AI Chatbot Ethics

AI Chatbot Ethics — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Symbolic regression

  Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity. No particular model is provided as a starting point for symbolic regression. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators, analytic functions, constants, and state variables. Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming, as well as more recent methods utilizing Bayesian methods and neural networks. Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy. Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system, as well as improving generalisability and extrapolation behaviour by preventing overfitting. Accuracy and simplicity may be left as two separate objectives of the regression—in which case the optimum solutions form a Pareto front—or they may be combined into a single objective by means of a model selection principle such as minimum description length. It has been proven that symbolic regression is an NP-hard problem. Nevertheless, if the sought-for equation is not too complex it is possible to solve the symbolic regression problem exactly by generating every possible function (built from some predefined set of operators) and evaluating them on the dataset in question. == Difference from classical regression == While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters. This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system. Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity. == Benchmarking == === SRBench === In 2021, SRBench was proposed as a large benchmark for symbolic regression. In its inception, SRBench featured 14 symbolic regression methods, 7 other ML methods, and 252 datasets from PMLB. The benchmark intends to be a living project: it encourages the submission of improvements, new datasets, and new methods, to keep track of the state of the art in SR. === SRBench Competition 2022 === In 2022, SRBench announced the competition Interpretable Symbolic Regression for Data Science, which was held at the GECCO conference in Boston, MA. The competition pitted nine leading symbolic regression algorithms against each other on a novel set of data problems and considered different evaluation criteria. The competition was organized in two tracks, a synthetic track and a real-world data track. ==== Synthetic Track ==== In the synthetic track, methods were compared according to five properties: re-discovery of exact expressions; feature selection; resistance to local optima; extrapolation; and sensitivity to noise. Rankings of the methods were: QLattice PySR (Python Symbolic Regression) uDSR (Deep Symbolic Optimization) ==== Real-world Track ==== In the real-world track, methods were trained to build interpretable predictive models for 14-day forecast counts of COVID-19 cases, hospitalizations, and deaths in New York State. These models were reviewed by a subject expert and assigned trust ratings and evaluated for accuracy and simplicity. The ranking of the methods was: uDSR (Deep Symbolic Optimization) QLattice geneticengine (Genetic Engine) == Non-standard methods == Most symbolic regression algorithms prevent combinatorial explosion by implementing evolutionary algorithms that iteratively improve the best-fit expression over many generations. Recently, researchers have proposed algorithms utilizing other tactics in AI. Silviu-Marian Udrescu and Max Tegmark developed the "AI Feynman" algorithm, which attempts symbolic regression by training a neural network to represent the mystery function, then runs tests against the neural network to attempt to break up the problem into smaller parts. For example, if f ( x 1 , . . . , x i , x i + 1 , . . . , x n ) = g ( x 1 , . . . , x i ) + h ( x i + 1 , . . . , x n ) {\displaystyle f(x_{1},...,x_{i},x_{i+1},...,x_{n})=g(x_{1},...,x_{i})+h(x_{i+1},...,x_{n})} , tests against the neural network can recognize the separation and proceed to solve for g {\displaystyle g} and h {\displaystyle h} separately and with different variables as inputs. This is an example of divide and conquer, which reduces the size of the problem to be more manageable. AI Feynman also transforms the inputs and outputs of the mystery function in order to produce a new function which can be solved with other techniques, and performs dimensional analysis to reduce the number of independent variables involved. The algorithm was able to "discover" 100 equations from The Feynman Lectures on Physics, while a leading software using evolutionary algorithms, Eureqa, solved only 71. AI Feynman, in contrast to classic symbolic regression methods, requires a very large dataset in order to first train the neural network and is naturally biased towards equations that are common in elementary physics.

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  Ilastik

  ilastik is free open source software for image classification and segmentation. No previous experience in image processing is required to run the software. Since 2018 ilastik is further developed and maintained by Anna Kreshuk's group at European Molecular Biology Laboratory. == Features == ilastik allows user to annotate an arbitrary number of classes in images with a mouse interface. Using these user annotations and the generic (nonlinear) image features, the user can train a random forest classifier. Trained ilastik classifiers can be applied new data not included in the training set in ilastik via its batch processing functionality, or without using the graphical user interface, in headless mode. ilastik can be integrated into various related tools: Pre-trained workflows can be executed directly from ImageJ/Fiji using the ilastik-ImageJ plugin. Pre-trained ilastik Pixel Classification workflows can be run directly in Python with the ilastik Python package, which is available via conda. ilastik has a CellProfiler module to use ilastik classifiers to process images within a CellProfiler framework. == History == ilastik was first released in 2011 by scientists at the Heidelberg Collaboratory for Image Processing (HCI), University of Heidelberg. == Application == The Interactive Learning and Segmentation Toolkit Carving Cell classification and neuron classification Synapse detection Cell tracking Neural Network Classification == Resources == ilastik project is hosted on GitHub. It is a collaborative project, any contributions such as comments, bug reports, bug fixes or code contributions are welcome. The ilastik team can be contacted for user support on the image.sc forum.

