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  Client honeypot

  Honeypots are security devices whose value lie in being probed and compromised. Traditional honeypots are servers (or devices that expose server services) that wait passively to be attacked. Client Honeypots are active security devices in search of malicious servers that attack clients. The client honeypot poses as a client and interacts with the server to examine whether an attack has occurred. Often the focus of client honeypots is on web browsers, but any client that interacts with servers can be part of a client honeypot (for example ftp, email, ssh, etc.). There are several terms that are used to describe client honeypots. Besides client honeypot, which is the generic classification, honeyclient is the other term that is generally used and accepted. However, there is a subtlety here, as "honeyclient" is actually a homograph that could also refer to the first known open source client honeypot implementation (see below), although this should be clear from the context. == Architecture == A client honeypot is composed of three components. The first component, a queuer, is responsible for creating a list of servers for the client to visit. This list can be created, for example, through crawling. The second component is the client itself, which is able to make a requests to servers identified by the queuer. After the interaction with the server has taken place, the third component, an analysis engine, is responsible for determining whether an attack has taken place on the client honeypot. In addition to these components, client honeypots are usually equipped with some sort of containment strategy to prevent successful attacks from spreading beyond the client honeypot. This is usually achieved through the use of firewalls and virtual machine sandboxes. Analogous to traditional server honeypots, client honeypots are mainly classified by their interaction level: high or low; which denotes the level of functional interaction the server can utilize on the client honeypot. In addition to this there are also newly hybrid approaches which denotes the usage of both high and low interaction detection techniques. == High interaction == High interaction client honeypots are fully functional systems comparable to real systems with real clients. As such, no functional limitations (besides the containment strategy) exist on high interaction client honeypots. Attacks on high interaction client honeypots are detected via inspection of the state of the system after a server has been interacted with. The detection of changes to the client honeypot may indicate the occurrence of an attack against that has exploited a vulnerability of the client. An example of such a change is the presence of a new or altered file. High interaction client honeypots are very effective at detecting unknown attacks on clients. However, the tradeoff for this accuracy is a performance hit from the amount of system state that has to be monitored to make an attack assessment. Also, this detection mechanism is prone to various forms of evasion by the exploit. For example, an attack could delay the exploit from immediately triggering (time bombs) or could trigger upon a particular set of conditions or actions (logic bombs). Since no immediate, detectable state change occurred, the client honeypot is likely to incorrectly classify the server as safe even though it did successfully perform its attack on the client. Finally, if the client honeypots are running in virtual machines, then an exploit may try to detect the presence of the virtual environment and cease from triggering or behave differently. === Capture-HPC === Capture [1] is a high interaction client honeypot developed by researchers at Victoria University of Wellington, NZ. Capture differs from existing client honeypots in various ways. First, it is designed to be fast. State changes are being detected using an event based model allowing to react to state changes as they occur. Second, Capture is designed to be scalable. A central Capture server is able to control numerous clients across a network. Third, Capture is supposed to be a framework that allows to utilize different clients. The initial version of Capture supports Internet Explorer, but the current version supports all major browsers (Internet Explorer, Firefox, Opera, Safari) as well as other HTTP aware client applications, such as office applications and media players. === HoneyClient === HoneyClient [2] is a web browser based (IE/FireFox) high interaction client honeypot designed by Kathy Wang in 2004 and subsequently developed at MITRE. It was the first open source client honeypot and is a mix of Perl, C++, and Ruby. HoneyClient is state-based and detects attacks on Windows clients by monitoring files, process events, and registry entries. It has integrated the Capture-HPC real-time integrity checker to perform this detection. HoneyClient also contains a crawler, so it can be seeded with a list of initial URLs from which to start and can then continue to traverse web sites in search of client-side malware. === HoneyMonkey (dead since 2010) === HoneyMonkey [3] is a web browser based (IE) high interaction client honeypot implemented by Microsoft in 2005. It is not available for download. HoneyMonkey is state based and detects attacks on clients by monitoring files, registry, and processes. A unique characteristic of HoneyMonkey is its layered approach to interacting with servers in order to identify zero-day exploits. HoneyMonkey initially crawls the web with a vulnerable configuration. Once an attack has been identified, the server is reexamined with a fully patched configuration. If the attack is still detected, one can conclude that the attack utilizes an exploit for which no patch has been publicly released yet and therefore is quite dangerous. === SHELIA (dead since 2009) === Shelia [4] is a high interaction client honeypot developed by Joan Robert Rocaspana at Vrije Universiteit Amsterdam. It integrates with an email reader and processes each email it receives (URLs & attachments). Depending on the type of URL or attachment received, it opens a different client application (e.g. browser, office application, etc.) It monitors whether executable instructions are executed in data area of memory (which would indicate a buffer overflow exploit has been triggered). With such an approach, SHELIA is not only able to detect exploits, but is able to actually ward off exploits from triggering. === UW Spycrawler === The Spycrawler [5] developed at the University of Washington is yet another browser based (Mozilla) high interaction client honeypot developed by Moshchuk et al. in 2005. This client honeypot is not available for download. The Spycrawler is state based and detects attacks on clients by monitoring files, processes, registry, and browser crashes. Spycrawlers detection mechanism is event based. Further, it increases the passage of time of the virtual machine the Spycrawler is operating in to overcome (or rather reduce the impact of) time bombs. === Web Exploit Finder === WEF [6] is an implementation of an automatic drive-by-download – detection in a virtualized environment, developed by Thomas Müller, Benjamin Mack and Mehmet Arziman, three students from the Hochschule der Medien (HdM), Stuttgart during the summer term in 2006. WEF can be used as an active HoneyNet with a complete virtualization architecture underneath for rollbacks of compromised virtualized machines. == Low interaction == Low interaction client honeypots differ from high interaction client honeypots in that they do not utilize an entire real system, but rather use lightweight or simulated clients to interact with the server. (in the browser world, they are similar to web crawlers). Responses from servers are examined directly to assess whether an attack has taken place. This could be done, for example, by examining the response for the presence of malicious strings. Low interaction client honeypots are easier to deploy and operate than high interaction client honeypots and also perform better. However, they are likely to have a lower detection rate since attacks have to be known to the client honeypot in order for it to detect them; new attacks are likely to go unnoticed. They also suffer from the problem of evasion by exploits, which may be exacerbated due to their simplicity, thus making it easier for an exploit to detect the presence of the client honeypot. === HoneyC === HoneyC [7] is a low interaction client honeypot developed at Victoria University of Wellington by Christian Seifert in 2006. HoneyC is a platform independent open source framework written in Ruby. It currently concentrates driving a web browser simulator to interact with servers. Malicious servers are detected by statically examining the web server's response for malicious strings through the usage of Snort signatures. === Monkey-Spider (dead since 2008) === Monkey-Spider [8] is a low-interaction client honeypot i

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  European Conference on Computer Vision

  The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conferences in computer vision, alongside CVPR and ICCV, with an 'A' rating from the Australian Ranking of ICT Conferences and an 'A1' rating from the Brazilian ministry of education. The acceptance rate for ECCV 2010 was 24.4% for posters and 3.3% for oral presentations. Like other top computer vision conferences, ECCV has tutorial talks, technical sessions, and poster sessions. The conference is usually spread over five to six days with the main technical program occupying three days in the middle, and tutorial and workshops, focused on specific topics, being held in the beginning and at the end. The ECCV presents the Koenderink Prize annually to recognize fundamental contributions in computer vision. == Location == The conference is usually held in autumn in Europe.

