A screen generator, also known as a screen painter, screen mapper, or forms generator is a software package (or component thereof) which enables data entry screens to be generated declaratively, by "painting" them on the screen WYSIWYG-style, or through filling-in forms, rather than requiring writing of code to display them manually. 4GLs commonly incorporate a screen generator feature. They are also commonly found bundled with database systems, especially entry-level databases. A screen generator is one aspect of an application generator, which can also include other functions such as report generation and a data dictionary. The earliest screen generators were character-based; by the 1990s, GUI support became common, and then support for generating HTML forms as well. Some screen generators work by generating code to display the screen in a high-level language (for example, COBOL); others store the screen definition in a data file or in database tables, and then have a runtime component responsible for actually displaying the form and receiving and validating user input. == Examples == Examples of screen generators include: IBM Screen Definition Facility II: generates screens for CICS BMS, IMS MFS, ISPF, GDDM and CSP/AD. Performix for Informix. Microsoft Visual Basic the forms component of Microsoft Access Oracle Developer, in particular its Oracle Forms component the QDesign component of PowerHouse SystemBuilder/SB+ the Screen Painter component of SAP's ABAP Workbench the FoxView component of FoxPro. FoxView was originally developed by Luis Castro as a dBASE screen generator named ViewGen; Fox purchased it and bundled it with FoxPro 1.0. Later, Fox replaced Castro's code with their own screen painter code. dBASE included a built-in screen generator in dBASE IV onwards; in dBASE III and earlier, third party screen generators were available, including the already mentioned ViewGen DPS 1100 for UNIVAC 1100 series mainframes.
Machine-learned interatomic potential
Machine-learned interatomic potentials (MLIPs), or simply machine learning potentials (MLPs), are interatomic potentials constructed using machine learning. Beginning in the 1990s, researchers have employed such programs to construct interatomic potentials by mapping atomic structures to their potential energies. These potentials are referred to as MLIPs or MLPs. Such machine learning potentials promised to fill the gap between density functional theory, a highly accurate but computationally intensive modelling method, and empirically derived or intuitively-approximated potentials, which were far lighter computationally but substantially less accurate. Improvements in artificial intelligence technology heightened the accuracy of MLPs while lowering their computational cost, increasing the role of machine learning in fitting potentials. Machine learning potentials began by using neural networks to tackle low-dimensional systems. While promising, these models could not systematically account for interatomic energy interactions; they could be applied to small molecules in a vacuum, or molecules interacting with frozen surfaces, but not much else – and even in these applications, the models often relied on force fields or potentials derived empirically or with simulations. These models thus remained confined to academia. Modern neural networks construct highly accurate and computationally light potentials, as theoretical understanding of materials science was increasingly built into their architectures and preprocessing. Almost all are local, accounting for all interactions between an atom and its neighbor up to some cutoff radius. There exist some nonlocal models, but these have been experimental for almost a decade. For most systems, reasonable cutoff radii enable highly accurate results. Almost all neural networks intake atomic coordinates and output potential energies. For some, these atomic coordinates are converted into atom-centered symmetry functions. From this data, a separate atomic neural network is trained for each element; each atomic network is evaluated whenever that element occurs in the given structure, and then the results are pooled together at the end. This process – in particular, the atom-centered symmetry functions which convey translational, rotational, and permutational invariances – has greatly improved machine learning potentials by significantly constraining the neural network search space. Other models use a similar process but emphasize bonds over atoms, using pair symmetry functions and training one network per atom pair. Other models to learn their own descriptors rather than using predetermined symmetry-dictating functions. These models, called message-passing neural networks (MPNNs), are graph neural networks. Treating molecules as three-dimensional graphs (where atoms are nodes and bonds are edges), the model takes feature vectors describing the atoms as input, and iteratively updates these vectors as information about neighboring atoms is processed through message functions and convolutions. These feature vectors are then used to predict the final potentials. The flexibility of this method often results in stronger, more generalizable models. In 2017, the first-ever MPNN model (a deep tensor neural network) was used to calculate the properties of small organic molecules. == Gaussian Approximation Potential (GAP) == One popular class of machine-learned interatomic potential is the Gaussian Approximation Potential (GAP), which combines compact descriptors of local atomic environments with Gaussian process regression to machine learn the potential energy surface of a given system. To date, the GAP framework has been used to successfully develop a number of MLIPs for various systems, including for elemental systems such as carbon, silicon, phosphorus, and tungsten, as well as for multicomponent systems such as Ge2Sb2Te5 and austenitic stainless steel, Fe7Cr2Ni. == Equivariant graph neural networks == A significant limitation of early MPNNs was that they were not inherently equivariant to rotations and