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  Template matching

  Template matching is a technique in digital image processing for finding small parts of an image which match a template image. It can be used for quality control in manufacturing, navigation of mobile robots, or edge detection in images. The main challenges in a template matching task are detection of occlusion, when a sought-after object is partly hidden in an image; detection of non-rigid transformations, when an object is distorted or imaged from different angles; sensitivity to illumination and background changes; background clutter; and scale changes. == Feature-based approach == The feature-based approach to template matching relies on the extraction of image features, such as shapes, textures, and colors, that match the target image or frame. This approach is usually achieved using neural networks and deep-learning classifiers such as VGG, AlexNet, and ResNet.Convolutional neural networks (CNNs), which many modern classifiers are based on, process an image by passing it through different hidden layers, producing a vector at each layer with classification information about the image. These vectors are extracted from the network and used as the features of the image. Feature extraction using deep neural networks, like CNNs, has proven extremely effective has become the standard in state-of-the-art template matching algorithms. This feature-based approach is often more robust than the template-based approach described below. As such, it has become the state-of-the-art method for template matching, as it can match templates with non-rigid and out-of-plane transformations, as well as high background clutter and illumination changes. == Template-based approach == For templates without strong features, or for when the bulk of a template image constitutes the matching image as a whole, a template-based approach may be effective. Since template-based matching may require sampling of a large number of data points, it is often desirable to reduce the number of sampling points by reducing the resolution of search and template images by the same factor before performing the operation on the resultant downsized images. This pre-processing method creates a multi-scale, or pyramid, representation of images, providing a reduced search window of data points within a search image so that the template does not have to be compared with every viable data point. Pyramid representations are a method of dimensionality reduction, a common aim of machine learning on data sets that suffer the curse of dimensionality. == Common challenges == In instances where the template may not provide a direct match, it may be useful to implement eigenspaces to create templates that detail the matching object under a number of different conditions, such as varying perspectives, illuminations, color contrasts, or object poses. For example, if an algorithm is looking for a face, its template eigenspaces may consist of images (i.e., templates) of faces in different positions to the camera, in different lighting conditions, or with different expressions (i.e., poses). It is also possible for a matching image to be obscured or occluded by an object. In these cases, it is unreasonable to provide a multitude of templates to cover each possible occlusion. For example, the search object may be a playing card, and in some of the search images, the card is obscured by the fingers of someone holding the card, or by another card on top of it, or by some other object in front of the camera. In cases where the object is malleable or poseable, motion becomes an additional problem, and problems involving both motion and occlusion become ambiguous. In these cases, one possible solution is to divide the template image into multiple sub-images and perform matching on each subdivision. == Deformable templates in computational anatomy == Template matching is a central tool in computational anatomy (CA). In this field, a deformable template model is used to model the space of human anatomies and their orbits under the group of diffeomorphisms, functions which smoothly deform an object. Template matching arises as an approach to finding the unknown diffeomorphism that acts on a template image to match the target image. Template matching algorithms in CA have come to be called large deformation diffeomorphic metric mappings (LDDMMs). Currently, there are LDDMM template matching algorithms for matching anatomical landmark points, curves, surfaces, volumes. == Template-based matching explained using cross correlation or sum of absolute differences == A basic method of template matching sometimes called "Linear Spatial Filtering" uses an image patch (i.e., the "template image" or "filter mask") tailored to a specific feature of search images to detect. This technique can be easily performed on grey images or edge images, where the additional variable of color is either not present or not relevant. Cross correlation techniques compare the similarities of the search and template images. Their outputs should be highest at places where the image structure matches the template structure, i.e., where large search image values get multiplied by large template image values. This method is normally implemented by first picking out a part of a search image to use as a template. Let S ( x , y ) {\displaystyle S(x,y)} represent the value of a search image pixel, where ( x , y ) {\displaystyle (x,y)} represents the coordinates of the pixel in the search image. For simplicity, assume pixel values are scalar, as in a greyscale image. Similarly, let T ( x t , y t ) {\textstyle T(x_{t},y_{t})} represent the value of a template pixel, where ( x t , y t ) {\textstyle (x_{t},y_{t})} represents the coordinates of the pixel in the template image. To apply the filter, simply move the center (or origin) of the template image over each point in the search image and calculate the sum of products, similar to a dot product, between the pixel values in the search and template images over the whole area spanned by the template. More formally, if ( 0 , 0 ) {\displaystyle (0,0)} is the center (or origin) of the template image, then the cross correlation T ⋆ S {\displaystyle T\star S} at each point ( x , y ) {\displaystyle (x,y)} in the search image can be computed as: ( T ⋆ S ) ( x , y ) = ∑ ( x t , y t ) ∈ T T ( x t , y t ) ⋅ S ( x t + x , y t + y ) {\displaystyle (T\star S)(x,y)=\sum _{(x_{t},y_{t})\in T}T(x_{t},y_{t})\cdot S(x_{t}+x,y_{t}+y)} For convenience, T {\displaystyle T} denotes both the pixel values of the template image as well as its domain, the bounds of the template. Note that all possible positions of the template with respect to the search image are considered. Since cross correlation values are greatest when the values of the search and template pixels align, the best matching position ( x m , y m ) {\displaystyle (x_{m},y_{m})} corresponds to the maximum value of T ⋆ S {\displaystyle T\star S} over S {\displaystyle S} . Another way to handle translation problems on images using template matching is to compare the intensities of the pixels, using the sum of absolute differences (SAD) measure. To formulate this, let I S ( x s , y s ) {\displaystyle I_{S}(x_{s},y_{s})} and I T ( x t , y t ) {\displaystyle I_{T}(x_{t},y_{t})} denote the light intensity of pixels in the search and template images with coordinates ( x s , y s ) {\displaystyle (x_{s},y_{s})} and ( x t , y t ) {\displaystyle (x_{t},y_{t})} , respectively. Then by moving the center (or origin) of the template to a point ( x , y ) {\displaystyle (x,y)} in the search image, as before, the sum of absolute differences between the template and search pixel intensities at that point is: S A D ( x , y ) = ∑ ( x t , y t ) ∈ T | I T ( x t , y t ) − I S ( x t + x , y t + y ) | {\displaystyle SAD(x,y)=\sum _{(x_{t},y_{t})\in T}\left\vert I_{T}(x_{t},y_{t})-I_{S}(x_{t}+x,y_{t}+y)\right\vert } With this measure, the lowest SAD gives the best position for the template, rather than the greatest as with cross correlation. SAD tends to be relatively simple to implement and understand, but it also tends to be relatively slow to execute. A simple C++ implementation of SAD template matching is given below. == Implementation == In this simple implementation, it is assumed that the above described method is applied on grey images: This is why Grey is used as pixel intensity. The final position in this implementation gives the top left location for where the template image best matches the search image. One way to perform template matching on color images is to decompose the pixels into their color components and measure the quality of match between the color template and search image using the sum of the SAD computed for each color separately. == Speeding up the process == In the past, this type of spatial filtering was normally only used in dedicated hardware solutions because of the computational complexity of the operation, however we can lessen this complexity b

