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Language resource

In linguistics and language technology, a language resource is a "[composition] of linguistic material used in the construction, improvement and/or evaluation of language processing applications, (...) in language and language-mediated research studies and applications." According to Bird & Simons (2003), this includes data, i.e. "any information that documents or describes a language, such as a published monograph, a computer data file, or even a shoebox full of handwritten index cards. The information could range in content from unanalyzed sound recordings to fully transcribed and annotated texts to a complete descriptive grammar", tools, i.e., "computational resources that facilitate creating, viewing, querying, or otherwise using language data", and advice, i.e., "any information about what data sources are reliable, what tools are appropriate in a given situation, what practices to follow when creating new data". The latter aspect is usually referred to as "best practices" or "(community) standards". In a narrower sense, language resource is specifically applied to resources that are available in digital form, and then, "encompassing (a) data sets (textual, multimodal/multimedia and lexical data, grammars, language models, etc.) in machine readable form, and (b) tools/technologies/services used for their processing and management". == Typology == As of May 2020, no widely used standard typology of language resources has been established (current proposals include the LREMap, METASHARE, and, for data, the LLOD classification). Important classes of language resources include data lexical resources, e.g., machine-readable dictionaries, linguistic corpora, i.e., digital collections of natural language data, linguistic data bases such as the Cross-Linguistic Linked Data collection, tools linguistic annotations and tools for creating such annotations in a manual or semiautomated fashion (e.g., tools for annotating interlinear glossed text such as Toolbox and FLEx, or other language documentation tools), applications for search and retrieval over such data (corpus management systems), for automated annotation (part-of-speech tagging, syntactic parsing, semantic parsing, etc.), metadata and vocabularies vocabularies, repositories of linguistic terminology and language metadata, e.g., MetaShare (for language resource metadata), the ISO 12620 data category registry (for linguistic features, data structures and annotations within a language resource), or the Glottolog database (identifiers for language varieties and bibliographical database). == Language resource publication, dissemination and creation == A major concern of the language resource community has been to develop infrastructures and platforms to present, discuss and disseminate language resources. Selected contributions in this regard include: a series of International Conferences on Language Resources and Evaluation (LREC), the European Language Resources Association (ELRA, EU-based), and the Linguistic Data Consortium (LDC, US-based), which represent commercial hosting and dissemination platforms for language resources, the Open Languages Archives Community (OLAC), which provides and aggregates language resource metadata, the Language Resources and Evaluation Journal (LREJ), the European Language Grid is a European platform for language technologies (eg services), data and resources. As for the development of standards and best practices for language resources, these are subject of several community groups and standardization efforts, including ISO Technical Committee 37: Terminology and other language and content resources (ISO/TC 37), developing standards for all aspects of language resources, W3C Community Group Best Practices for Multilingual Linked Open Data (BPMLOD), working on best practice recommendations for publishing language resources as Linked Data or in RDF, W3C Community Group Linked Data for Language Technology (LD4LT), working on linguistic annotations on the web and language resource metadata, W3C Community Group Ontology-Lexica (OntoLex), working on lexical resources, the Open Linguistics working group of the Open Knowledge Foundation, working on conventions for publishing and linking open language resources, developing the Linguistic Linked Open Data cloud, the Text Encoding Initiative (TEI), working on XML-based specifications for language resources and digitally edited text.
