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  Geometric primitive

  In vector computer graphics, CAD systems, and geographic information systems, a geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segments, which were all that early vector graphics systems had. In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus. Modern 2D computer graphics systems may operate with primitives which are curves (segments of straight lines, circles and more complicated curves), as well as shapes (boxes, arbitrary polygons, circles). A common set of two-dimensional primitives includes lines, points, and polygons, although some people prefer to consider triangles primitives, because every polygon can be constructed from triangles (polygon triangulation). All other graphic elements are built up from these primitives. In three dimensions, triangles or polygons positioned in three-dimensional space can be used as primitives to model more complex 3D forms. In some cases, curves (such as Bézier curves, circles, etc.) may be considered primitives; in other cases, curves are complex forms created from many straight, primitive shapes. == Common primitives == The set of geometric primitives is based on the dimension of the region being represented: Point (0-dimensional), a single location with no height, width, or depth. Line or curve (1-dimensional), having length but no width, although a linear feature may curve through a higher-dimensional space. Planar surface or curved surface (2-dimensional), having length and width. Volumetric region or solid (3-dimensional), having length, width, and depth. In GIS, the terrain surface is often spoken of colloquially as "2 1/2 dimensional," because only the upper surface needs to be represented. Thus, elevation can be conceptualized as a scalar field property or function of two-dimensional space, affording it a number of data modeling efficiencies over true 3-dimensional objects. A shape of any of these dimensions greater than zero consists of an infinite number of distinct points. Because digital systems are finite, only a sample set of the points in a shape can be stored. Thus, vector data structures typically represent geometric primitives using a strategic sample, organized in structures that facilitate the software interpolating the remainder of the shape at the time of analysis or display, using the algorithms of Computational geometry. A Point is a single coordinate in a Cartesian coordinate system. Some data models allow for Multipoint features consisting of several disconnected points. A Polygonal chain or Polyline is an ordered list of points (termed vertices in this context). The software is expected to interpolate the intervening shape of the line between adjacent points in the list as a parametric curve, most commonly a straight line, but other types of curves are frequently available, including circular arcs, cubic splines, and Bézier curves. Some of these curves require additional points to be defined that are not on the line itself, but are used for parametric control. A Polygon is a polyline that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior. Some data models allow for a single feature to consist of multiple polylines, which could collectively connect to form a single closed boundary, could represent a set of disjoint regions (e.g., the state of Hawaii), or could represent a region with holes (e.g., a lake with an island). A Parametric shape is a standardized two-dimensional or three-dimensional shape defined by a minimal set of parameters, such as an ellipse defined by two points at its foci, or three points at its center, vertex, and co-vertex. A Polyhedron or Polygon mesh is a set of polygon faces in three-dimensional space that are connected at their edges to completely enclose a volumetric region. In some applications, closure may not be required or may be implied, such as modeling terrain. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior. A triangle mesh is a subtype of polyhedron in which all faces must be triangles, the only polygon that will always be planar, including the Triangulated irregular network (TIN) commonly used in GIS. A parametric mesh represents a three-dimensional surface by a connected set of parametric functions, similar to a spline or Bézier curve in two dimensions. The most common structure is the Non-uniform rational B-spline (NURBS), supported by most CAD and animation software. == Application in GIS == A wide variety of vector data structures and formats have been developed during the history of Geographic information systems, but they share a fundamental basis of storing a core set of geometric primitives to represent the location and extent of geographic phenomena. Locations of points are almost always measured within a standard Earth-based coordinate system, whether the spherical Geographic coordinate system (latitude/longitude), or a planar coordinate system, such as the Universal Transverse Mercator. They also share the need to store a set of attributes of each geographic feature alongside its shape; traditionally, this has been accomplished using the data models, data formats, and even software of relational databases. Early vector formats, such as POLYVRT, the ARC/INFO Coverage, and the Esri shapefile support a basic set of geometric primitives: points, polylines, and polygons, only in two dimensional space and the latter two with only straight line interpolation. TIN data structures for representing terrain surfaces as triangle meshes were also added. Since the mid 1990s, new formats have been developed that extend the range of available primitives, generally standardized by the Open Geospatial Consortium's Simple Features specification. Common geometric primitive extensions include: three-dimensional coordinates for points, lines, and polygons; a fourth "dimension" to represent a measured attribute or time; curved segments in lines and polygons; text annotation as a form of geometry; and polygon meshes for three-dimensional objects. Frequently, a representation of the shape of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This dimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines). == In 3D modelling == In CAD software or 3D modelling, the interface may present the user with the ability to create primitives which may be further modified by edits. For example, in the practice of box modelling the user will start with a cuboid, then use extrusion and other operations to create the model. In this use the primitive is just a convenient starting point, rather than the fundamental unit of modelling. A 3D package may also include a list of extended primitives which are more complex shapes that come with the package. For example, a teapot is listed as a primitive in 3D Studio Max. == In graphics hardware == Various graphics accelerators exist with hardware acceleration for rendering specific primitives such as lines or triangles, frequently with texture mapping and shaders. Modern 3D accelerators typically accept sequences of triangles as triangle strips.