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  Rule-based machine learning

  Rule-based machine learning (RBML) is a term in computer science intended to encompass any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply. The defining characteristic of a rule-based machine learner is the identification and utilization of a set of relational rules that collectively represent the knowledge captured by the system. Rule-based machine learning approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge. While rule-based machine learning is conceptually a type of rule-based system, it is distinct from traditional rule-based systems, which are often hand-crafted, and other rule-based decision makers. This is because rule-based machine learning applies some form of learning algorithm such as Rough sets theory to identify and minimise the set of features and to automatically identify useful rules, rather than a human needing to apply prior domain knowledge to manually construct rules and curate a rule set. == Rules == Rules typically take the form of an '{IF:THEN} expression', (e.g. {IF 'condition' THEN 'result'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign}). An individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Therefore rule-based machine learning methods typically comprise a set of rules, or knowledge base, that collectively make up the prediction model usually known as decision algorithm. Rules can also be interpreted in various ways depending on the domain knowledge, data types(discrete or continuous) and in combinations. == RIPPER == Repeated incremental pruning to produce error reduction (RIPPER) is a propositional rule learner proposed by William W. Cohen as an optimized version of IREP.

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  Dynamic Bayesian network

  A dynamic Bayesian network (DBN) is a Bayesian network (BN) which relates variables to each other over adjacent time steps. == History == A dynamic Bayesian network (DBN) is often called a "two-timeslice" BN (2TBN) because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics. Dagum developed DBNs to unify and extend traditional linear state-space models such as Kalman filters, linear and normal forecasting models such as ARMA and simple dependency models such as hidden Markov models into a general probabilistic representation and inference mechanism for arbitrary nonlinear and non-normal time-dependent domains. Today, DBNs are common in robotics, and have shown potential for a wide range of data mining applications. For example, they have been used in speech recognition, digital forensics, protein sequencing, and bioinformatics. DBN is a generalization of hidden Markov models and Kalman filters. DBNs are conceptually related to probabilistic Boolean networks and can, similarly, be used to model dynamical systems at steady-state.

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  CN2 algorithm

  The CN2 induction algorithm is a learning algorithm for rule induction. It is designed to work even when the training data is imperfect. It is based on ideas from the AQ algorithm and the ID3 algorithm. As a consequence it creates a rule set like that created by AQ but is able to handle noisy data like ID3. == Description of algorithm == The algorithm must be given a set of examples, TrainingSet, which have already been classified in order to generate a list of classification rules. A set of conditions, SimpleConditionSet, which can be applied, alone or in combination, to any set of examples is predefined to be used for the classification. routine CN2(TrainingSet) let the ClassificationRuleList be empty repeat let the BestConditionExpression be Find_BestConditionExpression(TrainingSet) if the BestConditionExpression is not nil then let the TrainingSubset be the examples covered by the BestConditionExpression remove from the TrainingSet the examples in the TrainingSubset let the MostCommonClass be the most common class of examples in the TrainingSubset append to the ClassificationRuleList the rule 'if ' the BestConditionExpression ' then the class is ' the MostCommonClass until the TrainingSet is empty or the BestConditionExpression is nil return the ClassificationRuleList routine Find_BestConditionExpression(TrainingSet) let the ConditionalExpressionSet be empty let the BestConditionExpression be nil repeat let the TrialConditionalExpressionSet be the set of conditional expressions, {x and y where x belongs to the ConditionalExpressionSet and y belongs to the SimpleConditionSet}. remove all formulae in the TrialConditionalExpressionSet that are either in the ConditionalExpressionSet (i.e., the unspecialized ones) or null (e.g., big = y and big = n) for every expression, F, in the TrialConditionalExpressionSet if F is statistically significant and F is better than the BestConditionExpression by user-defined criteria when tested on the TrainingSet then replace the current value of the BestConditionExpression by F while the number of expressions in the TrialConditionalExpressionSet > user-defined maximum remove the worst expression from the TrialConditionalExpressionSet let the ConditionalExpressionSet be the TrialConditionalExpressionSet until the ConditionalExpressionSet is empty return the BestConditionExpression

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  Weak supervision

  Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log ⁡ p ( { x i , y i } i = 1 l | θ ) + λ log ⁡ p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign ⁡ f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst

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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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  Receptron