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  Causal Markov condition

  The Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. This is related to the Markov condition, an assumption made in Bayesian probability theory, that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents. Stated loosely, it is assumed that a node has no bearing on nodes which do not descend from it. In a DAG, this local Markov condition is equivalent to the global Markov condition, which states that d-separations in the graph also correspond to conditional independence relations. This also means that a node is conditionally independent of the entire network, given its Markov blanket. A network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. == Motivation == Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of probabilistic causation is that causes raise the probabilities of their effects, all else being equal. A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer. On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:). == Examples == In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall. A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the surface underneath the hammer affected its falling. This essentially states the Causal Markov Condition, that given the existence of gravity the release of the hammer, it will fall regardless of what is beneath it. == Implications == === Dependence and Causation === It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V. This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP). === Screening === It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.

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  Random projection

  In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. According to theoretical results, random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing. == Dimensionality reduction == Dimensionality reduction, as the name suggests, is reducing the number of random variables using various mathematical methods from statistics and machine learning. Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally use linear transformations in determining the intrinsic dimensionality of the manifold as well as extracting its principal directions. For this purpose there are various related techniques, including: principal component analysis, linear discriminant analysis, canonical correlation analysis, discrete cosine transform, random projection, etc. Random projection is a simple and computationally efficient way to reduce the dimensionality of data by trading a controlled amount of error for faster processing times and smaller model sizes. The dimensions and distribution of random projection matrices are controlled so as to approximately preserve the pairwise distances between any two samples of the dataset. == Method == The core idea behind random projection is given in the Johnson-Lindenstrauss lemma, which states that if points in a vector space are of sufficiently high dimension, then they may be projected into a suitable lower-dimensional space in a way which approximately preserves pairwise distances between the points with high probability. In random projection, the original d {\displaystyle d} -dimensional data is projected to a k {\displaystyle k} -dimensional subspace, by multiplying on the left by a random matrix R ∈ R k × d {\displaystyle R\in \mathbb {R} ^{k\times d}} . Using matrix notation: If X d × N {\displaystyle X_{d\times N}} is the original set of N d-dimensional observations, then X k × N R P = R k × d X d × N {\displaystyle X_{k\times N}^{RP}=R_{k\times d}X_{d\times N}} is the projection of the data onto a lower k-dimensional subspace. Random projection is computationally simple: form the random matrix "R" and project the d × N {\displaystyle d\times N} data matrix X onto K dimensions of order O ( d k N ) {\displaystyle O(dkN)} . If the data matrix X is sparse with about c nonzero entries per column, then the complexity of this operation is of order O ( c k N ) {\displaystyle O(ckN)} . === Orthogonal random projection === A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle v} be its projection. The norm-squared ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as projecting a random point, uniformly sampled on the unit sphere, to its first k {\displaystyle k} coordinates. This is equivalent to sampling a random point in the multivariate gaussian distribution x ∼ N ( 0 , I d × d ) {\displaystyle x\sim {\mathcal {N}}(0,I_{d\times d})} , then normalizing it. Therefore, ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as ∑ i = 1 k x i 2 ∑ i = 1 k x i 2 + ∑ i = k + 1 d x i 2 {\displaystyle {\frac {\sum _{i=1}^{k}x_{i}^{2}}{\sum _{i=1}^{k}x_{i}^{2}+\sum _{i=k+1}^{d}x_{i}^{2}}}} , which by the chi-squared construction of the Beta distribution, has distribution Beta ⁡ ( k / 2 , ( d − k ) / 2 ) {\displaystyle \operatorname {Beta} (k/2,(d-k)/2)} , with mean k / d {\displaystyle k/d} . We have a concentration inequality P r [ | ‖ v ‖ 2 − k d | ≥ ϵ k d ] ≤ 3 exp ⁡ ( − k ϵ 2 / 64 ) {\displaystyle Pr\left[\left|\|v\|_{2}-{\frac {k}{d}}\right|\geq \epsilon {\sqrt {\frac {k}{d}}}\right]\leq 3\exp \left(-k\epsilon ^{2}/64\right)} for any ϵ ∈ ( 0 , 1 ) {\displaystyle \epsilon \in (0,1)} . === Gaussian random projection === The random matrix R can be generated using a Gaussian distribution. The first row is a random unit vector uniformly chosen from S d − 1 {\displaystyle S^{d-1}} . The second row is a random unit vector from the space orthogonal to the first row, the third row is a random unit vector from the space orthogonal to the first two rows, and so on. In this way of choosing R, and the following properties are satisfied: Spherical symmetry: For any orthogonal matrix A ∈ O ( d ) {\displaystyle A\in O(d)} , RA and R have the same distribution. Orthogonality: The rows of R are orthogonal to each other. Normality: The rows of R are unit-length vectors. === More computationally efficient random projections === Achlioptas has shown that the random matrix can be sampled more efficiently. Either the full matrix can be sampled IID according to R i , j = 3 / k × { + 1 with probability 1 6 0 with probability 2 3 − 1 with probability 1 6 {\displaystyle R_{i,j}={\sqrt {3/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{6}}\\0&{\text{with probability }}{\frac {2}{3}}\\-1&{\text{with probability }}{\frac {1}{6}}\end{cases}}} or the full matrix can be sampled IID according to R i , j = 1 / k × { + 1 with probability 1 2 − 1 with probability 1 2 {\displaystyle R_{i,j}={\sqrt {1/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{2}}\\-1&{\text{with probability }}{\frac {1}{2}}\end{cases}}} Both are efficient for database applications because the computations can be performed using integer arithmetic. More related study is conducted in. It was later shown how to use integer arithmetic while making the distribution even sparser, having very few nonzeroes per column, in work on the Sparse JL Transform. This is advantageous since a sparse embedding matrix means being able to project the data to lower dimension even faster. === Random Projection with Quantization === Random projection can be further condensed by quantization (discretization), with 1-bit (sign random projection) or multi-bits. It is the building block of SimHash, RP tree, and other memory efficient estimation and learning methods. == Large quasiorthogonal bases == The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimensional Euclidean space. There are exponentially large (in dimension n) sets of almost orthogonal vectors (with small value of inner products) in n–dimensional Euclidean space. This observation is useful in indexing of high-dimensional data. Quasiorthogonality of large random sets is important for methods of random approximation in machine learning. In high dimensions, exponentially large numbers of randomly and independently chosen vectors from equidistribution on a sphere (and from many other distributions) are almost orthogonal with probability close to one. This implies that in order to represent an element of such a high-dimensional space by linear combinations of randomly and independently chosen vectors, it may often be necessary to generate samples of exponentially large length if we use bounded coefficients in linear combinations. On the other hand, if coefficients with arbitrarily large values are allowed, the number of randomly generated elements that are sufficient for approximation is even less than dimension of the data space. == Implementations == RandPro - An R package for random projection sklearn.random_projection - A module for random projection from the scikit-learn Python library Weka implementation [1]

