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  Turret lathe

  A turret lathe is a form of metalworking lathe that is used for repetitive production of duplicate parts, which by the nature of their cutting process are usually interchangeable. It evolved from earlier lathes with the addition of the turret, which is an indexable toolholder that allows multiple cutting operations to be performed, each with a different cutting tool, in easy, rapid succession, with no need for the operator to perform set-up tasks in between (such as installing or uninstalling tools) or to control the toolpath. The latter is due to the toolpath's being controlled by the machine, either in jig-like fashion, via the mechanical limits placed on it by the turret's slide and stops, or via digitally-directed servomechanisms for computer numerical control lathes. The name derives from the way early turrets took the general form of a flattened cylindrical block mounted to the lathe's cross-slide, capable of rotating about the vertical axis and with toolholders projecting out to all sides, and thus vaguely resembled a swiveling gun turret. Capstan lathe is the usual name in the UK and Commonwealth, though the two terms are also used in contrast: see below, Capstan versus turret. == History == Turret lathes became indispensable to the production of interchangeable parts and for mass production. The first turret lathe was built by Stephen Fitch in 1845 to manufacture screws for pistol percussion parts. In the mid-nineteenth century, the need for interchangeable parts for Colt revolvers enhanced the role of turret lathes in achieving this goal as part of the "American system" of manufacturing arms. Clock-making and bicycle manufacturing had similar requirements. Christopher Spencer invented the first fully automated turret lathe in 1873, which led to designs using cam action or hydraulic mechanisms. From the late-19th through mid-20th centuries, turret lathes, both manual and automatic (i.e., screw machines and chuckers), were one of the most important classes of machine tools for mass production. They were used extensively in the mass production for the war effort in World War II. The U.S. company Warner & Swasey was one of the premier brands in heavy turret lathes between the 1910s and 1960s; it became the world's largest manufacturer of such lathes by 1928. During World War II, it employed 7,000 people and produced half of the turret lathes manufactured in the United States. == Types == There are many variants of the turret lathe. They can be most generally classified by size (small, medium, or large); method of control (manual, automated mechanically, or automated via computer (numerical control (NC) or computer numerical control (CNC)); and bed orientation (horizontal or vertical). === Archetypical: horizontal, manual === In the late 1830s a "capstan lathe" with a turret was patented in Britain. The first American turret lathe was invented by Stephen Fitch in 1845. The archetypical turret lathe, and the first in order of historical appearance, is the horizontal-bed, manual turret lathe. The term "turret lathe" without further qualification is still understood to refer to this type. The formative decades for this class of machine were the 1840s through 1860s, when the basic idea of mounting an indexable turret on a bench lathe or engine lathe was born, developed, and disseminated from the originating shops to many other factories. Some important tool-builders in this development were Stephen Fitch; Gay, Silver & Co.; Elisha K. Root of Colt; J.D. Alvord of the Sharps Armory; Frederick W. Howe, Richard S. Lawrence, and Henry D. Stone of Robbins & Lawrence; J.R. Brown of Brown & Sharpe; and Francis A. Pratt of Pratt & Whitney. Various designers at these and other firms later made further refinements. === Semi-automatic === Sometimes machines similar to those above, but with power feeds and automatic turret-indexing at the end of the return stroke, are called "semi-automatic turret lathes". This nomenclature distinction is blurry and not consistently observed. The term "turret lathe" encompasses them all. During the 1860s, when semi-automatic turret lathes were developed, they were sometimes called "automatic". What we today would call "automatics", that is, fully automatic machines, had not been developed yet. During that era both manual and semi-automatic turret lathes were sometimes called "screw machines", although we today reserve that term for fully automatic machines. === Automatic === During the 1870s through 1890s, the mechanically automated "automatic" turret lathe was developed and disseminated. These machines can execute many part-cutting cycles without human intervention. Thus the duties of the operator, which were already greatly reduced by the manual turret lathe, were even further reduced, and productivity increased. These machines use cams to automate the sliding and indexing of the turret and the opening and closing of the chuck. Thus, they execute the part-cutting cycle somewhat analogously to the way in which an elaborate cuckoo clock performs an automated theater show. Small- to medium-sized automatic turret lathes are usually called "screw machines" or "automatic screw machines", while larger ones are usually called "automatic chucking lathes", "automatic chuckers", or "chuckers". Such machine tools of the "automatic" variety, which in the pre-computer era meant mechanically automated, had already reached a highly advanced state by World War I. === Computer numerical control === When World War II ended, the digital computer was poised to develop from a colossal laboratory curiosity into a practical technology that could begin to disseminate into business and industry. The advent of computer-based automation in machine tools via numerical control (NC) and then computer numerical control (CNC) displaced to a large extent, but not at all completely, the previously existing manual and mechanically automated machines. Numerically controlled turrets allow automated selection of tools on a turret. CNC lathes may be horizontal or vertical in orientation and mount six separate tools on one or more turrets. Such machine tools can work in two axes per turret, with up to six axes being feasible for complex work. === Vertical === Vertical turret lathes have the workpiece held vertically, which allows the headstock to sit on the floor and the faceplate to become a horizontal rotating table, analogous to a huge potter's wheel. This is useful for the handling of very large, heavy, short workpieces. Vertical lathes in general are also called "vertical boring mills" or often simply "boring mills"; therefore a vertical turret lathe is a vertical boring mill equipped with a turret. == Other variations == === Capstan versus turret === The term "capstan lathe" overlaps in sense with the term "turret lathe" to a large extent. In many times and places, it has been understood to be synonymous with "turret lathe". In other times and places it has been held in technical contradistinction to "turret lathe", with the difference being in whether the turret's slide is fixed to the bed (ram-type turret) or slides on the bed's ways (saddle-type turret). The difference in terminology is mostly a matter of United Kingdom and Commonwealth usage versus United States usage. === Flat === A subtype of horizontal turret lathe is the flat-turret lathe. Its turret is flat (and analogous to a rotary table), allowing the turret to pass beneath the part. Patented by James Hartness of Jones & Lamson, and first disseminated in the 1890s, it was developed to provide more rigidity via requiring less overhang in the tool setup, especially when the part is relatively long. === Hollow-hexagon === Hollow-hexagon turret lathes competed with flat-turret lathes by taking the conventional hexagon turret and making it hollow, allowing the part to pass into it during the cut, analogously to how the part would pass over the flat turret. In both cases, the main idea is to increase rigidity by allowing a relatively long part to be turned without the tool overhang that would be needed with a conventional turret, which is not flat or hollow. === Monitor lathe === The term "monitor lathe" formerly (1860s–1940s) referred to the class of small- to medium-sized manual turret lathes used on relatively small work. The name was inspired by the monitor-class warships, which the monitor lathe's turret resembled. Today, lathes of such appearance, such as the Hardinge DSM-59 and its many clones, are still common, but the name "monitor lathe" is no longer current in the industry. === Toolpost turrets and tailstock turrets === Turrets can be added to non-turret lathes (bench lathes, engine lathes, toolroom lathes, etc.) by mounting them on the toolpost, tailstock, or both. Often these turrets are not as large as a turret lathe's, and they usually do not offer the sliding and stopping that a turret lathe's turret does; but they do offer the ability to index through successive tool

