A grammar checker, in computing terms, is a program, or part of a program, that attempts to verify written text for grammatical correctness. Grammar checkers are most often implemented as a feature of a larger program, such as a word processor, but are also available as a stand-alone application that can be activated from within programs that work with editable text. The implementation of a grammar checker makes use of natural language processing. == History == The earliest "grammar checkers" were programs that checked for punctuation and style inconsistencies, rather than a complete range of possible grammatical errors. The first system was called Writer's Workbench, and was a set of writing tools included with Unix systems as far back as the 1970s. The whole Writer's Workbench package included several separate tools to check for various writing problems. The "diction" tool checked for wordy, trite, clichéd or misused phrases in a text. The tool would output a list of questionable phrases and provide suggestions for improving the writing. The "style" tool analyzed the writing style of a given text. It performed a number of readability tests on the text and output the results, and gave some statistical information about the sentences of the text. Aspen Software of Albuquerque, New Mexico released the earliest version of a diction and style checker for personal computers, Grammatik, in 1981. Grammatik was first available for a Radio Shack - TRS-80, and soon had versions for CP/M and the IBM PC. Reference Software International of San Francisco, California, acquired Grammatik in 1985. Development of Grammatik continued, and it became an actual grammar checker that could detect writing errors beyond simple style checking. Other early diction and style checking programs included Punctuation & Style, Correct Grammar, RightWriter and PowerEdit. While all the earliest programs started as simple diction and style checkers, all eventually added various levels of language processing, and developed some level of true grammar checking capability. Until 1992, grammar checkers were sold as add-on programs. There were a large number of different word processing programs available at that time, with WordPerfect and Microsoft Word the top two in market share. In 1992, Microsoft decided to add grammar checking as a feature of Word, and licensed CorrecText, a grammar checker from Houghton Mifflin that had not yet been marketed as a standalone product. WordPerfect answered Microsoft's move by acquiring Reference Software, and the direct descendant of Grammatik is still included with WordPerfect. As of 2019, grammar checkers are built into systems like Google Docs, browser extensions like Grammarly and Qordoba, desktop applications like Ginger, free and open-source software like LanguageTool, and text editor plugins like those available from WebSpellChecker Software. == Technical issues == The earliest writing style programs checked for wordy, trite, clichéd, or misused phrases in a text. This process was based on simple pattern matching. The heart of the program was a list of many hundreds or thousands of phrases that are considered poor writing by many experts. The list of questionable phrases included alternative wording for each phrase. The checking program would simply break text into sentences, check for any matches in the phrase dictionary, flag suspect phrases and show an alternative. These programs could also perform some mechanical checks. For example, they would typically flag doubled words, doubled punctuation, some capitalization errors, and other simple mechanical mistakes. True grammar checking is more complex. While a programming language has a very specific syntax and grammar, this is not so for natural languages. One can write a somewhat complete formal grammar for a natural language, but there are usually so many exceptions in real usage that a formal grammar is of minimal help in writing a grammar checker. One of the most important parts of a natural language grammar checker is a dictionary of all the words in the language, along with the part of speech of each word. The fact that a natural word may be used as any one of several parts of speech (such as "free" being used as an adjective, adverb, noun, or verb) greatly increases the complexity of any grammar checker. A grammar checker will find each sentence in a text, look up each word in the dictionary, and then attempt to parse the sentence into a form that matches a grammar. Using various rules, the program can then detect various errors, such as agreement in tense, number, word order, and so on. It is also possible to detect some stylistic problems with the text. For example, some popular style guides such as The Elements of Style deprecate excessive use of the passive voice. Grammar checkers may attempt to identify passive sentences and suggest an active-voice alternative. The software elements required for grammar checking are closely related to some of the development issues that need to be addressed for speech recognition software. In voice recognition, parsing can be used to help predict which word is most likely intended, based on part of speech and position in the sentence. In grammar checking, the parsing is used to detect words that fail to follow accepted grammar usage. Recently, research has focused on developing algorithms which can recognize grammar errors based on the context of the surrounding words. == Criticism == Grammar checkers are considered a type of foreign language writing aid which non-native speakers can use to proofread their writings as such programs endeavor to identify syntactical errors. However, as with other computerized writing aids such as spell checkers, popular grammar checkers are often criticized when they fail to spot errors and incorrectly flag correct text as erroneous. The linguist Geoffrey K. Pullum argued in 2007 that they were generally so inaccurate as to do more harm than good: "for the most part, accepting the advice of a computer grammar checker on your prose will make it much worse, sometimes hilariously incoherent."