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  Non-negative matrix factorization

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

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  SqueezeNet

  SqueezeNet is a deep neural network for image classification released in 2016. SqueezeNet was developed by researchers at DeepScale, University of California, Berkeley, and Stanford University. In designing SqueezeNet, the authors' goal was to create a smaller neural network with fewer parameters while achieving competitive accuracy. Their best-performing model achieved the same accuracy as AlexNet on ImageNet classification, but has a size 510x less than it. == Version history == SqueezeNet was originally released on February 22, 2016. This original version of SqueezeNet was implemented on top of the Caffe deep learning software framework. Shortly thereafter, the open-source research community ported SqueezeNet to a number of other deep learning frameworks. On February 26, 2016, Eddie Bell released a port of SqueezeNet for the Chainer deep learning framework. On March 2, 2016, Guo Haria released a port of SqueezeNet for the Apache MXNet framework. On June 3, 2016, Tammy Yang released a port of SqueezeNet for the Keras framework. In 2017, companies including Baidu, Xilinx, Imagination Technologies, and Synopsys demonstrated SqueezeNet running on low-power processing platforms such as smartphones, FPGAs, and custom processors. As of 2018, SqueezeNet ships "natively" as part of the source code of a number of deep learning frameworks such as PyTorch, Apache MXNet, and Apple CoreML. In addition, third party developers have created implementations of SqueezeNet that are compatible with frameworks such as TensorFlow. Below is a summary of frameworks that support SqueezeNet. == Relationship to other networks == === AlexNet === SqueezeNet was originally described in SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size. AlexNet is a deep neural network that has 240 MB of parameters, and SqueezeNet has just 5 MB of parameters. This small model size can more easily fit into computer memory and can more easily be transmitted over a computer network. However, it's important to note that SqueezeNet is not a "squeezed version of AlexNet." Rather, SqueezeNet is an entirely different DNN architecture than AlexNet. What SqueezeNet and AlexNet have in common is that both of them achieve approximately the same level of accuracy when evaluated on the ImageNet image classification validation dataset. === Model compression === Model compression (e.g. quantization and pruning of model parameters) can be applied to a deep neural network after it has been trained. In the SqueezeNet paper, the authors demonstrated that a model compression technique called Deep Compression can be applied to SqueezeNet to further reduce the size of the parameter file from 5 MB to 500 KB. Deep Compression has also been applied to other DNNs, such as AlexNet and VGG. == Variants == Some of the members of the original SqueezeNet team have continued to develop resource-efficient deep neural networks for a variety of applications. A few of these works are noted in the following table. As with the original SqueezeNet model, the open-source research community has ported and adapted these newer "squeeze"-family models for compatibility with multiple deep learning frameworks. In addition, the open-source research community has extended SqueezeNet to other applications, including semantic segmentation of images and style transfer.

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  GeWorkbench

  geWorkbench (genomics Workbench) is an open-source software platform for integrated genomic data analysis. It is a desktop application written in the programming language Java. geWorkbench uses a component architecture. As of 2016, there are more than 70 plug-ins available, providing for the visualization and analysis of gene expression, sequence, and structure data. geWorkbench is the Bioinformatics platform of MAGNet, the National Center for the Multi-scale Analysis of Genomic and Cellular Networks, one of the 8 National Centers for Biomedical Computing funded through the NIH Roadmap (NIH Common Fund). Many systems and structure biology tools developed by MAGNet investigators are available as geWorkbench plugins. == Features == Computational analysis tools such as t-test, hierarchical clustering, self-organizing maps, regulatory network reconstruction, BLAST searches, pattern-motif discovery, protein structure prediction, structure-based protein annotation, etc. Visualization of gene expression (heatmaps, volcano plot), molecular interaction networks (through Cytoscape), protein sequence and protein structure data (e.g., MarkUs). Integration of gene and pathway annotation information from curated sources as well as through Gene Ontology enrichment analysis. Component integration through platform management of inputs and outputs. Among data that can be shared between components are expression datasets, interaction networks, sample and marker (gene) sets and sequences. Dataset history tracking - complete record of data sets used and input settings. Integration with 3rd party tools such as GenePattern, Cytoscape, and Genomespace. Demonstrations of each feature described can be found at GeWorkbench-web Tutorials. == Versions == geWorkbench is open-source software that can be downloaded and installed locally. A zip file of the released version Java source is also available. Prepackaged installer versions also exist for Windows, Macintosh, and Linux.