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  Generative art

  Generative art is post-conceptual art that has been created (in whole or in part) with the use of an autonomous system. An autonomous system in this context is generally one that is non-human and can independently determine features of an artwork that would otherwise require decisions made directly by the artist. In some cases the human creator may claim that the generative system represents their own artistic idea, and in others that the system takes on the role of the creator. "Generative art" often refers to algorithmic art (algorithmically determined computer generated artwork) and synthetic media (general term for any algorithmically generated media), but artists can also make generative art using systems of chemistry, biology, mechanics and robotics, smart materials, manual randomization, mathematics, data mapping, symmetry, and tiling. Generative algorithms, algorithms programmed to produce artistic works through predefined rules, stochastic methods, or procedural logic, often yielding dynamic, unique, and contextually adaptable outputs—are central to many of these practices. == History == The use of the word "generative" in the discussion of art has developed over time. The use of "Artificial DNA" defines a generative approach to art focused on the construction of a system able to generate unpredictable events, all with a recognizable common character. The use of autonomous systems, required by some contemporary definitions, focuses a generative approach where the controls are strongly reduced. This approach is also named "emergent". Margaret Boden and Ernest Edmonds have noted the use of the term "generative art" in the broad context of automated computer graphics in the 1960s, beginning with artwork exhibited by Georg Nees and Frieder Nake in 1965: A. Michael Noll did his initial computer art, combining randomness with order, in 1962, and exhibited it along with works by Bell Julesz in 1965. The terms "generative art" and "computer art" have been used in tandem, and more or less interchangeably, since the very earliest days. The first such exhibition showed the work of Nees in February 1965, which some claim was titled "Generative Computergrafik". While Nees does not himself remember, this was the title of his doctoral thesis published a few years later. The correct title of the first exhibition and catalog was "computer-grafik". "Generative art" and related terms was in common use by several other early computer artists around this time, including Manfred Mohr and Ken Knowlton. Vera Molnár (born 1924) is a French media artist of Hungarian origin. Molnar is widely considered to be a pioneer of generative art, and is also one of the first women to use computers in her art practice. The term "Generative Art" with the meaning of dynamic artwork-systems able to generate multiple artwork-events was clearly used the first time for the "Generative Art" conference in Milan in 1998. The term has also been used to describe geometric abstract art where simple elements are repeated, transformed, or varied to generate more complex forms. Thus defined, generative art was practiced by the Argentinian artists Eduardo Mac Entyre and Miguel Ángel Vidal in the late 1960s. In 1972 the Romanian-born Paul Neagu created the Generative Art Group in Britain. It was populated exclusively by Neagu using aliases such as "Hunsy Belmood" and "Edward Larsocchi". In 1972 Neagu gave a lecture titled 'Generative Art Forms' at the Queen's University, Belfast Festival. In 1970 the School of the Art Institute of Chicago created a department called Generative Systems. As described by Sonia Landy Sheridan the focus was on art practices using the then new technologies for the capture, inter-machine transfer, printing and transmission of images, as well as the exploration of the aspect of time in the transformation of image information. Also noteworthy is John Dunn, first a student and then a collaborator of Sheridan. In 1988 Clauser identified the aspect of systemic autonomy as a critical element in generative art: It should be evident from the above description of the evolution of generative art that process (or structuring) and change (or transformation) are among its most definitive features, and that these features and the very term 'generative' imply dynamic development and motion. (the result) is not a creation by the artist but rather the product of the generative process - a self-precipitating structure. In 1989 Celestino Soddu defined the Generative Design approach to Architecture and Town Design in his book Citta' Aleatorie. In 1989 Franke referred to "generative mathematics" as "the study of mathematical operations suitable for generating artistic images." From the mid-1990s Brian Eno popularized the terms generative music and generative systems, making a connection with earlier experimental music by Terry Riley, Steve Reich and Philip Glass. From the end of the 20th century, communities of generative artists, designers, musicians and theoreticians began to meet, forming cross-disciplinary perspectives. The first meeting about generative Art was in 1998, at the inaugural International Generative Art conference at Politecnico di Milano University, Italy. In Australia, the Iterate conference on generative systems in the electronic arts followed in 1999. On-line discussion has centered around the eu-gene mailing list, which began late 1999, and has hosted much of the debate which has defined the field. These activities have more recently been joined by the Generator.x conference in Berlin starting in 2005. In 2012 the new journal GASATHJ, Generative Art Science and Technology Hard Journal was founded by Celestino Soddu and Enrica Colabella jointing several generative artists and scientists in the editorial board. Some have argued that as a result of this engagement across disciplinary boundaries, the community has converged on a shared meaning of the term. As Boden and Edmonds put it in 2011: Today, the term "Generative Art" is still current within the relevant artistic community. Since 1998 a series of conferences have been held in Milan with that title (Generativeart.com), and Brian Eno has been influential in promoting and using generative art methods (Eno, 1996). Both in music and in visual art, the use of the term has now converged on work that has been produced by the activation of a set of rules and where the artist lets a computer system take over at least some of the decision-making (although, of course, the artist determines the rules). In the call of the Generative Art conferences in Milan (annually starting from 1998), the definition of Generative Art by Celestino Soddu: Generative Art is the idea realized as genetic code of artificial events, as construction of dynamic complex systems able to generate endless variations. Each Generative Project is a concept-software that works producing unique and non-repeatable events, like music or 3D Objects, as possible and manifold expressions of the generating idea strongly recognizable as a vision belonging to an artist / designer / musician / architect /mathematician. Discussion on the eu-gene mailing list was framed by the following definition by Adrian Ward from 1999: Generative art is a term given to work which stems from concentrating on the processes involved in producing an artwork, usually (although not strictly) automated by the use of a machine or computer, or by using mathematic or pragmatic instructions to define the rules by which such artworks are executed. A similar definition is provided by Philip Galanter: Generative art refers to any art practice where the artist creates a process, such as a set of natural language rules, a computer program, a machine, or other procedural invention, which is then set into motion with some degree of autonomy contributing to or resulting in a completed work of art. Around the 2020s, generative AI models learned to imitate the distinct style of particular authors. For example, a generative image model such as Stable Diffusion is able to model the stylistic characteristics of an artist like Pablo Picasso (including his particular brush strokes, use of colour, perspective, and so on), and a user can engineer a prompt such as "an astronaut riding a horse, by Picasso" to cause the model to generate a novel image applying the artist's style to an arbitrary subject. Generative image models have received significant backlash from artists who object to their style being imitated without their permission, arguing that this harms their ability to profit from their own work. The emergence of text-to-image generative AI systems has expanded debates over authorship, copyright, and artistic labor. The main issues in these debates include the eligibility of AI-generated outputs for copyright protection and the legal and ethical questions of using existing copyrighted works as training data for generative AI systems. == Types == === Music === Johann Kirnberger's Mu