Pixelmator Pro
Pixelmator Pro is a photo, video, and vector graphic editor developed by Apple for macOS and iPadOS as part of its Pixelmator and pro apps platforms and as a part of their Apple Creator Studio suite of applications. Pixelmator Pro relies heavily on technologies from Apple platforms such as Metal, CoreML, Core Image, AVFoundation, GCD, and SwiftUI. == Features == GPU accelerated with Metal 50+ standard image editing tools Layer-based image editor Video editing support Vector graphic support (including SVG support) AI-powered editing features such as background removal ML Super Resolution and Smart Replace Supports a variety of media formats (JPEG, RAW, Apple ProRAW, PSD, PNG, GIF, MP4, HEIF, etc) == Reception == Pixelmator Pro was generally well-received by reviewers who praised its deep use of machine learning, fully macOS-native design, and relatively affordable one-time purchase compared to subscription software such as Adobe Photoshop. Some reviewers criticized that some features are hard to find or hard to use. It was awarded Apple's Mac App of the Year in 2018. Pixelmator Pro does not have support for panorama stitching. == Acquisition by Apple == On November 1, 2024, the Pixelmator Team announced that they were to be acquired by Apple, subject to regulatory approval. Their site promises "There will be no material changes to the Pixelmator Pro, Pixelmator for iOS, and Photomator apps at this time." The acquisition was completed in February 2025. On January 13, 2026, Apple announced that a new version of Pixelmator Pro with AI features would be included in its new Apple Creator Studio subscription, the app would be brought to the iPad and the Mac app would be redesigned with Liquid Glass. == Version history == == Applescript == In 2020 Pixelmator Pro added the ability to leverage Apple's automation language 'AppleScript' to automate many tasks in version 1.8 (Lynx). This enabled simple and advanced automation activities such as image resize, crop, color adjustments, format change, moving layers around, and more advanced actions like removing background, Gaussian blur, text replacement, shadows, color replacement, etc.
Multidimensional analysis
In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.
Blockmodeling
Blockmodeling is a set or a coherent framework, that is used for analyzing social structure and also for setting procedure(s) for partitioning (clustering) social network's units (nodes, vertices, actors), based on specific patterns, which form a distinctive structure through interconnectivity. It is primarily used in statistics, machine learning and network science. As an empirical procedure, blockmodeling assumes that all the units in a specific network can be grouped together to such extent to which they are equivalent. Regarding equivalency, it can be structural, regular or generalized. Using blockmodeling, a network can be analyzed using newly created blockmodels, which transforms large and complex network into a smaller and more comprehensible one. At the same time, the blockmodeling is used to operationalize social roles. While some contend that the blockmodeling is just clustering methods, Bonacich and McConaghy state that "it is a theoretically grounded and algebraic approach to the analysis of the structure of relations". Blockmodeling's unique ability lies in the fact that it considers the structure not just as a set of direct relations, but also takes into account all other possible compound relations that are based on the direct ones. The principles of blockmodeling were first introduced by Francois Lorrain and Harrison C. White in 1971. Blockmodeling is considered as "an important set of network analytic tools" as it deals with delineation of role structures (the well-defined places in social structures, also known as positions) and the discerning the fundamental structure of social networks. According to Batagelj, the primary "goal of blockmodeling is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily". Blockmodeling was at first used for analysis in sociometry and psychometrics, but has now spread also to other sciences. == Definition == A network as a system is composed of (or defined by) two different sets: one set of units (nodes, vertices, actors) and one set of links between the units. Using both sets, it is possible to create a graph, describing the structure of the network. During blockmodeling, the researcher is faced with two problems: how to partition the units (e.g., how to determine the clusters (or classes), that then form vertices in a blockmodel) and then how to determine the links in the blockmodel (and at the same time the values of these links). In the social sciences, the networks are usually social networks, composed of several individuals (units) and selected social relationships among them (links). Real-world networks can be large and complex; blockmodeling is used to simplify them into smaller structures that can be easier to interpret. Specifically, blockmodeling partitions the units into clusters and then determines the ties among the clusters. At the same time, blockmodeling can be used to explain the social roles existing in the network, as it is assumed that the created cluster of units mimics (or is closely associated with) the units' social roles. Blockmodeling can thus be defined as a set of approaches for partitioning units into clusters (also known as positions) and links into blocks, which are further defined