  The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j Read more →

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  Shattered set

  A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. == Definition == Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . {\displaystyle a=c\cap A.} Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = { c ∩ A | c ∈ C }. We employ the letter C to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set A is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points. == Example == We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C be the class of all discs. To test where C shatters A, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of A using intersections with class C and so C does not shatter A. As visualized below: Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that no set of four points is shattered by this C. However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below: With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots). == Shatter coefficient == To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω {\displaystyle s\subset \Omega } , Ω {\displaystyle \Omega } being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω card ⁡ { { x 1 , x 2 , … , x n } ∩ s , s ∈ C } {\displaystyle S_{C}(n)=\max _{\forall x_{1},x_{2},\dots ,x_{n}\in \Omega }\operatorname {card} \{\,\{\,x_{1},x_{2},\dots ,x_{n}\}\cap s,s\in C\}} where card {\displaystyle \operatorname {card} } denotes the cardinality of the set and x 1 , x 2 , … , x n ∈ Ω {\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega } is any set of n points,. S C ( n ) {\displaystyle S_{C}(n)} is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C. For example, if set A contains 3 points, its power set, P ( A ) {\displaystyle P(A)} , contains 2 3 = 8 {\displaystyle 2^{3}=8} elements. If C shatters A, its shattering coefficient(3) would be 8 and S C ( 2 ) {\displaystyle S_{C}(2)} would be 2 2 = 4 {\displaystyle 2^{2}=4} . However, if one of those sets in P ( A ) {\displaystyle P(A)} cannot be obtained through intersections in c, then S C ( 3 ) {\displaystyle S_{C}(3)} would only be 7. If none of those sets can be obtained, S C ( 3 ) {\displaystyle S_{C}(3)} would be 0. Additionally, if S C ( 2 ) = 3 {\displaystyle S_{C}(2)=3} , for example, then there is an element in the set of all 2-point sets from A that cannot be obtained from intersections with C. It follows from this that S C ( 3 ) {\displaystyle S_{C}(3)} would also be less than 8 (i.e. C would not shatter A) because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of S C ( n ) {\displaystyle S_{C}(n)} : S C ( n ) ≤ 2 n {\displaystyle S_{C}(n)\leq 2^{n}} for all n because { s ∩ A | s ∈ C } ⊆ P ( A ) {\displaystyle \{s\cap A|s\in C\}\subseteq P(A)} for any A ⊆ Ω {\displaystyle A\subseteq \Omega } . If S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} , that means there is a set of cardinality n, which can be shattered by C. If S C ( N ) < 2 N {\displaystyle S_{C}(N)<2^{N}} for some N > 1 {\displaystyle N>1} then S C ( n ) < 2 n {\displaystyle S_{C}(n)<2^{n}} for all n ≥ N {\displaystyle n\geq N} . The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities. == Vapnik–Chervonenkis class == If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n {\displaystyle 2^{n}} because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S C ( n ) {\displaystyle S_{C}(n)} is still 2 n {\displaystyle 2^{n}} , because as n gets smaller, there are fewer sets that could be omitted. The extreme of this is S C ( 0 ) {\displaystyle S_{C}(0)} (the shattering coefficient of the empty set), which must always be 2 0 = 1 {\displaystyle 2^{0}=1} . These statements lends themselves to defining the VC dimension of a class C as: V C ( C ) = min n { n : S C ( n ) < 2 n } {\displaystyle VC(C)={\underset {n}{\min }}\{n:S_{C}(n)<2^{n}\}\,} or, alternatively, as V C 0 ( C ) = max n { n : S C ( n ) = 2 n } . {\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,} Note that V C ( C ) = V C 0 ( C ) + 1. {\displaystyle VC(C)=VC_{0}(C)+1.} . The VC dimension is usually defined as V C 0 {\displaystyle VC_{0}} , the largest cardinality of points chosen that will still shatter A (i.e. n such that S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} ). Altneratively, if for any n there is a set of cardinality n which can be shattered by C, then S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} for all n and the VC dimension of this class C is infinite. A class with finite VC dimension is called a Vapnik–Chervonenkis class or VC class. A class C is uniformly Glivenko–Cantelli if and only if it is a VC class.

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  Scan line

  A scan line (also scanline) is one line, or row, in a raster scanning pattern, such as a line of video on a cathode-ray tube (CRT) display of a television set or computer monitor. On CRT screens the horizontal scan lines are visually discernible, even when viewed from a distance, as alternating colored lines and black lines, especially when a progressive scan signal with below maximum vertical resolution is displayed. This is sometimes used today as a visual effect in computer graphics. The term is used, by analogy, for a single row of pixels in a raster graphics image. Scan lines are important in representations of image data, because many image file formats have special rules for data at the end of a scan line. For example, there may be a rule that each scan line starts on a particular boundary (such as a byte or word; see for example BMP file format). This means that even otherwise compatible raster data may need to be analyzed at the level of scan lines in order to convert between formats.

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  Non-negative matrix factorization

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

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  ImageNets

  ImageNets is an open source framework for rapid prototyping of machine vision algorithms, developed by the Institute of Automation. == Description == ImageNets is an open source and platform independent (Windows & Linux) framework for rapid prototyping of machine vision algorithms. With the GUI ImageNet Designer, no programming knowledge is required to perform operations on images. A configured ImageNet can be loaded and executed from C++ code without the need for loading the ImageNet Designer GUI to achieve higher execution performance. == History == ImageNets was developed by the Institute of Automation, University of Bremen, Germany. The software was first publicly released in 2010. Originally, ImageNets was developed for the Care-Providing Robot FRIEND but it can be used for a wide range of computer vision applications.

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  Sigmoid function

  A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function". Special cases of sigmoid functions include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. There is also the Heaviside step function, which instantaneously transitions between 0 and 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function. == Theory == In mathematics, a unitary sigmoid function is a bounded sigmoid-type function normalized to the unit range, typically with lower and upper asymptotes at 0 and 1. The theory proposed by Grebenc distinguishes three kinds of unitary sigmoid functions according to their asymptotic behavior and the presence or absence of oscillation near the asymptotes. A general form of a unitary sigmoid function is y = A S ( f ( x ) ) + B , {\displaystyle y=A\,S(f(x))+B,} where S {\displaystyle S} is an increasing sigmoid function, f ( x ) {\displaystyle f(x)} is a transformation of the independent variable, and A {\displaystyle A} and B {\displaystyle B} are constants controlling scaling and translation. === Classification === ==== 1st kind ==== A unitary sigmoid function of the first kind is a bounded increasing function that approaches its lower and upper asymptotes monotonically, without oscillation. This class includes many of the standard sigmoid functions used in statistics, biomathematics, and engineering, such as the logistic function and related generalizations. ==== 2nd kind ==== A unitary sigmoid function of the second kind is a bounded increasing function that oscillates near the upper asymptote while preserving an overall sigmoid transition. ==== 3rd kind ==== A unitary sigmoid function of the third kind is a bounded increasing function that oscillates near both the lower and upper asymptotes. These functions retain the global shape of a sigmoid curve but exhibit oscillatory behavior in the vicinity of both limiting states. === Taxonomy === The tables below show the taxonomy of unitary sigmoid functions of all three kinds. Table 1. Taxonomy matrix with examples of sigmoid functions of the 1st kind Table 2. Taxonomy matrix with examples of sigmoid functions of the 2nd kind on the unbounded interval Table 3. Taxonomy matrix with examples of sigmoid functions of the 3rd kind === Construction methods === The same theory presents a list of 30 methods for constructing sigmoid functions.. These include algebraic transformations, integration and convolution methods, constructions from bell-shaped functions, solutions of ordinary and partial differential equations, recursive schemes, stochastic differential equations, feedback systems, and chaotic systems. M0: Construction method for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a sigmoid function to the power M4: Exponentiating a sigmoid function M5: Symmetric sigmoid functions derived from asymmetric ones M6: Sigmoid functions of the reciprocal independent variable M7: Embedding a sigmoid function into other function M8: Sum of sigmoid functions M9: Multiplication of sigmoid functions M10: Integral of the product of an increasing and a decreasing function M11: Derivation from lambda (bell-shaped) functions M12: Integration of lambda (bell-shaped) function M13: Integration of the sum of lambda (bell-shaped) functions M14: Integration of the product of two lambda (bell-shaped) functions M15: Integration of the difference of two shifted sigmoid functions M16: Integration of the product of two shifted sigmoid functions M17: Convolution of sigmoid functions M18: Integration of the product of lambda and sigmoid function M19: Solutions of ordinary differential equations M20: Solutions of partial differential equation (PDE) M21: Solutions of functional differential equation (FDE) M22: Sum of a sigmoid function and some derivatives M23: Combination of sigmoid functions, its derivative and integral M24: Filtering sigmoid functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward loops M29: Solutions of stochastic differential equation M30: Chaotic sigmoid functions Consult reference for more details. == Definition == A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point. == Properties == In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0. == Examples == Logistic function f ( x ) = 1 1 + e − x {\displaystyle f(x)={\frac {1}{1+e^{-x}}}} Hyperbolic tangent (shifted and scaled version of the logistic function, above) f ( x ) = tanh ⁡ x = e x − e − x e x + e − x {\displaystyle f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}} Arctangent function f ( x ) = arctan ⁡ x {\displaystyle f(x)=\arctan x} Gudermannian function f ( x ) = gd ⁡ ( x ) = ∫ 0 x d t cosh ⁡ t = 2 arctan ⁡ ( tanh ⁡ ( x 2 ) ) {\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {dt}{\cosh t}}=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)} Error function f ( x ) = erf ⁡ ( x ) = 2 π ∫ 0 x e − t 2 d t {\displaystyle f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt} Generalised logistic function f ( x ) = ( 1 + e − x ) − α , α > 0 {\displaystyle f(x)=\left(1+e^{-x}\right)^{-\alpha },\quad \alpha >0} Smoothstep function f ( x ) = { ( ∫ 0 1 ( 1 − u 2 ) N d u ) − 1 ∫ 0 x ( 1 − u 2 ) N d u , | x | ≤ 1 sgn ⁡ ( x ) | x | ≥ 1 N ∈ Z ≥ 1 {\displaystyle f(x)={\begin{cases}{\displaystyle \left(\int _{0}^{1}\left(1-u^{2}\right)^{N}du\right)^{-1}\int _{0}^{x}\left(1-u^{2}\right)^{N}\ du},&|x|\leq 1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\quad N\in \mathbb {Z} \geq 1} Some algebraic functions, for example f ( x ) = x 1 + x 2 {\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}} and in a more general form f ( x ) = x ( 1 + | x | k ) 1 / k {\displaystyle f(x)={\frac {x}{\left(1+|x|^{k}\right)^{1/k}}}} Up to shifts and scaling, many sigmoids are special cases of f ( x ) = φ ( φ ( x , β ) , α ) , {\displaystyle f(x)=\varphi (\varphi (x,\beta ),\alpha ),} where φ ( x , λ ) = { ( 1 − λ x ) 1 / λ λ ≠ 0 e − x λ = 0 {\displaystyle \varphi (x,\lambda )={\begin{cases}(1-\lambda x)^{1/\lambda }&\lambda \neq 0\\e^{-x}&\lambda =0\\\end{cases}}} is the inverse of the negative Box–Cox transformation, and α < 1 {\displaystyle \alpha <1} and β < 1 {\displaystyle \beta <1} are shape parameters. Smooth transition function normalized to (−1,1): f ( x ) = { 2 1 + e − 2 m x 1 − x 2 − 1 , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 = { tanh ⁡ ( m x 1 − x 2 ) , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 {\displaystyle {\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}} using the hyperbolic tangent mentioned above. Here, m {\displaystyle m} is a free parameter encoding the slope at x = 0 {\displaystyle x=0} , which must be great

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  Situated approach (artificial intelligence)

