AI Data Farms

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  Computational heuristic intelligence

  Computational heuristic intelligence (CHI) refers to specialized programming techniques in computational intelligence (also called artificial intelligence, or AI). These techniques have the express goal of avoiding complexity issues, also called NP-hard problems, by using human-like techniques. They are best summarized as the use of exemplar-based methods (heuristics), rather than rule-based methods (algorithms). Hence the term is distinct from the more conventional computational algorithmic intelligence, or symbolic AI. An example of a CHI technique is the encoding specificity principle of Tulving and Thompson. In general, CHI principles are problem solving techniques used by people, rather than programmed into machines. It is by drawing attention to this key distinction that the use of this term is justified in a field already replete with confusing neologisms. Note that the legal systems of all modern human societies employ both heuristics (generalisations of cases) from individual trial records as well as legislated statutes (rules) as regulatory guides. Another recent approach to the avoidance of complexity issues is to employ feedback control rather than feedforward modeling as a problem-solving paradigm. This approach has been called computational cybernetics, because (a) the term 'computational' is associated with conventional computer programming techniques which represent a strategic, compiled, or feedforward model of the problem, and (b) the term 'cybernetic' is associated with conventional system operation techniques which represent a tactical, interpreted, or feedback model of the problem. Of course, real programs and real problems both contain both feedforward and feedback components. A real example which illustrates this point is that of human cognition, which clearly involves both perceptual (bottom-up, feedback, sensor-oriented) and conceptual (top-down, feedforward, motor-oriented) information flows and hierarchies. The AI engineer must choose between mathematical and cybernetic problem solution and machine design paradigms. This is not a coding (program language) issue, but relates to understanding the relationship between the declarative and procedural programming paradigms. The vast majority of STEM professionals never get the opportunity to design or implement pure cybernetic solutions. When pushed, most responders will dismiss the importance of any difference by saying that all code can be reduced to a mathematical model anyway. Unfortunately, not only is this belief false, it fails most spectacularly in many AI scenarios. Mathematical models are not time agnostic, but by their very nature are pre-computed, i.e. feedforward. Dyer [2012] and Feldman [2004] have independently investigated the simplest of all somatic governance paradigms, namely control of a simple jointed limb by a single flexor muscle. They found that it is impossible to determine forces from limb positions- therefore, the problem cannot have a pre-computed (feedforward) mathematical solution. Instead, a top-down command bias signal changes the threshold feedback level in the sensorimotor loop, e.g. the loop formed by the afferent and efferent nerves, thus changing the so-called ‘equilibrium point’ of the flexor muscle/ elbow joint system. An overview of the arrangement reveals that global postures and limb position are commanded in feedforward terms, using global displacements (common coding), with the forces needed being computed locally by feedback loops. This method of sensorimotor unit governance, which is based upon what Anatol Feldman calls the ‘equilibrium Point’ theory, is formally equivalent to a servomechanism such as a car's ‘cruise control’.

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  Charge based boundary element fast multipole method

  The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t

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  Jackknife variance estimates for random forest

  In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects. == Jackknife variance estimates == The sampling variance of bagged learners is: V ( x ) = V a r [ θ ^ ∞ ( x ) ] {\displaystyle V(x)=Var[{\hat {\theta }}^{\infty }(x)]} Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as: V ^ j = n − 1 n ∑ i = 1 n ( θ ^ ( − i ) − θ ¯ ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\hat {\theta }}_{(-i)}-{\overline {\theta }})^{2}} In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as: V ^ j = n − 1 n ∑ i = 1 n ( t ¯ ( − i ) ⋆ ( x ) − t ¯ ⋆ ( x ) ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\overline {t}}_{(-i)}^{\star }(x)-{\overline {t}}^{\star }(x))^{2}} Here, t ⋆ {\displaystyle t^{\star }} denotes a decision tree after training, t ( − i ) ⋆ {\displaystyle t_{(-i)}^{\star }} denotes the result based on samples without i t h {\displaystyle ith} observation. == Examples == E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low. Here, accuracy is measured by error rate, which is defined as: E r r o r R a t e = 1 N ∑ i = 1 N ∑ j = 1 M y i j , {\displaystyle ErrorRate={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij},} Here N is also the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy: l o g l o s s = 1 N ∑ i = 1 N ∑ j = 1 M y i j l o g ( p i j ) {\displaystyle logloss={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij}log(p_{ij})} Here N is the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. p i j {\displaystyle p_{ij}} is the predicted probability of i t h {\displaystyle ith} observation in class j {\displaystyle j} .This method is used in Kaggle These two methods are very similar. == Modification for bias == When using Monte Carlo MSEs for estimating V I J ∞ {\displaystyle V_{IJ}^{\infty }} and V J ∞ {\displaystyle V_{J}^{\infty }} , a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large: E [ V ^ I J B ] − V ^ I J ∞ ≈ n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle E[{\hat {V}}_{IJ}^{B}]-{\hat {V}}_{IJ}^{\infty }\approx {\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} To eliminate this influence, bias-corrected modifications are suggested: V ^ I J − U B = V ^ I J B − n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{IJ-U}^{B}={\hat {V}}_{IJ}^{B}-{\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} V ^ J − U B = V ^ J B − ( e − 1 ) n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{J-U}^{B}={\hat {V}}_{J}^{B}-(e-1){\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}

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  Curriculum learning

  Curriculum learning is a technique in machine learning in which a model is trained on examples of increasing difficulty, where the definition of "difficulty" may be provided externally or discovered as part of the training process. This is intended to attain good performance more quickly, or to converge to a better local optimum if the global optimum is not found. == Approach == Most generally, curriculum learning is the technique of successively increasing the difficulty of examples in the training set that is presented to a model over multiple training iterations. This can produce better results than exposing the model to the full training set immediately under some circumstances; most typically, when the model is able to learn general principles from easier examples, and then gradually incorporate more complex and nuanced information as harder examples are introduced, such as edge cases. This has been shown to work in many domains, most likely as a form of regularization. There are several major variations in how the technique is applied: A concept of "difficulty" must be defined. This may come from human annotation or an external heuristic; for example in language modeling, shorter sentences might be classified as easier than longer ones. Another approach is to use the performance of another model, with examples accurately predicted by that model being classified as easier (providing a connection to boosting). Difficulty can be increased steadily or in distinct epochs, and in a deterministic schedule or according to a probability distribution. This may also be moderated by a requirement for diversity at each stage, in cases where easier examples are likely to be disproportionately similar to each other. Applications must also decide the schedule for increasing the difficulty. Simple approaches may use a fixed schedule, such as training on easy examples for half of the available iterations and then all examples for the second half. Other approaches use self-paced learning to increase the difficulty in proportion to the performance of the model on the current set. Since curriculum learning only concerns the selection and ordering of training data, it can be combined with many other techniques in machine learning. The success of the method assumes that a model trained for an easier version of the problem can generalize to harder versions, so it can be seen as a form of transfer learning. Some authors also consider curriculum learning to include other forms of progressively increasing complexity, such as increasing the number of model parameters. It is frequently combined with reinforcement learning, such as learning a simplified version of a game first. Some domains have shown success with anti-curriculum learning: training on the most difficult examples first. One example is the ACCAN method for speech recognition, which trains on the examples with the lowest signal-to-noise ratio first. == History == The term "curriculum learning" was introduced by Yoshua Bengio et al in 2009, with reference to the psychological technique of shaping in animals and structured education for humans: beginning with the simplest concepts and then building on them. The authors also note that the application of this technique in machine learning has its roots in the early study of neural networks such as Jeffrey Elman's 1993 paper Learning and development in neural networks: the importance of starting small. Bengio et al showed good results for problems in image classification, such as identifying geometric shapes with progressively more complex forms, and language modeling, such as training with a gradually expanding vocabulary. They conclude that, for curriculum strategies, "their beneficial effect is most pronounced on the test set", suggesting good generalization. The technique has since been applied to many other domains: Natural language processing: Part-of-speech tagging Intent detection Sentiment analysis Machine translation Speech recognition Language model pre-training Image recognition: Facial recognition Object detection Reinforcement learning: Game-playing Graph learning Matrix factorization

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  Legal information retrieval

  Legal information retrieval is the science of information retrieval applied to legal text, including legislation, case law, and scholarly works. Accurate legal information retrieval is important to provide access to the law to laymen and legal professionals. Its importance has increased because of the vast and quickly increasing amount of legal documents available through electronic means. Legal information retrieval is a part of the growing field of legal informatics. In a legal setting, it is frequently important to retrieve all information related to a specific query. However, commonly used boolean search methods (exact matches of specified terms) on full text legal documents have been shown to have an average recall rate as low as 20 percent, meaning that only 1 in 5 relevant documents are actually retrieved. In that case, researchers believed that they had retrieved over 75% of relevant documents. This may result in failing to retrieve important or precedential cases. In some jurisdictions this may be especially problematic, as legal professionals are ethically obligated to be reasonably informed as to relevant legal documents. Legal Information Retrieval attempts to increase the effectiveness of legal searches by increasing the number of relevant documents (providing a high recall rate) and reducing the number of irrelevant documents (a high precision rate). This is a difficult task, as the legal field is prone to jargon, polysemes (words that have different meanings when used in a legal context), and constant change. Techniques used to achieve these goals generally fall into three categories: boolean retrieval, manual classification of legal text, and natural language processing of legal text. == Problems == Application of standard information retrieval techniques to legal text can be more difficult than application in other subjects. One key problem is that the law rarely has an inherent taxonomy. Instead, the law is generally filled with open-ended terms, which may change over time. This can be especially true in common law countries, where each decided case can subtly change the meaning of a certain word or phrase. Legal information systems must also be programmed to deal with law-specific words and phrases. Though this is less problematic in the context of words which exist solely in law, legal texts also frequently use polysemes, words may have different meanings when used in a legal or common-speech manner, potentially both within the same document. The legal meanings may be dependent on the area of law in which it is applied. For example, in the context of European Union legislation, the term "worker" has four different meanings: Any worker as defined in Article 3(a) of Directive 89/391/EEC who habitually uses display screen equipment as a significant part of his normal work. Any person employed by an employer, including trainees and apprentices but excluding domestic servants; Any person carrying out an occupation on board a vessel, including trainees and apprentices, but excluding port pilots and shore personnel carrying out work on board a vessel at the quayside; Any person who, in the Member State concerned, is protected as an employee under national employment law and in accordance with national practice; It also has the common meaning: A person who works at a specific occupation. Though the terms may be similar, correct information retrieval must differentiate between the intended use and irrelevant uses in order to return the correct results. Even if a system overcomes the language problems inherent in law, it must still determine the relevancy of each result. In the context of judicial decisions, this requires determining the precedential value of the case. Case decisions from senior or superior courts may be more relevant than those from lower courts, even where the lower court's decision contains more discussion of the relevant facts. The opposite may be true, however, if the senior court has only a minor discussion of the topic (for example, if it is a secondary consideration in the case). An information retrieval system must also be aware of the authority of the jurisdiction. A case from a binding authority is most likely of more value than one from a non-binding authority. Additionally, the intentions of the user may determine which cases they find valuable. For instance, where a legal professional is attempting to argue a specific interpretation of law, he might find a minor court's decision which supports his position more valuable than a senior courts position which does not. He may also value similar positions from different areas of law, different jurisdictions, or dissenting opinions. Overcoming these problems can be made more difficult because of the large number of cases available. The number of legal cases available via electronic means is constantly increasing (in 2003, US appellate courts handed down approximately 500 new cases per day), meaning that an accurate legal information retrieval system must incorporate methods of both sorting past data and managing new data. == Techniques == === Boolean searches === Boolean searches, where a user may specify terms such as use of specific words or judgments by a specific court, are the most common type of search available via legal information retrieval systems. They are widely implemented but overcome few of the problems discussed above. The recall and precision rates of these searches vary depending on the implementation and searches analyzed. One study found a basic boolean search's recall rate to be roughly 20%, and its precision rate to be roughly 79%. Another study implemented a generic search (that is, not designed for legal uses) and found a recall rate of 56% and a precision rate of 72% among legal professionals. Both numbers increased when searches were run by non-legal professionals, to a 68% recall rate and 77% precision rate. This is likely explained because of the use of complex legal terms by the legal professionals. === Manual classification === In order to overcome the limits of basic boolean searches, information systems have attempted to classify case laws and statutes into more computer friendly structures. Usually, this results in the creation of an ontology to classify the texts, based on the way a legal professional might think about them. These attempt to link texts on the basis of their type, their value, and/or their topic areas. Most major legal search providers now implement some sort of classification search, such as Westlaw's “Natural Language” or LexisNexis' Headnote searches. Additionally, both of these services allow browsing of their classifications, via Westlaw's West Key Numbers or Lexis' Headnotes. Though these two search algorithms are proprietary and secret, it is known that they employ manual classification of text (though this may be computer-assisted). These systems can help overcome the majority of problems inherent in legal information retrieval systems, in that manual classification has the greatest chances of identifying landmark cases and understanding the issues that arise in the text. In one study, ontological searching resulted in a precision rate of 82% and a recall rate of 97% among legal professionals. The legal texts included, however, were carefully controlled to just a few areas of law in a specific jurisdiction. The major drawback to this approach is the requirement of using highly skilled legal professionals and large amounts of time to classify texts. As the amount of text available continues to increase, some have stated their belief that manual classification is unsustainable. === Natural language processing === In order to reduce the reliance on legal professionals and the amount of time needed, efforts have been made to create a system to automatically classify legal text and queries. Adequate translation of both would allow accurate information retrieval without the high cost of human classification. These automatic systems generally employ Natural Language Processing (NLP) techniques that are adapted to the legal domain, and also require the creation of a legal ontology. Though multiple systems have been postulated, few have reported results. One system, “SMILE,” which attempted to automatically extract classifications from case texts, resulted in an f-measure (which is a calculation of both recall rate and precision) of under 0.3 (compared to perfect f-measure of 1.0). This is probably much lower than an acceptable rate for general usage. Despite the limited results, many theorists predict that the evolution of such systems will eventually replace manual classification systems. === Citation-Based ranking === In the mid-90s the Room 5 case law retrieval project used citation mining for summaries and ranked its search results based on citation type and count. This slightly pre-dated the PageRank algorithm at Stanford which was also a citation-based ranking. Ranking of results was based

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  Winner-take-all (computing)

  Winner-take-all is a computational principle applied in computational models of neural networks by which neurons compete with each other for activation. In the classical form, only the neuron with the highest activation stays active while all other neurons shut down; however, other variations allow more than one neuron to be active, for example the soft winner take-all, by which a power function is applied to the neurons. == Neural networks == In the theory of artificial neural networks, winner-take-all networks are a case of competitive learning in recurrent neural networks. Output nodes in the network mutually inhibit each other, while simultaneously activating themselves through reflexive connections. After some time, only one node in the output layer will be active, namely the one corresponding to the strongest input. Thus the network uses nonlinear inhibition to pick out the largest of a set of inputs. Winner-take-all is a general computational primitive that can be implemented using different types of neural network models, including both continuous-time and spiking networks. Winner-take-all networks are commonly used in computational models of the brain, particularly for distributed decision-making or action selection in the cortex. Important examples include hierarchical models of vision, and models of selective attention and recognition. They are also common in artificial neural networks and neuromorphic analog VLSI circuits. It has been formally proven that the winner-take-all operation is computationally powerful compared to other nonlinear operations, such as thresholding. In many practical cases, there is not only one single neuron which becomes active but there are exactly k neurons which become active for a fixed number k. This principle is referred to as k-winners-take-all. === Example algorithm === Consider a single linear neuron, with inputs x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} . Each input has weight w i {\displaystyle w_{i}} , and the output of the neuron is ∑ i w i x i {\displaystyle \sum _{i}w_{i}x_{i}} . In the Instar learning rule, on each input vector, the weight vectors are modified according to Δ w i = η ( x i − w i ) {\displaystyle \Delta w_{i}=\eta (x_{i}-w_{i})} where η {\displaystyle \eta } is the learning rate. This rule is unsupervised, since we need just the input vector, not a reference output. Now, consider multiple linear neurons y 1 , … , y m {\displaystyle y_{1},\dots ,y_{m}} . The output of each satisfies y i = ∑ j w i j x j {\displaystyle y_{i}=\sum _{j}w_{ij}x_{j}} . In the winner-take-all algorithm, the weights are modified as follows. Given an input vector x {\displaystyle x} , each output is computed. The neuron with the largest output is selected, and the weights going into that neuron are modified according to the Instar learning rule. All other weights remain unchanged. The k-winners-take-all rule is similar, except that the Instar learning rule is applied to the weights going into the k neurons with the largest outputs. == Circuit example == A simple, but popular CMOS winner-take-all circuit is shown on the right. This circuit was originally proposed by Lazzaro et al. (1989) using MOS transistors biased to operate in the weak-inversion or subthreshold regime. In the particular case shown there are only two inputs (IIN,1 and IIN,2), but the circuit can be easily extended to multiple inputs in a straightforward way. It operates on continuous-time input signals (currents) in parallel, using only two transistors per input. In addition, the bias current IBIAS is set by a single global transistor that is common to all the inputs. The largest of the input currents sets the common potential VC. As a result, the corresponding output carries almost all the bias current, while the other outputs have currents that are close to zero. Thus, the circuit selects the larger of the two input currents, i.e., if IIN,1 > IIN,2, we get IOUT,1 = IBIAS and IOUT,2 = 0. Similarly, if IIN,2 > IIN,1, we get IOUT,1 = 0 and IOUT,2 = IBIAS. A SPICE-based DC simulation of the CMOS winner-take-all circuit in the two-input case is shown on the right. As shown in the top subplot, the input IIN,1 was fixed at 6nA, while IIN,2 was linearly increased from 0 to 10nA. The bottom subplot shows the two output currents. As expected, the output corresponding to the larger of the two inputs carries the entire bias current (10nA in this case), forcing the other output current nearly to zero. == Other uses == In stereo matching algorithms, following the taxonomy proposed by Scharstein and Szelliski, winner-take-all is a local method for disparity computation. Adopting a winner-take-all strategy, the disparity associated with the minimum or maximum cost value is selected at each pixel. It is axiomatic that in the electronic commerce market, early dominant players such as AOL or Yahoo! get most of the rewards. By 1998, one study found the top 5% of all web sites garnered more than 74% of all traffic. The winner-take-all hypothesis in economics suggests that once a technology or a firm gets ahead, it will do better and better over time, whereas lagging technology and firms will fall further behind. See First-mover advantage.

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  AdaBoost

  AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

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  Ordination (statistics)

  Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.

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  Shaded Picture System

  The Shaded Picture System was a 3D raster computer display processor introduced by Evans & Sutherland in October 1973. The Shaded Picture System was the first general-purpose, commercially available raster computer graphics display processor capable of real-time, shaded 3D graphics. It could only display black and white graphics at a resolution of 256 by 256. It was extremely expensive, and very few units were ever sold. == History == The principles of shaded, hidden-line true 3D graphics were pioneered at the University of Utah in 1967. However, this algorithm was slow and would take several minutes to produce an image. In 1970, Gary Watkins developed a FORTRAN simulator of a faster algorithm that would theoretically generate shaded 3D images in real-time, "if implemented in suitable hardware". The simulator itself was still not capable of real-time shaded 3D image rendering. Evans & Sutherland developed a functional prototype of this "suitable hardware", which was later sold as the Shaded Picture System in 1973. About a year earlier in 1972, Evans & Sutherland sold the first and only CT1 to Case Western Reserve University. The CT1, or Continuous Tone 1, was a specialized image generator, not meant as a marketable or mass-produced product. At the time, the CT1, along with G.E./NASA's upgraded Electronic Scene Generator from 1971, would have been the only real-time raster graphics systems sold to customers comparable to the Shaded Picture System, although both the CT1 and Electronic Scene Generator were intentionally produced as one-off products and specialized for the needs of their customers. The Shaded Picture System, in contrast, was intentionally marketed.In early 1975, Evans & Sutherland demonstrated a random-access video frame buffer using relatively low-cost semiconductor memory, which was much more capable than the Shaded Picture System. When interfaced with a (non-shaded) E&S Picture System, the frame buffer had a resolution of 512 by 512 in grayscale and partial color capabilities. By the end of 1975, this frame buffer was commercially available.

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  Linguamatics

  Linguamatics, headquartered in Cambridge, England, with offices in the United States and UK, is a provider of text mining systems through software licensing and services, primarily for pharmaceutical and healthcare applications. Founded in 2001, the company was purchased by IQVIA in January 2019. == Technology == The company develops enterprise search tools for the life sciences sector. The core natural language processing engine (I2E) uses a federated architecture to incorporate data from 3rd party resources. Initially developed to be used interactively through a graphic user interface, the core software also has an application programming interface that can be used to automate searches. LabKey, Penn Medicine, Atrius Health and Mercy all use Linguamatics software to extract electronic health record data into data warehouses. Linguamatics software is used by 17 of the top 20 global pharmaceutical companies, the US Food and Drug Administration, as well as healthcare providers. == Software community == The core software, "I2E", is used by a number of companies to either extend their own software or to publish their data. Copyright Clearance Center uses I2E to produce searchable indexes of material that would otherwise be unsearchable due to copyright. Thomson Reuters produces Cortellis Informatics Clinical Text Analytics, which depends on I2E to make clinical data accessible and searchable. Pipeline Pilot can integrate I2E as part of a workflow. ChemAxon can be used alongside I2E to allow named entity recognition of chemicals within unstructured data. Data sources include MEDLINE, ClinicalTrials.gov, FDA Drug Labels, PubMed Central, and Patent Abstracts.

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  Triplet loss

  Triplet loss is a machine learning loss function widely used in one-shot learning, a setting where models are trained to generalize effectively from limited examples. It was conceived by Google researchers for their prominent FaceNet algorithm for face detection. Triplet loss is designed to support metric learning. Namely, to assist training models to learn an embedding (mapping to a feature space) where similar data points are closer together and dissimilar ones are farther apart, enabling robust discrimination across varied conditions. In the context of face detection, data points correspond to images. == Definition == The loss function is defined using triplets of training points of the form ( A , P , N ) {\displaystyle (A,P,N)} . In each triplet, A {\displaystyle A} (called an "anchor point") denotes a reference point of a particular identity, P {\displaystyle P} (called a "positive point") denotes another point of the same identity in point A {\displaystyle A} , and N {\displaystyle N} (called a "negative point") denotes a point of an identity different from the identity in point A {\displaystyle A} and P {\displaystyle P} . Let x {\displaystyle x} be some point and let f ( x ) {\displaystyle f(x)} be the embedding of x {\displaystyle x} in the finite-dimensional Euclidean space. It shall be assumed that the L2-norm of f ( x ) {\displaystyle f(x)} is unity (the L2 norm of a vector X {\displaystyle X} in a finite dimensional Euclidean space is denoted by ‖ X ‖ {\displaystyle \Vert X\Vert } .) We assemble m {\displaystyle m} triplets of points from the training dataset. The goal of training here is to ensure that, after learning, the following condition (called the "triplet constraint") is satisfied by all triplets ( A ( i ) , P ( i ) , N ( i ) ) {\displaystyle (A^{(i)},P^{(i)},N^{(i)})} in the training data set: ‖ f ( A ( i ) ) − f ( P ( i ) ) ‖ 2 2 + α < ‖ f ( A ( i ) ) − f ( N ( i ) ) ‖ 2 2 {\displaystyle \Vert f(A^{(i)})-f(P^{(i)})\Vert _{2}^{2}+\alpha <\Vert f(A^{(i)})-f(N^{(i)})\Vert _{2}^{2}} The variable α {\displaystyle \alpha } is a hyperparameter called the margin, and its value must be set manually. In the FaceNet system, its value was set as 0.2. Thus, the full form of the function to be minimized is the following: L = ∑ i = 1 m max ( ‖ f ( A ( i ) ) − f ( P ( i ) ) ‖ 2 2 − ‖ f ( A ( i ) ) − f ( N ( i ) ) ‖ 2 2 + α , 0 ) {\displaystyle L=\sum _{i=1}^{m}\max {\Big (}\Vert f(A^{(i)})-f(P^{(i)})\Vert _{2}^{2}-\Vert f(A^{(i)})-f(N^{(i)})\Vert _{2}^{2}+\alpha ,0{\Big )}} == Intuition == A baseline for understanding the effectiveness of triplet loss is the contrastive loss, which operates on pairs of samples (rather than triplets). Training with the contrastive loss pulls embeddings of similar pairs closer together, and pushes dissimilar pairs apart. Its pairwise approach is greedy, as it considers each pair in isolation. Triplet loss innovates by considering relative distances. Its goal is that the embedding of an anchor (query) point be closer to positive points than to negative points (also accounting for the margin). It does not try to further optimize the distances once this requirement is met. This is approximated by simultaneously considering two pairs (anchor-positive and anchor-negative), rather than each pair in isolation. == Triplet "mining" == One crucial implementation detail when training with triplet loss is triplet "mining", which focuses on the smart selection of triplets for optimization. This process adds an additional layer of complexity compared to contrastive loss. A naive approach to preparing training data for the triplet loss involves randomly selecting triplets from the dataset. In general, the set of valid triplets of the form ( A ( i ) , P ( i ) , N ( i ) ) {\displaystyle (A^{(i)},P^{(i)},N^{(i)})} is very large. To speed-up training convergence, it is essential to focus on challenging triplets. In the FaceNet paper, several options were explored, eventually arriving at the following. For each anchor-positive pair, the algorithm considers only semi-hard negatives. These are negatives that violate the triplet requirement (i.e, are "hard"), but lie farther from the anchor than the positive (not too hard). Restated, for each A ( i ) {\displaystyle A^{(i)}} and P ( i ) {\displaystyle P^{(i)}} , they seek N ( i ) {\displaystyle N^{(i)}} such that: The rationale for this design choice is heuristic. It may appear puzzling that the mining process neglects "very hard" negatives (i.e., closer to the anchor than the positive). Experiments conducted by the FaceNet designers found that this often leads to a convergence to degenerate local minima. Triplet mining is performed at each training step, from within the sample points contained in the training batch (this is known as online mining), after embeddings were computed for all points in the batch. While ideally the entire dataset could be used, this is impractical in general. To support a large search space for triplets, the FaceNet authors used very large batches (1800 samples). Batches are constructed by selecting a large number of same-category sample points (40), and randomly selected negatives for them. == Extensions == Triplet loss has been extended to simultaneously maintain a series of distance orders by optimizing a continuous relevance degree with a chain (i.e., ladder) of distance inequalities. This leads to the Ladder Loss, which has been demonstrated to offer performance enhancements of visual-semantic embedding in learning to rank tasks. In Natural Language Processing, triplet loss is one of the loss functions considered for BERT fine-tuning in the SBERT architecture. Other extensions involve specifying multiple negatives (multiple negatives ranking loss).

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  Log-linear model

  A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form exp ⁡ ( c + ∑ i w i f i ( X ) ) {\displaystyle \exp \left(c+\sum _{i}w_{i}f_{i}(X)\right)} , in which the fi(X) are quantities that are functions of the variable X, in general a vector of values, while c and the wi stand for the model parameters. The term may specifically be used for: A log-linear plot or graph, which is a type of semi-log plot. Poisson regression for contingency tables, a type of generalized linear model. The specific applications of log-linear models are where the output quantity lies in the range 0 to ∞, for values of the independent variables X, or more immediately, the transformed quantities fi(X) in the range −∞ to +∞. This may be contrasted to logistic models, similar to the logistic function, for which the output quantity lies in the range 0 to 1. Thus the contexts where these models are useful or realistic often depends on the range of the values being modelled.

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  System context diagram

  A system context diagram in engineering is a diagram that defines the boundary between the system, or part of a system, and its environment, showing the entities that interact with it. This diagram is a high level view of a system. It is similar to a block diagram. == Overview == System context diagrams show a system, as a whole and its inputs and outputs from/to external factors. According to Kossiakoff and Sweet (2011): System Context Diagrams ... represent all external entities that may interact with a system ... Such a diagram pictures the system at the center, with no details of its interior structure, surrounded by all its interacting systems, environments and activities. The objective of the system context diagram is to focus attention on external factors and events that should be considered in developing a complete set of systems requirements and constraints. System context diagrams are used early in a project to get agreement on the scope under investigation. Context diagrams are typically included in a requirements document. These diagrams must be read by all project stakeholders and thus should be written in plain language, so the stakeholders can understand items within the document. == Building blocks == Context diagrams can be developed with the use of two types of building blocks: Entities (Actors): labeled boxes; one in the center representing the system, and around it multiple boxes for each external actor Relationships: labeled lines between the entities and system For example, "customer places order." Context diagrams can also use many different drawing types to represent external entities. They can use ovals, stick figures, pictures, clip art or any other representation to convey meaning. Decision trees and data storage are represented in system flow diagrams. A context diagram can also list the classifications of the external entities as one of a set of simple categories (Examples:), which add clarity to the level of involvement of the entity with regards to the system. These categories include: Active: Dynamic to achieve some goal or purpose (Examples: "Article readers" or "customers"). Passive: Static external entities which infrequently interact with the system (Examples: "Article editors" or "database administrator"). Cooperative: Predictable external entities which are used by the system to bring about some desired outcome (Examples: "Internet service providers" or "shipping companies"). Autonomous (Independent): External entities which are separated from the system, but affect the system indirectly, by means of imposed constraints or similar influences (Examples: "regulatory committees" or "standards groups"). == Alternatives == The best system context diagrams are used to display how a system interoperates at a very high level, or how systems operate and interact logically. The system context diagram is a necessary tool in developing a baseline interaction between systems and actors; actors and a system or systems and systems. Alternatives to the system context diagram are: Architecture Interconnect Diagram: The figure gives an example of an Architecture Interconnect Diagram: A representation of the Albuquerque regional ITS architecture interconnects for the Albuquerque Police Department that was generated using the Turbo Architecture tool is shown in the figure. Each block represents an ITS inventory element, including the name of the stakeholder in the top shaded portion. The interconnect lines between elements are solid or dashed, indicating existing or planned connections. Business Model Canvas, a strategic management template for developing new or documenting existing business models. It is a visual chart with elements describing a firm's value proposition, infrastructure, customers, and finances.[1] It assists firms in aligning their activities by illustrating potential trade-offs. Enterprise data model: this type of data model according to Simsion (2005) can contain up to 50 to 200 entity classes, which results from specific "high level of generalization in data modeling". IDEF0 Top Level Context Diagram: The IDEF0 process starts with the identification of the prime function to be decomposed. This function is identified on a "Top Level Context Diagram" that defines the scope of the particular IDEF0 analysis. Problem Diagrams (Problem Frames): In addition to the kinds of things shown on a context diagram, a problem diagram shows requirements and requirements references. Use case diagram: One of the Unified Modeling Language diagrams. They also represent the scope of the project at a similar level of abstraction. - Use Cases, however, tend to focus more on the goals of 'actors' who interact with the system, and do not specify any solution. Use Case diagrams represent a set of Use Cases, which are textual descriptions of how an actor achieves the goal of a use case. for Example Customer Places Order. ArchiMate: ArchiMate is an open and independent enterprise architecture modeling language to support the description, analysis and visualization of architecture within and across business domains in an unambiguous way. Most of these diagrams work well as long as a limited number of interconnects will be shown. Where twenty or more interconnects must be displayed, the diagrams become quite complex and can be difficult to read.

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  Modern Hopfield network

  Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons. The network still requires a sufficient number of hidden neurons. The key theoretical idea behind the modern Hopfield networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron’s configurations compared to the classical Hopfield network. == Classical Hopfield networks == Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons. In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous variable representing the output of neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the V i {\displaystyle V_{i}} (as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. == Discrete variables == A simple example of the Modern Hopfield network can be written in terms of binary variables V i {\displaystyle V_{i}} that represent the active V i = + 1 {\displaystyle V_{i}=+1} and inactive V i = − 1 {\displaystyle V_{i}=-1} state of the model neuron i {\displaystyle i} . E = − ∑ μ = 1 N mem F ( ∑ i = 1 N f ξ μ i V i ) {\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}} In this formula the weights ξ μ i {\textstyle \xi _{\mu i}} represent the matrix of memory vectors (index μ = 1... N mem {\displaystyle \mu =1...N_{\text{mem}}} enumerates different memories, and index i = 1... N f {\displaystyle i=1...N_{f}} enumerates the content of each memory corresponding to the i {\displaystyle i} -th feature neuron), and the function F ( x ) {\displaystyle F(x)} is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form V i ( t + 1 ) = sign ⁡ [ ∑ μ = 1 N mem ( F ( ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) − F ( − ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) ) ] {\displaystyle V_{i}^{(t+1)}=\operatorname {sign} {\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}} which states that in order to calculate the updated state of the i {\textstyle i} -th neuron the network compares two energies: the energy of the network with the i {\displaystyle i} -th neuron in the ON state and the energy of the network with the i {\displaystyle i} -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the i {\displaystyle i} -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic F ( x ) = x 2 {\displaystyle F(x)=x^{2}} these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function F ( x ) = x n {\displaystyle F(x)=x^{n}} the maximal number of memories that can be stored and retrieved from this network without errors is given by N mem max ≈ 1 2 ( 2 n − 3 ) ! ! N f n − 1 ln ⁡ ( N f ) {\displaystyle N_{\text{mem}}^{\max }\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}} For an exponential energy function F ( x ) = e x {\textstyle F(x)=e^{x}} the memory storage capacity is exponential in the number of feature neurons N mem max ≈ 2 N f / 2 {\displaystyle N_{\text{mem}}^{\max }\approx 2^{N_{f}/2}} == Continuous variables == Modern Hopfield networks or Dense Associative Memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for the neurons' state evolutionwhere the currents of the feature neurons are denoted by x i {\textstyle x_{i}} , and the currents of the memory neurons are denoted by h μ {\displaystyle h_{\mu }} ( h {\displaystyle h} stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix ξ μ i {\displaystyle \xi _{\mu i}} denotes the strength of synapses from a feature neuron i {\displaystyle i} to the memory neuron μ {\displaystyle \mu } . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron μ {\displaystyle \mu } to the feature neuron i {\displaystyle i} . The outputs of the memory neurons and the feature neurons are denoted by f μ {\displaystyle f_{\mu }} and g i {\displaystyle g_{i}} , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that f μ = f ( { h μ } ) {\displaystyle f_{\mu }=f(\{h_{\mu }\})} and g i = g ( { x i } ) {\textstyle g_{i}=g(\{x_{i}\})} . It is convenient to define these activation function as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by τ f {\displaystyle \tau _{f}} and τ h {\displaystyle \tau _{h}} , I i {\displaystyle I_{i}} is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state. In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons, τ h ≪ τ f {\textstyle \tau _{h}\ll \tau _{f}} . In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation). == Relationship to classical Hopfield network with continuous variables == Classical formulation of continuous Hopfield networks can be understood as a

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  Gremlin (query language)

  Gremlin is a graph traversal language and virtual machine developed by Apache TinkerPop of the Apache Software Foundation. Gremlin works for both OLTP-based graph databases as well as OLAP-based graph processors. Gremlin's automata and functional language foundation enable Gremlin to naturally support imperative and declarative querying, host language agnosticism, user-defined domain specific languages, an extensible compiler/optimizer, single- and multi-machine execution models, and hybrid depth- and breadth-first evaluation with Turing completeness. As an explanatory analogy, Apache TinkerPop and Gremlin are to graph databases what the JDBC and SQL are to relational databases. Likewise, the Gremlin traversal machine is to graph computing as what the Java virtual machine is to general purpose computing. == History == 2009-10-30 the project is born, and immediately named "TinkerPop" 2009-12-25 v0.1 is the first release 2011-05-21 v1.0 is released 2012-05-24 v2.0 is released 2015-01-16 TinkerPop becomes an Apache Incubator project 2015-07-09 v3.0.0-incubating is released 2016-05-23 Apache TinkerPop becomes a top-level project 2016-07-18 v3.1.3 and v3.2.1 are first releases as Apache TinkerPop 2017-12-17 v3.3.1 is released 2018-05-08 v3.3.3 is released 2019-08-05 v3.4.3 is released 2020-02-20 v3.4.6 is released 2021-05-01 v3.5.0 is released 2022-04-04 v3.6.0 is released 2023-07-31 v3.7.0 is released 2025-11-12 v3.8.0 is released == Vendor integration == Gremlin is an Apache2-licensed graph traversal language that can be used by graph system vendors. There are typically two types of graph system vendors: OLTP graph databases and OLAP graph processors. The table below outlines those graph vendors that support Gremlin. == Traversal examples == The following examples of Gremlin queries and responses in a Gremlin-Groovy environment are relative to a graph representation of the MovieLens dataset. The dataset includes users who rate movies. Users each have one occupation, and each movie has one or more categories associated with it. The MovieLens graph schema is detailed below. === Simple traversals === For each vertex in the graph, emit its label, then group and count each distinct label. What year was the oldest movie made? What is Die Hard's average rating? === Projection traversals === For each category, emit a map of its name and the number of movies it represents. For each movie with at least 11 ratings, emit a map of its name and average rating. Sort the maps in decreasing order by their average rating. Emit the first 10 maps (i.e. top 10). === Declarative pattern matching traversals === Gremlin supports declarative graph pattern matching similar to SPARQL. For instance, the following query below uses Gremlin's match()-step. What 80's action movies do 30-something programmers like? Group count the movies by their name and sort the group count map in decreasing order by value. Clip the map to the top 10 and emit the map entries. === OLAP traversal === Which movies are most central in the implicit 5-stars graph? == Gremlin graph traversal machine == Gremlin is a virtual machine composed of an instruction set as well as an execution engine. An analogy is drawn between Gremlin and Java. === Gremlin steps (instruction set) === The following traversal is a Gremlin traversal in the Gremlin-Java8 dialect. The Gremlin language (i.e. the fluent-style of expressing a graph traversal) can be represented in any host language that supports function composition and function nesting. Due to this simple requirement, there exists various Gremlin dialects including Gremlin-Groovy, Gremlin-Scala, Gremlin-Clojure, etc. The above Gremlin-Java8 traversal is ultimately compiled down to a step sequence called a traversal. A string representation of the traversal above provided below. The steps are the primitives of the Gremlin graph traversal machine. They are the parameterized instructions that the machine ultimately executes. The Gremlin instruction set is approximately 30 steps. These steps are sufficient to provide general purpose computing and what is typically required to express the common motifs of any graph traversal query. Given that Gremlin is a language, an instruction set, and a virtual machine, it is possible to design another traversal language that compiles to the Gremlin traversal machine (analogous to how Scala compiles to the JVM). For instance, the popular SPARQL graph pattern match language can be compiled to execute on the Gremlin machine. The following SPARQL query would compile to In Gremlin-Java8, the SPARQL query above would be represented as below and compile to the identical Gremlin step sequence (i.e. traversal). === Gremlin Machine (virtual machine) === The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).

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