by the newly obtained clusters. A block (also blockmodel) is defined as a submatrix, that shows interconnectivity (links) between nodes, present in the same or different clusters. Each of these positions in the cluster is defined by a set of (in)direct ties to and from other social positions. These links (connections) can be directed or undirected; there can be multiple links between the same pair of objects or they can have weights on them. If there are not any multiple links in a network, it is called a simple network. A matrix representation of a graph is composed of ordered units, in rows and columns, based on their names. The ordered units with similar patterns of links are partitioned together in the same clusters. Clusters are then arranged together so that units from the same clusters are placed next to each other, thus preserving interconnectivity. In the next step, the units (from the same clusters) are transformed into a blockmodel. With this, several blockmodels are usually formed, one being core cluster and others being cohesive; a core cluster is always connected to cohesive ones, while cohesive ones cannot be linked together. Clustering of nodes is based on the equivalence, such as structural and regular. The primary objective of the matrix form is to visually present relations between the persons included in the cluster. These ties are coded dichotomously (as present or absent), and the rows in the matrix form indicate the source of the ties, while the columns represent the destination of the ties. Equivalence can have two basic approaches: the equivalent units have the same connection pattern to the same neighbors or these units have same or similar connection pattern to different neighbors. If the units are connected to the rest of network in identical ways, then they are structurally equivalent. Units can also be regularly equivalent, when they are equivalently connected to equivalent others. With blockmodeling, it is necessary to consider the issue of results being affected by measurement errors in the initial stage of acquiring the data. == Different approaches == Regarding what kind of network is undergoing blockmodeling, a different approach is necessary. Networks can be one–mode or two–mode. In the former all units can be connected to any other unit and where units are of the same type, while in the latter the units are connected only to the unit(s) of a different type. Regarding relationships between units, they can be single–relational or multi–relational networks. Further more, the networks can be temporal or multilevel and also binary (only 0 and 1) or signed (allowing negative ties)/values (other values are possible) networks. Different approaches to blockmodeling can be grouped into two main classes: deterministic blockmodeling and stochastic blockmodeling approaches. Deterministic blockmodeling is then further divided into direct and indirect blockmodeling approaches. Among direct blockmodeling approaches are: structural equivalence and regular equivalence. Structural equivalence is a state, when units are connected to the rest of the network in an identical way(s), while regular equivalence occurs when units are equally related to equivalent others (units are not necessarily sharing neighbors, but have neighbour that are themselves similar). Indirect blockmodeling approaches, where partitioning is dealt with as a traditional cluster analysis problem (measuring (dis)similarity results in a (dis)similarity matrix), are: conventional blockmodeling, generalized blockmodeling: generalized blockmodeling of binary networks, generalized blockmodeling of valued networks and generalized homogeneity blockmodeling, prespecified blockmodeling. According to Brusco and Steinley (2011), the blockmodeling can be categorized (using a number of dimensions): deterministic or stochastic blockmodeling, one–mode or two–mode networks, signed or unsigned networks, exploratory or confirmatory blockmodeling. == Blockmodels == Blockmodels (sometimes also block models) are structures in which: vertices (e.g., units, nodes) are assembled within a cluster, with each cluster identified as a vertex; from such vertices a graph can be constructed; combinations of all the links (ties), represented in a block as a single link between positions, while at the same time constructing one tie for each block. In a case, when there are no ties in a block, there will be no ties between the two positions that define the block. Computer programs can partition the social network according to pre-set conditions. When empirical blocks can be reasonably approximated in terms of ideal blocks, such blockmodels can be reduced to a blockimage, which is a representation of the original network, capturing its underlying 'functional anatomy'. Thus, blockmodels can "permit the data to characterize their own structure", and at the same time not seek to manifest a preconceived structure imposed by the researcher. Blockmodels can be created indirectly or directly, based on the construction of the criterion function. Indirect construction refers to a function, based on "compatible (dis)similarity measure between paris of units", while the direct construction is "a function measuring the fit of real blocks induced by a given clustering to the corresponding ideal blocks with perfect relations within each cluster and between clusters according to the considered types of connections (equivalence)". === Types === Blockmodels can be specified regarding the intuition, substance or the insight into the nature of the studied network; this can result in such models as follows: parent-child role systems, organizational hierarchies, systems of
