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  Instance selection

  Instance selection (or dataset reduction, or dataset condensation) is an important data pre-processing step that can be applied in many machine learning (or data mining) tasks. Approaches for instance selection can be applied for reducing the original dataset to a manageable volume, leading to a reduction of the computational resources that are necessary for performing the learning process. Algorithms of instance selection can also be applied for removing noisy instances, before applying learning algorithms. This step can improve the accuracy in classification problems. Algorithm for instance selection should identify a subset of the total available data to achieve the original purpose of the data mining (or machine learning) application as if the whole data had been used. Considering this, the optimal outcome of IS would be the minimum data subset that can accomplish the same task with no performance loss, in comparison with the performance achieved when the task is performed using the whole available data. Therefore, every instance selection strategy should deal with a trade-off between the reduction rate of the dataset and the classification quality. == Instance selection algorithms == The literature provides several different algorithms for instance selection. They can be distinguished from each other according to several different criteria. Considering this, instance selection algorithms can be grouped in two main classes, according to what instances they select: algorithms that preserve the instances at the boundaries of classes and algorithms that preserve the internal instances of the classes. Within the category of algorithms that select instances at the boundaries it is possible to cite DROP3, ICF and LSBo. On the other hand, within the category of algorithms that select internal instances, it is possible to mention ENN and LSSm. In general, algorithm such as ENN and LSSm are used for removing harmful (noisy) instances from the dataset. They do not reduce the data as the algorithms that select border instances, but they remove instances at the boundaries that have a negative impact on the data mining task. They can be used by other instance selection algorithms, as a filtering step. For example, the ENN algorithm is used by DROP3 as the first step, and the LSSm algorithm is used by LSBo. There is also another group of algorithms that adopt different selection criteria. For example, the algorithms LDIS, CDIS and XLDIS select the densest instances in a given arbitrary neighborhood. The selected instances can include both, border and internal instances. The LDIS and CDIS algorithms are very simple and select subsets that are very representative of the original dataset. Besides that, since they search by the representative instances in each class separately, they are faster (in terms of time complexity and effective running time) than other algorithms, such as DROP3 and ICF. Besides that, there is a third category of algorithms that, instead of selecting actual instances of the dataset, select prototypes (that can be synthetic instances). In this category it is possible to include PSSA, PSDSP and PSSP. The three algorithms adopt the notion of spatial partition (a hyperrectangle) for identifying similar instances and extract prototypes for each set of similar instances. In general, these approaches can also be modified for selecting actual instances of the datasets. The algorithm ISDSP adopts a similar approach for selecting actual instances (instead of prototypes).

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  Learning classifier system

  Learning classifier systems, or LCS, are a paradigm of rule-based machine learning methods that combine a discovery component (e.g. typically a genetic algorithm in evolutionary computation) with a learning component (performing either supervised learning, reinforcement learning, or unsupervised learning). Learning classifier systems seek to identify a set of context-dependent rules that collectively store and apply knowledge in a piecewise manner in order to make predictions (e.g. behavior modeling, classification, data mining, regression, function approximation, or game strategy). This approach allows complex solution spaces to be broken up into smaller, simpler parts for the reinforcement learning that is inside artificial intelligence research. The founding concepts behind learning classifier systems came from attempts to model complex adaptive systems, using rule-based agents to form an artificial cognitive system (i.e. artificial intelligence). == Methodology == The architecture and components of a given learning classifier system can be quite variable. It is useful to think of an LCS as a machine consisting of several interacting components. Components may be added or removed, or existing components modified/exchanged to suit the demands of a given problem domain (like algorithmic building blocks) or to make the algorithm flexible enough to function in many different problem domains. As a result, the LCS paradigm can be flexibly applied to many problem domains that call for machine learning. The major divisions among LCS implementations are as follows: (1) Michigan-style architecture vs. Pittsburgh-style architecture, (2) reinforcement learning vs. supervised learning, (3) incremental learning vs. batch learning, (4) online learning vs. offline learning, (5) strength-based fitness vs. accuracy-based fitness, and (6) complete action mapping vs best action mapping. These divisions are not necessarily mutually exclusive. For example, XCS, the best known and best studied LCS algorithm, is Michigan-style, was designed for reinforcement learning but can also perform supervised learning, applies incremental learning that can be either online or offline, applies accuracy-based fitness, and seeks to generate a complete action mapping. === Elements of a generic LCS algorithm === Keeping in mind that LCS is a paradigm for genetic-based machine learning rather than a specific method, the following outlines key elements of a generic, modern (i.e. post-XCS) LCS algorithm. For simplicity let us focus on Michigan-style architecture with supervised learning. See the illustrations on the right laying out the sequential steps involved in this type of generic LCS. ==== Environment ==== The environment is the source of data upon which an LCS learns. It can be an offline, finite training dataset (characteristic of a data mining, classification, or regression problem), or an online sequential stream of live training instances. Each training instance is assumed to include some number of features (also referred to as attributes, or independent variables), and a single endpoint of interest (also referred to as the class, action, phenotype, prediction, or dependent variable). Part of LCS learning can involve feature selection, therefore not all of the features in the training data need to be informative. The set of feature values of an instance is commonly referred to as the state. For simplicity let's assume an example problem domain with Boolean/binary features and a Boolean/binary class. For Michigan-style systems, one instance from the environment is trained on each learning cycle (i.e. incremental learning). Pittsburgh-style systems perform batch learning, where rule sets are evaluated in each iteration over much or all of the training data. ==== Rule/classifier/population ==== A rule is a context dependent relationship between state values and some prediction. Rules typically take the form of an {IF:THEN} expression, (e.g. {IF 'condition' THEN 'action'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign'}). A critical concept in LCS and rule-based machine learning alike, is that an individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Think of a rule as a "local-model" of the solution space. Rules can be represented in many different ways to handle different data types (e.g. binary, discrete-valued, ordinal, continuous-valued). Given binary data LCS traditionally applies a ternary rule representation (i.e. rules can include either a 0, 1, or '#' for each feature in the data). The 'don't care' symbol (i.e. '#') serves as a wild card within a rule's condition allowing rules, and the system as a whole to generalize relationships between features and the target endpoint to be predicted. Consider the following rule (#1###0 ~ 1) (i.e. condition ~ action). This rule can be interpreted as: IF the second feature = 1 AND the sixth feature = 0 THEN the class prediction = 1. We would say that the second and sixth features were specified in this rule, while the others were generalized. This rule, and the corresponding prediction are only applicable to an instance when the condition of the rule is satisfied by the instance. This is more commonly referred to as matching. In Michigan-style LCS, each rule has its own fitness, as well as a number of other rule-parameters associated with it that can describe the number of copies of that rule that exist (i.e. the numerosity), the age of the rule, its accuracy, or the accuracy of its reward predictions, and other descriptive or experiential statistics. A rule along with its parameters is often referred to as a classifier. In Michigan-style systems, classifiers are contained within a population [P] that has a user defined maximum number of classifiers. Unlike most stochastic search algorithms (e.g. evolutionary algorithms), LCS populations start out empty (i.e. there is no need to randomly initialize a rule population). Classifiers will instead be initially introduced to the population with a covering mechanism. In any LCS, the trained model is a set of rules/classifiers, rather than any single rule/classifier. In Michigan-style LCS, the entire trained (and optionally, compacted) classifier population forms the prediction model. ==== Matching ==== One of the most critical and often time-consuming elements of an LCS is the matching process. The first step in an LCS learning cycle takes a single training instance from the environment and passes it to [P] where matching takes place. In step two, every rule in [P] is now compared to the training instance to see which rules match (i.e. are contextually relevant to the current instance). In step three, any matching rules are moved to a match set [M]. A rule matches a training instance if all feature values specified in the rule condition are equivalent to the corresponding feature value in the training instance. For example, assuming the training instance is (001001 ~ 0), these rules would match: (###0## ~ 0), (00###1 ~ 0), (#01001 ~ 1), but these rules would not (1##### ~ 0), (000##1 ~ 0), (#0#1#0 ~ 1). Notice that in matching, the endpoint/action specified by the rule is not taken into consideration. As a result, the match set may contain classifiers that propose conflicting actions. In the fourth step, since we are performing supervised learning, [M] is divided into a correct set [C] and an incorrect set [I]. A matching rule goes into the correct set if it proposes the correct action (based on the known action of the training instance), otherwise it goes into [I]. In reinforcement learning LCS, an action set [A] would be formed here instead, since the correct action is not known. ==== Covering ==== At this point in the learning cycle, if no classifiers made it into either [M] or [C] (as would be the case when the population starts off empty), the covering mechanism is applied (fifth step). Covering is a form of online smart population initialization. Covering randomly generates a rule that matches the current training instance (and in the case of supervised learning, that rule is also generated with the correct action. Assuming the training instance is (001001 ~ 0), covering might generate any of the following rules: (#0#0## ~ 0), (001001 ~ 0), (#010## ~ 0). Covering not only ensures that each learning cycle there is at least one correct, matching rule in [C], but that any rule initialized into the population will match at least one training instance. This prevents LCS from exploring the search space of rules that do not match any training instances. ==== Parameter updates/credit assignment/learning ==== In the sixth step, the rule parameters of any rule in [M] are updated to reflect the new experience gained from the current training instance. Depending on the LCS algorithm, a number of updates can take place at this step. For supervised learning, we can simply update the accuracy/error of a

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  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

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  Prefrontal cortex basal ganglia working memory

  Prefrontal cortex basal ganglia working memory (PBWM) is an algorithm that models working memory in the prefrontal cortex and the basal ganglia. It can be compared to long short-term memory (LSTM) in functionality, but is more biologically explainable. It uses the primary value learned value model to train prefrontal cortex working-memory updating system, based on the biology of the prefrontal cortex and basal ganglia. It is used as part of the Leabra framework and was implemented in Emergent in 2019. == Abstract == The prefrontal cortex has long been thought to subserve both working memory (the holding of information online for processing) and "executive" functions (deciding how to manipulate working memory and perform processing). Although many computational models of working memory have been developed, the mechanistic basis of executive function remains elusive. PBWM is a computational model of the prefrontal cortex to control both itself and other brain areas in a strategic, task-appropriate manner. These learning mechanisms are based on subcortical structures in the midbrain, basal ganglia and amygdala, which together form an actor/critic architecture. The critic system learns which prefrontal representations are task-relevant and trains the actor, which in turn provides a dynamic gating mechanism for controlling working memory updating. Computationally, the learning mechanism is designed to simultaneously solve the temporal and structural credit assignment problems. The model's performance compares favorably with standard backpropagation-based temporal learning mechanisms on the challenging 1-2-AX working memory task, and other benchmark working memory tasks. == Model == First, there are multiple separate stripes (groups of units) in the prefrontal cortex and striatum layers. Each stripe can be independently updated, such that this system can remember several different things at the same time, each with a different "updating policy" of when memories are updated and maintained. The active maintenance of the memory is in prefrontal cortex (PFC), and the updating signals (and updating policy more generally) come from the striatum units (a subset of basal ganglia units). PVLV provides reinforcement learning signals to train up the dynamic gating system in the basal ganglia. === Sensory input and motor output === The sensory input is connected to the posterior cortex which is connected to the motor output. The sensory input is also linked to the PVLV system. === Posterior cortex === The posterior cortex form the hidden layers of the input/output mapping. The PFC is connected with the posterior cortex to contextualize this input/output mapping. === PFC === The PFC (for output gating) has a localist one-to-one representation of the input units for every stripe. Thus, you can look at these PFC representations and see directly what the network is maintaining. The PFC maintains the working memory needed to perform the task. === Striatum === This is the dynamic gating system representing the striatum units of the basal ganglia. Every even-index unit within a stripe represents "Go", while the odd-index units represent "NoGo." The Go units cause updating of the PFC, while the NoGo units cause the PFC to maintain its existing memory representation. There are groups of units for every stripe. In the PBWM model in Emergent, the matrices represent the striatum. === PVLV === All of these layers are part of PVLV system. The PVLV system controls the dopaminergic modulation of the basal ganglia (BG). Thus, BG/PVLV form an actor-critic architecture where the PVLV system learns when to update. ==== SNrThal ==== SNrThal represents the substantia nigra pars reticulata (SNr) and the associated area of the thalamus, which produce a competition among the Go/NoGo units within a given stripe and mediates competition using k-winners-take-all dynamics. If there is more overall Go activity in a given stripe, then the associated SNrThal unit gets activated, and it drives updating in PFC. For every stripe, there is one unit in SNrThal. ==== VTA and SNc ==== Ventral tegmental area (VTA) and substantia nigra pars compacta (SNc) are part of the dopamine layer. This layer models midbrain dopamine neurons. They control the dopaminergic modulation of the basal ganglia.

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  Biomedical data science

  Biomedical data science is a multidisciplinary field which leverages large volumes of data to promote biomedical innovation and discovery. Biomedical data science draws from various fields including Biostatistics, Biomedical informatics, and machine learning, with the goal of understanding biological and medical data. It can be viewed as the study and application of data science to solve biomedical problems. Modern biomedical datasets often have specific features which make their analyses difficult, including: Large numbers of feature (sometimes billions), typically far larger than the number of samples (typically tens or hundreds) Noisy and missing data Privacy concerns (e.g., electronic health record confidentiality) Requirement of interpretability from decision makers and regulatory bodies Many biomedical data science projects apply machine learning to such datasets. These characteristics, while also present in many data science applications more generally, make biomedical data science a specific field. Examples of biomedical data science research include: Computational genomics Computational imaging Electronic health records data mining Biomedical network science Clinical Natural Language Processing (NLP) == Computational Imaging and Deep Learning == Computational imaging is a cornerstone of biomedical data science, focusing on the development of algorithms to enhance, analyze, and interpret medical imagery. In recent years, the field has been transformed by the integration of deep learning, particularly through the use of Convolutional Neural Networks. Deep learning started from researchers manually defining characteristics like edge detection or texture representation learning. In a more modern approach of computational imaging, models automatically learn a hierarchy of features directly from raw pixel data. This overlap between data science and deep learning is applied across several key tasks: Classification: Identifying the presence of specific diseases, such as distinguishing between benign and malignant tumors in histopathology slides or detecting pneumonia in chest X-rays. Segmentation: The precise delineation of anatomical structures or lesions. A notable example is the U-Net architecture, which is widely used for biomedical image segmentation to help clinicians quantify organ volume or track tumor growth. Detection: Automating the localization of small objects, such as identifying microcalcifications in mammograms or polyps during colonoscopies. Registration: The process of aligning multiple images to provide a comprehensive view of the patient's anatomy. Even with all of these enhancements, the application of deep learning in medical imaging requires accomplishing vigorous challenges. An example of these changes is building large, annotated datasets and creating the imperative for model interpretability in clinical decision-making. == Electronic Health Records == Electronic Health Records (EHRs) are a digital alternative to patient paper charts, usually including individual records or population health information. EHRs can be used in a wide variety of applications, including research and analysation as they often include demographics, diagnoses, medications, test results, and personal statistics. === History === ==== 1960s ==== The earliest precursor is considered Dr. Lawrence Weed's problem-oriented medical record (POMR) published in the 1968 which sorts and groups medical records by medical diagnoses and symptoms. The POMR was the first system to organize based off of patient information rather than the source (doctors, nurses, attendings, etc.). In 1969, the Regenstrief Institute developed and published the Regenstrief Medical Record System which established electronic writing, storage, and retrieval of records which served as the basis for modern EHR systems. ==== 2000s ==== In 2009, the Health Information Technology for Economic and Clinical Health Act (HITECH Act) was passed in the United States. This act standardized privacy and distribution of EHRs and increased the acceptance and utilization of EHRs within medical and academic settings. == Artificial Intelligence and Machine Learning Applications == Machine Learning and Artificial Intelligence have become central tools in biomedical data science. Recent advances in large language models (LLMs) have expanded their role beyond text, with models trained directly on genomic sequences enabling tasks such as gene function prediction, variant effect analysis, and drug discovery. In clinical settings, Natural Language Processing (NLP) models are applied to electronic health records to extract structured insights from unstructured clinical notes and data, supporting diagnosis and treatment planning. Beyond genomics, AI models have been applied to protein structure prediction. AlphaFold, developed by Google DeepMind, uses deep learning to predict three-dimensional protein structures from amino acid sequences with high accuracy. These predictions have been used to support drug target identification and the study of disease mechanisms. == Knowledge Graphs == Knowledge graphs (KGs) are widely used in biomedical data science to represent and analyze complex relationships among biological and medical entities. By structuring data as nodes (e.g., genes, diseases, drugs) and edges (relationships), KGs enable computational methods to extract insights and support decision-making. These biomedical relationships can be efficiently modeled and queried using technologies such as Neo4j. === Biomedical Research Applications === KGs provide biomedical researchers with a way to model complex biological systems. They have been used to identify the relationships between diseases and biomolecules, support drug repurposing, and to uncover new biological insights. Additional applications include: Identification of novel antibiotic resistance genes through graph-based link prediction. Finding associations between miRNA and diseases. Prediction of protein-protein interactions. === Clinical Applications === In clinical settings, KGs can be used to make visual representations of a patient's electronic health records. The data obtained from these graphs can assist healthcare providers in improving patient diagnoses and prescribing more effective drugs. Additionally, embeddings derived from resources like the Unified Medical Language System (UMLS) enable natural language processing of clinical text and similarity analysis between medical concepts. === Limitations === Despite their advantages, knowledge graphs face several challenges. Some of these include: High algorithmic complexity and large biological datasets make the process computationally expensive. KG construction can be a time-consuming process that requires careful attention to assign appropriate node types and vocabularies. Using data from a wide range of datasets in one KG requires them to be effectively integrated. == Privacy == A primary challenge in biomedical data science is maintaining medical privacy. Conducting research requires that data be collected on a number of people for training and testing purposes and is stored within biomedical datasets. This poses a risk for violating patient confidentiality and may dissuade people from participating in studies. The main sources of health statistics are surveys administrative and medical records health care claims data, vital records surveillance disease registries grey literature and peer-reviewed literature. Large data collection is a useful tool for researching various medical conditions. Researchers use these large datasets of information to identify factors that may make people more susceptible to certain diseases. Large amounts of collected data can help researchers identify patterns for disease probabilities. The findings can show a person is more likely for a condition, or identify environmental, social, and personal habits that may lead to adverse health issues. Institutions researching using personal medical information come with a moral and legal responsibility to protect the use of that information. Protection of the collected information has become a big concern. Sophisticated and coordinated attacks on certain medical systems happen more frequently. Medical companies, medical insurance and private businesses have invested a great deal into the protection of personal data. Despite this, data breaches continue to be documented. The chart below shows the top healthcare breaches in 2025. For these reasons, many people have reservations about giving up their personal data. Aside from the legitimate use of personal data there have been instances where companies have found methods to profit from brokering medical information. Concerns exist regarding unauthorized use of sensitive information within these data companies. If a person is identified within a dataset, then sensitive data can be used to discriminate against them. For example, insurance companies may charge a hi

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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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  BrownBoost

  BrownBoost is a boosting algorithm that may be robust to noisy datasets. BrownBoost is an adaptive version of the boost by majority algorithm. As is the case for all boosting algorithms, BrownBoost is used in conjunction with other machine learning methods. BrownBoost was introduced by Yoav Freund in 2001. == Motivation == AdaBoost performs well on a variety of datasets; however, it can be shown that AdaBoost does not perform well on noisy data sets. This is a result of AdaBoost's focus on examples that are repeatedly misclassified. In contrast, BrownBoost effectively "gives up" on examples that are repeatedly misclassified. The core assumption of BrownBoost is that noisy examples will be repeatedly mislabeled by the weak hypotheses and non-noisy examples will be correctly labeled frequently enough to not be "given up on." Thus only noisy examples will be "given up on," whereas non-noisy examples will contribute to the final classifier. In turn, if the final classifier is learned from the non-noisy examples, the generalization error of the final classifier may be much better than if learned from noisy and non-noisy examples. The user of the algorithm can set the amount of error to be tolerated in the training set. Thus, if the training set is noisy (say 10% of all examples are assumed to be mislabeled), the booster can be told to accept a 10% error rate. Since the noisy examples may be ignored, only the true examples will contribute to the learning process. == Algorithm description == BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods. The only parameter of BrownBoost ( c {\displaystyle c} in the algorithm) is the "time" the algorithm runs. The theory of BrownBoost states that each hypothesis takes a variable amount of time ( t {\displaystyle t} in the algorithm) which is directly related to the weight given to the hypothesis α {\displaystyle \alpha } . The time parameter in BrownBoost is analogous to the number of iterations T {\displaystyle T} in AdaBoost. A larger value of c {\displaystyle c} means that BrownBoost will treat the data as if it were less noisy and therefore will give up on fewer examples. Conversely, a smaller value of c {\displaystyle c} means that BrownBoost will treat the data as more noisy and give up on more examples. During each iteration of the algorithm, a hypothesis is selected with some advantage over random guessing. The weight of this hypothesis α {\displaystyle \alpha } and the "amount of time passed" t {\displaystyle t} during the iteration are simultaneously solved in a system of two non-linear equations ( 1. uncorrelated hypothesis w.r.t example weights and 2. hold the potential constant) with two unknowns (weight of hypothesis α {\displaystyle \alpha } and time passed t {\displaystyle t} ). This can be solved by bisection (as implemented in the JBoost software package) or Newton's method (as described in the original paper by Freund). Once these equations are solved, the margins of each example ( r i ( x j ) {\displaystyle r_{i}(x_{j})} in the algorithm) and the amount of time remaining s {\displaystyle s} are updated appropriately. This process is repeated until there is no time remaining. The initial potential is defined to be 1 m ∑ j = 1 m 1 − erf ( c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}({\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Since a constraint of each iteration is that the potential be held constant, the final potential is 1 m ∑ j = 1 m 1 − erf ( r i ( x j ) / c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}(r_{i}(x_{j})/{\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Thus the final error is likely to be near 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} . However, the final potential function is not the 0–1 loss error function. For the final error to be exactly 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} , the variance of the loss function must decrease linearly w.r.t. time to form the 0–1 loss function at the end of boosting iterations. This is not yet discussed in the literature and is not in the definition of the algorithm below. The final classifier is a linear combination of weak hypotheses and is evaluated in the same manner as most other boosting algorithms. == BrownBoost learning algorithm definition == Input: m {\displaystyle m} training examples ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x j ∈ X , y j ∈ Y = { − 1 , + 1 } {\displaystyle x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}} The parameter c {\displaystyle c} Initialise: s = c {\displaystyle s=c} . (The value of s {\displaystyle s} is the amount of time remaining in the game) r i ( x j ) = 0 {\displaystyle r_{i}(x_{j})=0} ∀ j {\displaystyle \forall j} . The value of r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin at iteration i {\displaystyle i} for example x j {\displaystyle x_{j}} . While s > 0 {\displaystyle s>0} : Set the weights of each example: W i ( x j ) = e − ( r i ( x j ) + s ) 2 c {\displaystyle W_{i}(x_{j})=e^{-{\frac {(r_{i}(x_{j})+s)^{2}}{c}}}} , where r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin of example x j {\displaystyle x_{j}} Find a classifier h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}} such that ∑ j W i ( x j ) h i ( x j ) y j > 0 {\displaystyle \sum _{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0} Find values α , t {\displaystyle \alpha ,t} that satisfy the equation: ∑ j h i ( x j ) y j e − ( r i ( x j ) + α h i ( x j ) y j + s − t ) 2 c = 0 {\displaystyle \sum _{j}h_{i}(x_{j})y_{j}e^{-{\frac {(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)^{2}}{c}}}=0} . (Note this is similar to the condition E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} set forth by Schapire and Singer. In this setting, we are numerically finding the W i + 1 = exp ⁡ ( ⋯ ⋯ ) {\displaystyle W_{i+1}=\exp \left({\frac {\cdots }{\cdots }}\right)} such that E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} .) This update is subject to the constraint ∑ ( Φ ( r i ( x j ) + α h ( x j ) y j + s − t ) − Φ ( r i ( x j ) + s ) ) = 0 {\displaystyle \sum \left(\Phi \left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi \left(r_{i}(x_{j})+s\right)\right)=0} , where Φ ( z ) = 1 − erf ( z / c ) {\displaystyle \Phi (z)=1-{\mbox{erf}}(z/{\sqrt {c}})} is the potential loss for a point with margin r i ( x j ) {\displaystyle r_{i}(x_{j})} Update the margins for each example: r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j {\displaystyle r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}} Update the time remaining: s = s − t {\displaystyle s=s-t} Output: H ( x ) = sign ( ∑ i α i h i ( x ) ) {\displaystyle H(x)={\textrm {sign}}\left(\sum _{i}\alpha _{i}h_{i}(x)\right)} == Empirical results == In preliminary experimental results with noisy datasets, BrownBoost outperformed AdaBoost's generalization error; however, LogitBoost performed as well as BrownBoost. An implementation of BrownBoost can be found in the open source software JBoost.

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  Multidimensional scaling

  Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n {\textstyle n} objects in a set into a configuration of n {\textstyle n} points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis. == Types == MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: === Classical multidimensional scaling === It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, which is given by Strain D ( x 1 , x 2 , . . . , x n ) = ( ∑ i , j ( b i j − x i T x j ) 2 ∑ i , j b i j 2 ) 1 / 2 , {\displaystyle {\text{Strain}}_{D}(x_{1},x_{2},...,x_{n})={\Biggl (}{\frac {\sum _{i,j}{\bigl (}b_{ij}-x_{i}^{T}x_{j}{\bigr )}^{2}}{\sum _{i,j}b_{ij}^{2}}}{\Biggr )}^{1/2},} where x i {\displaystyle x_{i}} denote vectors in N-dimensional space, x i T x j {\displaystyle x_{i}^{T}x_{j}} denotes the scalar product between x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} , and b i j {\displaystyle b_{ij}} are the elements of the matrix B {\displaystyle B} defined on step 2 of the following algorithm, which are computed from the distances. Steps of a Classical MDS algorithm: Classical MDS uses the fact that the coordinate matrix X {\displaystyle X} can be derived by eigenvalue decomposition from B = X X ′ {\textstyle B=XX'} . And the matrix B {\textstyle B} can be computed from proximity matrix D {\textstyle D} by using double centering. Set up the squared proximity matrix D ( 2 ) = [ d i j 2 ] {\textstyle D^{(2)}=[d_{ij}^{2}]} Apply double centering: B = − 1 2 C D ( 2 ) C {\textstyle B=-{\frac {1}{2}}CD^{(2)}C} using the centering matrix C = I − 1 n J n {\textstyle C=I-{\frac {1}{n}}J_{n}} , where n {\textstyle n} is the number of objects, I {\textstyle I} is the n × n {\textstyle n\times n} identity matrix, and J n {\textstyle J_{n}} is an n × n {\textstyle n\times n} matrix of all ones. Determine the m {\textstyle m} largest eigenvalues λ 1 , λ 2 , . . . , λ m {\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}} and corresponding eigenvectors e 1 , e 2 , . . . , e m {\textstyle e_{1},e_{2},...,e_{m}} of B {\textstyle B} (where m {\textstyle m} is the number of dimensions desired for the output). Now, X = E m Λ m 1 / 2 {\textstyle X=E_{m}\Lambda _{m}^{1/2}} , where E m {\textstyle E_{m}} is the matrix of m {\textstyle m} eigenvectors and Λ m {\textstyle \Lambda _{m}} is the diagonal matrix of m {\textstyle m} eigenvalues of B {\textstyle B} . Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings. === Metric multidimensional scaling (mMDS) === It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares: Stress D ( x 1 , x 2 , . . . , x n ) = ∑ i ≠ j = 1 , . . . , n ( d i j − ‖ x i − x j ‖ ) 2 . {\displaystyle {\text{Stress}}_{D}(x_{1},x_{2},...,x_{n})={\sqrt {\sum _{i\neq j=1,...,n}{\bigl (}d_{ij}-\|x_{i}-x_{j}\|{\bigr )}^{2}}}.} Metric scaling uses a power transformation with a user-controlled exponent p {\textstyle p} : d i j p {\textstyle d_{ij}^{p}} and − d i j 2 p {\textstyle -d_{ij}^{2p}} for distance. In classical scaling p = 1. {\textstyle p=1.} Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities. === Non-metric multidimensional scaling (NMDS) === In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d i j {\displaystyle d_{ij}} be the dissimilarity between points i , j {\displaystyle i,j} . Let d ^ i j = ‖ x i − x j ‖ {\displaystyle {\hat {d}}_{ij}=\|x_{i}-x_{j}\|} be the Euclidean distance between embedded points x i , x j {\displaystyle x_{i},x_{j}} . Now, for each choice of the embedded points x i {\displaystyle x_{i}} and is a monotonically increasing function f {\displaystyle f} , define the "stress" function: S ( x 1 , . . . , x n ; f ) = ∑ i < j ( f ( d i j ) − d ^ i j ) 2 ∑ i < j d ^ i j 2 . {\displaystyle S(x_{1},...,x_{n};f)={\sqrt {\frac {\sum _{i Read more →

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  Machine learning in video games

  Artificial intelligence and machine learning techniques are used in video games for a wide variety of applications such as non-player character (NPC) control, procedural content generation (PCG) and deep learning-based content generation. Machine learning is a subset of artificial intelligence that uses historical data to build predictive and analytical models. This is in sharp contrast to traditional methods of artificial intelligence such as search trees and expert systems. Information on machine learning techniques in the field of games is mostly known to public through research projects as most gaming companies choose not to publish specific information about their intellectual property. The most publicly known application of machine learning in games is likely the use of deep learning agents that compete with professional human players in complex strategy games. There has been a significant application of machine learning on games such as Atari/ALE, Doom, Minecraft, StarCraft, and car racing. Other games that did not originally exists as video games, such as chess and Go have also been affected by the machine learning. == Overview of relevant machine learning techniques == === Deep learning === Deep learning is a subset of machine learning which focuses heavily on the use of artificial neural networks (ANN) that learn to solve complex tasks. Deep learning uses multiple layers of ANN and other techniques to progressively extract information from an input. Due to this complex layered approach, deep learning models often require powerful machines to train and run on. ==== Convolutional neural networks ==== Convolutional neural networks (CNN) are specialized ANNs that are often used to analyze image data. These types of networks are able to learn translation invariant patterns, which are patterns that are not dependent on location. CNNs are able to learn these patterns in a hierarchy, meaning that earlier convolutional layers will learn smaller local patterns while later layers will learn larger patterns based on the previous patterns. A CNN's ability to learn visual data has made it a commonly used tool for deep learning in games. === Recurrent neural network === Recurrent neural networks are a type of ANN that are designed to process sequences of data in order, one part at a time rather than all at once. An RNN runs over each part of a sequence, using the current part of the sequence along with memory of previous parts of the current sequence to produce an output. These types of ANN are highly effective at tasks such as speech recognition and other problems that depend heavily on temporal order. There are several types of RNNs with different internal configurations; the basic implementation suffers from a lack of long term memory due to the vanishing gradient problem, thus it is rarely used over newer implementations. ==== Long short-term memory ==== A long short-term memory (LSTM) network is a specific implementation of a RNN that is designed to deal with the vanishing gradient problem seen in simple RNNs, which would lead to them gradually "forgetting" about previous parts of an inputted sequence when calculating the output of a current part. LSTMs solve this problem with the addition of an elaborate system that uses an additional input/output to keep track of long term data. LSTMs have achieved very strong results across various fields, and were used by several monumental deep learning agents in games. === Reinforcement learning === Reinforcement learning is the process of training an agent using rewards and/or punishments. The way an agent is rewarded or punished depends heavily on the problem; such as giving an agent a positive reward for winning a game or a negative one for losing. Reinforcement learning is used heavily in the field of machine learning and can be seen in methods such as Q-learning, policy search, Deep Q-networks and others. It has seen strong performance in both the field of games and robotics. === Neuroevolution === Neuroevolution involves the use of both neural networks and evolutionary algorithms. Instead of using gradient descent like most neural networks, neuroevolution models make use of evolutionary algorithms to update neurons in the network. Researchers claim that this process is less likely to get stuck in a local minimum and is potentially faster than state of the art deep learning techniques. == Deep learning agents == Machine learning agents have been used to take the place of a human player rather than function as NPCs, which are deliberately added into video games as part of designed gameplay. Deep learning agents have achieved impressive results when used in competition with both humans and other artificial intelligence agents. === Chess === Chess is a turn-based strategy game that is considered a difficult AI problem due to the computational complexity of its board space. Similar strategy games are often solved with some form of a Minimax Tree Search. These types of AI agents have been known to beat professional human players, such as the historic 1997 Deep Blue versus Garry Kasparov match. Since then, machine learning agents have shown ever greater success than previous AI agents. === Go === Go is another turn-based strategy game which is considered an even more difficult AI problem than chess. The state space of is Go is around 10^170 possible board states compared to the 10^120 board states for Chess. Prior to recent deep learning models, AI Go agents were only able to play at the level of a human amateur. ==== AlphaGo ==== Google's 2015 AlphaGo was the first AI agent to beat a professional Go player. AlphaGo used a deep learning model to train the weights of a Monte Carlo tree search (MCTS). The deep learning model consisted of 2 ANN, a policy network to predict the probabilities of potential moves by opponents, and a value network to predict the win chance of a given state. The deep learning model allows the agent to explore potential game states more efficiently than a vanilla MCTS. The network were initially trained on games of humans players and then were further trained by games against itself. ==== AlphaGo Zero ==== AlphaGo Zero, another implementation of AlphaGo, was able to train entirely by playing against itself. It was able to quickly train up to the capabilities of the previous agent. === StarCraft series === StarCraft and its sequel StarCraft II are real-time strategy (RTS) video games that have become popular environments for AI research. Blizzard and DeepMind have worked together to release a public StarCraft 2 environment for AI research to be done on. Various deep learning methods have been tested on both games, though most agents usually have trouble outperforming the default AI with cheats enabled or skilled players of the game. ==== Alphastar ==== Alphastar was the first AI agent to beat professional StarCraft 2 players without any in-game advantages. The deep learning network of the agent initially received input from a simplified zoomed out version of the gamestate, but was later updated to play using a camera like other human players. The developers have not publicly released the code or architecture of their model, but have listed several state of the art machine learning techniques such as relational deep reinforcement learning, long short-term memory, auto-regressive policy heads, pointer networks, and centralized value baseline. Alphastar was initially trained with supervised learning, it watched replays of many human games in order to learn basic strategies. It then trained against different versions of itself and was improved through reinforcement learning. The final version was hugely successful, but only trained to play on a specific map in a protoss mirror matchup. === Dota 2 === Dota 2 is a multiplayer online battle arena (MOBA) game. Like other complex games, traditional AI agents have not been able to compete on the same level as professional human player. The only widely published information on AI agents attempted on Dota 2 is OpenAI's deep learning Five agent. ==== OpenAI Five ==== OpenAI Five utilized separate long short-term memory networks to learn each hero. It trained using a reinforcement learning technique known as Proximal Policy Learning running on a system containing 256 GPUs and 128,000 CPU cores. Five trained for months, accumulating 180 years of game experience each day, before facing off with professional players. It was eventually able to beat the 2018 Dota 2 esports champion team in a 2019 series of games. === Planetary Annihilation === Planetary Annihilation is a real-time strategy game which focuses on massive scale war. The developers use ANNs in their default AI agent. === Supreme Commander 2 === Supreme Commander 2 is a real-time strategy (RTS) video game. The game uses Multilayer Perceptrons (MLPs) to control a platoon’s reaction to encountered enemy units. Total of four MLPs are used, one for each platoon type: land, naval

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  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.

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  Local tangent space alignment

  Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates. It is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned. It begins by computing the k-nearest neighbors of every point. It computes the tangent space at every point by computing the d-first principal components in each local neighborhood. It then optimizes to find an embedding that aligns the tangent spaces, but it ignores the label information conveyed by data samples, and thus can not be used for classification directly.

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  Vapnik–Chervonenkis dimension

  In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (x,x+2]} . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f {\displaystyle f} is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa

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  Pwnie Awards

  The Pwnie Awards are an annual awards ceremony that recognizes both excellence and incompetence in the field of information security, described by SecurityWeek as an event that "recognizes excellence and mocks incompetence in cybersecurity." Winners are selected by a committee of security industry professionals from nominations collected from the information security community. Nominees are announced yearly at Summercon, and the awards themselves are presented at the Black Hat Security Conference. == Origins == The name Pwnie Award is based on the word "pwn", which is hacker slang meaning to "compromise" or "control" based on the previous usage of the word "own" (and it is pronounced similarly). The name "The Pwnie Awards," pronounced as "Pony," is meant to sound like the Tony Awards, an awards ceremony for Broadway theater in New York City. == History == The Pwnie Awards were founded in 2007 by Alexander Sotirov and Dino Dai Zovi following discussions regarding Dino's discovery of a cross-platform QuickTime vulnerability (CVE-2007-2175) and Alexander's discovery of an ANI file processing vulnerability (CVE-2007-0038) in Internet Explorer. == Winners == === 2024 === Most Epic Fail: Crowdstrike for 2024 CrowdStrike incident Best Mobile Bug: Operation Triangulation Lamest Vendor Response: Xiaomi for obstructing Pwn2Own researchers from using their services Best Cryptographic Attack: GoFetch Best Desktop Bug: forcing realtime WebAudio playback in Chrome (CVE-2023-5996) Best Song: Touch Some Grass by UwU Underground Best Privilege Escalation: Windows Streaming Service UAF (CVE-2024-30089) by Valentina Palmiotti (chompie) Best Remote Code Execution: Microsoft Message Queuing (MSMQ) Remote Code Execution Vulnerability (CVE-2024-30080) Most Epic Achievement: Discovery and reverse engineering of the XZ Utils backdoor Most Innovative Research: Let the Cache Cache and Let the WebAssembly Assemble: Knocking’ on Chrome’s Shell by Edouard Bochin, Tao Yan, and Bo Qu Most Underhyped Research: See No Eval: Runtime Dynamic Code Execution in Objective-C === 2023 === Best Desktop Bug: CountExposure! by RyeLv(@b2ahex) Best Cryptographic Attack: Video-based cryptanalysis: Extracting Cryptographic Keys from Video Footage of a Device’s Power LED by Ben Nassi, Etay Iluz, Or Cohen, Ofek Vayner, Dudi Nassi, Boris Zadov, Yuval Elovici Best Song: Clickin’ Most Innovative Research: Inside Apple’s Lightning: Jtagging the iPhone for Fuzzing and Profit Most Under-Hyped Research: Activation Context Cache Poisoning Best Privilege Escalation Bug: URB Excalibur: Slicing Through the Gordian Knot of VMware VM Escapes Best Remote Code Execution Bug: ClamAV RCE Lamest Vendor Response: Three Lessons From Threema: Analysis of a Secure Messenger Most Epic Fail: “Holy fucking bingle, we have the no fly list,” Epic Achievement: Clement Lecigne: 0-days hunter world champion Lifetime Achievement Award: Mudge === 2022 === Lamest Vendor Response: Google's "TAG" response team for "unilaterally shutting down a counterterrorism operation." Epic Achievement: Yuki Chen’s Windows Server-Side RCE Bugs Most Epic Fail: HackerOne Employee Caught Stealing Vulnerability Reports for Personal Gains Best Desktop Bug: Pietro Borrello, Andreas Kogler, Martin Schwarzl, Moritz Lipp, Daniel Gruss, Michael Schwarz for Architecturally Leaking Data from the Microarchitecture Most Innovative Research: Pietro Borrello, Martin Schwarzl, Moritz Lipp, Daniel Gruss, Michael Schwarz for Custom Processing Unit: Tracing and Patching Intel Atom Microcode Best Cryptographic Attack: Hertzbleed: Turning Power Side-Channel Attacks Into Remote Timing Attacks on x86 by Yingchen Wang, Riccardo Paccagnella, Elizabeth Tang He, Hovav Shacham, Christopher Fletcher, David Kohlbrenner Best Remote Code Execution Bug: KunlunLab for Windows RPC Runtime Remote Code Execution (CVE-2022-26809) Best Privilege Escalation Bug: Qidan He of Dawnslab, for Mystique in the House: The Droid Vulnerability Chain That Owns All Your Userspace Best Mobile Bug: FORCEDENTRY Most Under-Hyped Research: Yannay Livneh for Spoofing IP with IPIP Best Song: Dialed Up by Project Mammoth === 2021 === Lamest Vendor Response: Cellebrite, for their response to Moxie, the creator of Signal, reverse-engineering their UFED and accompanying software and reporting a discovered exploit. Epic Achievement: Ilfak Guilfanov, in honor of IDA's 30th Anniversary. Best Privilege Escalation Bug: Baron Samedit of Qualys, for the discovery of a 10-year-old exploit in sudo. Best Song: The Ransomware Song by Forrest Brazeal Best Server-Side Bug: Orange Tsai, for his Microsoft Exchange Server ProxyLogon attack surface discoveries. Best Cryptographic Attack: The NSA for its disclosure of a bug in the verification of signatures in Windows which breaks the certificate trust chain. Most Innovative Research: Enes Göktaş, Kaveh Razavi, Georgios Portokalidis, Herbert Bos, and Cristiano Giuffrida at VUSec for their research on the "BlindSide" Attack. Most Epic Fail: Microsoft, for their failure to fix PrintNightmare. Best Client-Side Bug: Gunnar Alendal's discovery of a buffer overflow on the Samsung Galaxy S20's secure chip. Most Under-Hyped Research: The Qualys Research Team for 21Nails, 21 vulnerabilities in Exim, the Internet's most popular mail server. === 2020 === Best Server-Side Bug: BraveStarr (CVE-2020-10188) – A Fedora 31 netkit telnetd remote exploit (Ronald Huizer') Best Privilege Escalation Bug: checkm8 – A permanent unpatchable USB bootrom exploit for a billion iOS devices. (axi0mX) Epic Achievement: "Remotely Rooting Modern Android Devices" (Guang Gong) Best Cryptographic Attack: Zerologon vulnerability (Tom Tervoort, CVE-2020-1472) Best Client-Side Bug: RCE on Samsung Phones via MMS (CVE-2020-8899 and -16747), a zero click remote execution attack. (Mateusz Jurczyk) Most Under-Hyped Research: Vulnerabilities in System Management Mode (SMM) and Trusted Execution Technology (TXT) (CVE-2019-0151 and -0152) (Gabriel Negreira Barbosa, Rodrigo Rubira Branco, Joe Cihula) Most Innovative Research: TRRespass: When Memory Vendors Tell You Their Chips Are Rowhammer-free, They Are Not. (Pietro Frigo, Emanuele Vannacci, Hasan Hassan, Victor van der Veen, Onur Mutlu, Cristiano Giuffrida, Herbert Bos, Kaveh Razavi) Most Epic Fail: Microsoft; for the implementation of Elliptic-curve signatures which allowed attackers to generate private pairs for public keys of any signer, allowing HTTPS and signed binary spoofing. (CVE-2020-0601) Best Song: Powertrace by Rebekka Aigner, Daniel Gruss, Manuel Weber, Moritz Lipp, Patrick Radkohl, Andreas Kogler, Maria Eichlseder, ElTonno, tunefish, Yuki and Kater Lamest Vendor Response: Daniel J. Bernstein (CVE-2005-1513) === 2019 === Best Server-Side Bug: Orange Tsai and Meh Chang, for their SSL VPN research. Most Innovative Research: Vectorized Emulation Brandon Falk Best Cryptographic Attack: \m/ Dr4g0nbl00d \m/ Mathy Vanhoef, Eyal Ronen Lamest Vendor Response: Bitfi Most Over-hyped Bug: Allegations of Supermicro hardware backdoors, Bloomberg Most Under-hyped Bug: Thrangrycat, (Jatin Kataria, Red Balloon Security) === 2018 === Most Innovative Research: Spectre/Meltdown (Paul Kocher, Jann Horn, Anders Fogh, Daniel Genkin, Daniel Gruss, Werner Haas, Mike Hamburg, Moritz Lipp, Stefan Mangard, Thomas Prescher, Michael Schwarz, Yuval Yarom) Best Privilege Escalation Bug: Spectre/Meltdown (Paul Kocher, Jann Horn, Anders Fogh, Daniel Genkin, Daniel Gruss, Werner Haas, Mike Hamburg, Moritz Lipp, Stefan Mangard, Thomas Prescher, Michael Schwarz, Yuval Yarom) Lifetime Achievement: Michał Zalewski Best Cryptographic Attack: ROBOT - Return Of Bleichenbacher’s Oracle Threat Hanno Böck, Juraj Somorovsky, Craig Young Lamest Vendor Response: Bitfi hardware crypto-wallet, after the "unhackable" device was hacked to extract the keys required to steal coins and rooted to play Doom. === 2017 === Epic Achievement: Federico Bento for Finally getting TIOCSTI ioctl attack fixed Most Innovative Research: ASLR on the line Ben Gras, Kaveh Razavi, Erik Bosman, Herbert Bos, Cristiano Giuffrida Best Privilege Escalation Bug: DRAMMER Victor van der Veen, Yanick Fratantonio, Martina Lindorfer, Daniel Gruss, Clementine Maurice, Giovanni Vigna, Herbert Bos, Kaveh Razavi, Cristiano Giuffrida Best Cryptographic Attack: The first collision for full SHA-1 Marc Stevens, Elie Bursztein, Pierre Karpman, Ange Albertini, Yarik Markov Lamest Vendor Response: Lennart Poettering - for mishandling security vulnerabilities most spectacularly for multiple critical Systemd bugs Best Song: Hello (From the Other Side) - Manuel Weber, Michael Schwarz, Daniel Gruss, Moritz Lipp, Rebekka Aigner === 2016 === Most Innovative Research: Dedup Est Machina: Memory Deduplication as an Advanced Exploitation Vector Erik Bosman, Kaveh Razavi, Herbert Bos, Cristiano Giuffrida Lifetime Achievement: Peiter Zatko aka Mudge Best Cryptographic Attack: DROWN attack Nimrod Aviram et al. Best Song: Cyberlier - Katie Mous

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  Influence diagram

  An influence diagram (ID) (also called a relevance diagram, decision diagram or a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems (following the maximum expected utility criterion) can be modeled and solved. ID was first developed in the mid-1970s by decision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to the decision tree which typically suffers from exponential growth in number of branches with each variable modeled. ID is directly applicable in team decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extensions of ID also find their use in game theory as an alternative representation of the game tree. == Semantics == An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (or arrow) between nodes. Nodes: Decision node (corresponding to each decision to be made) is drawn as a rectangle. Uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval. Deterministic node (corresponding to special kind of uncertainty that its outcome is deterministically known whenever the outcome of some other uncertainties are also known) is drawn as a double oval. Value node (corresponding to each component of additively separable Von Neumann-Morgenstern utility function) is drawn as an octagon (or diamond). Arcs: Functional arcs (ending in value node) indicate that one of the components of additively separable utility function is a function of all the nodes at their tails. Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads is probabilistically conditioned on all the nodes at their tails. Conditional arcs (ending in deterministic node) indicate that the uncertainty at their heads is deterministically conditioned on all the nodes at their tails. Informational arcs (ending in decision node) indicate that the decision at their heads is made with the outcome of all the nodes at their tails known beforehand. Given a properly structured ID: Decision nodes and incoming information arcs collectively state the alternatives (what can be done when the outcome of certain decisions and/or uncertainties are known beforehand) Uncertainty/deterministic nodes and incoming conditional arcs collectively model the information (what are known and their probabilistic/deterministic relationships) Value nodes and incoming functional arcs collectively quantify the preference (how things are preferred over one another). Alternative, information, and preference are termed decision basis in decision analysis, they represent three required components of any valid decision situation. Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram. The specification is defined by the d {\displaystyle d} -separation criterion of Bayesian network. According to this semantic, every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes. Likewise, a missing arc between non-value node X {\displaystyle X} and non-value node Y {\displaystyle Y} implies that there exists a set of non-value nodes Z {\displaystyle Z} , e.g., the parents of Y {\displaystyle Y} , that renders Y {\displaystyle Y} independent of X {\displaystyle X} given the outcome of the nodes in Z {\displaystyle Z} . == Example == Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation. There is 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction). There are 2 functional arcs (ending in Satisfaction), 1 conditional arc (ending in Weather Forecast), and 1 informational arc (ending in Vacation Activity). Functional arcs ending in Satisfaction indicate that Satisfaction is a utility function of Weather Condition and Vacation Activity. In other words, their satisfaction can be quantified if they know what the weather is like and what their choice of activity is. (Note that they do not value Weather Forecast directly) Conditional arc ending in Weather Forecast indicates their belief that Weather Forecast and Weather Condition can be dependent. Informational arc ending in Vacation Activity indicates that they will only know Weather Forecast, not Weather Condition, when making their choice. In other words, actual weather will be known after they make their choice, and only forecast is what they can count on at this stage. It also follows semantically, for example, that Vacation Activity is independent on (irrelevant to) Weather Condition given Weather Forecast is known. == Applicability to value of information == The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as the value of information. Consider the following three scenarios; Scenario 1: The decision-maker could make their Vacation Activity decision while knowing what Weather Condition will be like. This corresponds to adding extra informational arc from Weather Condition to Vacation Activity in the above influence diagram. Scenario 2: The original influence diagram as shown above. Scenario 3: The decision-maker makes their decision without even knowing the Weather Forecast. This corresponds to removing informational arc from Weather Forecast to Vacation Activity in the above influence diagram. Scenario 1 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about (Weather Condition) when making their decision. Scenario 3, however, is the worst possible scenario for this decision situation since they need to make their decision without any hint (Weather Forecast) on what they care about (Weather Condition) will turn out to be. The decision-maker is usually better off (definitely no worse off, on average) to move from scenario 3 to scenario 2 through the acquisition of new information. The most they should be willing to pay for such move is called the value of information on Weather Forecast, which is essentially the value of imperfect information on Weather Condition. The applicability of this simple ID and the value of information concept is tremendous, especially in medical decision making when most decisions have to be made with imperfect information about their patients, diseases, etc. == Related concepts == Influence diagrams are hierarchical and can be defined either in terms of their structure or in greater detail in terms of the functional and numerical relation between diagram elements. An ID that is consistently defined at all levels—structure, function, and number—is a well-defined mathematical representation and is referred to as a well-formed influence diagram (WFID). WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence researchers concerning Bayesian network inference (belief propagation). An influence diagram having only uncertainty nodes (i.e., a Bayesian network) is also called a relevance diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A" (i.e., relevance is a symmetric relationship).

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  Multidimensional scaling

  Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n {\textstyle n} objects in a set into a configuration of n {\textstyle n} points mapped into an abstract Cartesian space. More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction. Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, N, an MDS algorithm places each object into N-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For N = 1, 2, and 3, the resulting points can be visualized on a scatter plot. Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis. == Types == MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: === Classical multidimensional scaling === It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, which is given by Strain D ( x 1 , x 2 , . . . , x n ) = ( ∑ i , j ( b i j − x i T x j ) 2 ∑ i , j b i j 2 ) 1 / 2 , {\displaystyle {\text{Strain}}_{D}(x_{1},x_{2},...,x_{n})={\Biggl (}{\frac {\sum _{i,j}{\bigl (}b_{ij}-x_{i}^{T}x_{j}{\bigr )}^{2}}{\sum _{i,j}b_{ij}^{2}}}{\Biggr )}^{1/2},} where x i {\displaystyle x_{i}} denote vectors in N-dimensional space, x i T x j {\displaystyle x_{i}^{T}x_{j}} denotes the scalar product between x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} , and b i j {\displaystyle b_{ij}} are the elements of the matrix B {\displaystyle B} defined on step 2 of the following algorithm, which are computed from the distances. Steps of a Classical MDS algorithm: Classical MDS uses the fact that the coordinate matrix X {\displaystyle X} can be derived by eigenvalue decomposition from B = X X ′ {\textstyle B=XX'} . And the matrix B {\textstyle B} can be computed from proximity matrix D {\textstyle D} by using double centering. Set up the squared proximity matrix D ( 2 ) = [ d i j 2 ] {\textstyle D^{(2)}=[d_{ij}^{2}]} Apply double centering: B = − 1 2 C D ( 2 ) C {\textstyle B=-{\frac {1}{2}}CD^{(2)}C} using the centering matrix C = I − 1 n J n {\textstyle C=I-{\frac {1}{n}}J_{n}} , where n {\textstyle n} is the number of objects, I {\textstyle I} is the n × n {\textstyle n\times n} identity matrix, and J n {\textstyle J_{n}} is an n × n {\textstyle n\times n} matrix of all ones. Determine the m {\textstyle m} largest eigenvalues λ 1 , λ 2 , . . . , λ m {\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}} and corresponding eigenvectors e 1 , e 2 , . . . , e m {\textstyle e_{1},e_{2},...,e_{m}} of B {\textstyle B} (where m {\textstyle m} is the number of dimensions desired for the output). Now, X = E m Λ m 1 / 2 {\textstyle X=E_{m}\Lambda _{m}^{1/2}} , where E m {\textstyle E_{m}} is the matrix of m {\textstyle m} eigenvectors and Λ m {\textstyle \Lambda _{m}} is the diagonal matrix of m {\textstyle m} eigenvalues of B {\textstyle B} . Classical MDS assumes metric distances. So this is not applicable for direct dissimilarity ratings. === Metric multidimensional scaling (mMDS) === It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called stress, which is often minimized using a procedure called stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares: Stress D ( x 1 , x 2 , . . . , x n ) = ∑ i ≠ j = 1 , . . . , n ( d i j − ‖ x i − x j ‖ ) 2 . {\displaystyle {\text{Stress}}_{D}(x_{1},x_{2},...,x_{n})={\sqrt {\sum _{i\neq j=1,...,n}{\bigl (}d_{ij}-\|x_{i}-x_{j}\|{\bigr )}^{2}}}.} Metric scaling uses a power transformation with a user-controlled exponent p {\textstyle p} : d i j p {\textstyle d_{ij}^{p}} and − d i j 2 p {\textstyle -d_{ij}^{2p}} for distance. In classical scaling p = 1. {\textstyle p=1.} Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities. === Non-metric multidimensional scaling (NMDS) === In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. Let d i j {\displaystyle d_{ij}} be the dissimilarity between points i , j {\displaystyle i,j} . Let d ^ i j = ‖ x i − x j ‖ {\displaystyle {\hat {d}}_{ij}=\|x_{i}-x_{j}\|} be the Euclidean distance between embedded points x i , x j {\displaystyle x_{i},x_{j}} . Now, for each choice of the embedded points x i {\displaystyle x_{i}} and is a monotonically increasing function f {\displaystyle f} , define the "stress" function: S ( x 1 , . . . , x n ; f ) = ∑ i < j ( f ( d i j ) − d ^ i j ) 2 ∑ i < j d ^ i j 2 . {\displaystyle S(x_{1},...,x_{n};f)={\sqrt {\frac {\sum _{i Read more →