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  Taguchi loss function

  The Taguchi loss function is graphical depiction of loss developed by the Japanese business statistician Genichi Taguchi to describe a phenomenon affecting the value of products produced by a company. Praised by Dr. W. Edwards Deming (the business guru of the 1980s American quality movement), it made clear the concept that quality does not suddenly plummet when, for instance, a machinist exceeds a rigid blueprint tolerance. Instead 'loss' in value progressively increases as variation increases from the intended condition. This was considered a breakthrough in describing quality, and helped fuel the continuous improvement movement. The concept of Taguchi's quality loss function was in contrast with the American concept of quality, popularly known as goal post philosophy, the concept given by American quality guru Phil Crosby. Goal post philosophy emphasizes that if a product feature doesn't meet the designed specifications it is termed as a product of poor quality (rejected), irrespective of amount of deviation from the target value (mean value of tolerance zone). This concept has similarity with the concept of scoring a 'goal' in the game of football or hockey, because a goal is counted 'one' irrespective of the location of strike of the ball in the 'goal post', whether it is in the center or towards the corner. This means that if the product dimension goes out of the tolerance limit the quality of the product drops suddenly. Through his concept of the quality loss function, Taguchi explained that from the customer's point of view this drop of quality is not sudden. The customer experiences a loss of quality the moment product specification deviates from the 'target value'. This 'loss' is depicted by a quality loss function and it follows a parabolic curve mathematically given by L = k(y–m)2, where m is the theoretical 'target value' or 'mean value' and y is the actual size of the product, k is a constant and L is the loss. This means that if the difference between 'actual size' and 'target value' i.e. (y–m) is large, loss would be more, irrespective of tolerance specifications. In Taguchi's view tolerance specifications are given by engineers and not by customers; what the customer experiences is 'loss'. This equation is true for a single product; if 'loss' is to be calculated for multiple products the loss function is given by L = k[S2 + ( y ¯ {\displaystyle {\bar {y}}} – m)2], where S2 is the 'variance of product size' and y ¯ {\displaystyle {\bar {y}}} is the average product size. == Overview == The Taguchi loss function is important for a number of reasons—primarily, to help engineers better understand the importance of designing for variation.

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  Self-organizing map

  A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. For example, a data set with p {\displaystyle p} variables measured in n {\displaystyle n} observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a two-dimensional "map" such that observations in proximal clusters have more similar values than observations in distal clusters. This can make high-dimensional data easier to visualize and analyze. A SOM is a type of artificial neural network but is trained using competitive learning rather than the error-correction learning (e.g., backpropagation with gradient descent) used by other artificial neural networks. The SOM was introduced by the Finnish professor Teuvo Kohonen in the 1980s and therefore is sometimes called a Kohonen map or Kohonen network. The Kohonen map or network is a computationally convenient abstraction building on biological models of neural systems from the 1970s and morphogenesis models dating back to Alan Turing in the 1950s. SOMs create internal representations reminiscent of the cortical homunculus, a distorted representation of the human body, based on a neurological "map" of the areas and proportions of the human brain dedicated to processing sensory functions, for different parts of the body. == Overview == Self-organizing maps, like most artificial neural networks, operate in two modes: training and mapping. First, training uses an input data set (the "input space") to generate a lower-dimensional representation of the input data (the "map space"). Second, mapping classifies additional input data using the generated map. The goal of training is to represent an input space with p dimensions as a map space with n dimensions, where p > n. Specifically, an input space with p variables is said to have p dimensions. A map space consists of components called "nodes" or "neurons", which are arranged as a hexagonal or rectangular grid with two dimensions. The number of nodes and their arrangement are specified beforehand based on the larger goals of the analysis and exploration of the data. Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After training, the map can be used to classify additional observations for the input space by finding the node with the closest weight vector (smallest distance metric) to the input space vector. == Learning algorithm == The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain. The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations. The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron whose weight vector is most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM grid are adjusted towards the input vector. The magnitude of the change decreases with time and with the grid-distance from the BMU. The update formula for a neuron v with weight vector Wv(s) is W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} , where s is the step index, t is an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s. Depending on the implementations, t can scan the training data set systematically (t is 0, 1, 2...T-1, then repeat, T being the training sample's size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function θ(u, v, s) (also called function of lateral interaction) depends on the grid-distance between the BMU (neuron u) and neuron v. In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and Mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates. In some implementations, the learning coefficient α and the neighborhood function θ decrease steadily with increasing s, in others (in particular those where t scans the training data set) they decrease in step-wise fashion, once every T steps. This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net. During mapping, there will be one single winning neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector. While representing input data as vectors has been emphasized in this article, any kind of object which can be represented digitally, which has an appropriate distance measure associated with it, and in which the necessary operations for training are possible can be used to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps. === Algorithm === Randomize the node weight vectors in a map For s = 0 , 1 , 2 , . . . , λ {\displaystyle s=0,1,2,...,\lambda } Randomly pick an input vector D ( t ) {\displaystyle {D}(t)} Find the node in the map closest to the input vector. This node is the best matching unit (BMU). Denote it by u {\displaystyle u} For each node v {\displaystyle v} , update its vector by pulling it closer to the input vector: W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} The variable names mean the following, with vectors in bold, s {\displaystyle s} is the current iteration λ {\displaystyle \lambda } is the iteration limit t {\displaystyle t} is the index of the target input data vector in the input data set D {\displaystyle \mathbf {D} } D ( t ) {\displaystyle {D}(t)} is a target input data vector v {\displaystyle v} is the index of the node in the map W v {\displaystyle \mathbf {W} _{v}} is the current weight vector of node v {\displaystyle v} u {\displaystyle u} is the index of the best matching unit (BMU) in the map θ ( u , v , s ) {\displaystyle \theta (u,v,s)} is the neighbourhood function, α ( s ) {\displaystyle \alpha (s)} is the learning rate schedule. The key design choices are the shape of the SOM, the neighbourhood function, and the learning rate schedule. The idea of the neighborhood function is to make it such that the BMU is updated the most, its immediate neighbors are updated a little less, and so on. The idea of the learning rate schedule is to make it so that the map updates are large at the start, and gradually stop updating. For example, if we want to learn a SOM using a square grid, we can index it using ( i , j ) {\displaystyle (i,j)} where both i , j ∈ 1 : N {\displaystyle i,j\in 1:N} . The neighborhood function can make it so that the BMU updates in full, the nearest neighbors update in half, and their neighbors update in half again, etc. θ ( ( i , j ) , ( i ′ , j ′ ) , s ) = 1 2 | i − i ′ | + | j − j ′ | = { 1 if i = i ′ , j = j ′ 1 / 2 if | i − i ′ | + | j − j ′ | = 1 1 / 4 if | i − i ′ | + | j − j ′ | = 2 ⋯ ⋯ {\displaystyle \theta ((i,j),(i',j'),s)={\frac {1}{2^{|i-i'|+|j-j'|}}}={\begin{cases}1&{\text{if }}i=i',j=j'\\1/2&{\text{if