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  Large margin nearest neighbor

  Large margin nearest neighbor (LMNN) classification is a statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based on semidefinite programming, a sub-class of convex optimization. The goal of supervised learning (more specifically classification) is to learn a decision rule that can categorize data instances into pre-defined classes. The k-nearest neighbor rule assumes a training data set of labeled instances (i.e. the classes are known). It classifies a new data instance with the class obtained from the majority vote of the k closest (labeled) training instances. Closeness is measured with a pre-defined metric. Large margin nearest neighbors is an algorithm that learns this global (pseudo-)metric in a supervised fashion to improve the classification accuracy of the k-nearest neighbor rule. == Setup == The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label. If this is achieved, the leave-one-out error (a special case of cross validation) is minimized. Let the training data consist of a data set D = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } ⊂ R d × C {\displaystyle D=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}\subset R^{d}\times C} , where the set of possible class categories is C = { 1 , … , c } {\displaystyle C=\{1,\dots ,c\}} . The algorithm learns a pseudometric of the type d ( x → i , x → j ) = ( x → i − x → j ) ⊤ M ( x → i − x → j ) {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})=({\vec {x}}_{i}-{\vec {x}}_{j})^{\top }\mathbf {M} ({\vec {x}}_{i}-{\vec {x}}_{j})} . For d ( ⋅ , ⋅ ) {\displaystyle d(\cdot ,\cdot )} to be well defined, the matrix M {\displaystyle \mathbf {M} } needs to be positive semi-definite. The Euclidean metric is a special case, where M {\displaystyle \mathbf {M} } is the identity matrix. This generalization is often (falsely) referred to as Mahalanobis metric. Figure 1 illustrates the effect of the metric under varying M {\displaystyle \mathbf {M} } . The two circles show the set of points with equal distance to the center x → i {\displaystyle {\vec {x}}_{i}} . In the Euclidean case this set is a circle, whereas under the modified (Mahalanobis) metric it becomes an ellipsoid. The algorithm distinguishes between two types of special data points: target neighbors and impostors. === Target neighbors === Target neighbors are selected before learning. Each instance x → i {\displaystyle {\vec {x}}_{i}} has exactly k {\displaystyle k} different target neighbors within D {\displaystyle D} , which all share the same class label y i {\displaystyle y_{i}} . The target neighbors are the data points that should become nearest neighbors under the learned metric. Let us denote the set of target neighbors for a data point x → i {\displaystyle {\vec {x}}_{i}} as N i {\displaystyle N_{i}} . === Impostors === An impostor of a data point x → i {\displaystyle {\vec {x}}_{i}} is another data point x → j {\displaystyle {\vec {x}}_{j}} with a different class label (i.e. y i ≠ y j {\displaystyle y_{i}\neq y_{j}} ) which is one of the nearest neighbors of x → i {\displaystyle {\vec {x}}_{i}} . During learning the algorithm tries to minimize the number of impostors for all data instances in the training set. == Algorithm == Large margin nearest neighbors optimizes the matrix M {\displaystyle \mathbf {M} } with the help of semidefinite programming. The objective is twofold: For every data point x → i {\displaystyle {\vec {x}}_{i}} , the target neighbors should be close and the impostors should be far away. Figure 1 shows the effect of such an optimization on an illustrative example. The learned metric causes the input vector x → i {\displaystyle {\vec {x}}_{i}} to be surrounded by training instances of the same class. If it was a test point, it would be classified correctly under the k = 3 {\displaystyle k=3} nearest neighbor rule. The first optimization goal is achieved by minimizing the average distance between instances and their target neighbors ∑ i , j ∈ N i d ( x → i , x → j ) {\displaystyle \sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})} . The second goal is achieved by penalizing distances to impostors x → l {\displaystyle {\vec {x}}_{l}} that are less than one unit further away than target neighbors x → j {\displaystyle {\vec {x}}_{j}} (and therefore pushing them out of the local neighborhood of x → i {\displaystyle {\vec {x}}_{i}} ). The resulting value to be minimized can be stated as: ∑ i , j ∈ N i , l , y l ≠ y i [ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ] + {\displaystyle \sum _{i,j\in N_{i},l,y_{l}\neq y_{i}}[d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})]_{+}} With a hinge loss function [ ⋅ ] + = max ( ⋅ , 0 ) {\textstyle [\cdot ]_{+}=\max(\cdot ,0)} , which ensures that impostor proximity is not penalized when outside the margin. The margin of exactly one unit fixes the scale of the matrix M {\displaystyle M} . Any alternative choice c > 0 {\displaystyle c>0} would result in a rescaling of M {\displaystyle M} by a factor of 1 / c {\displaystyle 1/c} . The final optimization problem becomes: min M ∑ i , j ∈ N i d ( x → i , x → j ) + λ ∑ i , j , l ξ i j l {\displaystyle \min _{\mathbf {M} }\sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})+\lambda \sum _{i,j,l}\xi _{ijl}} ∀ i , j ∈ N i , l , y l ≠ y i {\displaystyle \forall _{i,j\in N_{i},l,y_{l}\neq y_{i}}} d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ≤ ξ i j l {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})\leq \xi _{ijl}} ξ i j l ≥ 0 {\displaystyle \xi _{ijl}\geq 0} M ⪰ 0 {\displaystyle \mathbf {M} \succeq 0} The hyperparameter λ > 0 {\textstyle \lambda >0} is some positive constant (typically set through cross-validation). Here the variables ξ i j l {\displaystyle \xi _{ijl}} (together with two types of constraints) replace the term in the cost function. They play a role similar to slack variables to absorb the extent of violations of the impostor constraints. The last constraint ensures that M {\displaystyle \mathbf {M} } is positive semi-definite. The optimization problem is an instance of semidefinite programming (SDP). Although SDPs tend to suffer from high computational complexity, this particular SDP instance can be solved very efficiently due to the underlying geometric properties of the problem. In particular, most impostor constraints are naturally satisfied and do not need to be enforced during runtime (i.e. the set of variables ξ i j l {\displaystyle \xi _{ijl}} is sparse). A particularly well suited solver technique is the working set method, which keeps a small set of constraints that are actively enforced and monitors the remaining (likely satisfied) constraints only occasionally to ensure correctness. == Extensions and efficient solvers == LMNN was extended to multiple local metrics in the 2008 paper. This extension significantly improves the classification error, but involves a more expensive optimization problem. In their 2009 publication in the Journal of Machine Learning Research, Weinberger and Saul derive an efficient solver for the semi-definite program. It can learn a metric for the MNIST handwritten digit data set in several hours, involving billions of pairwise constraints. An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization.

