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  Character computing

  Character computing is a trans-disciplinary field of research at the intersection of computer science and psychology. It is any computing that incorporates the human character within its context. Character is defined as all features or characteristics defining an individual and guiding their behavior in a specific situation. It consists of stable trait markers (e.g., personality, background, history, socio-economic embeddings, culture,...) and variable state markers (emotions, health, cognitive state, ...). Character computing aims at providing a holistic psychologically driven model of human behavior. It models and predicts behavior based on the relationships between a situation and character. Three main research modules fall under the umbrella of character computing: character sensing and profiling, character-aware adaptive systems, and artificial characters. == Overview == Character computing can be viewed as an extension of the well-established field of affective computing. Based on the foundations of the different psychology branches, it advocates defining behavior as a compound attribute that is not driven by either personality, emotions, situation or cognition alone. It rather defines behavior as a function of everything that makes up an individual i.e., their character and the situation they are in. Affective computing aims at allowing machines to understand and translate the non-verbal cues of individuals into affect. Accordingly, character computing aims at understanding the character attributes of an individual and the situation to translate it to predicted behavior, and vice versa. ''In practical terms, depending on the application context, character computing is a branch of research that deals with the design of systems and interfaces that can observe, sense, predict, adapt to, affect, understand, or simulate the following: character based on behavior and situation, behavior based on character and situation, or situation based on character and behavior.'' The Character-Behavior-Situation (CBS) triad is at the core of character computing and defines each of the three edges based on the other two. Character computing relies on simultaneous development from a computational and psychological perspective and is intended to be used by researchers in both fields. Its main concept is aligning the computational model of character computing with empirical results from in-lab and in-the-wild psychology experiments. The model is to be continuously built and validated through the emergence of new data. Similar to affective and personality computing, the model is to be used as a base for different applications towards improving user experience. == History == Character computing as such was first coined in its first workshop in 2017. Since then it has had 3 international workshops and numerous publications. Despite its young age, it has already drawn some interest in the research community, leading to the publication of the first book under the same title in early 2020 published by Springer Nature. Research that can be categorized under the field dates much older than 2017. The notion of combining several factors towards the explanation of behavior or traits and states has long been investigated in both Psychology and Computer Science, for example. == Character == The word character originates from the Greek word meaning “stamping tool”, referring to distinctive features and traits. Over the years it has been given many different connotations, like the moral character in philosophy, the temperament in psychology, a person in literature or an avatar in various virtual worlds, including video games. According to character computing character is a unification of all the previous definitions, by referring back to the original meaning of the word. Character is defined as the holistic concept representing all interacting trait and state markers that distinguish an individual. Traits are characteristics that mainly remain stable over time. Traits include personality, affect, socio-demographics, and general health. States are characteristics that vary in short periods of time. They include emotions, well-being, health, cognitive state. Each characteristic has many representation methods and psychological models. The different models can be combined or one model can be preset for each characteristic. This depends on the use-case and the design choices. == Areas == Research into character computing can be divided into three areas, which complement each other but can each be investigated separately. The first area is sensing and predicting character states and traits or ensuing behavior. The second area is adapting applications to certain character states or traits and the behavior they predict. It also deals with trying to change or monitor such behavior. The final area deals with creating artificial agents e.g., chatbots or virtual reality avatars that exhibit certain characteristics. The three areas are investigated separately and build on existing findings in the literature. The results of each of the three areas can also be used as a stepping stone for the next area. Each of the three areas has already been investigated on its own in different research fields with focus on different subsets of character. For example, affective computing and personality computing both cover different areas with a focus on some character components without the others to account for human behavior. == The Character-Behavior-Situation triad == Character computing is based on a holistic psychologically driven model of human behavior. Human behavior is modeled and predicted based on the relationships between a situation and a human's character. To further define character in a more formal or holistic manner, we represent it in light of the Character–Behavior–Situation triad. This highlights that character not only determines who we are but how we are, i.e., how we behave. The triad investigated in Personality Psychology is extended through character computing to the Character–Behavior–Situation triad. Any member of the CBS triad is a function of the two other members, e.g., given the situation and personality, the behavior can be predicted. Each of the components in the triad can be further decomposed into smaller units and features that may best represent the human's behavior or character in a particular situation. Character is thus behind a person's behavior in any given situation. While this is a causality relation, the correlation between the three components is often more easily used to predict the components that are most difficult to measure from those measured more easily. There are infinitely many components to include in the representation of any of C, B, and S. The challenge is always to choose the smallest subset needed for prediction of a person's behavior in a particular situation.

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  Randomized weighted majority algorithm

  The randomized weighted majority algorithm is an algorithm in machine learning theory for aggregating expert predictions to a series of decision problems. It is a simple and effective method based on weighted voting which improves on the mistake bound of the deterministic weighted majority algorithm. In fact, in the limit, its prediction rate can be arbitrarily close to that of the best-predicting expert. == Example == Imagine that every morning before the stock market opens, we get a prediction from each of our "experts" about whether the stock market will go up or down. Our goal is to somehow combine this set of predictions into a single prediction that we then use to make a buy or sell decision for the day. The principal challenge is that we do not know which experts will give better or worse predictions. The RWMA gives us a way to do this combination such that our prediction record will be nearly as good as that of the single expert which, in hindsight, gave the most accurate predictions. == Motivation == In machine learning, the weighted majority algorithm (WMA) is a deterministic meta-learning algorithm for aggregating expert predictions. In pseudocode, the WMA is as follows: initialize all experts to weight 1 for each round: add each expert's weight to the option they predicted predict the option with the largest weighted sum multiply the weights of all experts who predicted wrongly by 1 2 {\displaystyle {\frac {1}{2}}} Suppose there are n {\displaystyle n} experts and the best expert makes m {\displaystyle m} mistakes. Then, the weighted majority algorithm (WMA) makes at most 2.4 ( log 2 ⁡ n + m ) {\displaystyle 2.4(\log _{2}n+m)} mistakes. This bound is highly problematic in the case of highly error-prone experts. Suppose, for example, the best expert makes a mistake 20% of the time; that is, in N = 100 {\displaystyle N=100} rounds using n = 10 {\displaystyle n=10} experts, the best expert makes m = 20 {\displaystyle m=20} mistakes. Then, the weighted majority algorithm only guarantees an upper bound of 2.4 ( log 2 ⁡ 10 + 20 ) ≈ 56 {\displaystyle 2.4(\log _{2}10+20)\approx 56} mistakes. As this is a known limitation of the weighted majority algorithm, various strategies have been explored in order to improve the dependence on m {\displaystyle m} . In particular, we can do better by introducing randomization. Drawing inspiration from the Multiplicative Weights Update Method algorithm, we will probabilistically make predictions based on how the experts have performed in the past. Similarly to the WMA, every time an expert makes a wrong prediction, we will decrement their weight. Mirroring the MWUM, we will then use the weights to make a probability distribution over the actions and draw our action from this distribution (instead of deterministically picking the majority vote as the WMA does). == Randomized weighted majority algorithm (RWMA) == The randomized weighted majority algorithm is an attempt to improve the dependence of the mistake bound of the WMA on m {\displaystyle m} . Instead of predicting based on majority vote, the weights, are used as probabilities for choosing the experts in each round and are updated over time (hence the name randomized weighted majority). Precisely, if w i {\displaystyle w_{i}} is the weight of expert i {\displaystyle i} , let W = ∑ i w i {\displaystyle W=\sum _{i}w_{i}} . We will follow expert i {\displaystyle i} with probability w i W {\displaystyle {\frac {w_{i}}{W}}} . This results in the following algorithm: initialize all experts to weight 1. for each round: add all experts' weights together to obtain the total weight W {\displaystyle W} choose expert i {\displaystyle i} randomly with probability w i W {\displaystyle {\frac {w_{i}}{W}}} predict as the chosen expert predicts multiply the weights of all experts who predicted wrongly by β {\displaystyle \beta } The goal is to bound the worst-case expected number of mistakes, assuming that the adversary has to select one of the answers as correct before we make our coin toss. This is a reasonable assumption in, for instance, the stock market example provided above: the variance of a stock price should not depend on the opinions of experts that influence private buy or sell decisions, so we can treat the price change as if it was decided before the experts gave their recommendations for the day. The randomized algorithm is better in the worst case than the deterministic algorithm (weighted majority algorithm): in the latter, the worst case was when the weights were split 50/50. But in the randomized version, since the weights are used as probabilities, there would still be a 50/50 chance of getting it right. In addition, generalizing to multiplying the weights of the incorrect experts by β < 1 {\displaystyle \beta <1} instead of strictly 1 2 {\displaystyle {\frac {1}{2}}} allows us to trade off between dependence on m {\displaystyle m} and log 2 ⁡ n {\displaystyle \log _{2}n} . This trade-off will be quantified in the analysis section. == Analysis == Let W t {\displaystyle W_{t}} denote the total weight of all experts at round t {\displaystyle t} . Also let F t {\displaystyle F_{t}} denote the fraction of weight placed on experts which predict the wrong answer at round t {\displaystyle t} . Finally, let N {\displaystyle N} be the total number of rounds in the process. By definition, F t {\displaystyle F_{t}} is the probability that the algorithm makes a mistake on round t {\displaystyle t} . It follows from the linearity of expectation that if M {\displaystyle M} denotes the total number of mistakes made during the entire process, E [ M ] = ∑ t = 1 N F t {\displaystyle E[M]=\sum _{t=1}^{N}F_{t}} . After round t {\displaystyle t} , the total weight is decreased by ( 1 − β ) F t W t {\displaystyle \ (1-\beta )F_{t}W_{t}} , since all weights corresponding to a wrong answer are multiplied by β < 1 {\displaystyle \ \beta <1} . It then follows that W t + 1 = W t ( 1 − ( 1 − β ) F t ) {\displaystyle W_{t+1}=W_{t}(1-(1-\beta )F_{t})} . By telescoping, since W 1 = n {\displaystyle W_{1}=n} , it follows that the total weight after the process concludes is On the other hand, suppose that m {\displaystyle \ m} is the number of mistakes made by the best-performing expert. At the end, this expert has weight β m {\displaystyle \ \beta ^{m}} . It follows, then, that the total weight is at least this much; in other words, W ≥ β m {\displaystyle \ W\geq \beta ^{m}} . This inequality and the above result imply Taking the natural logarithm of both sides yields Now, the Taylor series of the natural logarithm is In particular, it follows that ln ⁡ ( 1 − ( 1 − β ) F t ) < − ( 1 − β ) F t {\displaystyle \ \ln(1-(1-\beta )F_{t})<-(1-\beta )F_{t}} . Thus, Recalling that E [ M ] = ∑ t = 1 N F t {\displaystyle E[M]=\sum _{t=1}^{N}F_{t}} and rearranging, it follows that Now, as β → 1 {\displaystyle \beta \to 1} from below, the first constant tends to 1 {\displaystyle 1} ; however, the second constant tends to + ∞ {\displaystyle +\infty } . To quantify this tradeoff, define ε = 1 − β {\displaystyle \varepsilon =1-\beta } to be the penalty associated with getting a prediction wrong. Then, again applying the Taylor series of the natural logarithm, It then follows that the mistake bound, for small ε {\displaystyle \varepsilon } , can be written in the form ( 1 + ϵ 2 + O ( ε 2 ) ) m + ϵ − 1 ln ⁡ ( n ) {\displaystyle \ \left(1+{\frac {\epsilon }{2}}+O(\varepsilon ^{2})\right)m+\epsilon ^{-1}\ln(n)} . In English, the less that we penalize experts for their mistakes, the more that additional experts will lead to initial mistakes but the closer we get to capturing the predictive accuracy of the best expert as time goes on. In particular, given a sufficiently low value of ε {\displaystyle \varepsilon } and enough rounds, the randomized weighted majority algorithm can get arbitrarily close to the correct prediction rate of the best expert. In particular, as long as m {\displaystyle m} is sufficiently large compared to ln ⁡ ( n ) {\displaystyle \ln(n)} (so that their ratio is sufficiently small), we can assign we can obtain an upper bound on the number of mistakes equal to This implies that the "regret bound" on the algorithm (that is, how much worse it performs than the best expert) is sublinear, at O ( m ln ⁡ ( n ) ) {\displaystyle O({\sqrt {m\ln(n)}})} . == Revisiting the motivation == Recall that the motivation for the randomized weighted majority algorithm was given by an example where the best expert makes a mistake 20% of the time. Precisely, in N = 100 {\displaystyle N=100} rounds, with n = 10 {\displaystyle n=10} experts, where the best expert makes m = 20 {\displaystyle m=20} mistakes, the deterministic weighted majority algorithm only guarantees an upper bound of 2.4 ( log 2 ⁡ 10 + 20 ) ≈ 56 {\displaystyle 2.4(\log _{2}10+20)\approx 56} . By the analysis above, it follows that minimizing the number of worst-case expected mistakes is equivalent to minimizing the fun

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  Jpred

  Jpred v.4 is the latest version of the JPred Protein Secondary Structure Prediction Server which provides predictions by the JNet algorithm, one of the most accurate methods for secondary structure prediction, that has existed since 1998 in different versions. In addition to protein secondary structure, JPred also makes predictions of solvent accessibility and coiled-coil regions. The JPred service runs up to 134 000 jobs per month and has carried out over 2 million predictions in total for users in 179 countries. == JPred 2 == The static HTML pages of JPred 2 are still available for reference. == JPred 3 == The JPred v3 followed on from previous versions of JPred developed and maintained by James Cuff and Jonathan Barber (see JPred References). This release added new functionality and fixed many bugs. The highlights are: New, friendlier user interface Retrained and optimised version of Jnet (v2) - mean secondary structure prediction accuracy of >81% Batch submission of jobs Better error checking of input sequences/alignments Predictions now (optionally) returned via e-mail Users may provide their own query names for each submission JPred now makes a prediction even when there are no PSI-BLAST hits to the query PS/PDF output now incorporates all the predictions == JPred 4 == The current version of JPred (v4) has the following improvements and updates incorporated: Retrained on the latest UniRef90 and SCOPe/ASTRAL version of Jnet (v2.3.1) - mean secondary structure prediction accuracy of >82%. Upgraded the Web Server to the latest technologies (Bootstrap framework, JavaScript) and updating the web pages – improving the design and usability through implementing responsive technologies. Added RESTful API and mass-submission and results retrieval scripts - resulting in peak throughput above 20,000 predictions per day. Added prediction jobs monitoring tools. Upgraded the results reporting – both, on the web-site, and through the optional email summary reports: improved batch submission, added results summary preview through Jalview results visualization summary in SVG and adding full multiple sequence alignments into the reports. Improved help-pages, incorporating tool-tips, and adding one-page step-by-step tutorials. Sequence residues are categorised or assigned to one of the secondary structure elements, such as alpha-helix, beta-sheet and coiled-coil. Jnet uses two neural networks for its prediction. The first network is fed with a window of 17 residues over each amino acid in the alignment plus a conservation number. It uses a hidden layer of nine nodes and has three output nodes, one for each secondary structure element. The second network is fed with a window of 19 residues (the result of first network) plus the conservation number. It has a hidden layer with nine nodes and has three output nodes.

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  Oja's rule

  Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

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  List of speech recognition software

  Speech recognition software is available for many computing platforms, operating systems, use models, and software licenses. Here is a listing of such, grouped in various useful ways. == Acoustic models and speech corpus (compilation) == The following list presents notable speech recognition software engines with a brief synopsis of characteristics. == Macintosh == == Cross-platform web apps based on Chrome == The following list presents notable speech recognition software that operate in a Chrome browser as web apps. They make use of HTML5 Web-Speech-API. == Mobile devices and smartphones == Many mobile phone handsets, including feature phones and smartphones such as iPhones and BlackBerrys, have basic dial-by-voice features built in. Many third-party apps have implemented natural-language speech recognition support, including: == Windows == === Windows built-in speech recognition === The Windows Speech Recognition version 8.0 by Microsoft comes built into Windows Vista, Windows 7, Windows 8 and Windows 10. Speech Recognition is available only in English, French, Spanish, German, Japanese, Simplified Chinese, and Traditional Chinese and only in the corresponding version of Windows; meaning you cannot use the speech recognition engine in one language if you use a version of Windows in another language. Windows 7 Ultimate and Windows 8 Pro allow you to change the system language, and therefore change which speech engine is available. Windows Speech Recognition evolved into Cortana (software), a personal assistant included in Windows 10. === Windows 7, 8, 10, 11 third-party speech recognition === Braina – Dictate into third party software and websites, fill web forms and execute vocal commands. Dragon NaturallySpeaking from Nuance Communications – Successor to the older DragonDictate product. Focus on dictation. 64-bit Windows support since version 10.1. Tazti – Create speech command profiles to play PC games and control applications – programs. Create speech commands to open files, folders, webpages, applications. Windows 7, Windows 8 and Windows 8.1 versions. Voice Finger – software that improves the Windows speech recognition system by adding several extensions to it. The software enables controlling the mouse and the keyboard by only using the voice. It is especially useful for aiding users to overcome disabilities or to heal from computer injuries. === Microsoft Speech API === The first version of the Microsoft Speech API was released for Windows NT 3.51 and Windows 95 in 1995, it was then part of Windows up to Windows Vista. This initial version already contained Direct Speech Recognition and Direct Text To Speech APIs which applications could use to directly control engines, as well as simplified 'higher-level' Voice Command and Voice Talk APIs. Speech recognition functionality included as part of Microsoft Office and on Tablet PCs running Microsoft Windows XP Tablet PC Edition. It can also be downloaded as part of the Speech SDK 5.1 for Windows applications, but since that is aimed at developers building speech applications, the pure SDK form lacks any user interface (numerous applications were available), and thus is unsuitable for end users. == Built-in software == Microsoft Kinect includes built-in software which allows speech recognition of commands. Older generations of Nokia phones like Nokia N Series (before using Windows 7 mobile technology) used speech-recognition with family names from contact list and a few commands. Siri, originally implemented in the iPhone 4S, Apple's personal assistant for iOS, which uses technology from Nuance Communications. Cortana (software), Microsoft's personal assistant built into Windows Phone and Windows 10. == Interactive voice response == The following are interactive voice response (IVR) systems: CSLU Toolkit Genesys HTK – copyrighted by Microsoft, but allows altering software for licensee's internal use LumenVox ASR Tellme Networks; acquired by Microsoft == Unix-like x86 and x86-64 speech transcription software == Janus Recognition Toolkit (JRTk) Mozilla DeepSpeech was developing an open-source Speech-To-Text engine based on Baidu's deep speech research paper. Weesper Neon Flow – professional voice-dictation software that provides offline speech-to-text processing on macOS and Windows using local AI models. It is not open source and offers a paid subscription after a 15‑day free trial. Vocalinux – open-source speech transcription software for Linux. == Discontinued software == IBM VoiceType (formerly IBM Personal Dictation System) IBM ViaVoice – Embedded version still maintained by IBM. No longer supported for versions above Windows Vista. Untested above macOS 10.4 or on Macintoshes with an Intel chipset. Quack.com; acquired by AOL; the name has now been reused for an iPad search app. SpeechWorks from Nuance Communications. Yap Speech Cloud – Speech-to-text platform acquired by Amazon.com.

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  Extremal Ensemble Learning

  Extremal Ensemble Learning (EEL) is a machine learning algorithmic paradigm for graph partitioning. EEL creates an ensemble of partitions and then uses information contained in the ensemble to find new and improved partitions. The ensemble evolves and learns how to form improved partitions through extremal updating procedure. The final solution is found by achieving consensus among its member partitions about what the optimal partition is. == Reduced-Network Extremal Ensemble Learning (RenEEL) == A particular implementation of the EEL paradigm is the Reduced-Network Extremal Ensemble Learning (RenEEL) scheme for partitioning a graph. RenEEL uses consensus across many partitions in an ensemble to create a reduced network that can be efficiently analyzed to find more accurate partitions. These better quality partitions are subsequently used to update the ensemble. An algorithm that utilizes the RenEEL scheme is currently the best algorithm for finding the graph partition with maximum modularity, which is an NP-hard problem.

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  Huber loss

  In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used. == Definition == The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = { 1 2 a 2 for | a | ≤ δ , δ ⋅ ( | a | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\[4pt]\delta \cdot \left(|a|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} This function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections at the two points where | a | = δ {\displaystyle |a|=\delta } . The variable a often refers to the residuals, that is to the difference between the observed and predicted values a = y − f ( x ) {\displaystyle a=y-f(x)} , so the former can be expanded to L δ ( y , f ( x ) ) = { 1 2 ( y − f ( x ) ) 2 for | y − f ( x ) | ≤ δ , δ ⋅ ( | y − f ( x ) | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(y,f(x))={\begin{cases}{\frac {1}{2}}{\left(y-f(x)\right)}^{2}&{\text{for }}\left|y-f(x)\right|\leq \delta ,\\[4pt]\delta \ \cdot \left(\left|y-f(x)\right|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} The Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it "smoothens out" the former's corner at the origin. == Motivation == Two very commonly used loss functions are the squared loss, L ( a ) = a 2 {\displaystyle L(a)=a^{2}} , and the absolute loss, L ( a ) = | a | {\displaystyle L(a)=|a|} . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case). The squared loss has the disadvantage that it has the tendency to be dominated by outliers—when summing over a set of a {\displaystyle a} 's (as in ∑ i = 1 n L ( a i ) {\textstyle \sum _{i=1}^{n}L(a_{i})} ), the sample mean is influenced too much by a few particularly large a {\displaystyle a} -values when the distribution is heavy tailed: in terms of estimation theory, the asymptotic relative efficiency of the mean is poor for heavy-tailed distributions. As defined above, the Huber loss function is strongly convex in a uniform neighborhood of its minimum a = 0 {\displaystyle a=0} ; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points a = − δ {\displaystyle a=-\delta } and a = δ {\displaystyle a=\delta } . These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using the absolute value function). == Pseudo-Huber loss function == The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss function transitions from L2 loss for values close to the minimum to L1 loss for extreme values and the steepness at extreme values can be controlled by the δ {\displaystyle \delta } value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as L δ ( a ) = δ 2 ( 1 + ( a / δ ) 2 − 1 ) . {\displaystyle L_{\delta }(a)=\delta ^{2}\left({\sqrt {1+(a/\delta )^{2}}}-1\right).} As such, this function approximates a 2 / 2 {\displaystyle a^{2}/2} for small values of a {\displaystyle a} , and approximates a straight line with slope δ {\displaystyle \delta } for large values of a {\displaystyle a} . While the above is the most common form, other smooth approximations of the Huber loss function also exist. == Variant for classification == For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction f ( x ) {\displaystyle f(x)} (a real-valued classifier score) and a true binary class label y ∈ { + 1 , − 1 } {\displaystyle y\in \{+1,-1\}} , the modified Huber loss is defined as L ( y , f ( x ) ) = { max ( 0 , 1 − y f ( x ) ) 2 for y f ( x ) > − 1 , − 4 y f ( x ) otherwise. {\displaystyle L(y,f(x))={\begin{cases}\max(0,1-y\,f(x))^{2}&{\text{for }}\,\,y\,f(x)>-1,\\[4pt]-4y\,f(x)&{\text{otherwise.}}\end{cases}}} The term max ( 0 , 1 − y f ( x ) ) {\displaystyle \max(0,1-y\,f(x))} is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of L {\displaystyle L} . == Applications == The Huber loss function is used in robust statistics, M-estimation and additive modelling.

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  Softplus

  In mathematics and machine learning, the softplus function is f ( x ) = ln ⁡ ( 1 + e x ) . {\displaystyle f(x)=\ln(1+e^{x}).} It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known as the rectifier or ReLU (rectified linear unit) in machine learning. For large negative x {\displaystyle x} it is ln ⁡ ( 1 + e x ) = ln ⁡ ( 1 + ϵ ) ⪆ ln ⁡ 1 = 0 {\displaystyle \ln(1+e^{x})=\ln(1+\epsilon )\gtrapprox \ln 1=0} , so just above 0, while for large positive x {\displaystyle x} it is ln ⁡ ( 1 + e x ) ⪆ ln ⁡ ( e x ) = x {\displaystyle \ln(1+e^{x})\gtrapprox \ln(e^{x})=x} , so just above x {\displaystyle x} . The names softplus and SmoothReLU are used in machine learning. The name "softplus" (2000), by analogy with the earlier softmax (1989) is presumably because it is a smooth (soft) approximation of the positive part of x, which is sometimes denoted with a superscript plus, x + := max ( 0 , x ) {\displaystyle x^{+}:=\max(0,x)} . == Alternative forms == This function can be approximated as: ln ⁡ ( 1 + e x ) ≈ { ln ⁡ 2 , x = 0 , x 1 − e − x / ln ⁡ 2 , x ≠ 0 {\displaystyle \ln \left(1+e^{x}\right)\approx {\begin{cases}\ln 2,&x=0,\\[6pt]{\frac {x}{1-e^{-x/\ln 2}}},&x\neq 0\end{cases}}} By making the change of variables x = y ln ⁡ ( 2 ) {\displaystyle x=y\ln(2)} , this is equivalent to log 2 ⁡ ( 1 + 2 y ) ≈ { 1 , y = 0 , y 1 − e − y , y ≠ 0. {\displaystyle \log _{2}(1+2^{y})\approx {\begin{cases}1,&y=0,\\[6pt]{\frac {y}{1-e^{-y}}},&y\neq 0.\end{cases}}} A sharpness parameter k {\displaystyle k} may be included: f ( x ) = ln ⁡ ( 1 + e k x ) k , f ′ ( x ) = e k x 1 + e k x = 1 1 + e − k x . {\displaystyle f(x)={\frac {\ln(1+e^{kx})}{k}},\qquad \qquad f'(x)={\frac {e^{kx}}{1+e^{kx}}}={\frac {1}{1+e^{-kx}}}.} Additionally, the softplus function is equivalent to the log of the sigmoid function in the following way: − ln ⁡ ( sigmoid ( − x ) ) = − ln ⁡ ( 1 1 + e x ) = ln ⁡ ( 1 + e x ) = softplus ( x ) {\displaystyle -\ln({\text{sigmoid}}(-x))=-\ln \left({\frac {1}{1+e^{x}}}\right)=\ln \left(1+e^{x}\right)={\text{softplus}}(x)} == Related functions == The derivative of softplus is the standard logistic function: f ′ ( x ) = e x 1 + e x = 1 1 + e − x {\displaystyle f'(x)={\frac {e^{x}}{1+e^{x}}}={\frac {1}{1+e^{-x}}}} The logistic function or the sigmoid function is a smooth approximation of the rectifier, the Heaviside step function. === LogSumExp === The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero: L S E 0 + ⁡ ( x 1 , … , x n ) := LSE ⁡ ( 0 , x 1 , … , x n ) = ln ⁡ ( 1 + e x 1 + ⋯ + e x n ) . {\displaystyle \operatorname {LSE_{0}} ^{+}(x_{1},\dots ,x_{n}):=\operatorname {LSE} (0,x_{1},\dots ,x_{n})=\ln(1+e^{x_{1}}+\cdots +e^{x_{n}}).} The LogSumExp function is LSE ⁡ ( x 1 , … , x n ) = ln ⁡ ( e x 1 + ⋯ + e x n ) , {\displaystyle \operatorname {LSE} (x_{1},\dots ,x_{n})=\ln(e^{x_{1}}+\cdots +e^{x_{n}}),} and its gradient is the softmax; the softmax with the first argument set to zero is the multivariable generalization of the logistic function. Both LogSumExp and softmax are used in machine learning. === Convex conjugate === The convex conjugate (specifically, the Legendre transformation) of the softplus function is the negative binary entropy function (with base e). This is because (following the definition of the Legendre transformation: the derivatives are inverse functions) the derivative of softplus is the logistic function, whose inverse function is the logit, which is the derivative of negative binary entropy. Softplus can be interpreted as logistic loss (as a positive number), so, by duality, minimizing logistic loss corresponds to maximizing entropy. This justifies the principle of maximum entropy as loss minimization.

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  Memory color effect

  The memory color effect is the phenomenon that the canonical hue of a type of object acquired through experience (e.g. the sky, a leaf, or a strawberry) can directly modulate the appearance of the actual colors of objects. Human observers acquire memory colors through their experiences with instances of that type. For example, most human observers know that an apple typically has a reddish hue; this knowledge about the canonical color which is represented in memory constitutes a memory color. As an example of the effect, normal human trichromats, when presented with a gray banana, often perceive the gray banana as being yellow - the banana's memory color. In light of this, subjects typically adjust the color of the banana towards the color blue - the opponent color of yellow - when asked to adjust its surface to gray to cancel the subtle activation of banana's memory color. Subsequent empirical studies have also shown the memory color effect on man-made objects (e.g. smurfs, German mailboxes), the effect being especially pronounced for blue and yellow objects. To explain this, researchers have argued that because natural daylight shifts from short wavelengths of light (i.e., bluish hues) towards light of longer wavelengths (i.e., yellowish-orange hues) during the day, the memory colors for blue and yellow objects are recruited by the visual system to a higher degree to compensate for this fluctuation in illumination, thereby providing a stronger memory color effect. == Form identification == Memory color plays a role when detecting an object. In a study where participants were given objects, such as an apple, with two alternate forms for each, a crooked apple and a circular apple, researchers changed the colors of the alternate forms and asked if they could identify them. Most of the participants answered "unsure," suggesting that we use memory color when identifying an object. The research redefined memory color as a phenomenon when "a form's identity affects the phenomenal hue of that form." == Color effect on memorization == Memory color effect can be derived from the human instinct to memorize objects better. Comparing the effect of recognizing gray-scaled images and colored images, results showed that people were able to recall colored images 5% higher compared to gray-scaled images. An important factor was that higher level of contrast between the object and background color influences memory. In a specific study related to this, participants reported that colors were 5% to 10% easier to recognize compared to black and white. == Color constancy and memory color effect == Color constancy is the phenomenon where a surface to appear to be of the same color under a wide rage of illumination. A study tested two hypotheses with regards to color memory; the photoreceptor hypothesis and the surface reflectance hypothesis. The test color was surround either by various color patches forming a complex pattern or a uniform “grey” field at the same chromaticity as that of the illuminant. The test color was presented on a dark background for the control group. It was observed that complex surround results where in line with the surface-reflectance hypothesis and not the photoreceptor hypothesis, showing that the accuracy and precision of color memory are fundamentals to understanding the phenomenon of color constancy. == Significance to the evolution of trichromacy == While objects that possess canonical hues make up a small percentage of the objects which populate humans’ visual experience, the human visual system evolved in an environment populated with objects that possess canonical hues. This suggests that the memory color effect is related to the emergence of trichromacy because it has been argued that trichromacy evolved to optimize the ability to detect ripe fruits—objects that appear in canonical hues. == In perception research == In perception research, the memory color effect is cited as evidence for the opponent color theory, which states that four basic colors can be paired with its opponent color: red—green, blue—yellow. This explains why participants adjust the ripe banana color to a blueish tone to make its memory color yellow as gray. Researchers have also found empirical evidence that suggests memory color is recruited by the visual system to achieve color constancy. For example, participants had a lower percentage of color constancy when looking at a color incongruent scene, such as a purple banana, compared to a color diagnostical scene, a yellow banana. This suggests that color constancy is influenced by the color of objects that we are familiar with, which the memory color effect takes part.

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  Probabilistic latent semantic analysis

  Probabilistic latent semantic analysis (PLSA), also known as probabilistic latent semantic indexing (PLSI, especially in information retrieval circles) is a statistical technique for the analysis of two-mode and co-occurrence data. In effect, one can derive a low-dimensional representation of the observed variables in terms of their affinity to certain hidden variables, just as in latent semantic analysis, from which PLSA evolved. Compared to standard latent semantic analysis which stems from linear algebra and downsizes the occurrence tables (usually via a singular value decomposition), probabilistic latent semantic analysis is based on a mixture decomposition derived from a latent class model. == Model == Considering observations in the form of co-occurrences ( w , d ) {\displaystyle (w,d)} of words and documents, PLSA models the probability of each co-occurrence as a mixture of conditionally independent multinomial distributions: P ( w , d ) = ∑ c P ( d ) P ( c | d ) P ( w | c ) = P ( d ) ∑ c P ( c | d ) P ( w | c ) {\displaystyle P(w,d)=\sum _{c}P(d)P(c|d)P(w|c)=P(d)\sum _{c}P(c|d)P(w|c)} with c {\displaystyle c} being the words' topic. Note that the number of topics is a hyperparameter that must be chosen in advance and is not estimated from the data. The first formulation is the symmetric formulation, where w {\displaystyle w} and d {\displaystyle d} are both generated from the latent class c {\displaystyle c} in similar ways (using the conditional probabilities P ( d | c ) {\displaystyle P(d|c)} and P ( w | c ) {\displaystyle P(w|c)} ), whereas the second formulation is the asymmetric formulation, where, for each document d {\displaystyle d} , a latent class is chosen conditionally to the document according to P ( c | d ) {\displaystyle P(c|d)} , and a word is then generated from that class according to P ( w | c ) {\displaystyle P(w|c)} . Although we have used words and documents in this example, the co-occurrence of any couple of discrete variables may be modelled in exactly the same way. So, the number of parameters is equal to c d + w c {\displaystyle cd+wc} . The number of parameters grows linearly with the number of documents. In addition, although PLSA is a generative model of the documents in the collection it is estimated on, it is not a generative model of new documents. Their parameters are learned using the EM algorithm. == Application == PLSA may be used in a discriminative setting, via Fisher kernels. PLSA has applications in information retrieval and filtering, natural language processing, machine learning from text, bioinformatics, and related areas. It is reported that the aspect model used in the probabilistic latent semantic analysis has severe overfitting problems. == Extensions == Hierarchical extensions: Asymmetric: MASHA ("Multinomial ASymmetric Hierarchical Analysis") Symmetric: HPLSA ("Hierarchical Probabilistic Latent Semantic Analysis") Generative models: The following models have been developed to address an often-criticized shortcoming of PLSA, namely that it is not a proper generative model for new documents. Latent Dirichlet allocation – adds a Dirichlet prior on the per-document topic distribution Higher-order data: Although this is rarely discussed in the scientific literature, PLSA extends naturally to higher order data (three modes and higher), i.e. it can model co-occurrences over three or more variables. In the symmetric formulation above, this is done simply by adding conditional probability distributions for these additional variables. This is the probabilistic analogue to non-negative tensor factorisation. == History == This is an example of a latent class model (see references therein), and it is related to non-negative matrix factorization. The present terminology was coined in 1999 by Thomas Hofmann.

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  Quadratic classifier

  In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}} Read more →

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  Online machine learning

  In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., prediction of prices in the financial international markets. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches. Online machine learning algorithms find applications in a wide variety of fields such as sponsored search to maximize ad revenue, portfolio optimization, shortest path prediction (with stochastic weights, e.g. traffic on roads for a maps application), spam filtering, real-time fraud detection, dynamic pricing for e-commerce, etc. There is also growing interest in usage of online learning paradigms for LLMs to enable continuous, real-time adaptation after the initial training. == Introduction == In the setting of supervised learning, a function of f : X → Y {\displaystyle f:X\to Y} is to be learned, where X {\displaystyle X} is thought of as a space of inputs and Y {\displaystyle Y} as a space of outputs, that predicts well on instances that are drawn from a joint probability distribution p ( x , y ) {\displaystyle p(x,y)} on X × Y {\displaystyle X\times Y} . In reality, the learner never knows the true distribution p ( x , y ) {\displaystyle p(x,y)} over instances. Instead, the learner usually has access to a training set of examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} . In this setting, the loss function is given as V : Y × Y → R {\displaystyle V:Y\times Y\to \mathbb {R} } , such that V ( f ( x ) , y ) {\displaystyle V(f(x),y)} measures the difference between the predicted value f ( x ) {\displaystyle f(x)} and the true value y {\displaystyle y} . The ideal goal is to select a function f ∈ H {\displaystyle f\in {\mathcal {H}}} , where H {\displaystyle {\mathcal {H}}} is a space of functions called a hypothesis space, so that some notion of total loss is minimized. Depending on the type of model (statistical or adversarial), one can devise different notions of loss, which lead to different learning algorithms. == Statistical view of online learning == In statistical learning models, the training sample ( x i , y i ) {\displaystyle (x_{i},y_{i})} are assumed to have been drawn from the true distribution p ( x , y ) {\displaystyle p(x,y)} and the objective is to minimize the expected "risk" I [ f ] = E [ V ( f ( x ) , y ) ] = ∫ V ( f ( x ) , y ) d p ( x , y ) . {\displaystyle I[f]=\mathbb {E} [V(f(x),y)]=\int V(f(x),y)\,dp(x,y)\ .} A common paradigm in this situation is to estimate a function f ^ {\displaystyle {\hat {f}}} through empirical risk minimization or regularized empirical risk minimization (usually Tikhonov regularization). The choice of loss function here gives rise to several well-known learning algorithms such as regularized least squares and support vector machines. A purely online model in this category would learn based on just the new input ( x t + 1 , y t + 1 ) {\displaystyle (x_{t+1},y_{t+1})} , the current best predictor f t {\displaystyle f_{t}} and some extra stored information (which is usually expected to have storage requirements independent of training data size). For many formulations, for example nonlinear kernel methods, true online learning is not possible, though a form of hybrid online learning with recursive algorithms can be used where f t + 1 {\displaystyle f_{t+1}} is permitted to depend on f t {\displaystyle f_{t}} and all previous data points ( x 1 , y 1 ) , … , ( x t , y t ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{t},y_{t})} . In this case, the space requirements are no longer guaranteed to be constant since it requires storing all previous data points, but the solution may take less time to compute with the addition of a new data point, as compared to batch learning techniques. A common strategy to overcome the above issues is to learn using mini-batches, which process a small batch of b ≥ 1 {\displaystyle b\geq 1} data points at a time, this can be considered as pseudo-online learning for b {\displaystyle b} much smaller than the total number of training points. Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks. === Example: linear least squares === The simple example of linear least squares is used to explain a variety of ideas in online learning. The ideas are general enough to be applied to other settings, for example, with other convex loss functions. === Batch learning === Consider the setting of supervised learning with f {\displaystyle f} being a linear function to be learned: f ( x j ) = ⟨ w , x j ⟩ = w ⋅ x j {\displaystyle f(x_{j})=\langle w,x_{j}\rangle =w\cdot x_{j}} where x j ∈ R d {\displaystyle x_{j}\in \mathbb {R} ^{d}} is a vector of inputs (data points) and w ∈ R d {\displaystyle w\in \mathbb {R} ^{d}} is a linear filter vector. The goal is to compute the filter vector w {\displaystyle w} . To this end, a square loss function V ( f ( x j ) , y j ) = ( f ( x j ) − y j ) 2 = ( ⟨ w , x j ⟩ − y j ) 2 {\displaystyle V(f(x_{j}),y_{j})=(f(x_{j})-y_{j})^{2}=(\langle w,x_{j}\rangle -y_{j})^{2}} is used to compute the vector w {\displaystyle w} that minimizes the empirical loss I n [ w ] = ∑ j = 1 n V ( ⟨ w , x j ⟩ , y j ) = ∑ j = 1 n ( x j T w − y j ) 2 {\displaystyle I_{n}[w]=\sum _{j=1}^{n}V(\langle w,x_{j}\rangle ,y_{j})=\sum _{j=1}^{n}(x_{j}^{\mathsf {T}}w-y_{j})^{2}} where y j ∈ R . {\displaystyle y_{j}\in \mathbb {R} .} Let X {\displaystyle X} be the i × d {\displaystyle i\times d} data matrix and y ∈ R i {\displaystyle y\in \mathbb {R} ^{i}} is the column vector of target values after the arrival of the first i {\displaystyle i} data points. Assuming that the covariance matrix Σ i = X T X {\displaystyle \Sigma _{i}=X^{\mathsf {T}}X} is invertible (otherwise it is preferential to proceed in a similar fashion with Tikhonov regularization), the best solution f ∗ ( x ) = ⟨ w ∗ , x ⟩ {\displaystyle f^{}(x)=\langle w^{},x\rangle } to the linear least squares problem is given by w ∗ = ( X T X ) − 1 X T y = Σ i − 1 ∑ j = 1 i x j y j . {\displaystyle w^{}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y=\Sigma _{i}^{-1}\sum _{j=1}^{i}x_{j}y_{j}.} Now, calculating the covariance matrix Σ i = ∑ j = 1 i x j x j T {\displaystyle \Sigma _{i}=\sum _{j=1}^{i}x_{j}x_{j}^{\mathsf {T}}} takes time O ( i d 2 ) {\displaystyle O(id^{2})} , inverting the d × d {\displaystyle d\times d} matrix takes time O ( d 3 ) {\displaystyle O(d^{3})} , while the rest of the multiplication takes time O ( d 2 ) {\displaystyle O(d^{2})} , giving a total time of O ( i d 2 + d 3 ) {\displaystyle O(id^{2}+d^{3})} . When there are n {\displaystyle n} total points in the dataset, to recompute the solution after the arrival of every datapoint i = 1 , … , n {\displaystyle i=1,\ldots ,n} , the naive approach will have a total complexity O ( n 2 d 2 + n d 3 ) {\displaystyle O(n^{2}d^{2}+nd^{3})} . Note that when storing the matrix Σ i {\displaystyle \Sigma _{i}} , then updating it at each step needs only adding x i + 1 x i + 1 T {\displaystyle x_{i+1}x_{i+1}^{\mathsf {T}}} , which takes O ( d 2 ) {\displaystyle O(d^{2})} time, reducing the total time to O ( n d 2 + n d 3 ) = O ( n d 3 ) {\displaystyle O(nd^{2}+nd^{3})=O(nd^{3})} , but with an additional storage space of O ( d 2 ) {\displaystyle O(d^{2})} to store Σ i {\displaystyle \Sigma _{i}} . === Online learning: recursive least squares === The recursive least squares (RLS) algorithm considers an online approach to the least squares problem. It can be shown that by initialising w 0 = 0 ∈ R d {\displaystyle \textstyle w_{0}=0\in \mathbb {R} ^{d}} and Γ 0 = I ∈ R d × d {\displaystyle \textstyle \Gamma _{0}=I\in \mathbb {R} ^{d\times d}} , the solution of the linear least squares problem given in the previous section can be computed by the following iteration: Γ i = Γ i − 1 − Γ i − 1 x i x i T Γ i − 1 1 + x i T Γ i − 1 x i {\displaystyle \Gamma _{i}=\Gamma _{i-1}-{\frac {\Gamma _{i-1}x_{i}x_{i}^{\mathsf {T}}\Gamma _{i-1}}{1+x_{i}^{\mathsf {T}}\Gamma _{i-1}x_{i}}}} w i = w i − 1 − Γ i x i ( x i T w i − 1 − y i ) {\displaystyle w_{i}=w_{i-1}-\Gamma _{i}x_{i}\left(x_{i}^{\mathsf {T}}w_{

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  Adversarial machine learning

  Adversarial machine learning is the study of the attacks on machine learning algorithms, and of the defenses against such attacks. Machine learning techniques are mostly designed to work on specific problem sets, under the assumption that the training and test data are generated from the same statistical distribution (IID). However, this assumption is often violated in practical high-stake applications, where users may intentionally supply fabricated data that violates the statistical assumption. Most common attacks in adversarial machine learning include evasion attacks, data poisoning attacks, Byzantine attacks and model extraction. == History == At the MIT Spam Conference in January 2004, John Graham-Cumming showed that a machine-learning spam filter could be used to defeat another machine-learning spam filter by automatically learning which words to add to a spam email to get the email classified as not spam. In 2004, Nilesh Dalvi and others noted that linear classifiers used in spam filters could be defeated by simple "evasion attacks" as spammers inserted "good words" into their spam emails. (Around 2007, some spammers added random noise to fuzz words within "image spam" in order to defeat OCR-based filters.) In 2006, Marco Barreno and others published "Can Machine Learning Be Secure?", outlining a broad taxonomy of attacks. As late as 2013 many researchers continued to hope that non-linear classifiers (such as support vector machines and neural networks) might be robust to adversaries, until Battista Biggio and others demonstrated the first gradient-based attacks on such machine-learning models (2012–2013). In 2012, deep neural networks began to dominate computer vision problems; starting in 2014, Christian Szegedy and others demonstrated that deep neural networks could be fooled by adversaries, again using a gradient-based attack to craft adversarial perturbations. Further work would show that adversarial attacks are harder to produce in uncontrolled environments, due to the different environmental constraints that cancel out the effect of noise. For example, any small rotation or slight illumination on an adversarial image can destroy the adversariality. In addition, researchers such as Google Brain's Nick Frosst point out that it is much easier to make self-driving cars miss stop signs by physically removing the sign itself, rather than creating adversarial examples. Frosst also believes that the adversarial machine learning community incorrectly assumes models trained on a certain data distribution will also perform well on a completely different data distribution. He suggests that a new approach to machine learning should be explored, and is currently working on a unique neural network that has characteristics more similar to human perception than state-of-the-art approaches. While adversarial machine learning continues to be heavily rooted in academia, large tech companies such as Google, Microsoft, and IBM have begun curating documentation and open source code bases to allow others to concretely assess the robustness of machine learning models and minimize the risk of adversarial attacks. === Examples === Examples include attacks in spam filtering, where spam messages are obfuscated through the misspelling of "bad" words or the insertion of "good" words; attacks in computer security, such as obfuscating malware code within network packets or modifying the characteristics of a network flow to mislead intrusion detection; attacks in biometric recognition where fake biometric traits may be exploited to impersonate a legitimate user; or to compromise users' template galleries that adapt to updated traits over time. Researchers showed that by changing only one-pixel it was possible to fool deep learning algorithms. Others 3-D printed a toy turtle with a texture engineered to make Google's object detection AI classify it as a rifle regardless of the angle from which the turtle was viewed. Creating the turtle required only low-cost commercially available 3-D printing technology. A machine-tweaked image of a dog was shown to look like a cat to both computers and humans. A 2019 study reported that humans can guess how machines will classify adversarial images. Researchers discovered methods for perturbing the appearance of a stop sign such that an autonomous vehicle classified it as a merge or speed limit sign. A data poisoning filter called Nightshade was released in 2023 by researchers at the University of Chicago. It was created for use by visual artists to put on their artwork to corrupt the data set of text-to-image models, which usually scrape their data from the internet without the consent of the image creator. McAfee attacked Tesla's former Mobileye system, fooling it into driving 50 mph over the speed limit, simply by adding a two-inch strip of black tape to a speed limit sign. Adversarial patterns on glasses or clothing designed to deceive facial-recognition systems or license-plate readers, have led to a niche industry of "stealth streetwear". An adversarial attack on a neural network can allow an attacker to inject algorithms into the target system. Researchers can also create adversarial audio inputs to disguise commands to intelligent assistants in benign-seeming audio; a parallel literature explores human perception of such stimuli. Clustering algorithms are used in security applications. Malware and computer virus analysis aims to identify malware families, and to generate specific detection signatures. In the context of malware detection, researchers have proposed methods for adversarial malware generation that automatically craft binaries to evade learning-based detectors while preserving malicious functionality. Optimization-based attacks such as GAMMA use genetic algorithms to inject benign content (for example, padding or new PE sections) into Windows executables, framing evasion as a constrained optimization problem that balances misclassification success with the size of the injected payload and showing transferability to commercial antivirus products. Complementary work uses generative adversarial networks (GANs) to learn feature-space perturbations that cause malware to be classified as benign; Mal-LSGAN, for instance, replaces the standard GAN loss with a least-squares objective and modified activation functions to improve training stability and produce adversarial malware examples that substantially reduce true positive rates across multiple detectors. == Challenges in applying machine learning to security == Researchers have observed that the constraints under which machine-learning techniques function in the security domain are different from those of common benchmark domains. Security data may change over time, include mislabeled samples, or reflect adversarial behavior, which complicates evaluation and reproducibility. === Data collection issues === Security datasets vary across formats, including binaries, network traces, and log files. Studies have reported that the process of converting these sources into features can introduce bias or inconsistencies. In addition, time-based leakage can occur when related malware samples are not properly separated across training and testing splits, which may lead to overly optimistic results. === Labeling and ground truth challenges === Malware labels are often unstable because different antivirus engines may classify the same sample in conflicting ways. Ceschin et al. note that families may be renamed or reorganized over time, causing further discrepancies in ground truth and reducing the reliability of benchmarks. === Concept drift === Because malware creators continuously adapt their techniques, the statistical properties of malicious samples also change. This form of concept drift has been widely documented and may reduce model performance unless systems are updated regularly or incorporate mechanisms for incremental learning. === Feature robustness === Researchers differentiate between features that can be easily manipulated and those that are more resistant to modification. For example, simple static attributes, such as header fields, may be altered by attackers, while structural features, such as control-flow graphs, are generally more stable but computationally expensive to extract. === Class imbalance === In realistic deployment environments, the proportion of malicious samples can be extremely low, ranging from 0.01% to 2% of total data. This unbalanced distribution causes models to develop a bias towards the majority class, achieving high accuracy but failing to identify malicious samples. Prior approaches to this problem have included both data-level solutions and sequence-specific models. Methods like n-gram and Long Short-Term Memory (LSTM) networks can model sequential data, but their performance has been shown to decline significantly when malware samples are realistically proportioned in the training set, demonstrating the limitations in

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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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  Detrended correspondence analysis

  Detrended correspondence analysis (DCA) is a multivariate statistical technique widely used by ecologists to find the main factors or gradients in large, species-rich but usually sparse data matrices that typify ecological community data. DCA is frequently used to suppress artifacts inherent in most other multivariate analyses when applied to gradient data. == History == DCA was created in 1979 by Mark Hill of the United Kingdom's Institute for Terrestrial Ecology (now merged into Centre for Ecology and Hydrology) and implemented in FORTRAN code package called DECORANA (Detrended Correspondence Analysis), a correspondence analysis method. DCA is sometimes erroneously referred to as DECORANA; however, DCA is the underlying algorithm, while DECORANA is a tool implementing it. == Issues addressed == According to Hill and Gauch, DCA suppresses two artifacts inherent in most other multivariate analyses when applied to gradient data. An example is a time-series of plant species colonising a new habitat; early successional species are replaced by mid-successional species, then by late successional ones (see example below). When such data are analysed by a standard ordination such as a correspondence analysis: the ordination scores of the samples will exhibit the 'edge effect', i.e. the variance of the scores at the beginning and the end of a regular succession of species will be considerably smaller than that in the middle, when presented as a graph the points will be seen to follow a horseshoe shaped curve rather than a straight line ('arch effect'), even though the process under analysis is a steady and continuous change that human intuition would prefer to see as a linear trend. Outside ecology, the same artifacts occur when gradient data are analysed (e.g. soil properties along a transect running between 2 different geologies, or behavioural data over the lifespan of an individual) because the curved projection is an accurate representation of the shape of the data in multivariate space. Ter Braak and Prentice (1987, p. 121) cite a simulation study analysing two-dimensional species packing models resulting in a better performance of DCA compared to CA. == Method == DCA is an iterative algorithm that has shown itself to be a highly reliable and useful tool for data exploration and summary in community ecology (Shaw 2003). It starts by running a standard ordination (CA or reciprocal averaging) on the data, to produce the initial horse-shoe curve in which the 1st ordination axis distorts into the 2nd axis. It then divides the first axis into segments (default = 26), and rescales each segment to have mean value of zero on the 2nd axis - this effectively squashes the curve flat. It also rescales the axis so that the ends are no longer compressed relative to the middle, so that 1 DCA unit approximates to the same rate of turnover all the way through the data: the rule of thumb is that 4 DCA units mean that there has been a total turnover in the community. Ter Braak and Prentice (1987, p. 122) warn against the non-linear rescaling of the axes due to robustness issues and recommend using detrending-by-polynomials only. == Drawbacks == No significance tests are available with DCA, although there is a constrained (canonical) version called DCCA in which the axes are forced by Multiple linear regression to correlate optimally with a linear combination of other (usually environmental) variables; this allows testing of a null model by Monte-Carlo permutation analysis. == Example == The example shows an ideal data set: The species data is in rows, samples in columns. For each sample along the gradient, a new species is introduced but another species is no longer present. The result is a sparse matrix. Ones indicate the presence of a species in a sample. Except at the edges each sample contains five species. The plot of the first two axes of the correspondence analysis result on the right hand side clearly shows the disadvantages of this procedure: the edge effect, i.e. the points are clustered at the edges of the first axis, and the arch effect. == Software == An open source implementation of DCA, based on the original FORTRAN code, is available in the vegan R-package.

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