AI Analytics Usf

AI Analytics Usf — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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.
Read more →
Vapnik–Chervonenkis theory

Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. == Introduction == VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptotic theory of the rate of convergence of learning processes How fast is the rate of convergence of the learning process? Theory of controlling the generalization ability of learning processes How can one control the rate of convergence (the generalization ability) of the learning process? Theory of constructing learning machines How can one construct algorithms that can control the generalization ability? VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability, which is an alternative approach for characterizing generalization. In addition, VC theory and VC dimension are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to Statistics. == Overview of VC theory in empirical processes == === Background on empirical processes === Let ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} be a measurable space. For any measure Q {\displaystyle Q} on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} , and any measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } , define Q f = ∫ f d Q {\displaystyle Qf=\int fdQ} Measurability issues will be ignored here, for more technical detail see. Let F {\displaystyle {\mathcal {F}}} be a class of measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } and define: ‖ Q ‖ F = sup { | Q f | : f ∈ F } . {\displaystyle \|Q\|_{\mathcal {F}}=\sup\{\vert Qf\vert \ :\ f\in {\mathcal {F}}\}.} Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be independent, identically distributed random elements of ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Then define the empirical measure P n = n − 1 ∑ i = 1 n δ X i , {\displaystyle \mathbb {P} _{n}=n^{-1}\sum _{i=1}^{n}\delta _{X_{i}},} where δ here stands for the Dirac measure. The empirical measure induces a map F → R {\displaystyle {\mathcal {F}}\to \mathbf {R} } given by: f ↦ P n f = 1 n ( f ( X 1 ) + . . . + f ( X n ) ) {\displaystyle f\mapsto \mathbb {P} _{n}f={\frac {1}{n}}(f(X_{1})+...+f(X_{n}))} Now suppose P is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes F {\displaystyle {\mathcal {F}}} for which statements such as the following hold: uniform law of large numbers: ‖ P n − P ‖ F → n 0 , {\displaystyle \|\mathbb {P} _{n}-P\|_{\mathcal {F}}{\underset {n}{\to }}0,} That is, as n → ∞ {\displaystyle n\to \infty } , | 1 n ( f ( X 1 ) + . . . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigarrow \mathbb {G} ,\quad {\text{in }}\ell ^{\infty }({\mathcal {F}})} In the former case F {\displaystyle {\mathcal {F}}} is called Glivenko–Cantelli class, and in the latter case (under the assumption ∀ x , sup f ∈ F | f ( x ) − P f | < ∞ {\displaystyle \forall x,\sup \nolimits _{f\in {\mathcal {F}}}\vert f(x)-Pf\vert <\infty } ) the class F {\displaystyle {\mathcal {F}}} is called Donsker or P-Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem. These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal {F}}} cannot be too large, and as it turns out that the geometry of F {\displaystyle {\mathcal {F}}} plays a very important role. One way of measuring how big the function set F {\displaystyle {\mathcal {F}}} is to use the so-called covering numbers. The covering number N ( ε , F , ‖ ⋅ ‖ ) {\displaystyle N(\varepsilon ,{\mathcal {F}},\|\cdot \|)} is the minimal number of balls { g : ‖ g − f ‖ < ε } {\displaystyle \{g:\|g-f\|<\varepsilon \}} needed to cover the set F {\displaystyle {\mathcal {F}}} (here it is obviously assumed that there is an underlying norm on F {\displaystyle {\mathcal {F}}} ). The entropy is the logarithm of the covering number. Two sufficient conditions are provided below, under which it can be proved that the set F {\displaystyle {\mathcal {F}}} is Glivenko–Cantelli or Donsker. A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal {F}},L_{1}(Q))<\infty .} The next condition is a version of Dudley's theorem. If F {\displaystyle {\mathcal {F}}} is a class of functions such that ∫ 0 ∞ sup Q log ⁡ N ( ε ‖ F ‖ Q , 2 , F , L 2 ( Q ) ) d ε < ∞ {\displaystyle \int _{0}^{\infty }\sup \nolimits _{Q}{\sqrt {\log N\left(\varepsilon \|F\|_{Q,2},{\mathcal {F}},L_{2}(Q)\right)}}d\varepsilon <\infty } then F {\displaystyle {\mathcal {F}}} is P-Donsker for every probability measure P such that P ∗ F 2 < ∞ {\displaystyle P^{\ast }F^{2}<\infty } . In the last integral, the notation means ‖ f ‖ Q , 2 = ( ∫ | f | 2 d Q ) 1 2 {\displaystyle \|f\|_{Q,2}=\left(\int |f|^{2}dQ\right)^{\frac {1}{2}}} . === Symmetrization === The majority of the arguments about how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section). It is presented here: Consider the empirical process: f ↦ ( P n − P ) f = 1 n ∑ i = 1 n ( f ( X i ) − P f ) {\displaystyle f\mapsto (\mathbb {P} _{n}-P)f={\dfrac {1}{n}}\sum _{i=1}^{n}(f(X_{i})-Pf)} Turns out that there is a connection between the empirical and the following symmetrized process: f ↦ P n 0 f = 1 n ∑ i = 1 n ε i f ( X i ) {\displaystyle f\mapsto \mathbb {P} _{n}^{0}f={\dfrac {1}{n}}\sum _{i=1}^{n}\varepsilon _{i}f(X_{i})} The symmetrized process is a Rademacher process, conditionally on the data X i {\displaystyle X_{i}} . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions F {\displaystyle {\mathcal {F}}} , E Φ ( ‖ P n − P ‖ F ) ≤ E Φ ( 2 ‖ P n 0 ‖ F ) {\displaystyle \mathbb {E} \Phi (\|\mathbb {P} _{n}-P\|_{\mathcal {F}})\leq \mathbb {E} \Phi \left(2\left\|\mathbb {P} _{n}^{0}\right\|_{\mathcal {F}}\right)} The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X i {\displaystyle X_{i}} (sometimes referred to as a ghost sample) and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem. A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to P n 0 {\displaystyle \mathbb {P} _{n}^{0}} and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties. === VC Connection === It turns out that there is a fascinating connection between certain combinatorial properties of the set F {\displaystyle {\mathcal {F}}} and the entropy numbers. Uniform covering numbers can be controlled by the notion of Vapnik–Chervonenkis classes of sets – or shortly VC sets. Consider a collection C {\displaystyle {\mathcal {C}}} of subsets of the sample space X {\displaystyle
Read more →
Tucker decomposition

