Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes
Image
An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a projection on a surface, activation of electronic signals, or digital displays; they can also be reproduced through mechanical means, such as photography, printmaking, or photocopying. Images can also be animated through digital or physical processes. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term image (or optical image) refers specifically to the reproduction of an object formed by light waves coming from the object. A volatile image exists or is perceived only for a short period. This may be a reflection of an object by a mirror, a projection of a camera obscura, or a scene displayed on a cathode-ray tube. A fixed image, also called a hard copy, is one that has been recorded on a material object, such as paper or textile. A mental image exists in an individual's mind as something one remembers or imagines. The subject of an image does not need to be real; it may be an abstract concept such as a graph or function or an imaginary entity. For a mental image to be understood outside of an individual's mind, however, there must be a way of conveying that mental image through the words or visual productions of the subject. == Characteristics == === Two-dimensional images === The broader sense of the word 'image' also encompasses any two-dimensional figure, such as a map, graph, pie chart, painting, or banner. In this wider sense, images can also be rendered manually, such as by drawing, the art of painting, or the graphic arts (such as lithography or etching). Additionally, images can be rendered automatically through printing, computer graphics technology, or a combination of both methods. A two-dimensional image does not need to use the entire visual system to be a visual representation. An example of this is a grayscale ("black and white") image, which uses the visual system's sensitivity to brightness across all wavelengths without taking into account different colors. A black-and-white visual representation of something is still an image, even though it does not fully use the visual system's capabilities. On the other hand, some processes can be used to create visual representations of objects that are otherwise inaccessible to the human visual system. These include microscopy for the magnification of minute objects, telescopes that can observe objects at great distances, X-rays that can visually represent the interior structures of the human body (among other objects), magnetic resonance imaging (MRI), positron emission tomography (PET scans), and others. Such processes often rely on detecting electromagnetic radiation that occurs beyond the light spectrum visible to the human eye and converting such signals into recognizable images. === Three-dimensional images === Aside from sculpture and other physical activities that can create three-dimensional images from solid material, some modern techniques, such as holography, can create three-dimensional images that are reproducible but intangible to human touch. Some photographic processes can now render the illusion of depth in an otherwise "flat" image, but "3-D photography" (stereoscopy) or "3-D film" are optical illusions that require special devices such as eyeglasses to create the illusion of depth. === Moving images === "Moving" two-dimensional images are actually illusions of movement perceived when still images are displayed in sequence, each image lasting less, and sometimes much less, than a fraction of a second. The traditional standard for the display of individual frames by a motion picture projector has been 24 frames per second (FPS) since at least the commercial introduction of "talking pictures" in the late 1920s, which necessitated a standard for synchronizing images and sounds. Even in electronic formats such as television and digital image displays, the apparent "motion" is actually the result of many individual lines giving the impression of continuous movement. This phenomenon has often been described as "persistence of vision": a physiological effect of light impressions remaining on the retina of the eye for very brief periods. Even though the term is still sometimes used in popular discussions of movies, it is not a scientifically valid explanation. Other terms emphasize the complex cognitive operations of the brain and the human visual system. "Flicker fusion", the "phi phenomenon", and "beta movement" are among the terms that have replaced "persistence of vision", though no one term seems adequate to describe the process. == Cultural and other uses == Image-making seems to have been common to virtually all human cultures since at least the Paleolithic era. Prehistoric examples of rock art—including cave paintings, petroglyphs, rock reliefs, and geoglyphs—have been found on every inhabited continent. Many of these images seem to have served various purposes: as a form of record-keeping; as an element of spiritual, religious, or magical practice; or even as a form of communication. Early writing systems, including hieroglyphics, ideographic writing, and even the Roman alphabet, owe their origins in some respects to pictorial representations. === Meaning and signification === Images of any type may convey different meanings and sensations for individual viewers, regardless of whether the image's creator intended them. An image may be taken simply as a more or less "accurate" copy of a person, place, thing, or event. It may represent an abstract concept, such as the political power of a ruler or ruling class, a practical or moral lesson, an object for spiritual or religious veneration, or an object—human or otherwise—to be desired. It may also be regarded for its purely aesthetic qualities, rarity, or monetary value. Such reactions can depend on the viewer's context. A religious image in a church may be regarded differently than the same image mounted in a museum. Some might view it simply as an object to be bought or sold. Viewers' reactions will also be guided or shaped by their education, class, race, and other contexts. The study of emotional sensations and their relationship to any given image falls into the categories of aesthetics and the philosophy of art. While such studies inevitably deal with issues of meaning, another approach to signification was suggested by the American philosopher, logician, and semiotician Charles Sanders Peirce. "Images" are one type of the broad category of "signs" proposed by Peirce. Although his ideas are complex and have changed over time, the three categories of signs that he distinguished stand out: The "icon," which relates to an object by resemblance to some quality of the object. A painted or photographed portrait is an icon by virtue of its resemblance to