In computer graphics, graphics software refers to a program or collection of programs that enable a person to manipulate images or models visually on a computer. Computer graphics can be classified into two distinct categories: raster graphics and vector graphics, with further 2D and 3D variants. Many graphics programs focus exclusively on either vector or raster graphics, but there are a few that operate on both. It is simple to convert from vector graphics to raster graphics, but going the other way is harder. Some software attempts to do this. In addition to static graphics, there are animation and video editing software. Different types of software are often designed to edit different types of graphics such as video, photos, and vector-based drawings. The exact sources of graphics may vary for different tasks, but most can read and write files. Most graphics programs have the ability to import and export one or more graphics file formats, including those formats written for a particular computer graphics program. Such programs include, but are not limited to: GIMP, Adobe Photoshop, CorelDRAW, Microsoft Publisher, Picasa, etc. The use of a swatch is a palette of active colours that are selected and rearranged by the preference of the user. A swatch may be used in a program or be part of the universal palette on an operating system. It is used to change the colour of a text or image and in video editing. Vector graphics animation can be described as a series of mathematical transformations that are applied in sequence to one or more shapes in a scene. Raster graphics animation works in a similar fashion to film-based animation, where a series of still images produces the illusion of continuous movement. == History == SuperPaint was one of the earliest graphics software applications, first conceptualized in 1972 and achieving its first stable image in 1973 Fauve Matisse (later Macromedia xRes) was a pioneering program of the early 1990s, notably introducing layers in customer software. Currently Adobe Photoshop is one of the most used and best-known graphics programs in the Americas, having created more custom hardware solutions in the early 1990s, but was initially subject to various litigation. GIMP is a popular open-source alternative to Adobe Photoshop.
Core FTP
Core FTP LE is a freeware secure FTP client for Windows, developed by CoreFTP.com. Features include FTP, SSL/TLS, SFTP via SSH, and HTTP/HTTPS support. Secure FTP clients encrypt account information and data transferred across the internet, protecting data from being seen, or sniffed across networks. Core FTP is a traditional FTP client with local files displayed on the left, remote files on the right. Core FTP Server is a secure FTP server for Windows, developed by CoreFTP.com, starting in 2010. == Licensing == CoreFTP LE is free for personal, educational, non-profit, and business use.
Optical character recognition
Optical character recognition (OCR) or optical character reader is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a scene photo (for example the text on signs and billboards in a landscape photo) or from subtitle text superimposed on an image (for example: from a television broadcast). Widely used as a form of data entry from printed paper data records – whether passport documents, invoices, bank statements, computerized receipts, business cards, mail, printed data, or any suitable documentation – it is a common method of digitizing printed texts so that they can be electronically edited, searched, stored more compactly, displayed online, and used in machine processes such as cognitive computing, machine translation, (extracted) text-to-speech, key data and text mining. OCR is a field of research in pattern recognition, artificial intelligence and computer vision. Early versions needed to be trained with images of each character, and worked on one font at a time. Advanced systems capable of producing a high degree of accuracy for most fonts are now common, and with support for a variety of image file format inputs. Some systems are capable of reproducing formatted output that closely approximates the original page including images, columns, and other non-textual components. == History == Early optical character recognition may be traced to technologies involving telegraphy and creating reading devices for the blind. In 1914, Emanuel Goldberg developed a machine that read characters and converted them into standard telegraph code. Concurrently, Edmund Fournier d'Albe developed the Optophone, a handheld scanner that when moved across a printed page, produced tones that corresponded to specific letters or characters. In the late 1920s and into the 1930s, Emanuel Goldberg developed what he called a "Statistical Machine" for searching microfilm archives using an optical code recognition system. In 1931, he was granted US Patent number 1,838,389 for the invention. The patent was acquired by IBM. === Visually impaired users === In 1974, Ray Kurzweil started the company Kurzweil Computer Products, Inc. and continued development of omni-font OCR, which could recognize text printed in virtually any font. (Kurzweil is often credited with inventing omni-font OCR, but it was in use by companies, including CompuScan, in the late 1960s and 1970s.) Kurzweil used the technology to create a reading machine for blind people to have a computer read text to them out loud. The device included a CCD-type flatbed scanner and a text-to-speech synthesizer. On January 13, 1976, the finished product was unveiled during a widely reported news conference headed by Kurzweil and the leaders of the National Federation of the Blind. In 1978, Kurzweil Computer Products began selling a commercial version of the optical character recognition computer program. LexisNexis was one of the first customers, and bought the program to upload legal paper and news documents onto its nascent