reflections of atomic structures — meaning predictions could change depending on how a molecule was oriented in space. Beginning around 2021, a new class of models addressed this by incorporating equivariance directly into the message-passing layers using spherical harmonics and irreducible representations. Notable examples include NequIP (2021), MACE (2022), and GemNet-OC (2022). These equivariant architectures proved substantially more data-efficient and accurate than their predecessors, and became the dominant paradigm for high-accuracy MLIPs. == Universal MLIPs and large-scale datasets == Early MLIPs were system-specific, trained on a few thousand structures of a single material. A major shift occurred with the creation of large, chemically diverse datasets enabling models that generalize across many elements, bonding environments, and application domains — so-called universal MLIPs. A key driver was the Open Catalyst Project (OC20, OC22), a collaboration between Meta AI (FAIR) and Carnegie Mellon University launched in 2020. OC20 comprises approximately 1.3 million DFT relaxations across 82 elements, designed to accelerate the discovery of catalysts for renewable energy applications. It was among the first datasets large enough to train GNNs that generalize across diverse chemical systems, and established a widely-used benchmark for the field. A subsequent dataset, Open Direct Air Capture (OpenDAC 2023 and OpenDAC 2025), applied the same approach to carbon capture, providing a large computational database of metal-organic frameworks and sorbent candidates evaluated for CO₂ capture, generated using nearly 400 million CPU hours of quantum chemistry calculations in collaboration with Georgia Tech. These datasets revealed a new challenge: the GNN architectures most effective for atomic simulations were memory-intensive, as they model higher-order interactions between triplets or quadruplets of atoms, making it difficult to scale model size. Graph Parallelism, introduced by Sriram et al. (ICLR 2022), addressed this by distributing a single input graph across multiple GPUs — a distinct strategy from data parallelism (which distributes training examples) or model parallelism (which distributes layers). This enabled training GNNs with hundreds of millions to billions of parameters for the first time. Building on these foundations, Meta FAIR released the Universal Model for Atoms (UMA) in 2025, trained on approximately 500 million unique 3D atomic structures spanning molecules, materials, and catalysts — the largest training run to date for an MLIP. UMA introduced a Mixture of Linear Experts (MoLE) architecture, enabling one model to learn from datasets generated by different DFT codes and settings without significant inference overhead. It matches or surpasses specialized models across catalysis, materials, and molecular benchmarks without task-specific fine-tuning, and has been described as marking a "pre/post-UMA" divide in the field. == Applications == Catalyst discovery: MLIPs have significantly accelerated the computational screening of heterogeneous catalysts by replacing expensive DFT relaxations with fast neural network surrogates. The Open Catalyst Project explicitly targets this application, aiming to identify new catalysts for green hydrogen production and other renewable energy reactions. Carbon capture: The OpenDAC project applies universal MLIPs to screening sorbent materials for direct air capture of CO₂, a key technology for climate change mitigation. AI-accelerated screening allows evaluation of orders of magnitude more candidate materials than traditional DFT workflows. Drug discovery and molecular design: MLIPs are increasingly used in pharmaceutical research to model molecular conformations and binding energies. The Open Molecules 2025 (OMol25) dataset, released by Meta FAIR in 2025, provides high-accuracy calculations for a large set of molecular systems to support this use case. Materials discovery: Universal MLIPs enable high-throughput screening of novel inorganic materials, including battery electrolytes, semiconductors, and superconductors, by rapidly estimating stability and properties across large chemical spaces.
Farthest-first traversal
In computational geometry, the farthest-first traversal of a compact metric space is a sequence of points in the space, where the first point is selected arbitrarily and each successive point is as far as possible from the set of previously-selected points. The same concept can also be applied to a finite set of geometric points, by restricting the selected points to belong to the set or equivalently by considering the finite metric space generated by these points. For a finite metric space or finite set of geometric points, the resulting sequence forms a permutation of the points, also known as the greedy permutation. Every prefix of a farthest-first traversal provides a set of points that is widely spaced and close to all remaining points. More precisely, no other set of equally many points can be spaced more than twice as widely, and no other set of equally many points can be less than half as far to its farthest remaining point. In part because of these properties, farthest-point traversals have many applications, including the approximation of the traveling salesman problem and the metric k-center problem. They may be constructed in polynomial time, or (for low-dimensional Euclidean spaces) approximated in near-linear time. == Definition and properties == A farthest-first traversal is a sequence of points in a compact metric space, with each point appearing at most once. If the space is finite, each point appears exactly once, and the traversal is a permutation of all of the points in the space. The first point of the sequence may be any point in the space. Each point p after the first must have the maximum possible distance to the set of points earlier than p in the sequence, where the distance from a point to a set is defined as the minimum of the pairwise distances to points in the set. A given space may have many different