In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor T ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows, T = T × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) {\displaystyle T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}} where T ∈ F d 1 × d 2 × d 3 {\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}} is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of T {\displaystyle T} , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor T {\displaystyle {\mathcal {T}}} respectively. U ( 1 ) , U ( 2 ) , U ( 3 ) {\displaystyle U^{(1)},U^{(2)},U^{(3)}} are unitary matrices in F d 1 × n 1 , F d 2 × n 2 , F d 3 × n 3 {\displaystyle F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}} respectively. The k-mode product (k = 1, 2, 3) of T {\displaystyle {\mathcal {T}}} by U ( k ) {\displaystyle U^{(k)}} is denoted as T × U ( k ) {\displaystyle {\mathcal {T}}\times U^{(k)}} with entries as ( T × 1 U ( 1 ) ) ( i 1 , j 2 , j 3 ) = ∑ j 1 = 1 d 1 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) ( T × 2 U ( 2 ) ) ( j 1 , i 2 , j 3 ) = ∑ j 2 = 1 d 2 T ( j 1 , j 2 , j 3 ) U ( 2 ) ( j 2 , i 2 ) ( T × 3 U ( 3 ) ) ( j 1 , j 2 , i 3 ) = ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle {\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(i_{1},j_{2},j_{3})&=\sum _{j_{1}=1}^{d_{1}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(j_{1},i_{2},j_{3})&=\sum _{j_{2}=1}^{d_{2}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(2)}(j_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(j_{1},j_{2},i_{3})&=\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(3)}(j_{3},i_{3})\end{aligned}}} Altogether, the decomposition may also be written more directly as T ( i 1 , i 2 , i 3 ) = ∑ j 1 = 1 d 1 ∑ j 2 = 1 d 2 ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) U ( 2 ) ( j 2 , i 2 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})} Taking d i = n i {\displaystyle d_{i}=n_{i}} for all i {\displaystyle i} is always sufficient to represent T {\displaystyle T} exactly, but often T {\displaystyle T} can be compressed or efficiently approximately by choosing d i < n i {\displaystyle d_{i} Read more →
Ordination (statistics)

Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.
Read more →
Speech segmentation

Speech segmentation is the process of identifying the boundaries between words, syllables, or phonemes in spoken natural languages. The term applies both to the mental processes used by humans, and to artificial processes of natural language processing. In the field of automatic pronunciation assessment, the process of segmenting an utterance against expected word(s) is called forced alignment. Speech segmentation is a subfield of general speech perception and an important subproblem of the technologically focused field of speech recognition, and cannot be adequately solved in isolation. As in most natural language processing problems, one must take into account context, grammar, and semantics, and even so the result is often a probabilistic division (statistically based on likelihood) rather than a categorical one. Though it seems that coarticulation—a phenomenon which may happen between adjacent words just as easily as within a single word—presents the main challenge in speech segmentation across languages, some other problems and strategies employed in solving those problems can be seen in the following sections. This problem overlaps to some extent with the problem of text segmentation that occurs in some languages which are traditionally written without inter-word spaces, like Chinese and Japanese, compared to writing systems which indicate speech segmentation between words by a word divider, such as the space. However, even for those languages, text segmentation is often much easier than speech segmentation, because the written language usually has little interference between adjacent words, and often contains additional clues not present in speech (such as the use of Chinese characters for word stems in Japanese). == Lexical recognition == In natural languages, the meaning of a complex spoken sentence can be understood by decomposing it into smaller lexical segments (roughly, the words of the language), associating a meaning to each segment, and combining those meanings according to the grammar rules of the language. Though lexical recognition is not thought to be used by infants in their first year, due to their highly limited vocabularies, it is one of the major processes involved in speech segmentation for adults. Three main models of lexical recognition exist in current research: first, whole-word access, which argues that words have a whole-word representation in the lexicon; second, decomposition, which argues that morphologically complex words are broken down into their morphemes (roots, stems, inflections, etc.) and then interpreted and; third, the view that whole-word and decomposition models are both used, but that the whole-word model provides some computational advantages and is therefore dominant in lexical recognition. To give an example, in a whole-word model, the word "cats" might be stored and searched for by letter, first "c", then "ca", "cat", and finally "cats". The same word, in a decompositional model, would likely be stored under the root word "cat" and could be searched for after removing the "s" suffix. "Falling", similarly, would be stored as "fall" and suffixed with the "ing" inflection. Though proponents of the decompositional model recognize that a morpheme-by-morpheme analysis may require significantly more computation, they argue that the unpacking of morphological information is necessary for other processes (such as syntactic structure) which may occur parallel to lexical searches. As a whole, research into systems of human lexical recognition is limited due to little experimental evidence that fully discriminates between the three main models. In any case, lexical recognition likely contributes significantly to speech segmentation through the contextual clues it provides, given that it is a heavily probabilistic system—based on the statistical likelihood of certain words or constituents occurring together. For example, one can imagine a situation where a person might say "I bought my dog at a ____ shop" and the missing word's vowel is pronounced as in "net", "sweat", or "pet". While the probability of "netshop" is extremely low, since "netshop" isn't currently a compound or phrase in English, and "sweatshop" also seems contextually improbable, "pet shop" is a good fit because it is a common phrase and is also related to the word "dog". Moreover, an utterance can have different meanings depending on how it is split into words. A popular example, often quoted in the field, is the phrase "How to wreck a nice beach", which sounds very similar to "How to recognize speech". As this example shows, proper lexical segmentation depends on context and semantics which draws on the whole of human knowledge and experience, and would thus require advanced pattern recognition and artificial intelligence technologies to be implemented on a computer. Lexical recognition is of particular value in the field of computer speech recognition, since the ability to build and search a network of semantically connected ideas would greatly increase the effectiveness of speech-recognition software. Statistical models can be used to segment and align recorded speech to words or phones. Applications include automatic lip-synch timing for cartoon animation, follow-the-bouncing-ball video sub-titling, and linguistic research. Automatic segmentation and alignment software is commercially available. == Phonotactic cues == For most spoken languages, the boundaries between lexical units are difficult to identify; phonotactics are one answer to this issue. One might expect that the inter-word spaces used by many written languages like English or Spanish would correspond to pauses in their spoken version, but that is true only in very slow speech, when the speaker deliberately inserts those pauses. In normal speech, one typically finds many consecutive words being said with no pauses between them, and often the final sounds of one word blend smoothly or fuse with the initial sounds of the next word. The notion that speech is produced like writing, as a sequence of distinct vowels and consonants, may be a relic of alphabetic heritage for some language communities. In fact, the way vowels are produced depends on the surrounding consonants just as consonants are affected by surrounding vowels; this is called coarticulation. For example, in the word "kit", the [k] is farther forward than when we say 'caught'. But also, the vowel in "kick" is phonetically different from the vowel in "kit", though we normally do not hear this. In addition, there are language-specific changes which occur in casual speech which makes it quite different from spelling. For example, in English, the phrase "hit you" could often be more appropriately spelled "hitcha". From a decompositional perspective, in many cases, phonotactics play a part in letting speakers know where to draw word boundaries. In English, the word "strawberry" is perceived by speakers as consisting (phonetically) of two parts: "straw" and "berry". Other interpretations such as "stra" and "wberry" are inhibited by English phonotactics, which does not allow the cluster "wb" word-initially. Other such examples are "day/dream" and "mile/stone" which are unlikely to be interpreted as "da/ydream" or "mil/estone" due to the phonotactic probability or improbability of certain clusters. The sentence "Five women left", which could be phonetically transcribed as [faɪvwɪmɘnlɛft], is marked since neither /vw/ in /faɪvwɪmɘn/ nor /nl/ in /wɪmɘnlɛft/ are allowed as syllable onsets or codas in English phonotactics. These phonotactic cues often allow speakers to easily distinguish the boundaries in words. Vowel harmony in languages like Finnish can also serve to provide phonotactic cues. While the system does not allow front vowels and back vowels to exist together within one morpheme, compounds allow two morphemes to maintain their own vowel harmony while coexisting in a word. Therefore, in compounds such as "selkä/ongelma" ('back problem') where vowel harmony is distinct between two constituents in a compound, the boundary will be wherever the switch in harmony takes place—between the "ä" and the "ö" in this case. Still, there are instances where phonotactics may not aid in segmentation. Words with unclear clusters or uncontrasted vowel harmony as in "opinto/uudistus" ('student reform') do not offer phonotactic clues as to how they are segmented. From the perspective of the whole-word model, however, these words are thought be stored as full words, so the constituent parts would not necessarily be relevant to lexical recognition. == In infants and non-natives == Infants are one major focus of research in speech segmentation. Since infants have not yet acquired a lexicon capable of providing extensive contextual clues or probability-based word searches within their first year, as mentioned above, they must often rely primarily upon phonotactic and rhythmic cues (with prosody being the dominant cue), all
Read more →
Vanishing gradient problem