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  Kdb+

  kdb+ is a column-based relational time series database (TSDB) with in-memory (IMDB) abilities, developed and marketed by KX Systems. The database is commonly used in high-frequency trading (HFT) to store, analyze, process, and retrieve large data sets at high speed. kdb+ has the ability to handle billions of records and analyzes data within a database. The database is available in 32-bit and 64-bit versions for several operating systems. Financial institutions use kdb+ to analyze time series data such as stock or commodity exchange data. The database has also been used for other time-sensitive data applications including commodity markets such as energy trading, telecommunications, sensor data, log data, machine and computer network usage monitoring along with real time analytics in Formula One racing. == Overview == kdb+ is a high-performance column-store database that was designed to process and store large amounts of data. Commonly accessed data is pushed into random-access memory (RAM), which is faster to access than data in disk storage. Created with financial institutions in mind, the database was developed as a central repository to store time series data that supports real-time analysis of billions of records. kdb+ has the ability to analyze data over time and responds to queries similar to Structured Query Language (SQL). Columnar databases return answers to some queries in a more efficient way than row-based database management systems. kdb+ dictionaries, tables and nanosecond time stamps are native data types and are used to store time series data. At the core of kdb+ is the built-in programming language, q, a concise, expressive query array language, and dialect of the language APL. Q can manipulate streaming, real-time, and historical data. kdb+ uses q to aggregate and analyze data, perform statistical functions, and join data sets and supports SQL queries The vector language q was built for speed and expressiveness and eliminates most need for looping structures. kdb+ includes interfaces in C, C++, Java, C#, and Python. == History == In 1998, KX released kdb, a database built on the language K written by Arthur Whitney. In 2003, kdb+ was released as a 64-bit version of kdb. In 2004, the kdb+ tick market database framework was released along with kdb+ taq, a loader for the New York Stock Exchange (NYSE) taq data. kdb+ was created by Arthur Whitney, building on his prior work with array languages. In April 2007, KX announced that it was releasing a version of kdb+ for Mac OS X. Then, kdb+ was also available on the operating systems Linux, Windows, and Solaris. In September 2012, version 3.0 was released. It was optimized for Intel's upgraded processors with support for WebSockets, and universally unique identifiers (UUIDs, termed globally unique identifiers (GUID)s in Microsoft software). Intel's Advanced Vector Extensions (AVX) and Streaming SIMD Extensions 4 (SSE4) 4.2 on the Sandy Bridge processors of the time allowed for enhanced support of the kdb+ system. In June 2013, version 3.1 was released, with benchmarks up to 8 times faster than older versions. In March 2020, version 4.0 was released. New features included Multithreaded primitives, Intel Optane DC persistent memory support and Data at Rest Encryption.