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Correspondence analysis

Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ⁡ ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ⁡ ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ⁡ ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for
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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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Vowpal Wabbit

Vowpal Wabbit (VW) is an open-source fast online interactive machine learning system library and program developed originally at Yahoo! Research, and currently at Microsoft Research. It was started and is led by John Langford. Vowpal Wabbit's interactive learning support is particularly notable including Contextual Bandits, Active Learning, and forms of guided Reinforcement Learning. Vowpal Wabbit provides an efficient scalable out-of-core implementation with support for a number of machine learning reductions, importance weighting, and a selection of different loss functions and optimization algorithms. == Notable features == The VW program supports: Multiple supervised (and semi-supervised) learning problems: Classification (both binary and multi-class) Regression Active learning (partially labeled data) for both regression and classification Multiple learning algorithms (model-types / representations) OLS regression Matrix factorization (sparse matrix SVD) Single layer neural net (with user specified hidden layer node count) Searn (Search and Learn) Latent Dirichlet Allocation (LDA) Stagewise polynomial approximation Recommend top-K out of N One-against-all (OAA) and cost-sensitive OAA reduction for multi-class Weighted all pairs Contextual-bandit (with multiple exploration/exploitation strategies) Multiple loss functions: squared error quantile hinge logistic poisson Multiple optimization algorithms Stochastic gradient descent (SGD) BFGS Conjugate gradient Regularization (L1 norm, L2 norm, & elastic net regularization) Flexible input - input features may be: Binary Numerical Categorical (via flexible feature-naming and the hash trick) Can deal with missing values/sparse-features Other features On the fly generation of feature interactions (quadratic and cubic) On the fly generation of N-grams with optional skips (useful for word/language data-sets) Automatic test-set holdout and early termination on multiple passes bootstrapping User settable online learning progress report + auditing of the model Hyperparameter optimization == Scalability == Vowpal wabbit has been used to learn a tera-feature (1012) data-set on 1000 nodes in one hour. Its scalability is aided by several factors: Out-of-core online learning: no need to load all data into memory The hashing trick: feature identities are converted to a weight index via a hash (uses 32-bit MurmurHash3) Exploiting multi-core CPUs: parsing of input and learning are done in separate threads. Compiled C++ code
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Automation in construction

Automation in construction is the combination of methods, processes, and systems that allow for greater machine autonomy in construction activities. Construction automation may have multiple goals, including but not limited to, reducing jobsite injuries, decreasing activity completion times, and assisting with quality control and quality assurance. Some systems may be fielded as a direct response to increasing skilled labor shortages in some countries. Opponents claim that increased automation may lead to less construction jobs and that software leaves heavy equipment vulnerable to hackers. Research insights on this subject are today published in several journals such as Automation in Construction by Elsevier. == Uses of automation in construction == Equipment control and management: Automation can be used to control and monitor construction equipment, such as cranes, excavators, and bulldozers. Material handling: Automated systems can be used to handle, transport, and place materials such as concrete, bricks, and stones. Surveying: Automated survey equipment and drones can be used to collect and analyze data on construction sites. Quality control: Automated systems can be used to monitor and control the quality of materials and construction processes. Safety management: Automated systems can be used to monitor and control safety conditions on construction sites. Scheduling and planning: Automated systems can be used to manage schedules, resources, and costs. Waste management: Automated systems can be used to manage and dispose of waste materials generated during construction. 3D printing: Automated 3D printing can be used to create prototypes, models, and even full-scale building components. == Autonomous heavy equipment == Advances in sensors, machine learning, and autonomous vehicle technology have led to the development of self-operating construction equipment and retrofit systems designed to automate excavators, bulldozers, tracked loaders, skid steer loaders, and haul trucks, allowing them to perform tasks with limited human supervision. Since 2017, tech companies have developed autonomous or semi-autonomous retrofit kits that can be installed on existing construction machinery. Examples include Bedrock Robotics, Built Robotics, and SafeAI, which develop sensor and software systems that enable excavators and other earthmoving machines to operate with varying degrees of autonomy. Major equipment manufacturers have also introduced autonomous capabilities: Caterpillar and John Deere have developed autonomous or semi-autonomous systems for construction and mining equipment, including haul trucks and earthmoving machines. == Transportation сonstruction == Kratos Defense & Security Solutions fielded the world’s first Autonomous Truck-Mounted Attenuator (ATMA) in 2017, in conjunction with Royal Truck & Equipment. == Benefits of automation in construction == The use of automation in construction has become increasingly prevalent in recent years due to its numerous benefits. Automation in construction refers to the use of machinery, software, and other technologies to perform tasks that were previously done manually by workers. One of the most significant benefits of automation in construction is increased productivity. Automation can help speed up construction processes, reduce project completion times, and improve overall efficiency. For example, using automated machinery for tasks such as concrete pouring, bricklaying, and welding can significantly increase the speed and accuracy of these tasks, allowing for more work to be completed in a shorter amount of time. Another benefit of automation in construction is improved safety. By automating tasks that are hazardous to workers, such as demolition or working at height, companies can reduce the risk of accidents and injuries on site. Automation can also help to reduce worker fatigue, which can be a significant factor in accidents and mistakes. Overall, the use of automation in construction can improve productivity, reduce costs, increase safety, and improve the quality of construction projects. As technology continues to advance, the use of automation is likely to become even more prevalent in the construction industry.
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Kernel principal component analysis

In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. == Background: Linear PCA == Recall that conventional PCA operates on zero-centered data; that is, 1 N ∑ i = 1 N x i = 0 {\displaystyle {\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}=\mathbf {0} } , where x i {\displaystyle \mathbf {x} _{i}} is one of the N {\displaystyle N} multivariate observations. It operates by diagonalizing the covariance matrix, C = 1 N ∑ i = 1 N x i x i ⊤ {\displaystyle C={\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}\mathbf {x} _{i}^{\top }} in other words, it gives an eigendecomposition of the covariance matrix: λ v = C v {\displaystyle \lambda \mathbf {v} =C\mathbf {v} } which can be rewritten as λ x i ⊤ v = x i ⊤ C v for i = 1 , … , N {\displaystyle \lambda \mathbf {x} _{i}^{\top }\mathbf {v} =\mathbf {x} _{i}^{\top }C\mathbf {v} \quad {\textrm {for}}~i=1,\ldots ,N} . (See also: Covariance matrix as a linear operator) == Introduction of the Kernel to PCA == To understand the utility of kernel PCA, particularly for clustering, observe that, while N points cannot, in general, be linearly separated in d < N {\displaystyle d Read more →
Handwriting recognition

Handwriting recognition (HWR), also known as handwritten text recognition (HTR), is the ability of a computer to receive and interpret intelligible handwritten input from sources such as paper documents, photographs, touch-screens and other devices. The image of the written text may be sensed "off line" from a piece of paper by optical scanning (optical character recognition) or intelligent word recognition. Alternatively, the movements of the pen tip may be sensed "on line", for example by a pen-based computer screen surface, a generally easier task as there are more clues available. A handwriting recognition system handles formatting, performs correct segmentation into characters, and finds the most possible words. == Offline recognition == Offline handwriting recognition involves the automatic conversion of text in an image into letter codes that are usable within computer and text-processing applications. The data obtained by this form is regarded as a static representation of handwriting. Offline handwriting recognition is comparatively difficult, as different people have different handwriting styles. And, as of today, OCR engines are primarily focused on machine printed text and ICR for hand "printed" (written in capital letters) text. === Traditional techniques === ==== Character extraction ==== Offline character recognition often involves scanning a form or document. This means the individual characters contained in the scanned image will need to be extracted. Tools exist that are capable of performing this step. However, there are several common imperfections in this step. The most common is when characters that are connected are returned as a single sub-image containing both characters. This causes a major problem in the recognition stage. Yet many algorithms are available that reduce the risk of connected characters. ==== Character recognition ==== After individual characters have been extracted, a recognition engine is used to identify the corresponding computer character. Several different recognition techniques are currently available. ===== Feature extraction ===== Feature extraction works in a similar fashion to neural network recognizers. However, programmers must manually determine the properties they feel are important. This approach gives the recognizer more control over the properties used in identification. Yet any system using this approach requires substantially more development time than a neural network because the properties are not learned automatically. === Modern techniques === Where traditional techniques focus on segmenting individual characters for recognition, modern techniques focus on recognizing all the characters in a segmented line of text. Particularly they focus on machine learning techniques that are able to learn visual features, avoiding the limiting feature engineering previously used. State-of-the-art methods use convolutional networks to extract visual features over several overlapping windows of a text line image which a recurrent neural network uses to produce character probabilities. == Online recognition == Online handwriting recognition involves the automatic conversion of text as it is written on a special digitizer or PDA, where a sensor picks up the pen-tip movements as well as pen-up/pen-down switching. This kind of data is known as digital ink and can be regarded as a digital representation of handwriting. The obtained signal is converted into letter codes that are usable within computer and text-processing applications. The elements of an online handwriting recognition interface typically include: a pen or stylus for the user to write with a touch sensitive surface, which may be integrated with, or adjacent to, an output display. a software application which interprets the movements of the stylus across the writing surface, translating the resulting strokes into digital text. The process of online handwriting recognition can be broken down into a few general steps: preprocessing, feature extraction and classification The purpose of preprocessing is to discard irrelevant information in the input data, that can negatively affect the recognition. This concerns speed and accuracy. Preprocessing usually consists of binarization, normalization, sampling, smoothing and denoising. The second step is feature extraction. Out of the two- or higher-dimensional vector field received from the preprocessing algorithms, higher-dimensional data is extracted. The purpose of this step is to highlight important information for the recognition model. This data may include information like pen pressure, velocity or the changes of writing direction. The last big step is classification. In this step, various models are used to map the extracted features to different classes and thus identifying the characters or words the features represent. === Hardware === Commercial products incorporating handwriting recognition as a replacement for keyboard input were introduced in the early 1980s. Examples include handwriting terminals such as the Pencept Penpad and the Inforite point-of-sale terminal. With the advent of the large consumer market for personal computers, several commercial products were introduced to replace the keyboard and mouse on a personal computer with a single pointing/handwriting system, such as those from Pencept, CIC and others. The first commercially available tablet-type portable computer was the Write-Top from Linus Technologies, released in July 1988. Its operating system was based on MS-DOS. In the early 1990s, hardware makers including NCR, IBM and EO released tablet computers running the PenPoint operating system developed by GO Corp. PenPoint used handwriting recognition and gestures throughout and provided the facilities to third-party software. IBM's tablet computer was the first to use the ThinkPad name and used IBM's handwriting recognition. This recognition system was later ported to Microsoft Windows for Pen Computing, and IBM's Pen for OS/2. None of these were commercially successful. Advancements in electronics allowed the computing power necessary for handwriting recognition to fit into a smaller form factor than tablet computers, and handwriting recognition is often used as an input method for hand-held PDAs. The first PDA to provide written input was the Apple Newton, which exposed the public to the advantage of a streamlined user interface. However, the device was not a commercial success, owing to the unreliability of the software, which tried to learn a user's writing patterns. By the time of the release of the Newton OS 2.0, wherein the handwriting recognition was greatly improved, including unique features still not found in current recognition systems such as modeless error correction, the largely negative first impression had been made. After discontinuation of Apple Newton, the feature was incorporated in Mac OS X 10.2 and later as Inkwell. Palm later launched a successful series of PDAs based on the Graffiti recognition system. Graffiti improved usability by defining a set of "unistrokes", or one-stroke forms, for each character. This narrowed the possibility for erroneous input, although memorization of the stroke patterns did increase the learning curve for the user. The Graffiti handwriting recognition was found to infringe on a patent held by Xerox, and Palm replaced Graffiti with a licensed version of the CIC handwriting recognition which, while also supporting unistroke forms, pre-dated the Xerox patent. The court finding of infringement was reversed on appeal, and then reversed again on a later appeal. The parties involved subsequently negotiated a settlement concerning this and other patents. A Tablet PC is a notebook computer with a digitizer tablet and a stylus, which allows a user to handwrite text on the unit's screen. The operating system recognizes the handwriting and converts it into text. Windows Vista and Windows 7 include personalization features that learn a user's writing patterns or vocabulary for English, Japanese, Chinese Traditional, Chinese Simplified and Korean. The features include a "personalization wizard" that prompts for samples of a user's handwriting and uses them to retrain the system for higher accuracy recognition. This system is distinct from the less advanced handwriting recognition system employed in its Windows Mobile OS for PDAs. Although handwriting recognition is an input form that the public has become accustomed to, it has not achieved widespread use in either desktop computers or laptops. It is still generally accepted that keyboard input is both faster and more reliable. As of 2006, many PDAs offer handwriting input, sometimes even accepting natural cursive handwriting, but accuracy is still a problem, and some people still find even a simple on-screen keyboard more efficient. === Software === Early software could understand print handwriting where the characters were separated; however, cursive handwriting
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Win–stay, lose–switch

In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift or Pavlov, named after Ivan Pavlov) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to the prisoner's dilemma in order to model the evolution of altruism. In most versions, it starts either with a cooperate, then proceeds as always, or starts with a "probe" of cooperate-defect-cooperate to determine the other player's strategy. A mutual cooperation is regarded as a win. The learning rule bases its decision only on the outcome of the previous play. Outcomes are divided into successes (wins) and failures (losses). If the play on the previous round resulted in a success, then the agent plays the same strategy on the next round. Alternatively, if the play resulted in a failure the agent switches to another action. A large-scale empirical study of players of the game rock, paper, scissors shows that a variation of this strategy is adopted by real-world players of the game, instead of the Nash equilibrium strategy of choosing entirely at random between the three options.
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Foodsi

Foodsi is a Polish mobile application that connects customers with restaurants, convenience stores, bakeries and cafes that have a surplus of food, allowing its users to buy the surplus at a reduced price. The service launched in 2019 in Warsaw and has expanded to other major cities in Poland. In 2023, a new feature was introduced in the app, allowing users to buy packages not only with self-pickup but also with delivery. The products range has also been expanded to include unsold magazines, cosmetics or plants. == History == The company was created in 2019 in Poland by Mateusz Kowalczyk and Jakub Fryszczyn. During studies in their home country and abroad, when they made a living working in restaurants and bakeries, they recognized the problem and the scale of food waste. They launched the application by themselves, having previously raised PLN 100,000 on their own for the purpose. Initially, Foodsi was an Android-only app, but over time, an IOS version was developed. In 2022, the startup raised PLN 6 million in a seed round from VC companies including CofounderZone and Status Starter, as well as private investors such as founders of Pyszne.pl. As of December 2023, it claimed more than 5000 businesses, serving over 1,5 million users, have saved nearly 3 million bags of food. == Purpose == Foodsi aims to significantly reduce food waste, which contributes to the Sustainable Development Goals. The application bridges the gap between the customers who are looking for shopping deals and the companies that want to reduce surplus products but are unable to sell them at a normal price. This allows the customers to buy unsold products for as little as 30% of the normal price. The company claims that every 4 out of 5 packages are sold on average. As of 2019 Foodsi employed more than 30 people. By 2024 it was more than 50. For now, Foodsi operates in major Polish cities such as Warsaw, Kraków, Trójmiasto, Wrocław, Poznań etc. However, in the upcoming years, Foodsi plans to expand to other countries. == Use == To start selling surplus, a company must leave Foodsi its contact information to register in the system. Registration in the app is completely free of charge. Then, companies offer available packages anticipating what won’t be sold and post them in the app along with the price so that users can buy them and pick them up. Companies can put their packages in the app at any time during the day. Users can pick up packages from bakeries, grocery stores, restaurants, but also florists and beauty stores. Foodsi charges a small commission on each package from the cooperating companies. If a user wants to start ordering packages from Foodsi, he or she needs to install the app on their mobile phone (Android or IOS) and register an account. The app displays a list of restaurants and other venues available in a specific region set by the user's location. Customers can see the price, address, distance and time range for package pickup. Packages are usually in the form of so-called 'surprise-packages', meaning that customers do not know specifically what kind of food/product will be inside. Some restaurants offer a choice of different package sizes. Prices are up to 70% lower than those of the original products. Customers have to show up at the restaurant to pick up the package using their phone at a time specified in the app. == Awards == Auler All-Stars 2025 - 3rd place Deloitte Technology Fast 50 - 2025 Central Europe Executive Club - Innowacja Roku: Żywność i Rolnictwo - Wyróżnienie (2025) Stena Circular Economy Award - Lider Gospodarki Obiegu Zamkniętego (2025) - wyróżnienie w kategorii start-up wdrażający GOZ na rynku polskim 255th place in the international poll FoodTech 500 2025 Finalist for the EY Entrepreneur Of The Year™ 2025 Wpływowi 2024 - Laureat w kategorii “Zrównoważony rozwój” Supplier of the Year 2024 - XXII Food & Business Forum Supplier of the Year 2024 - VII Sweets & Coffee Forum Innovative Leader 2024 - Leader in Food / Food-Tech Category - Executive Summit “Orzeł Innowacji - Start-up z potencjałem Polska-Świat” (Rzeczpospolita, 2024) 102nd place in the international poll FoodTech 500 2024 Auler 2023 Startup of the Year 2023 according to money.pl Start(up) w zrównoważoną przyszłość Kongresu Kompas ESG 2023 Marka Godna Zaufania according to My Company Polska 2023 184th place in the international poll FoodTech 500 2023 In 2023, Foodsi co-founder Mateusz Kowalczyk was recognized by Forbes magazine and included in its "30 before 30" list.
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Non-negative matrix factorization

Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for o
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Stress majorization

Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n {\displaystyle n} m {\displaystyle m} -dimensional data items, a configuration X {\displaystyle X} of n {\displaystyle n} points in r {\displaystyle r} ( ≪ m ) {\displaystyle (\ll m)} -dimensional space is sought that minimizes the so-called stress function σ ( X ) {\displaystyle \sigma (X)} . Usually r {\displaystyle r} is 2 {\displaystyle 2} or 3 {\displaystyle 3} , i.e. the ( n × r ) {\displaystyle (n\times r)} matrix X {\displaystyle X} lists points in 2 − {\displaystyle 2-} or 3 − {\displaystyle 3-} dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function σ {\displaystyle \sigma } is a cost or loss function that measures the squared differences between ideal ( m {\displaystyle m} -dimensional) distances and actual distances in r-dimensional space. It is defined as: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 {\displaystyle \sigma (X)=\sum _{i Read more →
Weighted majority algorithm (machine learning)

In machine learning, weighted majority algorithm (WMA) is a meta learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well. Assume that the problem is a binary decision problem. To construct the compound algorithm, a positive weight is given to each of the algorithms in the pool. The compound algorithm then collects weighted votes from all the algorithms in the pool, and gives the prediction that has a higher vote. If the compound algorithm makes a mistake, the algorithms in the pool that contributed to the wrong predicting will be discounted by a certain ratio β where 0<β<1. It can be shown that the upper bounds on the number of mistakes made in a given sequence of predictions from a pool of algorithms A {\displaystyle \mathbf {A} } is O ( l o g | A | + m ) {\displaystyle \mathbf {O(log|A|+m)} } if one algorithm in x i {\displaystyle \mathbf {x} _{i}} makes at most m {\displaystyle \mathbf {m} } mistakes. There are many variations of the weighted majority algorithm to handle different situations, like shifting targets, infinite pools, or randomized predictions. The core mechanism remains similar, with the final performances of the compound algorithm bounded by a function of the performance of the specialist (best performing algorithm) in the pool.
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Thunderspy

Thunderspy is a type of security vulnerability, based on the Intel Thunderbolt 3 port, first reported publicly on 10 May 2020, that can result in an evil maid (i.e., attacker of an unattended device) attack gaining full access to a computer's information in about five minutes, and may affect millions of Apple, Linux and Windows computers, as well as any computers manufactured before 2019, and some after that. According to Björn Ruytenberg, the discoverer of the vulnerability, "All the evil maid needs to do is unscrew the backplate, attach a device momentarily, reprogram the firmware, reattach the backplate, and the evil maid gets full access to the laptop. All of this can be done in under five minutes." The malicious firmware is used to clone device identities which makes classical DMA attack possible. == History == The Thunderspy security vulnerabilities were first publicly reported by Björn Ruytenberg of Eindhoven University of Technology in the Netherlands on 10 May 2020. Thunderspy is similar to Thunderclap, another security vulnerability, reported in 2019, that also involves access to computer files through the Thunderbolt port. == Impact == The security vulnerability affects millions of Apple, Linux and Windows computers, as well as all computers manufactured before 2019, and some after that. However, this impact is restricted mainly to how precise a bad actor would have to be to execute the attack. Physical access to a machine with a vulnerable Thunderbolt controller is necessary, as well as a writable ROM chip for the Thunderbolt controller's firmware. Additionally, part of Thunderspy, specifically the portion involving re-writing the firmware of the controller, requires the device to be in sleep, or at least in some sort of powered-on state, to be effective. Machines that force power-off when the case is open may assist in resisting this attack to the extent that the feature (switch) itself resists tampering. Due to the nature of attacks that require extended physical access to hardware, it's unlikely the attack will affect users outside of a business or government environment. == Mitigation == The researchers claim there is no easy software solution, and may only be mitigated by disabling the Thunderbolt port altogether. However, the impacts of this attack (reading kernel level memory without the machine needing to be powered off) are largely mitigated by anti-intrusion features provided by many business machines. Intel claims enabling such features would substantially restrict the effectiveness of the attack. Microsoft's official security recommendations recommend disabling sleep mode while using BitLocker. Using hibernation in place of sleep mode turns the device off, mitigating potential risks of attack on encrypted data.
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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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Multiple correspondence analysis

In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables. == As an extension of correspondence analysis == MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display. In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues. == Details == Since MCA is adapted to draw statistical conclusions from categorical variables (such as multiple choice questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles). When the dataset is completely represented as categorical variables, one is able to build the corresponding so-called complete disjunctive table. We denote this table X {\displaystyle X} . If I {\displaystyle I} persons answered a survey with J {\displaystyle J} multiple choices questions with 4 answers each, X {\displaystyle X} will have I {\displaystyle I} rows and 4 J {\displaystyle 4J} columns. More theoretically, assume X {\displaystyle X} is the completely disjunctive table of I {\displaystyle I} observations of K {\displaystyle K} categorical variables. Assume also that the k {\displaystyle k} -th variable have J k {\displaystyle J_{k}} different levels (categories) and set J = ∑ k = 1 K J k {\displaystyle J=\sum _{k=1}^{K}J_{k}} . The table X {\displaystyle X} is then a I × J {\displaystyle I\times J} matrix with all coefficient being 0 {\displaystyle 0} or 1 {\displaystyle 1} . Set the sum of all entries of X {\displaystyle X} to be N {\displaystyle N} and introduce Z = X / N {\displaystyle Z=X/N} . In an MCA, there are also two special vectors: first r {\displaystyle r} , that contains the sums along the rows of Z {\displaystyle Z} , and c {\displaystyle c} , that contains the sums along the columns of Z {\displaystyle Z} . Note D r = diag ( r ) {\displaystyle D_{r}={\text{diag}}(r)} and D c = diag ( c ) {\displaystyle D_{c}={\text{diag}}(c)} , the diagonal matrices containing r {\displaystyle r} and c {\displaystyle c} respectively as diagonal. With these notations, computing an MCA consists essentially in the singular value decomposition of the matrix: M = D r − 1 / 2 ( Z − r c T ) D c − 1 / 2 {\displaystyle M=D_{r}^{-1/2}(Z-rc^{T})D_{c}^{-1/2}} The decomposition of M {\displaystyle M} gives you P {\displaystyle P} , Δ {\displaystyle \Delta } and Q {\displaystyle Q} such that M = P Δ Q T {\displaystyle M=P\Delta Q^{T}} with P, Q two unitary matrices and Δ {\displaystyle \Delta } is the generalized diagonal matrix of the singular values (with the same shape as Z {\displaystyle Z} ). The positive coefficients of Δ 2 {\displaystyle \Delta ^{2}} are the eigenvalues of Z {\displaystyle Z} . The interest of MCA comes from the way observations (rows) and variables (columns) in Z {\displaystyle Z} can be decomposed. This decomposition is called a factor decomposition. The coordinates of the observations in the factor space are given by F = D r − 1 / 2 P Δ {\displaystyle F=D_{r}^{-1/2}P\Delta } The i {\displaystyle i} -th rows of F {\displaystyle F} represent the i {\displaystyle i} -th observation in the factor space. And similarly, the coordinates of the variables (in the same factor space as observations!) are given by G = D c − 1 / 2 Q Δ {\displaystyle G=D_{c}^{-1/2}Q\Delta } == Recent works and extensions == In recent years, several students of Jean-Paul Benzécri have refined MCA and incorporated it into a more general framework of data analysis known as geometric data analysis. This involves the development of direct connections between simple correspondence analysis, principal component analysis and MCA with a form of cluster analysis known as Euclidean classification. Two extensions have great practical use. It is possible to include, as active elements in the MCA, several quantitative variables. This extension is called factor analysis of mixed data (see below). Very often, in questionnaires, the questions are structured in several issues. In the statistical analysis it is necessary to take into account this structure. This is the aim of multiple factor analysis which balances the different issues (i.e. the different groups of variables) within a global analysis and provides, beyond the classical results of factorial analysis (mainly graphics of individuals and of categories), several results (indicators and graphics) specific of the group structure. == Application fields == In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu, notably in his books La Distinction, Homo Academicus and The State Nobility. Bourdieu argued that there was an internal link between his vision of the social as spatial and relational --– captured by the notion of field, and the geometric properties of MCA. Sociologists following Bourdieu's work most often opt for the analysis of the indicator matrix, rather than the Burt table, largely because of the central importance accorded to the analysis of the 'cloud of individuals'. == Multiple correspondence analysis and principal component analysis == MCA can also be viewed as a PCA applied to the complete disjunctive table. To do this, the CDT must be transformed as follows. Let y i k {\displaystyle y_{ik}} denote the general term of the CDT. y i k {\displaystyle y_{ik}} is equal to 1 if individual i {\displaystyle i} possesses the category k {\displaystyle k} and 0 if not. Let denote p k {\displaystyle p_{k}} , the proportion of individuals possessing the category k {\displaystyle k} . The transformed CDT (TCDT) has as general term: x i k = y i k / p k − 1 {\displaystyle x_{ik}=y_{ik}/p_{k}-1} The unstandardized PCA applied to TCDT, the column k {\displaystyle k} having the weight p k {\displaystyle p_{k}} , leads to the results of MCA. This equivalence is fully explained in a book by Jérôme Pagès. It plays an important theoretical role because it opens the way to the simultaneous treatment of quantitative and qualitative variables. Two methods simultaneously analyze these two types of variables: factor analysis of mixed data and, when the active variables are partitioned in several groups: multiple factor analysis. This equivalence does not mean that MCA is a particular case of PCA as it is not a particular case of CA. It only means that these methods are closely linked to one another, as they belong to the same family: the factorial methods. == Software == There are numerous software of data analysis that include MCA, such as STATA and SPSS. The R package FactoMineR also features MCA. This software is related to a book describing the basic methods for performing MCA . There is also a Python package for [1] which works with numpy array matrices; the package has not been implemented yet for Spark dataframes.
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