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  Vladimir Batagelj

  Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks (blockmodeling). == Education and career == Vladimir Batagelj completed his Ph.D. at the University of Ljubljana in 1986 under the direction of Tomaž Pisanski. He stayed at the University of Ljubljana as a professor until his retirement, where he was a professor of sociology and statistics, while also being a chair of the Department of Sociology of the Faculty of Social Sciences. As visiting professor, he was taught at the University of Pittsburgh (1990-91) and at the University of Konstanz (2002). He was also a member of editorial boards of two journals: Informatica and Journal of Social Structure. His work has been cited over 11000 times. His book Exploratory Social Network Analysis with Pajek on blockmodeling, coauthored with Wouter de Nooy and Andrej Mrvar, is Batagelj's most cited work and has over 3300 citations. The book was translated into Chinese and Japanese. The revised and expanded third edition has been published by Cambridge University Press. In 1975, 11 years before completing his PhD, Batagelj published a solo paper in Communications of the ACM. Batagelj authored more than 20 textbooks in Slovenian, covering topics like TeX, combinatorics and discrete mathematics. He has also written extensively in the Slovenian popular science journal Presek. Batagelj has advised 9 Ph.D. students. == Pajek == Batagelj is particularly known for his work on Pajek, a freely available software for analysis and visualization of large networks. He began work on Pajek in 1996 with Andrej Mrvar, who was then his PhD student. == Awards and honors == First prizes for contributions (with Andrej Mrvar) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. In 2007 the book Generalized blockmodeling was awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association In 2007 he was awarded (together with Anuška Ferligoj) the Simmel Award by INSNA. In 2013, Vladimir Batagelj and Andrej Mrvar received the INSNA's William D. Richards Software award for their work on Pajek. == Selected bibliography == Vladimir Batagelj, Social Network Analysis, Large-Scale [1]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 8245–8265. Vladimir Batagelj, Complex Networks, Visualization of [2]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 1253–1268. Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press 2005 (ISBN 0-521-60262-9). ESNA in Japanese, TDU, 2010. Patrick Doreian, Vladimir Batagelj, Anuška Ferligoj, Mark Granovetter (Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)

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  NOMINATE (scaling method)

  NOMINATE (an acronym for nominal three-step estimation) is a multidimensional scaling application developed by US political scientists Keith T. Poole and Howard Rosenthal in the early 1980s to analyze preferential and choice data, such as legislative roll-call voting behavior. In its most well-known application, members of the US Congress are placed on a two-dimensional map, with politicians who are ideologically similar (i.e. who often vote the same) being close together. One of these two dimensions corresponds to the familiar left–right political spectrum (liberal–conservative in the United States). As computing capabilities grew, Poole and Rosenthal developed multiple iterations of their NOMINATE procedure: the original D-NOMINATE method, W-NOMINATE, and most recently DW-NOMINATE (for dynamic, weighted NOMINATE). In 2009, Poole and Rosenthal were the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE. In 2016, the society awarded Poole its Career Achievement Award, stating that "the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores." == Procedure == The main procedure is an application of multidimensional scaling techniques to political choice data. Though there are important technical differences between these types of NOMINATE scaling procedures, all operate under the same fundamental assumptions. First, that alternative choices can be projected on a basic, low-dimensional (often two-dimensional) Euclidean space. Second, within that space, individuals have utility functions which are bell-shaped (normally distributed), and maximized at their ideal point. Because individuals also have symmetric, single-peaked utility functions which center on their ideal point, ideal points represent individuals' most preferred outcomes. That is, individuals most desire outcomes closest their ideal point, and will choose/vote probabilistically for the closest outcome. Ideal points can be recovered from observing choices, with individuals exhibiting similar preferences placed more closely than those behaving dissimilarly. It is helpful to compare this procedure to producing maps based on driving distances between cities. For example, Los Angeles is about 1,800 miles from St. Louis; St. Louis is about 1,200 miles from Miami; and Miami is about 2,700 miles from Los Angeles. From this (dis)similarities data, any map of these three cities should place Miami far from Los Angeles, with St. Louis somewhere in between (though a bit closer to Miami than Los Angeles). Just as cities like Los Angeles and San Francisco would be clustered on a map, NOMINATE places ideologically similar legislators (e.g., liberal Senators Barbara Boxer (D-Calif.) and Al Franken (D-Minn.)) closer to each other, and farther from dissimilar legislators (e.g., conservative Senator Tom Coburn (R-Okla.)) based on the degree of agreement between their roll call voting records. At the heart of the NOMINATE procedures (and other multidimensional scaling methods, such as Poole's Optimal Classification method) are algorithms they utilize to arrange individuals and choices in low dimensional (usually two-dimensional) space. Thus, NOMINATE scores provide "maps" of legislatures. Using NOMINATE procedures to study congressional roll call voting behavior from the First Congress to the present-day, Poole and Rosenthal published Congress: A Political-Economic History of Roll Call Voting in 1997 and the revised edition Ideology and Congress in 2007. In 2009, Poole and Rosenthal were named the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE, a recognition conferred to "individual(s) for developing statistical software that makes a significant research contribution". In 2016, Keith T. Poole was awarded the Society for Political Methodology's Career Achievement Award. The citation for this award reads, in part, "One can say perfectly correctly, and without any hyperbole: the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores. NOMINATE has produced data that entire bodies of our discipline—and many in the press—have relied on to understand the U.S. Congress." == Dimensions == Poole and Rosenthal demonstrate that—despite the many complexities of congressional representation and politics—roll call voting in both the House and the Senate can be organized and explained by no more than two dimensions throughout the sweep of American history. The first dimension (horizontal or x-axis) is the familiar left-right (or liberal-conservative) spectrum on economic matters. The second dimension (vertical or y-axis) picks up attitudes on cross-cutting, salient issues of the day (which include or have included slavery, bimetallism, civil rights, regional, and social/lifestyle issues). Rosenthal and Poole have initially argued that the first dimension refers to socio-economic matters and the second dimension to race-relations. However, the often confusing and residual nature of the second dimension has led to the second dimension being largely ignored by other researchers. For the most part, congressional voting is uni-dimensional, with most of the variation in voting patterns explained by placement along the liberal-conservative first dimension. While the first dimension of the DW-NOMINATE score is able to predict results at 83% accuracy, the addition of the second dimension only increases accuracy to 85%. Furthermore, the second dimension only provided a significant increase in accuracy for Congresses 1-99. As late as the 1990s, the second dimension was able to measure partisan splits in abortion and gun rights issues. However, a 2017 analysis found that since 1987, the votes of the US Congress had best fit a one-dimensional model, suggesting increasing party polarization after 1987. == Interpretation of nominate scores == For illustrative purposes, consider the following plots which use W-NOMINATE scores to scale members of Congress and uses the probabilistic voting model (in which legislators farther from the "cutting line" between "yea" and "nay" outcomes become more likely to vote in the predicted manner) to illustrate some major Congressional votes in the 1990s. Some of these votes, like the House's vote on President Clinton's welfare reform package (the Personal Responsibility and Work Opportunity Act of 1996) are best modeled through the use of the first (economic liberal-conservative) dimension. On the welfare reform vote, nearly all Republicans joined the moderate-conservative bloc of House Democrats in voting for the bill, while opposition was virtually confined to the most liberal Democrats in the House. The errors (those representatives on the "wrong" side of the cutting line which separates predicted "yeas" and predicted "nays") are generally close to the cutting line, which is what we would expect. A legislator directly on the cutting line is indifferent between voting "yea" and "nay" on the measure. All members are shown on the left panel of the plot, while only errors are shown on the right panel: Economic ideology also dominates the Senate vote on the Balanced Budget Amendment of 1995: On other votes, however, a second dimension (which has recently come to represent attitudes on cultural and lifestyle issues) is important. For example, roll call votes on gun control routinely split party coalitions, with socially conservative "blue dog" Democrats joining most Republicans in opposing additional regulation and socially liberal Republicans joining most Democrats in supporting gun control. The addition of the second dimension accounts for these inter-party differences, and the cutting line is more horizontal than vertical (meaning the cleavage is found on the second dimension rather than the first dimension on these votes) This pattern was evident in the 1991 House vote to require waiting periods on handguns: == Political ideology == DW-NOMINATE scores have been used widely to describe the political ideology of political actors, political parties and political institutions. For instance, a score in the first dimension that is close to either pole means that such score is located at one of the extremes in the liberal-conservative scale. So, a score closer to 1 is described as conservative whereas a score closer to −1 can be described as liberal. Finally, a score at zero or close to zero is described as moderate. == Political polarization == Poole and Rosenthal (beginning with their 1984 article "The Polarization of American Politics") have also used NOMINATE data to show that, since the 1970s, party delegations in Congress have become ideologically homogeneous and distant from one another (a phenomenon known as "polarization"). Using DW-NOMINATE scores (which permit direct comparisons between members of different Congress

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  Modes of variation

  In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA). == Formulation == Modes of variation are a natural extension of PCA and FPCA. === Modes of variation in PCA === If a random vector X = ( X 1 , X 2 , ⋯ , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}} has the mean vector μ p {\displaystyle {\boldsymbol {\mu }}_{p}} , and the covariance matrix Σ p × p {\displaystyle \mathbf {\Sigma } _{p\times p}} with eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0} and corresponding orthonormal eigenvectors e 1 , e 2 , ⋯ , e p {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}} , by eigendecomposition of a real symmetric matrix, the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } can be decomposed as Σ = Q Λ Q T , {\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},} where Q {\displaystyle \mathbf {Q} } is an orthogonal matrix whose columns are the eigenvectors of Σ {\displaystyle \mathbf {\Sigma } } , and Λ {\displaystyle \mathbf {\Lambda } } is a diagonal matrix whose entries are the eigenvalues of Σ {\displaystyle \mathbf {\Sigma } } . By the Karhunen–Loève expansion for random vectors, one can express the centered random vector in the eigenbasis X − μ = ∑ k = 1 p ξ k e k , {\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},} where ξ k = e k T ( X − μ ) {\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})} is the principal component associated with the k {\displaystyle k} -th eigenvector e k {\displaystyle \mathbf {e} _{k}} , with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.} Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } is the set of vectors, indexed by α {\displaystyle \alpha } , m k , α = μ ± α λ k e k , α ∈ [ − A , A ] , {\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in [-A,A],} where A {\displaystyle A} is typically selected as 2 or 3 {\displaystyle 2\ {\text{or}}\ 3} . === Modes of variation in FPCA === For a square-integrable random function X ( t ) , t ∈ T ⊂ R p {\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}} , where typically p = 1 {\displaystyle p=1} and T {\displaystyle {\mathcal {T}}} is an interval, denote the mean function by μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),} where λ 1 ≥ λ 2 ≥ ⋯ ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0} are the eigenvalues and { φ 1 , φ 2 , ⋯ } {\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}} are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.} By the Karhunen–Loève theorem, one can express the centered function in the eigenbasis, X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),} where ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt} is the k {\displaystyle k} -th principal component with the properties E ⁡ ( ξ k ) = 0 , Var ⁡ ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ⁡ ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.} Then the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} is the set of functions, indexed by α {\displaystyle \alpha } , m k , α ( t ) = μ ( t ) ± α λ k φ k ( t ) , t ∈ T , α ∈ [ − A , A ] {\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in [-A,A]} that are viewed simultaneously over the range of α {\displaystyle \alpha } , usually for A = 2 or 3 {\displaystyle A=2\ {\text{or}}\ 3} . == Estimation == The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. === Modes of variation in PCA === Suppose the data x 1 , x 2 , ⋯ , x n {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}} represent n {\displaystyle n} independent drawings from some p {\displaystyle p} -dimensional population X {\displaystyle \mathbf {X} } with mean vector μ {\displaystyle {\boldsymbol {\mu }}} and covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . These data yield the sample mean vector x ¯ {\displaystyle {\overline {\mathbf {x} }}} , and the sample covariance matrix S {\displaystyle \mathbf {S} } with eigenvalue-eigenvector pairs ( λ ^ 1 , e ^ 1 ) , ( λ ^ 2 , e ^ 2 ) , ⋯ , ( λ ^ p , e ^ p ) {\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})} . Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } can be estimated by m ^ k , α = x ¯ ± α λ ^ k e ^ k , α ∈ [ − A , A ] . {\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in [-A,A].} === Modes of variation in FPCA === Consider n {\displaystyle n} realizations X 1 ( t ) , X 2 ( t ) , ⋯ , X n ( t ) {\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)} of a square-integrable random function X ( t ) , t ∈ T {\displaystyle X(t),t\in {\mathcal {T}}} with the mean function μ ( t ) = E ⁡ ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} and the covariance function G ( s , t ) = Cov ⁡ ( X ( s ) , X ( t ) ) {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))} . Functional principal component analysis provides methods for the estimation of μ ( t ) {\displaystyle \mu (t)} and G ( s , t ) {\displaystyle G(s,t)} in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} can be estimated by m ^ k , α ( t ) = μ ^ ( t ) ± α λ ^ k φ ^ k ( t ) , t ∈ T , α ∈ [ − A , A ] . {\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in [-A,A].} == Applications == Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , {\displaystyle \alpha =\pm 1,\pm 2,} and ± 3 {\displaystyle \pm 3} , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality

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  Superquadrics

  In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs. The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used in computer vision, robotics, and physical simulation. Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoids and the supertoroids. In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images and point clouds are covered in several computer vision literatures. == Formulas == === Implicit equation === The surface of the basic superquadric is given by | x | r + | y | s + | z | t = 1 {\displaystyle \left|x\right|^{r}+\left|y\right|^{s}+\left|z\right|^{t}=1} where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely: less than 1: a pointy octahedron modified to have concave faces and sharp edges. exactly 1: a regular octahedron. between 1 and 2: an octahedron modified to have convex faces, blunt edges and blunt corners. exactly 2: a sphere greater than 2: a cube modified to have rounded edges and corners. infinite (in the limit): a cube Each exponent can be varied independently to obtain combined shapes. For example, if r=s=2, and t=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) r = s. If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called super-hyperboloids. The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts A, B, C along each axis. Its general equation is | x A | r + | y B | s + | z C | t = 1. {\displaystyle \left|{\frac {x}{A}}\right|^{r}+\left|{\frac {y}{B}}\right|^{s}+\left|{\frac {z}{C}}\right|^{t}=1.} === Parametric description === Parametric equations in terms of surface parameters u and v (equivalent to longitude and latitude if m equals 2) are x ( u , v ) = A g ( v , 2 r ) g ( u , 2 r ) y ( u , v ) = B g ( v , 2 s ) f ( u , 2 s ) z ( u , v ) = C f ( v , 2 t ) − π 2 ≤ v ≤ π 2 , − π ≤ u < π , {\displaystyle {\begin{aligned}x(u,v)&{}=Ag\left(v,{\frac {2}{r}}\right)g\left(u,{\frac {2}{r}}\right)\\y(u,v)&{}=Bg\left(v,{\frac {2}{s}}\right)f\left(u,{\frac {2}{s}}\right)\\z(u,v)&{}=Cf\left(v,{\frac {2}{t}}\right)\\&-{\frac {\pi }{2}}\leq v\leq {\frac {\pi }{2}},\quad -\pi \leq u<\pi ,\end{aligned}}} where the auxiliary functions are f ( ω , m ) = sgn ⁡ ( sin ⁡ ω ) | sin ⁡ ω | m g ( ω , m ) = sgn ⁡ ( cos ⁡ ω ) | cos ⁡ ω | m {\displaystyle {\begin{aligned}f(\omega ,m)&{}=\operatorname {sgn}(\sin \omega )\left|\sin \omega \right|^{m}\\g(\omega ,m)&{}=\operatorname {sgn}(\cos \omega )\left|\cos \omega \right|^{m}\end{aligned}}} and the sign function sgn(x) is sgn ⁡ ( x ) = { − 1 , x < 0 0 , x = 0 + 1 , x > 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&x<0\\0,&x=0\\+1,&x>0.\end{cases}}} === Spherical product === Barr introduces the spherical product which given two plane curves produces a 3D surface. If f ( μ ) = ( f 1 ( μ ) f 2 ( μ ) ) , g ( ν ) = ( g 1 ( ν ) g 2 ( ν ) ) {\displaystyle f(\mu )={\begin{pmatrix}f_{1}(\mu )\\f_{2}(\mu )\end{pmatrix}},\quad g(\nu )={\begin{pmatrix}g_{1}(\nu )\\g_{2}(\nu )\end{pmatrix}}} are two plane curves then the spherical product is h ( μ , ν ) = f ( μ ) ⊗ g ( ν ) = ( f 1 ( μ ) g 1 ( ν ) f 1 ( μ ) g 2 ( ν ) f 2 ( μ ) ) {\displaystyle h(\mu ,\nu )=f(\mu )\otimes g(\nu )={\begin{pmatrix}f_{1}(\mu )\ g_{1}(\nu )\\f_{1}(\mu )\ g_{2}(\nu )\\f_{2}(\mu )\end{pmatrix}}} This is similar to the typical parametric equation of a sphere: x = x 0 + r sin ⁡ θ cos ⁡ φ y = y 0 + r sin ⁡ θ sin ⁡ φ ( 0 ≤ θ ≤ π , 0 ≤ φ < 2 π ) z = z 0 + r cos ⁡ θ {\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \end{aligned}}} which give rise to the name spherical product. Barr uses the spherical product to define quadric surfaces, like ellipsoids, and hyperboloids as well as the torus, superellipsoid, superquadric hyperboloids of one and two sheets, and supertoroids. == Plotting code == The following GNU Octave code generates a mesh approximation of a superquadric:

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  Logic learning machine

  Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems), financial services and supply chain management. == History == The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy. Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed. == General == Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if premise then consequence where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} . for ordered variables the condition is written as an inequality or an interval: x 2 ≤ α {\displaystyle x_{2}\leq \alpha } or β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } A possible rule is therefore in the form if x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} AND x 2 ≤ α {\displaystyle x_{2}\leq \alpha } AND β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } then y = y ¯ {\displaystyle y={\bar {y}}} == Types == According to the output type, different versions of the Logic Learning Machine have been developed: Logic Learning Machine for classification, when the output is a categorical variable, which can assume values in a finite set Logic Learning Machine for regression, when the output is an integer or real number.

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  Correlation clustering

  Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the

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  Time-aware long short-term memory

  Time-aware LSTM (T-LSTM) is a long short-term memory (LSTM) unit capable of handling irregular time intervals in longitudinal patient records. T-LSTM was developed by researchers from Michigan State University, IBM Research, and Cornell University and was first presented in the Knowledge Discovery and Data Mining (KDD) conference. Experiments using real and synthetic data proved that T-LSTM auto-encoder outperformed widely used frameworks including LSTM and MF1-LSTM auto-encoders.

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  Time series

  In mathematics, a time series is a sequence of data points indexed, listed, or graphed in chronological order. Most commonly, a time series consists of observations recorded at successive equally spaced points in time. Thus, it represents a form of discrete-time data. A time series may describe measurements collected over seconds, days, years, or even centuries. Common examples include heights of ocean tides, counts of sunspots, daily temperature readings, and the closing values of stock market indices such as the Dow Jones Industrial Average. A time series is often visualized using a run chart (a type of temporal line chart), which helps identify patterns such as trends, seasonal effects, and irregular fluctuations. Time series are widely used in statistics, actuarial science, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and many other areas of applied science and engineering that involve temporal measurements. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. Generally, time series data is modeled as a stochastic process. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series. Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility). Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language). == Methods for analysis == Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, thereby mitigating the need to operate in the frequency domain. Additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving-average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure. Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate. == Panel data == A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a cross-sectional dataset). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to ask what makes one data record unique from the other records. If the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a time data field and an additional identifier which is unrelated to time (e.g. student ID, stock symbol, country code), then it is panel data candidate. If the differentiation lies on the non-time identifier, then the data set is a cross-sectional data set candidate. == Analysis == There are several types of motivation and data analysis available for time series which are appropriate for different purposes. === Motivation === In the context of statistics, econometrics, quantitative finance, seismology, meteorology, and geophysics the primary goal of time series analysis is forecasting. In the context of signal processing, control engineering and communication engineering it is used for signal detection. Other applications are in data mining, pattern recognition and machine learning, where time series analysis can be used for clustering, classification, query by content, anomaly detection as well as forecasting. === Exploratory analysis === A simple way to examine a regular time series is manually with a line chart. The datagraphic shows tuberculosis deaths in the United States, along with the yearly change and the percentage change from year to year. The total number of deaths declined in every year until the mid-1980s, after which there were occasional increases, often proportionately - but not absolutely - quite large. A study of corporate data analysts found two challenges to exploratory time series analysis: discovering the shape of interesting patterns, and finding an explanation for these patterns. Visual tools that represent time series data as heat map matrices can help overcome these challenges. === Estimation, filtering, and smoothing === This approach may be based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation. Its development was significantly accelerated during World War II by mathematician Norbert Wiener, electrical engineers Rudolf E. Kálmán, Dennis Gabor and others for filtering signals from noise and predicting signal values at a certain point in time. An equivalent effect may be achieved in the time domain, as in a Kalman filter; see filtering and smoothing for more techniques. Other related techniques include: Autocorrelation analysis to examine serial dependence Spectral analysis to examine cyclic behavior which need not be related to seasonality. For example, sunspot activity varies over 11 year cycles. Other common examples include celestial phenomena, weather patterns, neural activity, commodity prices, and economic activity. Separation into components representing trend, seasonality, slow and fast variation, and cyclical irregularity: see trend estimation and decomposition of time series === Curve fitting === Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data. For processes that are expected to generally grow in magnitude one of the curves in the graphic (and many others) can be fitted by estimating their parameters. The construction of economic time series involves the estimation of some components for some dates by interpolation between values ("benchmarks") for earlier and later dates. Interpolation is estimation of an unknown quantity between two known quantities (historical data), or drawing conclusions about missing information from the available information ("reading between the lines"). Interpolation is useful where the data surrounding the missing data is available and its trend, seasonality, and longer-term cycles are known. This is often done by using a relat

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  KNIME

  KNIME ( ), the Konstanz Information Miner, is a data analytics, reporting and integrating platform. KNIME integrates various components for machine learning and data mining through its modular data pipelining "Building Blocks of Analytics" concept. A graphical user interface and use of Java Database Connectivity (JDBC) allows assembly of nodes blending different data sources, including preprocessing (extract, transform, load, or ETL), for modeling, data analysis and visualization with minimal, or no, programming. It is free and open-source software released under a GNU General Public License. Since 2006, KNIME has been used in pharmaceutical research, and in other areas including customer relationship management (CRM) and data analysis, business intelligence, text mining and financial data analysis. Recently, attempts were made to use KNIME as robotic process automation (RPA) tool. KNIME's headquarters are based in Zurich, with other offices in Konstanz, Berlin, and Austin (USA). == History == Development of KNIME began in January 2004, with a team of software engineers at the University of Konstanz, as an open-source platform. The original team, headed by Michael Berthold, came from a Silicon Valley pharmaceutical industry software company. The initial goal was to create a modular, highly scalable and open data processing platform that allows easy integration of different data loading, processing, transforming, analyzing, and visual exploring modules, without focus on any one application area. The platform was intended for collaborating, research, and for integrating various other data analysis projects. In 2006, the first version of KNIME was released. Several pharmaceutical companies began using KNIME, and several life science software vendors began integrating their tools into the platform. Later that year, after an article in the German magazine c't, users from a number of other areas joined ship. As of 2012, KNIME is in use by over 15,000 actual users (i.e. not counting downloads, but users regularly retrieving updates) in the life sciences and at banks, publishers, car manufacturer, telcos, consulting firms, and various other industries, and a large number of research groups, worldwide. Latest updates to KNIME Server and KNIME Big Data Extensions, provide support for Apache Spark 2.3, Parquet and HDFS-type storage. For the sixth year in a row, KNIME has been placed as a leader for data science and machine learning platforms in Gartner's Magic Quadrant. == Design philosophy, features == These are the design principles and features that KNIME software follows: Visual, Interactive Framework: KNIME Software prioritizes a user-friendly and intuitive approach to data analysis. This is achieved through a visual and interactive framework where data flows can be combined using a drag-and-drop interface. Users can develop customized and interactive applications by creating simple to advanced and highly-automated data pipelines. These may include, for example, access to databases, machine learning libraries, logic for workflow control (e.g., loops, switches, etc.), abstraction (e.g., interactive widgets), invocation, dynamic data apps, integrated deployment, or error handling. Modularity: processing units and data containers should remain independent of each other. This design choice enables easy distribution of computation and allows for the independent development of different algorithms. Data types within KNIME are encapsulated, meaning no types are predefined. This design choice facilitates adding new data types, and integrating them with extant types, while including type-specific renderers and comparators. This principle also enables inspecting results at the end of each single data operation. Extensibility: KNIME Software is designed to be extensible. Adding new processing nodes or views is made simple through a plug-in mechanism. This mechanism ensures that users can distribute their custom functionalities without the need for complicated install or uninstall procedures. Interleaving No-Code with Code: the platform supports integrating both visual programming (no-code) and script-based programming (e.g., Python, R, JavaScript) approaches to data analysis. This design principle is termed low-code. Automation and Scalability: for example, the use of parameterization via flow variables, or the encapsulation of workflow segments in components contribute to reduce manual work and errors in analyses. Further, the scheduling of workflow execution (available in KNIME Business Hub and KNIME Community Hub for Teams) reduces dependency on human resources. In terms of scalability, a few examples include the ability to handle large datasets (millions of rows), execute multiple processes simultaneously out of the box and reuse workflow segments. Full Usability: due to the open source nature, KNIME Analytics Platform provides free full usability with no limited trial periods. == Internals == KNIME allows users to visually create data flows (or pipelines), selectively execute some or all analysis steps, and later inspect the results, models, using interactive widgets and views. KNIME is written in Java and based on Eclipse. It makes use of an extension mechanism to add plug-ins providing added functions. The core version includes hundreds of modules for data integration (file input/output (I/O), database nodes supporting all common database management systems through JDBC or native connectors: SQLite, MS-Access, SQL Server, MySQL, Oracle, PostgreSQL, Vertica and H2), data transformation (filter, converter, splitter, combiner, joiner), and the commonly used methods of statistics, data mining, analysis and text analytics. Visualization is supported with the Report Designer extension. KNIME workflows can be used as data sets to create report templates that can be exported to document formats such as doc, ppt, xls, pdf and others. Other KNIME abilities are: KNIMEs core-architecture allows processing of large data volumes that are only limited by the available hard disk space (not limited to the available RAM). E.g., KNIME allows analyzing 300 million customer addresses, 20 million cell images, and 10 million molecular structures. Added plug-ins allow integrating methods for text mining, image mining, time series analysis, and networking. KNIME integrates various other open-source projects, e.g., machine learning algorithms from Weka, H2O, Keras, Spark, the R project and LIBSVM; plotly, JFreeChart, ImageJ, and the Chemistry Development Kit. KNIME is implemented in Java, allows for wrappers calling other code, in addition to providing nodes that allow it to run Java, Python, R, Ruby and other code fragments. Since 2021, KNIME's Python Integration utilizes Anaconda for Python distribution and environment management. == License == In 2024, KNIME version 5.3 is released under the same GPLv3 license as previous versions. As of version 2.1, KNIME is released under the GPLv3 license, with an exception that allow commercial software vendors to use the well-defined node application programming interface (API) to add proprietary extensions, or wrappers calling their tools from KNIME. == Courses == KNIME allows the performance of data analysis without programming skills. Several free, online courses are provided.

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  Sammon mapping

  Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications. Denote the distance between ith and jth objects in the original space by d i j ∗ {\displaystyle \scriptstyle d_{ij}^{}} , and the distance between their projections by d i j {\displaystyle \scriptstyle d_{ij}^{}} . Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error: E = 1 ∑ i < j d i j ∗ ∑ i < j ( d i j ∗ − d i j ) 2 d i j ∗ . {\displaystyle E={\frac {1}{\sum \limits _{i Read more →

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  Principal component analysis

  Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector

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  MyChild App

  MyChild App is an Android app that helps parents screen developmental disorders in their children between the age of 1 and 24 months. The app contains information for parents about the different stages of a child's development. == Background == Launched in 2015 on Google PlayStore, the app is a consumer product of the parent company, Time Ahead, Inc. Its office is based in Bhopal, Madhya Pradesh, India. As of August 2016, the app had been downloaded by 11,000+ users in 140+ countries and is a part of fbstart case study. == Funding == In 2015, MyChild App raised a seed round of $100k led by 500 Startups, followed by angel investors Samir Bangara, Anisha Mittal, Pallav Nadhani, Deobrat Singh, Lalit Mangal, Arihant Patni, Amit Gupta, Dr. Ritesh Malik, Saurab Paruthi, and Singapore Angel Network.

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  Latent Dirichlet allocation

  In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can be described by a set of unobserved "topics." For example, given a set of news articles, LDA might discover that one topic is characterized by words like "president", "government", and "election", while another is characterized by "team", "game", and "score". It is one of the most common topic models. The LDA model was first presented as a graphical model for population genetics by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. The model was subsequently applied to machine learning by David Blei, Andrew Ng, and Michael I. Jordan in 2003. Although its most frequent application is in modeling text corpora, it has also been used for other problems, such as in clinical psychology, social science, and computational musicology. The core assumption of LDA is that documents are represented as a random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic latent semantic analysis (pLSA), differing primarily in that LDA treats the topic mixture as a Dirichlet prior, leading to more reasonable mixtures and less susceptibility to overfitting. Learning the latent topics and their associated probabilities from a corpus is typically done using Bayesian inference, often with methods like Gibbs sampling or variational Bayes. == History == In the context of population genetics, LDA was proposed by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. LDA was applied in machine learning by David Blei, Andrew Ng and Michael I. Jordan in 2003. == Overview == === Population genetics === In population genetics, the model is used to detect the presence of structured genetic variation in a group of individuals. The model assumes that alleles carried by individuals under study have origin in various extant or past populations. The model and various inference algorithms allow scientists to estimate the allele frequencies in those source populations and the origin of alleles carried by individuals under study. The source populations can be interpreted ex-post in terms of various evolutionary scenarios. In association studies, detecting the presence of genetic structure is considered a necessary preliminary step to avoid confounding. === Clinical psychology, mental health, and social science === In clinical psychology research, LDA has been used to identify common themes of self-images experienced by young people in social situations. Other social scientists have used LDA to examine large sets of topical data from discussions on social media (e.g., tweets about prescription drugs). Additionally, supervised Latent Dirichlet Allocation with covariates (SLDAX) has been specifically developed to combine latent topics identified in texts with other manifest variables. This approach allows for the integration of text data as predictors in statistical regression analyses, improving the accuracy of mental health predictions. One of the main advantages of SLDAX over traditional two-stage approaches is its ability to avoid biased estimates and incorrect standard errors, allowing for a more accurate analysis of psychological texts. In the field of social sciences, LDA has proven to be useful for analyzing large datasets, such as social media discussions. For instance, researchers have used LDA to investigate tweets discussing socially relevant topics, like the use of prescription drugs and cultural differences in China. By analyzing these large text corpora, it is possible to uncover patterns and themes that might otherwise go unnoticed, offering valuable insights into public discourse and perception in real time. === Musicology === In the context of computational musicology, LDA has been used to discover tonal structures in different corpora. === Machine learning === One application of LDA in machine learning – specifically, topic discovery, a subproblem in natural language processing – is to discover topics in a collection of documents, and then automatically classify any individual document within the collection in terms of how "relevant" it is to each of the discovered topics. A topic is considered to be a set of terms (i.e., individual words or phrases) that, taken together, suggest a shared theme. For example, in a document collection related to pet animals, the terms dog, spaniel, beagle, golden retriever, puppy, bark, and woof would suggest a DOG_related theme, while the terms cat, siamese, Maine coon, tabby, manx, meow, purr, and kitten would suggest a CAT_related theme. There may be many more topics in the collection – e.g., related to diet, grooming, healthcare, behavior, etc. that we do not discuss for simplicity's sake. (Very common, so called stop words in a language – e.g., "the", "an", "that", "are", "is", etc., – would not discriminate between topics and are usually filtered out by pre-processing before LDA is performed. Pre-processing also converts terms to their "root" lexical forms – e.g., "barks", "barking", and "barked" would be converted to "bark".) If the document collection is sufficiently large, LDA will discover such sets of terms (i.e., topics) based upon the co-occurrence of individual terms, though the task of assigning a meaningful label to an individual topic (i.e., that all the terms are DOG_related) is up to the user, and often requires specialized knowledge (e.g., for collection of technical documents). The LDA approach assumes that: The semantic content of a document is composed by combining one or more terms from one or more topics. Certain terms are ambiguous, belonging to more than one topic, with different probability. (For example, the term training can apply to both dogs and cats, but are more likely to refer to dogs, which are used as work animals or participate in obedience or skill competitions.) However, in a document, the accompanying presence of specific neighboring terms (which belong to only one topic) will disambiguate their usage. Most documents will contain only a relatively small number of topics. In the collection, e.g., individual topics will occur with differing frequencies. That is, they have a probability distribution, so that a given document is more likely to contain some topics than others. Within a topic, certain terms will be used much more frequently than others. In other words, the terms within a topic will also have their own probability distribution. When LDA machine learning is employed, both sets of probabilities are computed during the training phase, using Bayesian methods and an expectation–maximization algorithm. LDA is a generalization of older approach of probabilistic latent semantic analysis (pLSA), The pLSA model is equivalent to LDA under a uniform Dirichlet prior distribution. pLSA relies on only the first two assumptions above and does not care about the remainder. While both methods are similar in principle and require the user to specify the number of topics to be discovered before the start of training (as with k-means clustering) LDA has the following advantages over pLSA: LDA yields better disambiguation of words and a more precise assignment of documents to topics. Computing probabilities allows a "generative" process by which a collection of new "synthetic documents" can be generated that would closely reflect the statistical characteristics of the original collection. Unlike LDA, pLSA is vulnerable to overfitting especially when the size of corpus increases. The LDA algorithm is more readily amenable to scaling up for large data sets using the MapReduce approach on a computing cluster. == Model == With plate notation, which is often used to represent probabilistic graphical models (PGMs), the dependencies among the many variables can be captured concisely. The boxes are "plates" representing replicates, which are repeated entities. The outer plate represents documents, while the inner plate represents the repeated word positions in a given document; each position is associated with a choice of topic and word. The variable names are defined as follows: M denotes the number of documents N is number of words in a given document (document i has N i {\displaystyle N_{i}} words) α is the parameter of the Dirichlet prior on the per-document topic distributions β is the parameter of the Dirichlet prior on the per-topic word distribution θ i {\displaystyle \theta _{i}} is the topic distribution for document i φ k {\displaystyle \varphi _{k}} is the word distribution for topic k z i j {\displaystyle z_{ij}} is the topic for the j-th word in document i w i j {\displaystyle w_{ij}} is the specific word. The fact that W is grayed out means that words w i j {\displaystyle w_{ij}} are the only observable variables, and the other variables are latent variables. As proposed in the original paper, a sparse Dirichlet prior can be used to model the to

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  Vapnik–Chervonenkis dimension

  In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (x,x+2]} . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f {\displaystyle f} is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa

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