In machine learning, the vanishing gradient problem is the problem of greatly diverging gradient magnitudes between earlier and later layers encountered when training neural networks with backpropagation. In such methods, neural network weights are updated proportional to their partial derivative of the loss function. As the number of forward propagation steps in a network increases, for instance due to greater network depth, the gradients of earlier weights are calculated with increasingly many multiplications. These multiplications shrink the gradient magnitude. Consequently, the gradients of earlier weights will be exponentially smaller than the gradients of later weights. This difference in gradient magnitude might introduce instability in the training process, slow it, or halt it entirely. For instance, consider the hyperbolic tangent activation function. The gradients of this function are in range [0,1]. The product of repeated multiplication with such gradients decreases exponentially. The inverse problem, when weight gradients at earlier layers get exponentially larger, is called the exploding gradient problem. Backpropagation allowed researchers to train supervised deep artificial neural networks from scratch, initially with little success. Hochreiter's diplom thesis of 1991 formally identified the reason for this failure in the "vanishing gradient problem", which not only affects many-layered feedforward networks, but also recurrent networks. The latter are trained by unfolding them into very deep feedforward networks, where a new layer is created for each time-step of an input sequence processed by the network (the combination of unfolding and backpropagation is termed backpropagation through time). == Prototypical models == This section is based on the paper On the difficulty of training Recurrent Neural Networks by Pascanu, Mikolov, and Bengio. === Recurrent network model === A generic recurrent network has hidden states h 1 , h 2 , … {\displaystyle h_{1},h_{2},\dots } , inputs u 1 , u 2 , … {\displaystyle u_{1},u_{2},\dots } , and outputs x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } . Let it be parameterized by θ {\displaystyle \theta } , so that the system evolves as ( h t , x t ) = F ( h t − 1 , u t , θ ) {\displaystyle (h_{t},x_{t})=F(h_{t-1},u_{t},\theta )} Often, the output x t {\displaystyle x_{t}} is a function of h t {\displaystyle h_{t}} , as some x t = G ( h t ) {\displaystyle x_{t}=G(h_{t})} . The vanishing gradient problem already presents itself clearly when x t = h t {\displaystyle x_{t}=h_{t}} , so we simplify our notation to the special case with: x t = F ( x t − 1 , u t , θ ) {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )} Now, take its differential: d x t = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) d x t − 1 = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) [ ∇ θ F ( x t − 2 , u t − 1 , θ ) d θ + ∇ x F ( x t − 2 , u t − 1 , θ ) d x t − 2 ] ⋮ = [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] d θ {\displaystyle {\begin{aligned}dx_{t}&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )dx_{t-1}\\&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )\left[\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )d\theta +\nabla _{x}F(x_{t-2},u_{t-1},\theta )dx_{t-2}\right]\\&\;\;\vdots \\&=\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]d\theta \end{aligned}}} Training the network requires us to define a loss function to be minimized. Let it be L ( x T , u 1 , … , u T ) {\displaystyle L(x_{T},u_{1},\dots ,u_{T})} , then minimizing it by gradient descent gives Δ θ = − η ⋅ [ ∇ x L ( x T ) ( ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ) ] T {\displaystyle \Delta \theta =-\eta \cdot \left[\nabla _{x}L(x_{T})\left(\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right)\right]^{T}} where η {\displaystyle \eta } is the learning rate. The vanishing/exploding gradient problem appears because there are repeated multiplications, of the form ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ∇ x F ( x t − 3 , u t − 2 , θ ) ⋯ {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\nabla _{x}F(x_{t-3},u_{t-2},\theta )\cdots } ==== Example: recurrent network with sigmoid activation ==== For a concrete example, consider a typical recurrent network defined by x t = F ( x t − 1 , u t , θ ) = W rec σ ( x t − 1 ) + W in u t + b {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\sigma (x_{t-1})+W_{\text{in}}u_{t}+b} where θ = ( W rec , W in ) {\displaystyle \theta =(W_{\text{rec}},W_{\text{in}})} is the network parameter, σ {\displaystyle \sigma } is the sigmoid activation function, applied to each vector coordinate separately, and b {\displaystyle b} is the bias vector. Then, ∇ x F ( x t − 1 , u t , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))} , and so ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ⋯ ∇ x F ( x t − k , u t − k + 1 , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) W rec diag ⁡ ( σ ′ ( x t − 2 ) ) ⋯ W rec diag ⁡ ( σ ′ ( x t − k ) ) {\displaystyle {\begin{aligned}&\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\cdots \nabla _{x}F(x_{t-k},u_{t-k+1},\theta )\\&=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-2}))\cdots W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-k}))\end{aligned}}} Since | σ ′ | ≤ 1 {\displaystyle \left|\sigma '\right|\leq 1} , the operator norm of the above multiplication is bounded above by ‖ W rec ‖ k {\displaystyle \left\|W_{\text{rec}}\right\|^{k}} . So if the spectral radius of W rec {\displaystyle W_{\text{rec}}} is γ < 1 {\displaystyle \gamma <1} , then at large k {\displaystyle k} , the above multiplication has operator norm bounded above by γ k → 0 {\displaystyle \gamma ^{k}\to 0} . This is the prototypical vanishing gradient problem. The effect of a vanishing gradient is that the network cannot learn long-range effects. Recall Equation (loss differential): ∇ θ L = ∇ x L ( x T , u 1 , … , u T ) [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] {\displaystyle \nabla _{\theta }L=\nabla _{x}L(x_{T},u_{1},\dots ,u_{T})\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]} The components of ∇ θ F ( x , u , θ ) {\displaystyle \nabla _{\theta }F(x,u,\theta )} are just components of σ ( x ) {\displaystyle \sigma (x)} and u {\displaystyle u} , so if u t , u t − 1 , … {\displaystyle u_{t},u_{t-1},\dots } are bounded, then ‖ ∇ θ F ( x t − k − 1 , u t − k , θ ) ‖ {\displaystyle \left\|\nabla _{\theta }F(x_{t-k-1},u_{t-k},\theta )\right\|} is also bounded by some M > 0 {\displaystyle M>0} , and so the terms in ∇ θ L {\displaystyle \nabla _{\theta }L} decay as M γ k {\displaystyle M\gamma ^{k}} . This means that, effectively, ∇ θ L {\displaystyle \nabla _{\theta }L} is affected only by the first O ( γ − 1 ) {\displaystyle O(\gamma ^{-1})} terms in the sum. If γ ≥ 1 {\displaystyle \gamma \geq 1} , the above analysis does not quite work. For the prototypical exploding gradient problem, the next model is clearer. === Dynamical systems model === Following (Doya, 1993), consider this one-neuron recurrent network with sigmoid activation: x t + 1 = ( 1 − ε ) x t + ε σ ( w x t + b ) + ε w ′ u t {\displaystyle x_{t+1}=(1-\varepsilon )x_{t}+\varepsilon \sigma (wx_{t}+b)+\varepsilon w'u_{t}} At the small ε {\displaystyle \varepsilon } limit, the dynamics of the network becomes d x d t = − x ( t ) + σ ( w x ( t ) + b ) + w ′ u ( t ) {\displaystyle {\frac {dx}{dt}}=-x(t)+\sigma (wx(t)+b)+w'u(t)} Consider first the autonomous case, with u = 0 {\displaystyle u=0} . Set w = 5.0 {\displaystyle w=5.0} , and vary b {\displaystyle b} in [ − 3 , − 2 ] {\displaystyle [-3,-2]} . As b {\displaystyle b} decreases, the system has 1 stable point, then has 2 stable points and 1 unstable point, and finally has 1 stable point again. Explicitly, the stable points are ( x , b ) = ( x , ln ⁡ ( x 1 − x ) − 5 x ) {\displaystyle (x,b)=\left(x,\ln \left({\frac {x}{1-x}}\right)-5x\right)} . Now consider Δ x ( T ) Δ x ( 0 ) {\displaystyle {\frac {\Delta x(T)}{\Delta x(0)}}} and Δ x ( T ) Δ b {\displaystyle {\frac {\Delta x(T)}{\Delta b}}} , where T {\displaystyle T} is large enough that the system has settled into one of the stable points. If ( x ( 0 ) , b ) {\displaystyle (x(0),b)} puts the system very close to an unstable point, then a tiny variation in x ( 0 ) {\displaystyle x(0)} or b {\displaystyle b} wo
Read more →
NSynth