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  Mixture model

  In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information. Mixture models are used for clustering, under the name model-based clustering, and also for density estimation. Mixture models should not be confused with models for compositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). However, compositional models can be thought of as mixture models, where members of the population are sampled at random. Conversely, mixture models can be thought of as compositional models, where the total size reading population has been normalized to 1. == Structure == === General mixture model === A typical finite-dimensional mixture model is a hierarchical model consisting of the following components: N random variables that are observed, each distributed according to a mixture of K components, with the components belonging to the same parametric family of distributions (e.g., all normal, all Zipfian, etc.) but with different parameters. However, it is also possible to have a finite mixture model where each component belongs to a different parametric family of distributions, for example, a mixture of a multivariate normal distribution and a generalized hyperbolic distribution. N random latent variables specifying the identity of the mixture component of each observation, each distributed according to a K-dimensional categorical distribution A set of K mixture weights, which are probabilities that sum to 1. A set of K parameters, each specifying the parameter of the corresponding mixture component. In many cases, each "parameter" is actually a set of parameters. For example, if the mixture components are Gaussian distributions, there will be a mean and variance for each component. If the mixture components are categorical distributions (e.g., when each observation is a token from a finite alphabet of size V), there will be a vector of V probabilities summing to 1. In addition, in a Bayesian setting, the mixture weights and parameters will themselves be random variables, and prior distributions will be placed over the variables. In such a case, the weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors. Mathematically, a basic parametric mixture model can be described as follows: K = number of mixture components N = number of observations θ i = 1 … K = parameter of distribution of observation associated with component i ϕ i = 1 … K = mixture weight, i.e., prior probability of a particular component i ϕ = K -dimensional vector composed of all the individual ϕ 1 … K ; must sum to 1 z i = 1 … N = component of observation i x i = 1 … N = observation i F ( x | θ ) = probability distribution of an observation, parametrized on θ z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N | z i = 1 … N ∼ F ( θ z i ) {\displaystyle {\begin{array}{lcl}K&=&{\text{number of mixture components}}\\N&=&{\text{number of observations}}\\\theta _{i=1\dots K}&=&{\text{parameter of distribution of observation associated with component }}i\\\phi _{i=1\dots K}&=&{\text{mixture weight, i.e., prior probability of a particular component }}i\\{\boldsymbol {\phi }}&=&K{\text{-dimensional vector composed of all the individual }}\phi _{1\dots K}{\text{; must sum to 1}}\\z_{i=1\dots N}&=&{\text{component of observation }}i\\x_{i=1\dots N}&=&{\text{observation }}i\\F(x|\theta )&=&{\text{probability distribution of an observation, parametrized on }}\theta \\z_{i=1\dots N}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}|z_{i=1\dots N}&\sim &F(\theta _{z_{i}})\end{array}}} In a Bayesian setting, all parameters are associated with random variables, as follows: K , N = as above θ i = 1 … K , ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N , F ( x | θ ) = as above α = shared hyperparameter for component parameters β = shared hyperparameter for mixture weights H ( θ | α ) = prior probability distribution of component parameters, parametrized on α θ i = 1 … K ∼ H ( θ | α ) ϕ ∼ S y m m e t r i c - D i r i c h l e t K ⁡ ( β ) z i = 1 … N | ϕ ∼ Categorical ⁡ ( ϕ ) x i = 1 … N | z i = 1 … N , θ i = 1 … K ∼ F ( θ z i ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\theta _{i=1\dots K},\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N},F(x|\theta )&=&{\text{as above}}\\\alpha &=&{\text{shared hyperparameter for component parameters}}\\\beta &=&{\text{shared hyperparameter for mixture weights}}\\H(\theta |\alpha )&=&{\text{prior probability distribution of component parameters, parametrized on }}\alpha \\\theta _{i=1\dots K}&\sim &H(\theta |\alpha )\\{\boldsymbol {\phi }}&\sim &\operatorname {Symmetric-Dirichlet} _{K}(\beta )\\z_{i=1\dots N}|{\boldsymbol {\phi }}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}|z_{i=1\dots N},\theta _{i=1\dots K}&\sim &F(\theta _{z_{i}})\end{array}}} This characterization uses F and H to describe arbitrary distributions over observations and parameters, respectively. Typically H will be the conjugate prior of F. The two most common choices of F are Gaussian aka "normal" (for real-valued observations) and categorical (for discrete observations). Other common possibilities for the distribution of the mixture components are: Binomial distribution, for the number of "positive occurrences" (e.g., successes, yes votes, etc.) given a fixed number of total occurrences Multinomial distribution, similar to the binomial distribution, but for counts of multi-way occurrences (e.g., yes/no/maybe in a survey) Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs Poisson distribution, for the number of occurrences of an event in a given period of time, for an event that is characterized by a fixed rate of occurrence Exponential distribution, for the time before the next event occurs, for an event that is characterized by a fixed rate of occurrence Log-normal distribution, for positive real numbers that are assumed to grow exponentially, such as incomes or prices Multivariate normal distribution (aka multivariate Gaussian distribution), for vectors of correlated outcomes that are individually Gaussian-distributed Multivariate Student's t-distribution, for vectors of heavy-tailed correlated outcomes A vector of Bernoulli-distributed values, corresponding, e.g., to a black-and-white image, with each value representing a pixel; see the handwriting-recognition example below === Specific examples === ==== Gaussian mixture model ==== A typical non-Bayesian Gaussian mixture model looks like this: K , N = as above ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N = as above θ i = 1 … K = { μ i = 1 … K , σ i = 1 … K 2 } μ i = 1 … K = mean of component i σ i = 1 … K 2 = variance of component i z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N ∼ N ( μ z i , σ z i 2 ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N}&=&{\text{as above}}\\\theta _{i=1\dots K}&=&\{\mu _{i=1\dots K},\sigma _{i=1\dots K}^{2}\}\\\mu _{i=1\dots K}&=&{\text{mean of component }}i\\\sigma _{i=1\dots K}^{2}&=&{\text{variance of component }}i\\z_{i=1\dots N}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}&\sim &{\mathcal {N}}(\mu _{z_{i}},\sigma _{z_{i}}^{2})\end{array}}} A Bayesian version of a Gaussian mixture model is as follows: K , N = as above ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N = as above θ i = 1 … K = { μ i = 1 … K , σ i = 1 … K 2 } μ i = 1 … K = mean of component i σ i = 1 … K 2 = variance of component i μ 0 , λ , ν , σ 0 2 = shared hyperparameters μ i = 1 … K ∼ N ( μ 0 , λ σ i 2 ) σ i = 1 … K 2 ∼ I n v e r s e - G a m m a ⁡ ( ν , σ 0 2 ) ϕ ∼ S y m m e t r i c - D i r i c h l e t K ⁡ ( β ) z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N ∼ N ( μ z i , σ z i 2 ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N}&=&{\text{as above}}\\\theta _{i=1\

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  Ordinal regression

  In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning. == Linear models for ordinal regression == Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by length-p vectors x1 through xn, with associated responses y1 through yn, where each yi is an ordinal variable on a scale 1, ..., K. For simplicity, and without loss of generality, we assume y is a non-decreasing vector, that is, yi ≤ {\displaystyle \leq } yi+1. To this data, one fits a length-p coefficient vector w and a set of thresholds θ1, ..., θK−1 with the property that θ1 < θ2 < ... < θK−1. This set of thresholds divides the real number line into K disjoint segments, corresponding to the K response levels. The model can now be formulated as Pr ( y ≤ i ∣ x ) = σ ( θ i − w ⋅ x ) {\displaystyle \Pr(y\leq i\mid \mathbf {x} )=\sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )} or, the cumulative probability of the response y being at most i is given by a function σ (the inverse link function) applied to a linear function of x. Several choices exist for σ; the logistic function σ ( θ i − w ⋅ x ) = 1 1 + e − ( θ i − w ⋅ x ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )={\frac {1}{1+e^{-(\theta _{i}-\mathbf {w} \cdot \mathbf {x} )}}}} gives the ordered logit model, while using the CDF of the standard normal distribution gives the ordered probit model. A third option is to use an exponential function σ ( θ i − w ⋅ x ) = 1 − exp ⁡ ( − exp ⁡ ( θ i − w ⋅ x ) ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )=1-\exp(-\exp(\theta _{i}-\mathbf {w} \cdot \mathbf {x} ))} which gives the proportional hazards model. === Latent variable model === The probit version of the above model can be justified by assuming the existence of a real-valued latent variable (unobserved quantity) y, determined by y ∗ = w ⋅ x + ε {\displaystyle y^{}=\mathbf {w} \cdot \mathbf {x} +\varepsilon } where ε is normally distributed with zero mean and unit variance, conditioned on x. The response variable y results from an "incomplete measurement" of y, where one only determines the interval into which y falls: y = { 1 if y ∗ ≤ θ 1 , 2 if θ 1 < y ∗ ≤ θ 2 , 3 if θ 2 < y ∗ ≤ θ 3 ⋮ K if θ K − 1 < y ∗ . {\displaystyle y={\begin{cases}1&{\text{if}}~~y^{}\leq \theta _{1},\\2&{\text{if}}~~\theta _{1} Read more →

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  Information gain ratio

  In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information. == Information gain calculation == Information gain is the reduction in entropy produced from partitioning a set with attributes a {\displaystyle a} and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle {\text{IG}}(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where T {\displaystyle T} is a random variable and H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0. == Split information calculation == The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) {\displaystyle {\text{SplitInformation}}(X)=-\sum _{i=1}^{n}{{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}\log {_{2}}{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}}} where X {\displaystyle X} is a discrete random variable with possible values x 1 , x 2 , . . . , x i {\displaystyle {x_{1},x_{2},...,x_{i}}} and N ( x i ) {\displaystyle N(x_{i})} being the number of times that x i {\displaystyle x_{i}} occurs divided by the total count of events N ( x ) {\displaystyle N(x)} where x {\displaystyle x} is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation. == Information gain ratio calculation == The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) {\displaystyle {\text{IGR}}(T,a)={\text{IG}}(T,a)/{\text{SplitInformation}}(T)} IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T | a ) log ⁡ P ( T | a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) {\displaystyle {\text{IGR}}(T,a)={\frac {-\sum _{i=1}^{n}{\mathrm {P} (T)\log \mathrm {P} (T)}-(-\sum _{i=1}^{n}{\mathrm {P} (T|a)\log \mathrm {P} (T|a)})}{-\sum _{i=1}^{n}{{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}\log {_{2}}{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}}}}} == Example == Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more. == Advantages == Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root. == Disadvantages == Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values. == Difference from information gain == Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.

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  International Clinical Trials Registry Platform

  The International Clinical Trials Registry Platform (ICTRP) is a platform for the registration of clinical trials operated by the World Health Organization. The ICTRP combines data from multiple cooperating clinical trials registries to generate a global view of clinical trials worldwide, with a search portal that allows access to the entire dataset. It requires a minimum standard set of database fields, the WHO Trial Registration Data Set, to be present for a trial to be registered. All entries are given a Universal Trial Number (UTN) that identifies them uniquely. The organization has sought to assist various national governments in establishing their own clinical trials databases. It combines data from the following primary registries and data providers: Australian New Zealand Clinical Trials Registry (ANZCTR) Brazilian Clinical Trials Registry (ReBec) Chinese Clinical Trial Registry (ChiCTR) Clinical Research Information Service (CRiS), Republic of Korea ClinicalTrials.gov Clinical Trials Information System (CTIS), European Medicines Agency Clinical Trials Registry - India (CTRI) Cuban Public Registry of Clinical Trials (RPCEC) EU Clinical Trials Register (EU-CTR) German Clinical Trials Register (DRKS) Iranian Registry of Clinical Trials (IRCT) ISRCTN (UK) International Traditional Medicine Clinical Trial Registry (ITMCTR) Japan Registry of Clinical Trials (jRCT) Japan Primary Registries Network (JPRN) Lebanese Clinical Trials Registry (LBCTR) Overview of Medical Research in the Netherlands (OMON) Thai Clinical Trials Registry (TCTR) Pan African Clinical Trial Registry (PACTR) Peruvian Clinical Trial Registry (REPEC) Sri Lanka Clinical Trials Registry (SLCTR)

