A data stream management system (DSMS) is a computer software system to manage continuous data streams. It is similar to a database management system (DBMS), which is, however, designed for static data in conventional databases. A DBMS also offers a flexible query processing so that the information needed can be expressed using queries. However, in contrast to a DBMS, a DSMS executes a continuous query that is not only performed once, but is permanently installed. Therefore, the query is continuously executed until it is explicitly uninstalled. Since most DSMS are data-driven, a continuous query produces new results as long as new data arrive at the system. This basic concept is similar to complex event processing so that both technologies are partially coalescing. == Functional principle == One important feature of a DSMS is the possibility to handle potentially infinite and rapidly changing data streams by offering flexible processing at the same time, although there are only limited resources such as main memory. The following table provides various principles of DSMS and compares them to traditional DBMS. == Processing and streaming models == One of the biggest challenges for a DSMS is to handle potentially infinite data streams using a fixed amount of memory and no random access to the data. There are different approaches to limit the amount of data in one pass, which can be divided into two classes. For the one hand, there are compression techniques that try to summarize the data and for the other hand there are window techniques that try to portion the data into (finite) parts. === Synopses === The idea behind compression techniques is to maintain only a synopsis of the data, but not all (raw) data points of the data stream. The algorithms range from selecting random data points called sampling to summarization using histograms, wavelets or sketching. One simple example of a compression is the continuous calculation of an average. Instead of memorizing each data point, the synopsis only holds the sum and the number of items. The average can be calculated by dividing the sum by the number. However, it should be mentioned that synopses cannot reflect the data accurately. Thus, a processing that is based on synopses may produce inaccurate results. === Windows === Instead of using synopses to compress the characteristics of the whole data streams, window techniques only look on a portion of the data. This approach is motivated by the idea that only the most recent data are relevant. Therefore, a window continuously cuts out a part of the data stream, e.g. the last ten data stream elements, and only considers these elements during the processing. There are different kinds of such windows like sliding windows that are similar to FIFO lists or tumbling windows that cut out disjoint parts. Furthermore, the windows can also be differentiated into element-based windows, e.g., to consider the last ten elements, or time-based windows, e.g., to consider the last ten seconds of data. There are also different approaches to implementing windows. There are, for example, approaches that use timestamps or time intervals for system-wide windows or buffer-based windows for each single processing step. Sliding-window query processing is also suitable to being implemented in parallel processors by exploiting parallelism between different windows and/or within each window extent. == Query processing == Since there are a lot of prototypes, there is no standardized architecture. However, most DSMS are based on the query processing in DBMS by using declarative languages to express queries, which are translated into a plan of operators. These plans can be optimized and executed. A query processing often consists of the following steps. === Formulation of continuous queries === The formulation of queries is mostly done using declarative languages like SQL in DBMS. Since there are no standardized query languages to express continuous queries, there are a lot of languages and variations. However, most of them are based on SQL, such as the Continuous Query Language (CQL), StreamSQL and ESP. There are also graphical approaches where each processing step is a box and the processing flow is expressed by arrows between the boxes. The language strongly depends on the processing model. For example, if windows are used for the processing, the definition of a window has to be expressed. In StreamSQL, a query with a sliding window for the last 10 elements looks like follows: This stream continuously calculates the average value of "price" of the last 10 tuples, but only considers those tuples whose prices are greater than 100.0. In the next step, the declarative query is translated into a logical query plan. A query plan is a directed graph where the nodes are operators and the edges describe the processing flow. Each operator in the query plan encapsulates the semantic of a specific operation, such as filtering or aggregation. In DSMSs that process relational data streams, the operators are equal or similar to the operators of the Relational algebra, so that there are operators for selection, projection, join, and set operations. This operator concept allows the very flexible and versatile processing of a DSMS. === Optimization of queries === The logical query plan can be optimized, which strongly depends on the streaming model. The basic concepts for optimizing continuous queries are equal to those from database systems. If there are relational data streams and the logical query plan is based on relational operators from the Relational algebra, a query optimizer can use the algebraic equivalences to optimize the plan. These may be, for example, to push selection operators down to the sources, because they are not so computationally intensive like join operators. Furthermore, there are also cost-based optimization techniques like in DBMS, where a query plan with the lowest costs is chosen from different equivalent query plans. One example is to choose the order of two successive join operators. In DBMS this decision is mostly done by certain statistics of the involved databases. But, since the data of a data streams is unknown in advance, there are no such statistics in a DSMS. However, it is possible to observe a data stream for a certain time to obtain some statistics. Using these statistics, the query can also be optimized later. So, in contrast to a DBMS, some DSMS allows to optimize the query even during runtime. Therefore, a DSMS needs some plan migration strategies to replace a running query plan with a new one. === Transformation of queries === Since a logical operator is only responsible for the semantics of an operation but does not consist of any algorithms, the logical query plan must be transformed into an executable counterpart. This is called a physical query plan. The distinction between a logical and a physical operator plan allows more than one implementation for the same logical operator. The join, for example, is logically the same, although it can be implemented by different algorithms like a Nested loop join or a Sort-merge join. Notice, these algorithms also strongly depend on the used stream and processing model. Finally, the query is available as a physical query plan. === Execution of queries === Since the physical query plan consists of executable algorithms, it can be directly executed. For this, the physical query plan is installed into the system. The bottom of the graph (of the query plan) is connected to the incoming sources, which can be everything like connectors to sensors. The top of the graph is connected to the outgoing sinks, which may be for example a visualization. Since most DSMSs are data-driven, a query is executed by pushing the incoming data elements from the source through the query plan to the sink. Each time when a data element passes an operator, the operator performs its specific operation on the data element and forwards the result to all successive operators. == Examples == AURORA, StreamBase Systems, Inc. Archived 23 March 2009 at the Wayback Machine Hortonworks DataFlow IBM Streams NIAGARA Query Engine NiagaraST: A Research Data Stream Management System at Portland State University Odysseus, an open source Java-based framework for Data Stream Management Systems Pipeline DB PIPES Archived 24 December 2016 at the Wayback Machine, webMethods Business Events QStream SAS Event Stream Processing SQLstream STREAM StreamGlobe StreamInsight TelegraphCQ WSO2 Stream Processor
Bitstrips
Bitstrips, Inc. was a Canadian media and technology company based in Toronto, founded in 2007 by Jacob Blackstock, David Kennedy, Shahan Panth, Dorian Baldwin, and Jesse Brown. The company created and offered a web application, Bitstrips.com, which allowed users to create comic strips using personalized avatars, and preset templates and poses. Brown and Blackstock explained that the service was meant to enable self-expression without the need to have artistic skills. Bitstrips was first presented in 2008 at South by Southwest in Austin, Texas, and the service later piloted and launched a version designed for use as educational software. The service achieved increasing prominence following the launch of versions for Facebook and mobile platforms. In 2014, Bitstrips launched a spin-off app known as Bitmoji, which allows users to create personalized stickers for use in instant messaging. In July 2016, Snapchat Inc. announced that it had acquired the company; the Bitstrips comic service was shut down, but Bitmoji remains operational, and has subsequently been given greater prominence within Snapchat's overall platform. == History == Bitstrips was co-developed by Toronto-based comic artist Jacob Blackstock and his high school friend, journalist Jesse Brown. The service was originally envisioned as a means to allow anyone to create their own comic strip without needing artistic skills. Brown explained that "it's so difficult and time-consuming to tell a story in comic book form, drawing the same characters again and again in these tiny little panels, and just the amount of craftsmanship required. And even if you can do it well, which I never could, it takes years to make a story." Brown stated that the service would be "groundwork for a whole new way to communicate", and went as far as describing the service as being a "YouTube for comics". Blackstock explained that the concept of Bitstrips was influenced by his own use of comics as a form of socialization; a student, Blackstock and his friends drew comics featuring each other and shared them during classes. He felt that Bitstrips was a "medium for self-expression", stating that "It's not just about you making the comics, but since you and your friends star in these comics, it's like you're the medium. The visual nature of comics just speaks so much louder than text." The service was publicly unveiled at South by Southwest in 2008. In 2009, the service introduced a version oriented towards the educational market, Bitstrips for Schools, which was initially piloted at a number of schools in Ontario. The service was praised by educators for being engaging to students, especially within language classes. Brown noted that students were using the service to create comics outside of class as well, stating that it was "so gratifying and shocking what people do with your tool to make their own stories in ways that you never would have anticipated. Some of them are just brilliant." In December 2012, Bitstrips launched a version for Facebook; by July 2013, Bitstrips had 10 million unique users on Facebook, having created over 50 million comics. In October 2013, Bitstrips launched a mobile app; in two months, Bitstrips became a top-downloaded app in 40 countries, and over 30 million avatars had been created with it. In November 2013, Bitstrips secured a round of funding from Horizons Ventures and Li Ka-shing. In October 2014, Bitstrips launched Bitmoji, a spin-off app that allows users to create stickers featuring Bitstrips characters in various templates. In July 2016, following unconfirmed reports earlier in the year, Snapchat Inc. announced that it had acquired Bitstrips. The company's staff continue to operate out of Toronto, but the original Bitstrips comic service was shut down in favour of focusing exclusively on Bitmoji, leaving many Bitstrips users to call for a reboot of the comic service.
Elastic map
Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let S {\displaystyle {\mathcal {S}}} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j {\displaystyle {\bf {w}}_{j}} in the same space. Each datapoint s ∈ S {\displaystyle s\in {\mathcal {S}}} has a host node, namely the closest node w j {\displaystyle {\bf {w}}_{j}} (if there are several closest nodes then one takes the node with the smallest number). The data set S {\displaystyle {\mathcal {S}}} is divided into classes K j = { s | w j is a host of s } {\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}} . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 {\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}} , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points { s i } {\displaystyle \{s_{i}\}} . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})} , are connected by elastic edges. Call this set of pairs E {\displaystyle E} . Some triplets of nodes, ( w i , w j , w k ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})} , form bending ribs. Call this set of triplets G {\displaystyle G} . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 {\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}} , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 {\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}} , where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . {\displaystyle U=D+U_{E}+U_{G}.} The position of the nodes { w j } {\displaystyle \{{\bf {w}}_{j}\}} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U {\displaystyle U} . == Expectation-maximization algorithm == For a given splitting of dataset S {\displaystyle {\mathcal {S}}} in classes K j {\displaystyle K_{j}} , minimization of the quadratic functional U {\displaystyle U} is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given { w j } {\displaystyle \{{\bf {w}}_{j}\}} find { K j } {\displaystyle \{K_{j}\}} ; For given { K j } {\displaystyle \{K_{j}\}} minimize U {\displaystyle U} and find { w j } {\displaystyle \{{\bf {w}}_{j}\}} ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U {\displaystyle U} . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ {\displaystyle \lambda } and μ {\displaystyle \mu } coefficients) and finishes with soft grids (small λ {\displaystyle \lambda } and μ {\displaystyle \mu } ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes. == Applications == Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.
TabPFN
TabPFN (Tabular Prior-data Fitted Network) is a machine learning model for tabular datasets proposed in 2022. It uses a transformer architecture. It is intended for supervised classification and regression analysis on tabular datasets, particularly focusing on small- to medium-sized datasets. The latest version, TabPFN-3, was released in May 2026 and supports datasets with up to one million rows and 200 features. == History == TabPFN was first introduced in a 2022 pre-print and presented at ICLR 2023. TabPFN v2 was published in 2025 in Nature by Hollmann and co-authors. The source code is published on GitHub under a modified Apache License and on PyPi. Writing for ICLR blogs, McCarter states that the model has attracted attention due to its performance on small dataset benchmarks. TabPFN v2.5 was released on November 6, 2025. TabPFN-3 was released on May 12, 2026. Prior Labs, founded in 2024, aims to commercialize TabPFN. As of April 2026, the open-source TabPFN repository had more than 6,000 stars on GitHub. == Overview and pre-training == TabPFN supports classification, regression and generative tasks. It leverages "Prior-Data Fitted Networks" models to model tabular data. By using a transformer pre-trained on synthetic tabular datasets, TabPFN avoids benchmark contamination and costs of curating real-world data. TabPFN v2 was pre-trained on approximately 130 million such datasets. Synthetic datasets are generated using causal models or Bayesian neural networks; this can include simulating missing values, imbalanced data, and noise. Random inputs are passed through these models to generate outputs, with a bias towards simpler causal structures. During pre-training, TabPFN predicts the masked target values of new data points given training data points and their known targets, effectively learning a generic learning algorithm that is executed by running a neural network forward pass. The new dataset is then processed in a single forward pass without retraining. The model's transformer encoder processes features and labels by alternating attention across rows and columns. TabPFN v2 handles numerical and categorical features, missing values, and supports tasks like regression and synthetic data generation, while TabPFN-2.5 scales this approach to datasets with up to 50,000 rows and 2,000 features. TabPFN-3 introduced a redesigned architecture with row-compression, an attention-based many-class decoder, native missing-value handling, and inference optimizations such as row chunking and a reduced key-value cache, with benchmark-validated regimes of up to 1 million rows with 200 features, 100,000 rows with 2,000 features, or 1,000 rows with 20,000 features. Since TabPFN is pre-trained, in contrast to other deep learning methods, it does not require costly hyperparameter optimization. == Research == TabPFN is the subject of on-going research. Applications for TabPFN have been investigated for domains such as chemoproteomics, insurance risk classification, and metagenomics. In clinical research, TabPFN was used in a study on the early detection of pancreatic cancer from blood samples, where it was combined with metabolomic data and reported high diagnostic performance. == Applications == TabPFN has been used in industrial and biomedical contexts. Hitachi Ltd. has been reported to use the model for predictive maintenance in rail networks, with its use described as helping to identify track issues earlier and reduce manual inspections. In the biomedical domain, Oxford Cancer Analytics has used TabPFN in the analysis of proteomic data in lung disease research. A 2025 ML Contests report noted that the winners of DrivenData's PREPARE challenge used TabPFN to generate features for gradient-boosted decision tree models. == Limitations == TabPFN has been criticized for its "one large neural network is all you need" approach to modeling problems. Further, its performance is limited in high-dimensional and large-scale datasets. == Scaling Mode == In late November 2025, Prior Labs introduced ‘‘Scaling Mode’’, an operating mode for TabPFN designed to remove the fixed upper bound on training set size. Earlier versions of TabPFN had been optimized and validated primarily for datasets of up to 100,000 rows, whereas Scaling Mode was reported to extend support to substantially larger datasets, with benchmarked experiments on datasets containing up to 10 million rows. According to Prior Labs, Scaling Mode preserves the existing TabPFN architecture, including its alternating row-attention and column-attention design, as well as the same feature-count limits as prior releases. Inference remains based on a single forward pass rather than dataset-specific gradient-descent training, while scalability is described as being constrained primarily by available compute and memory resources. Prior Labs reported benchmark results on an internal collection of datasets ranging from 1 million to 10 million rows across industry and scientific applications. In these benchmarks, Scaling Mode was compared with CatBoost, XGBoost, LightGBM, and TabPFN 2.5 using 50,000-row subsampling. The company stated that predictive performance improved monotonically with increasing training set size and that no diminishing returns from scaling were observed within the tested range. Prior Labs also announced the release through company and executive social media channels. TabPFN-3 later incorporated related scaling improvements, including row chunking and a reduced key-value cache, into the model architecture and inference pipeline.
Clustering illusion
The clustering illusion is the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random. The illusion is caused by a human tendency to underpredict the amount of variability likely to appear in a small sample of random or pseudorandom data. Thomas Gilovich, an early author on the subject, argued that the effect occurs for different types of random dispersions. Some might perceive patterns in stock market price fluctuations over time, or clusters in two-dimensional data such as the locations of impact of World War II V-1 flying bombs on maps of London. Although Londoners developed specific theories about the pattern of impacts within London, a statistical analysis by R. D. Clarke originally published in 1946 showed that the impacts of V-2 rockets on London were a close fit to a random distribution. == Similar biases == Using this cognitive bias in causal reasoning may result in the Texas sharpshooter fallacy, in which differences in data are ignored and similarities are overemphasized. More general forms of erroneous pattern recognition are pareidolia and apophenia. Related biases are the illusion of control which the clustering illusion could contribute to, and insensitivity to sample size in which people don't expect greater variation in smaller samples. A different cognitive bias involving misunderstanding of chance streams is the gambler's fallacy. == Possible causes == Daniel Kahneman and Amos Tversky explained this kind of misprediction as being caused by the representativeness heuristic (which itself they also first proposed).
Autognostics
Autognostics is a new paradigm that describes the capacity for computer networks to be self-aware. It is considered one of the major components of Autonomic Networking. == Introduction == One of the most important characteristics of today's Internet that has contributed to its success is its basic design principle: a simple and transparent core with intelligence at the edges (the so-called "end-to-end principle"). Based on this principle, the network carries data without knowing the characteristics of that data (e.g., voice, video, etc.) - only the end-points have application-specific knowledge. If something goes wrong with the data, only the edge may be able to recognize that since it knows about the application and what the expected behavior is. The core has no information about what should happen with that data - it only forwards packets. Although an effective and beneficial attribute, this design principle has also led to many of today's problems, limitations, and frustrations. Currently, it is almost impossible for most end-users to know why certain network-based applications do not work well and what they need to do to make it better. Also, network operators who interact with the core in low-level terms such as router configuration have problems expressing their high-level goals into low-level actions. In high-level terms, this may be summarized as a weak coupling between the network and application layers of the overall system. As a consequence of the Internet end-to-end principle, the network performance experienced by a particular application is difficult to attribute based on the behavior of the individual elements. At any given moment, the measure of performance between any two points is typically unknown and applications must operate blindly. As a further consequence, changes to the configuration of given element, or changes in the end-to-end path, cannot easily be validated. Optimization and provisioning cannot then be automated except against only the simplest design specifications. There is an increasing interest in Autonomic Networking research, and a strong conviction that an evolution from the current networking status quo is necessary. Although to date there have not been any practical implementations demonstrating the benefits of an effective autonomic networking paradigm, there seems to be a consensus as to the characteristics which such implementations would need to demonstrate. These specifically include continuous monitoring, identifying, diagnosing and fixing problems based on high-level policies and objectives. Autognostics, as a major part of the autonomic networking concept, intends to bring networks to a new level of awareness and eliminate the lack of visibility which currently exists in today's networks. == Definition == Autognostics is a new paradigm that describes the capacity for computer networks to be self-aware, in part and as a whole, and dynamically adapt to the applications running on them by autonomously monitoring, identifying, diagnosing, resolving issues, subsequently verifying that any remediation was successful, and reporting the impact with respect to the application's use (i.e., providing visibility into the changes to networks and their effects). Although similar to the concept of network awareness, i.e., the capability of network devices and applications to be aware of network characteristics (see References section below), it is noteworthy that autognostics takes that concept one step further. The main difference is the auto part of autognostics, which entails that network devices are self-aware of network characteristics, and have the capability to adapt themselves as a result of continuous monitoring and diagnostics. == Path to autognostics == Autognostics, or in other words deep self-knowledge, can be best described as the ability of a network to know itself and the applications that run on it. This knowledge is used to autonomously adapt to dynamic network and application conditions such as utilization, capacity, quality of service/application/user experience, etc. In order to achieve autognosis, networks need a means to: Continuously monitor/test the network for application-specific performance Analyze the monitoring/test data to detect problems (e.g., performance degradation) Diagnose, identify and localize sources of degradation Automatically take actions to resolve problems via remediation/provisioning Verify the problems have been resolved (potentially rolling back changes if ineffective) Subsequently, continue to monitor/test for performance
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