Nathalie Japkowicz is a Canadian computer scientist specializing in machine learning. She is a professor and department chair of computer science at the American University College of Arts and Sciences. == Life == Nathalie Japkowicz completed a B.Sc. at McGill University in 1988. She earned an M.Sc. from the University of Toronto in 1990. She completed a Ph.D. at Rutgers University in 1999. Her dissertation was titled Concept-learning in the absence of counter-examples: an autoassociation-based approach to classification. Stephen José Hanson and Casimir Alexander Kulikowski were her doctoral advisors. Japkowicz worked at the University of Ottawa in the school of electrical engineering and computer science. She was the lead of its laboratory for research on machine learning for defense security. From 2003 to 2005, Japkowicz was the secretary of the Canadian Artificial Intelligence Association (CAIAC). She was CAIAC vice president from 2009 to 2014 and president from 2013 to 2015, and part-president from 2015 to 2017. Japkowicz is a professor and department chair of computer science at the American University College of Arts and Sciences. She researches artificial intelligence, machine learning, data mining, and big data analysis. == Selected works == Gao, Yong; Japkowicz, Nathalie, eds. (2009). Advances in Artificial Intelligence: 22nd Canadian Conference on Artificial Intelligence, Canadian AI 2009 Kelowna, Canada, May 25–27, 2009 Proceedings. Lecture Notes in Computer Science. Vol. 5549. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-01818-3. ISBN 978-3-642-01817-6. S2CID 27083226. Japkowicz, Nathalie; Shah, Mohak (2011). Evaluating Learning Algorithms: A Classification Perspective (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511921803. ISBN 978-0-511-92180-3. Japkowicz, Nathalie; Matwin, Stan, eds. (2015). Discovery Science: 18th International Conference, DS 2015, Banff, AB, Canada, October 4–6, 2015. Proceedings. Lecture Notes in Computer Science. Vol. 9356. Cham: Springer International Publishing. doi:10.1007/978-3-319-24282-8. ISBN 978-3-319-24281-1. S2CID 1302223. Japkowicz, Nathalie; Stefanowski, Jerzy, eds. (2016). Big Data Analysis: New Algorithms for a New Society. Studies in Big Data. Vol. 16. Cham: Springer International Publishing. doi:10.1007/978-3-319-26989-4. ISBN 978-3-319-26987-0. Ceci, Michelangelo; Japkowicz, Nathalie; Liu, Jiming; Papadopoulos, George A.; Raś, Zbigniew W., eds. (2018). Foundations of Intelligent Systems: 24th International Symposium, ISMIS 2018, Limassol, Cyprus, October 29–31, 2018, Proceedings. Lecture Notes in Computer Science. Vol. 11177. Cham: Springer International Publishing. doi:10.1007/978-3-030-01851-1. ISBN 978-3-030-01850-4. S2CID 53038780.
Common Image Generator Interface
The Common Image Generator Interface (CIGI) (pronounced sig-ee), is an on-the-wire data protocol that allows communication between an Image Generator and its host simulation. The interface is designed to promote a standard way for a host device to communicate with an image generator (IG) within the industry. CIGI enables plug-and-play by standard-compliant image generator vendors and reduces integration costs when upgrading visual systems. == Background == Most high-end simulators do not have everything running on a single machine the way popular home software flight simulators are currently implemented. The airplane model is run on one machine, normally referred to as the host, and the out the window visuals or scene graph program is run on another, usually referred to as an Image Generator (IG). Frequently there are multiple IGs required to display the surrounding environment created by a host. CIGI is the interface between the 'host' and the IGs. The main goal of CIGI is to capitalize on previous investments through the use of a common interface. CIGI is designed to assist suppliers and integrators of IG systems with ease of integration, code reuse, and overall cost reduction. In the past most image generators provided their own proprietary interface; every host had to implement that interface making changing image generators a costly ordeal. CIGI was created to standardize the interface between the host and the image generator so that little modification would be needed to switch image generators. The CIGI initiative was largely spearheaded by The Boeing Company during the early 21st century. The latest version of CIGI (CIGI 4.0) was developed by the Simulation Interoperability Standards Organization (SISO) in the form of SISO-STD-013-2014, Standard for Common Image Generator Interface (CIGI), Version 4.0, dated 22 August 2014. SISO-STD-013-2014 is freely available from SISO. == Definitions == Image generator – In this context an image generator consists of one or more rendering channels that produce an image that can be used to visualize an “Out-The-Window” scene, or images produced by various sensor simulations such as Infra-red, Day TV, electro-optical, and night vision. Host simulation – In this context a “Host” is the computational system that provides information about the device being simulated so that the image generator can portray the correct scenery to the user. This information is passed via CIGI to the image generator. == Maturation == CIGI 4 is the latest version of the standard as was approved by the Simulation Interoperability Standards Organization on August 22, 2014. CIGI became an international SISO standard known as SISO-STD-013-2014; which contains the CIGI version 4.0 Interface Control Document (ICD). CIGI 4.0 is the official standard, published by SISO. Previous versions of CIGI were spearheaded by Boeing include CIGI v3.3, in November 2008, v3.2 April 2006, v3.1 June 2004, v3 November 2003, v2 in March 2002, and the original (v1) in March 2001 == Protocol dependencies == Typically, CIGI uses UDP as its transport protocol, but CIGI does not require a specific transport mechanism, only packet definition conformance. CIGI traffic does not have a well known port; however, the use of ports 8004-8005 has been widely adopted by commercial image generator vendors implementations. == Development tools == === Host Emulator === The Host Emulator can be used as a surrogate to manipulate the interface when a simulation Host is not available. It is a Windows-based image generator Host application used to develop, integrate and test image generators that use the CIGI protocol. It provides a graphical user interface (GUI) for the creation, modification and deletion of entities; manipulation of views; control of environmental attributes and phenomena; and other host functions. The Host Emulator has several features that are useful for integration and testing. A free-flight mode allows for fixed-wing and rotorcraft flight, movement along entity axes and free rotation using a joystick or a joystick-like widget. Scripting and record/playback features support regression testing, demonstrations and other tasks needing exact reproduction of certain sequences of events. A packet-level snoop feature allows the user to examine the contents of CIGI messages, image generator response times and latencies. A Heartbeat Monitor Window shows a graphical timing history of the Image Generator's data frame rate. Other features include explicit packet creation, animation control, missile flyouts and a situation display window (Host Emulator 3.x only). === Multi-Purpose Viewer === The Multi-Purpose Viewer (MPV) provides the basic functionality expected of an Image Generator, such as loading and displaying a terrain database, displaying entities and so forth. The Multi-Purpose Viewer can be used as a surrogate to manipulate the interface when a real Image Generator is not available. The MPV is capable of operating with both the Windows and Linux operating systems. === CIGI Class Library === The CCL is an object-oriented software interface that automatically handles message composition and decomposition (i.e. packing, unpacking and byte swapping to the ICD specification) on both the Host and Image Generator sides of the interface. The CCL interprets Host or Image Generator messages based on compile time parameters. It also performs error handling and translation between different versions of CIGI. Each packet type has its own class. The individual packet members are accessed through packet class accessors. Outgoing messages are constructed by placing each packet into the outgoing buffer using a streaming operator. Incoming messages are parsed using callback or event-based mechanisms that supply the using program with fully populated packet objects. === Current tool suite === A set of CIGI development tools are managed and maintained by the SISO CIGI Product Support Group. The latest packages are available on SourceForge. Comments/Suggestions to the package can be directed to the SISO discussion board at: https://discussions.sisostds.org/index.htm?A0=SAC-PSG-CIGI Archived 2017-09-13 at the Wayback Machine === Wireshark === Wireshark is a free and open source packet analyzer. It is used for network troubleshooting, analysis, software and communications protocol development, and education. Wireshark provides a dissector for CIGI packets. As of October 2016, “The CIGI dissector is fully functional for CIGI version 2 and 3. Version 1 is not yet implemented.” === Older versions of CIGI === A CIGI Interface Control Document (ICD) and development suite is available in open source format. The tools, ICD, and accompanying user documentation can be found and downloaded from the CIGI sourceforge web site. The SourceForge version of the MPV is limited in its support of CIGI data packets and is intended to grow as needs arise. The MPV uses CIGI 3 as its interface, but the MPV is backward-compatible with earlier CIGI versions through the use of the CCL. The MPV uses the Open Scene Graph library to render a scene. The scene graph is manipulated according to the CIGI commands received from the Host via the CCL. The MPV itself is an application layer that consists of a small kernel leveraging heavily on a plug-in architecture for ease of maintainability and flexibility. An implementer can implement the interface from scratch, however a full suite of integration tools is available. These tools consist of three elements. The Host Emulator (HE), the Multi-Purpose Viewer (MPV), and the CIGI Class Library (CCL).
Averaged one-dependence estimators
Averaged one-dependence estimators (AODE) is a probabilistic classification learning technique. It was developed to address the attribute-independence problem of the popular naive Bayes classifier. It frequently develops substantially more accurate classifiers than naive Bayes at the cost of a modest increase in the amount of computation. == The AODE classifier == AODE seeks to estimate the probability of each class y given a specified set of features x1, ... xn, P(y | x1, ... xn). To do so it uses the formula P ^ ( y ∣ x 1 , … x n ) = ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y , x i ) ∏ j = 1 n P ^ ( x j ∣ y , x i ) ∑ y ′ ∈ Y ∑ i : 1 ≤ i ≤ n ∧ F ( x i ) ≥ m P ^ ( y ′ , x i ) ∏ j = 1 n P ^ ( x j ∣ y ′ , x i ) {\displaystyle {\hat {P}}(y\mid x_{1},\ldots x_{n})={\frac {\sum _{i:1\leq i\leq n\wedge F(x_{i})\geq m}{\hat {P}}(y,x_{i})\prod _{j=1}^{n}{\hat {P}}(x_{j}\mid y,x_{i})}{\sum _{y^{\prime }\in Y}\sum _{i:1\leq i\leq n\wedge F(x_{i})\geq m}{\hat {P}}(y^{\prime },x_{i})\prod _{j=1}^{n}{\hat {P}}(x_{j}\mid y^{\prime },x_{i})}}} where P ^ ( ⋅ ) {\displaystyle {\hat {P}}(\cdot )} denotes an estimate of P ( ⋅ ) {\displaystyle P(\cdot )} , F ( ⋅ ) {\displaystyle F(\cdot )} is the frequency with which the argument appears in the sample data and m is a user specified minimum frequency with which a term must appear in order to be used in the outer summation. In recent practice m is usually set at 1. == Derivation of the AODE classifier == We seek to estimate P(y | x1, ... xn). By the definition of conditional probability P ( y ∣ x 1 , … x n ) = P ( y , x 1 , … x n ) P ( x 1 , … x n ) . {\displaystyle P(y\mid x_{1},\ldots x_{n})={\frac {P(y,x_{1},\ldots x_{n})}{P(x_{1},\ldots x_{n})}}.} For any 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} , P ( y , x 1 , … x n ) = P ( y , x i ) P ( x 1 , … x n ∣ y , x i ) . {\displaystyle P(y,x_{1},\ldots x_{n})=P(y,x_{i})P(x_{1},\ldots x_{n}\mid y,x_{i}).} Under an assumption that x1, ... xn are independent given y and xi, it follows that P ( y , x 1 , … x n ) = P ( y , x i ) ∏ j = 1 n P ( x j ∣ y , x i ) . {\displaystyle P(y,x_{1},\ldots x_{n})=P(y,x_{i})\prod _{j=1}^{n}P(x_{j}\mid y,x_{i}).} This formula defines a special form of One Dependence Estimator (ODE), a variant of the naive Bayes classifier that makes the above independence assumption that is weaker (and hence potentially less harmful) than the naive Bayes' independence assumption. In consequence, each ODE should create a less biased estimator than naive Bayes. However, because the base probability estimates are each conditioned by two variables rather than one, they are formed from less data (the training examples that satisfy both variables) and hence are likely to have more variance. AODE reduces this variance by averaging the estimates of all such ODEs. == Features of the AODE classifier == Like naive Bayes, AODE does not perform model selection and does not use tuneable parameters. As a result, it has low variance. It supports incremental learning whereby the classifier can be updated efficiently with information from new examples as they become available. It predicts class probabilities rather than simply predicting a single class, allowing the user to determine the confidence with which each classification can be made. Its probabilistic model can directly handle situations where some data are missing. AODE has computational complexity O ( l n 2 ) {\displaystyle O(ln^{2})} at training time and O ( k n 2 ) {\displaystyle O(kn^{2})} at classification time, where n is the number of features, l is the number of training examples and k is the number of classes. This makes it infeasible for application to high-dimensional data. However, within that limitation, it is linear with respect to the number of training examples and hence can efficiently process large numbers of training examples. == Implementations == The free Weka machine learning suite includes an implementation of AODE.
Linear genetic programming
"Linear genetic programming" is unrelated to "linear programming". Linear genetic programming (LGP) is a particular method of genetic programming wherein computer programs in a population are represented as a sequence of register-based instructions from an imperative programming language or machine language. The adjective "linear" stems from the fact that each LGP program is a sequence of instructions and the sequence of instructions is normally executed sequentially. Like in other programs, the data flow in LGP can be modeled as a graph that will visualize the potential multiple usage of register contents and the existence of structurally noneffective code (introns) which are two main differences of this genetic representation from the more common tree-based genetic programming (TGP) variant. Like other Genetic Programming methods, Linear genetic programming requires the input of data to run the program population on. Then, the output of the program (its behaviour) is judged against some target behaviour, using a fitness function. However, LGP is generally more efficient than tree genetic programming due to its two main differences mentioned above: Intermediate results (stored in registers) can be reused and a simple intron removal algorithm exists that can be executed to remove all non-effective code prior to programs being run on the intended data. These two differences often result in compact solutions and substantial computational savings compared to the highly constrained data flow in trees and the common method of executing all tree nodes in TGP. Furthermore, LGP naturally has multiple outputs by defining multiple output registers and easily cooperates with control flow operations. Linear genetic programming has been applied in many domains, including system modeling and system control with considerable success. Linear genetic programming should not be confused with linear tree programs in tree genetic programming, program composed of a variable number of unary functions and a single terminal. Note that linear tree GP differs from bit string genetic algorithms since a population may contain programs of different lengths and there may be more than two types of functions or more than two types of terminals. == Examples of LGP programs == Because LGP programs are basically represented by a linear sequence of instructions, they are simpler to read and to operate on than their tree-based counterparts. For example, a simple program written to solve a Boolean function problem with 3 inputs (in R1, R2, R3) and one output (in R0), could read like this: R1, R2, R3 have to be declared as input (read-only) registers, while R0 and R4 are declared as calculation (read-write) registers. This program is very simple, having just 5 instructions. But mutation and crossover operators could work to increase the length of the program, as well as the content of each of its instructions. Note that one instruction is non-effective or an intron (marked), since it does not impact the output register R0. Recognition of those instructions is the basis for the intron removal algorithm which is used analyze code prior to execution. Technically, this happens by copying an individual and then run the intron removal once. The copy with removed introns is then executed as many times as dictated by the number of training cases. Notably, the original individual is left intact, so as to continue participating in the evolutionary process. It is only the copy that is executed that is compressed by removing these "structural" introns. Another simple program, this one written in the LGP language Slash/A looks like a series of instructions separated by a slash: By representing such code in bytecode format, i.e. as an array of bytes each representing a different instruction, one can make mutation operations simply by changing an element of such an array.
Joseph Nechvatal
Joseph Nechvatal (born January 15, 1951) is an American post-conceptual digital artist and art theoretician who creates computer-assisted paintings and computer animations, often using custom computer viruses. == Life and work == Joseph Nechvatal was born in Chicago. He studied fine art and philosophy at Southern Illinois University Carbondale, Cornell University, and Columbia University. He earned a Doctor of Philosophy in Philosophy of Art and Technology at the Planetary Collegium at University of Wales, Newport and has taught art theory and art history at the School of Visual Arts. He has had many solo exhibitions and is one of five artists that art historian Patrick Frank examines in his 2024 book Art of the 1980s: As If the Digital Mattered. His work in the late 1970s and early 1980s chiefly consisted of postminimal gray palimpsest-like drawings that were often photo-mechanically enlarged. Beginning in 1979 he became associated with the artist group Colab, organized the Public Arts International/Free Speech series, and helped established the non-profit group ABC No Rio. In 1983 he co-founded the avant-garde electronic art music audio project Tellus Audio Cassette Magazine. In 1984, Nechvatal began work on an opera called XS: The Opera Opus (1984-6) with the no wave musical composer Rhys Chatham. He began using computers and robotics to make post-conceptual paintings in 1986 and later, in his signature work, began to employ self-created computer viruses. From 1991 to 1993, he was artist-in-residence at the Louis Pasteur Atelier in Arbois, France and at the Saline Royale/Ledoux Foundation's computer lab. There he worked on The Computer Virus Project, his first artistic experiment with computer viruses and computer virus animation. He exhibited computer-robotic paintings at Documenta 8 in 1987. In 2002 he extended his experimentation into viral artificial life through a collaboration with the programmer Stephane Sikora of music2eye in a work called the Computer Virus Project II. Nechvatal has also created a noise music work called viral symphOny, a collaborative sound symphony created by using his computer virus software at the Institute for Electronic Arts at Alfred University. In 2021 Pentiments released Nechvatal's retrospective audio cassette called Selected Sound Works (1981-2021) and in 2022 his The Viral Tempest, a double vinyl LP of new audio work. In 2025, he joined the roster of artists/musicians at Table of the Elements with two CD/book releases: Selected Sound Works (1981-2021) and The Marriage of Orlando and Artaud, Even. From 1999 to 2013, Nechvatal taught art theories of immersive virtual reality and the viractual at the School of Visual Arts in New York City (SVA). A book of his collected essays entitled Towards an Immersive Intelligence: Essays on the Work of Art in the Age of Computer Technology and Virtual Reality (1993–2006) was published by Edgewise Press in 2009. Also in 2009, his virtual reality art theory and art history book Immersive Ideals / Critical Distances was published. In 2011, his book Immersion Into Noise was published by Open Humanities Press in conjunction with the University of Michigan Library's Scholarly Publishing Office. Nechvatal has also published three books with Punctum Books: Minóy (noise music—ed.—2014), Destroyer of Naivetés (poetry—2015), and Styling Sagaciousness (poetry—2022). In 2023 his art theory cybersex farce novella venus©~Ñ~vibrator, even was published by Orbis Tertius Press The Joseph Nechvatal archive is housed at The Fales Library Downtown Collection at the NYU Special Collections Library in New York City. === Viractualism === Viractualism is an art theory concept developed by Nechvatal in 1999 from Ph.D. research Nechvatal conducted at the Planetary Collegium at University of Wales, Newport. There he developed his concept of the viractual, which strives to create an interface between the actual and the virtual.
Contrast-to-noise ratio
Contrast-to-noise ratio (CNR) is a measure used to determine image quality. CNR is similar to the metric signal-to-noise ratio (SNR), but subtracts a term before taking the ratio. This is important when there is a significant bias in an image, such as from haze. As can be seen in the picture at right, the intensity is rather high even though the features of the image are washed out by the haze. Thus this image may have a high SNR metric, but will have a low CNR metric. One way to define contrast-to-noise ratio is: C = | S A − S B | σ o {\displaystyle C={\frac {|S_{A}-S_{B}|}{\sigma _{o}}}} where SA and SB are signal intensities for signal producing structures A and B in the region of interest and σo is the standard deviation of the pure image noise.
Kernel method
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing. Kernel methods owe their name to the use of kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. This operation is often computationally cheaper than the explicit computation of the coordinates. This approach is called the "kernel trick". Kernel functions have been introduced for sequence data, graphs, text, images, as well as vectors. Algorithms capable of operating with kernels include the kernel perceptron, support-vector machines (SVM), Gaussian processes, principal components analysis (PCA), canonical correlation analysis, ridge regression, spectral clustering, linear adaptive filters and many others. Most kernel algorithms are based on convex optimization or eigenproblems and are statistically well-founded. Typically, their statistical properties are analyzed using statistical learning theory (for example, using Rademacher complexity). == Motivation and informal explanation == Kernel methods can be thought of as instance-based learners: rather than learning some fixed set of parameters corresponding to the features of their inputs, they instead "remember" the i {\displaystyle i} -th training example ( x i , y i ) {\displaystyle (\mathbf {x} _{i},y_{i})} and learn for it a corresponding weight w i {\displaystyle w_{i}} . Prediction for unlabeled inputs, i.e., those not in the training set, are treated by the application of a similarity function k {\displaystyle k} , called a kernel, between the unlabeled input x ′ {\displaystyle \mathbf {x'} } and each of the training inputs x i {\displaystyle \mathbf {x} _{i}} . For instance, a kernelized binary classifier typically computes a weighted sum of similarities y ^ = sgn ∑ i = 1 n w i y i k ( x i , x ′ ) , {\displaystyle {\hat {y}}=\operatorname {sgn} \sum _{i=1}^{n}w_{i}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ),} where y ^ ∈ { − 1 , + 1 } {\displaystyle {\hat {y}}\in \{-1,+1\}} is the kernelized binary classifier's predicted label for the unlabeled input x ′ {\displaystyle \mathbf {x'} } whose hidden true label y {\displaystyle y} is of interest; k : X × X → R {\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } is the kernel function that measures similarity between any pair of inputs x , x ′ ∈ X {\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}} ; the sum ranges over the n labeled examples { ( x i , y i ) } i = 1 n {\displaystyle \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n}} in the classifier's training set, with y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} ; the w i ∈ R {\displaystyle w_{i}\in \mathbb {R} } are the weights for the training examples, as determined by the learning algorithm; the sign function sgn {\displaystyle \operatorname {sgn} } determines whether the predicted classification y ^ {\displaystyle {\hat {y}}} comes out positive or negative. Kernel classifiers were described as early as the 1960s, with the invention of the kernel perceptron. They rose to great prominence with the popularity of the support-vector machine (SVM) in the 1990s, when the SVM was found to be competitive with neural networks on tasks such as handwriting recognition. == Mathematics: the kernel trick == The kernel trick avoids the explicit mapping that is needed to get linear learning algorithms to learn a nonlinear function or decision boundary. For all x {\displaystyle \mathbf {x} } and x ′ {\displaystyle \mathbf {x'} } in the input space X {\displaystyle {\mathcal {X}}} , certain functions k ( x , x ′ ) {\displaystyle k(\mathbf {x} ,\mathbf {x'} )} can be expressed as an inner product in another space V {\displaystyle {\mathcal {V}}} . The function k : X × X → R {\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } is often referred to as a kernel or a kernel function. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or integral. Certain problems in machine learning have more structure than an arbitrary weighting function k {\displaystyle k} . The computation is made much simpler if the kernel can be written in the form of a "feature map" φ : X → V {\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}} which satisfies k ( x , x ′ ) = ⟨ φ ( x ) , φ ( x ′ ) ⟩ V . {\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle _{\mathcal {V}}.} The key restriction is that ⟨ ⋅ , ⋅ ⟩ V {\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}} must be a proper inner product. On the other hand, an explicit representation for φ {\displaystyle \varphi } is not necessary, as long as V {\displaystyle {\mathcal {V}}} is an inner product space. The alternative follows from Mercer's theorem: an implicitly defined function φ {\displaystyle \varphi } exists whenever the space X {\displaystyle {\mathcal {X}}} can be equipped with a suitable measure ensuring the function k {\displaystyle k} satisfies Mercer's condition. Mercer's theorem is similar to a generalization of the result from linear algebra that associates an inner product to any positive-definite matrix. In fact, Mercer's condition can be reduced to this simpler case. If we choose as our measure the counting measure μ ( T ) = | T | {\displaystyle \mu (T)=|T|} for all T ⊂ X {\displaystyle T\subset X} , which counts the number of points inside the set T {\displaystyle T} , then the integral in Mercer's theorem reduces to a summation ∑ i = 1 n ∑ j = 1 n k ( x i , x j ) c i c j ≥ 0. {\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.} If this summation holds for all finite sequences of points ( x 1 , … , x n ) {\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})} in X {\displaystyle {\mathcal {X}}} and all choices of n {\displaystyle n} real-valued coefficients ( c 1 , … , c n ) {\displaystyle (c_{1},\dots ,c_{n})} (cf. positive definite kernel), then the function k {\displaystyle k} satisfies Mercer's condition. Some algorithms that depend on arbitrary relationships in the native space X {\displaystyle {\mathcal {X}}} would, in fact, have a linear interpretation in a different setting: the range space of φ {\displaystyle \varphi } . The linear interpretation gives us insight about the algorithm. Furthermore, there is often no need to compute φ {\displaystyle \varphi } directly during computation, as is the case with support-vector machines. Some cite this running time shortcut as the primary benefit. Researchers also use it to justify the meanings and properties of existing algorithms. Theoretically, a Gram matrix K ∈ R n × n {\displaystyle \mathbf {K} \in \mathbb {R} ^{n\times n}} with respect to { x 1 , … , x n } {\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}} (sometimes also called a "kernel matrix"), where K i j = k ( x i , x j ) {\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})} , must be positive semi-definite (PSD). Empirically, for machine learning heuristics, choices of a function k {\displaystyle k} that do not satisfy Mercer's condition may still perform reasonably if k {\displaystyle k} at least approximates the intuitive idea of similarity. Regardless of whether k {\displaystyle k} is a Mercer kernel, k {\displaystyle k} may still be referred to as a "kernel". If the kernel function k {\displaystyle k} is also a covariance function as used in Gaussian processes, then the Gram matrix K {\displaystyle \mathbf {K} } can also be called a covariance matrix. == Applications == Application areas of kernel methods are diverse and include geostatistics, kriging, inverse distance weighting, 3D reconstruction, bioinformatics, cheminformatics, information extraction and handwriting recognition. == Popular kernels == Fisher kernel Graph kernels Kernel smoother Polynomial kernel Radial basis function kern