Winner-take-all (computing)
Winner-take-all is a computational principle applied in computational models of neural networks by which neurons compete with each other for activation. In the classical form, only the neuron with the highest activation stays active while all other neurons shut down; however, other variations allow more than one neuron to be active, for example the soft winner take-all, by which a power function is applied to the neurons. == Neural networks == In the theory of artificial neural networks, winner-take-all networks are a case of competitive learning in recurrent neural networks. Output nodes in the network mutually inhibit each other, while simultaneously activating themselves through reflexive connections. After some time, only one node in the output layer will be active, namely the one corresponding to the strongest input. Thus the network uses nonlinear inhibition to pick out the largest of a set of inputs. Winner-take-all is a general computational primitive that can be implemented using different types of neural network models, including both continuous-time and spiking networks. Winner-take-all networks are commonly used in computational models of the brain, particularly for distributed decision-making or action selection in the cortex. Important examples include hierarchical models of vision, and models of selective attention and recognition. They are also common in artificial neural networks and neuromorphic analog VLSI circuits. It has been formally proven that the winner-take-all operation is computationally powerful compared to other nonlinear operations, such as thresholding. In many practical cases, there is not only one single neuron which becomes active but there are exactly k neurons which become active for a fixed number k. This principle is referred to as k-winners-take-all. === Example algorithm === Consider a single linear neuron, with inputs x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} . Each input has weight w i {\displaystyle w_{i}} , and the output of the neuron is ∑ i w i x i {\displaystyle \sum _{i}w_{i}x_{i}} . In the Instar learning rule, on each input vector, the weight vectors are modified according to Δ w i = η ( x i − w i ) {\displaystyle \Delta w_{i}=\eta (x_{i}-w_{i})} where η {\displaystyle \eta } is the learning rate. This rule is unsupervised, since we need just the input vector, not a reference output. Now, consider multiple linear neurons y 1 , … , y m {\displaystyle y_{1},\dots ,y_{m}} . The output of each satisfies y i = ∑ j w i j x j {\displaystyle y_{i}=\sum _{j}w_{ij}x_{j}} . In the winner-take-all algorithm, the weights are modified as follows. Given an input vector x {\displaystyle x} , each output is computed. The neuron with the largest output is selected, and the weights going into that neuron are modified according to the Instar learning rule. All other weights remain unchanged. The k-winners-take-all rule is similar, except that the Instar learning rule is applied to the weights going into the k neurons with the largest outputs. == Circuit example == A simple, but popular CMOS winner-take-all circuit is shown on the right. This circuit was originally proposed by Lazzaro et al. (1989) using MOS transistors biased to operate in the weak-inversion or subthreshold regime. In the particular case shown there are only two inputs (IIN,1 and IIN,2), but the circuit can be easily extended to multiple inputs in a straightforward way. It operates on continuous-time input signals (currents) in parallel, using only two transistors per input. In addition, the bias current IBIAS is set by a single global transistor that is common to all the inputs. The largest of the input currents sets the common potential VC. As a result, the corresponding output carries almost all the bias current, while the other outputs have currents that are close to zero. Thus, the circuit selects the larger of the two input currents, i.e., if IIN,1 > IIN,2, we get IOUT,1 = IBIAS and IOUT,2 = 0. Similarly, if IIN,2 > IIN,1, we get IOUT,1 = 0 and IOUT,2 = IBIAS. A SPICE-based DC simulation of the CMOS winner-take-all circuit in the two-input case is shown on the right. As shown in the top subplot, the input IIN,1 was fixed at 6nA, while IIN,2 was linearly increased from 0 to 10nA. The bottom subplot shows the two output currents. As expected, the output corresponding to the larger of the two inputs carries the entire bias current (10nA in this case), forcing the other output current nearly to zero. == Other uses == In stereo matching algorithms, following the taxonomy proposed by Scharstein and Szelliski, winner-take-all is a local method for disparity computation. Adopting a winner-take-all strategy, the disparity associated with the minimum or maximum cost value is selected at each pixel. It is axiomatic that in the electronic commerce market, early dominant players such as AOL or Yahoo! get most of the rewards. By 1998, one study found the top 5% of all web sites garnered more than 74% of all traffic. The winner-take-all hypothesis in economics suggests that once a technology or a firm gets ahead, it will do better and better over time, whereas lagging technology and firms will fall further behind. See First-mover advantage.