farthest-first traversals, depending both on the choice of the first point in the sequence (which may be any point in the space) and on ties for the maximum distance among later choices. Farthest-point traversals may be characterized by the following properties. Fix a number k, and consider the prefix formed by the first k points of the farthest-first traversal of any metric space. Let r be the distance between the final point of the prefix and the other points in the prefix. Then this subset has the following two properties: All pairs of the selected points are at distance at least r from each other, and All points of the metric space are at distance at most r from the subset. Conversely any sequence having these properties, for all choices of k, must be a farthest-first traversal. These are the two defining properties of a Delone set, so each prefix of the farthest-first traversal forms a Delone set. == Applications == Rosenkrantz, Stearns & Lewis (1977) used the farthest-first traversal to define the farthest-insertion heuristic for the travelling salesman problem. This heuristic finds approximate solutions to the travelling salesman problem by building up a tour on a subset of points, adding one point at a time to the tour in the ordering given by a farthest-first traversal. To add each point to the tour, one edge of the previous tour is broken and replaced by a pair of edges through the added point, in the cheapest possible way. Although Rosenkrantz et al. prove only a logarithmic approximation ratio for this method, they show that in practice it often works better than other insertion methods with better provable approximation ratios. Later, the same sequence of points was popularized by Gonzalez (1985), who used it as part of greedy approximation algorithms for two problems in clustering, in which the goal is to partition a set of points into k clusters. One of the two problems that Gonzalez solve in this way seeks to minimize the maximum diameter of a cluster, while the other, known as the metric k-center problem, seeks to minimize the maximum radius, the distance from a chosen central point of a cluster to the farthest point from it in the same cluster. For instance, the k-center problem can be used to model the placement of fire stations within a city, in order to ensure that every address within the city can be reached quickly by a fire truck. For both clustering problems, Gonzalez chooses a set of k cluster centers by selecting the first k points of a farthest-first traversal, and then creates clusters by assigning each input point to the nearest cluster center. If r is the distance from the set of k selected centers to the next point at position k + 1 in the traversal, then with this clustering every point is within distance r of its center and every cluster has diameter at most 2r. However, the subset of k centers together with the next point are all at distance at least r from each other, and any k-clustering would put some two of these points into a single cluster, with one of them at distance at least r/2 from its center and with diameter at least r. Thus, Gonzalez's heuristic gives an approximation ratio of 2 for both clustering problems. Gonzalez's heuristic was independently rediscovered for the metric k-center problem by Dyer & Frieze (1985), who applied it more generally to weighted k-center problems. Another paper on the k-center problem from the same time, Hochbaum & Shmoys (1985), achieves the same approximation ratio of 2, but its techniques are different. Nevertheless, Gonzalez's heuristic, and the name "farthest-first traversal", are often incorrectly attributed to Hochbaum and Shmoys. For both the min-max diameter clustering problem and the metric k-center problem, these approximations are optimal: the existence of a polynomial-time heuristic with any constant approximation ratio less than 2 would imply that P = NP. As well as for clustering, the farthest-first traversal can also be used in another type of facility location problem, the max-min facility dispersion problem, in which the goal is to choose the locations of k different facilities so that they are as far apart from each other as possible. More precisely, the goal in this problem is to choose k points from a given metric space or a given set of candidate points, in such a way as to maximize the minimum pairwise distance between the selected points. Again, this can be approximated by choosing the first k points of a farthest-first traversal. If r denotes the distance of the kth point from all previous points, then every point of the metric space or the candidate set is within distance r of the first k − 1 points. By the pigeonhole principle, some two points of the optimal solution (whatever it is) must both be within distance r of the same point among these first k − 1 chosen points, and (by the triangle inequality) within distance 2r of each other. Therefore, the heuristic solution given by the farthest-first traversal is within a factor of two of optimal. Other applications of the farthest-first traversal include color quantization (clustering the colors in an image to a smaller set of representative colors), progressive scanning of images (choosing an order to display the pixels of an image so that prefixes of the ordering produce good lower-resolution versions of the whole image rather than filling in the image from top to bottom), point selection in the probabilistic roadmap method for motion planning, simplification of point clouds, generating masks for halftone images, hierarchical clustering, finding the similarities between polygon meshes of similar surfaces, choosing diverse and high-value observation targets for underwater robot exploration, fault detection in sensor networks, modeling phylogenetic diversity, matching vehicles in a heterogenous fleet to customer delivery requests, uniform distribution of geodetic observatories on the Earth's surface or of other types of sensor network, generation of virtual point lights in the instant radiosity computer graphics rendering method, and geometric