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  Lazy learning

  (Not to be confused with the lazy learning regime, see Neural tangent kernel). In machine learning, lazy learning is a learning method in which generalization of the training data is, in theory, delayed until a query is made to the system, as opposed to eager learning, where the system tries to generalize the training data before receiving queries. The primary motivation for employing lazy learning, as in the K-nearest neighbors algorithm, used by online recommendation systems ("people who viewed/purchased/listened to this movie/item/tune also ...") is that the data set is continuously updated with new entries (e.g., new items for sale at Amazon, new movies to view at Netflix, new clips at YouTube, new music at Spotify or Pandora). Because of the continuous update, the "training data" would be rendered obsolete in a relatively short time especially in areas like books and movies, where new best-sellers or hit movies/music are published/released continuously. Therefore, one cannot really talk of a "training phase". Lazy classifiers are most useful for large, continuously changing datasets with few attributes that are commonly queried. Specifically, even if a large set of attributes exist - for example, books have a year of publication, author/s, publisher, title, edition, ISBN, selling price, etc. - recommendation queries rely on far fewer attributes - e.g., purchase or viewing co-occurrence data, and user ratings of items purchased/viewed. == Advantages == The main advantage gained in employing a lazy learning method is that the target function will be approximated locally, such as in the k-nearest neighbor algorithm. Because the target function is approximated locally for each query to the system, lazy learning systems can simultaneously solve multiple problems and deal successfully with changes in the problem domain. At the same time they can reuse a lot of theoretical and applied results from linear regression modelling (notably PRESS statistic) and control. It is said that the advantage of this system is achieved if the predictions using a single training set are only developed for few objects. This can be demonstrated in the case of the k-NN technique, which is instance-based and function is only estimated locally. == Disadvantages == Theoretical disadvantages with lazy learning include: The large space requirement to store the entire training dataset. In practice, this is not an issue because of advances in hardware and the relatively small number of attributes (e.g., as co-occurrence frequency) that need to be stored. Particularly noisy training data increases the case base unnecessarily, because no abstraction is made during the training phase. In practice, as stated earlier, lazy learning is applied to situations where any learning performed in advance soon becomes obsolete because of changes in the data. Also, for the problems for which lazy learning is optimal, "noisy" data does not really occur - the purchaser of a book has either bought another book or hasn't. Lazy learning methods are usually slower to evaluate. In practice, for very large databases with high concurrency loads, the queries are not postponed until actual query time, but recomputed in advance on a periodic basis - e.g., nightly, in anticipation of future queries, and the answers stored. This way, the next time new queries are asked about existing entries in the database, the answers are merely looked up rapidly instead of having to be computed on the fly, which would almost certainly bring a high-concurrency multi-user system to its knees. Larger training data also entail increased cost. Particularly, there is the fixed amount of computational cost, where a processor can only process a limited amount of training data points. There are standard techniques to improve re-computation efficiency so that a particular answer is not recomputed unless the data that impact this answer has changed (e.g., new items, new purchases, new views). In other words, the stored answers are updated incrementally. This approach, used by large e-commerce or media sites, has long been used in the Entrez portal of the National Center for Biotechnology Information (NCBI) to precompute similarities between the different items in its large datasets: biological sequences, 3-D protein structures, published-article abstracts, etc. Because "find similar" queries are asked so frequently, the NCBI uses highly parallel hardware to perform nightly recomputation. The recomputation is performed only for new entries in the datasets against each other and against existing entries: the similarity between two existing entries need not be recomputed. == Examples of Lazy Learning Methods == K-nearest neighbors, which is a special case of instance-based learning. Local regression. Lazy naive Bayes rules, which are extensively used in commercial spam detection software. Here, the spammers keep getting smarter and revising their spamming strategies, and therefore the learning rules must also be continually updated.

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  Evolutionary algorithm

  Evolutionary algorithms (EA) reproduce essential elements of biological evolution in a computer algorithm in order to solve "difficult" problems, at least approximately, for which no exact or satisfactory solution methods are known. They are metaheuristics and population-based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, recombination and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolution (microevolutionary processes) and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity. == Generic definition == The following is an example of a generic evolutionary algorithm: Randomly generate the initial population of individuals, the first generation. Evaluate the fitness of each individual in the population. Check, if the goal is reached and the algorithm can be terminated. Select individuals as parents, preferably of higher fitness. Produce offspring with optional crossover (mimicking reproduction). Apply mutation operations on the offspring. Select individuals preferably of lower fitness for replacement with new individuals (mimicking natural selection). Return to 2 == Types == Similar techniques differ in genetic representation and other implementation details, and the nature of the particular applied problem. Genetic algorithm – This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved), by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in optimization problems. Genetic programming – Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming: Cartesian genetic programming Gene expression programming Grammatical evolution Linear genetic programming Multi expression programming Evolutionary programming – Similar to evolution strategy, but with a deterministic selection of all parents. Evolution strategy (ES) – Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks. CMA-ES Natural evolution strategy Differential evolution – Based on vector differences and is therefore primarily suited for numerical optimization problems. Coevolutionary algorithm – Similar to genetic algorithms and evolution strategies, but the created solutions are compared on the basis of their outcomes from interactions with other solutions. Solutions can either compete or cooperate during the search process. Coevolutionary algorithms are often used in scenarios where the fitness landscape is dynamic, complex, or involves competitive interactions. Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect. Learning classifier system – Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression types. Fitness is typically determined with either a strength or accuracy based reinforcement learning or supervised learning approach. Quality–Diversity algorithms – QD algorithms simultaneously aim for high-quality and diverse solutions. Unlike traditional optimization algorithms that solely focus on finding the best solution to a problem, QD algorithms explore a wide variety of solutions across a problem space and keep those that are not just high performing, but also diverse and unique. == Theoretical background == The following theoretical principles apply to all or almost all EAs. === No free lunch theorem === The no free lunch theorem of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust. === Convergence === For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence under the condition that an optimum exists. Without loss of generality, a maximum search is assumed for the proof: From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation k {\displaystyle k} an improvement of the fitness F {\displaystyle F} of the respective best individual x ′ {\displaystyle x'} will occur with a probability P > 0 {\displaystyle P>0} . Thus: F ( x 1 ′ ) ≤ F ( x 2 ′ ) ≤ F ( x 3 ′ ) ≤ ⋯ ≤ F ( x k ′ ) ≤ ⋯ {\displaystyle F(x'_{1})\leq F(x'_{2})\leq F(x'_{3})\leq \cdots \leq F(x'_{k})\leq \cdots } I.e., the fitness values represent a monotonically non-decreasing sequence, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum. Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (see step 4 of the generic definition) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection. === Virtual alphabets === With the theory of virtual alphabets, David E. Goldberg showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion. == Comparison to other concepts == === Biological processes === A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis to become a mature p

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  Soft independent modelling of class analogies

  Soft independent modelling by class analogy (SIMCA) is a statistical method for supervised classification of data. The method requires a training data set consisting of samples (or objects) with a set of attributes and their class membership. The term soft refers to the fact the classifier can identify samples as belonging to multiple classes and not necessarily producing a classification of samples into non-overlapping classes. == Method == In order to build the classification models, the samples belonging to each class need to be analysed using principal component analysis (PCA); only the significant components are retained. For a given class, the resulting model then describes either a line (for one Principal Component or PC), plane (for two PCs) or hyper-plane (for more than two PCs). For each modelled class, the mean orthogonal distance of training data samples from the line, plane, or hyper-plane (calculated as the residual standard deviation) is used to determine a critical distance for classification. This critical distance is based on the F-distribution and is usually calculated using 95% or 99% confidence intervals. New observations are projected into each PC model and the residual distances calculated. An observation is assigned to the model class when its residual distance from the model is below the statistical limit for the class. The observation may be found to belong to multiple classes and a measure of goodness of the model can be found from the number of cases where the observations are classified into multiple classes. The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant. == Application == SIMCA as a method of classification has gained widespread use especially in applied statistical fields such as chemometrics and spectroscopic data analysis.

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  Aleph (ILP)

  Aleph (A Learning Engine for Proposing Hypotheses) is an inductive logic programming system introduced by Ashwin Srinivasan in 2001. As of 2022 it is still one of the most widely used inductive logic programming systems. It is based on the earlier system Progol. == Learning task == The input to Aleph is background knowledge, specified as a logic program, a language bias in the form of mode declarations, as well as positive and negative examples specified as ground facts. As output it returns a logic program which, together with the background knowledge, entails all of the positive examples and none of the negative examples. == Basic algorithm == Starting with an empty hypothesis, Aleph proceeds as follows: It chooses a positive example to generalise; if none are left, it aborts and outputs the current hypothesis. Then it constructs the bottom clause, that is, the most specific clause that is allowed by the mode declarations and covers the example. It then searches for a generalisation of the bottom clause that scores better on the chosen metric. It then adds the new clause to the hypothesis program and removes all examples that are covered by the new clause. == Search algorithm == Aleph searches for clauses in a top-down manner, using the bottom clause constructed in the preceding step to bound the search from below. It searches the refinement graph in a breadth-first manner, with tunable parameters to bound the maximal clause size and proof depth. It scores each clause using one of 13 different evaluation metrics, as chosen in advance by the user.

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  Kindara

  Kindara is a femtech company headquartered in Colorado that develops apps that help women identify their fertile window. The products are used for women trying to get pregnant, or women who want to track their menstrual cycle for overall health. Their latest product, Priya Fertility and Ovulation Monitor, maximizes a woman's chance of getting pregnancy by identifying her most fertile days. == Overview == Kindara was founded in 2011 by husband-and-wife team Will Sacks and Kati Bicknell. The company launched its free mobile application in 2012. Kindara's mobile application allows women to track signs of fertility, such as basal body temperature, cervical fluid, and the position of the cervix to determine when ovulation is occurring. Kindara also sells a thermometer, Wink, which records basal body temperature and syncs automatically to the Kindara fertility application. In 2018, Kindara was acquired by the company Prima-Temp.

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  Constrained clustering

  In computer science, constrained clustering is a class of semi-supervised learning algorithms. Typically, constrained clustering incorporates either a set of must-link constraints, cannot-link constraints, or both, with a data clustering algorithm. A cluster in which the members conform to all must-link and cannot-link constraints is called a chunklet. == Types of constraints == Both a must-link and a cannot-link constraint define a relationship between two data instances. Together, the sets of these constraints act as a guide for which a constrained clustering algorithm will attempt to find chunklets (clusters in the dataset which satisfy the specified constraints). A must-link constraint is used to specify that the two instances in the must-link relation should be associated with the same cluster. A cannot-link constraint is used to specify that the two instances in the cannot-link relation should not be associated with the same cluster. Some constrained clustering algorithms will abort if no such clustering exists which satisfies the specified constraints. Others will try to minimize the amount of constraint violation should it be impossible to find a clustering which satisfies the constraints. Constraints could also be used to guide the selection of a clustering model among several possible solutions. == Examples == Examples of constrained clustering algorithms include: COP K-means PCKmeans (Pairwise Constrained K-means) CMWK-Means (Constrained Minkowski Weighted K-Means)

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  Characteristic samples

  Characteristic samples is a concept in the field of grammatical inference, related to passive learning. In passive learning, an inference algorithm I {\displaystyle I} is given a set of pairs of strings and labels S {\displaystyle S} , and returns a representation R {\displaystyle R} that is consistent with S {\displaystyle S} . Characteristic samples consider the scenario when the goal is not only finding a representation consistent with S {\displaystyle S} , but finding a representation that recognizes a specific target language. A characteristic sample of language L {\displaystyle L} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} where: l ( s ) = 1 {\displaystyle l(s)=1} if and only if s ∈ L {\displaystyle s\in L} l ( s ) = − 1 {\displaystyle l(s)=-1} if and only if s ∉ L {\displaystyle s\notin L} Given the characteristic sample S {\displaystyle S} , I {\displaystyle I} 's output on it is a representation R {\displaystyle R} , e.g. an automaton, that recognizes L {\displaystyle L} . == Formal Definition == === The Learning Paradigm associated with Characteristic Samples === There are three entities in the learning paradigm connected to characteristic samples, the adversary, the teacher and the inference algorithm. Given a class of languages C {\displaystyle \mathbb {C} } and a class of representations for the languages R {\displaystyle \mathbb {R} } , the paradigm goes as follows: The adversary A {\displaystyle A} selects a language L ∈ C {\displaystyle L\in \mathbb {C} } and reports it to the teacher The teacher T {\displaystyle T} then computes a set of strings and label them correctly according to L {\displaystyle L} , trying to make sure that the inference algorithm will compute L {\displaystyle L} The adversary can add correctly labeled words to the set in order to confuse the inference algorithm The inference algorithm I {\displaystyle I} gets the sample and computes a representation R ∈ R {\displaystyle R\in \mathbb {R} } consistent with the sample. The goal is that when the inference algorithm receives a characteristic sample for a language L {\displaystyle L} , or a sample that subsumes a characteristic sample for L {\displaystyle L} , it will return a representation that recognizes exactly the language L {\displaystyle L} . === Sample === Sample S {\displaystyle S} is a set of pairs of the form ( s , l ( s ) ) {\displaystyle (s,l(s))} such that l ( s ) ∈ { − 1 , 1 } {\displaystyle l(s)\in \{-1,1\}} ==== Sample consistent with a language ==== We say that a sample S {\displaystyle S} is consistent with language L {\displaystyle L} if for every pair ( s , l ( s ) ) {\displaystyle (s,l(s))} in S {\displaystyle S} : l ( s ) = 1 if and only if s ∈ L {\displaystyle l(s)=1{\text{ if and only if }}s\in L} l ( s ) = − 1 if and only if s ∉ L {\displaystyle l(s)=-1{\text{ if and only if }}s\notin L} === Characteristic sample === Given an inference algorithm I {\displaystyle I} and a language L {\displaystyle L} , a sample S {\displaystyle S} that is consistent with L {\displaystyle L} is called a characteristic sample of L {\displaystyle L} for I {\displaystyle I} if: I {\displaystyle I} 's output on S {\displaystyle S} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . For every sample D {\displaystyle D} that is consistent with L {\displaystyle L} and also fulfils S ⊆ D {\displaystyle S\subseteq D} , I {\displaystyle I} 's output on D {\displaystyle D} is a representation R {\displaystyle R} that recognizes L {\displaystyle L} . A Class of languages C {\displaystyle \mathbb {C} } is said to have charistaristic samples if every L ∈ C {\displaystyle L\in \mathbb {C} } has a characteristic sample. == Related Theorems == === Theorem === If equivalence is undecidable for a class C {\textstyle \mathbb {C} } over Σ {\textstyle \Sigma } of cardinality bigger than 1, then C {\textstyle \mathbb {C} } doesn't have characteristic samples. ==== Proof ==== Given a class of representations C {\textstyle \mathbb {C} } such that equivalence is undecidable, for every polynomial p ( x ) {\displaystyle p(x)} and every n ∈ N {\displaystyle n\in \mathbb {N} } , there exist two representations r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} of sizes bounded by n {\displaystyle n} , that recognize different languages but are inseparable by any string of size bounded by p ( n ) {\displaystyle p(n)} . Assuming this is not the case, we can decide if r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} are equivalent by simulating their run on all strings of size smaller than p ( n ) {\displaystyle p(n)} , contradicting the assumption that equivalence is undecidable. === Theorem === If S 1 {\displaystyle S_{1}} is a characteristic sample for L 1 {\displaystyle L_{1}} and is also consistent with L 2 {\displaystyle L_{2}} , then every characteristic sample of L 2 {\displaystyle L_{2}} , is inconsistent with L 1 {\displaystyle L_{1}} . ==== Proof ==== Given a class C {\textstyle \mathbb {C} } that has characteristic samples, let R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} be representations that recognize L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} respectively. Under the assumption that there is a characteristic sample for L 1 {\displaystyle L_{1}} , S 1 {\displaystyle S_{1}} that is also consistent with L 2 {\displaystyle L_{2}} , we'll assume falsely that there exist a characteristic sample for L 2 {\displaystyle L_{2}} , S 2 {\displaystyle S_{2}} that is consistent with L 1 {\displaystyle L_{1}} . By the definition of characteristic sample, the inference algorithm I {\displaystyle I} must return a representation which recognizes the language if given a sample that subsumes the characteristic sample itself. But for the sample S 1 ∪ S 2 {\displaystyle S_{1}\cup S_{2}} , the answer of the inferring algorithm needs to recognize both L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , in contradiction. === Theorem === If a class is polynomially learnable by example based queries, it is learnable with characteristic samples. == Polynomialy characterizable classes == === Regular languages === The proof that DFA's are learnable using characteristic samples, relies on the fact that every regular language has a finite number of equivalence classes with respect to the right congruence relation, ∼ L {\displaystyle \sim _{L}} (where x ∼ L y {\displaystyle x\sim _{L}y} for x , y ∈ Σ ∗ {\displaystyle x,y\in \Sigma ^{}} if and only if ∀ z ∈ Σ ∗ : x z ∈ L ↔ y z ∈ L {\displaystyle \forall z\in \Sigma ^{}:xz\in L\leftrightarrow yz\in L} ). Note that if x {\displaystyle x} , y {\displaystyle y} are not congruent with respect to ∼ L {\displaystyle \sim _{L}} , there exists a string z {\displaystyle z} such that x z ∈ L {\displaystyle xz\in L} but y z ∉ L {\displaystyle yz\notin L} or vice versa, this string is called a separating suffix. ==== Constructing a characteristic sample ==== The construction of a characteristic sample for a language L {\displaystyle L} by the teacher goes as follows. Firstly, by running a depth first search on a deterministic automaton A {\displaystyle A} recognizing L {\displaystyle L} , starting from its initial state, we get a suffix closed set of words, W {\displaystyle W} , ordered in shortlex order. From the fact above, we know that for every two states in the automaton, there exists a separating suffix that separates between every two strings that the run of A {\displaystyle A} on them ends in the respective states. We refer to the set of separating suffixes as S {\displaystyle S} . The labeled set (sample) of words the teacher gives the adversary is { ( w , l ( w ) ) | w ∈ W ⋅ S ∪ W ⋅ Σ ⋅ S } {\displaystyle \{(w,l(w))|w\in W\cdot S\cup W\cdot \Sigma \cdot S\}} where l ( w ) {\displaystyle l(w)} is the correct label of w {\displaystyle w} (whether it is in L {\displaystyle L} or not). We may assume that ϵ ∈ S {\displaystyle \epsilon \in S} . ==== Constructing a deterministic automata ==== Given the sample from the adversary W {\displaystyle W} , the construction of the automaton by the inference algorithm I {\displaystyle I} starts with defining P = prefix ( W ) {\displaystyle P={\text{prefix}}(W)} and S = suffix ( W ) {\displaystyle S={\text{suffix}}(W)} , which are the set of prefixes and suffixes of W {\displaystyle W} respectively. Now the algorithm constructs a matrix M {\displaystyle M} where the elements of P {\displaystyle P} function as the rows, ordered by the shortlex order, and the elements of S {\displaystyle S} function as the columns, ordered by the shortlex order. Next, the cells in the matrix are filled in the following manner for prefix p i {\displaystyle p_{i}} and suffix s j {\displaystyle s_{j}} : If p i s j ∈ W → M i j = l ( p i s j ) {\displaystyle p_{i}s_{j}\in W\rightarrow M_{ij}=l(p_{i}s_{j})} else, M i j = 0 {\displaystyle M_{ij}=0} Now, we say row i {\displaystyle i} and t {\displaystyle t} are distinguishable if there exi

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  Random projection

  In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. According to theoretical results, random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing. == Dimensionality reduction == Dimensionality reduction, as the name suggests, is reducing the number of random variables using various mathematical methods from statistics and machine learning. Dimensionality reduction is often used to reduce the problem of managing and manipulating large data sets. Dimensionality reduction techniques generally use linear transformations in determining the intrinsic dimensionality of the manifold as well as extracting its principal directions. For this purpose there are various related techniques, including: principal component analysis, linear discriminant analysis, canonical correlation analysis, discrete cosine transform, random projection, etc. Random projection is a simple and computationally efficient way to reduce the dimensionality of data by trading a controlled amount of error for faster processing times and smaller model sizes. The dimensions and distribution of random projection matrices are controlled so as to approximately preserve the pairwise distances between any two samples of the dataset. == Method == The core idea behind random projection is given in the Johnson-Lindenstrauss lemma, which states that if points in a vector space are of sufficiently high dimension, then they may be projected into a suitable lower-dimensional space in a way which approximately preserves pairwise distances between the points with high probability. In random projection, the original d {\displaystyle d} -dimensional data is projected to a k {\displaystyle k} -dimensional subspace, by multiplying on the left by a random matrix R ∈ R k × d {\displaystyle R\in \mathbb {R} ^{k\times d}} . Using matrix notation: If X d × N {\displaystyle X_{d\times N}} is the original set of N d-dimensional observations, then X k × N R P = R k × d X d × N {\displaystyle X_{k\times N}^{RP}=R_{k\times d}X_{d\times N}} is the projection of the data onto a lower k-dimensional subspace. Random projection is computationally simple: form the random matrix "R" and project the d × N {\displaystyle d\times N} data matrix X onto K dimensions of order O ( d k N ) {\displaystyle O(dkN)} . If the data matrix X is sparse with about c nonzero entries per column, then the complexity of this operation is of order O ( c k N ) {\displaystyle O(ckN)} . === Orthogonal random projection === A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle v} be its projection. The norm-squared ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as projecting a random point, uniformly sampled on the unit sphere, to its first k {\displaystyle k} coordinates. This is equivalent to sampling a random point in the multivariate gaussian distribution x ∼ N ( 0 , I d × d ) {\displaystyle x\sim {\mathcal {N}}(0,I_{d\times d})} , then normalizing it. Therefore, ‖ v ‖ 2 2 {\displaystyle \|v\|_{2}^{2}} has the same distribution as ∑ i = 1 k x i 2 ∑ i = 1 k x i 2 + ∑ i = k + 1 d x i 2 {\displaystyle {\frac {\sum _{i=1}^{k}x_{i}^{2}}{\sum _{i=1}^{k}x_{i}^{2}+\sum _{i=k+1}^{d}x_{i}^{2}}}} , which by the chi-squared construction of the Beta distribution, has distribution Beta ⁡ ( k / 2 , ( d − k ) / 2 ) {\displaystyle \operatorname {Beta} (k/2,(d-k)/2)} , with mean k / d {\displaystyle k/d} . We have a concentration inequality P r [ | ‖ v ‖ 2 − k d | ≥ ϵ k d ] ≤ 3 exp ⁡ ( − k ϵ 2 / 64 ) {\displaystyle Pr\left[\left|\|v\|_{2}-{\frac {k}{d}}\right|\geq \epsilon {\sqrt {\frac {k}{d}}}\right]\leq 3\exp \left(-k\epsilon ^{2}/64\right)} for any ϵ ∈ ( 0 , 1 ) {\displaystyle \epsilon \in (0,1)} . === Gaussian random projection === The random matrix R can be generated using a Gaussian distribution. The first row is a random unit vector uniformly chosen from S d − 1 {\displaystyle S^{d-1}} . The second row is a random unit vector from the space orthogonal to the first row, the third row is a random unit vector from the space orthogonal to the first two rows, and so on. In this way of choosing R, and the following properties are satisfied: Spherical symmetry: For any orthogonal matrix A ∈ O ( d ) {\displaystyle A\in O(d)} , RA and R have the same distribution. Orthogonality: The rows of R are orthogonal to each other. Normality: The rows of R are unit-length vectors. === More computationally efficient random projections === Achlioptas has shown that the random matrix can be sampled more efficiently. Either the full matrix can be sampled IID according to R i , j = 3 / k × { + 1 with probability 1 6 0 with probability 2 3 − 1 with probability 1 6 {\displaystyle R_{i,j}={\sqrt {3/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{6}}\\0&{\text{with probability }}{\frac {2}{3}}\\-1&{\text{with probability }}{\frac {1}{6}}\end{cases}}} or the full matrix can be sampled IID according to R i , j = 1 / k × { + 1 with probability 1 2 − 1 with probability 1 2 {\displaystyle R_{i,j}={\sqrt {1/k}}\times {\begin{cases}+1&{\text{with probability }}{\frac {1}{2}}\\-1&{\text{with probability }}{\frac {1}{2}}\end{cases}}} Both are efficient for database applications because the computations can be performed using integer arithmetic. More related study is conducted in. It was later shown how to use integer arithmetic while making the distribution even sparser, having very few nonzeroes per column, in work on the Sparse JL Transform. This is advantageous since a sparse embedding matrix means being able to project the data to lower dimension even faster. === Random Projection with Quantization === Random projection can be further condensed by quantization (discretization), with 1-bit (sign random projection) or multi-bits. It is the building block of SimHash, RP tree, and other memory efficient estimation and learning methods. == Large quasiorthogonal bases == The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimensional Euclidean space. There are exponentially large (in dimension n) sets of almost orthogonal vectors (with small value of inner products) in n–dimensional Euclidean space. This observation is useful in indexing of high-dimensional data. Quasiorthogonality of large random sets is important for methods of random approximation in machine learning. In high dimensions, exponentially large numbers of randomly and independently chosen vectors from equidistribution on a sphere (and from many other distributions) are almost orthogonal with probability close to one. This implies that in order to represent an element of such a high-dimensional space by linear combinations of randomly and independently chosen vectors, it may often be necessary to generate samples of exponentially large length if we use bounded coefficients in linear combinations. On the other hand, if coefficients with arbitrarily large values are allowed, the number of randomly generated elements that are sufficient for approximation is even less than dimension of the data space. == Implementations == RandPro - An R package for random projection sklearn.random_projection - A module for random projection from the scikit-learn Python library Weka implementation [1]

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  Terminology model

  A terminology model is a refinement of a concept system. Within a terminology model the concepts (object types) of a specific problem or subject area are defined by subject-matter experts in terms of concept (object type) definitions and definitions of subordinated concepts or characteristics (properties). Besides object types, the terminology model allows defining hierarchical classifications, definitions for object type and property behavior and definition of casual relations. The terminology model is a means for subject-matter experts to express their knowledge about the subject in subject-specific terms. Since the terminology model is structured rather similar to an object-oriented database schema, is can be transformed without loss of information into an object-oriented database schema. Thus, the terminology model is a method for problem analysis on the one side and a mean of defining database schema on the other side. Several terminology models have been developed and published in the field of statistics: Terminology model for classifications Terminology model for statistical variables Reference model for statistical metadata

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  Training, validation, and test data sets

  In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from input data. These input data used to build the model are usually divided into multiple data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and testing sets. The model is initially fit on a training data set, which is a set of examples used to fit the parameters (e.g. weights of connections between neurons in artificial neural networks) of the model. The model (e.g. a naive Bayes classifier) is trained on the training data set using a supervised learning method, for example using optimization methods such as gradient descent or stochastic gradient descent. In practice, the training data set often consists of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), where the answer key is commonly denoted as the target (or label). The current model is run with the training data set and produces a result, which is then compared with the target, for each input vector in the training data set. Based on the result of the comparison and the specific learning algorithm being used, the parameters of the model are adjusted. The model fitting can include both variable selection and parameter estimation. Successively, the fitted model is used to predict the responses for the observations in a second data set called the validation data set. The validation data set provides an unbiased evaluation of a model fit on the training data set while tuning the model's hyperparameters (e.g. the number of hidden units—layers and layer widths—in a neural network). Validation data sets can be used for regularization by early stopping (stopping training when the error on the validation data set increases, as this is a sign of over-fitting to the training data set). This simple procedure is complicated in practice by the fact that the validation data set's error may fluctuate during training, producing multiple local minima. This complication has led to the creation of many ad-hoc rules for deciding when over-fitting has truly begun. Finally, the test data set is a data set used to provide an unbiased evaluation of a model fit on the training data set. When the data in the test data set has never been used (for example in cross-validation), the test data set is called a holdout data set. The term "validation set" is sometimes used instead of "test set" in some literature (e.g., if the original data set was partitioned into only two subsets, the test set might be referred to as the validation set). Deciding the sizes and strategies for data set division in training, test and validation sets is very dependent on the problem and data available. == Training data set == A training data set is a data set of examples used during the learning process and is used to fit the parameters (e.g., weights) of, for example, a classifier. For classification tasks, a supervised learning algorithm looks at the training data set to determine, or learn, the optimal combinations of variables that will generate a good predictive model. The goal is to produce a trained (fitted) model that generalizes well to new, unknown data. The fitted model is evaluated using “new” examples from the held-out data sets (validation and test data sets) to estimate the model’s accuracy in classifying new data. To reduce the risk of issues such as over-fitting, the examples in the validation and test data sets should not be used to train the model. Most approaches that search through training data for empirical relationships tend to overfit the data, meaning that they can identify and exploit apparent relationships in the training data that do not hold in general. When a training set is continuously expanded with new data, then this is incremental learning. == Validation data set == A validation data set is a data set of examples used to tune the hyperparameters (i.e. the architecture) of a model. It is sometimes also called the development set or the "dev set". An example of a hyperparameter for artificial neural networks includes the number of hidden units in each layer. It, as well as the testing set (as mentioned below), should follow the same probability distribution as the training data set. In order to avoid overfitting, when any classification parameter needs to be adjusted, it is necessary to have a validation data set in addition to the training and test data sets. For example, if the most suitable classifier for the problem is sought, the training data set is used to train the different candidate classifiers, the validation data set is used to compare their performances and decide which one to take and, finally, the test data set is used to obtain the performance characteristics such as accuracy, sensitivity, specificity, F-measure, and so on. The validation data set functions as a hybrid: it is training data used for testing, but neither as part of the low-level training nor as part of the final testing. The basic process of using a validation data set for model selection (as part of training data set, validation data set, and test data set) is: Since our goal is to find the network having the best performance on new data, the simplest approach to the comparison of different networks is to evaluate the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function defined with respect to a training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network having the smallest error with respect to the validation set is selected. This approach is called the hold out method. Since this procedure can itself lead to some overfitting to the validation set, the performance of the selected network should be confirmed by measuring its performance on a third independent set of data called a test set. An application of this process is in early stopping, where the candidate models are successive iterations of the same network, and training stops when the error on the validation set grows, choosing the previous model (the one with minimum error). == Test data set == A test data set is a data set that is independent of the training data set, but that follows the same probability distribution as the training data set. A test set is therefore a set of examples used only to assess the performance (i.e. generalization) of a specified classifier on unseen data. To do this, the model is used to predict classifications of examples in the test set. Those predictions are compared to the examples' true classifications to assess the model's accuracy. If a model fit to the training and validation data set also fits the test data set well, minimal overfitting has taken place (see figure below). A better fitting of the training or validation data sets as opposed to the test data set usually points to overfitting. In the scenario where a data set has a low number of samples, it is usually partitioned into a training set and a validation data set, where the model is trained on the training set and refined using the validation set to improve accuracy, but this approach will lead to overfitting. The holdout method can also be employed, where the test set is used at the end, after training on the training set. Other techniques, such as cross-validation and bootstrapping, are used on small data sets. The bootstrap method generates numerous simulated data sets of the same size by randomly sampling with replacement from the original data, allowing the random data points to serve as test sets for evaluating model performance. Cross-validation splits the data set into multiple folds, with a single sub-fold used as test data; the model is trained on the remaining folds, and all folds are cross-validated (with results averaged and models consolidated) to estimate final model performance. Note that some sources advise against using a single split, as it can lead to overfitting as well as biased model performance estimates. For this reason, data sets are split into three partitions: training, validation and test data sets. The standard machine learning practice is to train on the training set and tune hyperparameters using the validation set, where the validation process selects the model with the lowest validation loss, which is then tested on the test data set (normally held out) to assess the final model. The holdout method for the test set reduces computation by avoiding using the test set after each epoch. The test data set should never be used for validating the training model or fine-tuning hyperparameters, as it provides an accurate and honest evaluation of the model's final performance on unseen dat

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  Correlation clustering

  Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the

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