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  Naive Bayes classifier

  In statistics, naive (sometimes simple or idiot's) Bayes classifiers are a family of "probabilistic classifiers" which assume that the features are conditionally independent, given the target class. In other words, a naive Bayes model assumes the information about the class provided by each variable is unrelated to the information from the others, with no information shared between the predictors. The highly unrealistic nature of this assumption, called the naive independence assumption, is what gives the classifier its name. These classifiers are some of the simplest Bayesian network models. Naive Bayes classifiers generally perform worse than more advanced models like logistic regressions, especially at quantifying uncertainty (with naive Bayes models often producing wildly overconfident probabilities). However, they are highly scalable, requiring only one parameter for each feature or predictor in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression (simply by counting observations in each group), rather than the expensive iterative approximation algorithms required by most other models. Despite the use of Bayes' theorem in the classifier's decision rule, naive Bayes is not (necessarily) a Bayesian method, and naive Bayes models can be fit to data using either Bayesian or frequentist methods. == Introduction == Naive Bayes is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. There is not a single algorithm for training such classifiers, but a family of algorithms based on a common principle: all naive Bayes classifiers assume that the value of a particular feature is independent of the value of any other feature, given the class variable. For example, a fruit may be considered to be an apple if it is red, round, and about 10 cm in diameter. A naive Bayes classifier considers each of these features to contribute independently to the probability that this fruit is an apple, regardless of any possible correlations between the color, roundness, and diameter features. In many practical applications, parameter estimation for naive Bayes models uses the method of maximum likelihood; in other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods. Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex real-world situations. In 2004, an analysis of the Bayesian classification problem showed that there are sound theoretical reasons for the apparently implausible efficacy of naive Bayes classifiers. Still, a comprehensive comparison with other classification algorithms in 2006 showed that Bayes classification is outperformed by other approaches, such as boosted trees or random forests. An advantage of naive Bayes is that it only requires a small amount of training data to estimate the parameters necessary for classification. == Probabilistic model == Abstractly, naive Bayes is a conditional probability model: it assigns probabilities p ( C k ∣ x 1 , … , x n ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})} for each of the K possible outcomes or classes C k {\displaystyle C_{k}} given a problem instance to be classified, represented by a vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} encoding some n features (independent variables). The problem with the above formulation is that if the number of features n is large or if a feature can take on a large number of values, then basing such a model on probability tables is infeasible. The model must therefore be reformulated to make it more tractable. Using Bayes' theorem, the conditional probability can be decomposed as: p ( C k ∣ x ) = p ( C k ) p ( x ∣ C k ) p ( x ) {\displaystyle p(C_{k}\mid \mathbf {x} )={\frac {p(C_{k})\ p(\mathbf {x} \mid C_{k})}{p(\mathbf {x} )}}\,} In plain English, using Bayesian probability terminology, the above equation can be written as posterior = prior × likelihood evidence {\displaystyle {\text{posterior}}={\frac {{\text{prior}}\times {\text{likelihood}}}{\text{evidence}}}\,} In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on C {\displaystyle C} and the values of the features x i {\displaystyle x_{i}} are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model p ( C k , x 1 , … , x n ) {\displaystyle p(C_{k},x_{1},\ldots ,x_{n})\,} which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability: p ( C k , x 1 , … , x n ) = p ( x 1 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) p ( x 3 , … , x n , C k ) = ⋯ = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) ⋯ p ( x n − 1 ∣ x n , C k ) p ( x n ∣ C k ) p ( C k ) {\displaystyle {\begin{aligned}p(C_{k},x_{1},\ldots ,x_{n})&=p(x_{1},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\ p(x_{3},\ldots ,x_{n},C_{k})\\&=\cdots \\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\cdots p(x_{n-1}\mid x_{n},C_{k})\ p(x_{n}\mid C_{k})\ p(C_{k})\\\end{aligned}}} Now the "naive" conditional independence assumptions come into play: assume that all features in x {\displaystyle \mathbf {x} } are mutually independent, conditional on the category C k {\displaystyle C_{k}} . Under this assumption, p ( x i ∣ x i + 1 , … , x n , C k ) = p ( x i ∣ C k ) . {\displaystyle p(x_{i}\mid x_{i+1},\ldots ,x_{n},C_{k})=p(x_{i}\mid C_{k})\,.} Thus, the joint model can be expressed as p ( C k ∣ x 1 , … , x n ) ∝ p ( C k , x 1 , … , x n ) = p ( C k ) p ( x 1 ∣ C k ) p ( x 2 ∣ C k ) p ( x 3 ∣ C k ) ⋯ = p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) , {\displaystyle {\begin{aligned}p(C_{k}\mid x_{1},\ldots ,x_{n})\varpropto \ &p(C_{k},x_{1},\ldots ,x_{n})\\&=p(C_{k})\ p(x_{1}\mid C_{k})\ p(x_{2}\mid C_{k})\ p(x_{3}\mid C_{k})\ \cdots \\&=p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})\,,\end{aligned}}} where ∝ {\displaystyle \varpropto } denotes proportionality since the denominator p ( x ) {\displaystyle p(\mathbf {x} )} is omitted. This means that under the above independence assumptions, the conditional distribution over the class variable C {\displaystyle C} is: p ( C k ∣ x 1 , … , x n ) = 1 Z p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})={\frac {1}{Z}}\ p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})} where the evidence Z = p ( x ) = ∑ k p ( C k ) p ( x ∣ C k ) {\displaystyle Z=p(\mathbf {x} )=\sum _{k}p(C_{k})\ p(\mathbf {x} \mid C_{k})} is a scaling factor dependent only on x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , that is, a constant if the values of the feature variables are known. Often, it is only necessary to discriminate between classes. In that case, the scaling factor is irrelevant, and it is sufficient to calculate the log-probability up to a factor: ln ⁡ p ( C k ∣ x 1 , … , x n ) = ln ⁡ p ( C k ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C k ) − ln ⁡ Z ⏟ irrelevant {\displaystyle \ln p(C_{k}\mid x_{1},\ldots ,x_{n})=\ln p(C_{k})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{k})\underbrace {-\ln Z} _{\text{irrelevant}}} The scaling factor is irrelevant, since discrimination subtracts it away: ln ⁡ p ( C k ∣ x 1 , … , x n ) p ( C l ∣ x 1 , … , x n ) = ( ln ⁡ p ( C k ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C k ) ) − ( ln ⁡ p ( C l ) + ∑ i = 1 n ln ⁡ p ( x i ∣ C l ) ) {\displaystyle \ln {\frac {p(C_{k}\mid x_{1},\ldots ,x_{n})}{p(C_{l}\mid x_{1},\ldots ,x_{n})}}=\left(\ln p(C_{k})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{k})\right)-\left(\ln p(C_{l})+\sum _{i=1}^{n}\ln p(x_{i}\mid C_{l})\right)} There are two benefits of using log-probability. One is that it allows an interpretation in information theory, where log-probabilities are units of information in nats. Another is that it avoids arithmetic underflow. === Constructing a classifier from the probability model === The discussion so far has derived the independent feature model, that is, the naive Bayes probability model. The naive Bayes classifier combines this model with a decision rule. One common rule is to pick the hypothesis that is most probable so as to minimize the probability of misclassification; this is known as the maximum a posteriori or MAP decision rule. The corresponding classifier, a Bayes classifier, is the function that assigns a class label y ^ = C k {\displaystyle {\hat {y}}=C_{k}} for some k as follows: y ^ = argmax k ∈ { 1 , … , K } p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) . {\displaystyle {\hat {y}}={\underset {k\in \{1,\ldots ,K\}}{\operatorname {argmax} }}\ p(C_{k})\displays