Transfer function matrix
In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (MIMO) systems. The matrix relates the outputs of the system to its inputs. It is a particularly useful construction for linear time-invariant (LTI) systems because it can be expressed in terms of the s-plane. In some systems, especially ones consisting entirely of passive components, it can be ambiguous which variables are inputs and which are outputs. In electrical engineering, a common scheme is to gather all the voltage variables on one side and all the current variables on the other regardless of which are inputs or outputs. This results in all the elements of the transfer matrix being in units of impedance. The concept of impedance (and hence impedance matrices) has been borrowed into other energy domains by analogy, especially mechanics and acoustics. Many control systems span several different energy domains. This requires transfer matrices with elements in mixed units. This is needed both to describe transducers that make connections between domains and to describe the system as a whole. If the matrix is to properly model energy flows in the system, compatible variables must be chosen to allow this. == General == A MIMO system with m outputs and n inputs is represented by a m × n matrix. Each entry in the matrix is in the form of a transfer function relating an output to an input. For example, for a three-input, two-output system, one might write, [ y 1 y 2 ] = [ g 11 g 12 g 13 g 21 g 22 g 23 ] [ u 1 u 2 u 3 ] {\displaystyle {\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}g_{11}&g_{12}&g_{13}\\g_{21}&g_{22}&g_{23}\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}} where the un are the inputs, the ym are the outputs, and the gmn are the transfer functions. This may be written more succinctly in matrix operator notation as, Y = G U {\displaystyle \mathbf {Y} =\mathbf {G} \mathbf {U} } where Y is a column vector of the outputs, G is a matrix of the transfer functions, and U is a column vector of the inputs. In many cases, the system under consideration is a linear time-invariant (LTI) system. In such cases, it is convenient to express the transfer matrix in terms of the Laplace transform (in the case of continuous time variables) or the z-transform (in the case of discrete time variables) of the variables. This may be indicated by writing, for instance, Y ( s ) = G ( s ) U ( s ) {\displaystyle \mathbf {Y} (s)=\mathbf {G} (s)\mathbf {U} (s)} which indicates that the variables and matrix are in terms of s, the complex frequency variable of the s-plane arising from Laplace transforms, rather than time. The examples in this article are all assumed to be in this form, although that is not explicitly indicated for brevity. For discrete time systems s is replaced by z from the z-transform, but this makes no difference to subsequent analysis. The matrix is particularly useful when it is a proper rational matrix, that is, all its elements are proper rational functions. In this case, the state-space representation can be applied. In systems engineering, the overall system transfer matrix G (s) is decomposed into two parts: H (s) representing the system being controlled, and C(s) representing the control system. C (s) takes as its inputs the inputs of G (s) and the outputs of H (s). The outputs of C (s) form the inputs for H (s). == Electrical systems == In electrical systems, it is often the case that the distinction between input and output variables is ambiguous. They can be either, depending on circumstance and point of view. In such cases, the concept of port (a place where energy is transferred from one system to another) can be more useful than input and output. It is customary to define two variables for each port (p): the voltage across it (Vp) and the current entering it (Ip). For instance, the transfer matrix of a two-port network can be defined as follows, [ V 1 V 2 ] = [ z 11 z 12 z 21 z 22 ] [ I 1 I 2 ] {\displaystyle {\begin{bmatrix}V_{1}\\V_{2}\end{bmatrix}}={\begin{bmatrix}z_{11}&z_{12}\\z_{21}&z_{22}\\\end{bmatrix}}{\begin{bmatrix}I_{1}\\I_{2}\end{bmatrix}}} where the zmn are called the impedance parameters, or z-parameters. They are so-called because they are in units of impedance and relate port currents to a port voltage. The z-parameters are not the only way that transfer matrices are defined for two-port networks. Six basic matrices relate voltages and currents, each