural human chat. LaMDA, a family of conversational neural language models developed by Google. LLaMA, a 2023 language model family developed by Meta that includes 7, 13, 33 and 65 billion parameter models.[1] Mycroft, a free and open-source intelligent personal assistant that uses a natural language user interface. PARRY, another early chatterbot, written in 1972 by Kenneth Colby, attempting to simulate a paranoid schizophrenic. SHRDLU, an early natural language processing computer program developed by Terry Winograd at MIT from 1968 to 1970. SYSTRAN, a machine translation technology by the company of the same name, used by Yahoo!, AltaVista and Google, among others. === Speech recognition === CMU Sphinx, a group of speech recognition systems developed at Carnegie Mellon University. DeepSpeech, an open-source Speech-To-Text engine based on Baidu's deep speech research paper. Whisper, an open-source speech recognition system developed at OpenAI. === Speech synthesis === 15.ai, a real-time artificial intelligence text-to-speech tool developed by an anonymous researcher from MIT. Amazon Polly, a speech synthesis software by Amazon. Festival Speech Synthesis System, a general multi-lingual speech synthesis system developed at the Centre for Speech Technology Research (CSTR) at the University of Edinburgh. WaveNet, a deep neural network for generating raw audio. === Video === CapCut is a video editor tool, developed
Texture artist
A texture artist is an individual who develops textures for digital media, usually for video games, movies, web sites and television shows or things like 3D posters. These textures can be in the form of 2D or (rarely) 3D art that may be overlaid onto a polygon mesh to create a realistic 3D model. Texture artists often take advantage of web sites for the purposes of marketing their art and self-promotion of their skills with the goal of gaining employment from a professional game studio or to join a team working on a "mod" (modification) of an existing game in hopes of establishing industry or trade credentials.
Higuchi dimension
In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about the Higuchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves in seismograms, clinical neurophysiology and analyzing changes in the electroencephalogram in Alzheimer's disease. == Formulation of the method == The original formulation of the method is due to T. Higuchi. Given a time series X : { 1 , … , N } → R {\displaystyle X:\{1,\dots ,N\}\to \mathbb {R} } consisting of N {\displaystyle N} data points and a parameter k m a x ≥ 2 {\displaystyle k_{\mathrm {max} }\geq 2} the Higuchi Fractal dimension (HFD) of X {\displaystyle X} is calculated in the following way: For each k ∈ { 1 , … , k m a x } {\displaystyle k\in \{1,\dots ,k_{\mathrm {max} }}\} and m ∈ { 1 , … , k } {\displaystyle m\in \{1,\dots ,k}\} define the length L m ( k ) {\displaystyle L_{m}(k)} by L m ( k ) = N − 1 ⌊ N − m k ⌋ k 2 ∑ i = 1 ⌊ N − m k ⌋ | X N ( m + i k ) − X N ( m + ( i − 1 ) k ) | . {\displaystyle L_{m}(k)={\frac {N-1}{\lfloor {\frac {N-m}{k}}\rfloor k^{2}}}\sum _{i=1}^{\lfloor {\frac {N-m}{k}}\rfloor }|X_{N}(m+ik)-X_{N}(m+(i-1)k)|.} The length L ( k ) {\displaystyle L(k)} is defined by the average value of the k {\displaystyle k} lengths L 1 ( k ) , … , L k ( k ) {\displaystyle L_{1}(k),\dots ,L_{k}(k)} , L ( k ) = 1 k ∑ m = 1 k L m ( k ) . {\displaystyle L(k)={\frac {1}{k}}\sum _{m=1}^{k}L_{m}(k).} The slope of the best-fitting linear function through the data points { ( log 1 k , log L ( k ) ) } {\displaystyle \left\{\left(\log {\frac {1}{k}},\log L(k)\right)\right\}} is defined to be the Higuchi fractal dimension of the time-series X {\displaystyle X} . == Application to functions == For a real-valued function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } one can partition the unit interval [ 0 , 1 ] {\displaystyle [0,1]} into N {\displaystyle N} equidistantly intervals [ t j , t j + 1 ) {\displaystyle [t_{j},t_{j+1})} and apply the Higuchi algorithm to the times series X ( j ) = f ( t j ) {\displaystyle X(j)=f(t_{j})} . This results into the Higuchi fractal dimension of the function f {\displaystyle f} . It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of f {\displaystyle f} as it follows a geometrical approach (see Liehr & Massopust 2020). == Robustness and stability == Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension. On the other hand, the method can be unstable in the case where the data X ( 1 ) , … , X ( N ) {\displaystyle X(1),\dots ,X(N)} are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020).