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  Oja's rule

  Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

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  LamaH

  LamaH (Large-Sample Data for Hydrology and Environmental Sciences) is a cross-state initiative for unified data preparation and collection in the field of catchment hydrology. Hydrological datasets, for example, are an integral component for creating flood forecasting models. == Features == LamaH datasets always consist of a combination of meteorological time series (e.g., precipitation, temperature) and hydrologically relevant catchment attributes (e.g., elevation, slope, forest area, soil, bedrock) aggregated over the respective catchment as well as associated hydrological time series at the catchment outlet (discharge). By evaluating the large and heterogeneous sample (large-sample) of catchments, it is possible to gain insights into the hydrological cycle that would probably not be achievable with local and small-scale studies. The structure of the dataset allows an evaluation based on machine learning methods (deep learning). The accompanying paper explains not only the data preparation but also any limitations, uncertainties and possible applications. == Difference to CAMELS == The LamaH datasets are quite similar to the CAMELS datasets, but additionally feature: Further basin delineations (based on intermediate catchments) and attributes (e.g. flow distance and altitude difference between two topologically adjacent discharge gauges), enabling the setup of an interconnected hydrological network Attributes for classifying catchments and runoff gauges according to the degree and type of (anthropogenic) influence == Availability == LamaH datasets are available for the following regions: Central Europe (Austria and its hydrological upstream areas in Germany, Czech Republic, Switzerland, Slovakia, Italy, Liechtenstein, Slovenia and Hungary) / 859 catchments CAMELS datasets are available for (ranked by publication date): Contiguous USA (exclusive Alaska and Hawaii) / 671 catchments Chile / 516 catchments Brazil / 897 catchments Great Britain / 671 catchments Australia / 222 catchments Both the CAMELS and LamaH datasets are licensed with Creative Commons and are therefore available barrier-free for the public.

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  Imaging

  Imaging is the process of creating visual representations of objects, scenes, or phenomena. The term encompasses both the formation of images through physical processes and the technologies used to capture, store, process, and display them. While traditional imaging relies on visible light, modern imaging systems can visualize information across the electromagnetic spectrum and through other physical phenomena such as sound waves, magnetic fields, and particle emissions, enabling the visualization of subjects invisible to the human eye. Imaging science is the multidisciplinary field concerned with the theoretical foundations and practical applications of image creation and analysis. The field draws on physics, mathematics, electrical engineering, computer science, computer vision, and perceptual psychology to develop systems that generate, collect, duplicate, analyze, modify, and visualize images. == Principles == === The imaging chain === The imaging chain is a conceptual framework describing the interconnected components of any imaging system. Understanding each link in this chain allows engineers and scientists to optimize system performance for specific applications. The chain begins with the subject and its observable properties, typically energy that is emitted, reflected, or transmitted. A light source or other energy source may illuminate the subject to make these properties detectable. The capture device then collects this energy using appropriate sensors: optical systems for electromagnetic radiation, transducers for acoustic waves, or antenna arrays for radio frequencies. In digital systems, a processor converts the captured signals into a format suitable for rendering, applying algorithms for noise reduction, enhancement, or reconstruction. Finally, a display renders the processed information as a visible image on media such as paper, screens, or projection surfaces. Throughout this process, the characteristics of the human visual system inform design decisions, as the ultimate purpose of most imaging systems is to convey information to human observers. === Coherent and non-coherent imaging === Imaging systems are often classified by whether they use coherent or non-coherent illumination. Coherent imaging employs an active source that produces waves with a consistent phase relationship, as in radar, synthetic aperture radar, medical ultrasound, and optical coherence tomography. These systems can capture phase information in addition to amplitude, enabling techniques such as holography and interferometry. Non-coherent imaging systems, including conventional photography, fluorescence microscopy, and telescopes, rely on illumination sources where light waves have random phase relationships. == Methods and applications == Imaging methods span a wide range of physical principles, each suited to particular applications. Optical imaging encompasses photography, cinematography, microscopy, and telescopic observation. These methods capture electromagnetic radiation in or near the visible spectrum and form the basis of most consumer and scientific imaging. Extensions include thermography, which visualizes infrared radiation to reveal temperature distributions, and multispectral imaging, which captures data across multiple wavelength bands for applications in remote sensing and materials analysis. Medical imaging comprises techniques designed to visualize the interior of the human body for diagnostic and therapeutic purposes. Radiography and computed tomography use X-rays to image dense structures such as bone. Magnetic resonance imaging exploits nuclear magnetic properties to produce detailed soft-tissue images without ionizing radiation. Ultrasound imaging uses high-frequency sound waves and is particularly valuable for real-time imaging and fetal monitoring. Nuclear medicine techniques such as positron emission tomography track radioactive tracers to reveal metabolic activity. Emerging modalities include photoacoustic imaging, which combines optical and acoustic principles, and Magneto-acousto-electrical tomography, which maps electrical conductivity in biological tissues. Acoustic imaging uses sound waves to create images. Beyond medical ultrasound, applications include sonar for underwater navigation and mapping, seismic imaging for geological exploration, and industrial non-destructive testing. Radar and microwave imaging employ radio waves to detect and image objects. Synthetic aperture radar produces high-resolution images from aircraft or satellites regardless of weather or lighting conditions, making it essential for Earth observation and reconnaissance. Ground-penetrating radar images subsurface structures for archaeological and engineering applications. Electron and particle imaging use beams of electrons or other particles to achieve resolutions far beyond the diffraction limit of visible light. Electron microscopes can image individual atoms, enabling advances in materials science and structural biology. Chemical imaging combines spectroscopy with spatial imaging to map the chemical composition of samples, with applications in pharmaceutical development, food safety, and forensics. LIDAR (Light Detection and Ranging) measures distances using laser pulses to create three-dimensional representations of surfaces and objects, widely used in autonomous vehicles, topographic mapping, and forestry. Computational and digital imaging encompasses image processing, computer graphics, three-dimensional rendering, and digital image restoration. Computer vision applies algorithmic analysis to extract information from images automatically. == History == Photography and imaging have always been intertwined. When Joseph Nicéphore Niépce created the first permanent photograph using heliography in 1826, and Louis Daguerre refined the process into the daguerreotype a decade later, they weren't just inventing a new art form, they were laying the groundwork for an entire scientific discipline built on silver halide chemistry. For most of the nineteenth century, photography remained the province of specialists. That changed with George Eastman's Kodak camera, introduced in 1888 with the slogan "You press the button, we do the rest." Suddenly, anyone could take pictures. Around the same time, Wilhelm Röntgen stumbled onto X-rays in 1895, an accident that would spawn the entire field of medical imaging. World War II proved to be a turning point. Radar technology, developed frantically on both sides of the conflict, introduced concepts that engineers would later adapt for synthetic aperture radar and medical ultrasound. Then the charge-coupled device came: Willard Boyle and George E. Smith built the first one at Bell Labs in 1969, and within a few decades it had made film nearly obsolete. Magnetic resonance imaging arrived in the 1970s, offering doctors something X-rays never could, detailed views of soft tissue without any radiation. Digital cameras took over fast. By the 2000s, film was already in decline; by the 2010s, smartphones had put a surprisingly capable camera in nearly every pocket. Features that once required real skill, proper exposure, sharp focus, accurate color, became automatic. Today, billions of photos get uploaded to social media every day. As a result, a growing issue is that generative artificial intelligence can fabricate photorealistic images from scratch. What counts as a "real" photograph is no longer necessarily obvious.

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  Effective fitness

  In natural evolution and artificial evolution (e.g. artificial life and evolutionary computation) the fitness (or performance or objective measure) of a schema is rescaled to give its effective fitness which takes into account crossover and mutation. Effective fitness is used in Evolutionary Computation to understand population dynamics. While a biological fitness function only looks at reproductive success, an effective fitness function tries to encompass things that are needed to be fulfilled for survival on population level. In homogeneous populations, reproductive fitness and effective fitness are equal. When a population moves away from homogeneity a higher effective fitness is reached for the recessive genotype. This advantage will decrease while the population moves toward an equilibrium. The deviation from this equilibrium displays how close the population is to achieving a steady state. When this equilibrium is reached, the maximum effective fitness of the population is achieved. Problem solving with evolutionary computation is realized with a cost function. If cost functions are applied to swarm optimization they are called a fitness function. Strategies like reinforcement learning and NEAT neuroevolution are creating a fitness landscape which describes the reproductive success of cellular automata. The effective fitness function models the number of fit offspring and is used in calculations that include evolutionary processes, such as mutation and crossover, important on the population level. The effective fitness model is superior to its predecessor, the standard reproductive fitness model. It advances in the qualitatively and quantitatively understanding of evolutionary concepts like bloat, self-adaptation, and evolutionary robustness. While reproductive fitness only looks at pure selection, effective fitness describes the flow of a population and natural selection by taking genetic operators into account. A normal fitness function fits to a problem, while an effective fitness function is an assumption if the objective was reached. The difference is important for designing fitness functions with algorithms like novelty search in which the objective of the agents is unknown. In the case of bacteria effective fitness could include production of toxins and rate of mutation of different plasmids, which are mostly stochastically determined == Applications == When evolutionary equations of the studied population dynamics are available, one can algorithmically compute the effective fitness of a given population. Though the perfect effective fitness model is yet to be found, it is already known to be a good framework to the better understanding of the moving of the genotype-phenotype map, population dynamics, and the flow on fitness landscapes. Models using a combination of Darwinian fitness functions and effective functions are better at predicting population trends. Effective models could be used to determine therapeutic outcomes of disease treatment. Other models could determine effective protein engineering and works towards finding novel or heightened biochemistry.

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  Silhouette (clustering)