  In artificial intelligence research, the situated approach builds agents that are designed to behave effectively successfully in their environment. This requires designing AI "from the bottom-up" by focussing on the basic perceptual and motor skills required to survive. The situated approach gives a much lower priority to abstract reasoning or problem-solving skills. The approach was originally proposed as an alternative to traditional approaches (that is, approaches popular before 1985 or so). After several decades, classical AI technologies started to face intractable issues (e.g. combinatorial explosion) when confronted with real-world modeling problems. All approaches to address these issues focus on modeling intelligences situated in an environment. They have become known as the situated approach to AI. == Emergence of a concept == === From traditional AI to Nouvelle AI === During the late 1980s, the approach now known as Nouvelle AI (Nouvelle means new in French) was pioneered at the MIT Artificial Intelligence Laboratory by Rodney Brooks. As opposed to classical or traditional artificial intelligence, Nouvelle AI purposely avoided the traditional goal of modeling human-level performance, but rather tries to create systems with intelligence at the level of insects, closer to real-world robots. But eventually, at least at MIT new AI did lead to an attempt for humanoid AI in the Cog Project. === From Nouvelle AI to behavior-based and situated AI === The conceptual shift introduced by nouvelle AI flourished in the robotics area, given way to behavior-based robotics (BBR), a methodology for developing AI based on a modular decomposition of intelligence. It was made famous by Rodney Brooks: his subsumption architecture was one of the earliest attempts to describe a mechanism for developing BBAI. It is extremely popular in robotics and to a lesser extent to implement intelligent virtual agents because it allows the successful creation of real-time dynamic systems that can run in complex environments. For example, it underlies the intelligence of the Sony Aibo and many RoboCup robot teams. Realizing that in fact all these approaches were aiming at building not an abstract intelligence, but rather an intelligence situated in a given environment, they have come to be known as the situated approach. In fact, this approach stems out from early insights of Alan Turing, describing the need to build machines equipped with sense organs to learn directly from the real-world instead of focusing on abstract activities, such as playing chess. == Definitions == Classically, a software entity is defined as a simulated element, able to act on itself and on its environment, and which has an internal representation of itself and of the outside world. An entity can communicate with other entities, and its behavior is the consequence of its perceptions, its representations, and its interactions with the other entities. === AI loop === Simulating entities in a virtual environment requires simulating the entire process that goes from a perception of the environment, or more generally from a stimulus, to an action on the environment. This process is called the AI loop and technology used to simulate it can be subdivided in two categories. Sensorimotor or low-level AI deals with either the perception problem (what is perceived?) or the animation problem (how are actions executed?). Decisional or high-level AI deals with the action selection problem (what is the most appropriate action in response to a given perception, i.e. what is the most appropriate behavior?). === Traditional or symbolic AI === There are two main approaches in decisional AI. The vast majority of the technologies available on the market, such as planning algorithms, finite-state machines (FSA), or expert systems, are based on the traditional or symbolic AI approach. Its main characteristics are: It is top-down: it subdivides, in a recursive manner, a given problem into a series of sub-problems that are supposedly easier to solve. It is knowledge-based: it relies on a symbolic description of the world, such as a set of rules. However, the limits of traditional AI, which goal is to build systems that mimic human intelligence, are well-known: inevitably, a combinatorial explosion of the number of rules occurs due to the complexity of the environment. In fact, it is impossible to predict all the situations that will be encountered by an autonomous entity. === Situated or behavioral AI === In order to address these issues, another approach to decisional AI, also known as situated or behavioral AI, has been proposed. It does not attempt to model systems that produce deductive reasoning processes, but rather systems that behave realistically in their environment. The main characteristics of this approach are the following: It is bottom-up: it relies on elementary behaviors, which can be combined to implement more complex behaviors. It is behavior-based: it does not rely on a symbolic description of the environment, but rather on a model of the interactions of the entities with their environment. The goal of situated AI is to model entities that are autonomous in their environment. This is achieved thanks to both the intrinsic robustness of the control architecture, and its adaptation capabilities to unforeseen situations. === Situated agents === In artificial intelligence and cognitive science, the term situated refers to an agent which is embedded in an environment. The term situated is commonly used to refer to robots, but some researchers argue that software agents can also be situated if: they exist in a dynamic (rapidly changing) environment, which they can manipulate or change through their actions, and which they can sense or perceive. Examples might include web-based agents, which can alter data or trigger processes (such as purchases) over the Internet, or virtual-reality bots which inhabit and change virtual worlds, such as Second Life. Being situated is generally considered to be part of being embodied, but it is useful to consider each perspective individually. The situated perspective emphasizes that intelligent behavior derives from the environment and the agent's interactions with it. The nature of these interactions are defined by an agent's embodiment. == Implementation principles == === Modular decomposition === The most important attribute of a system driven by situated AI is that the intelligence is controlled by a set of independent semi-autonomous modules. In the original systems, each module was actually a separate device or was at least conceived of as running on its own processing thread. Generally, though, the modules are just abstractions. In this respect, situated AI may be seen as a software engineering approach to AI, perhaps akin to object oriented design. Situated AI is often associated with reactive planning, but the two are not synonymous. Brooks advocated an extreme version of cognitive minimalism which required initially that the behavior modules were finite-state machines and thus contained no conventional memory or learning. This is associated with reactive AI because reactive AI requires reacting to the current state of the world, not to an agent's memory or preconception of that world. However, learning is obviously key to realistic strong AI, so this constraint has been relaxed, though not entirely abandoned. === Action selection mechanism === The situated AI community has presented several solutions to modeling decision-making processes, also known as action selection mechanisms. The first attempt to solve this problem goes back to subsumption architectures, which were in fact more an implementation technique than an algorithm. However, this attempt paved the way to several others, in particular the free-flow hierarchies and activation networks. A comparison of the structure and performances of these two mechanisms demonstrated the advantage of using free-flow hierarchies in solving the action selection problem. However, motor schemas and process description languages are two other approaches that have been used with success for autonomous robots. == Notes and references == Arsenio, Artur M. (2004) Towards an embodied and situated AI, In: Proceedings of the International FLAIRS conference, 2004. (online) The Artificial Life Route To Artificial Intelligence: Building Embodied, Situated Agents, Luc Steels and Rodney Brooks Eds., Lawrence Erlbaum Publishing, 1995. (ISBN 978-0805815184) Rodney A. Brooks Cambrian Intelligence (MIT Press, 1999) ISBN 0-262-52263-2; collection of early papers including "Intelligence without representation" and "Intelligence without reason", from 1986 & 1991 respectively. Ronald C. Arkin Behavior-Based Robotics (MIT Press, 1998) ISBN 0-262-01165-4 Hendriks-Jansen, Horst (1996) Catching Ourselves in the Act: Situated Activity, Interactive Emergence, Evolution, and Human Thought. Cambridge, Mass.: MIT Press.

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  K-nearest neighbors algorithm

  In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method. It was first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. In classification, a new example is assigned a label based on the labels of its k nearest training examples; in regression, the prediction is computed from the values of those neighbors. Most often, it is used for classification, as a k-NN classifier, the output of which is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest neighbor. The k-NN algorithm can also be generalized for regression. In k-NN regression, also known as nearest neighbor smoothing, the output is the property value for the object. This value is the average of the values of k nearest neighbors. If k = 1, then the output is simply assigned to the value of that single nearest neighbor, also known as nearest neighbor interpolation. For both classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that nearer neighbors contribute more to the average than distant ones. For example, a common weighting scheme consists of giving each neighbor a weight of 1/d, where d is the distance to the neighbor. The input consists of the k closest training examples in a data set. The neighbors are taken from a set of objects for which the class (for k-NN classification) or the object property value (for k-NN regression) is known. This can be thought of as the training set for the algorithm, though no explicit training step is required. A peculiarity (sometimes even a disadvantage) of the k-NN algorithm is its sensitivity to the local structure of the data. In k-NN classification the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance, if the features represent different physical units or come in vastly different scales, then feature-wise normalizing of the training data can greatly improve its accuracy. == Statistical setting == Suppose we have pairs ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … , ( X n , Y n ) {\displaystyle (X_{1},Y_{1}),(X_{2},Y_{2}),\dots ,(X_{n},Y_{n})} taking values in R d × { 1 , 2 } {\displaystyle \mathbb {R} ^{d}\times \{1,2\}} , where Y is the class label of X, so that X | Y = r ∼ P r {\displaystyle X|Y=r\sim P_{r}} for r = 1 , 2 {\displaystyle r=1,2} (and probability distributions P r {\displaystyle P_{r}} ). Given some norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R d {\displaystyle \mathbb {R} ^{d}} and a point x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} , let ( X ( 1 ) , Y ( 1 ) ) , … , ( X ( n ) , Y ( n ) ) {\displaystyle (X_{(1)},Y_{(1)}),\dots ,(X_{(n)},Y_{(n)})} be a reordering of the training data such that ‖ X ( 1 ) − x ‖ ≤ ⋯ ≤ ‖ X ( n ) − x ‖ {\displaystyle \|X_{(1)}-x\|\leq \dots \leq \|X_{(n)}-x\|} . == Algorithm == The training examples are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training samples. In the classification phase, k is a user-defined constant, and an unlabeled vector (a query or test point) is classified by assigning the label which is most frequent among the k training samples nearest to that query point. A commonly used distance metric for continuous variables is Euclidean distance. For discrete variables, such as for text classification, another metric can be used, such as the overlap metric (or Hamming distance). In the context of gene expression microarray data, for example, k-NN has been employed with correlation coefficients, such as Pearson and Spearman, as a metric. Often, the classification accuracy of k-NN can be improved significantly if the distance metric is learned with specialized algorithms such as large margin nearest neighbor or neighborhood components analysis. A drawback of the basic "majority voting" classification occurs when the class distribution is skewed. That is, examples of a more frequent class tend to dominate the prediction of the new example, because they tend to be common among the k nearest neighbors due to their large number. One way to overcome this problem is to weight the classification, taking into account the distance from the test point to each of its k nearest neighbors. The class (or value, in regression problems) of each of the k nearest points is multiplied by a weight proportional to the inverse of the distance from that point to the test point. Another way to overcome skew is by abstraction in data representation. For example, in a self-organizing map (SOM), each node is a representative (a center) of a cluster of similar points, regardless of their density in the original training data. k-NN can then be applied to the SOM. == Parameter selection == The best choice of k depends upon the data; generally, larger values of k reduces effect of the noise on the classification, but make boundaries between classes less distinct. A good k can be selected by various heuristic techniques (see hyperparameter optimization). The special case where the class is predicted to be the class of the closest training sample (i.e. when k = 1) is called the nearest neighbor algorithm. The accuracy of the k-NN algorithm can be severely degraded by the presence of noisy or irrelevant features, or if the feature scales are not consistent with their importance. Much research effort has been put into selecting or scaling features to improve classification. A particularly popular approach is the use of evolutionary algorithms to optimize feature scaling. Another popular approach is to scale features by the mutual information of the training data with the training classes. In binary (two class) classification problems, it is helpful to choose k to be an odd number as this avoids tied votes. One popular way of choosing the empirically optimal k in this setting is via bootstrap method. == The 1-nearest neighbor classifier == The most intuitive nearest neighbour type classifier is the one nearest neighbour classifier that assigns a point x to the class of its closest neighbour in the feature space, that is C n 1 n n ( x ) = Y ( 1 ) {\displaystyle C_{n}^{1nn}(x)=Y_{(1)}} . As the size of training data set approaches infinity, the one nearest neighbour classifier guarantees an error rate of no worse than twice the Bayes error rate (the minimum achievable error rate given the distribution of the data). == The weighted nearest neighbour classifier == The k-nearest neighbour classifier can be viewed as assigning the k nearest neighbours a weight 1 / k {\displaystyle 1/k} and all others 0 weight. This can be generalised to weighted nearest neighbour classifiers. That is, where the ith nearest neighbour is assigned a weight w n i {\displaystyle w_{ni}} , with ∑ i = 1 n w n i = 1 {\textstyle \sum _{i=1}^{n}w_{ni}=1} . An analogous result on the strong consistency of weighted nearest neighbour classifiers also holds. Let C n w n n {\displaystyle C_{n}^{wnn}} denote the weighted nearest classifier with weights { w n i } i = 1 n {\displaystyle \{w_{ni}\}_{i=1}^{n}} . Subject to regularity conditions, which in asymptotic theory are conditional variables which require assumptions to differentiate among parameters with some criteria. On the class distributions the excess risk has the following asymptotic expansion R R ( C n w n n ) − R R ( C Bayes ) = ( B 1 s n 2 + B 2 t n 2 ) { 1 + o ( 1 ) } , {\displaystyle {\mathcal {R}}_{\mathcal {R}}(C_{n}^{wnn})-{\mathcal {R}}_{\mathcal {R}}(C^{\text{Bayes}})=\left(B_{1}s_{n}^{2}+B_{2}t_{n}^{2}\right)\{1+o(1)\},} for constants B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} where s n 2 = ∑ i = 1 n w n i 2 {\displaystyle s_{n}^{2}=\sum _{i=1}^{n}w_{ni}^{2}} and t n = n − 2 / d ∑ i = 1 n w n i { i 1 + 2 / d − ( i − 1 ) 1 + 2 / d } {\displaystyle t_{n}=n^{-2/d}\sum _{i=1}^{n}w_{ni}\left\{i^{1+2/d}-(i-1)^{1+2/d}\right\}} . The optimal weighting scheme { w n i ∗ } i = 1 n {\displaystyle \{w_{ni}^{}\}_{i=1}^{n}} , that balances the two terms in the display above, is given as follows: set k ∗ = ⌊ B n 4 d + 4 ⌋ {\displaystyle k^{}=\lfloor Bn^{\frac {4}{d+4}}\rfloor } , w n i ∗ = 1 k ∗ [ 1 + d 2 − d 2 k ∗ 2 / d { i 1 + 2 / d − ( i − 1 ) 1 + 2 / d } ] {\displaystyle w_{ni}^{}={\frac {1}{k^{}}}\left[1+{\frac {d}{2}}-{\frac {d}{2{k^{}}^{2/d}}}\{i^{1+2/d}-(i-1)^{1+2/d}\}\right]} for i = 1 , 2 , … , k ∗ {\displaystyle i=1,2,\dots ,k^{}} and w n i ∗ = 0 {\displaystyle w_{ni}^{}=0} for i = k ∗ + 1 , … , n {\displaystyle i=k^{}+1,\dots ,n} . With optimal weights the dominant term in the asymptotic expansion of the excess risk is O ( n − 4 d + 4 ) {\displaystyle {\mathcal {O}}(n^{-{\frac {4}{d+4}}})}

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  Sliced inverse regression

  Sliced inverse regression (SIR) is a tool for dimensionality reduction in the field of multivariate statistics. In statistics, regression analysis is a method of studying the relationship between a response variable y and its input variable x _ {\displaystyle {\underline {x}}} , which is a p-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing. As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, E ( x _ | y ) {\displaystyle E({\underline {x}}\,|\,y)} , to perform a weighted principal component analysis. == Model == Given a response variable Y {\displaystyle \,Y} and a (random) vector X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} of explanatory variables, SIR is based on the model Y = f ( β 1 ⊤ X , … , β k ⊤ X , ε ) ( 1 ) {\displaystyle Y=f(\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X,\varepsilon )\quad \quad \quad \quad \quad (1)} where β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} are unknown projection vectors, k {\displaystyle \,k} is an unknown number smaller than p {\displaystyle \,p} , f {\displaystyle \;f} is an unknown function on R k + 1 {\displaystyle \mathbb {R} ^{k+1}} as it only depends on k {\displaystyle \,k} arguments, and ε {\displaystyle \varepsilon } is a random variable representing error with E [ ε | X ] = 0 {\displaystyle E[\varepsilon |X]=0} and a finite variance of σ 2 {\displaystyle \sigma ^{2}} . The model describes an ideal solution, where Y {\displaystyle \,Y} depends on X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} only through a k {\displaystyle \,k} dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from p {\displaystyle \,p} to a smaller number k {\displaystyle \,k} without losing any information. An equivalent version of ( 1 ) {\displaystyle \,(1)} is: the conditional distribution of Y {\displaystyle \,Y} given X {\displaystyle \,X} depends on X {\displaystyle \,X} only through the k {\displaystyle \,k} dimensional random vector ( β 1 ⊤ X , … , β k ⊤ X ) {\displaystyle (\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X)} . It is assumed that this reduced vector is as informative as the original X {\displaystyle \,X} in explaining Y {\displaystyle \,Y} . The unknown β i ′ s {\displaystyle \,\beta _{i}'s} are called the effective dimension reducing directions (EDR-directions). The space that is spanned by these vectors is denoted by the effective dimension reducing space (EDR-space). == Relevant linear algebra background == Given a _ 1 , … , a _ r ∈ R n {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}\in \mathbb {R} ^{n}} , then V := L ( a _ 1 , … , a _ r ) {\displaystyle V:=L({\underline {a}}_{1},\ldots ,{\underline {a}}_{r})} , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors a _ 1 , … , a _ r {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}} span V {\displaystyle \,V} , but the vectors that span space V {\displaystyle \,V} are not unique. The dimension of V ( ∈ R n ) {\displaystyle \,V(\in \mathbb {R} ^{n})} is equal to the maximum number of linearly independent vectors in V {\displaystyle \,V} . A set of n {\displaystyle \,n} linear independent vectors of R n {\displaystyle \mathbb {R} ^{n}} makes up a basis of R n {\displaystyle \mathbb {R} ^{n}} . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line. == Inverse regression == Computing the inverse regression curve (IR) means instead of looking for E [ Y | X = x ] {\displaystyle \,E[Y|X=x]} , which is a curve in R p {\displaystyle \mathbb {R} ^{p}} it is actually E [ X | Y = y ] {\displaystyle \,E[X|Y=y]} , which is also a curve in R p {\displaystyle \mathbb {R} ^{p}} , but consisting of p {\displaystyle \,p} one-dimensional regressions. The center of the inverse regression curve is located at E [ E [ X | Y ] ] = E [ X ] {\displaystyle \,E[E[X|Y]]=E[X]} . Therefore, the centered inverse regression curve is E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} which is a p {\displaystyle \,p} dimensional curve in R p {\displaystyle \mathbb {R} ^{p}} . == Inverse regression versus dimension reduction == The centered inverse regression curve lies on a k {\displaystyle \,k} -dimensional subspace spanned by Σ x x β i ′ s {\displaystyle \,\Sigma _{xx}\beta _{i}\,'s} . This is a connection between the model and inverse regression. Given this condition and ( 1 ) {\displaystyle \,(1)} , the centered inverse regression curve E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} is contained in the linear subspace spanned by Σ x x β k ( k = 1 , … , K ) {\displaystyle \,\Sigma _{xx}\beta _{k}(k=1,\ldots ,K)} , where Σ x x = C o v ( X ) {\displaystyle \,\Sigma _{xx}=Cov(X)} . == Estimation of the EDR-directions == After having had a look at all the theoretical properties, the aim now is to estimate the EDR-directions. For that purpose, weighted principal component analyses are needed. If the sample means m ^ h ′ s {\displaystyle \,{\hat {m}}_{h}\,'s} , X {\displaystyle \,X} would have been standardized to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} . Corresponding to the theorem above, the IR-curve m 1 ( y ) = E [ Z | Y = y ] {\displaystyle \,m_{1}(y)=E[Z|Y=y]} lies in the space spanned by ( η 1 , … , η k ) {\displaystyle \,(\eta _{1},\ldots ,\eta _{k})} , where η i = Σ x x 1 / 2 β i {\displaystyle \,\eta _{i}=\Sigma _{xx}^{1/2}\beta _{i}} . As a consequence, the covariance matrix c o v [ E [ Z | Y ] ] {\displaystyle \,cov[E[Z|Y]]} is degenerate in any direction orthogonal to the η i ′ s {\displaystyle \,\eta _{i}\,'s} . Therefore, the eigenvectors η k ( k = 1 , … , K ) {\displaystyle \,\eta _{k}(k=1,\ldots ,K)} associated with the largest K {\displaystyle \,K} eigenvalues are the standardized EDR-directions. == Algorithm == === SIR algorithm === The algorithm from Li, K-C. (1991) to estimate the EDR-directions via SIR is as follows. 1. Let Σ x x {\displaystyle \,\Sigma _{xx}} be the covariance matrix of X {\displaystyle \,X} . Standardize X {\displaystyle \,X} to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} ( 1 ) {\displaystyle \,(1)} can also be rewritten as Y = f ( η 1 ⊤ Z , … , η k ⊤ Z , ε ) {\displaystyle Y=f(\eta _{1}^{\top }Z,\ldots ,\eta _{k}^{\top }Z,\varepsilon )} where η k = β k Σ x x 1 / 2 ∀ k {\displaystyle \,\eta _{k}=\beta _{k}\Sigma _{xx}^{1/2}\quad \forall \;k} .) 2. Divide the range of y i {\displaystyle \,y_{i}} into S {\displaystyle \,S} non-overlapping slices H s ( s = 1 , … , S ) . n s {\displaystyle \,H_{s}(s=1,\ldots ,S).\;n_{s}} is the number of observations within each slice and I H s {\displaystyle \,I_{H_{s}}} is the indicator function for the slice: n s = ∑ i = 1 n I H s ( y i ) {\displaystyle n_{s}=\sum _{i=1}^{n}I_{H_{s}}(y_{i})} 3. Compute the mean of z i {\displaystyle \,z_{i}} over all slices, which is a crude estimate m ^ 1 {\displaystyle \,{\hat {m}}_{1}} of the inverse regression curve m 1 {\displaystyle \,m_{1}} : z ¯ s = n s − 1 ∑ i = 1 n z i I H s ( y i ) {\displaystyle \,{\bar {z}}_{s}=n_{s}^{-1}\sum _{i=1}^{n}z_{i}I_{H_{s}}(y_{i})} 4. Calculate the estimate for C o v { m 1 ( y ) } {\displaystyle \,Cov\{m_{1}(y)\}} : V ^ = n − 1 ∑ i = 1 S n s z ¯ s z ¯ s ⊤ {\displaystyle \,{\hat {V}}=n^{-1}\sum _{i=1}^{S}n_{s}{\bar {z}}_{s}{\bar {z}}_{s}^{\top }} 5. Identify the eigenvalues λ ^ i {\displaystyle \,{\hat {\lambda }}_{i}} and the eigenvectors η ^ i {\displaystyle \,{\hat {\eta }}_{i}} of V ^ {\displaystyle \,{\hat {V}}} , which are the standardized EDR-directions. 6. Transform the standardized EDR-directions back to the original scale. The estimates for the EDR-directions are given by: β ^ i = Σ ^ x x − 1 / 2 η ^ i {\displaystyle \,{\hat {\beta }}_{i}={\hat {\Sigma }}_{xx}^{-1/2}{\hat {\eta }}_{i}} (which are not necessarily orthogonal.)

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