online databases. Two years later, Kurzweil sold his company to Xerox, which eventually spun it off as Scansoft, which merged with Nuance Communications. In the 2000s, OCR was made available online as a service (WebOCR), in a cloud computing environment, and in mobile applications like real-time translation of foreign-language signs on a smartphone. With the advent of smartphones and smartglasses, OCR can be used in internet connected mobile device applications that extract text captured using the device's camera. These devices that do not have built-in OCR functionality will typically use an OCR API to extract the text from the image file captured by the device. The OCR API returns the extracted text, along with information about the location of the detected text in the original image back to the device app for further processing (such as text-to-speech) or display. Various commercial and open source OCR systems are available for most common writing systems, including Latin, Cyrillic, Arabic, Hebrew, Indic, Bengali (Bangla), Devanagari, Tamil, Chinese, Japanese, and Korean characters. == Applications == OCR engines have been developed into software applications specializing in various subjects such as receipts, invoices, checks, and legal billing documents. The software can be used for: Entering data for business documents, e.g. checks, passports, invoices, bank statements and receipts Automatic number-plate recognition Passport recognition and information extraction in airports Automatically extracting key information from insurance documents Traffic-sign recognition Extracting business card information into a contact list Creating textual versions of printed documents, e.g. book scanning for Project Gutenberg Making electronic images of printed documents searchable, e.g. Google Books Converting handwriting in real-time to control a computer (pen computing) Defeating or testing the robustness of CAPTCHA anti-bot systems, though these are specifically designed to prevent OCR. Assistive technology for blind and visually impaired users Writing instructions for vehicles by identifying CAD images in a database that are appropriate to the vehicle design as it changes in real time Making scanned documents searchable by converting them to PDFs == Types == Optical character recognition (OCR) – targets typewritten text, one glyph or character at a time. Optical word recognition – targets typewritten text, one word at a time (for languages that use a space as a word divider). Usually just called "OCR". Intelligent character recognition (ICR) – also targets handwritten printscript or cursive text one glyph or character at a time, usually involving machine learning. Intelligent word recognition (IWR) – also targets handwritten printscript or cursive text, one word at a time. This is especially useful for languages where glyphs are not separated in cursive script. OCR is generally an offline process, which analyses a static document. There are cloud based services which provide an online OCR API service. Handwriting movement analysis can be used as input to handwriting recognition. Instead of merely using the shapes of glyphs and words, this technique is able to capture motion, such as the order in which segments are drawn, the direction, and the pattern of putting the pen down and lifting it. This additional information can make the process more accurate. This technology is also known as "online character recognition", "dynamic character recognition", "real-time character recognition", and "intelligent character recognition". == Techniques == === Pre-processing === OCR software often pre-processes images to improve the chances of successful recognition. Techniques include: De-skewing – if the document was not aligned properly when scanned, it may need to be tilted a few degrees clockwise or counterclockwise in order to make lines of text perfectly horizontal or vertical. Despeckling – removal of positive and negative spots, smoothing edges Binarization – conversion of an image from color or greyscale to black-and-white (called a binary image because there are two colors). The task is performed as a simple way of separating the text (or any other desired image component) from the background. The task of binarization is necessary since most commercial recognition algorithms work only on binary images, as it is simpler to do so. In addition, the effectiveness of binarization influences to a significant extent the quality of character recognition, and careful decisions are made in the choice of the binarization employed for a given input image type; since the quality of the method used to obtain the binary result depends on the type of image (scanned document, scene text image, degraded historical document, etc.). Line removal – Cleaning up non-glyph boxes and lines Layout analysis or zoning – Identification of columns, paragraphs, captions, etc. as distinct blocks. Especially important in multi-column layouts and tables. Line and word detection – Establishment of a baseline for word and character shapes, separating words as necessary. Script recognition – In multilingual documents, the script may change at the level of the words and hence, identification of the script is necessary, before the right OCR can be invoked to handle the specific script. Character isolation or segmentation – For per-character OCR, multiple characters that are connected due to image artifacts must be separated; single characters that are broken into multiple pieces due to artifacts must be connected. Normalization of aspect ratio and scale Segmentation of fixed-pitch fonts is accomplished relatively simply by aligning the image to a uniform grid based on where vertical grid lines will least often intersect black areas. For proportional fonts, more sophisticated techniques are needed because whitespace bet
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.
Memtransistor
The memtransistor (a blend word from Memory Transfer Resistor) is an experimental multi-terminal passive electronic component that might be used in the construction of artificial neural networks. It is a combination of the memristor and transistor technology. This technology is different from the 1T-1R approach since the devices are merged into one single entity. Multiple memristors can be embedded with a single transistor, enabling it to more accurately model a neuron with its multiple synaptic connections. A neural network produced from these would provide hardware-based artificial intelligence with a good foundation. == Applications == These types of devices would allow for a synapse model that could realise a learning rule, by which the synaptic efficacy is altered by voltages applied to the terminals of the device. An example of such a learning rule is spike-timing-dependant-plasticty by which the weight of the synapse, in this case the conductivity, could be modulated based on the timing of pre and post synaptic spikes arriving at each terminal. The advantage of this approach over two terminal memristive devices is that read and write protocols have the possibility to occur simultaneously and distinctly.
Virtual assistant
A virtual assistant (VA) is a software agent that can perform a range of tasks or services for a user based on user input, such as commands or questions, including verbal ones. Such technologies often incorporate chatbot capabilities to streamline task execution. The interaction may be via text, graphical interface, or voice, as some virtual assistants are able to interpret human speech and respond via synthesized voices. In many cases, users can ask their virtual assistants questions, control home automation devices and media playback, and manage other basic tasks such as email, to-do lists, and calendars – all with verbal commands. In recent years, prominent virtual assistants for direct consumer use have included Apple Siri, Amazon Alexa, Google Assistant (Gemini), Microsoft Copilot and Samsung Bixby. Also, companies in various industries often incorporate some kind of virtual assistant technology into their customer service or support. Into the 2020s, the emergence of artificial intelligence based chatbots, such as ChatGPT, has brought increased capability and interest to the field of virtual assistant products and services. == History == === Experimental decades: 1910s–1980s === Radio Rex was the first voice-activated toy, patented in 1916 and released in 1922. It was a wooden toy in the shape of a dog that would come out of its house when its name is called. In 1952, Bell Labs presented "Audrey", the Automatic Digit Recognition machine. It occupied a six-foot-high relay rack, consumed substantial power, had streams of cables and exhibited the myriad maintenance problems associated with complex vacuum-tube circuitry. It could recognize the fundamental units of speech, phonemes. It was limited to the accurate recognition of digits spoken by designated talkers. It could therefore be used for voice dialing, but in most cases, push-button dialing was cheaper and faster, rather than speaking the consecutive digits. Another early tool which was enabled to perform digital speech recognition was the IBM Shoebox voice-activated calculator, presented to the general public during the 1962 Seattle World's Fair after its initial market launch in 1961. This early computer, developed almost 20 years before the introduction of the first IBM Personal Computer in 1981, was able to recognize 16 spoken words and the digits 0 to 9. The first natural language processing computer program or the chatbot ELIZA was developed by MIT professor Joseph Weizenbaum in the 1960s. It was created to "demonstrate that the communication between man and machine was superficial". ELIZA used pattern matching and substitution methodology into scripted responses to simulate conversation, which gave an illusion of understanding on the part of the program. Weizenbaum's own secretary reportedly asked Weizenbaum to leave the room so that she and ELIZA could have a real conversation. Weizenbaum was surprised by this, later writing: "I had not realized ... that extremely short exposures to a relatively simple computer program could induce powerful delusional thinking in quite normal people. This gave name to the ELIZA effect, the tendency to unconsciously assume computer behaviors are analogous to human behaviors; that is, anthropomorphisation, a phenomenon present in human interactions with virtual assistants. The next milestone in the development of voice recognition technology was achieved in the 1970s at the Carnegie Mellon University in Pittsburgh, Pennsylvania with substantial support of the United States Department of Defense and its DARPA agency, funded five years of a Speech Understanding Research program, aiming to reach a minimum vocabulary of 1,000 words. Companies and academia including IBM, Carnegie Mellon University (CMU) and Stanford Research Institute took part in the program. The result was "Harpy", it mastered about 1000 words, the vocabulary of a three-year-old and it could understand sentences. It could process speech that followed pre-programmed vocabulary, pronunciation, and grammar structures to determine which sequences of words made sense together, and thus reducing speech recognition errors. In 1986, Tangora was an upgrade of the Shoebox, it was a voice recognizing typewriter. Named after the world's fastest typist at the time, it had a vocabulary of 20,000 words and used prediction to decide the most likely result based on what was said in the past. IBM's approach was based on a hidden Markov model, which adds statistics to digital signal processing techniques. The method makes it possible to predict the most likely phonemes to follow a given phoneme. Still each speaker had to individually train the typewriter to recognize their voice, and pause between each word. In 1983, Gus Searcy invented the "Butler in a Box", an electronic voice home controller system. === Birth of smart virtual assistants: 1990s–2010s === In the 1990s, digital speech recognition technology became a feature of the personal computer with IBM, Philips and Lernout & Hauspie fighting for customers. Much later the market launch of the first smartphone IBM Simon in 1994 laid the foundation for smart virtual assistants as we know them today. In 1997, Dragon's NaturallySpeaking software could recognize and transcribe natural human speech without pauses between each word into a document at a rate of 100 words per minute. A version of Naturally Speaking is still available for download and it is still used today, for instance, by many doctors in the US and the UK to document their medical records. In 2001 Colloquis publicly launched SmarterChild, on platforms like AIM and MSN Messenger. While entirely text-based SmarterChild was able to play games, check the weather, look up facts, and converse with users to an extent. The first modern digital virtual assistant installed on a smartphone was Siri, which was introduced as a feature of the iPhone 4S on 4 October 2011. Apple Inc. developed Siri following the 2010 acquisition of Siri Inc., a spin-off of SRI International, which is a research institute financed by DARPA and the United States Department of Defense. Its aim was to aid in tasks such as sending a text message, making phone calls, checking the weather or setting up an alarm. Over time, it has developed to provide restaurant recommendations, search the internet, and provide driving directions. In November 2014, Amazon announced Alexa alongside the Echo. In 2016, Salesforce debuted Einstein, developed from a set of technologies underlying the Salesforce platform. Einstein was replaced by Agentforce, an agentic AI, in September 2024. In April 2017 Amazon released a service for building conversational interfaces for any type of virtual assistant or interface. === Large Language Models: 2020s-present === In the 2020s, artificial intelligence (AI) systems like ChatGPT have gained popularity for their ability to generate human-like responses to text-based conversations. In February 2020, Microsoft introduced its Turing Natural Language Generation (T-NLG), which was then the "largest language model ever published at 17 billion parameters." On November 30, 2022, ChatGPT was launched as a prototype and quickly garnered attention for its detailed responses and articulate answers across many domains of knowledge. The advent of ChatGPT and its introduction to the wider public increased interest and competition in the space. In February 2023, Google began introducing an experimental service called "Bard" which is based on its LaMDA program to generate text responses to questions asked based on information gathered from the web. While ChatGPT and other generalized chatbots based on the latest generative AI are capable of performing various tasks associated with virtual assistants, there are also more specialized forms of such technology that are designed to target more specific situations or needs. == Method of interaction == Virtual assistants work via: Text, including: online chat (especially in an instant messaging application or other application ), SMS text, e-mail or other text-based communication channel, for example Conversica's intelligent virtual assistants for business. Voice: for example with Amazon Alexa on Amazon Echo devices, Siri on an iPhone, Google Assistant on Google-enabled Android devices, or Bixby on Samsung devices. Images: some assistants, such as Google Assistant (which includes Google Lens) and Bixby on the Samsung Galaxy series, have the added capability of performing image processing to recognize objects in images. Many virtual assistants are accessible via multiple methods, offering versatility in how users can interact with them, whether through chat, voice commands, or other integrated technologies. Virtual assistants use natural language processing (NLP) to match user text or voice input to executable commands. Some continually learn using artificial intelligence techniques including machine learning and ambient intelligence. To activate a virtual assistant u
Latent class model
In statistics, a latent class model (LCM) is a model for clustering multivariate discrete data. It assumes that the data arise from a mixture of discrete distributions, within each of which the variables are independent. It is called a latent class model because the class to which each data point belongs is unobserved (or latent). Latent class analysis (LCA) is a subset of structural equation modeling used to find groups or subtypes of cases in multivariate categorical data. These groups or subtypes of cases are called "latent classes". When faced with the following situation, a researcher might opt to use LCA to better understand the data: Symptoms a, b, c, and d have been recorded in a variety of patients diagnosed with diseases X, Y, and Z. Disease X is associated with symptoms a, b, and c; disease Y is linked to symptoms b, c, and d; and disease Z is connected to symptoms a, c, and d. In this context, the LCA would attempt to detect the presence of latent classes (i.e., the disease entities), thus creating patterns of association in the symptoms. As in factor analysis, LCA can also be used to classify cases according to their maximum likelihood class membership probability. The key criterion for resolving the LCA is identifying latent classes in which the observed symptom associations are effectively rendered null. This is because within each class, the diseases responsible for the symptoms create a structure of dependencies. As a result, the symptoms become conditionally independent, meaning that, given the class a case belongs to, the symptoms are no longer related to one another. == Model == Within each latent class, the observed variables are statistically independent—an essential aspect of latent class modeling. Usually, the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes, variables are independent (local independence). Therefore, the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987). In one form, the LCM is written as p i 1 , i 2 , … , i N ≈ ∑ t T p t ∏ n N p i n , t n , {\displaystyle p_{i_{1},i_{2},\ldots ,i_{N}}\approx \sum _{t}^{T}p_{t}\,\prod _{n}^{N}p_{i_{n},t}^{n},} where T {\displaystyle T} is the number of latent classes and p t {\displaystyle p_{t}} are the so-called recruitment or unconditional probabilities that should sum to one. p i n , t n {\displaystyle p_{i_{n},t}^{n}} are the marginal or conditional probabilities. For a two-way latent class model, the form is p i j ≈ ∑ t T p t p i t p j t . {\displaystyle p_{ij}\approx \sum _{t}^{T}p_{t}\,p_{it}\,p_{jt}.} This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization. The probability model used in LCA is closely related to the Naive Bayes classifier. The main difference is that in LCA, the class membership of an individual is a latent variable, whereas in Naive Bayes classifiers, the class membership is an observed label. == Related methods == There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data and assumes that such data arise from a mixture of distributions, such as a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent profile analysis. Modified to handle discrete data, this constrained analysis is known as LCA. Discrete latent trait models further constrain the classes to form from segments of a single dimension, allocating members to classes based on that dimension. An example would be assigning cases to social classes based on ability or merit. In a practical instance, the variables could be multiple choice items of a political questionnaire. In this case, the data consists of an N-way contingency table with answers to the items for a number of respondents. In this example, the latent variable refers to political opinion, and the latent classes to political groups. Given group membership, the conditional probabilities specify the chance that certain answers are chosen. == Application == LCA may be used in many fields, such as: collaborative filtering, Behavior Genetics and Evaluation of diagnostic tests.