In theoretical computer science and formal language theory, a tree transducer (TT) is an abstract machine taking as input a tree, and generating output – generally other trees, but models producing words or other structures exist. Roughly speaking, tree transducers extend tree automata in the same way that word transducers extend word automata. Manipulating tree structures instead of words enable TT to model syntax-directed transformations of formal or natural languages. However, TT are not as well-behaved as their word counterparts in terms of algorithmic complexity, closure properties, etcetera. In particular, most of the main classes are not closed under composition. The main classes of tree transducers are: == Top-Down Tree Transducers (TOP) == A TOP T is a tuple (Q, Σ, Γ, I, δ) such that: Q is a finite set, the set of states; Σ is a finite ranked alphabet, called the input alphabet; Γ is a finite ranked alphabet, called the output alphabet; I is a subset of Q, the set of initial states; and δ is a set of rules of the form q ( f ( x 1 , … , x n ) ) → u {\displaystyle q(f(x_{1},\dots ,x_{n}))\to u} , where f is a symbol of Σ, n is the arity of f, q is a state, and u is a tree on Γ and Q × 1.. n {\displaystyle Q\times 1..n} , such pairs being nullary. === Examples of rules and intuitions on semantics === For instance, q ( f ( x 1 , … , x 3 ) ) → g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle q(f(x_{1},\dots ,x_{3}))\to g(a,q'(x_{1}),h(q''(x_{3})))} is a rule – one customarily writes q ( x i ) {\displaystyle q(x_{i})} instead of the pair ( q , x i ) {\displaystyle (q,x_{i})} – and its intuitive semantics is that, under the action of q, a tree with f at the root and three children is transformed into g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle g(a,q'(x_{1}),h(q''(x_{3})))} where, recursively, q ′ ( x 1 ) {\displaystyle q'(x_{1})} and q ″ ( x 3 ) {\displaystyle q''(x_{3})} are replaced, respectively, with the application of q ′ {\displaystyle q'} on the first child and with the application of q ″ {\displaystyle q''} on the third. === Semantics as term rewriting === The semantics of each state of the transducer T, and of T itself, is a binary relation between input trees (on Σ) and output trees (on Γ). A way of defining the semantics formally is to see δ {\displaystyle \delta } as a term rewriting system, provided that in the right-hand sides the calls are written in the form q ( x i ) {\displaystyle q(x_{i})} , where states q are unary symbols. Then the semantics [ [ q ] ] {\displaystyle [\![q]\!]} of a state q is given by [ [ q ] ] = { u ↦ v ∣ u is a tree on Σ , v is a tree on Γ , and q ( u ) → δ ∗ v } . {\displaystyle [\![q]\!]=\{u\mapsto v\mid u{\text{ is a tree on }}\Sigma ,\ v{\text{ is a tree on }}\Gamma {\text{, and }}q(u)\to _{\delta }^{}v\}.} The semantics of T is then defined as the union of the semantics of its initial states: [ [ T ] ] = ⋃ q ∈ I [ [ q ] ] . {\displaystyle [\![T]\!]=\bigcup _{q\in I}[\![q]\!].} === Determinism and domain === As with tree automata, a TOP is said to be deterministic (abbreviated DTOP) if no two rules of δ share the same left-hand side, and there is at most one initial state. In that case, the semantics of the DTOP is a partial function from input trees (on Σ) to output trees (on Γ), as are the semantics of each of the DTOP's states. The domain of a transducer is the domain of its semantics. Likewise, the image of a transducer is the image of its semantics. === Properties of DTOP === DTOP are not closed under union: this is already the case for deterministic word transducers. The domain of a DTOP is a regular tree language. Furthermore, the domain is recognisable by a deterministic top-down tree automaton (DTTA) of size at most exponential in that of the initial DTOP. That the domain is DTTA-recognizable is not surprising, considering that the left-hand sides of DTOP rules are the same as for DTTA. As for the reason for the exponential explosion in the worst case (that does not exist in the word case), consider the rule q ( f ( x 1 , x 2 ) ) → g ( p 1 ( x 1 ) , p 2 ( x 1 ) , p 3 ( x 2 ) ) {\displaystyle q(f(x_{1},x_{2}))\to g(p_{1}(x_{1}),p_{2}(x_{1}),p_{3}(x_{2}))} . In order for the computation to succeed, it must succeed for both children. That means that the right child must be in the domain of p 3 {\displaystyle p_{3}} . As for the left child, it must be in the domain of both p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} . Generally, since subtrees can be copied, a single subtree can be evaluated by multiple states during a run, despite the determinism, and unlike DTTA. Thus the construction of the DTTA recognising the domain of a DTOP must account for sets of states and compute the intersections of their domains, hence the exponential. In the special case of linear DTOP, that is to say DTOP where each x i {\displaystyle x_{i}} appears at most once in the right-hand side of each rule, the construction is linear in time and space. The image of a DTOP is not a regular tree language. Consider the transducer coding the transformation f ( x ) → g ( x , x ) {\displaystyle f(x)\to g(x,x)} ; that is, duplicate the child of the input. This is easily done by a rule q ( f ( x 1 ) ) → g ( p ( x 1 ) , p ( x 1 ) ) {\displaystyle q(f(x_{1}))\to g(p(x_{1}),p(x_{1}))} , where p encodes the identity. Then, absent any restrictions on the first child of the input, the image is a classical non-regular tree language. However, the domain of a DTOP cannot be restricted to a regular tree language. That is to say, given a DTOP T and a language L, one cannot in general build a DTOP T ′ {\displaystyle T'} such that the semantics of T ′ {\displaystyle T'} is that of T, restricted to L. This property is linked to the reason deterministic top-down tree automata are less expressive than bottom-up automata: once you go down a given path, information from other paths is inaccessible. Consider the transducer coding the transformation f ( x , y ) → y {\displaystyle f(x,y)\to y} ; that is, output the right child of the input. This is easily done by a rule q ( f ( x 1 , x 2 ) ) → p ( x 2 ) {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2})} , where p encodes the identity. Now let's say we want to restrict this transducer to the finite (and thus, in particular, regular) domain { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} . We must use the rules q ( f ( x 1 , x 2 ) ) → p ( x 2 ) , p ( a ) → a , p ( b ) → b {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2}),\ p(a)\to a,\ p(b)\to b} . But in the first rule, x 1 {\displaystyle x_{1}} does not appear at all, since nothing is produced from the left child. Thus, it is not possible to test that the left child is c. In contrast, since we produce from the right child, we can test that it is a or b. In general, the criterion is that DTOP cannot test properties of subtrees from which they do not produce output. DTOP are not closed under composition. However this problem can be solved by the addition of a lookahead: a tree automaton, coupled to the transducer, that can perform tests on the domain which the transducer is incapable of. This follows from the point about domain restriction: composing the DTOP encoding identity on { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} with the one encoding f ( x , y ) → y {\displaystyle f(x,y)\to y} must yield a transducer with the semantics { f ( c , a ) ↦ a , f ( c , b ) ↦ b } {\displaystyle \{f(c,a)\mapsto a,\ f(c,b)\mapsto b\}} , which we know is not expressible by a DTOP. The typechecking problem—testing whether the image of a regular tree language is included in another regular tree language—is decidable. The equivalence problem—testing whether two DTOP define the same functions—is decidable. == Bottom-Up Tree Transducers (BOT) == As in the simpler case of tree automata, bottom-up tree transducers are defined similarly to their top-down counterparts, but proceed from the leaves of the tree to the root, instead of from the root to the leaves. Thus the main difference is in the form of the rules, which are of the form f ( q 1 ( x 1 ) , … , q n ( x n ) ) → q ( u ) {\displaystyle f(q_{1}(x_{1}),\dots ,q_{n}(x_{n}))\to q(u)} .
Automated negotiation
Automated negotiation is a form of interaction in systems that are composed of multiple autonomous agents, in which the aim is to reach agreements through an iterative process of making offers. Automated negotiation can be employed for many tasks human negotiators regularly engage in, such as bargaining and joint decision making. The main topics in automated negotiation revolve around the design of protocols and negotiating strategies. == History == Through digitization, the beginning of the 21st century has seen a growing interest in the automation of negotiation and e-negotiation systems, for example in the setting of e-commerce. This interest is fueled by the promise of automated agents being able to negotiate on behalf of human negotiators, and to find better outcomes than human negotiators. == Examples == Examples of automated negotiation include: Online dispute resolution, in which disagreements between parties are settled. Sponsored search auction, where bids are placed on advertisement keywords. Content negotiation, in which user agents negotiate over HTTP about how to best represent a web resource. Negotiation support systems, in which negotiation decision-making activities are supported by an information system.
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
NETtalk (artificial neural network)
NETtalk is an artificial neural network that learns to pronounce written English text by supervised learning. It takes English text as input, and produces a matching phonetic transcriptions as output. It is the result of research carried out in the mid-1980s by Terrence Sejnowski and Charles Rosenberg. The intent behind NETtalk was to construct simplified models that might shed light on the complexity of learning human level cognitive tasks, and their implementation as a connectionist model that could also learn to perform a comparable task. The authors trained it by backpropagation. The network was trained on a large amount of English words and their corresponding pronunciations, and is able to generate pronunciations for unseen words with a high level of accuracy. The output of the network was a stream of phonemes, which fed into DECtalk to produce audible speech. It achieved popular success, appearing on the Today show. From the point of view of modeling human cognition, NETtalk does not specifically model the image processing stages and letter recognition of the visual cortex. Rather, it assumes that the letters have been pre-classified and recognized. It is NETtalk's task to learn proper associations between the correct pronunciation with a given sequence of letters based on the context in which the letters appear. A similar architecture was subsequently used for the opposite task, that of converting continuous speech signal to a phoneme sequence. == Training == The training dataset was a 20,008-word subset of the Brown Corpus, with manually annotated phoneme and stress for each letter. The development process was described in a 1993 interview. It took three months -- 250 person-hours -- to create the training dataset, but only a few days to train the network. After it was run successfully on this, the authors tried it on a phonological transcription of an interview with a young Latino boy from a barrio in Los Angeles. This resulted in a network that reproduced his Spanish accent. The original NETtalk was implemented on a Ridge 32, which took 0.275 seconds per learning step (one forward and one backward pass). Training NETtalk became a benchmark to test for the efficiency of backpropagation programs. For example, an implementation on Connection Machine-1 (with 16384 processors) ran at 52x speedup. An implementation on a 10-cell Warp ran at 340x speedup. The following table compiles the benchmark scores as of 1988. Speed is measured in "millions of connections per second" (MCPS). For example, the original NETtalk on Ridge 32 took 0.275 seconds per forward-backward pass, giving 18629 / 10 6 0.275 = 0.068 {\displaystyle {\frac {18629/10^{6}}{0.275}}=0.068} MCPS. Relative times are normalized to the MicroVax. == Architecture == The network had three layers and 18,629 adjustable weights, large by the standards of 1986. There were worries that it would overfit the dataset, but it was trained successfully. The input of the network has 203 units, divided into 7 groups of 29 units each. Each group is a one-hot encoding of one character. There are 29 possible characters: 26 letters, comma, period, and word boundary (whitespace). To produce the pronunciation of a single character, the network takes the character itself, as well as 3 characters before and 3 characters after it. The hidden layer has 80 units. The output has 26 units. 21 units encode for articulatory features (point of articulation, voicing, vowel height, etc.) of phonemes, and 5 units encode for stress and syllable boundaries. Sejnowski studied the learned representation in the network, and found that phonemes that sound similar are clustered together in representation space. The output of the network degrades, but remains understandable, when some hidden neurons are removed.
Multiple discriminant analysis
Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure. MDA is not directly used to perform classification. It merely supports classification by yielding a compressed signal amenable to classification. The method described in Duda et al. (2001) §3.8.3 projects the multivariate signal down to an M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curse of dimensionality. In other words, when signals are represented in very-high-dimensional spaces, the classifier's performance is catastrophically impaired by the overfitting problem. This problem is reduced by compressing the signal down to a lower-dimensional space as MDA does. MDA has been used to reveal neural codes.
Mobile cloud computing
Mobile Cloud Computing (MCC) is the combination of cloud computing and mobile computing to bring rich computational resources to mobile users, network operators, as well as cloud computing providers. The ultimate goal of MCC is to enable execution of rich mobile applications on a plethora of mobile devices, with a rich user experience. MCC provides business opportunities for mobile network operators as well as cloud providers. More comprehensively, MCC can be defined as "a rich mobile computing technology that leverages unified elastic resources of varied clouds and network technologies toward unrestricted functionality, storage, and mobility to serve a multitude of mobile devices anywhere, anytime through the channel of Ethernet or Internet regardless of heterogeneous environments and platforms based on the pay-as-you-use principle." == Architecture == MCC uses computational augmentation approaches (computations are executed remotely instead of on the device) by which resource-constraint mobile devices can utilize computational resources of varied cloud-based resources. In MCC, there are four types of cloud-based resources, namely distant immobile clouds, proximate immobile computing entities, proximate mobile computing entities, and hybrid (combination of the other three model). Giant clouds such as Amazon EC2 are in the distant immobile groups whereas cloudlet or surrogates are member of proximate immobile computing entities. Smartphones, tablets, handheld devices, and wearable computing devices are part of the third group of cloud-based resources which is proximate mobile computing entities. Vodafone, Orange and Verizon have started to offer cloud computing services for companies. == Challenges == In the MCC landscape, an amalgam of mobile computing, cloud computing, and communication networks (to augment smartphones) creates several complex challenges such as Mobile Computation Offloading, Seamless Connectivity, Long WAN Latency, Mobility Management, Context-Processing, Energy Constraint, Vendor/data Lock-in, Security and Privacy, Elasticity that hinder MCC success and adoption. === Open research issues === Although significant research and development in MCC is available in the literature, efforts in the following domains is still lacking: Architectural issues: A reference architecture for heterogeneous MCC environment is a crucial requirement for unleashing the power of mobile computing towards unrestricted ubiquitous computing. Energy-efficient transmission: MCC requires frequent transmissions between cloud platform and mobile devices, due to the stochastic nature of wireless networks, the transmission protocol should be carefully designed. Context-awareness issues: Context-aware and socially-aware computing are inseparable traits of contemporary handheld computers. To achieve the vision of mobile computing among heterogeneous converged networks and computing devices, designing resource-efficient environment-aware applications is an essential need. Live VM migration issues: Executing resource-intensive mobile application via Virtual Machine (VM) migration-based application offloading involves encapsulation of application in VM instance and migrating it to the cloud, which is a challenging task due to additional overhead of deploying and managing VM on mobile devices. Mobile communication congestion issues: Mobile data traffic is tremendously hiking by ever increasing mobile user demands for exploiting cloud resources which impact on mobile network operators and demand future efforts to enable smooth communication between mobile and cloud endpoints. Trust, security, and privacy issues: Trust is an essential factor for the success of the burgeoning MCC paradigm. It is because the data along with code/component/application/complete VM is offloaded to the cloud for execution. Moreover, just like software and mobile application piracy, the MCC application development models are also affected by the piracy issue. Pirax is known to be the first specialized framework for controlling application piracy in MCC requirements == MCC research groups and activities == Several academic and industrial research groups in MCC have been emerging since last few years. Some of the MCC research groups in academia with large number of researchers and publications include: MDC, Mobile and Distributed Computing research group is at Faculty of Computer and Information Science, King Saud University. MDC research group focuses on architectures, platforms, and protocols for mobile and distributed computing. The group has developed algorithms, tools, and technologies which offer energy efficient, fault tolerant, scalable, secure, and high performance computing on mobile devices. MobCC lab, Faculty of Computer Science and Information Technology, University Malaya. The lab was established in 2010 under the High Impact Research Grant, Ministry of Higher Education, Malaysia. It has 17 researchers and has track of 22 published articles in international conference and peer-reviewed CS journals. ICCLAB, Zürich University of Applied Sciences has a segment working on MCC. The InIT Cloud Computing Lab is a research lab within the Institute of Applied Information Technology (InIT) of Zürich University of Applied Sciences (ZHAW). It covers topic areas across the entire cloud computing technology stack. Mobile & Cloud Lab, Institute of Computer Science, University of Tartu. Mobile & Cloud Lab conducts research and teaching in the mobile computing and cloud computing domains. The research topics of the group include cloud computing, mobile application development, mobile cloud, mobile web services and migrating scientific computing and enterprise applications to the cloud. SmartLab, Data Management Systems Laboratory, Department of Computer Science, University of Cyprus. SmartLab is a first-of-a-kind open cloud of smartphones that enables a new line of systems-oriented mobile computing research. Mobile Cloud Networking: Mobile Cloud Networking (MCN) was an EU FP7 Large-scale Integrating Project (IP, 15m Euro) funded by the European Commission. The MCN project was launched in November 2012 for the period of 36 month. The project was coordinated by SAP Research and the ICCLab at the Zurich University of Applied Science. In total 19 partners from industry and academia established the first vision of Mobile Cloud Computing. The project was primarily motivated by an ongoing transformation that drives the convergence between the Mobile Communications and Cloud Computing industry enabled by the Internet and is considered the first pioneer in the area of Network Function Virtualization.
Canonical correspondence analysis
In multivariate analysis, canonical correspondence analysis (CCA) is an ordination technique that determines axes from the response data as a unimodal combination of measured predictors. CCA is commonly used in ecology in order to extract gradients that drive the composition of ecological communities. CCA extends correspondence analysis (CA) with regression, in order to incorporate predictor variables. == History == CCA was developed in 1986 by Cajo ter Braak and implemented in the program CANOCO, an extension of DECORANA. To date, CCA is one of the most popular multivariate methods in ecology, despite the availability of contemporary alternatives. CCA was originally derived and implemented using an algorithm of weighted averaging, though Legendre & Legendre (1998) derived an alternative algorithm. == Assumptions == The requirements of a CCA are that the samples are random and independent. Also, the data are categorical and that the independent variables are consistent within the sample site and error-free. The original publication states the need for equal species tolerances, equal species maxima, and equispaced or uniformly distributed species optima and site scores.