the painting's or photograph's subject. A more abstract representation, such as a map or diagram, can also be an icon. The "index," which relates to an object by some real connection. For example, smoke may be an index of fire, or the temperature recorded on a thermometer may be an index of a patient's illness or health. The "symbol," which lacks direct resemblance or connection to an object but whose association is arbitrarily assigned by the creator or dictated by cultural and historical habit, convention, etc. The color red, for example, may connote rage, beauty, prosperity, political affiliation, or other meanings within a given culture or context; the Swedish film director Ingmar Bergman claimed that his use of the color in his 1972 film Cries and Whispers came from his personal visualization of the human soul. A single image may exist in all three categories at the same time. The Statue of Liberty provides an example. While there have been countless two-dimensional and three-dimensional "reproductions" of the statue (i.e., "icons" themselves), the statue itself exists as an "icon" by virtue of its resemblance to a human woman (or, more specifically, previous representations of the Roman goddess Libertas or the female model used by the artist Frederic-Auguste Bartholdi). an "index" representing New York City or the United States of America in general due to its placement in New York Harbor, or with "immigration" from its proximity to the immigration center at Ellis Island. a "symbol" as a visualization of the abstract concept of "liberty" or "freedom" or even "opportunity" or "diversity". === Critiques of imagery === The nature of images, whether three-dimensional or two-dimensional, created for a specific purpose or only for aesthetic pleasure, has continued to provoke questions and even condemnation at different times and places. In his dialogue, The Republic, the Greek philosopher Plato described our apparent reality as a copy of a higher order of universal forms.
Multidimensional analysis
In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.
Geographical cluster
A geographical cluster is a localized anomaly, usually an excess of something given the distribution or variation of something else. Often it is considered as an incidence rate that is unusual in that there is more of some variable than might be expected. Examples would include: a local excess disease rate, a crime hot spot, areas of high unemployment, accident blackspots, unusually high positive residuals from a model, high concentrations of flora or fauna, physical features or events like earthquake epicenters etc... Identifying these extreme regions may be useful in that there could be implicit geographical associations with other variables that can be identified and would be of interest. Pattern detection via the identification of such geographical clusters is a very simple and generic form of geographical analysis that has many applications in many different contexts. The emphasis is on localized clustering or patterning because this may well contain the most useful information. A geographical cluster is different from a high concentration as it is generally second order, involving the factoring in of the distribution of something else. == Geographical cluster detection == Identifying geographical clusters can be an important stage in a geographical analysis. Mapping the locations of unusual concentrations may help identify causes of these. Some techniques include the Geographical Analysis Machine and Besag and Newell's cluster detection method.
IDistance
In pattern recognition, iDistance is an indexing and query processing technique for k-nearest neighbor queries on point data in multi-dimensional metric spaces. The kNN query is one of the hardest problems on multi-dimensional data, especially when the dimensionality of the data is high. iDistance is designed to process kNN queries in high-dimensional spaces efficiently and performs extremely well for skewed data distributions, which usually occur in real-life data sets. iDistance employs a two-phase search strategy involving an initial filtering of candidate regions and a subsequent refinement of results, an approach aligned with the Filter and Refine Principle (FRP). This means that the index first prunes the search space to eliminate unlikely candidates, then verifies the true nearest neighbors in a refinement step, following the general FRP paradigm used in database search algorithms. The iDistance index can also be augmented with machine learning models to learn data distributions for improved searching and storage of multi-dimensional data. == Indexing == Building the iDistance index has two steps: A number of reference points in the data space are chosen. There are various ways of choosing reference points. Using cluster centers as reference points is the most efficient way. The data points are partitioned into Voronoi cells based on well-chosen reference points. The distance between a data point and its closest reference point is calculated. This distance plus a scaling value is called the point's iDistance. By this means, points in a multi-dimensional space are mapped to one-dimensional values, and then a B+-tree can be adopted to index the points using the iDistance as the key. The figure on the right shows an example where three reference points (O1, O2, O3) are chosen. The data points are then mapped to a one-dimensional space and indexed in a B+-tree. Various extensions have been proposed to make the selection of reference points for effective query performance, including employing machine learning to learn the identification of reference points. == Query processing == To process a kNN query, the query is mapped to a number of one-dimensional range queries, which can be processed efficiently on a B+-tree. In the above figure, the query Q is mapped to a value in the B+-tree while the kNN search ``sphere" is mapped to a range in the B+-tree. The search sphere expands gradually until the k NNs are found. This corresponds to gradually expanding range searches in the B+-tree. The iDistance technique can be viewed as a way of accelerating the sequential scan. Instead of scanning records from the beginning to the end of the data file, the iDistance starts the scan from spots where the nearest neighbors can be obtained early with a very high probability. == Applications == The iDistance has been used in many applications including Image retrieval Video indexing Similarity search in P2P systems Mobile computing Recommender system == Historical background == The iDistance was first proposed by Cui Yu, Beng Chin Ooi, Kian-Lee Tan and H. V. Jagadish in 2001. Later, together with Rui Zhang, they improved the technique and performed a more comprehensive study on it in 2005.
Baby Bundle (app)
Baby Bundle is a parenting mobile app for iPhone and iPad. It was designed to help new parents through pregnancy and the first two years of parenthood. Developed in collaboration with medical experts, it helps track and record the child's development and growth, offers parental advice, manages vaccinations and health check-ups, stores photos and provides baby monitoring services. == History == Baby Bundle was founded in the United Kingdom by brothers, Nick and Anthony von Christierson. Each worked in investment banking prior to developing Baby Bundle, Nick at Greenhill & Co., and Anthony at Goldman Sachs. The idea for the app came when a friend's wife voiced her frustration over having multiple parenting apps on her smartphone. Nick and Anthony left their jobs to create a single app that would include all those features. They conducted market research by interviewing more than 500 parents in the UK and US. It took them a year to build the app, which was named by their mother. Looking for endorsement, they first went to the US in 2013 and partnered with parenting expert and pediatrician Dr. Jennifer Trachtenberg. Baby Bundle was launched in the US and Canadian App Stores in April 2014. In the same month, it became the #1 parenting app in iTunes and was featured by Apple as the #1 Editor's pick across all categories. Mashable called it one of the "Top 5 Can’t Miss Apps." Baby Bundle raised $1.8m seed round in March 2015 to fund development. The money came from a range of angel investors from across the US, UK and Asia. The von Christierson brothers have signed a deal to co-brand the app in the Middle East and expect to launch in Europe and Africa. == Features == Baby Bundle is an app for both the iPhone or iPad and provides smart monitoring tools and trackers for pregnancy and child development. It acts as a growth and daily activity tracker and offers parental advice, manages vaccinations and health check-ups. It has a parenting guide with tips and advice on what to expect when the baby arrives. An interactive forum also lets parents ask questions from others in the community. The app is free and also include paid premium features like the ability to turn two iPhones running into a baby monitor, a cloud service to share the child's data with a spouse and the ability to store data on more than one baby.
Ho–Kashyap algorithm
The Ho–Kashyap algorithm is an iterative method in machine learning for finding a linear decision boundary that separates two linearly separable classes. It was developed by Yu-Chi Ho and Rangasami L. Kashyap in 1965, and usually presented as a problem in linear programming. == Setup == Given a training set consisting of samples from two classes, the Ho–Kashyap algorithm seeks to find a weight vector w {\displaystyle \mathbf {w} } and a margin vector b {\displaystyle \mathbf {b} } such that: Y w = b {\displaystyle \mathbf {Yw} =\mathbf {b} } where Y {\displaystyle \mathbf {Y} } is the augmented data matrix with samples from both classes (with appropriate sign conventions, e.g., samples from class 2 are negated), w {\displaystyle \mathbf {w} } is the weight vector to be determined, and b {\displaystyle \mathbf {b} } is a positive margin vector. The algorithm minimizes the criterion function: J ( w , b ) = | | Y w − b | | 2 {\displaystyle J(\mathbf {w} ,\mathbf {b} )=||\mathbf {Yw} -\mathbf {b} ||^{2}} subject to the constraint that b > 0 {\displaystyle \mathbf {b} >\mathbf {0} } (element-wise). Given a problem of linearly separating two classes, we consider a dataset of elements { ( x i , y i ) } i ∈ 1 : N {\displaystyle \{(\mathbf {x_{i}} ,y_{i})\}_{i\in 1:N}} where y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} . Linearly separating them by a perceptron is equivalent to finding weight and bias w , b {\displaystyle \mathbf {w} ,b} for a perceptron, such that: [ y 1 x 1 1 ⋮ ⋮ y N x N 1 ] [ w b ] > 0 {\displaystyle {\begin{bmatrix}y_{1}\mathbf {x} _{1}&1\\\vdots &\vdots \\y_{N}\mathbf {x} _{N}&1\\\end{bmatrix}}{\begin{bmatrix}\mathbf {w} \\b\end{bmatrix}}>0} == Algorithm == The idea of the Ho–Kashyap algorithm is as follows: Given any b {\displaystyle \mathbf {b} } , the corresponding w {\displaystyle \mathbf {w} } is known: It is simply w = Y + b {\displaystyle \mathbf {w} =\mathbf {Y} ^{+}\mathbf {b} } , where Y + {\displaystyle \mathbf {Y} ^{+}} denotes the Moore–Penrose pseudoinverse of Y {\displaystyle \mathbf {Y} } . Therefore, it only remains to find b {\displaystyle \mathbf {b} } by gradient descent. However, the gradient descent may sometimes decrease some of the coordinates of b {\displaystyle \mathbf {b} } , which may cause some coordinates of b {\displaystyle \mathbf {b} } to become negative, which is undesirable. Therefore, whenever some coordinates of b {\displaystyle \mathbf {b} } would have decreased, those coordinates are unchanged instead. As for the coordinates of b {\displaystyle \mathbf {b} } that would increase, those would increase without issue. Formally, the algorithm is as follows: Initialization: Set b ( 0 ) {\displaystyle \mathbf {b} (0)} to an arbitrary positive vector, typically b ( 0 ) = 1 {\displaystyle \mathbf {b} (0)=\mathbf {1} } (a vector of ones). Set the iteration counter k = 0 {\displaystyle k=0} . Set w ( 0 ) = Y + b ( 0 ) {\displaystyle \mathbf {w} (0)=\mathbf {Y} ^{+}\mathbf {b} (0)} Loop until convergence, or until iteration counter exceeds some k m a x {\displaystyle k_{max}} . Error calculation: Compute the error vector: e ( k ) = Y w ( k ) − b ( k ) {\displaystyle \mathbf {e} (k)=\mathbf {Yw} (k)-\mathbf {b} (k)} . Margin update: Update the margin vector: b ( k + 1 ) = b ( k ) + 2 η k ( e ( k ) + | e ( k ) | ) {\displaystyle \mathbf {b} (k+1)=\mathbf {b} (k)+2\eta _{k}(\mathbf {e} (k)+|\mathbf {e} (k)|)} where η k {\displaystyle \eta _{k}} is a positive learning rate parameter, and | e ( k ) | {\displaystyle |\mathbf {e} (k)|} denotes the element-wise absolute value. Weight calculation: Compute the weight vector using the pseudoinverse: w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} . Convergence check: If | | e ( k ) | | ≤ θ {\displaystyle ||\mathbf {e} (k)||\leq \theta } for some predetermined threshold θ {\displaystyle \theta } (close to zero), then return b ( k + 1 ) , w ( k + 1 ) {\displaystyle \mathbf {b} (k+1),\mathbf {w} (k+1)} . if e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } (all components non-positive), return "Samples not separable.". Return "Algorithm failed to converge in time.". == Properties == If the training data is linearly separable, the algorithm converges to a solution (where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } ) in a finite number of iterations. If the data is not linearly separable, the algorithm may or may not ever reach the point where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } . However, if it does happen that e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } at some iteration, this proves non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability of the data. == Relationship to other algorithms == Perceptron algorithm: Both seek linear separators. The perceptron updates weights incrementally based on individual misclassified samples, while Ho–Kashyap is a batch method that processes all samples to compute the pseudoinverse and updates based on an overall error vector. Linear discriminant analysis (LDA): LDA assumes underlying Gaussian distributions with equal covariances for the classes and derives the decision boundary from these statistical assumptions. Ho–Kashyap makes no explicit distributional assumptions and instead tries to solve a system of linear inequalities directly. Support vector machines (SVM): For linearly separable data, SVMs aim to find the maximum-margin hyperplane. The Ho–Kashyap algorithm finds a separating hyperplane but not necessarily the one with the maximum margin. If the data is not separable, soft-margin SVMs allow for some misclassifications by optimizing a trade-off between margin size and misclassification penalty, while Ho–Kashyap provides a least-squares solution. == Variants == Modified Ho–Kashyap algorithm changes weight calculation step w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} to w ( k + 1 ) = w ( k ) + η k Y + | e ( k ) | {\displaystyle \mathbf {w} (k+1)=\mathbf {w} (k)+\eta _{k}\mathbf {Y} ^{+}|\mathbf {e} (k)|} . Kernel Ho–Kashyap algorithm: Applies kernel methods (the "kernel trick") to the Ho–Kashyap framework to enable non-linear classification by implicitly mapping data to a higher-dimensional feature space.