with advantages for particular system network topologies. However, only two of these can be extended beyond two ports to an arbitrary number of ports. These two are the z-parameters and their inverse, the admittance parameters or y-parameters. To understand the relationship between port voltages and currents and inputs and outputs, consider the simple voltage divider circuit. If we only wish to consider the output voltage (V2) resulting from applying the input voltage (V1) then the transfer function can be expressed as, [ V 2 ] = [ R 2 R 1 + R 2 ] [ V 1 ] {\displaystyle {\begin{bmatrix}V_{2}\end{bmatrix}}={\begin{bmatrix}{\dfrac {R_{2}}{R_{1}+R_{2}}}\end{bmatrix}}{\begin{bmatrix}V_{1}\end{bmatrix}}} which can be considered the trivial case of a 1×1 transfer matrix. The expression correctly predicts the output voltage if there is no current leaving port 2, but is increasingly inaccurate as the load increases. If, however, we attempt to use the circuit in reverse, driving it with a voltage at port 2 and calculate the resulting voltage at port 1 the expression gives completely the wrong result even with no load on port 1. It predicts a greater voltage at port 1 than was applied at port 2, an impossibility with a purely resistive circuit like this one. To correctly predict the behaviour of the circuit, the currents entering or leaving the ports must also be taken into account, which is what the transfer matrix does. The impedance matrix for the voltage divider circuit is, [ V 1 V 2 ] = [ R 1 + R 2 R 2 R 2 R 2 ] [ I 1 I 2 ] {\displaystyle {\begin{bmatrix}V_{1}\\V_{2}\end{bmatrix}}={\begin{bmatrix}R_{1}+R_{2}&R_{2}\\R_{2}&R_{2}\end{bmatrix}}{\begin{bmatrix}I_{1}\\I_{2}\end{bmatrix}}} which fully describes its behaviour under all input and output conditions. At microwave frequencies, none of the transfer matrices based on port voltages and currents are convenient to use in practice. Voltage is difficult to measure directly, current next to impossible, and the open circuits and short circuits required by the measurement technique cannot be achieved with any accuracy. For waveguide implementations, circuit voltage and current are entirely meaningless. Transfer matrices using different sorts of variables are used instead. These are the powers transmitted into, and reflected from a port, which are readily measured in the transmission line technology used in distributed-element circuits in the microwave band. The most well-known and widely used of these sorts of parameters is the scattering parameters, or s-parameters. == Mechanical and other systems == The concept of impedance can be extended into the mechanical and other domains through a mechanical-electrical analogy, hence the impedance parameters and other forms of 2-port network parameters can also be extended to the mechanical domain. To do this, an effort variable and a flow variable are made analogues of voltage and current, respectively. For mechanical systems under translation these variables are force and velocity respectively. Expressing the behaviour of a mechanical component as a two-port or multi-port with a transfer matrix is a useful thing to do because, like electrical circuits, the component can often be operated in reverse and its behaviour is dependent on the loads at the inputs and outputs. For instance, a gear train is often characterised simply by its gear ratio, a SISO transfer function. However, the gearbox output shaft can be driven around to turn the input shaft, requiring a MIMO analysis. In this example, the effort and flow variables are torque T and angular velocity ω, respectively. The transfer matrix in terms of z-parameters will look like, [ T 1 T 2 ] = [ z 11 z 12 z 21 z 22 ] [ ω 1 ω 2 ] {\displaystyle {\begin{bmatrix}T_{1}\\T_{2}\end{bmatrix}}={\begin{bmatrix}z_{11}&z_{12}\\z_{21}&z_{22}\end{bmatrix}}{\begin{bmatrix}\omega _{1}\\\omega _{2}\end{bmatrix}}} However, the z-parameters are not necessarily the most convenient for characterising gear trains. A gear train is the analogue of an electrical transformer and the h-parameters (hybrid parameters) better describe transformers because they directly include the turns ratios (the analogue of gear ratios). The gearbox transfer matrix in h-parameter format is, [ T 1 ω 2 ] = [ h 11 h 12 h 21 h 22 ] [ ω 1 T 2 ] {\displaystyle {\begin{bmatrix}T_{1}\\\omega _{2}\end{bm
Bayesian hierarchical modeling
Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis