AI Artists On Spotify

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  Matchbox Educable Noughts and Crosses Engine

  The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Michie and his colleague Roger Chambers, in 1961. It was designed to play human opponents in games of noughts and crosses (tic-tac-toe) by returning a move for any given state of play and to refine its strategy through reinforcement learning. This was one of the first types of artificial intelligence. Michie and Chambers did not have immediate access to a computer; they worked around this by building the engine out of matchboxes. The matchboxes they used each represented a single possible layout of a noughts and crosses grid. When the computer first played, it would randomly choose moves based on the current layout. As it played more games, through a reinforcement loop, it disqualified strategies that led to losing games, and supplemented strategies that led to winning games. Michie held a tournament against MENACE in 1961, wherein he experimented with different openings. Following MENACE's maiden tournament against Michie, it demonstrated successful artificial intelligence in its strategy. Michie's essays on MENACE's weight initialisation and the BOXES algorithm used by MENACE became popular in the field of computer science research. Michie was honoured for his contribution to machine learning research, and was twice commissioned to program a MENACE simulation on an actual computer. == Origin == Donald Michie (1923–2007) had been on the team decrypting the German Tunny Code during World War II. Fifteen years later, he wanted to further display his mathematical and computational prowess with an early convolutional neural network. Since computer equipment was not obtainable for such uses, and Michie did not have a computer readily available, he decided to display and demonstrate artificial intelligence in a more esoteric format and constructed a functional mechanical computer out of matchboxes and beads. MENACE was constructed as the result of a bet with a computer science colleague who postulated that such a machine was impossible. Michie undertook the task of collecting and defining each matchbox as a "fun project", later turned into a demonstration tool. Michie completed his essay on MENACE in 1963, "Experiments on the mechanization of game-learning", as well as his essay on the BOXES Algorithm, written with R. A. Chambers and had built up an AI research unit in Hope Park Square, Edinburgh, Scotland. MENACE learned by playing successive matches of noughts and crosses. Each time, it would eliminate a losing strategy by the human player confiscating the beads that corresponded to each move. It reinforced winning strategies by making the moves more likely, by supplying extra beads. This was one of the earliest versions of the Reinforcement Loop, the schematic algorithm of looping the algorithm, dropping unsuccessful strategies until only the winning ones remain. This model starts as completely random, and gradually learns. == Composition == MENACE was made from 304 matchboxes glued together in an arrangement similar to a chest of drawers. Each box had a code number, which was keyed into a chart. This chart had drawings of tic-tac-toe game grids with various configurations of X, O, and empty squares, corresponding to all possible permutations a game could go through as it progressed. After removing duplicate arrangements (ones that were simply rotations or mirror images of other configurations), MENACE used 304 permutations in its chart and thus that many matchboxes. Each individual matchbox tray contained a collection of coloured beads. Each colour represented a move on a square on the game grid, and so matchboxes with arrangements where positions on the grid were already taken would not have beads for that position. Additionally, at the front of the tray were two extra pieces of card in a "V" shape, the point of the "V" pointing at the front of the matchbox. Michie and his artificial intelligence team called MENACE's algorithm "Boxes", after the apparatus used for the machine. The first stage "Boxes" operated in five phases, each setting a definition and a precedent for the rules of the algorithm in relation to the game. == Operation == MENACE played first, as O, since all matchboxes represented permutations only relevant to the "X" player. To retrieve MENACE's choice of move, the opponent or operator located the matchbox that matched the current game state, or a rotation or mirror image of it. For example, at the start of a game, this would be the matchbox for an empty grid. The tray would be removed and lightly shaken so as to move the beads around. Then, the bead that had rolled into the point of the "V" shape at the front of the tray was the move MENACE had chosen to make. Its colour was then used as the position to play on, and, after accounting for any rotations or flips needed based on the chosen matchbox configuration's relation to the current grid, the O would be placed on that square. Then the player performed their move, the new state was located, a new move selected, and so on, until the game was finished. When the game had finished, the human player observed the game's outcome. As a game was played, each matchbox that was used for MENACE's turn had its tray returned to it ajar, and the bead used kept aside, so that MENACE's choice of moves and the game states they belonged to were recorded. Michie described his reinforcement system with "reward" and "punishment". Once the game was finished, if MENACE had won, it would then receive a "reward" for its victory. The removed beads showed the sequence of the winning moves. These were returned to their respective trays, easily identifiable since they were slightly open, as well as three bonus beads of the same colour. In this way, in future games MENACE would become more likely to repeat those winning moves, reinforcing winning strategies. If it lost, the removed beads were not returned, "punishing" MENACE, and meaning that in future it would be less likely, and eventually incapable if that colour of bead became absent, to repeat the moves that cause a loss. If the game was a draw, one additional bead was added to each box. == Results in practice == === Optimal strategy === Noughts and crosses has a well-known optimal strategy. A player must place their symbol in a way that blocks the other player from achieving any rows while simultaneously making a row themself. However, if both players use this strategy, the game always ends in a draw. If the human player is familiar with the optimal strategy, and MENACE can quickly learn it, then the games will eventually only end in draws. The likelihood of the computer winning increases quickly when the computer plays against a random-playing opponent. When playing against a player using optimal strategy, the odds of a draw grow to 100%. In Donald Michie's official tournament against MENACE in 1961 he used optimal strategy, and he and the computer began to draw consistently after twenty games. Michie's tournament had the following milestones: Michie began by consistently opening with "Variant 0", the middle square. At 15 games, MENACE abandoned all non-corner openings. At just over 20, Michie switched to consistently using "Variant 1", the bottom-right square. At 60, he returned to Variant 0. As he neared 80 games, he moved to "Variant 2", the top-middle. At 110, he switched to "Variant 3", the top right. At 135, he switched to "Variant 4", middle-right. At 190, he returned to Variant 1, and at 210, he returned to Variant 0. The trend in changes of beads in the "2" boxes runs: === Correlation === Depending on the strategy employed by the human player, MENACE produces a different trend on scatter graphs of wins. Using a random turn from the human player results in an almost-perfect positive trend. Playing the optimal strategy returns a slightly slower increase. The reinforcement does not create a perfect standard of wins; the algorithm will draw random uncertain conclusions each time. After the j-th round, the correlation of near-perfect play runs: 1 − D D − D ( j + 2 ) ∑ i = 0 j D ( j i + 1 ) V i {\displaystyle {1-D \over D-D^{(j+2)}}\sum _{i=0}^{j}D^{(ji+1)}V_{i}} Where Vi is the outcome (+1 is win, 0 is draw and -1 is loss) and D is the decay factor (average of past values of wins and losses). Below, Mn is the multiplier for the n-th round of the game. == Legacy == Donald Michie's MENACE proved that a computer could learn from failure and success to become good at a task. It used what would become core principles within the field of machine learning before they had been properly theorised. For example, the combination of how MENACE starts with equal numbers of types of beads in each matchbox, and how these are then selected at random, creates a learning behaviour similar to weight initialisation

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  Silhouette (clustering)

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

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  Rule-based machine learning

  Rule-based machine learning (RBML) is a term in computer science intended to encompass any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply. The defining characteristic of a rule-based machine learner is the identification and utilization of a set of relational rules that collectively represent the knowledge captured by the system. Rule-based machine learning approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge. While rule-based machine learning is conceptually a type of rule-based system, it is distinct from traditional rule-based systems, which are often hand-crafted, and other rule-based decision makers. This is because rule-based machine learning applies some form of learning algorithm such as Rough sets theory to identify and minimise the set of features and to automatically identify useful rules, rather than a human needing to apply prior domain knowledge to manually construct rules and curate a rule set. == Rules == Rules typically take the form of an '{IF:THEN} expression', (e.g. {IF 'condition' THEN 'result'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign}). An individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Therefore rule-based machine learning methods typically comprise a set of rules, or knowledge base, that collectively make up the prediction model usually known as decision algorithm. Rules can also be interpreted in various ways depending on the domain knowledge, data types(discrete or continuous) and in combinations. == RIPPER == Repeated incremental pruning to produce error reduction (RIPPER) is a propositional rule learner proposed by William W. Cohen as an optimized version of IREP.

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  Teaching dimension

  In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).

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  InciWeb

  InciWeb is an interagency all-risk incident web information management system provided by the United States Forest Service released in 2004. It was originally developed for wildland fire emergencies, but can be also used for other emergency incidents (natural disasters, such as earthquakes, floods, hurricanes, and tornadoes). == Introduction == It was developed with two primary missions: 1. Provide the public a single source of incident related information 2. Provide a standardized reporting tool for the Public Affairs community Official announcements include evacuations, road closures, news releases, maps, photographs, and basic info and current situation about the incident. Incident information can be accessed by: web browser at https://inciweb.wildfire.gov/ Twitter RSS web feed == Technical == The original application was hosted at the United States Forest Service - Wildland Fire Training and Conference Center, at McClellan Airfield, California, comprising three servers: Database server Administrative server Load balancer for the public content which routes traffic to a pool of eight servers. Web traffic averages 2 million plus hits daily during the fire season with the ability to handle 3.5 million hits. The servers were moved to the National information Technology Center (NITC), Kansas City, Missouri on July 16, 2008, along with the release of version 2.0; the current version is 2.2. == Availability issues == InciWeb was having technical difficulties due to the high volume of Internet users trying to access the site during the September–October 2006 Day Fire and the Summer 2008 California wildfires. == Participating agencies == United States Forest Service Bureau of Land Management Bureau of Indian Affairs Fish and Wildlife Service National Park Service National Oceanic & Atmospheric Administration Department of the Interior Office of Aircraft Services National Association of State Foresters United States Fire Administration These same agencies are also in the National Interagency Fire Center.

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  Spatial Analysis of Principal Components

  Spatial Principal Component Analysis (sPCA) is a multivariate statistical technique that complements the traditional Principal Component Analysis (PCA) by incorporating spatial information into the analysis of genetic variation. While traditional PCA can be used to find spatial patterns, it focuses on reducing data dimensionality by identifying uncorrelated principal components that capture maximum variance, thus often lacking power to identify non-trivial spatial genetic patterns. By accounting for spatial autocorrelation, sPCA is able to uncover spatial patterns in the data and find the spatial structure of datasets where observations are either geographically or topologically linked. This statistical power improvement allows the investigation of cryptic spatial patterns of genetic variability otherwise overlooked. sPCA has been applied in various fields, including geography, ecology and genetics. == History == sPCA was introduced in 2008 by Thibaut Jombart, Sébastien Devillard, Anne-Béatrice Dufour, and D. Pontier as a spatially explicit method to investigate the spatial pattern of genetic variation among individuals or populations. In 2017, Valeria Montano and Thibaut Jombart published an alternative non-parametric test to evaluate the significance of global and local spatial genetic patterns with improved statistical power. == Details == sPCA modifies the PCA framework by integrating spatial weights, typically in the form of connectivity matrices or spatial adjacency graphs. It identifies principal components (PCs) that maximize both genentic variance and spatial autocorreation, as measured by Moran's I. These weights represent relationships between observations based on geographic distance or other spatial criteria. The method decomposes variance into two components: Global structures, correspond to positive autocorrelation, that is, reflect broad-scale spatial patterns where similar values cluster over large regions. Local structures, correspond to negative autocorrelation, that is, capture fine-scale spatial variations or localized patterns. The core of sPCA relies on the eigenanalysis of a spatially weighted covariance or correlation matrix. The spatial weight matrix can be constructed using techniques such as Delaunay triangulation, nearest-neighbor graphs, or distance-based criteria. Applications of sPCA should be used only as an explorative tool. == Applications == sPCA has been widely used in many fields, including: Ecology: To find spatial patterns in species distributions and environmental gradients. Genetics: Population structure and gene flow analysis while allowing for spatial autocorrelation considerations. Biogeography: To identify historical dispersal routes, and barriers to gene flow, providing insights into species distribution patterns and evolutionary history. == Software/Source Code == sPCA implementations are available in R in adegenet and ntbox . These tools facilitate the application of sPCA by providing functions for constructing spatial weight matrices, performing eigenanalysis, and obtaining spatial principal components in an easy-to-read form.

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  Consensus clustering

  Consensus clustering is a method of aggregating (potentially conflicting) results from multiple clustering algorithms. Also called cluster ensembles or aggregation of clustering (or partitions), it refers to the situation in which a number of different (input) clusterings have been obtained for a particular dataset and it is desired to find a single (consensus) clustering which is a better fit in some sense than the existing clusterings. Consensus clustering is thus the problem of reconciling clustering information about the same data set coming from different sources or from different runs of the same algorithm. When cast as an optimization problem, consensus clustering is known as median partition, and has been shown to be NP-complete, even when the number of input clusterings is three. Consensus clustering for unsupervised learning is analogous to ensemble learning in supervised learning. == Issues with existing clustering techniques == Current clustering techniques do not address all the requirements adequately. Dealing with large number of dimensions and large number of data items can be problematic because of time complexity; Effectiveness of the method depends on the definition of "distance" (for distance-based clustering) If an obvious distance measure doesn't exist, we must "define" it, which is not always easy, especially in multidimensional spaces. The result of the clustering algorithm (that, in many cases, can be arbitrary itself) can be interpreted in different ways. == Justification for using consensus clustering == There are potential shortcomings for all existing clustering techniques. This may cause interpretation of results to become difficult, especially when there is no knowledge about the number of clusters. Clustering methods are also very sensitive to the initial clustering settings, which can cause non-significant data to be amplified in non-reiterative methods. An extremely important issue in cluster analysis is the validation of the clustering results, that is, how to gain confidence about the significance of the clusters provided by the clustering technique (cluster numbers and cluster assignments). Lacking an external objective criterion (the equivalent of a known class label in supervised analysis), this validation becomes somewhat elusive. Iterative descent clustering methods, such as the SOM and k-means clustering circumvent some of the shortcomings of hierarchical clustering by providing for univocally defined clusters and cluster boundaries. Consensus clustering provides a method that represents the consensus across multiple runs of a clustering algorithm, to determine the number of clusters in the data, and to assess the stability of the discovered clusters. The method can also be used to represent the consensus over multiple runs of a clustering algorithm with random restart (such as K-means, model-based Bayesian clustering, SOM, etc.), so as to account for its sensitivity to the initial conditions. It can provide data for a visualization tool to inspect cluster number, membership, and boundaries. However, they lack the intuitive and visual appeal of hierarchical clustering dendrograms, and the number of clusters must be chosen a priori. == The Monti consensus clustering algorithm == The Monti consensus clustering algorithm is one of the most popular consensus clustering algorithms and is used to determine the number of clusters, K {\displaystyle K} . Given a dataset of N {\displaystyle N} total number of points to cluster, this algorithm works by resampling and clustering the data, for each K {\displaystyle K} and a N × N {\displaystyle N\times N} consensus matrix is calculated, where each element represents the fraction of times two samples clustered together. A perfectly stable matrix would consist entirely of zeros and ones, representing all sample pairs always clustering together or not together over all resampling iterations. The relative stability of the consensus matrices can be used to infer the optimal K {\displaystyle K} . More specifically, given a set of points to cluster, D = { e 1 , e 2 , . . . e N } {\displaystyle D=\{e_{1},e_{2},...e_{N}\}} , let D 1 , D 2 , . . . , D H {\displaystyle D^{1},D^{2},...,D^{H}} be the list of H {\displaystyle H} perturbed (resampled) datasets of the original dataset D {\displaystyle D} , and let M h {\displaystyle M^{h}} denote the N × N {\displaystyle N\times N} connectivity matrix resulting from applying a clustering algorithm to the dataset D h {\displaystyle D^{h}} . The entries of M h {\displaystyle M^{h}} are defined as follows: M h ( i , j ) = { 1 , if points i and j belong to the same cluster 0 , otherwise {\displaystyle M^{h}(i,j)={\begin{cases}1,&{\text{if}}{\text{ points i and j belong to the same cluster}}\\0,&{\text{otherwise}}\end{cases}}} Let I h {\displaystyle I^{h}} be the N × N {\displaystyle N\times N} identicator matrix where the ( i , j ) {\displaystyle (i,j)} -th entry is equal to 1 if points i {\displaystyle i} and j {\displaystyle j} are in the same perturbed dataset D h {\displaystyle D^{h}} , and 0 otherwise. The indicator matrix is used to keep track of which samples were selected during each resampling iteration for the normalisation step. The consensus matrix C {\displaystyle C} is defined as the normalised sum of all connectivity matrices of all the perturbed datasets and a different one is calculated for every K {\displaystyle K} . C ( i , j ) = ( ∑ h = 1 H M h ( i , j ) ∑ h = 1 H I h ( i , j ) ) {\displaystyle C(i,j)=\left({\frac {\textstyle \sum _{h=1}^{H}M^{h}(i,j)\displaystyle }{\sum _{h=1}^{H}I^{h}(i,j)}}\right)} That is the entry ( i , j ) {\displaystyle (i,j)} in the consensus matrix is the number of times points i {\displaystyle i} and j {\displaystyle j} were clustered together divided by the total number of times they were selected together. The matrix is symmetric and each element is defined within the range [ 0 , 1 ] {\displaystyle [0,1]} . A consensus matrix is calculated for each K {\displaystyle K} to be tested, and the stability of each matrix, that is how far the matrix is towards a matrix of perfect stability (just zeros and ones) is used to determine the optimal K {\displaystyle K} . One way of quantifying the stability of the K {\displaystyle K} th consensus matrix is examining its CDF curve (see below). == Over-interpretation potential of the Monti consensus clustering algorithm == Monti consensus clustering can be a powerful tool for identifying clusters, but it needs to be applied with caution as shown by Şenbabaoğlu et al. It has been shown that the Monti consensus clustering algorithm is able to claim apparent stability of chance partitioning of null datasets drawn from a unimodal distribution, and thus has the potential to lead to over-interpretation of cluster stability in a real study. If clusters are not well separated, consensus clustering could lead one to conclude apparent structure when there is none, or declare cluster stability when it is subtle. Identifying false positive clusters is a common problem throughout cluster research, and has been addressed by methods such as SigClust and the GAP-statistic. However, these methods rely on certain assumptions for the null model that may not always be appropriate. Şenbabaoğlu et al demonstrated the original delta K metric to decide K {\displaystyle K} in the Monti algorithm performed poorly, and proposed a new superior metric for measuring the stability of consensus matrices using their CDF curves. In the CDF curve of a consensus matrix, the lower left portion represents sample pairs rarely clustered together, the upper right portion represents those almost always clustered together, whereas the middle segment represent those with ambiguous assignments in different clustering runs. The proportion of ambiguous clustering (PAC) score measure quantifies this middle segment; and is defined as the fraction of sample pairs with consensus indices falling in the interval (u1, u2) ∈ [0, 1] where u1 is a value close to 0 and u2 is a value close to 1 (for instance u1=0.1 and u2=0.9). A low value of PAC indicates a flat middle segment, and a low rate of discordant assignments across permuted clustering runs. One can therefore infer the optimal number of clusters by the K {\displaystyle K} value having the lowest PAC. == Related work == Clustering ensemble (Strehl and Ghosh): They considered various formulations for the problem, most of which reduce the problem to a hyper-graph partitioning problem. In one of their formulations they considered the same graph as in the correlation clustering problem. The solution they proposed is to compute the best k-partition of the graph, which does not take into account the penalty for merging two nodes that are far apart. Clustering aggregation (Fern and Brodley): They applied the clustering aggregation idea to a collection of soft clusterings they obtained by random projections. They used an agglomerative algorithm

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  Bootstrap aggregating

  Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the ensemble averaging approach. == Description of the technique == Given a standard training set D {\displaystyle D} of size n {\displaystyle n} , bagging generates m {\displaystyle m} new training sets D i {\displaystyle D_{i}} , each of size n ′ {\displaystyle n'} , by sampling from D {\displaystyle D} uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D i {\displaystyle D_{i}} . If n ′ = n {\displaystyle n'=n} , then for large n {\displaystyle n} the set D i {\displaystyle D_{i}} is expected to have the fraction (1 - 1/e) (~63.2%) of the unique samples of D {\displaystyle D} , the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as k-nearest neighbors. == Process of the algorithm == === Key Terms === There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. === Creating the bootstrap dataset === The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal. By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. === Creating the out-of-bag dataset === The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: === Application === Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of ensemble learning algorithms like random forest. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. === Creation of Decision Trees === The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those that possess the top feature, and those that do not. The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. === Random Forests === The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. == Improving Random Forests and Bagging == While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: Specify the maximum depth of trees: Instead of allowing the random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting. Prune the dataset: Using an extremely large dataset may create results that are less indicative of the data provided than a smaller set that more accurately represents what is being focused on. Continue pruning the data at each

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  Automated negotiation

  Automated negotiation is a form of interaction in systems that are composed of multiple autonomous agents, in which the aim is to reach agreements through an iterative process of making offers. Automated negotiation can be employed for many tasks human negotiators regularly engage in, such as bargaining and joint decision making. The main topics in automated negotiation revolve around the design of protocols and negotiating strategies. == History == Through digitization, the beginning of the 21st century has seen a growing interest in the automation of negotiation and e-negotiation systems, for example in the setting of e-commerce. This interest is fueled by the promise of automated agents being able to negotiate on behalf of human negotiators, and to find better outcomes than human negotiators. == Examples == Examples of automated negotiation include: Online dispute resolution, in which disagreements between parties are settled. Sponsored search auction, where bids are placed on advertisement keywords. Content negotiation, in which user agents negotiate over HTTP about how to best represent a web resource. Negotiation support systems, in which negotiation decision-making activities are supported by an information system.

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  Discrete diffusion model

  In machine learning, discrete diffusion models are a class of diffusion models, which themselves are a class of latent variable generative models. Each discrete diffusion model consists of two major components: the forward jump diffusion process, and the reverse jump diffusion process. The goal of diffusion modeling is, given a given dataset and a forward process, to learn a model for the reverse process, such that the reverse process can generate new elements that are distributed similarly as the original dataset. A trained discrete diffusion model can be sampled in many ways, which trades off computational efficiency and sample quality. In general, higher quality data can be obtained, but at the price of higher computational cost. In standard diffusion modeling, the diffusion process takes place over a state space that is continuous space of R n {\displaystyle \mathbb {R} ^{n}} , but over a discrete set S {\displaystyle S} . A discrete set is simply a set where one cannot speak of "infinitesimally close" points. Points can be more or less separated from each other, but the separation is always a finite number. This in particular means the standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced. Discrete diffusion is usually used for language modeling. In practice, the state space S {\displaystyle S} is not only discrete, but finite, so this is what we will assume from now on. == Continuous time Markov process == In the case of continuous state space, during the forward discrete diffusion process, at each step t → t + d t {\displaystyle t\to t+dt} , we mix in an infinitesimal amount of gaussian noise d x t = − 1 2 β ( t ) x t d t + β ( t ) d W t {\displaystyle dx_{t}=-{\frac {1}{2}}\beta (t)x_{t}dt+{\sqrt {\beta (t)}}dW_{t}} . This changes the probability density function, by first a convolution with the density of a gaussian, followed by a scaling. In the case of discrete state space, the gaussian noise must be replaced by a noise that takes values over a finite set. For example, if the noise is the uniform distribution over S {\displaystyle S} , then the probability distribution at time t + d t {\displaystyle t+dt} satisfies q t + d t ( x ) = ( 1 − d t ) q t ( x ) + d t ( 1 | S | ∑ y ∈ S q t ( y ) ) {\displaystyle q_{t+dt}(x)=(1-dt)q_{t}(x)+dt\left({\frac {1}{|S|}}\sum _{y\in S}q_{t}(y)\right)} More succinctly, ∂ t q t ( x ) = − ( 1 − 1 | S | ) q t ( x ) + ∑ y ∈ S , y ≠ x 1 | S | q t ( y ) {\displaystyle \partial _{t}q_{t}(x)=-\left(1-{\frac {1}{|S|}}\right)q_{t}(x)+\sum _{y\in S,y\neq x}{\frac {1}{|S|}}q_{t}(y)} In general, we do not need to convolve with a uniformly distributed noise, but with an arbitrary noise process. That is, we use an arbitrary matrix Q t {\displaystyle Q_{t}} such that ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} where Q t {\displaystyle Q_{t}} is called the rate matrix. Any matrix may be used as a rate matrix if it has non-negative off-diagonals, and each column sums to 0: Q t ( y , x ) ≥ 0 ∀ y ≠ x , ∑ y ∈ S Q t ( y , x ) = 0 ∀ x {\displaystyle Q_{t}(y,x)\geq 0\quad \forall y\neq x,\quad \sum _{y\in S}Q_{t}(y,x)=0\quad \forall x} A continuous time Markov chain (CTMC) is defined by a continuous function Q {\displaystyle Q} that maps any time t ∈ [ 0 , T ) {\displaystyle t\in [0,T)} to a rate matrix Q t {\displaystyle Q_{t}} . Given the function Q {\displaystyle Q} , time-evolution under the CTMC is done as follows: Given state x t {\displaystyle x_{t}} at time t {\displaystyle t} , and given an infinitesimal d t {\displaystyle dt} , the state at t + d t {\displaystyle t+dt} is x t + d t {\displaystyle x_{t+dt}} , such that Pr ( x t + d t | x t ) = { 1 + Q t ( x t + d t , x t ) d t if x t + d t = x t Q t ( x t + d t , x t ) d t else {\displaystyle \Pr(x_{t+dt}|x_{t})={\begin{cases}1+Q_{t}(x_{t+dt},x_{t})dt&{\text{if }}x_{t+dt}=x_{t}\\Q_{t}(x_{t+dt},x_{t})dt&{\text{else}}\end{cases}}} This implies that the probability distribution function evolves according to ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} which is what we previously specified. === Backward process === Similarly to the case of continuous diffusion, in discrete diffusion, there exists a backward diffusion process Q ¯ t {\displaystyle {\bar {Q}}_{t}} : s ( x , t ) y := q t ( y ) q t ( x ) , Q ¯ t ( y , x ) := { s ( x , t ) y Q t ( x , y ) if y ≠ x − ∑ y : y ≠ x Q ¯ t ( y , x ) if y = x {\displaystyle s(x,t)_{y}:={\frac {q_{t}(y)}{q_{t}(x)}},\quad {\bar {Q}}_{t}(y,x):={\begin{cases}s(x,t)_{y}Q_{t}(x,y)&{\text{if }}y\neq x\\-\sum _{y:y\neq x}{\bar {Q}}_{t}(y,x)&{\text{if }}y=x\end{cases}}} where s ( x , t ) y {\displaystyle s(x,t)_{y}} should be interpreted as the discrete score or concrete score, since, abusing notation a bit, the score function is ∇ ln ⁡ ρ t ( x ) = 1 d x ( ρ t ( x + d x ) ρ t ( x ) − 1 ) {\displaystyle \nabla \ln \rho _{t}(x)={\frac {1}{dx}}\left({\frac {\rho _{t}(x+dx)}{\rho _{t}(x)}}-1\right)} . If we picture the distribution q t {\displaystyle q_{t}} as a bunch of point-masses, one per state x ∈ S {\displaystyle x\in S} , then the forward diffusion from time t {\displaystyle t} to t + d t {\displaystyle t+dt} is performed by removing Q t ( x , y ) q t ( y ) d t {\displaystyle Q_{t}(x,y)q_{t}(y)dt} from the mass at y {\displaystyle y} and moving it to the mass at x {\displaystyle x} , for each pair x ≠ y {\displaystyle x\neq y} . Thus, the process is reversed in detail by the CTMC defined by Q ¯ {\displaystyle {\bar {Q}}} , since Q ¯ t ( y , x ) q t ( x ) = Q t ( x , y ) q t ( y ) {\displaystyle {\bar {Q}}_{t}(y,x)q_{t}(x)=Q_{t}(x,y)q_{t}(y)} . Given Q ¯ t {\displaystyle {\bar {Q}}_{t}} , if we have a way to sample from q t {\displaystyle q_{t}} , then we can sample from q t − d t {\displaystyle q_{t-dt}} by first sampling x t ∼ q t {\displaystyle x_{t}\sim q_{t}} , then sampling x t − d t {\displaystyle x_{t-dt}} according to Pr ( x t − d t | x t ) = { 1 + Q ¯ t ( x t − d t , x t ) d t if x t − d t = x t Q ¯ t ( x t − d t , x t ) d t else {\displaystyle \Pr(x_{t-dt}|x_{t})={\begin{cases}1+{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{if }}x_{t-dt}=x_{t}\\{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{else}}\end{cases}}} === Overall plan of score-matching discrete diffusion modeling === Similar to score-matching continuous diffusion, score-matching discrete diffusion is a method to sample an initial distribution. If we have a certain function s θ {\displaystyle s_{\theta }} that approximates the true score function s θ ( x , t ) y ≈ s ( x , t ) y {\displaystyle s_{\theta }(x,t)_{y}\approx s(x,t)_{y}} , then it allows a corresponding Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} to be defined in the same way. If we also have a base distribution q base {\displaystyle q_{\text{base}}} such that it is easy to sample from, and approximately equal to the true terminal distribution q base ≈ q T {\displaystyle q_{\text{base}}\approx q_{T}} , then we can perform the backward CTMC with Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} and q T θ := q terminal {\displaystyle q_{T}^{\theta }:=q_{\text{terminal}}} . When both approximations are good, the backward CTMC would give q 0 θ ≈ q 0 {\displaystyle q_{0}^{\theta }\approx q_{0}} . This is the idea of score-matching discrete diffusion modeling. If q data {\displaystyle q_{\text{data}}} is sharp, in the sense that for some x , x ′ {\displaystyle x,x'} , we have q data ( x ) ≫ q data ( x ′ ) {\displaystyle q_{\text{data}}(x)\gg q_{\text{data}}(x')} , then the score function would diverge as 1 / t {\displaystyle 1/t} at the t → 0 {\displaystyle t\to 0} limit. To avoid this in practice, it is common to use early stopping, which is to stop the backward process at some time δ > 0 {\displaystyle \delta >0} , and sample from q δ θ {\displaystyle q_{\delta }^{\theta }} instead of q 0 θ {\displaystyle q_{0}^{\theta }} . === Tractable forward processes === The theory of CTMC works for any continuous choice of rate matrices Q {\displaystyle Q} . However, most choices are computationally expensive and cannot be used in practice. In the case of continuous diffusion, the gaussian noise is used for the simple reason that the sum of any number of gaussians is still a gaussian. This allows one to sample any x t ∼ ρ t {\displaystyle x_{t}\sim \rho _{t}} by sampling a single x 0 ∼ ρ 0 {\displaystyle x_{0}\sim \rho _{0}} , followed by a single gaussian noise z ∼ N ( 0 , I ) {\displaystyle z\sim {\mathcal {N}}(0,I)} , and let x t = α ¯ t x 0 + σ t z {\displaystyle x_{t}={\sqrt {{\bar {\alpha }}_{t}}}x_{0}+\sigma _{t}z} , without needing any x s {\displaystyle x_{s}} for any 0 < s < t {\displaystyle 0 Read more →

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  Absorbing Markov chain

  In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case. == Formal definition == A Markov chain is an absorbing chain if there is at least one absorbing state and it is possible to go from any state to at least one absorbing state in a finite number of steps. In an absorbing Markov chain, a state that is not absorbing is called transient. === Canonical form === Let an absorbing Markov chain with transition matrix P have t transient states and r absorbing states. The rows of P represent sources, while columns represent destinations. By ordering the transient states before the absorbing states, it can be assumed that P has the form P = [ Q R 0 I r ] , {\displaystyle P={\begin{bmatrix}Q&R\\\mathbf {0} &I_{r}\end{bmatrix}},} where Q is a t-by-t matrix, R is a nonzero t-by-r matrix, 0 is an r-by-t zero matrix, and Ir is the r-by-r identity matrix. Thus, Q describes the probability of transitioning from some transient state to another while R describes the probability of transitioning from some transient state to some absorbing state. The probability of transitioning from i to j in exactly k steps is the (i,j)-entry of Pk, further computed below. When considering only transient states, the probability is found in the upper left of Pk, the (i,j)-entry of Qk. == Fundamental matrix == === Expected number of visits to a transient state === A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Qk for all k (from 0 to ∞). It can be proven that N := ∑ k = 0 ∞ Q k = ( I t − Q ) − 1 , {\displaystyle N:=\sum _{k=0}^{\infty }Q^{k}=(I_{t}-Q)^{-1},} where It is the t-by-t identity matrix. The computation of this formula is the matrix equivalent of the geometric series of scalars, ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle {\textstyle \sum }_{k=0}^{\infty }q^{k}={\tfrac {1}{1-q}}} . With the matrix N in hand, also other properties of the Markov chain are easy to obtain. === Expected number of steps before being absorbed === The expected number of steps before being absorbed in any absorbing state, when starting in transient state i can be computed via a sum over transient states. The value is given by the ith entry of the vector t := N 1 , {\displaystyle \mathbf {t} :=N\mathbf {1} ,} where 1 is a length-t column vector whose entries are all 1. === Absorbing probabilities === By induction, P k = [ Q k ( I t − Q k ) N R 0 I r ] . {\displaystyle P^{k}={\begin{bmatrix}Q^{k}&(I_{t}-Q^{k})NR\\\mathbf {0} &I_{r}\end{bmatrix}}.} The probability of eventually being absorbed in the absorbing state j when starting from transient state i is given by the (i,j)-entry of the matrix B := N R {\displaystyle B:=NR} . The number of columns of this matrix equals the number of absorbing states r. An approximation of those probabilities can also be obtained directly from the (i,j)-entry of P k {\displaystyle P^{k}} for a large enough value of k, when i is the index of a transient, and j the index of an absorbing state. This is because ( lim k → ∞ P k ) i , t + j = B i , j {\displaystyle \left(\lim _{k\to \infty }P^{k}\right)_{i,t+j}=B_{i,j}} . === Transient visiting probabilities === The probability of visiting transient state j when starting at a transient state i is the (i,j)-entry of the matrix H := ( N − I t ) ( N dg ) − 1 , {\displaystyle H:=(N-I_{t})(N_{\operatorname {dg} })^{-1},} where Ndg is the diagonal matrix with the same diagonal as N. === Variance on number of transient visits === The variance on the number of visits to a transient state j with starting at a transient state i (before being absorbed) is the (i,j)-entry of the matrix N 2 := N ( 2 N dg − I t ) − N sq , {\displaystyle N_{2}:=N(2N_{\operatorname {dg} }-I_{t})-N_{\operatorname {sq} },} where Nsq is the Hadamard product of N with itself (i.e. each entry of N is squared). === Variance on number of steps === The variance on the number of steps before being absorbed when starting in transient state i is the ith entry of the vector ( 2 N − I t ) t − t sq , {\displaystyle (2N-I_{t})\mathbf {t} -\mathbf {t} _{\operatorname {sq} },} where tsq is the Hadamard product of t with itself (i.e., as with Nsq, each entry of t is squared). == Examples == === String generation === Consider the process of repeatedly flipping a fair coin until the sequence (heads, tails, heads) appears. This process is modeled by an absorbing Markov chain with transition matrix P = [ 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 0 1 / 2 0 0 1 / 2 0 0 0 1 ] . {\displaystyle P={\begin{bmatrix}1/2&1/2&0&0\\0&1/2&1/2&0\\1/2&0&0&1/2\\0&0&0&1\end{bmatrix}}.} The first state represents the empty string, the second state the string "H", the third state the string "HT", and the fourth state the string "HTH". Although in reality, the coin flips cease after the string "HTH" is generated, the perspective of the absorbing Markov chain is that the process has transitioned into the absorbing state representing the string "HTH" and, therefore, cannot leave. For this absorbing Markov chain, the fundamental matrix is N = ( I − Q ) − 1 = ( [ 1 0 0 0 1 0 0 0 1 ] − [ 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 0 ] ) − 1 = [ 1 / 2 − 1 / 2 0 0 1 / 2 − 1 / 2 − 1 / 2 0 1 ] − 1 = [ 4 4 2 2 4 2 2 2 2 ] . {\displaystyle {\begin{aligned}N&=(I-Q)^{-1}=\left({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}-{\begin{bmatrix}1/2&1/2&0\\0&1/2&1/2\\1/2&0&0\end{bmatrix}}\right)^{-1}\\[4pt]&={\begin{bmatrix}1/2&-1/2&0\\0&1/2&-1/2\\-1/2&0&1\end{bmatrix}}^{-1}={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}.\end{aligned}}} The expected number of steps starting from each of the transient states is t = N 1 = [ 4 4 2 2 4 2 2 2 2 ] [ 1 1 1 ] = [ 10 8 6 ] . {\displaystyle \mathbf {t} =N\mathbf {1} ={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}10\\8\\6\end{bmatrix}}.} Therefore, the expected number of coin flips before observing the sequence (heads, tails, heads) is 10, the entry for the state representing the empty string. === Games of chance === Games based entirely on chance can be modeled by an absorbing Markov chain. A classic example of this is the ancient Indian board game Snakes and Ladders. The graph on the left plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To determine the expected number of turns to complete the game, compute the vector t as described above and examine tstart, which is approximately 39.2. === Infectious disease testing === Infectious disease testing, either of blood products or in medical clinics, is often taught as an example of an absorbing Markov chain. The public U.S. Centers for Disease Control and Prevention (CDC) model for HIV and for hepatitis B, for example, illustrates the property that absorbing Markov chains can lead to the detection of disease, versus the loss of detection through other means. In the standard CDC model, the Markov chain has five states, a state in which the individual is uninfected, then a state with infected but undetectable virus, a state with detectable virus, and absorbing states of having quit/been lost from the clinic, or of having been detected (the goal). The typical rates of transition between the Markov states are the probability p per unit time of being infected with the virus, w for the rate of window period removal (time until virus is detectable), q for quit/loss rate from the system, and d for detection, assuming a typical rate λ {\displaystyle \lambda } at which the health system administers tests of the blood product or patients in question. It follows that we can "walk along" the Markov model to identify the overall probability of detection for a person starting as undetected, by multiplying the probabilities of transition to each next state of the model as: p ( p + q ) w ( w + q ) d ( d + q ) {\displaystyle {\frac {p}{(p+q)}}{\frac {w}{(w+q)}}{\frac {d}{(d+q)}}} . The subsequent total absolute number of false negative tests—the primary CDC concern—would then be the rate of tests, multiplied by the probability of reaching the infected but undetectable state, times the duration of staying in the infected undetectable state: p ( p + q ) 1 ( w + q ) λ {\displaystyle {\frac {p}{(p+q)}}{\frac {1}{(w+q)}}\lambda } .

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  Online machine learning

  In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., prediction of prices in the financial international markets. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches. Online machine learning algorithms find applications in a wide variety of fields such as sponsored search to maximize ad revenue, portfolio optimization, shortest path prediction (with stochastic weights, e.g. traffic on roads for a maps application), spam filtering, real-time fraud detection, dynamic pricing for e-commerce, etc. There is also growing interest in usage of online learning paradigms for LLMs to enable continuous, real-time adaptation after the initial training. == Introduction == In the setting of supervised learning, a function of f : X → Y {\displaystyle f:X\to Y} is to be learned, where X {\displaystyle X} is thought of as a space of inputs and Y {\displaystyle Y} as a space of outputs, that predicts well on instances that are drawn from a joint probability distribution p ( x , y ) {\displaystyle p(x,y)} on X × Y {\displaystyle X\times Y} . In reality, the learner never knows the true distribution p ( x , y ) {\displaystyle p(x,y)} over instances. Instead, the learner usually has access to a training set of examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} . In this setting, the loss function is given as V : Y × Y → R {\displaystyle V:Y\times Y\to \mathbb {R} } , such that V ( f ( x ) , y ) {\displaystyle V(f(x),y)} measures the difference between the predicted value f ( x ) {\displaystyle f(x)} and the true value y {\displaystyle y} . The ideal goal is to select a function f ∈ H {\displaystyle f\in {\mathcal {H}}} , where H {\displaystyle {\mathcal {H}}} is a space of functions called a hypothesis space, so that some notion of total loss is minimized. Depending on the type of model (statistical or adversarial), one can devise different notions of loss, which lead to different learning algorithms. == Statistical view of online learning == In statistical learning models, the training sample ( x i , y i ) {\displaystyle (x_{i},y_{i})} are assumed to have been drawn from the true distribution p ( x , y ) {\displaystyle p(x,y)} and the objective is to minimize the expected "risk" I [ f ] = E [ V ( f ( x ) , y ) ] = ∫ V ( f ( x ) , y ) d p ( x , y ) . {\displaystyle I[f]=\mathbb {E} [V(f(x),y)]=\int V(f(x),y)\,dp(x,y)\ .} A common paradigm in this situation is to estimate a function f ^ {\displaystyle {\hat {f}}} through empirical risk minimization or regularized empirical risk minimization (usually Tikhonov regularization). The choice of loss function here gives rise to several well-known learning algorithms such as regularized least squares and support vector machines. A purely online model in this category would learn based on just the new input ( x t + 1 , y t + 1 ) {\displaystyle (x_{t+1},y_{t+1})} , the current best predictor f t {\displaystyle f_{t}} and some extra stored information (which is usually expected to have storage requirements independent of training data size). For many formulations, for example nonlinear kernel methods, true online learning is not possible, though a form of hybrid online learning with recursive algorithms can be used where f t + 1 {\displaystyle f_{t+1}} is permitted to depend on f t {\displaystyle f_{t}} and all previous data points ( x 1 , y 1 ) , … , ( x t , y t ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{t},y_{t})} . In this case, the space requirements are no longer guaranteed to be constant since it requires storing all previous data points, but the solution may take less time to compute with the addition of a new data point, as compared to batch learning techniques. A common strategy to overcome the above issues is to learn using mini-batches, which process a small batch of b ≥ 1 {\displaystyle b\geq 1} data points at a time, this can be considered as pseudo-online learning for b {\displaystyle b} much smaller than the total number of training points. Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks. === Example: linear least squares === The simple example of linear least squares is used to explain a variety of ideas in online learning. The ideas are general enough to be applied to other settings, for example, with other convex loss functions. === Batch learning === Consider the setting of supervised learning with f {\displaystyle f} being a linear function to be learned: f ( x j ) = ⟨ w , x j ⟩ = w ⋅ x j {\displaystyle f(x_{j})=\langle w,x_{j}\rangle =w\cdot x_{j}} where x j ∈ R d {\displaystyle x_{j}\in \mathbb {R} ^{d}} is a vector of inputs (data points) and w ∈ R d {\displaystyle w\in \mathbb {R} ^{d}} is a linear filter vector. The goal is to compute the filter vector w {\displaystyle w} . To this end, a square loss function V ( f ( x j ) , y j ) = ( f ( x j ) − y j ) 2 = ( ⟨ w , x j ⟩ − y j ) 2 {\displaystyle V(f(x_{j}),y_{j})=(f(x_{j})-y_{j})^{2}=(\langle w,x_{j}\rangle -y_{j})^{2}} is used to compute the vector w {\displaystyle w} that minimizes the empirical loss I n [ w ] = ∑ j = 1 n V ( ⟨ w , x j ⟩ , y j ) = ∑ j = 1 n ( x j T w − y j ) 2 {\displaystyle I_{n}[w]=\sum _{j=1}^{n}V(\langle w,x_{j}\rangle ,y_{j})=\sum _{j=1}^{n}(x_{j}^{\mathsf {T}}w-y_{j})^{2}} where y j ∈ R . {\displaystyle y_{j}\in \mathbb {R} .} Let X {\displaystyle X} be the i × d {\displaystyle i\times d} data matrix and y ∈ R i {\displaystyle y\in \mathbb {R} ^{i}} is the column vector of target values after the arrival of the first i {\displaystyle i} data points. Assuming that the covariance matrix Σ i = X T X {\displaystyle \Sigma _{i}=X^{\mathsf {T}}X} is invertible (otherwise it is preferential to proceed in a similar fashion with Tikhonov regularization), the best solution f ∗ ( x ) = ⟨ w ∗ , x ⟩ {\displaystyle f^{}(x)=\langle w^{},x\rangle } to the linear least squares problem is given by w ∗ = ( X T X ) − 1 X T y = Σ i − 1 ∑ j = 1 i x j y j . {\displaystyle w^{}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y=\Sigma _{i}^{-1}\sum _{j=1}^{i}x_{j}y_{j}.} Now, calculating the covariance matrix Σ i = ∑ j = 1 i x j x j T {\displaystyle \Sigma _{i}=\sum _{j=1}^{i}x_{j}x_{j}^{\mathsf {T}}} takes time O ( i d 2 ) {\displaystyle O(id^{2})} , inverting the d × d {\displaystyle d\times d} matrix takes time O ( d 3 ) {\displaystyle O(d^{3})} , while the rest of the multiplication takes time O ( d 2 ) {\displaystyle O(d^{2})} , giving a total time of O ( i d 2 + d 3 ) {\displaystyle O(id^{2}+d^{3})} . When there are n {\displaystyle n} total points in the dataset, to recompute the solution after the arrival of every datapoint i = 1 , … , n {\displaystyle i=1,\ldots ,n} , the naive approach will have a total complexity O ( n 2 d 2 + n d 3 ) {\displaystyle O(n^{2}d^{2}+nd^{3})} . Note that when storing the matrix Σ i {\displaystyle \Sigma _{i}} , then updating it at each step needs only adding x i + 1 x i + 1 T {\displaystyle x_{i+1}x_{i+1}^{\mathsf {T}}} , which takes O ( d 2 ) {\displaystyle O(d^{2})} time, reducing the total time to O ( n d 2 + n d 3 ) = O ( n d 3 ) {\displaystyle O(nd^{2}+nd^{3})=O(nd^{3})} , but with an additional storage space of O ( d 2 ) {\displaystyle O(d^{2})} to store Σ i {\displaystyle \Sigma _{i}} . === Online learning: recursive least squares === The recursive least squares (RLS) algorithm considers an online approach to the least squares problem. It can be shown that by initialising w 0 = 0 ∈ R d {\displaystyle \textstyle w_{0}=0\in \mathbb {R} ^{d}} and Γ 0 = I ∈ R d × d {\displaystyle \textstyle \Gamma _{0}=I\in \mathbb {R} ^{d\times d}} , the solution of the linear least squares problem given in the previous section can be computed by the following iteration: Γ i = Γ i − 1 − Γ i − 1 x i x i T Γ i − 1 1 + x i T Γ i − 1 x i {\displaystyle \Gamma _{i}=\Gamma _{i-1}-{\frac {\Gamma _{i-1}x_{i}x_{i}^{\mathsf {T}}\Gamma _{i-1}}{1+x_{i}^{\mathsf {T}}\Gamma _{i-1}x_{i}}}} w i = w i − 1 − Γ i x i ( x i T w i − 1 − y i ) {\displaystyle w_{i}=w_{i-1}-\Gamma _{i}x_{i}\left(x_{i}^{\mathsf {T}}w_{

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  PropBank

  PropBank is a corpus that is annotated with verbal propositions and their arguments—a "proposition bank". Although "PropBank" refers to a specific corpus produced by Martha Palmer et al., the term propbank is also coming to be used as a common noun referring to any corpus that has been annotated with propositions and their arguments. The PropBank project has played a role in research in natural language processing, and has been used in semantic role labelling. == Comparison == PropBank differs from FrameNet, the resource to which it is most frequently compared, in several ways. PropBank is a verb-oriented resource, while FrameNet is centered on the more abstract notion of frames, which generalizes descriptions across similar verbs (e.g. "describe" and "characterize") as well as nouns and other words (e.g. "description"). PropBank does not annotate events or states of affairs described using nouns. PropBank commits to annotating all verbs in a corpus, whereas the FrameNet project chooses sets of example sentences from a large corpus and only in a few cases has annotated longer continuous stretches of text. PropBank-style annotations often remain close to the syntactic level, while FrameNet-style annotations are sometimes more semantically motivated. From the start, PropBank was developed with the idea of serving as training data for machine learning-based semantic role labeling systems in mind. It requires that all arguments to a verb be syntactic constituents and different senses of a word are only distinguished if the differences bear on the arguments. Due to such differences, semantic role labeling with respect to PropBank is often a somewhat easier task than producing FrameNet-style annotations.

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  Information gain ratio

  In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information. == Information gain calculation == Information gain is the reduction in entropy produced from partitioning a set with attributes a {\displaystyle a} and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle {\text{IG}}(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where T {\displaystyle T} is a random variable and H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0. == Split information calculation == The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) {\displaystyle {\text{SplitInformation}}(X)=-\sum _{i=1}^{n}{{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}\log {_{2}}{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}}} where X {\displaystyle X} is a discrete random variable with possible values x 1 , x 2 , . . . , x i {\displaystyle {x_{1},x_{2},...,x_{i}}} and N ( x i ) {\displaystyle N(x_{i})} being the number of times that x i {\displaystyle x_{i}} occurs divided by the total count of events N ( x ) {\displaystyle N(x)} where x {\displaystyle x} is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation. == Information gain ratio calculation == The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) {\displaystyle {\text{IGR}}(T,a)={\text{IG}}(T,a)/{\text{SplitInformation}}(T)} IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T | a ) log ⁡ P ( T | a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) {\displaystyle {\text{IGR}}(T,a)={\frac {-\sum _{i=1}^{n}{\mathrm {P} (T)\log \mathrm {P} (T)}-(-\sum _{i=1}^{n}{\mathrm {P} (T|a)\log \mathrm {P} (T|a)})}{-\sum _{i=1}^{n}{{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}\log {_{2}}{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}}}}} == Example == Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more. == Advantages == Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root. == Disadvantages == Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values. == Difference from information gain == Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.

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  Calibration (statistics)

  There are two main uses of the term calibration in statistics that denote special types of statistical inference problems. Calibration can mean a reverse process to regression, where instead of a future dependent variable being predicted from known explanatory variables, a known observation of the dependent variables is used to predict a corresponding explanatory variable; procedures in statistical classification to determine class membership probabilities which assess the uncertainty of a given new observation belonging to each of the already established classes. In addition, calibration is used in statistics with the usual general meaning of calibration. For example, model calibration can be also used to refer to Bayesian inference about the value of a model's parameters, given some data set, or more generally to any type of fitting of a statistical model. As Philip Dawid puts it, "a forecaster is well calibrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent." == In classification == Calibration in classification means transforming classifier scores into class membership probabilities. An overview of calibration methods for two-class and multi-class classification tasks is given by Gebel (2009). A classifier might separate the classes well, but be poorly calibrated, meaning that the estimated class probabilities are far from the true class probabilities. In this case, a calibration step may help improve the estimated probabilities. A variety of metrics exist that are aimed to measure the extent to which a classifier produces well-calibrated probabilities. Foundational work includes the Expected Calibration Error (ECE). Into the 2020s, variants include the Adaptive Calibration Error (ACE) and the Test-based Calibration Error (TCE), which address limitations of the ECE metric that may arise when classifier scores concentrate on narrow subset of the [0,1] range. A 2020s advancement in calibration assessment is the introduction of the Estimated Calibration Index (ECI). The ECI extends the concepts of the Expected Calibration Error (ECE) to provide a more nuanced measure of a model's calibration, particularly addressing overconfidence and underconfidence tendencies. Originally formulated for binary settings, the ECI has been adapted for multiclass settings, offering both local and global insights into model calibration. This framework aims to overcome some of the theoretical and interpretative limitations of existing calibration metrics. Through a series of experiments, Famiglini et al. demonstrate the framework's effectiveness in delivering a more accurate understanding of model calibration levels and discuss strategies for mitigating biases in calibration assessment. An online tool has been proposed to compute both ECE and ECI. The following univariate calibration methods exist for transforming classifier scores into class membership probabilities in the two-class case: Assignment value approach, see Garczarek (2002) Bayes approach, see Bennett (2002) Isotonic regression, see Zadrozny and Elkan (2002) Platt scaling (a form of logistic regression), see Lewis and Gale (1994) and Platt (1999) Bayesian Binning into Quantiles (BBQ) calibration, see Naeini, Cooper, Hauskrecht (2015) Beta calibration, see Kull, Filho, Flach (2017) === In probability prediction and forecasting === In prediction and forecasting, a Brier score is sometimes used to assess prediction accuracy of a set of predictions, specifically that the magnitude of the assigned probabilities track the relative frequency of the observed outcomes. Philip E. Tetlock employs the term "calibration" in this sense in his 2015 book Superforecasting. This differs from accuracy and precision. For example, as expressed by Daniel Kahneman, "if you give all events that happen a probability of .6 and all the events that don't happen a probability of .4, your discrimination is perfect but your calibration is miserable". In meteorology, in particular, as concerns weather forecasting, a related mode of assessment is known as forecast skill. == In regression == The calibration problem in regression is the use of known data on the observed relationship between a dependent variable and an independent variable to make estimates of other values of the independent variable from new observations of the dependent variable. This can be known as "inverse regression"; there is also sliced inverse regression. The following multivariate calibration methods exist for transforming classifier scores into class membership probabilities in the case with classes count greater than two: Reduction to binary tasks and subsequent pairwise coupling, see Hastie and Tibshirani (1998) Dirichlet calibration, see Gebel (2009) === Example === One example is that of dating objects, using observable evidence such as tree rings for dendrochronology or carbon-14 for radiometric dating. The observation is caused by the age of the object being dated, rather than the reverse, and the aim is to use the method for estimating dates based on new observations. The problem is whether the model used for relating known ages with observations should aim to minimise the error in the observation, or minimise the error in the date. The two approaches will produce different results, and the difference will increase if the model is then used for extrapolation at some distance from the known results.

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