AI Headshot Free Reddit

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  Ulead MediaStudio Pro

  Ulead MediaStudio Pro (MSP) is real-time, timeline based prosumer level video editing software by Ulead Systems. It is a suite of 5 digital video and audio applications, including: Video Capture, Video Paint, CG Infinity, Audio Editor and Video Editor. MSP is only available on the Windows platform. Since version 8.0, CG Infinity and Video Paint are separate from the MSP suite, and are being sold as a combination product called VideoGraphics Lab (VGL). On June 18, 2008, Corel formally announced that MediaStudio Pro would be discontinued. The final MediaStudio Pro version was 8.10.0039 (Pro 8 Service Pack 1) released June 2, 2006. Corel discontinued support for MediaStudio Pro in June 2009. Version 6.0 is last version to support Windows 95, although recent versions are not compatible with Windows Vista or Windows 7. == Modules == There are 5 stand-alone modules in MSP before version 8.0, they are: Video Capture – The video capturing module in MSP. Video Paint – A frame-by-frame editor which can let user to make some image or hand-drawing effects on video frames. CG Infinity – A vector-based video editing tool which allows user to create logo animation or vector graphics on video frames. Audio Editor – The audio editing tool in MSP. It can utilize DirectX audio filters and Ulead audio filters to do audio effect processing. Video Editor – The module that users do video editing with audio/video effects. It can also utilize DirectX audio filters and 3rd party video filters to do the video editing. Since version 8.0, CG Infinity and Video Paint have been separated from the MSP suite and are being sold as a combination product called VideoGraphics Lab (VGL). == Editions == Ulead MediaStudio Pro had several editions before version 7.0. They are: Full edition: this edition includes all 5 modules. Director's Cut edition: this edition has 3 modules including Video Capture, Video Editor and Audio Editor. SE edition: SE means Simple Edition or Special Edition and is an OEM bundle version. It also includes the 3 modules as Director's Cut, however, is feature limited. Sometimes it will be given freely in video magazines. After version 7.0 only Full edition is available in the MSP suite. On June 18, 2008, Corel formally announced that MediaStudio Pro would be discontinued. == Release history ==

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  Jackknife variance estimates for random forest

  In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects. == Jackknife variance estimates == The sampling variance of bagged learners is: V ( x ) = V a r [ θ ^ ∞ ( x ) ] {\displaystyle V(x)=Var[{\hat {\theta }}^{\infty }(x)]} Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as: V ^ j = n − 1 n ∑ i = 1 n ( θ ^ ( − i ) − θ ¯ ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\hat {\theta }}_{(-i)}-{\overline {\theta }})^{2}} In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as: V ^ j = n − 1 n ∑ i = 1 n ( t ¯ ( − i ) ⋆ ( x ) − t ¯ ⋆ ( x ) ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\overline {t}}_{(-i)}^{\star }(x)-{\overline {t}}^{\star }(x))^{2}} Here, t ⋆ {\displaystyle t^{\star }} denotes a decision tree after training, t ( − i ) ⋆ {\displaystyle t_{(-i)}^{\star }} denotes the result based on samples without i t h {\displaystyle ith} observation. == Examples == E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low. Here, accuracy is measured by error rate, which is defined as: E r r o r R a t e = 1 N ∑ i = 1 N ∑ j = 1 M y i j , {\displaystyle ErrorRate={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij},} Here N is also the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy: l o g l o s s = 1 N ∑ i = 1 N ∑ j = 1 M y i j l o g ( p i j ) {\displaystyle logloss={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij}log(p_{ij})} Here N is the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. p i j {\displaystyle p_{ij}} is the predicted probability of i t h {\displaystyle ith} observation in class j {\displaystyle j} .This method is used in Kaggle These two methods are very similar. == Modification for bias == When using Monte Carlo MSEs for estimating V I J ∞ {\displaystyle V_{IJ}^{\infty }} and V J ∞ {\displaystyle V_{J}^{\infty }} , a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large: E [ V ^ I J B ] − V ^ I J ∞ ≈ n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle E[{\hat {V}}_{IJ}^{B}]-{\hat {V}}_{IJ}^{\infty }\approx {\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} To eliminate this influence, bias-corrected modifications are suggested: V ^ I J − U B = V ^ I J B − n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{IJ-U}^{B}={\hat {V}}_{IJ}^{B}-{\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} V ^ J − U B = V ^ J B − ( e − 1 ) n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{J-U}^{B}={\hat {V}}_{J}^{B}-(e-1){\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}

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  Linear classifier

  In machine learning, a linear classifier makes a classification decision for each object based on a linear combination of its features. A simpler definition is to say that a linear classifier is one whose decision boundaries are linear. Such classifiers work well for practical problems such as document classification, and more generally for problems with many variables (features), reaching accuracy levels comparable to non-linear classifiers while taking less time to train and use. == Definition == If the input feature vector to the classifier is a real vector x → {\displaystyle {\vec {x}}} , then the output score is y = f ( w → ⋅ x → ) = f ( ∑ j w j x j ) , {\displaystyle y=f({\vec {w}}\cdot {\vec {x}})=f\left(\sum _{j}w_{j}x_{j}\right),} where w → {\displaystyle {\vec {w}}} is a real vector of weights and f is a function that converts the dot product of the two vectors into the desired output. (In other words, w → {\displaystyle {\vec {w}}} is a one-form or linear functional mapping x → {\displaystyle {\vec {x}}} onto R.) The weight vector w → {\displaystyle {\vec {w}}} is learned from a set of labeled training samples. Often f is a threshold function, which maps all values of w → ⋅ x → {\displaystyle {\vec {w}}\cdot {\vec {x}}} above a certain threshold to the first class and all other values to the second class; e.g., f ( x ) = { 1 if w T ⋅ x > θ , 0 otherwise {\displaystyle f(\mathbf {x} )={\begin{cases}1&{\text{if }}\ \mathbf {w} ^{T}\cdot \mathbf {x} >\theta ,\\0&{\text{otherwise}}\end{cases}}} The superscript T indicates the transpose and θ {\displaystyle \theta } is a scalar threshold. A more complex f might give the probability that an item belongs to a certain class. For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a hyperplane: all points on one side of the hyperplane are classified as "yes", while the others are classified as "no". A linear classifier is often used in situations where the speed of classification is an issue, since it is often the fastest classifier, especially when x → {\displaystyle {\vec {x}}} is sparse. Also, linear classifiers often work very well when the number of dimensions in x → {\displaystyle {\vec {x}}} is large, as in document classification, where each element in x → {\displaystyle {\vec {x}}} is typically the number of occurrences of a word in a document (see document-term matrix). In such cases, the classifier should be well-regularized. == Generative models vs. discriminative models == There are two broad classes of methods for determining the parameters of a linear classifier w → {\displaystyle {\vec {w}}} . They can be generative and discriminative models. Methods of the former model joint probability distribution, whereas methods of the latter model conditional density functions P ( c l a s s | x → ) {\displaystyle P({\rm {class}}|{\vec {x}})} . Examples of such algorithms include: Linear Discriminant Analysis (LDA)—assumes Gaussian conditional density models Naive Bayes classifier with multinomial or multivariate Bernoulli event models. The second set of methods includes discriminative models, which attempt to maximize the quality of the output on a training set. Additional terms in the training cost function can easily perform regularization of the final model. Examples of discriminative training of linear classifiers include: Logistic regression—maximum likelihood estimation of w → {\displaystyle {\vec {w}}} assuming that the observed training set was generated by a binomial model that depends on the output of the classifier. Perceptron—an algorithm that attempts to fix all errors encountered in the training set Fisher's Linear Discriminant Analysis—an algorithm (different than "LDA") that maximizes the ratio of between-class scatter to within-class scatter, without any other assumptions. It is in essence a method of dimensionality reduction for binary classification. Support vector machine—an algorithm that maximizes the margin between the decision hyperplane and the examples in the training set. Note: Despite its name, LDA does not belong to the class of discriminative models in this taxonomy. However, its name makes sense when we compare LDA to the other main linear dimensionality reduction algorithm: principal components analysis (PCA). LDA is a supervised learning algorithm that utilizes the labels of the data, while PCA is an unsupervised learning algorithm that ignores the labels. To summarize, the name is a historical artifact. Discriminative training often yields higher accuracy than modeling the conditional density functions. However, handling missing data is often easier with conditional density models. All of the linear classifier algorithms listed above can be converted into non-linear algorithms operating on a different input space φ ( x → ) {\displaystyle \varphi ({\vec {x}})} , using the kernel trick. === Discriminative training === Discriminative training of linear classifiers usually proceeds in a supervised way, by means of an optimization algorithm that is given a training set with desired outputs and a loss function that measures the discrepancy between the classifier's outputs and the desired outputs. Thus, the learning algorithm solves an optimization problem of the form arg ⁡ min w R ( w ) + C ∑ i = 1 N L ( y i , w T x i ) {\displaystyle {\underset {\mathbf {w} }{\arg \min }}\;R(\mathbf {w} )+C\sum _{i=1}^{N}L(y_{i},\mathbf {w} ^{\mathsf {T}}\mathbf {x} _{i})} where w is a vector of classifier parameters, L(yi, wTxi) is a loss function that measures the discrepancy between the classifier's prediction and the true output yi for the i'th training example, R(w) is a regularization function that prevents the parameters from getting too large (causing overfitting), and C is a scalar constant (set by the user of the learning algorithm) that controls the balance between the regularization and the loss function. Popular loss functions include the hinge loss (for linear SVMs) and the log loss (for linear logistic regression). If the regularization function R is convex, then the above is a convex problem. Many algorithms exist for solving such problems; popular ones for linear classification include (stochastic) gradient descent, L-BFGS, coordinate descent and Newton methods.

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  Nonlinear dimensionality reduction

  Nonlinear dimensionality reduction (NLDR), also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. == Applications of NLDR == High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while keeping its essential features relatively intact, can make algorithms more efficient and allow analysts to visualize trends and patterns. The reduced-dimensional representations of data are often referred to as "intrinsic variables". This description implies that these are the values from which the data was produced. For example, consider a dataset that contains images of a letter 'A', which has been scaled and rotated by varying amounts. Each image has 32×32 pixels. Each image can be represented as a vector of 1024 pixel values. Each row is a sample on a two-dimensional manifold in 1024-dimensional space (a Hamming space). The intrinsic dimensionality is two, because two variables (rotation and scale) were varied in order to produce the data. Information about the shape or look of a letter 'A' is not part of the intrinsic variables because it is the same in every instance. Nonlinear dimensionality reduction will discard the correlated information (the letter 'A') and recover only the varying information (rotation and scale). By comparison, if principal component analysis, which is a linear dimensionality reduction algorithm, is used to reduce this same dataset into two dimensions, the resulting values are not so well organized. This demonstrates that the high-dimensional vectors (each representing a letter 'A') that sample this manifold vary in a non-linear manner. It should be apparent, therefore, that NLDR has several applications in the field of computer-vision. For example, consider a robot that uses a camera to navigate in a closed static environment. The images obtained by that camera can be considered to be samples on a manifold in high-dimensional space, and the intrinsic variables of that manifold will represent the robot's position and orientation. Invariant manifolds are of general interest for model order reduction in dynamical systems. In particular, if there is an attracting invariant manifold in the phase space, nearby trajectories will converge onto it and stay on it indefinitely, rendering it a candidate for dimensionality reduction of the dynamical system. While such manifolds are not guaranteed to exist in general, the theory of spectral submanifolds (SSM) gives conditions for the existence of unique attracting invariant objects in a broad class of dynamical systems. Active research in NLDR seeks to unfold the observation manifolds associated with dynamical systems to develop modeling techniques. Some of the more prominent nonlinear dimensionality reduction techniques are listed below. == Important concepts == === Sammon's mapping === Sammon's mapping is one of the first and most popular NLDR techniques. === Self-organizing map === The self-organizing map (SOM, also called Kohonen map) and its probabilistic variant generative topographic mapping (GTM) use a point representation in the embedded space to form a latent variable model based on a non-linear mapping from the embedded space to the high-dimensional space. These techniques are related to work on density networks, which also are based around the same probabilistic model. === Kernel principal component analysis === Perhaps the most widely used algorithm for dimensional reduction is kernel PCA. PCA begins by computing the covariance matrix of the m × n {\displaystyle m\times n} matrix X {\displaystyle \mathbf {X} } C = 1 m ∑ i = 1 m x i x i T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\mathbf {x} _{i}\mathbf {x} _{i}^{\mathsf {T}}}.} It then projects the data onto the first k eigenvectors of that matrix. By comparison, KPCA begins by computing the covariance matrix of the data after being transformed into a higher-dimensional space, C = 1 m ∑ i = 1 m Φ ( x i ) Φ ( x i ) T . {\displaystyle C={\frac {1}{m}}\sum _{i=1}^{m}{\Phi (\mathbf {x} _{i})\Phi (\mathbf {x} _{i})^{\mathsf {T}}}.} It then projects the transformed data onto the first k eigenvectors of that matrix, just like PCA. It uses the kernel trick to factor away much of the computation, such that the entire process can be performed without actually computing Φ ( x ) {\displaystyle \Phi (\mathbf {x} )} . Of course Φ {\displaystyle \Phi } must be chosen such that it has a known corresponding kernel. Unfortunately, it is not trivial to find a good kernel for a given problem, so KPCA does not yield good results with some problems when using standard kernels. For example, it is known to perform poorly with these kernels on the Swiss roll manifold. However, one can view certain other methods that perform well in such settings (e.g., Laplacian Eigenmaps, LLE) as special cases of kernel PCA by constructing a data-dependent kernel matrix. KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time. === Principal curves and manifolds === Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold. This approach was originally proposed by Trevor Hastie in his 1984 thesis, which he formally introduced in 1989. This idea has been explored further by many authors. How to define the "simplicity" of the manifold is problem-dependent, however, it is commonly measured by the intrinsic dimensionality and/or the smoothness of the manifold. Usually, the principal manifold is defined as a solution to an optimization problem. The objective function includes a quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and Kohonen's SOM. === Laplacian eigenmaps === Laplacian eigenmaps uses spectral techniques to perform dimensionality reduction. This technique relies on the basic assumption that the data lies in a low-dimensional manifold in a high-dimensional space. This algorithm cannot embed out-of-sample points, but techniques based on Reproducing kernel Hilbert space regularization exist for adding this capability. Such techniques can be applied to other nonlinear dimensionality reduction algorithms as well. Traditional techniques like principal component analysis do not consider the intrinsic geometry of the data. Laplacian eigenmaps builds a graph from neighborhood information of the data set. Each data point serves as a node on the graph and connectivity between nodes is governed by the proximity of neighboring points (using e.g. the k-nearest neighbor algorithm). The graph thus generated can be considered as a discrete approximation of the low-dimensional manifold in the high-dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low-dimensional space, preserving local distances. The eigenfunctions of the Laplace–Beltrami operator on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold (compare to Fourier series on the unit circle manifold). Attempts to place Laplacian eigenmaps on solid theoretical ground have met with some success, as under certain nonrestrictive assumptions, the graph Laplacian matrix has been shown to converge to the Laplace–Beltrami operator as the number of points goes to infinity. === Isomap === Isomap is a combination of the Floyd–Warshall algorithm with classic Multidimensional Scaling (MDS). Classic MDS takes a matrix of pair-wise distances between all points and computes a position for each point. Isomap assumes that the pair-wise distances are only known between neighboring points, and uses the Floyd–Warshall algorithm to compute the pair-wise distances between all other points. This effectively estimates the full matrix of pair-wise geodesic distances between all of the points. Isomap th

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  Instance (computer science)

  In computer science, an instance or token (from metalogic and metamathematics) is a specific occurrence of a software element that is based on a type definition. When created, an occurrence is said to have been instantiated, and both the creation process and the result of creation are called instantiation. == Examples == Chat AI instance In chat-based AI systems, an assistant can be invoked across many independent conversation sessions (often called a thread), each with its own message history. A specific execution of the assistant over that session may be represented as a run (an execution on a thread). Class instance In object-oriented programming, an object created from a class type. Each instance of a class shares the class-defined structure and behavior but has its own identity and state. Procedural instance In some contexts (including Simula), each procedure call can be viewed as an instance of that procedure—an activation with its own parameters and local variables. Computer instance In cloud computing and virtualization, an instance commonly refers to a provisioned virtual machine or virtual server with an allocated combination of compute, memory, network, and storage resources. Polygonal model In computer graphics, a model may be instanced so it can be drawn multiple times with different transforms and parameters, improving performance by reusing shared geometry data. Program instance In a POSIX-oriented operating system, a running process is an instance of a program. It can be instantiated via system calls such as fork() and exec(). Each executing process is an instance of a program it has been instantiated from.

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  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.

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  Crossover (evolutionary algorithm)

  Crossover in evolutionary algorithms and evolutionary computation, also called recombination, is a genetic operator used to combine the genetic information of two parents to generate new offspring. It is one way to stochastically generate new solutions from an existing population, and is analogous to the crossover that happens during sexual reproduction in biology. New solutions can also be generated by cloning an existing solution, which is analogous to asexual reproduction. Newly generated solutions may be mutated before being added to the population. The aim of recombination is to transfer good characteristics from two different parents to one child. Different algorithms in evolutionary computation may use different data structures to store genetic information, and each genetic representation can be recombined with different crossover operators. Typical data structures that can be recombined with crossover are bit arrays, vectors of real numbers, or trees. The list of operators presented below is by no means complete and serves mainly as an exemplary illustration of this dyadic genetic operator type. More operators and more details can be found in the literature. == Crossover for binary arrays == Traditional genetic algorithms store genetic information in a chromosome represented by a bit array. Crossover methods for bit arrays are popular and an illustrative example of genetic recombination. === One-point crossover === A point on both parents' chromosomes is picked randomly, and designated a 'crossover point'. Bits to the right of that point are swapped between the two parent chromosomes. This results in two offspring, each carrying some genetic information from both parents. === Two-point and k-point crossover === In two-point crossover, two crossover points are picked randomly from the parent chromosomes. The bits in between the two points are swapped between the parent organisms. Two-point crossover is equivalent to performing two single-point crossovers with different crossover points. This strategy can be generalized to k-point crossover for any positive integer k, picking k crossover points. === Uniform crossover === In uniform crossover, typically, each bit is chosen from either parent with equal probability. Other mixing ratios are sometimes used, resulting in offspring which inherit more genetic information from one parent than the other. In a uniform crossover, we don’t divide the chromosome into segments, rather we treat each gene separately. In this, we essentially flip a coin for each chromosome to decide whether or not it will be included in the off-spring. == Crossover for integer or real-valued genomes == For the crossover operators presented above and for most other crossover operators for bit strings, it holds that they can also be applied accordingly to integer or real-valued genomes whose genes each consist of an integer or real-valued number. Instead of individual bits, integer or real-valued numbers are then simply copied into the child genome. The offspring lie on the remaining corners of the hyperbody spanned by the two parents P 1 = ( 1.5 , 6 , 8 ) {\displaystyle P_{1}=(1.5,6,8)} and P 2 = ( 7 , 2 , 1 ) {\displaystyle P_{2}=(7,2,1)} , as exemplified in the accompanying image for the three-dimensional case. === Discrete recombination === If the rules of the uniform crossover for bit strings are applied during the generation of the offspring, this is also called discrete recombination. === Intermediate recombination === In this recombination operator, the allele values of the child genome a i {\displaystyle a_{i}} are generated by mixing the alleles of the two parent genomes a i , P 1 {\displaystyle a_{i,P_{1}}} and a i , P 2 {\displaystyle a_{i,P_{2}}} : α i = α i , P 1 ⋅ β i + α i , P 2 ⋅ ( 1 − β i ) w i t h β i ∈ [ − d , 1 + d ] {\displaystyle \alpha _{i}=\alpha _{i,P_{1}}\cdot \beta _{i}+\alpha _{i,P_{2}}\cdot \left(1-\beta _{i}\right)\quad {\mathsf {with}}\quad \beta _{i}\in \left[-d,1+d\right]} randomly equally distributed per gene i {\displaystyle i} The choice of the interval [ − d , 1 + d ] {\displaystyle [-d,1+d]} causes that besides the interior of the hyperbody spanned by the allele values of the parent genes additionally a certain environment for the range of values of the offspring is in question. A value of 0.25 {\displaystyle 0.25} is recommended for d {\displaystyle d} to counteract the tendency to reduce the allele values that otherwise exists at d = 0 {\displaystyle d=0} . The adjacent figure shows for the two-dimensional case the range of possible new alleles of the two exemplary parents P 1 = ( 3 , 6 ) {\displaystyle P_{1}=(3,6)} and P 2 = ( 9 , 2 ) {\displaystyle P_{2}=(9,2)} in intermediate recombination. The offspring of discrete recombination C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are also plotted. Intermediate recombination satisfies the arithmetic calculation of the allele values of the child genome required by virtual alphabet theory. Discrete and intermediate recombination are used as a standard in the evolution strategy. == Crossover for permutations == For combinatorial tasks, permutations are usually used that are specifically designed for genomes that are themselves permutations of a set. The underlying set is usually a subset of N {\displaystyle \mathbb {N} } or N 0 {\displaystyle \mathbb {N} _{0}} . If 1- or n-point or uniform crossover for integer genomes is used for such genomes, a child genome may contain some values twice and others may be missing. This can be remedied by genetic repair, e.g. by replacing the redundant genes in positional fidelity for missing ones from the other child genome. In order to avoid the generation of invalid offspring, special crossover operators for permutations have been developed which fulfill the basic requirements of such operators for permutations, namely that all elements of the initial permutation are also present in the new one and only the order is changed. It can be distinguished between combinatorial tasks, where all sequences are admissible, and those where there are constraints in the form of inadmissible partial sequences. A well-known representative of the first task type is the traveling salesman problem (TSP), where the goal is to visit a set of cities exactly once on the shortest tour. An example of the constrained task type is the scheduling of multiple workflows. Workflows involve sequence constraints on some of the individual work steps. For example, a thread cannot be cut until the corresponding hole has been drilled in a workpiece. Such problems are also called order-based permutations. In the following, two crossover operators are presented as examples, the partially mapped crossover (PMX) motivated by the TSP and the order crossover (OX1) designed for order-based permutations. A second offspring can be produced in each case by exchanging the parent chromosomes. === Partially mapped crossover (PMX) === The PMX operator was designed as a recombination operator for TSP like Problems. The explanation of the procedure is illustrated by an example: === Order crossover (OX1) === The order crossover goes back to Davis in its original form and is presented here in a slightly generalized version with more than two crossover points. It transfers information about the relative order from the second parent to the offspring. First, the number and position of the crossover points are determined randomly. The resulting gene sequences are then processed as described below: Among other things, order crossover is well suited for scheduling multiple workflows, when used in conjunction with 1- and n-point crossover. === Further crossover operators for permutations === Over time, a large number of crossover operators for permutations have been proposed, so the following list is only a small selection. For more information, the reader is referred to the literature. cycle crossover (CX) order-based crossover (OX2) position-based crossover (POS) edge recombination voting recombination (VR) alternating-positions crossover (AP) maximal preservative crossover (MPX) merge crossover (MX) sequential constructive crossover operator (SCX) The usual approach to solving TSP-like problems by genetic or, more generally, evolutionary algorithms, presented earlier, is either to repair illegal descendants or to adjust the operators appropriately so that illegal offspring do not arise in the first place. Alternatively, Riazi suggests the use of a double chromosome representation, which avoids illegal offspring.

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  Bootstrap aggregating

  Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the ensemble averaging approach. == Description of the technique == Given a standard training set D {\displaystyle D} of size n {\displaystyle n} , bagging generates m {\displaystyle m} new training sets D i {\displaystyle D_{i}} , each of size n ′ {\displaystyle n'} , by sampling from D {\displaystyle D} uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D i {\displaystyle D_{i}} . If n ′ = n {\displaystyle n'=n} , then for large n {\displaystyle n} the set D i {\displaystyle D_{i}} is expected to have the fraction (1 - 1/e) (~63.2%) of the unique samples of D {\displaystyle D} , the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as k-nearest neighbors. == Process of the algorithm == === Key Terms === There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. === Creating the bootstrap dataset === The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal. By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. === Creating the out-of-bag dataset === The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: === Application === Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of ensemble learning algorithms like random forest. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. === Creation of Decision Trees === The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those that possess the top feature, and those that do not. The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. === Random Forests === The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. == Improving Random Forests and Bagging == While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: Specify the maximum depth of trees: Instead of allowing the random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting. Prune the dataset: Using an extremely large dataset may create results that are less indicative of the data provided than a smaller set that more accurately represents what is being focused on. Continue pruning the data at each

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  Rapid application development

  Rapid application development (RAD), also called rapid application building (RAB), is both a general term for adaptive software development approaches, and the name for James Martin's method of rapid development. In general, RAD approaches to software development put less emphasis on planning and more emphasis on an adaptive process. Prototypes are often used in addition to or sometimes even instead of design specifications. RAD is especially well suited for (although not limited to) developing software that is driven by user interface requirements. Graphical user interface builders are often called rapid application development tools. Other approaches to rapid development include the adaptive, agile, spiral, and unified models. == History == Rapid application development was a response to plan-driven waterfall processes, developed in the 1970s and 1980s, such as the Structured Systems Analysis and Design Method (SSADM). One of the problems with these methods is that they were based on a traditional engineering model used to design and build things like bridges and buildings. Software is an inherently different kind of artifact. Software can change the process used to solve a problem. As a result, knowledge gained from the development process itself can feed back to the requirements and design of the solution. Plan-driven approaches attempt to define requirements, the solution, and the implementation plan, and have a process that discourages changes. RAD approaches, on the other hand, recognize that software development is a knowledge intensive process and provide flexible processes that help take advantage of knowledge gained during the project to improve or adapt the solution. The first such RAD alternative was developed by Barry Boehm and was known as the spiral model. Boehm and other subsequent RAD approaches emphasized developing prototypes as well as or instead of rigorous design specifications. Prototypes had several advantages over traditional specifications: Risk reduction. A prototype could test some of the most difficult potential parts of the system early on in the life-cycle. This can provide valuable information as to the feasibility of a design and can prevent the team from pursuing solutions that turn out to be too complex or time-consuming to implement. This benefit of finding problems earlier in the life-cycle rather than later was a key benefit of the RAD approach. The earlier a problem can be found the cheaper it is to address. Users are better at using and reacting than at creating specifications. In the waterfall model it was common for a user to sign off on a set of requirements but then when presented with an implemented system to suddenly realize that a given design lacked some critical features or was too complex. In general most users give much more useful feedback when they can experience a prototype of the running system rather than abstractly define what that system should be. Prototypes can be usable and can evolve into the completed product. One approach used in some RAD methods was to build the system as a series of prototypes that evolve from minimal functionality to moderately useful to the final completed system. The advantage of this besides the two advantages above was that the users could get useful business functionality much earlier in the process. Starting with the ideas of Barry Boehm and others, James Martin developed the rapid application development approach during the 1980s at IBM and finally formalized it by publishing a book in 1991, Rapid Application Development. This has resulted in some confusion over the term RAD even among IT professionals. It is important to distinguish between RAD as a general alternative to the waterfall model and RAD as the specific method created by Martin. The Martin method was tailored toward knowledge intensive and UI intensive business systems. These ideas were further developed and improved upon by RAD pioneers like James Kerr and Richard Hunter, who together wrote the seminal book on the subject, Inside RAD, which followed the journey of a RAD project manager as he drove and refined the RAD Methodology in real-time on an actual RAD project. These practitioners, and those like them, helped RAD gain popularity as an alternative to traditional systems project life cycle approaches. The RAD approach also matured during the period of peak interest in business re-engineering. The idea of business process re-engineering was to radically rethink core business processes such as sales and customer support with the new capabilities of Information Technology in mind. RAD was often an essential part of larger business re engineering programs. The rapid prototyping approach of RAD was a key tool to help users and analysts "think out of the box" about innovative ways that technology might radically reinvent a core business process. Much of James Martin's comfort with RAD stemmed from Dupont's Information Engineering division and its leader Scott Schultz and their respective relationships with John Underwood who headed up a bespoke RAD development company that pioneered many successful RAD projects in Australia and Hong Kong. Successful projects that included ANZ Bank, Lendlease, BHP, Coca-Cola Amatil, Alcan, Hong Kong Jockey Club and numerous others. Success that led to both Scott Shultz and James Martin both spending time in Australia with John Underwood to understand the methods and details of why Australia was disproportionately successful in implementing significant mission critical RAD projects. == James Martin approach == The James Martin approach to RAD divides the process into four distinct phases: Requirements planning phase – combines elements of the system planning and systems analysis phases of the systems development life cycle (SDLC). Users, managers, and IT staff members discuss and agree on business needs, project scope, constraints, and system requirements. It ends when the team agrees on the key issues and obtains management authorization to continue. User design phase – during this phase, users interact with systems analysts and develop models and prototypes that represent all system processes, inputs, and outputs. The RAD groups or subgroups typically use a combination of joint application design (JAD) techniques and CASE tools to translate user needs into working models. User design is a continuous interactive process that allows users to understand, modify, and eventually approve a working model of the system that meets their needs. Construction phase – focuses on program and application development task similar to the SDLC. In RAD, however, users continue to participate and can still suggest changes or improvements as actual screens or reports are developed. Its tasks are programming and application development, coding, unit-integration and system testing. Cutover phase – resembles the final tasks in the SDLC implementation phase, including data conversion, testing, changeover to the new system, and user training. Compared with traditional methods, the entire process is compressed. As a result, the new system is built, delivered, and placed in operation much sooner. == Advantages == In modern Information Technology environments, many systems are now built using some degree of Rapid Application Development (not necessarily the James Martin approach). In addition to Martin's method, agile methods and the Rational Unified Process are often used for RAD development. The purported advantages of RAD include: Better quality. By having users interact with evolving prototypes the business functionality from a RAD project can often be much higher than that achieved via a waterfall model. The software can be more usable and has a better chance to focus on business problems that are critical to end users rather than technical problems of interest to developers. However, this excludes other categories of what are usually known as Non-functional requirements (AKA constraints or quality attributes) including security and portability. Risk control. Although much of the literature on RAD focuses on speed and user involvement a critical feature of RAD done correctly is risk mitigation. It's worth remembering that Boehm initially characterized the spiral model as a risk based approach. A RAD approach can focus in early on the key risk factors and adjust to them based on empirical evidence collected in the early part of the process. E.g., the complexity of prototyping some of the most complex parts of the system. More projects completed on time and within budget. By focusing on the development of incremental units the chances for catastrophic failures that have dogged large waterfall projects is reduced. In the Waterfall model it was common to come to a realization after six months or more of analysis and development that required a radical rethinking of the entire system. With RAD this kind of information can be discovered and acted upon earlier in the proces

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  Gaussian adaptation

  Gaussian adaptation (GA), also called normal or natural adaptation (NA) is an evolutionary algorithm designed for the maximization of manufacturing yield due to statistical deviation of component values of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n-dimensional vector x[xT = (x1, x2, ..., xn)] are taken from a multivariate Gaussian distribution, N(m, M), having mean m and moment matrix M. The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass samples are m and M. The outcome of x as a pass sample is determined by a function s(x), 0 < s(x) < q ≤ 1, such that s(x) is the probability that x will be selected as a pass sample. The average probability of finding pass samples (yield) is P ( m ) = ∫ s ( x ) N ( x − m ) d x {\displaystyle P(m)=\int s(x)N(x-m)\,dx} Then the theorem of GA states: For any s(x) and for any value of P < q, there always exist a Gaussian p. d. f. [ probability density function ] that is adapted for maximum dispersion. The necessary conditions for a local optimum are m = m and M proportional to M. The dual problem is also solved: P is maximized while keeping the dispersion constant (Kjellström, 1991). Proofs of the theorem may be found in the papers by Kjellström, 1970, and Kjellström & Taxén, 1981. Since dispersion is defined as the exponential of entropy/disorder/average information it immediately follows that the theorem is valid also for those concepts. Altogether, this means that Gaussian adaptation may carry out a simultaneous maximisation of yield and average information (without any need for the yield or the average information to be defined as criterion functions). The theorem is valid for all regions of acceptability and all Gaussian distributions. It may be used by cyclic repetition of random variation and selection (like the natural evolution). In every cycle a sufficiently large number of Gaussian distributed points are sampled and tested for membership in the region of acceptability. The centre of gravity of the Gaussian, m, is then moved to the centre of gravity of the approved (selected) points, m. Thus, the process converges to a state of equilibrium fulfilling the theorem. A solution is always approximate because the centre of gravity is always determined for a limited number of points. It was used for the first time in 1969 as a pure optimization algorithm making the regions of acceptability smaller and smaller (in analogy to simulated annealing, Kirkpatrick 1983). Since 1970 it has been used for both ordinary optimization and yield maximization. == Natural evolution and Gaussian adaptation == It has also been compared to the natural evolution of populations of living organisms. In this case s(x) is the probability that the individual having an array x of phenotypes will survive by giving offspring to the next generation; a definition of individual fitness given by Hartl 1981. The yield, P, is replaced by the mean fitness determined as a mean over the set of individuals in a large population. Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it may push the centre of gravity of the Gaussian to the centre of gravity of the selected individuals. This may be accomplished by the Hardy–Weinberg law. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of the structure (Kjellström, 1996). In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense Gaussian adaptation may be seen as a genetic algorithm. == How to climb a mountain == Mean fitness may be calculated provided that the distribution of parameters and the structure of the landscape is known. The real landscape is not known, but figure below shows a fictitious profile (blue) of a landscape along a line (x) in a room spanned by such parameters. The red curve is the mean based on the red bell curve at the bottom of figure. It is obtained by letting the bell curve slide along the x-axis, calculating the mean at every location. As can be seen, small peaks and pits are smoothed out. Thus, if evolution is started at A with a relatively small variance (the red bell curve), then climbing will take place on the red curve. The process may get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small. If the mutation rate is sufficiently high, the disorder or variance may increase and the parameter(s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue up to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides — dependent on the landscape — the process may become very jerky, and if the ratio between the time spent by the process at a local peak and the time of transition to the next peak is very high, it may as well look like a punctuated equilibrium as suggested by Gould (see Ridley). == Computer simulation of Gaussian adaptation == Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically. The implementation of normal adaptation on a computer is a fairly simple task. The adaptation of m may be done by one sample (individual) at a time, for example m(i + 1) = (1 – a) m(i) + ax where x is a pass sample, and a < 1 a suitable constant so that the inverse of a represents the number of individuals in the population. M may in principle be updated after every step y leading to a feasible point x = m + y according to: M(i + 1) = (1 – 2b) M(i) + 2byyT, where yT is the transpose of y and b << 1 is another suitable constant. In order to guarantee a suitable increase of average information, y should be normally distributed with moment matrix μ2M, where the scalar μ > 1 is used to increase average information (information entropy, disorder, diversity) at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WWT = M. Thus, we have y = Wg, where g is normally distributed with the moment matrix μU, and U is the unit matrix. W and WT may be updated by the formulas W = (1 – b)W + bygT and WT = (1 – b)WT + bgyT because multiplication gives M = (1 – 2b)M + 2byyT, where terms including b2 have been neglected. Thus, M will be indirectly adapted with good approximation. In practice it will suffice to update W only W(i + 1) = (1 – b)W(i) + bygT. This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999). The figure below illustrates the effect of increased average information in a Gaussian p.d.f. used to climb a mountain Crest (the two lines represent the contour line). Both the red and green cluster have equal mean fitness, about 65%, but the green cluster has a much higher average information making the green process much more efficient. The effect of this adaptation is not very salient in a 2-dimensional case, but in a high-dimensional case, the efficiency of the search process may be increased by many orders of magnitude. == The evolution in the brain == In the brain the evolution of DNA-messages is supposed to be replaced by an evolution of signal patterns and the phenotypic landscape is replaced by a mental landscape, the complexity of which will hardly be second to the former. The metaphor with the mental landscape is based on the assumption that certain signal patterns give rise to a better well-being or performance. For instance, the control of a group of muscles leads to a better pronunciation of a word or performance of a piece of music. In this simple model it is assumed that the brain consists of interconnected components that may add, multiply and delay signal values. A nerve cell kernel may add signal values, a synapse may multiply with a constant and An axon may delay values. This is a basis of the theory of digital filters and neural networks consisting of components that may add, multiply and delay signalvalues and also of many brain models, Levine 1991. In the figure below the brain stem is supposed to deliver Gaussian distributed signal patterns. This may be possible since certai

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  UIMA

  UIMA ( yoo-EE-mə), short for Unstructured Information Management Architecture, is an OASIS standard for content analytics, originally developed at IBM. It provides a component software architecture for the development, discovery, composition, and deployment of multi-modal analytics for the analysis of unstructured information and integration with search technologies. == Structure == The UIMA architecture can be thought of in four dimensions: It specifies component interfaces in an analytics pipeline. It describes a set of design patterns. It suggests two data representations: an in-memory representation of annotations for high-performance analytics and an XML representation of annotations for integration with remote web services. It suggests development roles allowing tools to be used by users with diverse skills. == Implementations and uses == Apache UIMA, a reference implementation of UIMA, is maintained by the Apache Software Foundation. UIMA is used in a number of software projects: IBM Research's Watson uses UIMA for analyzing unstructured data. The Clinical Text Analysis and Knowledge Extraction System (Apache cTAKES) is a UIMA-based system for information extraction from medical records. DKPro Core is a collection of reusable UIMA components for general-purpose natural language processing.

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  Absorbing Markov chain

  In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case. == Formal definition == A Markov chain is an absorbing chain if there is at least one absorbing state and it is possible to go from any state to at least one absorbing state in a finite number of steps. In an absorbing Markov chain, a state that is not absorbing is called transient. === Canonical form === Let an absorbing Markov chain with transition matrix P have t transient states and r absorbing states. The rows of P represent sources, while columns represent destinations. By ordering the transient states before the absorbing states, it can be assumed that P has the form P = [ Q R 0 I r ] , {\displaystyle P={\begin{bmatrix}Q&R\\\mathbf {0} &I_{r}\end{bmatrix}},} where Q is a t-by-t matrix, R is a nonzero t-by-r matrix, 0 is an r-by-t zero matrix, and Ir is the r-by-r identity matrix. Thus, Q describes the probability of transitioning from some transient state to another while R describes the probability of transitioning from some transient state to some absorbing state. The probability of transitioning from i to j in exactly k steps is the (i,j)-entry of Pk, further computed below. When considering only transient states, the probability is found in the upper left of Pk, the (i,j)-entry of Qk. == Fundamental matrix == === Expected number of visits to a transient state === A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Qk for all k (from 0 to ∞). It can be proven that N := ∑ k = 0 ∞ Q k = ( I t − Q ) − 1 , {\displaystyle N:=\sum _{k=0}^{\infty }Q^{k}=(I_{t}-Q)^{-1},} where It is the t-by-t identity matrix. The computation of this formula is the matrix equivalent of the geometric series of scalars, ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle {\textstyle \sum }_{k=0}^{\infty }q^{k}={\tfrac {1}{1-q}}} . With the matrix N in hand, also other properties of the Markov chain are easy to obtain. === Expected number of steps before being absorbed === The expected number of steps before being absorbed in any absorbing state, when starting in transient state i can be computed via a sum over transient states. The value is given by the ith entry of the vector t := N 1 , {\displaystyle \mathbf {t} :=N\mathbf {1} ,} where 1 is a length-t column vector whose entries are all 1. === Absorbing probabilities === By induction, P k = [ Q k ( I t − Q k ) N R 0 I r ] . {\displaystyle P^{k}={\begin{bmatrix}Q^{k}&(I_{t}-Q^{k})NR\\\mathbf {0} &I_{r}\end{bmatrix}}.} The probability of eventually being absorbed in the absorbing state j when starting from transient state i is given by the (i,j)-entry of the matrix B := N R {\displaystyle B:=NR} . The number of columns of this matrix equals the number of absorbing states r. An approximation of those probabilities can also be obtained directly from the (i,j)-entry of P k {\displaystyle P^{k}} for a large enough value of k, when i is the index of a transient, and j the index of an absorbing state. This is because ( lim k → ∞ P k ) i , t + j = B i , j {\displaystyle \left(\lim _{k\to \infty }P^{k}\right)_{i,t+j}=B_{i,j}} . === Transient visiting probabilities === The probability of visiting transient state j when starting at a transient state i is the (i,j)-entry of the matrix H := ( N − I t ) ( N dg ) − 1 , {\displaystyle H:=(N-I_{t})(N_{\operatorname {dg} })^{-1},} where Ndg is the diagonal matrix with the same diagonal as N. === Variance on number of transient visits === The variance on the number of visits to a transient state j with starting at a transient state i (before being absorbed) is the (i,j)-entry of the matrix N 2 := N ( 2 N dg − I t ) − N sq , {\displaystyle N_{2}:=N(2N_{\operatorname {dg} }-I_{t})-N_{\operatorname {sq} },} where Nsq is the Hadamard product of N with itself (i.e. each entry of N is squared). === Variance on number of steps === The variance on the number of steps before being absorbed when starting in transient state i is the ith entry of the vector ( 2 N − I t ) t − t sq , {\displaystyle (2N-I_{t})\mathbf {t} -\mathbf {t} _{\operatorname {sq} },} where tsq is the Hadamard product of t with itself (i.e., as with Nsq, each entry of t is squared). == Examples == === String generation === Consider the process of repeatedly flipping a fair coin until the sequence (heads, tails, heads) appears. This process is modeled by an absorbing Markov chain with transition matrix P = [ 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 0 1 / 2 0 0 1 / 2 0 0 0 1 ] . {\displaystyle P={\begin{bmatrix}1/2&1/2&0&0\\0&1/2&1/2&0\\1/2&0&0&1/2\\0&0&0&1\end{bmatrix}}.} The first state represents the empty string, the second state the string "H", the third state the string "HT", and the fourth state the string "HTH". Although in reality, the coin flips cease after the string "HTH" is generated, the perspective of the absorbing Markov chain is that the process has transitioned into the absorbing state representing the string "HTH" and, therefore, cannot leave. For this absorbing Markov chain, the fundamental matrix is N = ( I − Q ) − 1 = ( [ 1 0 0 0 1 0 0 0 1 ] − [ 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 0 ] ) − 1 = [ 1 / 2 − 1 / 2 0 0 1 / 2 − 1 / 2 − 1 / 2 0 1 ] − 1 = [ 4 4 2 2 4 2 2 2 2 ] . {\displaystyle {\begin{aligned}N&=(I-Q)^{-1}=\left({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}-{\begin{bmatrix}1/2&1/2&0\\0&1/2&1/2\\1/2&0&0\end{bmatrix}}\right)^{-1}\\[4pt]&={\begin{bmatrix}1/2&-1/2&0\\0&1/2&-1/2\\-1/2&0&1\end{bmatrix}}^{-1}={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}.\end{aligned}}} The expected number of steps starting from each of the transient states is t = N 1 = [ 4 4 2 2 4 2 2 2 2 ] [ 1 1 1 ] = [ 10 8 6 ] . {\displaystyle \mathbf {t} =N\mathbf {1} ={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}10\\8\\6\end{bmatrix}}.} Therefore, the expected number of coin flips before observing the sequence (heads, tails, heads) is 10, the entry for the state representing the empty string. === Games of chance === Games based entirely on chance can be modeled by an absorbing Markov chain. A classic example of this is the ancient Indian board game Snakes and Ladders. The graph on the left plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To determine the expected number of turns to complete the game, compute the vector t as described above and examine tstart, which is approximately 39.2. === Infectious disease testing === Infectious disease testing, either of blood products or in medical clinics, is often taught as an example of an absorbing Markov chain. The public U.S. Centers for Disease Control and Prevention (CDC) model for HIV and for hepatitis B, for example, illustrates the property that absorbing Markov chains can lead to the detection of disease, versus the loss of detection through other means. In the standard CDC model, the Markov chain has five states, a state in which the individual is uninfected, then a state with infected but undetectable virus, a state with detectable virus, and absorbing states of having quit/been lost from the clinic, or of having been detected (the goal). The typical rates of transition between the Markov states are the probability p per unit time of being infected with the virus, w for the rate of window period removal (time until virus is detectable), q for quit/loss rate from the system, and d for detection, assuming a typical rate λ {\displaystyle \lambda } at which the health system administers tests of the blood product or patients in question. It follows that we can "walk along" the Markov model to identify the overall probability of detection for a person starting as undetected, by multiplying the probabilities of transition to each next state of the model as: p ( p + q ) w ( w + q ) d ( d + q ) {\displaystyle {\frac {p}{(p+q)}}{\frac {w}{(w+q)}}{\frac {d}{(d+q)}}} . The subsequent total absolute number of false negative tests—the primary CDC concern—would then be the rate of tests, multiplied by the probability of reaching the infected but undetectable state, times the duration of staying in the infected undetectable state: p ( p + q ) 1 ( w + q ) λ {\displaystyle {\frac {p}{(p+q)}}{\frac {1}{(w+q)}}\lambda } .

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  YaDICs

  YaDICs is a program written to perform digital image correlation on 2D and 3D tomographic images. The program was designed to be both modular, by its plugin strategy and efficient, by it multithreading strategy. It incorporates different transformations (Global, Elastic, Local), optimizing strategy (Gauss-Newton, Steepest descent), Global and/or local shape functions (Rigid-body motions, homogeneous dilatations, flexural and Brazilian test models)... == Theoretical background == === Context === In solid mechanics, digital image correlation is a tool that allows to identify the displacement field to register a reference image (called herein fixed image) to images during an experiment (mobile image). For example, it is possible to observe the face of a specimen with a painted speckle on it in order to determine its displacement fields during a tensile test. Before the appearance of such methods, researchers usually used strain gauges to measure the mechanical state of the material but strain gauges only measure the strain on a point and don't allow to understand material with an heterogeneous behavior. One can obtain a full in plane strain tensor by derivation of the displacement fields. Many methods are based upon the optical flow. In fluid mechanics a similar method is used, called Particle Image Velocimetry (PIV); the algorithms are similar to those of DIC but it is impossible to ensure that the optical flow is conserved so a vast majority of the software used the normalized cross correlation metric. In mechanics the displacement or velocity fields are the only concern, registering images is just a side effect. There is another process called image registration using the same algorithms (on monomodal images) but where the goal is to register images and thereby identifying the displacement field is just a side effect. YaDICs uses the general principle of image registration with a particular attention to the displacement fields basis. === Image registration principle === YaDICs can be explained using the classical image registration framework: === Image registration general scheme === The common idea of image registration and digital image correlation is to find the transformation between a fixed image and a moving one for a given metric using an optimization scheme. While there are many methods to achieve such a goal, Yadics focuses on registering images with the same modality. The idea behind the creation of this software is to be able to process data that comes from a μ-tomograph; i.e.: data cube over 10003 voxels. With such a size it is not possible to use naive approach usually used in a two-dimensional context. In order to get sufficient performances OpenMP parallelism is used and data are not globally stored in memory. As an extensive description of the different algorithms is given in. === Sampling === Contrary to image registration, Digital Image Correlation targets the transformation, one wants to extracted the most accurate transformation from the two images and not just match the images. Yadics uses the whole image as a sampling grid: it is thus a total sampling. === Interpolator === It is possible to choose between bilinear interpolation and bicubic interpolation for the grey level evaluation at non integer coordinates. The bi-cubic interpolation is the recommended one. === Metrics === ==== Sum of squared differences (SSD) ==== The SSD is also known as mean squared error. The equation below defines the SSD metric: S S D ( μ , I F , I M ) = 1 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) 2 , {\displaystyle SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {1}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)^{2},} where I F {\displaystyle {\mathcal {I_{F}}}} is the fixed image, I M {\displaystyle {\mathcal {I_{M}}}} the moving one, Ω F {\displaystyle \Omega _{F}} the integration area | Ω F | {\displaystyle \left|\Omega _{F}\right|} the number of pi(vo)xels (cardinal) and T μ {\displaystyle {T}_{\mu }} the transformation parametrized by μ The transformation can be written as: T μ ( x ) = x + { Φ ( x ) } t { μ } . {\displaystyle T_{\mu }(x)=x+\left\{\Phi (x)\right\}^{t}\left\{\mu \right\}.} This metric is the main one used in the YaDICs as it works well with same modality images. One has to find the minimum of this metric ==== Normalized cross-correlation ==== The normalized cross-correlation (NCC) is used when one cannot assure the optical flow conservation; it happens in case of change of lighting or if particles disappear from the scene can occur in particle images velocimetry (PIV). The NCC is defined by: N C C ( μ , I F , I M ) = ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) ( I M ( T μ ( x i ) ) − I M ¯ ) ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) 2 ∑ x i ∈ Ω F ( I M ( T μ ( x i ) ) − I M ¯ ) 2 , {\displaystyle NCC(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)}{\sqrt {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)^{2}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)^{2}}}},} where I F ¯ {\displaystyle {\overline {\mathcal {I_{F}}}}} and I M ¯ {\displaystyle {\overline {\mathcal {I_{M}}}}} are the mean values of the fixed and mobile images. This metric is only used to find local translation in Yadics. This metric with translation transform can be solved using cross-correlation methods, which are non iterative and can be accelerated using Fast Fourier Transform . === Classification of transformations === There are three categories of parametrization: elastic, global and local transformation. The elastic transformations respect the partition of unity, there are no holes created or surfaces counted several times. This is commonly used in Image Registration by the use of B-Spline functions and in solid mechanics with finite element basis. The global transformations are defined on the whole picture using rigid body or affine transformation (which is equivalent to homogeneous strain transformation). More complex transformations can be defined such as mechanically based one. These transformations have been used for stress intensity factor identification by and for rod strain by. The local transformation can be considered as the same global transformation defined on several Zone Of Interest (ZOI) of the fixed image. ==== Global ==== Several global transforms have been implemented: Rigid and homogeneous (Tx,Ty,Rz in 2D; Tx,Ty,Tz,Rx,Ry,Rz,Exx,Eyy,Ezz,Eyz,Exz,Exy in 3D) Brazilian (Only in 2D), Dynamic Flexion, ==== Elastic ==== First-order quadrangular finite elements Q4P1 are used in Yadics. ===== Local ===== Every global transform can be used on a local mesh. === Optimization === The YaDICs optimization process follows a gradient descent scheme. The first step is to compute the gradient of the metric regarding the transform parameters ∂ S S D ( μ , I F , I M ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ∂ I M ( T μ ( x i ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ( ∂ T μ ( x i ) ∂ μ ) t ∂ I M ( T μ ( x i ) ) ∂ x {\displaystyle {\begin{array}{lcl}{\dfrac {\partial SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right){\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i})}{\partial \mu }}\\&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)\left({\dfrac {\partial {T}_{\mu }(x_{i})}{\partial \mu }}\right)^{t}{\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i}))}{\partial x}}\\\end{array}}} ==== Gradient method ==== Once the metric gradient has been computed, one has to find an optimization strategy The gradient method principle is explained below: μ k + 1 = μ k + α k d k {\displaystyle \mu _{k+1}=\mu _{k}+\alpha _{k}d_{k}} The gradient step can be constant or updated at every iteration. d k = − γ k ∂ C ( μ , I F , I M ) ∂ μ {\displaystyle d_{k}=-\gamma _{k}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}} , γ k {\displaystyle \gamma _{k}} allows one to choose between the following methods : γ k {\displaystyle \gamma _{k}} ⟹ {\displaystyle \Longrightarrow } steepest descent, γ k = [ ∂ C ( μ , I F , I M ) ∂ μ ∂ C ( μ , I F , I M ) ∂ μ t ] − 1 {\displaystyle \gamma _{k}=\left[{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}^{t}\right]^{-1}} ⟹ {\displaystyle \Longrightarrow } Gauss-Newto

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  AdaBoost

  AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

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  Differential evolution

  Differential evolution (DE) is an evolutionary algorithm to optimize a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as they make few or no assumptions about the optimized problem and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found. DE is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means DE does not require the optimization problem to be differentiable, as is required by classic optimization methods such as gradient descent and quasi-newton methods. DE can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple formulae, and then keeping whichever candidate solution has the best score or fitness on the optimization problem at hand. In this way, the optimization problem is treated as a black box that merely provides a measure of quality given a candidate solution and the gradient is therefore not needed. == History == Storn and Price introduced Differential Evolution in 1995. Books have been published on theoretical and practical aspects of using DE in parallel computing, multiobjective optimization, constrained optimization, and the books also contain surveys of application areas. Surveys on the multi-faceted research aspects of DE can be found in journal articles. == Algorithm == A basic variant of the DE algorithm works by having a population of candidate solutions (called agents). These agents are moved around in the search-space by using simple mathematical formulae to combine the positions of existing agents from the population. If the new position of an agent is an improvement then it is accepted and forms part of the population, otherwise the new position is simply discarded. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered. Formally, let f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be the fitness function which must be minimized (note that maximization can be performed by considering the function h := − f {\displaystyle h:=-f} instead). The function takes a candidate solution as argument in the form of a vector of real numbers. It produces a real number as output which indicates the fitness of the given candidate solution. The gradient of f {\displaystyle f} is not known. The goal is to find a solution m {\displaystyle \mathbf {m} } for which f ( m ) ≤ f ( p ) {\displaystyle f(\mathbf {m} )\leq f(\mathbf {p} )} for all p {\displaystyle \mathbf {p} } in the search-space, which means that m {\displaystyle \mathbf {m} } is the global minimum. Let x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} designate a candidate solution (agent) in the population. The basic DE algorithm can then be described as follows: Choose the parameters NP ≥ 4 {\displaystyle {\text{NP}}\geq 4} , CR ∈ [ 0 , 1 ] {\displaystyle {\text{CR}}\in [0,1]} , and F ∈ [ 0 , 2 ] {\displaystyle F\in [0,2]} . NP : NP {\displaystyle {\text{NP}}} is the population size, i.e. the number of candidate agents or "parents". CR : The parameter CR ∈ [ 0 , 1 ] {\displaystyle {\text{CR}}\in [0,1]} is called the crossover probability. F : The parameter F ∈ [ 0 , 2 ] {\displaystyle F\in [0,2]} is called the differential weight. Typical settings are N P = 10 n {\displaystyle NP=10n} , C R = 0.9 {\displaystyle CR=0.9} and F = 0.8 {\displaystyle F=0.8} . Optimization performance may be greatly impacted by these choices; see below. Initialize all agents x {\displaystyle \mathbf {x} } with random positions in the search-space. Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following: For each agent x {\displaystyle \mathbf {x} } in the population do: Pick three agents a , b {\displaystyle \mathbf {a} ,\mathbf {b} } , and c {\displaystyle \mathbf {c} } from the population at random, they must be distinct from each other as well as from agent x {\displaystyle \mathbf {x} } . ( a {\displaystyle \mathbf {a} } is called the "base" vector.) Pick a random index R ∈ { 1 , … , n } {\displaystyle R\in \{1,\ldots ,n\}} where n {\displaystyle n} is the dimensionality of the problem being optimized. Compute the agent's potentially new position y = [ y 1 , … , y n ] {\displaystyle \mathbf {y} =[y_{1},\ldots ,y_{n}]} as follows: For each i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , pick a uniformly distributed random number r i ∼ U ( 0 , 1 ) {\displaystyle r_{i}\sim U(0,1)} If r i < C R {\displaystyle r_{i} Read more →