VHS (Video Home System) is a discontinued standard for consumer-level analog video recording on tape cassettes, introduced in 1976 by JVC. It was the dominant home video format throughout the tape media period of the 1980s and 1990s. Magnetic tape video recording was adopted by the television industry in the 1950s in the form of the first commercialized video tape recorders (VTRs), but the devices were expensive and used only in professional environments. In the 1970s, videotape technology became affordable for home use, and widespread adoption of videocassette recorders (VCRs) began; the VHS became the most popular media format for VCRs as it would win the "format war" against Betamax (backed by Sony) and a number of other competing tape standards. The cassettes themselves use a 0.5-inch (12.7 mm) magnetic tape between two spools and typically offer a capacity of at least two hours. The popularity of VHS was intertwined with the rise of the video rental market, when films were released on pre-recorded videotapes for home viewing. Newer improved tape formats such as S-VHS were later developed, as well as the earliest optical disc format, LaserDisc; the lack of global adoption of these formats increased VHS's lifetime, which eventually peaked and started to decline in the late 1990s after the introduction of DVD, a digital optical disc format. VHS rentals were surpassed by DVD in the United States in 2003, which eventually became the preferred low-end method of movie distribution. For home recording purposes, VHS and VCRs were surpassed by (typically hard disk–based) digital video recorders (DVR) in the 2000s. Production of all VHS equipment ceased by 2016, although the format has since gained some popularity amongst collectors. A niche revival of VHS has taken place with This Is How The World Ends becoming the first straight-to-VHS release in 20 years. == History == === Before VHS === In 1956, after several attempts by other companies, the first commercially successful VTR, the Ampex VRX-1000, was introduced by Ampex Corporation. At a price of US$50,000 in 1956 (equivalent to $592,000 in 2025) and US$300 (equivalent to $3,600 in 2025) for a 90-minute reel of tape, it was intended only for the professional market. Kenjiro Takayanagi, a television broadcasting pioneer then working for JVC as its vice president, saw the need for his company to produce VTRs for the Japanese market at a more affordable price. In 1959, JVC developed a two-head video tape recorder and, by 1960, a color version for professional broadcasting. In 1964, JVC released the DV220, which would be the company's standard VTR until the mid-1970s. In 1969, JVC collaborated with Sony and Matsushita Electric (Matsushita was the majority stockholder of JVC until 2011) to build a video recording standard for the Japanese consumer. The effort produced the U-matic format in 1971, which was the first cassette format to become a unified standard for different companies. It was preceded by the reel-to-reel 1⁄2-inch EIAJ format. The U-matic format was successful in businesses and some broadcast television applications, such as electronic news-gathering, and was produced by all three companies until the late 1980s, but because of cost and limited recording time, very few of the machines were sold for home use. Therefore, soon after the U-Matic release, all three companies started working on new consumer-grade video recording formats of their own. Sony started working on Betamax, Matsushita started working on VX, and JVC released the CR-6060 in 1975, based on the U-matic format. === VHS development === In 1971, JVC engineers Yuma Shiraishi and Shizuo Takano put together a team to develop a VTR for consumers. By the end of 1971, they created an internal diagram, "VHS Development Matrix", which established twelve objectives for JVC's new VTR; among them: The system must be compatible with any ordinary television set. Picture quality must be similar to a normal air broadcast. The tape must have at least a two-hour recording capacity. Tapes must be interchangeable between machines. The overall system should be versatile, meaning it can be scaled and expanded, such as connecting a video camera, or dubbing between two recorders. Recorders should be affordable, easy to operate, and have low maintenance costs. Recorders must be capable of being produced in high volume, their parts must be interchangeable, and they must be easy to service. In early 1972, the commercial video recording industry in Japan took a financial hit. JVC cut its budgets and restructured its video division, shelving the VHS project. However, despite the lack of funding, Takano and Shiraishi continued to work on the project in secret. By 1973, the two engineers had produced a functional prototype. === Competition with Betamax === In 1974, the Japanese Ministry of International Trade and Industry (MITI), desiring to avoid consumer confusion, attempted to force the Japanese video industry to standardize on just one home video recording format. Later, Sony had a functional prototype of the Betamax format, and was very close to releasing a finished product. With this prototype, Sony persuaded the MITI to adopt Betamax as the standard, and allow it to license the technology to other companies. JVC believed that an open standard, with the format shared among competitors without licensing the technology, was better for the consumer. To prevent the MITI from adopting Betamax, JVC worked to convince other companies, in particular Matsushita (Japan's largest electronics manufacturer at the time, marketing its products under the National brand in most territories and the Panasonic brand in North America, and JVC's majority stockholder), to accept VHS, and thereby work against Sony and the MITI. Matsushita agreed, fearing Sony would dominate the market with a Betamax monopoly. Matsushita also regarded Betamax's one-hour recording time limit as a disadvantage. Matsushita's backing of JVC persuaded Hitachi, Mitsubishi, and Sharp to back the VHS standard as well. Sony's release of its Betamax unit to the Japanese market in 1975 placed further pressure on the MITI to side with the company. However, the collaboration of JVC and its partners was much stronger, which eventually led the MITI to drop its push for an industry standard. JVC released the first VHS machines in Japan in late 1976, and in the United States in mid-1977. Sony's Betamax competed with VHS throughout the late 1970s and into the 1980s (see Videotape format war). Betamax's major advantages were its smaller cassette size, theoretical higher video quality, and earlier availability, but its shorter recording time proved to be a major shortcoming. Originally, Beta I machines using the NTSC television standard were able to record one hour of programming at their standard tape speed of 1.5 inches per second (ips). The first VHS machines could record for two hours, due to both a slightly slower tape speed (1.31 ips) and significantly longer tape. Betamax's smaller cassette limited the size of the reel of tape, and could not compete with VHS's two-hour capability by extending the tape length. Instead, Sony had to slow the tape down to 0.787 ips (Beta II) in order to achieve two hours of recording in the same cassette size. Sony eventually created a Beta III speed of 0.524 ips, which allowed NTSC Betamax to break the two-hour limit, but by then VHS had already won the format battle. Additionally, VHS had a "far less complex tape transport mechanism" than Betamax, and VHS machines were faster at rewinding and fast-forwarding than their Sony counterparts. VHS eventually won the war, gaining 60% of the North American market by 1980. == Initial releases of VHS-based devices == The first VCR to use VHS was the Victor HR-3300, and was introduced by the president of JVC in Japan on September 9, 1976. JVC started selling the HR-3300 in Akihabara, Tokyo, Japan, on October 31, 1976. Region-specific versions of the JVC HR-3300 were also distributed later on, such as the HR-3300U in the United States, and the HR-3300EK in the United Kingdom. The United States received its first VHS-based VCR, the RCA VBT200, on August 23, 1977. The RCA unit was designed by Matsushita and was the first VHS-based VCR manufactured by a company other than JVC. It was also capable of recording four hours in LP (long play) mode. The UK received its first VHS-based VCR, the Victor HR-3300EK, in 1978. Quasar and General Electric followed-up with VHS-based VCRs – all designed by Matsushita. By 1999, Matsushita alone produced just over half of all Japanese VCRs. TV/VCR combos, combining a TV set with a VHS mechanism, were also once available for purchase. Combo units containing both a VHS mechanism and a DVD player were introduced in the late 1990s, and at least one combo unit, the Panasonic DMP-BD70V, included a Blu-ray player. == Technical details == VHS has been standardized in IEC 60774–1. === Cassette and
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = 1/h K(x/h). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ( h ) = AMISE ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the
World Congress of Universal Documentation
The World Congress of Universal Documentation was held from 16 to 21 August 1937 in Paris, France. Delegates from 45 countries met to discuss means by which all of the world's information, in print, in manuscript, and in other forms, could be efficiently organized and made accessible. == The Congress in the history of information science == The Congress, held at the Trocadéro under "the auspices" of the Institut International de Bibliographie, was "the apotheosis" of a general movement in the 1930s towards the classification of the growing mass of information and the improvement of access to that information. For the first time in the history of information science, technological means were beginning to catch up with theoretical ends, and the discussions at the conference reflected that fact. Its participation in the Congress was one of the first projects of the American Documentation Institute (ADI). Participants in the conference discussed what has been more recently called "a continuously updated hypertext encyclopedia." Joseph Reagle sees many of the ideas considered at the conference as forerunners of some of the key goals and norms of Wikipedia. == Microfilm == The main resolution adopted by the congress proposed that microfilm be used to make information universally available. Watson Davis, chairman of the American delegation and president of the ADI, stated that the volume of information being produced created difficult problems of access and preservation, but that these could be solved by the use of microfilm. In his address to the Congress, Davis said: Most immediate and practical to put into operation is the microfilming of material in libraries upon demand. It will become fashionable and economical to send a potential book borrower a little strip of microfilm for his permanent possession instead of the book and