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  Arabic Speech Corpus

  The Arabic Speech Corpus is a Modern Standard Arabic (MSA) speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of more than 3.7 hours of MSA speech aligned with recorded speech on the phoneme level. The annotations include word stress marks on the individual phonemes. The Arabic Speech Corpus was built as part of a doctoral project by Nawar Halabi at the University of Southampton funded by MicroLinkPC who own an exclusive license to commercialise the corpus, but the corpus is available for strictly non-commercial purposes through the official Arabic Speech Corpus website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. == Purpose == The corpus was mainly built for speech synthesis purposes, specifically Speech Synthesis, but the corpus has been used for building HMM based voices in Arabic. It was also used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The package contains the following: 1813 .wav files containing spoken utterances. 1813 .lab files containing text utterances. 1813 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line. orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line. Orthography is in Buckwalter Format which is friendlier where there is software that does not read Arabic script. It can be easily converted back to Arabic. There is an extra 18 minutes of fully annotated corpus (separate from above but with the same structure as above) which was used to evaluated the corpus (see PhD thesis). The corpus was also used to prove that using automatically extracted, orthography-based stress marks improve the quality of speech synthesis in MSA.

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  Causal Markov condition

  The Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. This is related to the Markov condition, an assumption made in Bayesian probability theory, that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents. Stated loosely, it is assumed that a node has no bearing on nodes which do not descend from it. In a DAG, this local Markov condition is equivalent to the global Markov condition, which states that d-separations in the graph also correspond to conditional independence relations. This also means that a node is conditionally independent of the entire network, given its Markov blanket. A network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. == Motivation == Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of probabilistic causation is that causes raise the probabilities of their effects, all else being equal. A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer. On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:). == Examples == In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall. A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the surface underneath the hammer affected its falling. This essentially states the Causal Markov Condition, that given the existence of gravity the release of the hammer, it will fall regardless of what is beneath it. == Implications == === Dependence and Causation === It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V. This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP). === Screening === It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.

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  Structural synthesis of programs

  Structural synthesis of programs (SSP) is a special form of (automatic) program synthesis that is based on propositional calculus. More precisely, it uses intuitionistic logic for describing the structure of a program in such a detail that the program can be automatically composed from pieces like subroutines or even computer commands. It is assumed that these pieces have been implemented correctly, hence no correctness verification of these pieces is needed. SSP is well suited for automatic composition of services for service-oriented architectures and for synthesis of large simulation programs. == History == Automatic program synthesis began in the artificial intelligence field, with software intended for automatic problem solving. The first program synthesizer was developed by Cordell Green in 1969. At about the same time, mathematicians including R. Constable, Z. Manna, and R. Waldinger explained the possible use of formal logic for automatic program synthesis. Practically applicable program synthesizers appeared considerably later. The idea of structural synthesis of programs was introduced at a conference on algorithms in modern mathematics and computer science organized by Andrey Ershov and Donald Knuth in 1979. The idea originated from G. Pólya’s well-known book on problem solving. The method for devising a plan for solving a problem in SSP was presented as a formal system. The inference rules of the system were restructured and justified in logic by G. Mints and E. Tyugu in 1982. A programming tool PRIZ that uses SSP was developed in the 1980s. A recent Integrated development environment that supports SSP is CoCoViLa — a model-based software development platform for implementing domain specific languages and developing large Java programs. == The logic of SSP == Structural synthesis of programs is a method for composing programs from already implemented components (e.g. from computer commands or software object methods) that can be considered as functions. A specification for synthesis is given in intuitionistic propositional logic by writing axioms about the applicability of functions. An axiom about the applicability of a function f is a logical implication X1 ∧ X2 ∧ ... ∧ Xm → Y1 ∧ Y2 ... Yn, where X1, X2, ... Xm are preconditions and Y1, Y2, ... Yn are postconditions of the application of the function f. In intuitionistic logic, the function f is called a realization of this formula. A precondition can be a proposition stating that input data exists, e.g. Xi may have the meaning “variable xi has received a value”, but it may denote also some other condition, e.g. that resources needed for using the function f are available, etc. A precondition may also be an implication of the same form as the axiom given above; then it is called a subtask. A subtask denotes a function that must be available as an input when the function f is applied. This function itself must be synthesized in the process of SSP. In this case, realization of the axiom is a higher order function, i.e., a function that uses another function as an input. For instance, the formula (state → nextState) ∧ initialState → result can specify a higher order function with two inputs and an output result. The first input is a function that has to be synthesized for computing nextState from state, and the second input is initialState. Higher order functions give generality to the SSP – any control structure needed in a synthesized program can be preprogrammed and used then automatically with a respective specification. In particular, the last axiom presented here is a specification of a complex program – a simulation engine for simulating dynamic systems on models where nextState can be computed from state of the system.

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  Synaptic weight

  In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.

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  Santa Fe Trail problem

  The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.

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