range searching data structures. == Algorithms == === Greedy exact algorithm === The farthest-first traversal of a finite point set may be computed by a greedy algorithm that maintains the distance of each point from the previously selected points, performing the following steps: Initialize the sequence of selected points to the empty sequence, and the distances of each point to the selected points to infinity. While not all points have been selected, repeat the following steps: Scan the list of not-yet-selected points to find a point p that has the maximum distance from the selected points. Remove p from the not-yet-selected points and add it to the end of the sequence of selected points. For each remaining not-yet-selected point q, replace the distance stored for q by the minimum of its old value and the distance from p to q. For a set of n points, this algorithm takes O(n2) steps and O(n2) distance computations. === Approximations === A faster approximation algorithm, given by Har-Peled & Mendel (2006), applie
Multiclass classification
In machine learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification). For example, deciding on whether an image is showing a banana, peach, orange, or an apple is a multiclass classification problem, with four possible classes (banana, peach, orange, apple), while deciding on whether an image contains an apple or not is a binary classification problem (with the two possible classes being: apple, no apple). While many classification algorithms (e.g., decision trees, k-NN, neural networks and multinomial logistic regression) naturally permit the use of more than two classes, some are by nature binary algorithms (e.g., classical binary support vector machine) and require decomposition strategies such as one-vs-all, one-vs-one, or ECOC to solve multiclass problems. Multiclass classification should not be confused with multi-label classification, where multiple labels are to be predicted for each instance (e.g., predicting that an image contains both an apple and an orange, in the previous example). == Better-than-random multiclass models == From the confusion matrix of a multiclass model, we can determine whether a model does better than chance. Let K ≥ 3 {\displaystyle K\geq 3} be the number of classes, O {\displaystyle {\mathcal {O}}} a set of observations, y ^ : O → { 1 , . . . , K } {\displaystyle {\hat {y}}:{\mathcal {O}}\to \{1,...,K\}} a model of the target variable y : O → { 1 , . . . , K } {\displaystyle y:{\mathcal {O}}\to \{1,...,K\}} and n i , j {\displaystyle n_{i,j}} be the number of observations in the set { y = i } ∩ { y ^ = j } {\displaystyle \{y=i\}\cap \{{\hat {y}}=j\}} . We note n i . = ∑ j n i , j {\displaystyle n_{i.}=\sum _{j}n_{i,j}} , n . j = ∑ i n i , j {\displaystyle n_{.j}=\sum _{i}n_{i,j}} , n = ∑ j n . j = ∑ i n i . {\displaystyle n=\sum _{j}n_{.j}=\sum _{i}n_{i.}} , λ i = n i . n {\displaystyle \lambda _{i}={\frac {n_{i.}}{n}}} and μ j = n . j n {\displaystyle \mu _{j}={\frac {n_{.j}}{n}}} . It is assumed that the confusion matrix ( n i , j ) i , j {\displaystyle (n_{i,j})_{i,j}} contains at least one non-zero entry in each row, that is λ i > 0 {\displaystyle \lambda _{i}>0} for any i {\displaystyle i} . Finally we call "normalized confusion matrix" the matrix of conditional probabilities ( P ( y ^ = j ∣ y = i ) ) i , j = ( n i , j n i . ) i , j {\displaystyle (\mathbb {P} ({\hat {y}}=j\mid y=i))_{i,j}=\left({\frac {n_{i,j}}{n_{i.}}}\right)_{i,j}} . === Intuitive explanation === The lift is a way of measuring the deviation from independence of two events A {\displaystyle A} and B {\displaystyle B} : L i f t ( A , B ) = P ( A ∩ B ) P ( A ) P ( B ) = P ( A ∣ B ) P ( A ) = P ( B ∣ A ) P ( B ) {\displaystyle \mathrm {Lift} (A,B)={\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (A)\mathbb {P} (B)}}={\frac {\mathbb {P} (A\mid B)}{\mathbb {P} (A)}}={\frac {\mathbb {P} (B\mid A)}{\mathbb {P} (B)}}} We have L i f t ( A , B ) > 1 {\displaystyle \mathrm {Lift} (A,B)>1} if and only if events A {\displaystyle A} and B {\displaystyle B} occur simultaneously with a greater probability than if they were independent. In other words, if one of the two events occurs, the probability of observing the other event increases. A first condition to satisfy is to have L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} . And the quality of a model (better or worse than chance) does not change if we over- or undersample the dataset, that is if we multiply each row R i {\displaystyle R_{i}} of the confusion matrix by a constant c i {\displaystyle c_{i}} . Thus the second condition is that the necessary and sufficient conditions for doing better than chance need only depend on the normalized confusion matrix. The condition on lifts can be reformulated with One versus Rest binary models : for any i {\displaystyle i} , we define the binary target variable y i {\displaystyle y_{i}} which is the indicator of event { y = i } {\displaystyle \{y=i\}} , and the binary model y ^ i {\displaystyle {\hat {y}}_{i}} of y i {\displaystyle y_{i}} which is the indicator of event { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} . Each of the y ^ i {\displaystyle {\hat {y}}_{i}} models is a "One versus Rest" model. L i f t ( y = i , y ^ = i ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)} only depends on the events { y = i } {\displaystyle \{y=i\}} and { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} , so merging or not merging the other classes doesn't change its value. We therefore have L i f t ( y = i , y ^ = i ) = L i f t ( y i = 1 , y ^ i = 1 ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)=\mathrm {Lift} (y_{i}=1,{\hat {y}}_{i}=1)} and the first condition is that all binary One versus Rest models are better than chance. ==== Example ==== If K = 2 {\displaystyle K=2} and 2 is the class of interest , the normalized confusion matrix is ( s p e c i f i c i t y 1 − s p e c i f i c i t y 1 − s e n s i t i v i t y s e n s i t i v i t y ) {\displaystyle {\begin{pmatrix}\mathrm {specificity} &1-\mathrm {specificity} \\1-\mathrm {sensitivity} &\mathrm {sensitivity} \end{pmatrix}}} and we have L i f t ( y = 1 , y ^ = 1 ) − 1 = P ( y = y ^ = 1 ) λ 1 μ 1 − 1 = n 1 , 1 n n 1. n .1 − 1 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)-1={\frac {\mathbb {P} (y={\hat {y}}=1)}{\lambda _{1}\mu _{1}}}-1={\frac {n_{1,1}n}{n_{1.}n_{.1}}}-1} = n 1 , 1 ( n 1 , 1 + n 1 , 2 + n 2 , 1 + n 2 , 2 ) − ( n 1 , 1 + n 1 , 2 ) ( n 1 , 1 + n 2 , 1 ) n 1. n .1 = n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 n 1. n .1 {\displaystyle ={\frac {n_{1,1}(n_{1,1}+n_{1,2}+n_{2,1}+n_{2,2})-(n_{1,1}+n_{1,2})(n_{1,1}+n_{2,1})}{n_{1.}n_{.1}}}={\frac {n_{1,1}n_{2,2}-n_{1,2}n_{2,1}}{n_{1.}n_{.1}}}} . Thus L i f t ( y = 1 , y ^ = 1 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Similarly, by swapping the roles of 1 and 2, we find that L i f t ( y = 2 , y ^ = 2 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=2,{\hat {y}}=2)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Dividing by n 1. n 2. {\displaystyle n_{1.}n_{2.}} we find that the necessary and sufficient condition on the normalized confusion matrix is s e n s i t i v i t y s p e c i f i c i t y − ( 1 − s e n s i t i v i t y ) ( 1 − s p e c i f i c i t y ) ≥ 0 ⟺ s e n s i t i v i t y + s p e c i f i c i t y − 1 ≥ 0 ⟺ J ≥ 0 {\displaystyle \mathrm {sensitivity} \ \mathrm {specificity} -(1-\mathrm {sensitivity} )(1-\mathrm {specificity} )\geq 0\iff \mathrm {sensitivity} +\mathrm {specificity} -1\geq 0\iff J\geq 0} . This brings us back to the classical binary condition: Youden's J must be positive (or zero for random models). === Random models === A random model is a model that is independent of the target variable. This property is easily reformulated with the confusion matrix. This proposition shows that the model y ^ {\displaystyle {\hat {y}}} of y {\displaystyle y} is uninformative if and only if there are two families of numbers ( α i ) i {\displaystyle (\alpha _{i})_{i}} and ( β j ) j {\displaystyle (\beta _{j})_{j}} such that P ( { y = i } ∩ { y ^ = j } ) = α i β j {\displaystyle \mathbb {P} (\{y=i\}\cap \{{\hat {y}}=j\})=\alpha _{i}\beta _{j}} for any i {\displaystyle i} and j {\displaystyle j} . === Multiclass likelihood ratios and diagnostic odds ratios === We define generalized likelihood ratios calculated from the normalized confusion matrix: for any i {\displaystyle i} and j ≠ i {\displaystyle j\not =i} , let L R i , j = P ( y ^ = j ∣ y = j ) P ( y ^ = j ∣ y = i ) {\displaystyle \mathrm {LR} _{i,j}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)}}} . When K = 2 {\displaystyle K=2} , if 2 is the class of interest,, we find the classical likelihood ratios L R 1 , 2 = L R + {\displaystyle \mathrm {LR} _{1,2}=\mathrm {LR} _{+}} and L R 2 , 1 = 1 L R − {\displaystyle \mathrm {LR} _{2,1}={\frac {1}{\mathrm {LR} _{-}}}} . Multiclass diagnostic odds ratios can also be defined using the formula D O R i , j = D O R j , i = L R i , j L R j , i = n i , i n j , j n i , j n j , i = P ( y ^ = j ∣ y = j ) / P ( y ^ = i ∣ y = j ) P ( y ^ = j ∣ y = i ) / P ( y ^ = i ∣ y = i ) {\displaystyle \mathrm {DOR} _{i,j}=\mathrm {DOR} _{j,i}=\mathrm {LR} _{i,j}\mathrm {LR} _{j,i}={\frac {n_{i,i}n_{j,j}}{n_{i,j}n_{j,i}}}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)/\mathbb {P} ({\hat {y}}=i\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)/\mathbb {P} ({\hat {y}}=i\mid y=i)}}} We saw above that a better-than-chance model (or a random model) must verify L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} and λ i {\displaystyle \lambda _{i}} . According to the previous corollary, likelihood ratios are thus greater
Alternating decision tree
An alternating decision tree (ADTree) is a machine learning method for classification. It generalizes decision trees and has connections to boosting. An ADTree consists of an alternation of decision nodes, which specify a predicate condition, and prediction nodes, which contain a single number. An instance is classified by an ADTree by following all paths for which all decision nodes are true, and summing any prediction nodes that are traversed. == History == ADTrees were introduced by Yoav Freund and Llew Mason. However, the algorithm as presented had several typographical errors. Clarifications and optimizations were later presented by Bernhard Pfahringer, Geoffrey Holmes and Richard Kirkby. Implementations are available in Weka and JBoost. == Motivation == Original boosting algorithms typically used either decision stumps or decision trees as weak hypotheses. As an example, boosting decision stumps creates a set of T {\displaystyle T} weighted decision stumps (where T {\displaystyle T} is the number of boosting iterations), which then vote on the final classification according to their weights. Individual decision stumps are weighted according to their ability to classify the data. Boosting a simple learner results in an unstructured set of T {\displaystyle T} hypotheses, making it difficult to infer correlations between attributes. Alternating decision trees introduce structure to the set of hypotheses by requiring that they build off a hypothesis that was produced in an earlier iteration. The resulting set of hypotheses can be visualized in a tree based on the relationship between a hypothesis and its "parent." Another important feature of boosted algorithms is that the data is given a different distribution at each iteration. Instances that are misclassified are given a larger weight while accurately classified instances are given reduced weight. == Alternating decision tree structure == An alternating decision tree consists of decision nodes and prediction nodes. Decision nodes specify a predicate condition. Prediction nodes contain a single number. ADTrees always have prediction nodes as both root and leaves. An instance is classified by an ADTree by following all paths for which all decision nodes are true and summing any prediction nodes that are traversed. This is different from binary classification trees such as CART (Classification and regression tree) or C4.5 in which an instance follows only one path through the