then badgering him to return it before he has had a chance to use it effectively. I believe that reading machines for microfilm will become as common as typewriters in studies and laboratories. If the principal libraries and information centers of the world will cooperate in such "bibliofilm services," as they are called, if they exchange orders and have essentially uniform methods, forms for ordering, standard microfilm format and production methods and comparable if not uniform prices, the resources of any library will be placed at the disposal of any scholar or scientist anywhere in the world. All the libraries cooperating will merge into one world library without loss of identity or individuality. The world's documentation will become available to even the most isolated and individualistic scholar. The Congress included two separate exhibits on microfilm. One was of the equipment used at the Bibliothèque nationale de France and the other, coordinated by Herman H. Fussler of the University of Chicago, consisting of "an entire microfilm laboratory," complete with cameras, a darkroom, and various kinds of reading machines. Emanuel Goldberg presented a paper on an early copying camera he had invented. Other resolutions passed by the Congress concerned uniform standards for the preparation of articles, for classifying books and other documents, for indexing newspapers and periodicals, and for cooperation between libraries. == H. G. Wells == In his address to the Congress, H. G. Wells said that he thought that his idea of the "world brain" was a precursor to the ideas other delegates were proposing, and explicitly linked the projects being discussed to the work of the encyclopédistes: I am speaking of a process of mental organization throughout the world which I believe to be as inevitable as anything can be in human affairs. All the distresses and horrors of the present time are fundamentally intellectual. The world has to pull its mind together, and this [Congress] is the beginning of its efforts. Civilization is a Phoenix. It perishes in flames and even as it dies it is born again. This synthesis of knowledge upon which you are working is the necessary beginning of a new world. It is good to be meeting here in Paris where the first encyclopedia of power was made. It would be impossible to overrate our debt to Diderot and his associates. == Other participants == Participants in the Congress included authors, librarians, scholars, archivists, scientists, and editors. Some of the notable people in attendance not mentioned above were:
Run-to-completion scheduling
Run-to-completion scheduling or nonpreemptive scheduling is a scheduling model in which each task runs until it either finishes, or explicitly yields control back to the scheduler. Run-to-completion systems typically have an event queue which is serviced either in strict order of admission by an event loop, or by an admission scheduler which is capable of scheduling events out of order, based on other constraints such as deadlines. Some preemptive multitasking scheduling systems behave as run-to-completion schedulers in regard to scheduling tasks at one particular process priority level, at the same time as those processes still preempt other lower priority tasks and are themselves preempted by higher priority tasks.
Known-item search
Known-item search is a specialization of information exploration which represents the activities carried out by searchers who have a particular item in mind. In the context of library catalogs, known‐item search means a search for an item for which the author or title is known. Although the concept of known-item search originated in library science, it is now applied in the context of web search and other online search activities. Known-item search is distinguished from exploratory search, in which a searcher is unfamiliar with the domain of their search goal, unsure about the ways to achieve their goal, and/or unsure about what their goal is.
Computer vision dazzle
Computer vision dazzle, also known as CV dazzle, dazzle makeup, or anti-surveillance makeup, is a type of camouflage used to hamper facial recognition software, inspired by dazzle camouflage used by vehicles such as ships and planes. == Methods == CV dazzle combines stylized makeup, asymmetric hair, and sometimes infrared lights built in to glasses or clothing to break up detectable facial patterns recognized by computer vision algorithms in much the same way that warships contrasted color and used sloping lines and curves to distort the structure of a vessel. It has been shown to be somewhat successful at defeating face detection software in common use, including that employed by Facebook. CV dazzle attempts to block detection by facial recognition technologies such as DeepFace "by creating an 'anti-face'". It uses occlusion, covering certain facial features; transformation, altering the shape or colour of parts of the face; and a combination of the two. Prominent artists employing this technique include Adam Harvey and Jillian Mayer. == Use in protests == Computer vision dazzle makeup has been used by protestors in several different protest movements. Its use as a protesting aid has often been found ineffective. It may be effective to thwart computer technology, but draws human attention, is easy for human monitors to spot on security cameras, and makes it hard for protestors to blend in within a crowd. Advances in facial recognition technology make dazzle makeup increasingly ineffective.
Time Warp Edit Distance
In the data analysis of time series, Time Warp Edit Distance (TWED) is a measure of similarity (or dissimilarity) between pairs of discrete time series, controlling the relative distortion of the time units of the two series using the physical notion of elasticity. In comparison to other distance measures, (e.g. DTW (dynamic time warping) or LCS (longest common subsequence problem)), TWED is a metric. Its computational time complexity is O ( n 2 ) {\displaystyle O(n^{2})} , but can be drastically reduced in some specific situations by using a corridor to reduce the search space. Its memory space complexity can be reduced to O ( n ) {\displaystyle O(n)} . It was first proposed in 2009 by P.-F. Marteau. == Definition == δ λ , ν ( A 1 p , B 1 q ) = M i n { δ λ , ν ( A 1 p − 1 , B 1 q ) + Γ ( a p ′ → Λ ) d e l e t e i n A δ λ , ν ( A 1 p − 1 , B 1 q − 1 ) + Γ ( a p ′ → b q ′ ) m a t c h o r s u b s t i t u t i o n δ λ , ν ( A 1 p , B 1 q − 1 ) + Γ ( Λ → b q ′ ) d e l e t e i n B {\displaystyle \delta _{\lambda ,\nu }(A_{1}^{p},B_{1}^{q})=Min{\begin{cases}\delta _{\lambda ,\nu }(A_{1}^{p-1},B_{1}^{q})+\Gamma (a_{p}^{'}\to \Lambda )&{\rm {delete\ in\ A}}\\\delta _{\lambda ,\nu }(A_{1}^{p-1},B_{1}^{q-1})+\Gamma (a_{p}^{'}\to b_{q}^{'})&{\rm {match\ or\ substitution}}\\\delta _{\lambda ,\nu }(A_{1}^{p},B_{1}^{q-1})+\Gamma (\Lambda \to b_{q}^{'})&{\rm {delete\ in\ B}}\end{cases}}} whereas Γ ( α p ′ → Λ ) = d L P ( a p ′ , a p − 1 ′ ) + ν ⋅ ( t a p − t a p − 1 ) + λ {\displaystyle \Gamma (\alpha _{p}^{'}\to \Lambda )=d_{LP}(a_{p}^{'},a_{p-1}^{'})+\nu \cdot (t_{a_{p}}-t_{a_{p-1}})+\lambda } Γ ( α p ′ → b q ′ ) = d L P ( a p ′ , b q ′ ) + d L P ( a p − 1 ′ , b q − 1 ′ ) + ν ⋅ ( | t a p − t b q | + | t a p − 1 − t b q − 1 | ) {\displaystyle \Gamma (\alpha _{p}^{'}\to b_{q}^{'})=d_{LP}(a_{p}^{'},b_{q}^{'})+d_{LP}(a_{p-1}^{'},b_{q-1}^{'})+\nu \cdot (|t_{a_{p}}-t_{b_{q}}|+|t_{a_{p-1}}-t_{b_{q-1}}|)} Γ ( Λ → b q ′ ) = d L P ( b p ′ , b p − 1 ′ ) + ν ⋅ ( t b q − t b q − 1 ) + λ {\displaystyle \Gamma (\Lambda \to b_{q}^{'})=d_{LP}(b_{p}^{'},b_{p-1}^{'})+\nu \cdot (t_{b_{q}}-t_{b_{q-1}})+\lambda } Whereas the recursion δ λ , ν {\displaystyle \delta _{\lambda ,\nu }} is initialized as: δ λ , ν ( A 1 0 , B 1 0 ) = 0 , {\displaystyle \delta _{\lambda ,\nu }(A_{1}^{0},B_{1}^{0})=0,} δ λ , ν ( A 1 0 , B 1 j ) = ∞ f o r j ≥ 1 {\displaystyle \delta _{\lambda ,\nu }(A_{1}^{0},B_{1}^{j})=\infty \ {\rm {{for\ }j\geq 1}}} δ λ , ν ( A 1 i , B 1 0 ) = ∞ f o r i ≥ 1 {\displaystyle \delta _{\lambda ,\nu }(A_{1}^{i},B_{1}^{0})=\infty \ {\rm {{for\ }i\geq 1}}} with a 0 ′ = b 0 ′ = 0 {\displaystyle a'_{0}=b'_{0}=0} === Implementations === An implementation of the TWED algorithm in C with a Python wrapper is available at TWED is also implemented into the Time Series Subsequence Search Python package (TSSEARCH for short) available at [1]. An R implementation of TWED has been integrated into the TraMineR, a R package for mining, describing and visualizing sequences of states or events, and more generally discrete sequence data. Additionally, cuTWED is a CUDA- accelerated implementation of TWED which uses an improved algorithm due to G. Wright (2020). This method is linear in memory and massively parallelized. cuTWED is written in CUDA C/C++, comes with Python bindings, and also includes Python bindings for Marteau's reference C implementation. ==== Python ==== Backtracking, to find the most cost-efficient path: ==== MATLAB ==== Backtracking, to find the most cost-efficient path: