AI Generator With Image

AI Generator With Image — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  NationBuilder

  NationBuilder is a Los Angeles-based technology start-up that develops content management and customer relationship management (CRM) software. Although the company initially targeted political campaigns and nonprofit organizations, it later expanded its marketing efforts to include other people and organizations trying to build an online following, such as artists, musicians and restaurants. The software uses voter data such as names, addresses and other information, such as previous voting records in the case of political campaigns, to allow users to centralize, build and manage campaigns by integrating various communication tools like websites, newsletters, text messaging and social media channels under one platform. Among other features, the software enables users to quickly create websites, build databases through registrations, send targeted newsletters, analyse data from multiple sources and leverage micro-donations. The software's appeal towards political campaigns comes from the combination of a number of previously separate campaigning services, channels and data sources into a single platform that was presented as a facile solution for non-technical users and which enabled political campaigners to quickly deploy campaigns by convincing numerous people to donate. == History == NationBuilder was founded in 2009 in Los Angeles by Jim Gilliam and launched in 2011. In 2012 Joe Green joined NationBuilder as co-founder and president. He left that role 11 months later in February 2013. Gilliam was previously a movie-maker who co-founded Brave New Films with Robert Greenwald and had sought funding for his films through crowd-sourcing. Green, who studied organizing at Harvard and was Mark Zuckerberg's roommate, is also the co-founder of the Causes Facebook app; he left NationBuilder in 2013. Since its founding, the company has helped campaigns raise $1.2 billion. In 2012, NationBuilder announced that 1,000 subscribers have used its software to amass 2.5 million supporters and raise $12 million in campaign donations. In 2015 it has helped raise $264 million, recruit over one million volunteers and coordinate some 129,000 events. By 2016, the company said its software was used by about 40 percent of all contested elections at the state and national level in the U.S., which included 3,000 political campaigns. Using such software is easier in the U.S. than Europe, where comprehensive data protection and privacy laws are in effect since 2018. The Scottish National Party was the first political party to use NationBuilder, harvesting vast amounts of data pertaining to voter activity via websites such as Facebook and Twitter. This revelation prompted outrage over privacy concerns. Guy Herbert of the No2ID campaign called the use of such data harvesting tools by the SNP "utterly hypocritical". == Funding == Investors in NationBuilder include Chris Hughes - the Facebook co-founder, Sean Parker - first president of Facebook and co-founder of Napster and Causes, Dan Senor - the former Republican foreign-policy adviser and Ben Horowitz, co-founder of Andreessen Horowitz. In 2012, it has raised $6.3 million in funding from a number of investors. == Notable implementations == The software is reported to have played a role in some public elections in Europe, the US and New Zealand, as well as non-profit initiatives, and political parties in Australia. Notable users include Bernie Sanders, Mitch McConnell, Andrew Yang, Theresa May, Amnesty International, the NAACP and Donald Trump. === France === La République En Marche used NationBuilder in their campaign for the 2017 National Assembly. === New Zealand === NationBuilder's services are used by New Zealand political parties, including in the campaigns of both the National and Labour parties in the 2017 general election. === United Kingdom === Despite stricter data protection and privacy laws in the UK and EU, NationBuilder was used to significant impact in a number of UK elections, most notably in the 2016 campaign for withdrawal of the United Kingdom from the European Union. The company later made a public announcement that both sides in that campaign had used its software. === United States === NationBuilder was used in the Donald Trump presidential campaign to advance his election efforts and eventually win the 2016 presidential race. Jill Stein of the Green Party, Republican Rick Santorum, and independent supporters of various candidates all used NationBuilder during their 2016 runs for president. During the 2018 US election cycle, political entities paid more than $1 million for the use of NationBuilder. Among the entities paying the most were Donald J. Trump for President, Prosperity Action and the Republican Party of Tennessee.

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  Sum of absolute transformed differences

  The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.

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  PVLV

  The primary value learned value (PVLV) model is a possible explanation for the reward-predictive firing properties of dopamine (DA) neurons. It simulates behavioral and neural data on Pavlovian conditioning and the midbrain dopaminergic neurons that fire in proportion to unexpected rewards. It is an alternative to the temporal-differences (TD) algorithm. It is used as part of Leabra.

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  Determining the number of clusters in a data set

  Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision. == Elbow method == The elbow method looks at the percentage of explained variance as a function of the number of clusters: One should choose a number of clusters so that adding another cluster does not give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". In most datasets, this "elbow" is ambiguous, making this method subjective and unreliable. Because the scale of the axes is arbitrary, the concept of an angle is not well-defined, and even on uniform random data, the curve produces an "elbow", making the method rather unreliable. Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance. The method can be traced to speculation by Robert L. Thorndike in 1953. While the idea of the elbow method sounds simple and straightforward, other methods (as detailed below) give better results. == X-means clustering == In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached. == Information criterion approach == Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values. == Information–theoretic approach == Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice. The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of K cluster centers, with c X {\displaystyle c_{X}} the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is: d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] {\displaystyle d_{K}={\frac {1}{p}}\min _{c_{1}\ldots c_{K}}{E[(X-c_{X})^{T}\Gamma ^{-1}(X-c_{X})]}} This is also the average Mahalanobis distance per dimension between X and the closest cluster center c X {\displaystyle c_{X}} . Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is: JumpMethod(X): Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k) The choice of the transform power Y = ( p / 2 ) {\displaystyle Y=(p/2)} is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 {\displaystyle \alpha ^{-2}} . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) {\displaystyle (-p/2)} is proportional to α p {\displaystyle \alpha ^{p}} , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components. Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 {\displaystyle \alpha ^{-2}} . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K. Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 {\displaystyle d_{K}^{-p/2}} is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters. Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions. == Silhouette method == The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is match

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  SimSimi

  SimSimi is an artificial intelligence conversation program created in 2002 by ISMaker. It grows its artificial intelligence day by day assisted by a feature that allows users to teach it to respond correctly. SimSimi, pronounced as "shim-shimi", is from a Korean word simsim (심심) which means "bored". It has an application designed for Android, Windows Phone and iOS. The application was banned in Thailand in 2012 after users taught it to make responses containing profanity, and to criticise leading politicians. In April 2018, SimSimi was suspended in Brazil due to accusations of sending inappropriate messages, such as sexual language, bullying and even death threats, being labeled as "dangerous" mainly due to its popularity among children, and according to its developer, the suspension of the app in the country "was inevitable because the SimSimi app, at least in the last few days, had a significant negative social impact in Brazil.”

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  Hinge loss

  In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs). For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as ℓ ( y ) = max ( 0 , 1 − t ⋅ y ) {\displaystyle \ell (y)=\max(0,1-t\cdot y)} Note that y {\displaystyle y} should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, y = w ⋅ x + b {\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b} , where ( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and y have the same sign (meaning y predicts the right class) and | y | ≥ 1 {\displaystyle |y|\geq 1} , the hinge loss ℓ ( y ) = 0 {\displaystyle \ell (y)=0} . When they have opposite signs, ℓ ( y ) {\displaystyle \ell (y)} increases linearly with y, and similarly if | y | < 1 {\displaystyle |y|<1} , even if it has the same sign (correct prediction, but not by enough margin). The Hinge loss is not a proper scoring rule. == Extensions == While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion, it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer defined it for a linear classifier as ℓ ( y ) = max ( 0 , 1 + max y ≠ t w y x − w t x ) {\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} , where t {\displaystyle t} is the target label, w t {\displaystyle \mathbf {w} _{t}} and w y {\displaystyle \mathbf {w} _{y}} are the model parameters. Weston and Watkins provided a similar definition, but with a sum rather than a max: ℓ ( y ) = ∑ y ≠ t max ( 0 , 1 + w y x − w t x ) {\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} . In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where w denotes the SVM's parameters, y the SVM's predictions, φ the joint feature function, and Δ the Hamming loss: ℓ ( y ) = max ( 0 , Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ − ⟨ w , ϕ ( x , t ) ⟩ ) = max ( 0 , max y ∈ Y ( Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ ) − ⟨ w , ϕ ( x , t ) ⟩ ) {\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}} . == Optimization == The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters w of a linear SVM with score function y = w ⋅ x {\displaystyle y=\mathbf {w} \cdot \mathbf {x} } that is given by ∂ ℓ ∂ w i = { − t ⋅ x i if t ⋅ y < 1 , 0 otherwise . {\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}} However, since the derivative of the hinge loss at t y = 1 {\displaystyle ty=1} is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's ℓ ( y ) = { 1 2 − t y if t y ≤ 0 , 1 2 ( 1 − t y ) 2 if 0 < t y < 1 , 0 if 1 ≤ t y {\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0 Read more →

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  Ni1000

  The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.

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  Representer theorem

  For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. == Formal statement == The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } on a non-empty set X {\displaystyle {\mathcal {X}}} with a corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} . Let there be given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{n}\to \mathbb {R} \cup \lbrace \infty \rbrace } , which together define the following regularized empirical risk functional on H k {\displaystyle H_{k}} : f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . {\displaystyle f\mapsto E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right).} Then, any minimizer of the empirical risk f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) } , ( ∗ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad ()} admits a representation of the form: f ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } for all 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Proof: Define a mapping φ : X → H k φ ( x ) = k ( ⋅ , x ) {\displaystyle {\begin{aligned}\varphi \colon {\mathcal {X}}&\to H_{k}\\\varphi (x)&=k(\cdot ,x)\end{aligned}}} (so that φ ( x ) = k ( ⋅ , x ) {\displaystyle \varphi (x)=k(\cdot ,x)} is itself a map X → R {\displaystyle {\mathcal {X}}\to \mathbb {R} } ). Since k {\displaystyle k} is a reproducing kernel, then φ ( x ) ( x ′ ) = k ( x ′ , x ) = ⟨ φ ( x ′ ) , φ ( x ) ⟩ , {\displaystyle \varphi (x)(x')=k(x',x)=\langle \varphi (x'),\varphi (x)\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product on H k {\displaystyle H_{k}} . Given any x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , one can use orthogonal projection to decompose any f ∈ H k {\displaystyle f\in H_{k}} into a sum of two functions, one lying in span ⁡ { φ ( x 1 ) , … , φ ( x n ) } {\displaystyle \operatorname {span} \left\lbrace \varphi (x_{1}),\ldots ,\varphi (x_{n})\right\rbrace } , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , {\displaystyle f=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,} where ⟨ v , φ ( x i ) ⟩ = 0 {\displaystyle \langle v,\varphi (x_{i})\rangle =0} for all i {\displaystyle i} . The above orthogonal decomposition and the reproducing property together show that applying f {\displaystyle f} to any training point x j {\displaystyle x_{j}} produces f ( x j ) = ⟨ ∑ i = 1 n α i φ ( x i ) + v , φ ( x j ) ⟩ = ∑ i = 1 n α i ⟨ φ ( x i ) , φ ( x j ) ⟩ , {\displaystyle f(x_{j})=\left\langle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,\varphi (x_{j})\right\rangle =\sum _{i=1}^{n}\alpha _{i}\langle \varphi (x_{i}),\varphi (x_{j})\rangle ,} which we observe is independent of v {\displaystyle v} . Consequently, the value of the error function E {\displaystyle E} in () is likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ ( x i ) {\displaystyle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})} and g {\displaystyle g} is strictly monotonic, we have g ( ‖ f ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) + v ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ 2 + ‖ v ‖ 2 ) ≥ g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ ) . {\displaystyle {\begin{aligned}g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v\rVert \right)\\&=g\left({\sqrt {\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert ^{2}+\lVert v\rVert ^{2}}}\right)\\&\geq g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert \right).\end{aligned}}} Therefore, setting v = 0 {\displaystyle v=0} does not affect the first term of (), while it strictly decreases the second term. Consequently, any minimizer f ∗ {\displaystyle f^{}} in () must have v = 0 {\displaystyle v=0} , i.e., it must be of the form f ∗ ( ⋅ ) = ∑ i = 1 n α i φ ( x i ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} which is the desired result. == Generalizations == The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) = 1 n ∑ i = 1 n ( f ( x i ) − y i ) 2 , g ( ‖ f ‖ ) = λ ‖ f ‖ 2 {\displaystyle {\begin{aligned}E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)&={\frac {1}{n}}\sum _{i=1}^{n}(f(x_{i})-y_{i})^{2},\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^{2}\end{aligned}}} for λ > 0 {\displaystyle \lambda >0} . Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function g ( ⋅ ) {\displaystyle g(\cdot )} of the Hilbert space norm. It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization f ~ ∗ = argmin ⁡ { E ( ( x 1 , y 1 , f ~ ( x 1 ) ) , … , ( x n , y n , f ~ ( x n ) ) ) + g ( ‖ f ‖ ) ∣ f ~ = f + h ∈ H k ⊕ span ⁡ { ψ p ∣ 1 ≤ p ≤ M } } , ( † ) {\displaystyle {\tilde {f}}^{}=\operatorname {argmin} \left\lbrace E\left((x_{1},y_{1},{\tilde {f}}(x_{1})),\ldots ,(x_{n},y_{n},{\tilde {f}}(x_{n}))\right)+g\left(\lVert f\rVert \right)\mid {\tilde {f}}=f+h\in H_{k}\oplus \operatorname {span} \lbrace \psi _{p}\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger )} i.e., we consider functions of the form f ~ = f + h {\displaystyle {\tilde {f}}=f+h} , where f ∈ H k {\displaystyle f\in H_{k}} and h {\displaystyle h} is an unpenalized function lying in the span of a finite set of real-valued functions { ψ p : X → R ∣ 1 ≤ p ≤ M } {\displaystyle \lbrace \psi _{p}\colon {\mathcal {X}}\to \mathbb {R} \mid 1\leq p\leq M\rbrace } . Under the assumption that the n × M {\displaystyle n\times M} matrix ( ψ p ( x i ) ) i p {\displaystyle \left(\psi _{p}(x_{i})\right)_{ip}} has rank M {\displaystyle M} , they show that the minimizer f ~ ∗ {\displaystyle {\tilde {f}}^{}} in ( † ) {\displaystyle (\dagger )} admits a representation of the form f ~ ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) + ∑ p = 1 M β p ψ p ( ⋅ ) {\displaystyle {\tilde {f}}^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i})+\sum _{p=1}^{M}\beta _{p}\psi _{p}(\cdot )} where α i , β p ∈ R {\displaystyle \alpha _{i},\beta _{p}\in \mathbb {R} } and the β p {\displaystyle \beta _{p}} are all uniquely determined. The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following: Theorem: Let X {\displaystyle {\mathcal {X}}} be a nonempty set, k {\displaystyle k} a positive-definite real-valued kernel on X × X {\displaystyle {\mathcal {X}}\times {\mathcal {X}}} with corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} , and let R : H k → R {\displaystyle R\colon H_{k}\to \mathbb {R} } be a differentiable regularization function. Then given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } and an arbitrary error function E : ( X × R 2 ) m → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace } , a minimizer f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + R ( f ) } ( ‡ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\rbrace \quad (\ddagger )} of the regularized empirical risk admits a repr

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  Blobotics

  Blobotics is a term describing research into chemical-based computer processors based on ions rather than electrons. Andrew Adamatzky, a computer scientist at the University of the West of England, Bristol used the term in an article in New Scientist March 28, 2005 [1]. The aim is to create 'liquid logic gates' which would be 'infinitely reconfigurable and self-healing'. The process relies on the Belousov–Zhabotinsky reaction, a repeating cycle of three separate sets of reactions. Such a processor could form the basis of a robot which, using artificial sensors, interact with its surroundings in a way which mimics living creatures. The coining of the term was featured by ABC radio in Australia [2].

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  Optical character recognition

  Optical character recognition (OCR) or optical character reader is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a scene photo (for example the text on signs and billboards in a landscape photo) or from subtitle text superimposed on an image (for example: from a television broadcast). Widely used as a form of data entry from printed paper data records – whether passport documents, invoices, bank statements, computerized receipts, business cards, mail, printed data, or any suitable documentation – it is a common method of digitizing printed texts so that they can be electronically edited, searched, stored more compactly, displayed online, and used in machine processes such as cognitive computing, machine translation, (extracted) text-to-speech, key data and text mining. OCR is a field of research in pattern recognition, artificial intelligence and computer vision. Early versions needed to be trained with images of each character, and worked on one font at a time. Advanced systems capable of producing a high degree of accuracy for most fonts are now common, and with support for a variety of image file format inputs. Some systems are capable of reproducing formatted output that closely approximates the original page including images, columns, and other non-textual components. == History == Early optical character recognition may be traced to technologies involving telegraphy and creating reading devices for the blind. In 1914, Emanuel Goldberg developed a machine that read characters and converted them into standard telegraph code. Concurrently, Edmund Fournier d'Albe developed the Optophone, a handheld scanner that when moved across a printed page, produced tones that corresponded to specific letters or characters. In the late 1920s and into the 1930s, Emanuel Goldberg developed what he called a "Statistical Machine" for searching microfilm archives using an optical code recognition system. In 1931, he was granted US Patent number 1,838,389 for the invention. The patent was acquired by IBM. === Visually impaired users === In 1974, Ray Kurzweil started the company Kurzweil Computer Products, Inc. and continued development of omni-font OCR, which could recognize text printed in virtually any font. (Kurzweil is often credited with inventing omni-font OCR, but it was in use by companies, including CompuScan, in the late 1960s and 1970s.) Kurzweil used the technology to create a reading machine for blind people to have a computer read text to them out loud. The device included a CCD-type flatbed scanner and a text-to-speech synthesizer. On January 13, 1976, the finished product was unveiled during a widely reported news conference headed by Kurzweil and the leaders of the National Federation of the Blind. In 1978, Kurzweil Computer Products began selling a commercial version of the optical character recognition computer program. LexisNexis was one of the first customers, and bought the program to upload legal paper and news documents onto its nascent online databases. Two years later, Kurzweil sold his company to Xerox, which eventually spun it off as Scansoft, which merged with Nuance Communications. In the 2000s, OCR was made available online as a service (WebOCR), in a cloud computing environment, and in mobile applications like real-time translation of foreign-language signs on a smartphone. With the advent of smartphones and smartglasses, OCR can be used in internet connected mobile device applications that extract text captured using the device's camera. These devices that do not have built-in OCR functionality will typically use an OCR API to extract the text from the image file captured by the device. The OCR API returns the extracted text, along with information about the location of the detected text in the original image back to the device app for further processing (such as text-to-speech) or display. Various commercial and open source OCR systems are available for most common writing systems, including Latin, Cyrillic, Arabic, Hebrew, Indic, Bengali (Bangla), Devanagari, Tamil, Chinese, Japanese, and Korean characters. == Applications == OCR engines have been developed into software applications specializing in various subjects such as receipts, invoices, checks, and legal billing documents. The software can be used for: Entering data for business documents, e.g. checks, passports, invoices, bank statements and receipts Automatic number-plate recognition Passport recognition and information extraction in airports Automatically extracting key information from insurance documents Traffic-sign recognition Extracting business card information into a contact list Creating textual versions of printed documents, e.g. book scanning for Project Gutenberg Making electronic images of printed documents searchable, e.g. Google Books Converting handwriting in real-time to control a computer (pen computing) Defeating or testing the robustness of CAPTCHA anti-bot systems, though these are specifically designed to prevent OCR. Assistive technology for blind and visually impaired users Writing instructions for vehicles by identifying CAD images in a database that are appropriate to the vehicle design as it changes in real time Making scanned documents searchable by converting them to PDFs == Types == Optical character recognition (OCR) – targets typewritten text, one glyph or character at a time. Optical word recognition – targets typewritten text, one word at a time (for languages that use a space as a word divider). Usually just called "OCR". Intelligent character recognition (ICR) – also targets handwritten printscript or cursive text one glyph or character at a time, usually involving machine learning. Intelligent word recognition (IWR) – also targets handwritten printscript or cursive text, one word at a time. This is especially useful for languages where glyphs are not separated in cursive script. OCR is generally an offline process, which analyses a static document. There are cloud based services which provide an online OCR API service. Handwriting movement analysis can be used as input to handwriting recognition. Instead of merely using the shapes of glyphs and words, this technique is able to capture motion, such as the order in which segments are drawn, the direction, and the pattern of putting the pen down and lifting it. This additional information can make the process more accurate. This technology is also known as "online character recognition", "dynamic character recognition", "real-time character recognition", and "intelligent character recognition". == Techniques == === Pre-processing === OCR software often pre-processes images to improve the chances of successful recognition. Techniques include: De-skewing – if the document was not aligned properly when scanned, it may need to be tilted a few degrees clockwise or counterclockwise in order to make lines of text perfectly horizontal or vertical. Despeckling – removal of positive and negative spots, smoothing edges Binarization – conversion of an image from color or greyscale to black-and-white (called a binary image because there are two colors). The task is performed as a simple way of separating the text (or any other desired image component) from the background. The task of binarization is necessary since most commercial recognition algorithms work only on binary images, as it is simpler to do so. In addition, the effectiveness of binarization influences to a significant extent the quality of character recognition, and careful decisions are made in the choice of the binarization employed for a given input image type; since the quality of the method used to obtain the binary result depends on the type of image (scanned document, scene text image, degraded historical document, etc.). Line removal – Cleaning up non-glyph boxes and lines Layout analysis or zoning – Identification of columns, paragraphs, captions, etc. as distinct blocks. Especially important in multi-column layouts and tables. Line and word detection – Establishment of a baseline for word and character shapes, separating words as necessary. Script recognition – In multilingual documents, the script may change at the level of the words and hence, identification of the script is necessary, before the right OCR can be invoked to handle the specific script. Character isolation or segmentation – For per-character OCR, multiple characters that are connected due to image artifacts must be separated; single characters that are broken into multiple pieces due to artifacts must be connected. Normalization of aspect ratio and scale Segmentation of fixed-pitch fonts is accomplished relatively simply by aligning the image to a uniform grid based on where vertical grid lines will least often intersect black areas. For proportional fonts, more sophisticated techniques are needed because whitespace bet

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  Sufficient dimension reduction

  In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} . For example, R ( x ) {\displaystyle R({\textbf {x}})} may be one or more linear combinations of x {\displaystyle {\textbf {x}}} . A dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} is said to be sufficient if the distribution of y ∣ R ( x ) {\displaystyle y\mid R({\textbf {x}})} is the same as that of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, no information about the regression is lost in reducing the dimension of x {\displaystyle {\textbf {x}}} if the reduction is sufficient. == Graphical motivation == In a regression setting, it is often useful to summarize the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} graphically. For instance, one may consider a scatterplot of y {\displaystyle y} versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot. When x {\displaystyle {\textbf {x}}} is high-dimensional, particularly when p ≥ 3 {\displaystyle p\geq 3} , it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} with small enough dimension, a sufficient summary plot of y {\displaystyle y} versus R ( x ) {\displaystyle R({\textbf {x}})} may be constructed and visually interpreted with relative ease. Hence sufficient dimension reduction allows for graphical intuition about the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , which might not have otherwise been available for high-dimensional data. Most graphical methodology focuses primarily on dimension reduction involving linear combinations of x {\displaystyle {\textbf {x}}} . The rest of this article deals only with such reductions. == Dimension reduction subspace == Suppose R ( x ) = A T x {\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}} is a sufficient dimension reduction, where A {\displaystyle A} is a p × k {\displaystyle p\times k} matrix with rank k ≤ p {\displaystyle k\leq p} . Then the regression information for y ∣ x {\displaystyle y\mid {\textbf {x}}} can be inferred by studying the distribution of y ∣ A T x {\displaystyle y\mid A^{T}{\textbf {x}}} , and the plot of y {\displaystyle y} versus A T x {\displaystyle A^{T}{\textbf {x}}} is a sufficient summary plot. Without loss of generality, only the space spanned by the columns of A {\displaystyle A} need be considered. Let η {\displaystyle \eta } be a basis for the column space of A {\displaystyle A} , and let the space spanned by η {\displaystyle \eta } be denoted by S ( η ) {\displaystyle {\mathcal {S}}(\eta )} . It follows from the definition of a sufficient dimension reduction that F y ∣ x = F y ∣ η T x , {\displaystyle F_{y\mid x}=F_{y\mid \eta ^{T}x},} where F {\displaystyle F} denotes the appropriate distribution function. Another way to express this property is y ⊥ ⊥ x ∣ η T x , {\displaystyle y\perp \!\!\!\perp {\textbf {x}}\mid \eta ^{T}{\textbf {x}},} or y {\displaystyle y} is conditionally independent of x {\displaystyle {\textbf {x}}} , given η T x {\displaystyle \eta ^{T}{\textbf {x}}} . Then the subspace S ( η ) {\displaystyle {\mathcal {S}}(\eta )} is defined to be a dimension reduction subspace (DRS). === Structural dimensionality === For a regression y ∣ x {\displaystyle y\mid {\textbf {x}}} , the structural dimension, d {\displaystyle d} , is the smallest number of distinct linear combinations of x {\displaystyle {\textbf {x}}} necessary to preserve the conditional distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, the smallest dimension reduction that is still sufficient maps x {\displaystyle {\textbf {x}}} to a subset of R d {\displaystyle \mathbb {R} ^{d}} . The corresponding DRS will be d-dimensional. === Minimum dimension reduction subspace === A subspace S {\displaystyle {\mathcal {S}}} is said to be a minimum DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} if it is a DRS and its dimension is less than or equal to that of all other DRSs for y ∣ x {\displaystyle y\mid {\textbf {x}}} . A minimum DRS S {\displaystyle {\mathcal {S}}} is not necessarily unique, but its dimension is equal to the structural dimension d {\displaystyle d} of y ∣ x {\displaystyle y\mid {\textbf {x}}} , by definition. If S {\displaystyle {\mathcal {S}}} has basis η {\displaystyle \eta } and is a minimum DRS, then a plot of y versus η T x {\displaystyle \eta ^{T}{\textbf {x}}} is a minimal sufficient summary plot, and it is (d + 1)-dimensional. == Central subspace == If a subspace S {\displaystyle {\mathcal {S}}} is a DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} , and if S ⊂ S drs {\displaystyle {\mathcal {S}}\subset {\mathcal {S}}_{\text{drs}}} for all other DRSs S drs {\displaystyle {\mathcal {S}}_{\text{drs}}} , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . In other words, a central subspace for y ∣ x {\displaystyle y\mid {\textbf {x}}} exists if and only if the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . The central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does not necessarily exist because the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} is not necessarily a DRS. However, if S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does exist, then it is also the unique minimum dimension reduction subspace. === Existence of the central subspace === While the existence of the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998): Let S 1 {\displaystyle {\mathcal {S}}_{1}} and S 2 {\displaystyle {\mathcal {S}}_{2}} be dimension reduction subspaces for y ∣ x {\displaystyle y\mid {\textbf {x}}} . If x {\displaystyle {\textbf {x}}} has density f ( a ) > 0 {\displaystyle f(a)>0} for all a ∈ Ω x {\displaystyle a\in \Omega _{x}} and f ( a ) = 0 {\displaystyle f(a)=0} everywhere else, where Ω x {\displaystyle \Omega _{x}} is convex, then the intersection S 1 ∩ S 2 {\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}} is also a dimension reduction subspace. It follows from this proposition that the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} exists for such x {\displaystyle {\textbf {x}}} . == Methods for dimension reduction == There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction, estimating the central subspace based on the inverse third moment (or kth moment), estimating the central solution space, graphical regression, envelope model, and the principal support vector machine. For more details on these and other methods, consult the statistical literature. Principal components analysis (PCA) and similar methods for dimension reduction are not based on the sufficiency principle. === Example: linear regression === Consider the regression model y = α + β T x + ε , where ε ⊥ ⊥ x . {\displaystyle y=\alpha +\beta ^{T}{\textbf {x}}+\varepsilon ,{\text{ where }}\varepsilon \perp \!\!\!\perp {\textbf {x}}.} Note that the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} is the same as the distribution of y ∣ β T x {\displ

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  Jpred

  Jpred v.4 is the latest version of the JPred Protein Secondary Structure Prediction Server which provides predictions by the JNet algorithm, one of the most accurate methods for secondary structure prediction, that has existed since 1998 in different versions. In addition to protein secondary structure, JPred also makes predictions of solvent accessibility and coiled-coil regions. The JPred service runs up to 134 000 jobs per month and has carried out over 2 million predictions in total for users in 179 countries. == JPred 2 == The static HTML pages of JPred 2 are still available for reference. == JPred 3 == The JPred v3 followed on from previous versions of JPred developed and maintained by James Cuff and Jonathan Barber (see JPred References). This release added new functionality and fixed many bugs. The highlights are: New, friendlier user interface Retrained and optimised version of Jnet (v2) - mean secondary structure prediction accuracy of >81% Batch submission of jobs Better error checking of input sequences/alignments Predictions now (optionally) returned via e-mail Users may provide their own query names for each submission JPred now makes a prediction even when there are no PSI-BLAST hits to the query PS/PDF output now incorporates all the predictions == JPred 4 == The current version of JPred (v4) has the following improvements and updates incorporated: Retrained on the latest UniRef90 and SCOPe/ASTRAL version of Jnet (v2.3.1) - mean secondary structure prediction accuracy of >82%. Upgraded the Web Server to the latest technologies (Bootstrap framework, JavaScript) and updating the web pages – improving the design and usability through implementing responsive technologies. Added RESTful API and mass-submission and results retrieval scripts - resulting in peak throughput above 20,000 predictions per day. Added prediction jobs monitoring tools. Upgraded the results reporting – both, on the web-site, and through the optional email summary reports: improved batch submission, added results summary preview through Jalview results visualization summary in SVG and adding full multiple sequence alignments into the reports. Improved help-pages, incorporating tool-tips, and adding one-page step-by-step tutorials. Sequence residues are categorised or assigned to one of the secondary structure elements, such as alpha-helix, beta-sheet and coiled-coil. Jnet uses two neural networks for its prediction. The first network is fed with a window of 17 residues over each amino acid in the alignment plus a conservation number. It uses a hidden layer of nine nodes and has three output nodes, one for each secondary structure element. The second network is fed with a window of 19 residues (the result of first network) plus the conservation number. It has a hidden layer with nine nodes and has three output nodes.

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  Semantic analysis (machine learning)

  In machine learning, semantic analysis of a text corpus is the task of building structures that approximate concepts from a large set of documents. It generally does not involve prior semantic understanding of the documents. Semantic analysis strategies include: Metalanguages based on first-order logic, which can analyze the speech of humans. Understanding the semantics of a text is symbol grounding: if language is grounded, it is equal to recognizing a machine-readable meaning. For the restricted domain of spatial analysis, a computer-based language understanding system was demonstrated. Latent semantic analysis (LSA), a class of techniques where documents are represented as vectors in a term space. A prominent example is probabilistic latent semantic analysis (PLSA). Latent Dirichlet allocation, which involves attributing document terms to topics. n-grams and hidden Markov models, which work by representing the term stream as a Markov chain, in which each term is derived from preceding terms. == Stochastic semantic analysis ==

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  Operational taxonomic unit

  An operational taxonomic unit (OTU) is an operational definition used to classify groups of closely related individuals. The term was originally introduced in 1963 by Robert R. Sokal and Peter H. A. Sneath in the context of numerical taxonomy, where an "operational taxonomic unit" is simply the group of organisms currently being studied. In this sense, an OTU is a pragmatic definition to group individuals by similarity, equivalent to but not necessarily in line with classical Linnaean taxonomy or modern evolutionary taxonomy. Nowadays, however, the term is commonly used in a different context and refers to clusters of (uncultivated or unknown) organisms, grouped by DNA sequence similarity of a specific taxonomic marker gene (originally coined as mOTU; molecular OTU). In other words, OTUs are pragmatic proxies for "species" at different taxonomic levels, in the absence of traditional systems of biological classification as are available for macroscopic organisms. For several years, OTUs have been the most commonly used units of diversity, especially when analysing small subunit 16S (for prokaryotes) or 18S rRNA (for eukaryotes) marker gene sequence datasets. == Molecular OTU by clustering of marker gene sequences == In the approach represented by DNA barcoding, a particular locus is chosen to be used as the marker gene for classification. This locus should be universally present in the scope selected, variable enough to be different among close-related species, and be flanked by conservative sequences that allow for easy amplification and detection. There are databases containing sequences for such marker genes from many different species, allowing for comparison. (Sometimes only using one locus does not provide sufficient resolution, so multiple marker genes are used. This is the case for plants, where rbcL+matK is common.) Sequences obtained this way can be clustered according to their similarity to one another, and operational taxonomic units are defined based on the similarity threshold set by the researcher. The exact threshold depends on the taxa in question and the mutational rates of the selected locus in the taxon. 97–99% are commonly used, but "it is now recognized to be somewhat arbitrary as sequence variation within and among species varies across taxa". 100% similarity (fully identical) is also common, also known as single variants. It remains debatable how well this commonly used method recapitulates true microbial species phylogeny or ecology. Although OTUs can be calculated differently when using different algorithms or thresholds, research by Schmidt et al. (2014) demonstrated that 16S-derived microbial OTUs were generally ecologically consistent across habitats and several clustering approaches. The number of OTUs defined may be inflated due to errors in DNA sequencing. === OTU clustering approaches === There are three main approaches to clustering OTUs: De novo, for which the clustering is based on similarities between sequencing reads. Closed-reference, for which the clustering is performed against a reference database of sequences. Open-reference, where clustering is first performed against a reference database of sequences, then any remaining sequences that could not be mapped to the reference are clustered de novo. Using a reference provides taxonomic context for the OTUs found. Alternatively, taxonomic context can be found after the construction of clusters by comparing representative sequences from clusters against a reference database. There are also specialized classifiers for this purpose which are much faster than naive comparison using BLAST. === OTU clustering algorithms === Hierarchical clustering algorithms (HCA): uclust & cd-hit & ESPRIT Bayesian clustering: CROP == Molecular OTU by other methods == In addition to similarity-based grouping, marker gene sequences can be sorted into OTUs using molecular phylogeny, k-mer composition, or hybrid methods combining these methods with similarity. There are also Bayesian tree-less methods and machine learning approaches. Using phylogeny often involves manually assigning terminal clades or single nodes to an OTU, so this is usually only done for refinement. Genome skimming can be used to obtain high-copy DNA without the need to choose marker genes or to design PCR primers for the chosen genes. It can provide fairly good coverage of organelle DNA and repetitive elements such as ribosomal DNA, both of which can be used like marker genes in OTU analysis. Whole-genome sequencing is more expensive and involves the production and processing of more data. By considering the entire genome, many (sometimes over 100) marker genes can be used at the same time, producing highly resolved phylogenies that correctly identify problematic taxa. It is also possible to use entire genomes for OTU assignment. For example, genomes from different bacterial species almost always have an average nucleotide identity lower than 95%, a fact that can be used to define new OTUs (and likely new species).

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  Determining the number of clusters in a data set

  Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision. == Elbow method == The elbow method looks at the percentage of explained variance as a function of the number of clusters: One should choose a number of clusters so that adding another cluster does not give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". In most datasets, this "elbow" is ambiguous, making this method subjective and unreliable. Because the scale of the axes is arbitrary, the concept of an angle is not well-defined, and even on uniform random data, the curve produces an "elbow", making the method rather unreliable. Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance. The method can be traced to speculation by Robert L. Thorndike in 1953. While the idea of the elbow method sounds simple and straightforward, other methods (as detailed below) give better results. == X-means clustering == In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached. == Information criterion approach == Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values. == Information–theoretic approach == Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice. The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of K cluster centers, with c X {\displaystyle c_{X}} the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is: d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] {\displaystyle d_{K}={\frac {1}{p}}\min _{c_{1}\ldots c_{K}}{E[(X-c_{X})^{T}\Gamma ^{-1}(X-c_{X})]}} This is also the average Mahalanobis distance per dimension between X and the closest cluster center c X {\displaystyle c_{X}} . Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is: JumpMethod(X): Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k) The choice of the transform power Y = ( p / 2 ) {\displaystyle Y=(p/2)} is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 {\displaystyle \alpha ^{-2}} . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) {\displaystyle (-p/2)} is proportional to α p {\displaystyle \alpha ^{p}} , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components. Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 {\displaystyle \alpha ^{-2}} . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K. Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 {\displaystyle d_{K}^{-p/2}} is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters. Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions. == Silhouette method == The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is match

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