  Silhouette is a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette value ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable", and over 0.25 "weak". However, with an increasing dimensionality of the data, it becomes difficult to achieve such high values because of the curse of dimensionality, as the distances become more similar. The silhouette score is specialized for measuring cluster quality when the clusters are convex-shaped, and may not perform well if the data clusters have irregular shapes or are of varying sizes. The silhouette value can be calculated with any distance metric, such as Euclidean distance or Manhattan distance. == Definition == Assume the data have been clustered via any technique, such as k-medoids or k-means, into k {\displaystyle k} clusters. For data point i ∈ C i {\displaystyle i\in C_{i}} (data point i {\displaystyle i} in the cluster C i {\displaystyle C_{i}} ), calculate a ( i ) {\displaystyle a(i)} , the average distance that i {\displaystyle i} is from all other points in that cluster: a ( i ) = 1 | C i | − 1 ∑ j ∈ C i , i ≠ j d ( i , j ) {\displaystyle a(i)={\frac {1}{|C_{i}|-1}}\sum _{j\in C_{i},i\neq j}d(i,j)} where | C i | {\displaystyle |C_{i}|} is the number of points belonging to cluster C i {\displaystyle C_{i}} , and d ( i , j ) {\displaystyle d(i,j)} is the distance between data points i {\displaystyle i} and j {\displaystyle j} in the cluster C i {\displaystyle C_{i}} (we divide by | C i | − 1 {\displaystyle |C_{i}|-1} because the distance d ( i , i ) {\displaystyle d(i,i)} is not included in the sum). a ( i ) {\displaystyle a(i)} can be interpreted as a measure of how well i {\displaystyle i} is assigned to its cluster (the smaller the value, the better the assignment). We then define the mean dissimilarity of point i {\displaystyle i} to some cluster C j {\displaystyle C_{j}} as the mean of the distance from i {\displaystyle i} to all points in C j {\displaystyle C_{j}} (where C j ≠ C i {\displaystyle C_{j}\neq C_{i}} ). For each data point i ∈ C i {\displaystyle i\in C_{i}} , we now define b ( i ) {\displaystyle b(i)} as the average distance between i {\displaystyle i} and the points in the closest cluster (hence: "min") that i {\displaystyle i} does not belong to: b ( i ) = min j ≠ i 1 | C j | ∑ l ∈ C j d ( i , l ) {\displaystyle b(i)=\min _{j\neq i}{\frac {1}{|C_{j}|}}\sum _{l\in C_{j}}d(i,l)} The cluster with the smallest mean dissimilarity is said to be the "neighboring cluster" of i {\displaystyle i} because it is the next best fit cluster for point i {\displaystyle i} . We now define a silhouette (value) of one data point i {\displaystyle i} s ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } {\displaystyle s(i)={\frac {b(i)-a(i)}{\max\{a(i),b(i)\}}}} , if | C i | > 1 {\displaystyle |C_{i}|>1} and s ( i ) = 0 {\displaystyle s(i)=0} , if | C i | = 1 {\displaystyle |C_{i}|=1} , which can also be written as s ( i ) = { 1 − a ( i ) b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) a ( i ) − 1 , if a ( i ) > b ( i ) {\displaystyle s(i)={\begin{cases}1-{\frac {a(i)}{b(i)}},&{\mbox{ if }}a(i)b(i)\\\end{cases}}} From the above definition, s ( i ) {\displaystyle s(i)} is bounded to the interval [ − 1 , 1 ] {\displaystyle [-1,1]} , i.e. − 1 ≤ s ( i ) ≤ 1. {\displaystyle -1\leq s(i)\leq 1.} Note that a ( i ) {\displaystyle a(i)} is not clearly defined for clusters with size = 1, in which case we set s ( i ) = 0 {\displaystyle s(i)=0} . This choice is arbitrary, but neutral in the sense that it is at the midpoint of the bounds, -1 and 1. For s ( i ) {\displaystyle s(i)} to be close to 1 we require a ( i ) ≪ b ( i ) {\displaystyle a(i)\ll b(i)} . As a ( i ) {\displaystyle a(i)} is a measure of how dissimilar i {\displaystyle i} is to its own cluster, a small value means it is well matched. Furthermore, a large b ( i ) {\displaystyle b(i)} implies that i {\displaystyle i} is badly matched to its neighbouring cluster. Thus an s ( i ) {\displaystyle s(i)} close to 1 means that the data is appropriately clustered. If s ( i ) {\displaystyle s(i)} is close to -1, then by the same logic we see that i {\displaystyle i} would be more appropriate if it was clustered in its neighbouring cluster. An s ( i ) {\displaystyle s(i)} near zero means that the datum is on the border of two natural clusters. The mean s ( i ) {\displaystyle s(i)} over all points of a cluster is a measure of how tightly grouped all the points in the cluster are. Thus the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset is a measure of how appropriately the data have been clustered. If there are too many or too few clusters, as may occur when a poor choice of k {\displaystyle k} is used in the clustering algorithm (e.g., k-means), some of the clusters will typically display much narrower silhouettes than the rest. Thus silhouette plots and means may be used to determine the natural number of clusters within a dataset. One can also increase the likelihood of the silhouette being maximized at the correct number of clusters by re-scaling the data using feature weights that are cluster specific. Kaufman et al. introduced the term silhouette coefficient for the maximum value of the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset, i.e., S C = max k s ~ ( k ) , {\displaystyle SC=\max _{k}{\tilde {s}}\left(k\right),} where s ~ ( k ) {\displaystyle {\tilde {s}}\left(k\right)} represents the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset for a specific number of clusters k {\displaystyle k} . The silhouette coefficient describes the best possible clustering possible for a given number of clusters, as measured by the highest average silhouette score for all points in the dataset. == Simplified and medoid silhouette == Computing the silhouette coefficient needs all O ( N 2 ) {\displaystyle {\mathcal {O}}(N^{2})} pairwise distances, making this evaluation much more costly than clustering with k-means. For a clustering with centers μ C I {\displaystyle \mu _{C_{I}}} for each cluster C I {\displaystyle C_{I}} , we can use the following simplified Silhouette for each point i ∈ C I {\displaystyle i\in C_{I}} instead, which can be computed using only O ( N k ) {\displaystyle {\mathcal {O}}(Nk)} distances: a ′ ( i ) = d ( i , μ C I ) {\displaystyle a'(i)=d(i,\mu _{C_{I}})} and b ′ ( i ) = min C J ≠ C I d ( i , μ C J ) {\displaystyle b'(i)=\min _{C_{J}\neq C_{I}}d(i,\mu _{C_{J}})} , which has the additional benefit that a ′ ( i ) {\displaystyle a'(i)} is always defined, then define accordingly the simplified silhouette and simplified silhouette coefficient s ′ ( i ) = b ′ ( i ) − a ′ ( i ) max { a ′ ( i ) , b ′ ( i ) } {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{\max\{a'(i),b'(i)\}}}} S C ′ = max k 1 N ∑ i s ′ ( i ) {\displaystyle SC'=\max _{k}{\frac {1}{N}}\sum _{i}s'\left(i\right)} . If the cluster centers are medoids (as in k-medoids clustering) instead of arithmetic means (as in k-means clustering), this is also called the medoid-based silhouette or medoid silhouette. If every object is assigned to the nearest medoid (as in k-medoids clustering), we know that a ′ ( i ) ≤ b ′ ( i ) {\displaystyle a'(i)\leq b'(i)} , and hence s ′ ( i ) = b ′ ( i ) − a ′ ( i ) b ′ ( i ) = 1 − a ′ ( i ) b ′ ( i ) {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{b'(i)}}=1-{\frac {a'(i)}{b'(i)}}} . == Silhouette clustering == Instead of using the average silhouette to evaluate a clustering obtained from, e.g., k-medoids or k-means, we can try to directly find a solution that maximizes the Silhouette. We do not have a closed form solution to maximize this, but it will usually be best to assign points to the nearest cluster as done by these methods. Van der Laan et al. proposed to adapt the standard algorithm for k-medoids, PAM, for this purpose and call this algorithm PAMSIL: Choose initial medoids by using PAM Compute the average silhouette of this initial solution For each pair of a medoid m and a non-medoid x swap m and x compute the average silhouette of the resulting solution remember the best swap un-swap m and x for the next iteration Perform the best swap and return to

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  Optimal discriminant analysis and classification tree analysis

  Optimal Discriminant Analysis (ODA) and the related classification tree analysis (CTA) are exact statistical methods that maximize predictive accuracy. For any specific sample and exploratory or confirmatory hypothesis, optimal discriminant analysis (ODA) identifies the statistical model that yields maximum predictive accuracy, assesses the exact Type I error rate, and evaluates potential cross-generalizability. Optimal discriminant analysis may be applied to > 0 dimensions, with the one-dimensional case being referred to as UniODA and the multidimensional case being referred to as MultiODA. Optimal discriminant analysis is an alternative to ANOVA (analysis of variance) and regression analysis.

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  Data augmentation

  Data augmentation is a statistical technique which allows maximum likelihood estimation from incomplete data. Data augmentation has important applications in Bayesian analysis, and the technique is widely used in machine learning to reduce overfitting when training machine learning models, achieved by training models on several slightly-modified copies of existing data. == Synthetic oversampling techniques for traditional machine learning == Synthetic Minority Over-sampling Technique (SMOTE) is a method used to address imbalanced datasets in machine learning. In such datasets, the number of samples in different classes varies significantly, leading to biased model performance. For example, in a medical diagnosis dataset with 90 samples representing healthy individuals and only 10 samples representing individuals with a particular disease, traditional algorithms may struggle to accurately classify the minority class. SMOTE rebalances the dataset by generating synthetic samples for the minority class. For instance, if there are 100 samples in the majority class and 10 in the minority class, SMOTE can create synthetic samples by randomly selecting a minority class sample and its nearest neighbors, then generating new samples along the line segments joining these neighbors. This process helps increase the representation of the minority class, improving model performance. == Data augmentation for image classification == When convolutional neural networks grew larger in mid-1990s, there was a lack of data to use, especially considering that some part of the overall dataset should be spared for later testing. It was proposed to perturb existing data with affine transformations to create new examples with the same labels, which were complemented by so-called elastic distortions in 2003, and the technique was widely used as of 2010s. Data augmentation can enhance CNN performance and acts as a countermeasure against CNN profiling attacks. Data augmentation has become fundamental in image classification, enriching training dataset diversity to improve model generalization and performance. The evolution of this practice has introduced a broad spectrum of techniques, including geometric transformations, color space adjustments, and noise injection. === Geometric Transformations === Geometric transformations alter the spatial properties of images to simulate different perspectives, orientations, and scales. Common techniques include: Affine Transformation Rotation: Rotating images by a specified degree to help models recognize objects at various angles. Reflection: Reflecting images horizontally or vertically to introduce variability in orientation. Translation: Shifting images in different directions to teach models positional invariance. Scaling Shear Mapping Cropping: Removing sections of the image to focus on particular features or simulate closer views. Elastic Distortion Morphing within the same class: Generating new samples by applying morphing techniques between two images belonging to the same class, thereby increasing intra-class diversity. === Color Space Transformations === Color space transformations modify the color properties of images, addressing variations in lighting, color saturation, and contrast. Techniques include: Brightness Adjustment: Varying the image's brightness to simulate different lighting conditions. Contrast Adjustment: Changing the contrast to help models recognize objects under various clarity levels. Saturation Adjustment: Altering saturation to prepare models for images with diverse color intensities. Color Jittering: Randomly adjusting brightness, contrast, saturation, and hue to introduce color variability. === Noise Injection === Injecting noise into images simulates real-world imperfections, teaching models to ignore irrelevant variations. Techniques involve: Gaussian Noise: Adding Gaussian noise mimics sensor noise or graininess. Salt and Pepper Noise: Introducing black or white pixels at random simulates sensor dust or dead pixels. == Data augmentation for signal processing == Residual or block bootstrap can be used for time series augmentation. === Biological signals === Synthetic data augmentation is of paramount importance for machine learning classification, particularly for biological data, which tend to be high dimensional and scarce. The applications of robotic control and augmentation in disabled and able-bodied subjects still rely mainly on subject-specific analyses. Data scarcity is notable in signal processing problems such as for Parkinson's Disease Electromyography signals, which are difficult to source - Zanini, et al. noted that it is possible to use a generative adversarial network (in particular, a DCGAN) to perform style transfer in order to generate synthetic electromyographic signals that corresponded to those exhibited by sufferers of Parkinson's Disease. The approaches are also important in electroencephalography (brainwaves). Wang, et al. explored the idea of using deep convolutional neural networks for EEG-Based Emotion Recognition, results show that emotion recognition was improved when data augmentation was used. A common approach is to generate synthetic signals by re-arranging components of real data. Lotte proposed a method of "Artificial Trial Generation Based on Analogy" where three data examples x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} provide examples and an artificial x s y n t h e t i c {\displaystyle x_{synthetic}} is formed which is to x 3 {\displaystyle x_{3}} what x 2 {\displaystyle x_{2}} is to x 1 {\displaystyle x_{1}} . A transformation is applied to x 1 {\displaystyle x_{1}} to make it more similar to x 2 {\displaystyle x_{2}} , the same transformation is then applied to x 3 {\displaystyle x_{3}} which generates x s y n t h e t i c {\displaystyle x_{synthetic}} . This approach was shown to improve performance of a Linear Discriminant Analysis classifier on three different datasets. Current research shows great impact can be derived from relatively simple techniques. For example, Freer observed that introducing noise into gathered data to form additional data points improved the learning ability of several models which otherwise performed relatively poorly. Tsinganos et al. studied the approaches of magnitude warping, wavelet decomposition, and synthetic surface EMG models (generative approaches) for hand gesture recognition, finding classification performance increases of up to +16% when augmented data was introduced during training. More recently, data augmentation studies have begun to focus on the field of deep learning, more specifically on the ability of generative models to create artificial data which is then introduced during the classification model training process. In 2018, Luo et al. observed that useful EEG signal data could be generated by Conditional Wasserstein Generative Adversarial Networks (GANs) which was then introduced to the training set in a classical train-test learning framework. The authors found classification performance was improved when such techniques were introduced. === Mechanical signals === The prediction of mechanical signals based on data augmentation brings a new generation of technological innovations, such as new energy dispatch, 5G communication field, and robotics control engineering. In 2022, Yang et al. integrate constraints, optimization and control into a deep network framework based on data augmentation and data pruning with spatio-temporal data correlation, and improve the interpretability, safety and controllability of deep learning in real industrial projects through explicit mathematical programming equations and analytical solutions.

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  Nonlinear dimensionality reduction

  Nonlinear dimensionality reduction (NLDR), also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. == Applications of NLDR == High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keeping its essential features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns. The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32×32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information (the letter 'A') and recover only the varying information (rotation and scale). By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner. It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold will represent the robot's position and orientation. Invariant manifolds are of general interest for model order reduction in dynamical systems. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of spectral submanifolds (SSM) gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques. Some of the more prominent nonlinear dimensionality reduction techniques are listed below. == Important concepts == === Sammon's mapping === Sammon's mapping is one of the first and most popular NLDR techniques. === Self-organizing map === The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear mapping from the embedded space to the high-dimensional space. These techniques are related to work on density networks, which also are based around the same probabilistic model. === Kernel principal component analysis === Perhaps the most widely used algorithm for dimensional reduction is kernel PCA. PCA begins by computing the covariance matrix of the m × n {\displaystyle m\times n} matrix X {\displaystyle \mathbf {X} } C = 1 m ∑ i = 1 m x i x i T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\mathbf {x} _{i}\mathbf {x} _{i}^{\mathsf {T}}}.} It then projects the data onto the first k eigenvectors of that matrix. By comparison, KPCA begins by computing the covariance matrix of the data after being transformed into a higher-dimensional space, C = 1 m ∑ i = 1 m Φ ( x i ) Φ ( x i ) T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\Phi (\mathbf {x} _{i})\Phi (\mathbf {x} _{i})^{\mathsf {T}}}.} It then projects the transformed data onto the first k eigenvectors of that matrix, just like PCA. It uses the kernel trick to factor away much of the computation, such that the entire process can be performed without actually computing Φ ( x ) {\displaystyle \Phi (\mathbf {x} )} . Of course Φ {\displaystyle \Phi } must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to find a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can view certain other methods that perform well in such settings (e.g., Laplacian Eigenmaps, LLE) as special cases of kernel PCA by constructing a data-dependent kernel matrix. KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time. === Principal curves and manifolds === Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was originally proposed by Trevor Hastie in his 1984 thesis, which he formally introduced in 1989. This idea has been explored further by many authors. How to define the "simplicity" of the manifold is problem-dependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold. Usually, the principal manifold is defined as a solution to an optimization problem. The objective function includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and Kohonen's SOM. === Laplacian eigenmaps === Laplacian eigenmaps uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. This algorithm cannot embed out-of-sample points, but techniques based on Reproducing kernel Hilbert space regularization exist for adding this capability. Such techniques can be applied to other nonlinear dimensionality reduction algorithms as well. Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the graph and connectivity between nodes is governed by the proximity of neighboring points (using e.g. the k-nearest neighbor algorithm). The graph thus generated can be considered as a discrete approximation of the low-dimensional manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances. The eigenfunctions of the Laplace–Beltrami operator on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold (compare to Fourier series on the unit circle manifold). Attempts to place Laplacian eigenmaps on solid theoretical ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has been shown to converge to the Laplace–Beltrami operator as the number of points goes to infinity. === Isomap === Isomap is a combination of the Floyd–Warshall algorithm with classic Multidimensional Scaling (MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise geodesic distances between all of the points. Isomap th

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  Santa Fe Trail problem

  The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.

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