tree. === Example === The following tree was constructed using JBoost on the spambase dataset (available from the UCI Machine Learning Repository). In this example, spam is coded as 1 and regular email is coded as −1. The following table contains part of the information for a single instance. The instance is scored by summing all of the prediction nodes through which it passes. In the case of the instance above, the score is calculated as The final score of 0.657 is positive, so the instance is classified as spam. The magnitude of the value is a measure of confidence in the prediction. The original authors list three potential levels of interpretation for the set of attributes identified by an ADTree: Individual nodes can be evaluated for their own predictive ability. Sets of nodes on the same path may be interpreted as having a joint effect The tree can be interpreted as a whole. Care must be taken when interpreting individual nodes as the scores reflect a re weighting of the data in each iteration. == Description of the algorithm == The inputs to the alternating decision tree algorithm are: A set of inputs ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x i {\displaystyle x_{i}} is a vector of attributes and y i {\displaystyle y_{i}} is either -1 or 1. Inputs are also called instances. A set of weights w i {\displaystyle w_{i}} corresponding to each instance. The fundamental element of the ADTree algorithm is the rule. A single rule consists of a precondition, a condition, and two scores. A condition is a predicate of the form "attribute
Automatic summarization
Automatic summarization is the process of shortening a set of data computationally, to create a subset (a summary) that represents the most important or relevant information within the original content. Artificial intelligence (AI) algorithms are commonly developed and employed to achieve this, specialized for different types of data. Text summarization is usually implemented by natural language processing methods, designed to locate the most informative sentences in a given document. On the other hand, visual content can be summarized using computer vision algorithms. Image summarization is the subject of ongoing research; existing approaches typically attempt to display the most representative images from a given image collection, or generate a video that only includes the most important content from the entire collection. Video summarization algorithms identify and extract from the original video content the most important frames (key-frames), and/or the most important video segments (key-shots), normally in a temporally ordered fashion. Video summaries simply retain a carefully selected subset of the original video frames and, therefore, are not identical to the output of video synopsis algorithms, where new video frames are being synthesized based on the original video content. == Commercial products == In 2022 Google Docs released an automatic summarization feature. == Approaches == There are two general approaches to automatic summarization: extraction and abstraction. === Extraction-based summarization === Here, content is extracted from the original data, but the extracted content is not modified in any way. Examples of extracted content include key-phrases that can be used to "tag" or index a text document, or key sentences (including headings) that collectively comprise an abstract, and representative images or video segments, as stated above. For text, extraction is analogous to the process of skimming, where the summary (if available), headings and subheadings, figures, the first and last paragraphs of a section, and optionally the first and last sentences in a paragraph are read before one chooses to read the entire document in detail. Other examples of extraction that include key sequences of text in terms of clinical relevance (including patient/problem, intervention, and outcome). === Abstractive-based summarization === Abstractive summarization methods generate new text that did not exist in the original text. This has been applied mainly for text. Abstractive methods build an internal semantic representation of the original content (often called a language model), and then use this representation to create a summary that is closer to what a human might express. Abstraction may transform the extracted content by paraphrasing sections of the source document, to condense a text more strongly than extraction. Such transformation, however, is computationally much more challenging than extraction, involving both natural language processing and often a deep understanding of the domain of the original text in cases where the original document relates to a special field of knowledge. "Paraphrasing" is even more difficult to apply to images and videos, which is why most summarization systems are extractive. === Aided summarization === Approaches aimed at higher summarization quality rely on combined software and human effort. In Machine Aided Human Summarization, extractive techniques highlight candidate passages for inclusion (to which the human adds or removes text). In Human Aided Machine Summarization, a human post-processes software output, in the same way that one edits the output of automatic translation by Google Translate. == Applications and systems for summarization == There are broadly two types of extractive summarization tasks depending on what the summarization program focuses on. The first is generic summarization, which focuses on obtaining a generic summary or abstract of the collection (whether documents, or sets of images, or videos, news stories etc.). The second is query relevant summarization, sometimes called query-based summarization, which summarizes objects specific to a query. Summarization systems are able to create both query relevant text summaries and generic machine-generated summaries depending on what the user needs. An example of a summarization problem is document summarization, which attempts to automatically produce an abstract from a given document. Sometimes one might be interested in generating a summary from a single source document, while others can use multiple source documents (for example, a cluster of articles on the same topic). This problem is called multi-document summarization. A related application is summarizing news articles. Imagine a system, which automatically pulls together news articles on a given topic (from the web), and concisely represents the latest news as a summary. Image collection summarization is another application example of automatic summarization. It consists in selecting a representative set of images from a larger set of images. A summary in this context is useful to show the most representative images of results in an image collection exploration system. Video summarization is a related domain, where the system automatically creates a trailer of a long video. This also has applications in consumer or personal videos, where one might want to skip the boring or repetitive actions. Similarly, in surveillance videos, one would want to extract important and suspicious activity, while ignoring all the boring and redundant frames captured. At a very high level, summarization algorithms try to find subsets of objects (like set of sentences, or a set of images), which cover information of the entire set. This is also called the core-set. These algorithms model notions like diversity, coverage, information and representativeness of the summary. Query based summarization techniques, additionally model for relevance of the summary with the query. Some techniques and algorithms which naturally model summarization problems are TextRank and PageRank, Submodular set function, Determinantal point process, maximal marginal relevance (MMR) etc. === Keyphrase extraction === The task is the following. You are given a piece of text, such as a journal article, and you must produce a list of keywords or key[phrase]s that capture the primary topics discussed in the text. In the case of research articles, many authors provide manually assigned keywords, but most text lacks pre-existing keyphrases. For example, news articles rarely have keyphrases attached, but it would be useful to be able to automatically do so for a number of applications discussed below. Consider the example text from a news article: "The Army Corps of Engineers, rushing to meet President Bush's promise to protect New Orleans by the start of the 2006 hurricane season, installed defective flood-control pumps last year despite warnings from its own expert that the equipment would fail during a storm, according to documents obtained by The Associated Press". A keyphrase extractor might select "Army Corps of Engineers", "President Bush", "New Orleans", and "defective flood-control pumps" as keyphrases. These are pulled directly from the text. In contrast, an abstractive keyphrase system would somehow internalize the content and generate keyphrases that do not appear in the text, but more closely resemble what a human might produce, such as "political negligence" or "inadequate protection from floods". Abstraction requires a deep understanding of the text, which makes it difficult for a computer system. Keyphrases have many applications. They can enable document browsing by providing a short summary, improve information retrieval (if documents have keyphrases assigned, a user could search by keyphrase to produce more reliable hits than a full-text search), and be employed in generating index entries for a large text corpus. Depending on the different literature and the definition of key terms, words or phrases, keyword extraction is a highly related theme. ==== Supervised learning approaches ==== Beginning with the work of Turney, many researchers have approached keyphrase extraction as a supervised machine learning problem. Given a document, we construct an example for each unigram, bigram, and trigram found in the text (though other text units are also possible, as discussed below). We then compute various features describing each example (e.g., does the phrase begin with an upper-case letter?). We assume there are known keyphrases available for a set of training documents. Using the known keyphrases, we can assign positive or negative labels to the examples. Then we learn a classifier that can discriminate between positive and negative examples as a function of the features. Some classifiers make a binary classification for a test example, while others assign a probability of being a keyphrase. For ins
Sliced inverse regression
Sliced inverse regression (SIR) is a tool for dimensionality reduction in the field of multivariate statistics. In statistics, regression analysis is a method of studying the relationship between a response variable y and its input variable x _ {\displaystyle {\underline {x}}} , which is a p-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing. As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, E ( x _ | y ) {\displaystyle E({\underline {x}}\,|\,y)} , to perform a weighted principal component analysis. == Model == Given a response variable Y {\displaystyle \,Y} and a (random) vector X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} of explanatory variables, SIR is based on the model Y = f ( β 1 ⊤ X , … , β k ⊤ X , ε ) ( 1 ) {\displaystyle Y=f(\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X,\varepsilon )\quad \quad \quad \quad \quad (1)} where β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} are unknown projection vectors, k {\displaystyle \,k} is an unknown number smaller than p {\displaystyle \,p} , f {\displaystyle \;f} is an unknown function on R k + 1 {\displaystyle \mathbb {R} ^{k+1}} as it only depends on k {\displaystyle \,k} arguments, and ε {\displaystyle \varepsilon } is a random variable representing error with E [ ε | X ] = 0 {\displaystyle E[\varepsilon |X]=0} and a finite variance of σ 2 {\displaystyle \sigma ^{2}} . The model describes an ideal solution, where Y {\displaystyle \,Y} depends on X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} only through a k {\displaystyle \,k} dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from p {\displaystyle \,p} to a smaller number k {\displaystyle \,k} without losing any information. An equivalent version of ( 1 ) {\displaystyle \,(1)} is: the conditional distribution of Y {\displaystyle \,Y} given X {\displaystyle \,X} depends on X {\displaystyle \,X} only through the k {\displaystyle \,k} dimensional random vector ( β 1 ⊤ X , … , β k ⊤ X ) {\displaystyle (\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X)} . It is assumed that this reduced vector is as informative as the original X {\displaystyle \,X} in explaining Y {\displaystyle \,Y} . The unknown β i ′ s {\displaystyle \,\beta _{i}'s} are called the effective dimension reducing directions (EDR-directions). The space that is spanned by these vectors is denoted by the effective dimension reducing space (EDR-space). == Relevant linear algebra background == Given a _ 1 , … , a _ r ∈ R n {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}\in \mathbb {R} ^{n}} , then V := L ( a _ 1 , … , a _ r ) {\displaystyle V:=L({\underline {a}}_{1},\ldots ,{\underline {a}}_{r})} , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors a _ 1 , … , a _ r {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}} span V {\displaystyle \,V} , but the vectors that span space V {\displaystyle \,V} are not unique. The dimension of V ( ∈ R n ) {\displaystyle \,V(\in \mathbb {R} ^{n})} is equal to the maximum number of linearly independent vectors in V {\displaystyle \,V} . A set of n {\displaystyle \,n} linear independent vectors of R n {\displaystyle \mathbb {R} ^{n}} makes up a basis of R n {\displaystyle \mathbb {R} ^{n}} . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line. == Inverse regression == Computing the inverse regression curve (IR) means instead of looking for E [ Y | X = x ] {\displaystyle \,E[Y|X=x]} , which is a curve in R p {\displaystyle \mathbb {R} ^{p}} it is actually E [ X | Y = y ] {\displaystyle \,E[X|Y=y]} , which is also a curve in R p {\displaystyle \mathbb {R} ^{p}} , but consisting of p {\displaystyle \,p} one-dimensional regressions. The center of the inverse regression curve is located at E [ E [ X | Y ] ] = E [ X ] {\displaystyle \,E[E[X|Y]]=E[X]} . Therefore, the centered inverse regression curve is E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} which is a p {\displaystyle \,p} dimensional curve in R p {\displaystyle \mathbb {R} ^{p}} . == Inverse regression versus dimension reduction == The centered inverse regression curve lies on a k {\displaystyle \,k} -dimensional subspace spanned by Σ x x β i ′ s {\displaystyle \,\Sigma _{xx}\beta _{i}\,'s} . This is a connection between the model and inverse regression. Given this condition and ( 1 ) {\displaystyle \,(1)} , the centered inverse regression curve E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} is contained in the linear subspace spanned by Σ x x β k ( k = 1 , … , K ) {\displaystyle \,\Sigma _{xx}\beta _{k}(k=1,\ldots ,K)} , where Σ x x = C o v ( X ) {\displaystyle \,\Sigma _{xx}=Cov(X)} . == Estimation of the EDR-directions == After having had a look at all the theoretical properties, the aim now is to estimate the EDR-directions. For that purpose, weighted principal component analyses are needed. If the sample means m ^ h ′ s {\displaystyle \,{\hat {m}}_{h}\,'s} , X {\displaystyle \,X} would have been standardized to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} . Corresponding to the theorem above, the IR-curve m 1 ( y ) = E [ Z | Y = y ] {\displaystyle \,m_{1}(y)=E[Z|Y=y]} lies in the space spanned by ( η 1 , … , η k ) {\displaystyle \,(\eta _{1},\ldots ,\eta _{k})} , where η i = Σ x x 1 / 2 β i {\displaystyle \,\eta _{i}=\Sigma _{xx}^{1/2}\beta _{i}} . As a consequence, the covariance matrix c o v [ E [ Z | Y ] ] {\displaystyle \,cov[E[Z|Y]]} is degenerate in any direction orthogonal to the η i ′ s {\displaystyle \,\eta _{i}\,'s} . Therefore, the eigenvectors η k ( k = 1 , … , K ) {\displaystyle \,\eta _{k}(k=1,\ldots ,K)} associated with the largest K {\displaystyle \,K} eigenvalues are the standardized EDR-directions. == Algorithm == === SIR algorithm === The algorithm from Li, K-C. (1991) to estimate the EDR-directions via SIR is as follows. 1. Let Σ x x {\displaystyle \,\Sigma _{xx}} be the covariance matrix of X {\displaystyle \,X} . Standardize X {\displaystyle \,X} to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} ( 1 ) {\displaystyle \,(1)} can also be rewritten as Y = f ( η 1 ⊤ Z , … , η k ⊤ Z , ε ) {\displaystyle Y=f(\eta _{1}^{\top }Z,\ldots ,\eta _{k}^{\top }Z,\varepsilon )} where η k = β k Σ x x 1 / 2 ∀ k {\displaystyle \,\eta _{k}=\beta _{k}\Sigma _{xx}^{1/2}\quad \forall \;k} .) 2. Divide the range of y i {\displaystyle \,y_{i}} into S {\displaystyle \,S} non-overlapping slices H s ( s = 1 , … , S ) . n s {\displaystyle \,H_{s}(s=1,\ldots ,S).\;n_{s}} is the number of observations within each slice and I H s {\displaystyle \,I_{H_{s}}} is the indicator function for the slice: n s = ∑ i = 1 n I H s ( y i ) {\displaystyle n_{s}=\sum _{i=1}^{n}I_{H_{s}}(y_{i})} 3. Compute the mean of z i {\displaystyle \,z_{i}} over all slices, which is a crude estimate m ^ 1 {\displaystyle \,{\hat {m}}_{1}} of the inverse regression curve m 1 {\displaystyle \,m_{1}} : z ¯ s = n s − 1 ∑ i = 1 n z i I H s ( y i ) {\displaystyle \,{\bar {z}}_{s}=n_{s}^{-1}\sum _{i=1}^{n}z_{i}I_{H_{s}}(y_{i})} 4. Calculate the estimate for C o v { m 1 ( y ) } {\displaystyle \,Cov\{m_{1}(y)\}} : V ^ = n − 1 ∑ i = 1 S n s z ¯ s z ¯ s ⊤ {\displaystyle \,{\hat {V}}=n^{-1}\sum _{i=1}^{S}n_{s}{\bar {z}}_{s}{\bar {z}}_{s}^{\top }} 5. Identify the eigenvalues λ ^ i {\displaystyle \,{\hat {\lambda }}_{i}} and the eigenvectors η ^ i {\displaystyle \,{\hat {\eta }}_{i}} of V ^ {\displaystyle \,{\hat {V}}} , which are the standardized EDR-directions. 6. Transform the standardized EDR-directions back to the original scale. The estimates for the EDR-directions are given by: β ^ i = Σ ^ x x − 1 / 2 η ^ i {\displaystyle \,{\hat {\beta }}_{i}={\hat {\Sigma }}_{xx}^{-1/2}{\hat {\eta }}_{i}} (which are not necessarily orthogonal.)