NSynth (a portmanteau of "Neural Synthesis") is a WaveNet-based autoencoder for synthesizing audio, outlined in a paper in April 2017. == Overview == The model generates sounds through a neural network based synthesis, employing a WaveNet-style autoencoder to learn its own temporal embeddings from four different sounds. Google then released an open source hardware interface for the algorithm called NSynth Super, used by notable musicians such as Grimes and YACHT to generate experimental music using artificial intelligence. The research and development of the algorithm was part of a collaboration between Google Brain, Magenta and DeepMind. == Technology == === Dataset === The NSynth dataset is composed of 305,979 one-shot instrumental notes featuring a unique pitch, timbre, and envelope, sampled from 1,006 instruments from commercial sample libraries. For each instrument the dataset contains four-second 16 kHz audio snippets by ranging over every pitch of a standard MIDI piano, as well as five different velocities. The dataset is made available under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. === Machine learning model === A spectral autoencoder model and a WaveNet autoencoder model are publicly available on GitHub. The baseline model uses a spectrogram with fft_size 1024 and hop_size 256, MSE loss on the magnitudes, and the Griffin-Lim algorithm for reconstruction. The WaveNet model trains on mu-law encoded waveform chunks of size 6144. It learns embeddings with 16 dimensions that are downsampled by 512 in time. == NSynth Super == In 2018 Google released a hardware interface for the NSynth algorithm, called NSynth Super, designed to provide an accessible physical interface to the algorithm for musicians to use in their artistic production. Design files, source code and internal components are released under an open source Apache License 2.0, enabling hobbyists and musicians to freely build and use the instrument. At the core of the NSynth Super there is a Raspberry Pi, extended with a custom printed circuit board to accommodate the interface elements. == Influence == Despite not being publicly available as a commercial product, NSynth Super has been used by notable artists, including Grimes and YACHT. Grimes reported using the instrument in her 2020 studio album Miss Anthropocene. YACHT announced an extensive use of NSynth Super in their album Chain Tripping. Claire L. Evans compared the potential influence of the instrument to the Roland TR-808. The NSynth Super design was honored with a D&AD Yellow Pencil award in 2018.
Read more →
Variational message passing

Variational message passing (VMP) is an approximate inference technique for continuous- or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as latent Dirichlet allocation, and works by updating an approximate distribution at each node through messages in the node's Markov blanket. == Likelihood lower bound == Given some set of hidden variables H {\displaystyle H} and observed variables V {\displaystyle V} , the goal of approximate inference is to maximize a lower-bound on the probability that a graphical model is in the configuration V {\displaystyle V} . Over some probability distribution Q {\displaystyle Q} (to be defined later), ln ⁡ P ( V ) = ∑ H Q ( H ) ln ⁡ P ( H , V ) P ( H | V ) = ∑ H Q ( H ) [ ln ⁡ P ( H , V ) Q ( H ) − ln ⁡ P ( H | V ) Q ( H ) ] {\displaystyle \ln P(V)=\sum _{H}Q(H)\ln {\frac {P(H,V)}{P(H|V)}}=\sum _{H}Q(H){\Bigg [}\ln {\frac {P(H,V)}{Q(H)}}-\ln {\frac {P(H|V)}{Q(H)}}{\Bigg ]}} . So, if we define our lower bound to be L ( Q ) = ∑ H Q ( H ) ln ⁡ P ( H , V ) Q ( H ) {\displaystyle L(Q)=\sum _{H}Q(H)\ln {\frac {P(H,V)}{Q(H)}}} , then the likelihood is simply this bound plus the relative entropy between P {\displaystyle P} and Q {\displaystyle Q} . Because the relative entropy is non-negative, the function L {\displaystyle L} defined above is indeed a lower bound of the log likelihood of our observation V {\displaystyle V} . The distribution Q {\displaystyle Q} will have a simpler character than that of P {\displaystyle P} because marginalizing over P {\displaystyle P} is intractable for all but the simplest of graphical models. In particular, VMP uses a factorized distribution Q ( H ) = ∏ i Q i ( H i ) , {\displaystyle Q(H)=\prod _{i}Q_{i}(H_{i}),} where H i {\displaystyle H_{i}} is a disjoint part of the graphical model. == Determining the update rule == The likelihood estimate needs to be as large as possible; because it's a lower bound, getting closer log ⁡ P {\displaystyle \log P} improves the approximation of the log likelihood. By substituting in the factorized version of Q {\displaystyle Q} , L ( Q ) {\displaystyle L(Q)} , parameterized over the hidden nodes H i {\displaystyle H_{i}} as above, is simply the negative relative entropy between Q j {\displaystyle Q_{j}} and Q j ∗ {\displaystyle Q_{j}^{}} plus other terms independent of Q j {\displaystyle Q_{j}} if Q j ∗ {\displaystyle Q_{j}^{}} is defined as Q j ∗ ( H j ) = 1 Z e E − j { ln ⁡ P ( H , V ) } {\displaystyle Q_{j}^{}(H_{j})={\frac {1}{Z}}e^{\mathbb {E} _{-j}\{\ln P(H,V)\}}} , where E − j { ln ⁡ P ( H , V ) } {\displaystyle \mathbb {E} _{-j}\{\ln P(H,V)\}} is the expectation over all distributions Q i {\displaystyle Q_{i}} except Q j {\displaystyle Q_{j}} . Thus, if we set Q j {\displaystyle Q_{j}} to be Q j ∗ {\displaystyle Q_{j}^{}} , the bound L {\displaystyle L} is maximized. == Messages in variational message passing == Parents send their children the expectation of their sufficient statistic while children send their parents their natural parameter, which also requires messages to be sent from the co-parents of the node. == Relationship to exponential families == Because all nodes in VMP come from exponential families and all parents of nodes are conjugate to their children nodes, the expectation of the sufficient statistic can be computed from the normalization factor. == VMP algorithm == The algorithm begins by computing the expected value of the sufficient statistics for that vector. Then, until the likelihood converges to a stable value (this is usually accomplished by setting a small threshold value and running the algorithm until it increases by less than that threshold value), do the following at each node: Get all messages from parents. Get all messages from children (this might require the children to get messages from the co-parents). Compute the expected value of the nodes sufficient statistics. == Constraints == Because every child must be conjugate to its parent, this has limited the types of distributions that can be used in the model. For example, the parents of a Gaussian distribution must be a Gaussian distribution (corresponding to the Mean) and a gamma distribution (corresponding to the precision, or one over σ {\displaystyle \sigma } in more common parameterizations). Discrete variables can have Dirichlet parents, and Poisson and exponential nodes must have gamma parents. More recently, VMP has been extended to handle models that violate this conditional conjugacy constraint. == Literature == John Winn; Christopher M. Bishop (2005). "Variational Message Passing" (PDF). Journal of Machine Learning Research. 6: 661–694. ISSN 1533-7928. Wikidata Q139488859. Beal, M.J. (2003). Variational Algorithms for Approximate Bayesian Inference (PDF) (PhD). Gatsby Computational Neuroscience Unit, University College London. Archived from the original (PDF) on 2005-04-28. Retrieved 2007-02-15.
Read more →
Automated negotiation

Automated negotiation is a form of interaction in systems that are composed of multiple autonomous agents, in which the aim is to reach agreements through an iterative process of making offers. Automated negotiation can be employed for many tasks human negotiators regularly engage in, such as bargaining and joint decision making. The main topics in automated negotiation revolve around the design of protocols and negotiating strategies. == History == Through digitization, the beginning of the 21st century has seen a growing interest in the automation of negotiation and e-negotiation systems, for example in the setting of e-commerce. This interest is fueled by the promise of automated agents being able to negotiate on behalf of human negotiators, and to find better outcomes than human negotiators. == Examples == Examples of automated negotiation include: Online dispute resolution, in which disagreements between parties are settled. Sponsored search auction, where bids are placed on advertisement keywords. Content negotiation, in which user agents negotiate over HTTP about how to best represent a web resource. Negotiation support systems, in which negotiation decision-making activities are supported by an information system.
Read more →
Ho–Kashyap algorithm

The Ho–Kashyap algorithm is an iterative method in machine learning for finding a linear decision boundary that separates two linearly separable classes. It was developed by Yu-Chi Ho and Rangasami L. Kashyap in 1965, and usually presented as a problem in linear programming. == Setup == Given a training set consisting of samples from two classes, the Ho–Kashyap algorithm seeks to find a weight vector w {\displaystyle \mathbf {w} } and a margin vector b {\displaystyle \mathbf {b} } such that: Y w = b {\displaystyle \mathbf {Yw} =\mathbf {b} } where Y {\displaystyle \mathbf {Y} } is the augmented data matrix with samples from both classes (with appropriate sign conventions, e.g., samples from class 2 are negated), w {\displaystyle \mathbf {w} } is the weight vector to be determined, and b {\displaystyle \mathbf {b} } is a positive margin vector. The algorithm minimizes the criterion function: J ( w , b ) = | | Y w − b | | 2 {\displaystyle J(\mathbf {w} ,\mathbf {b} )=||\mathbf {Yw} -\mathbf {b} ||^{2}} subject to the constraint that b > 0 {\displaystyle \mathbf {b} >\mathbf {0} } (element-wise). Given a problem of linearly separating two classes, we consider a dataset of elements { ( x i , y i ) } i ∈ 1 : N {\displaystyle \{(\mathbf {x_{i}} ,y_{i})\}_{i\in 1:N}} where y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} . Linearly separating them by a perceptron is equivalent to finding weight and bias w , b {\displaystyle \mathbf {w} ,b} for a perceptron, such that: [ y 1 x 1 1 ⋮ ⋮ y N x N 1 ] [ w b ] > 0 {\displaystyle {\begin{bmatrix}y_{1}\mathbf {x} _{1}&1\\\vdots &\vdots \\y_{N}\mathbf {x} _{N}&1\\\end{bmatrix}}{\begin{bmatrix}\mathbf {w} \\b\end{bmatrix}}>0} == Algorithm == The idea of the Ho–Kashyap algorithm is as follows: Given any b {\displaystyle \mathbf {b} } , the corresponding w {\displaystyle \mathbf {w} } is known: It is simply w = Y + b {\displaystyle \mathbf {w} =\mathbf {Y} ^{+}\mathbf {b} } , where Y + {\displaystyle \mathbf {Y} ^{+}} denotes the Moore–Penrose pseudoinverse of Y {\displaystyle \mathbf {Y} } . Therefore, it only remains to find b {\displaystyle \mathbf {b} } by gradient descent. However, the gradient descent may sometimes decrease some of the coordinates of b {\displaystyle \mathbf {b} } , which may cause some coordinates of b {\displaystyle \mathbf {b} } to become negative, which is undesirable. Therefore, whenever some coordinates of b {\displaystyle \mathbf {b} } would have decreased, those coordinates are unchanged instead. As for the coordinates of b {\displaystyle \mathbf {b} } that would increase, those would increase without issue. Formally, the algorithm is as follows: Initialization: Set b ( 0 ) {\displaystyle \mathbf {b} (0)} to an arbitrary positive vector, typically b ( 0 ) = 1 {\displaystyle \mathbf {b} (0)=\mathbf {1} } (a vector of ones). Set the iteration counter k = 0 {\displaystyle k=0} . Set w ( 0 ) = Y + b ( 0 ) {\displaystyle \mathbf {w} (0)=\mathbf {Y} ^{+}\mathbf {b} (0)} Loop until convergence, or until iteration counter exceeds some k m a x {\displaystyle k_{max}} . Error calculation: Compute the error vector: e ( k ) = Y w ( k ) − b ( k ) {\displaystyle \mathbf {e} (k)=\mathbf {Yw} (k)-\mathbf {b} (k)} . Margin update: Update the margin vector: b ( k + 1 ) = b ( k ) + 2 η k ( e ( k ) + | e ( k ) | ) {\displaystyle \mathbf {b} (k+1)=\mathbf {b} (k)+2\eta _{k}(\mathbf {e} (k)+|\mathbf {e} (k)|)} where η k {\displaystyle \eta _{k}} is a positive learning rate parameter, and | e ( k ) | {\displaystyle |\mathbf {e} (k)|} denotes the element-wise absolute value. Weight calculation: Compute the weight vector using the pseudoinverse: w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} . Convergence check: If | | e ( k ) | | ≤ θ {\displaystyle ||\mathbf {e} (k)||\leq \theta } for some predetermined threshold θ {\displaystyle \theta } (close to zero), then return b ( k + 1 ) , w ( k + 1 ) {\displaystyle \mathbf {b} (k+1),\mathbf {w} (k+1)} . if e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } (all components non-positive), return "Samples not separable.". Return "Algorithm failed to converge in time.". == Properties == If the training data is linearly separable, the algorithm converges to a solution (where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } ) in a finite number of iterations. If the data is not linearly separable, the algorithm may or may not ever reach the point where e ( k ) = 0 {\displaystyle \mathbf {e} (k)=\mathbf {0} } . However, if it does happen that e ( k ) ≤ 0 {\displaystyle \mathbf {e} (k)\leq \mathbf {0} } at some iteration, this proves non-separability. The convergence rate depends on the choice of the learning rate parameter ρ {\displaystyle \rho } and the degree of linear separability of the data. == Relationship to other algorithms == Perceptron algorithm: Both seek linear separators. The perceptron updates weights incrementally based on individual misclassified samples, while Ho–Kashyap is a batch method that processes all samples to compute the pseudoinverse and updates based on an overall error vector. Linear discriminant analysis (LDA): LDA assumes underlying Gaussian distributions with equal covariances for the classes and derives the decision boundary from these statistical assumptions. Ho–Kashyap makes no explicit distributional assumptions and instead tries to solve a system of linear inequalities directly. Support vector machines (SVM): For linearly separable data, SVMs aim to find the maximum-margin hyperplane. The Ho–Kashyap algorithm finds a separating hyperplane but not necessarily the one with the maximum margin. If the data is not separable, soft-margin SVMs allow for some misclassifications by optimizing a trade-off between margin size and misclassification penalty, while Ho–Kashyap provides a least-squares solution. == Variants == Modified Ho–Kashyap algorithm changes weight calculation step w ( k + 1 ) = Y + b ( k + 1 ) {\displaystyle \mathbf {w} (k+1)=\mathbf {Y} ^{+}\mathbf {b} (k+1)} to w ( k + 1 ) = w ( k ) + η k Y + | e ( k ) | {\displaystyle \mathbf {w} (k+1)=\mathbf {w} (k)+\eta _{k}\mathbf {Y} ^{+}|\mathbf {e} (k)|} . Kernel Ho–Kashyap algorithm: Applies kernel methods (the "kernel trick") to the Ho–Kashyap framework to enable non-linear classification by implicitly mapping data to a higher-dimensional feature space.
Read more →
Memtransistor

The memtransistor (a blend word from Memory Transfer Resistor) is an experimental multi-terminal passive electronic component that might be used in the construction of artificial neural networks. It is a combination of the memristor and transistor technology. This technology is different from the 1T-1R approach since the devices are merged into one single entity. Multiple memristors can be embedded with a single transistor, enabling it to more accurately model a neuron with its multiple synaptic connections. A neural network produced from these would provide hardware-based artificial intelligence with a good foundation. == Applications == These types of devices would allow for a synapse model that could realise a learning rule, by which the synaptic efficacy is altered by voltages applied to the terminals of the device. An example of such a learning rule is spike-timing-dependant-plasticty by which the weight of the synapse, in this case the conductivity, could be modulated based on the timing of pre and post synaptic spikes arriving at each terminal. The advantage of this approach over two terminal memristive devices is that read and write protocols have the possibility to occur simultaneously and distinctly.
Read more →
Radial basis function kernel

In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two samples x , x ′ ∈ R k {\displaystyle \mathbf {x} ,\mathbf {x'} \in \mathbb {R} ^{k}} , represented as feature vectors in some input space, is defined as K ( x , x ′ ) = exp ⁡ ( − ‖ x − x ′ ‖ 2 2 σ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp \left(-{\frac {\|\mathbf {x} -\mathbf {x'} \|^{2}}{2\sigma ^{2}}}\right)} ‖ x − x ′ ‖ 2 {\displaystyle \textstyle \|\mathbf {x} -\mathbf {x'} \|^{2}} may be recognized as the squared Euclidean distance between the two feature vectors. σ {\displaystyle \sigma } is a free parameter. An equivalent definition involves a parameter γ = 1 2 σ 2 {\displaystyle \textstyle \gamma ={\tfrac {1}{2\sigma ^{2}}}} : K ( x , x ′ ) = exp ⁡ ( − γ ‖ x − x ′ ‖ 2 ) {\displaystyle K(\mathbf {x} ,\mathbf {x'} )=\exp(-\gamma \|\mathbf {x} -\mathbf {x'} \|^{2})} Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ⁡ ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ⁡ ( 2 2 x ⊤ x ′ − 1 2 ‖ x ‖ 2 − 1 2 ‖ x ′ ‖ 2 ) = exp ⁡ ( x ⊤ x ′ ) exp ⁡ ( − 1 2 ‖ x ‖ 2 ) exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ( x ⊤ x ′ ) j j ! exp ⁡ ( − 1 2 ‖ x ‖ 2 ) exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ∞ ∑ n 1 + n 2 + ⋯ + n k = j exp ⁡ ( − 1 2 ‖ x ‖ 2 ) x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! exp ⁡ ( − 1 2 ‖ x ′ ‖ 2 ) x ′ 1 n 1 ⋯ x ′ k n k n 1 ! ⋯ n k ! = ⟨ φ ( x ) , φ ( x ′ ) ⟩ {\displaystyle {\begin{alignedat}{2}\exp \left(-{\frac {1}{2}}\|\mathbf {x} -\mathbf {x'} \|^{2}\right)&=\exp \left({\frac {2}{2}}\mathbf {x} ^{\top }\mathbf {x'} -{\frac {1}{2}}\|\mathbf {x} \|^{2}-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\exp \left(\mathbf {x} ^{\top }\mathbf {x'} \right)\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }{\frac {(\mathbf {x} ^{\top }\mathbf {x'} )^{j}}{j!}}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right)\\[5pt]&=\sum _{j=0}^{\infty }\quad \sum _{n_{1}+n_{2}+\dots +n_{k}=j}\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right){\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\exp \left(-{\frac {1}{2}}\|\mathbf {x'} \|^{2}\right){\frac {{x'}_{1}^{n_{1}}\cdots {x'}_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\\[5pt]&=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle \end{alignedat}}} φ ( x ) = exp ⁡ ( − 1 2 ‖ x ‖ 2 ) ( a ℓ 0 ( 0 ) , a 1 ( 1 ) , … , a ℓ 1 ( 1 ) , … , a 1 ( j ) , … , a ℓ j ( j ) , … ) {\displaystyle \varphi (\mathbf {x} )=\exp \left(-{\frac {1}{2}}\|\mathbf {x} \|^{2}\right)\left(a_{\ell _{0}}^{(0)},a_{1}^{(1)},\dots ,a_{\ell _{1}}^{(1)},\dots ,a_{1}^{(j)},\dots ,a_{\ell _{j}}^{(j)},\dots \right)} where ℓ j = ( k + j − 1 j ) {\displaystyle \ell _{j}={\tbinom {k+j-1}{j}}} , a ℓ ( j ) = x 1 n 1 ⋯ x k n k n 1 ! ⋯ n k ! | n 1 + n 2 + ⋯ + n k = j ∧ 1 ≤ ℓ ≤ ℓ j {\displaystyle a_{\ell }^{(j)}={\frac {x_{1}^{n_{1}}\cdots x_{k}^{n_{k}}}{\sqrt {n_{1}!\cdots n_{k}!}}}\quad |\quad n_{1}+n_{2}+\dots +n_{k}=j\wedge 1\leq \ell \leq \ell _{j}} == Approximations == Because support vector machines and other models employing the kernel trick do not scale well to large numbers of training samples or large numbers of features in the input space, several approximations to the RBF kernel (and similar kernels) have been introduced. Typically, these take the form of a function z that maps a single vector to a vector of higher dimensionality, approximating the kernel: ⟨ z ( x ) , z ( x ′ ) ⟩ ≈ ⟨ φ ( x ) , φ ( x ′ ) ⟩ = K ( x , x ′ ) {\displaystyle \langle z(\mathbf {x} ),z(\mathbf {x'} )\rangle \approx \langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle =K(\mathbf {x} ,\mathbf {x'} )} where φ {\displaystyle \textstyle \varphi } is the implicit mapping embedded in the RBF kernel. === Fourier random features === One way to construct such a z is to randomly sample from the Fourier transformation of the kernel φ ( x ) = 1 D [ cos ⁡ ⟨ w 1 , x ⟩ , sin ⁡ ⟨ w 1 , x ⟩ , … , cos ⁡ ⟨ w D , x ⟩ , sin ⁡ ⟨ w D , x ⟩ ] T {\displaystyle \varphi (x)={\frac {1}{\sqrt {D}}}[\cos \langle w_{1},x\rangle ,\sin \langle w_{1},x\rangle ,\ldots ,\cos \langle w_{D},x\rangle ,\sin \langle w_{D},x\rangle ]^{T}} where w 1 , . . . , w D {\displaystyle w_{1},...,w_{D}} are independent samples from the normal distribution N ( 0 , σ − 2 I ) {\displaystyle N(0,\sigma ^{-2}I)} . Theorem: E ⁡ [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = e ‖ x − y ‖ 2 / ( 2 σ 2 ) . {\displaystyle \operatorname {E} [\langle \varphi (x),\varphi (y)\rangle ]=e^{\|x-y\|^{2}/(2\sigma ^{2})}.} Proof: It suffices to prove the case of D = 1 {\displaystyle D=1} . Use the trigonometric identity cos ⁡ ( a − b ) = cos ⁡ ( a ) cos ⁡ ( b ) + sin ⁡ ( a ) sin ⁡ ( b ) {\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)} , the spherical symmetry of Gaussian distribution, then evaluate the integral ∫ − ∞ ∞ cos ⁡ ( k x ) e − x 2 / 2 2 π d x = e − k 2 / 2 . {\displaystyle \int _{-\infty }^{\infty }{\frac {\cos(kx)e^{-x^{2}/2}}{\sqrt {2\pi }}}dx=e^{-k^{2}/2}.} Theorem: Var ⁡ [ ⟨ φ ( x ) , φ ( y ) ⟩ ] = O ( D − 1 ) {\displaystyle \operatorname {Var} [\langle \varphi (x),\varphi (y)\rangle ]=O(D^{-1})} . (Appendix A.2). === Nyström method === Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix K, using only a random sample of the training set.
Read more →
Online OS

The Online Operating System was a fully multi-lingual and free to use web desktop written in JavaScript using Ajax. It was a Windows-based desktop environment with open-source applications and system utilities developed upon the reBOX web application framework by iCUBE Network Solutions, an Austrian company located in Vienna. == About the project == OOS.cc, which is short for Online Operating System, was a web application platform that mimicked the look and feel of classic desktop operating systems such as Microsoft Windows, Mac OS X or KDE. It consisted of various open source applications built upon the so-called reBOX web application framework. As applications could be executed in an integrated and parallel way, the OOS could have been considered a web desktop or webtop. It provided basic services such as a GUI, a virtual file system, access control management and possibilities to develop and deploy applications online. As the Online Operating System was executed within a web browser, it was no real operating system but rather a portal to various web applications, offering a high usability and flexibility. The project was partly funded by grants from the Internetprivatstiftung Austria (IPA). As at 01.08.2008 almost 20.000 users have joined the oos.cc community, using the offered featured and applications. == History == The development of the web desktop was started by iCUBE Network Solutions in 2005, followed by the first beta releases in 2006. Hence, together with YouOS and eyeOS, it can be considered to be one of the first publicly available systems of its kind. The first full version including core-level multi-language support, the file system and a basic set of applications was released to the public in March 2007 on the occasion of a national exhibition (ITnT Austria Archived 2007-06-30 at the Wayback Machine) and has left beta state half a year later in October 2007. The first release considered stable (1.0.0) was published in July 2007. The project itself and the contained applications have received several national innovation awards (see,) and have gained attention mainly due to the comprehensive approach taken (see,). OOS.cc started as a national project. The full platform including all offered applications are currently available in three languages (German, English as well as Spanish) and is receiving increasing coverage around the world (for examples see, or). The current version is 1.3.01 from 01.08.2008. == Technical overview == The project is fully written in JavaScript, exclusively using DHTML techniques to run in any web browser without any additional software installation needed. The system implements a modern kind of web application model, excessively using Ajax for communicating between client components and the Java server backend in an exclusively asynchronous manner. Aim is to offer users the unique interaction behavior following the desktop metaphor, which is the main idea of any web desktop. Also typical for this sort of web application is the broadly use of Javascript-on-demand techniques, cutting the complete project source into pieces and loading them instantly when needed. Based on this technical basis, reBOX was the framework library all applications in oos.cc were built of. It is a fully flexible and extensible API, including a GUI widget set, communication mechanisms and server services offering general and framework specific services. The Online Operating System itself consisted of a basic framework, which was able to launch any JavaScript application using the reBOX library. The user interface was based on the behavior of the Windows desktop with a start menu, a task bar and a desktop background. All applications were running in this environment. At server side, there were Java based web services that ran to serve the client processes and to provide data from the relational database in the backend. oos.cc also provided an integrated development environment called Developer Suite, which allowed the community to build own applications for the desktop environment based on reBOX (see development section below). == License == All applications available in oos.cc were open source under the European Union Public Licence (EUPL). The reBOX development toolkit is free to use developing any applications for the webtop. == Features == As mentioned above, all applications published on oos.cc are open source based on the EUPL, and can be "installed" or "deinstalled" to what-ever preferences the user has. Besides global services like the multi-language support or the global theme support, as well as some minor tools and games, oos.cc offered four major services that could be used completely free of charge. Integrated and fully flexible file storage (1 GB per user) HTTP as well as FTP file transfer from and to local file system User-based file-shares within the oos-community WebDAV access Document Management (including Version Control and File Locking mechanisms) Image publishing, organization and post-processing A free sub domain (user.oos.cc) for web- or image publishing, directly integrated in the desktop Groupware applications, including free mail, fetchmail and contact management An integrated development environment where oos-applications can be created directly from within the system (see development section below) Next releases were planned to focus on an extensive security and privacy suite, dealing with challenges like anonymous communication (browsing as well as temporary mail-addresses) as well as offering encrypted password and file storage and connectivity services. Since its initial stable release, OOS.cc could have been accessed using https to ensure secure communication. == Limitations and drawbacks == Limited number of applications: no commercial applications can be hosted. Only reviewed applications are being published No processing of popular office formats (.doc, .odt, etc.) Limited language support: Only English, German and Spanish Dependence on foreign infrastructure: No possibility to extend storage, no additional/guaranteed bandwidth, etc. == Development == One of the key focuses of the team was right from the beginning to offer a very flexible and comprehensive API, that can be used to develop not only custom applications within oos.cc, but also stand-alone web-applications or to integrate single components in existing web-sites. By decoupling the development from web-related "problems" using the reBOX API web-applications can be development in a similar fashion to any Java program: Elements can be positioned and can interact like in high-level object oriented programming languages, without taking care of divs, browser specific behavior or communication handling. The framework also offers multi-language and theme support for existing as well as newly created applications, allowing changing almost every aspect of the look and feel of the used components according to the preferences of its users. For taking advantage of this approach, one of the applications offered in the OOS was an integrated Development Suite, allowing directly writing and executing code and hence creating new programs within the boundaries of the web computer. All applications on oos.cc were released as open source, thus all existing programs were offered to be imported, reviewed or changed and then locally deployed. Following this idea, every user was free to submit changed or newly created applications to be included in the globally offered application set. The last release offered features like auto-completion and an outline-window.
Read more →
Plate notation

In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The assumptions are that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition. == Example == In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate; α is the parameter of the uniform Dirichlet prior on the per-document topic distributions, and β is the parameter of the uniform Dirichlet prior on the per-topic word distribution. The outermost plate represents all the variables related to a specific document, including θ i {\displaystyle \theta _{i}} , the topic distribution for document i. The M in the corner of the plate indicates that the variables inside are repeated M times, once for each document. The inner plate represents the variables associated with each of the N i {\displaystyle N_{i}} words in document i: z i j {\displaystyle z_{ij}} is the topic distribution for the jth word in document i, and w i j {\displaystyle w_{ij}} is the actual word used. The N in the corner represents the repetition of the variables in the inner plate N j {\displaystyle N_{j}} times, once for each word in document i. The circle representing the individual words is shaded, indicating that each w i j {\displaystyle w_{ij}} is observable, and the other circles are empty, indicating that the other variables are latent variables. The directed edges between variables indicate dependencies between the variables: for example, each w i j {\displaystyle w_{ij}} depends on z i j {\displaystyle z_{ij}} and β. == Extensions == A number of extensions have been created by various authors to express more information than simply the conditional relationships. However, few of these have become standard. Perhaps the most commonly used extension is to use rectangles in place of circles to indicate non-random variables—either parameters to be computed, hyperparameters given a fixed value (or computed through empirical Bayes), or variables whose values are computed deterministically from a random variable. The diagram on the right shows a few more non-standard conventions used in some articles in Wikipedia (e.g. variational Bayes): Variables that are actually random vectors are indicated by putting the vector size in brackets in the middle of the node. Variables that are actually random matrices are similarly indicated by putting the matrix size in brackets in the middle of the node, with commas separating row size from column size. Categorical variables are indicated by placing their size (without a bracket) in the middle of the node. Categorical variables that act as "switches", and which pick one or more other random variables to condition on from a large set of such variables (e.g. mixture components), are indicated with a special type of arrow containing a squiggly line and ending in a T junction. Boldface is consistently used for vector or matrix nodes (but not categorical nodes). == Software implementation == Plate notation has been implemented in various TeX/LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab and PyMC.
Read more →
C4.5 algorithm

C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date". It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008. == Algorithm == C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set S = s 1 , s 2 , . . . {\displaystyle S={s_{1},s_{2},...}} of already classified samples. Each sample s i {\displaystyle s_{i}} consists of a p-dimensional vector ( x 1 , i , x 2 , i , . . . , x p , i ) {\displaystyle (x_{1,i},x_{2,i},...,x_{p,i})} , where the x j {\displaystyle x_{j}} represent attribute values or features of the sample, as well as the class in which s i {\displaystyle s_{i}} falls. At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists. This algorithm has a few base cases. All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class. None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class. Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value. === Pseudocode === In pseudocode, the general algorithm for building decision trees is: Check for the above base cases. For each attribute a, find the normalized information gain ratio from splitting on a. Let a_best be the attribute with the highest normalized information gain. Create a decision node that splits on a_best. Recurse on the sublists obtained by splitting on a_best, and add those nodes as children of node. == Improvements from ID3 algorithm == C4.5 made a number of improvements to ID3. Some of these are: Handling both continuous and discrete attributes: In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it. Handling training data with missing attribute values: C4.5 allows attribute values to be marked as missing. Missing attribute values are simply not used in gain and entropy calculations. Handling attributes with differing costs. Pruning trees after creation: C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes. == Improvements in C5.0/See5 algorithm == Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are: Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude) Memory usage - C5.0 is more memory efficient than C4.5 Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees. Support for boosting - Boosting improves the trees and gives them more accuracy. Weighting - C5.0 allows you to weight different cases and misclassification types. Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful. Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL).
Read more →