Class activation mapping

Class activation mapping methods are explainable AI (XAI) techniques used to visualize the regions of an input image that are the most relevant for a particular task, especially image classification, in convolutional neural networks (CNNs). These methods generate heatmaps by weighting the feature maps from a convolutional layer according to their relevance to the target class. In the field of artificial intelligence, generically defined as "the effort to automate intellectual tasks normally performed by humans", machine learning and deep learning were created. They both use statistical and computational methods to learn patterns from data, reducing the need for manually coded rules. Machine learning models are trained on input data and the known respective answers, learning the underlying patterns or structures present in the data. Traditional Machine learning algorithms employ manually designed feature sets, posing a direct link between machine learning designers and employed features. Deep learning is a subfield of machine learning, based on the concept of successive layers of representation, in which the data is progressively unfolded in different ways, to extract relevant and informative patterns in data analysis. Deep learning algorithms are defined as feature learning algorithms automatically learning hierarchical feature representations from raw data, extracting increasingly abstract features through multiple layers. CNNs are a specific architecture of deep learning models, designed to process spatially structured data, such as images, exploiting a series of convolution, non-linear activation and pooling operations to extract relevant features, contained in the so-called feature maps from input data. CNNs have demonstrated to be highly effective in a variety of computer vision and image processing tasks. CNNs (and deep learning models more broadly) are described as black boxes due to their complex and non-transparent internal layers of representation. The need for clearer indications on its internal working and decision-making process gave birth to XAI techniques. Among the proposed XAI techniques for computer vision tasks, Class activation mapping methods can show which pixels in an input image are important to the predicted logit for a class of interest, in a classification task. Class activation mapping methods were originally developed for class-discriminative scenarios to visualize which parts of the input image influenced the classification decision, namely to visually highlight the regions of those feature maps that contribute most strongly to the prediction of a given class. More advanced versions of these methods are not limited to image classification tasks, but have been extended also to several vision-related tasks, such as object detection, image captioning, visual question answering and image segmentation. == Background == The following methods laid the groundwork for the class activation maps approaches, forming the conceptual basis of using gradients to highlight class-discriminative regions. === Class model visualization and saliency maps for convolutional neural networks === The class model visualization and image-specific saliency maps approaches have been presented in the foundational work "Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps" by Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman and it generalizes the deconvnet method by Zeiler and Fergus. Class model visualization synthesizes an artificial input image that strongly activates the output neurons associated with a target class. Given a trained, fixed model, this method starts with a zero-initialized image, backpropagates the gradients from the class score to the image pixels, updates the image pixels increasing the specific class scores and it repeats the pixel updating process, showing an encoded (idealized version) prototype of the class of interest. Image-specific class saliency visualization method provides a visual explanation by highlighting the most relevant pixels in an image for predicting a certain class C of interest. This is done by computing the gradient of the class score with respect to the input image, I 0 , {\displaystyle I_{0},} w = ∂ S C ∂ I | I 0 {\displaystyle w=\left.{\frac {\partial S_{C}}{\partial I}}\right|_{I_{0}}} approximating the model locally (around I 0 {\displaystyle I_{0}} ) as linear, using a first-order Taylor expansion: S C ( I ) ≈ w C T I + b {\displaystyle S_{C}(I)\approx w_{C}^{T}I+b} . The magnitude of w C {\displaystyle w_{C}} , the gradient, indicates the importancy of the pixels: larger gradients suggest greater influence on the prediction. Once the gradient is known, the saliency map is defined as the maximum absolute gradient across the color channels: M i j = m a x C | ∂ S C ∂ I i j C | {\displaystyle M_{ij}=max_{C}\left|{\frac {\partial S_{C}}{\partial I_{ij}^{C}}}\right|} resulting in an saliency map (i.e. heatmap). === Guided backpropagation === The concept of guided backpropagation can be traced for the first time in the paper by Springenberg et al. "Striving For Simplicity: The All Convolutional Net" and also this method builds upon the work by Zeiler and Fergus "Visualizing and Understanding Convolutional Networks". Guided backpropagation core is to understand what a CNN is learning, by visualizing the patterns that activate more strongly individual neurons (or filters), in architectures which do not rely on max-pooling layer. When propagating gradients back through a rectified linear unit (ReLU), guided backpropagation passes the gradient if and only if the input to the ReLU was positive (forward pass) and the output gradient is positive (backward signal), tackling both inactive neurons, negative gradients and suppressing the noise. The result displays sharper, high-resolution visualizations of what each neuron is responding to. Guided backpropagation represents a simple and practical method for model interpretability, helping understand how and where neural networks detect semantic concepts across layers. Moreover, it can be applied to any network architecture, due to its working principle. == Base versions == Class activation mapping and gradient-weighted class activation mapping are the original and most widely used methods for visual explanations in convolutional neural networks. These methods serve as the foundation for many later developments in explainable AI. Notation: In this article, the symbols i and j represent integer indices that disappear inside sums or averages, while x and y are the continuous (or up-sampled integer) coordinates of the final heat-map that is plotted. === Class activation mapping (CAM) === Class activation mapping (CAM) was the first, and the original, version of CAM methods, and it gave the name to the whole category. The approach was firstly introduced by Zhou et al. in their seminal work "Learning Deep Features for Discriminative Localization". This approach achieves class-specific heatmaps by modifying image classification CNN architectures, replacing fully-connected layers with convolutional layers and a final global average pooling layer. Its main scope is to localize and highlight discriminative regions of an input image that a CNN uses to identify a particular class, without needing explicit bounding box annotations. ==== Global average pooling (GAP) ==== Global average pooling (GAP) represents the key element in the original CAM approach. It is a dimensionality reduction technique and, similarly to other pooling layers, it allows the downsampling of the feature maps, calculating representative values for a specific region of the feature map. The particularity of GAP is that it calculates a single value for an entire feature map, significantly reducing the model dimensions. ==== Mathematical description ==== The mathematical description considers as its key the combination of convolutional and GAP layers. In CAM, it is mandatory to have the GAP layer after the last convolutional layer and before the final linear classifier layer. This last element of the architecture connects the output logits (the network predictions) y C {\displaystyle y^{C}} , to the GAP values, with its respective fine-tuned weights, w k C {\displaystyle w_{k}^{C}} . Considering A k {\displaystyle A^{k}} as the last feature maps of the last convolutional layer, GAP produces one value for each feature map, by averaging all the matrix elements (i, j) of the feature map: F k = 1 m n ∑ i = 1 m ∑ j = 1 n A i j k {\displaystyle F^{k}={\frac {1}{mn}}\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}^{k}} with A k = [ A 11 k A 12 k ⋯ A 1 n k A 21 k A 22 k ⋯ A 2 n k ⋮ ⋮ ⋱ ⋮ A m 1 k A m 2 k ⋯ A m n k ] = { A i j k ∣ 1 ≤ i ≤ m , 1 ≤ j ≤ n } {\displaystyle A^{k}={\begin{bmatrix}A_{11}^{k}&A_{12}^{k}&\cdots &A_{1n}^{k}\\A_{21}^{k}&A_{22}^{k}&\cdots &A_{2n}^{k}\\\vdots &\vdots &\ddots &\vdots \\A_{m1}^{k}&A_{m2}^{k}&\cdots &A_{mn}^{k}\end{bmatrix}}=\left\{A_{

JSGF

JSGF stands for Java Speech Grammar Format or the JSpeech Grammar Format (in a W3C Note). Developed by Sun Microsystems, it is a textual representation of grammars for use in speech recognition for technologies like XHTML+Voice. JSGF adopts the style and conventions of the Java programming language in addition to use of traditional grammar notations. The Speech Recognition Grammar Specification was derived from this specification. == Example == The following JSGF grammar will recognize the words coffee, tea, and milk.

Neural gas

Neural gas is an artificial neural network, inspired by the self-organizing map and introduced in 1991 by Thomas Martinetz and Klaus Schulten. The neural gas is a simple algorithm for finding optimal data representations based on feature vectors. The algorithm was coined "neural gas" because of the dynamics of the feature vectors during the adaptation process, which distribute themselves like a gas within the data space. It is applied where data compression or vector quantization is an issue, for example speech recognition, image processing or pattern recognition. As a robustly converging alternative to the k-means clustering it is also used for cluster analysis. == Algorithm == Suppose we want to model a probability distribution P ( x ) {\displaystyle P(x)} of data vectors x {\displaystyle x} using a finite number of feature vectors w i {\displaystyle w_{i}} , where i = 1 , ⋯ , N {\displaystyle i=1,\cdots ,N} . For each time step t {\displaystyle t} Sample data vector x {\displaystyle x} from P ( x ) {\displaystyle P(x)} Compute the distance between x {\displaystyle x} and each feature vector. Rank the distances. Let i 0 {\displaystyle i_{0}} be the index of the closest feature vector, i 1 {\displaystyle i_{1}} the index of the second closest feature vector, and so on. Update each feature vector by: w i k t + 1 = w i k t + ε ⋅ e − k / λ ⋅ ( x − w i k t ) , k = 0 , ⋯ , N − 1 {\displaystyle w_{i_{k}}^{t+1}=w_{i_{k}}^{t}+\varepsilon \cdot e^{-k/\lambda }\cdot (x-w_{i_{k}}^{t}),k=0,\cdots ,N-1} In the algorithm, ε {\displaystyle \varepsilon } can be understood as the learning rate, and λ {\displaystyle \lambda } as the neighborhood range. ε {\displaystyle \varepsilon } and λ {\displaystyle \lambda } are reduced with increasing t {\displaystyle t} so that the algorithm converges after many adaptation steps. The adaptation step of the neural gas can be interpreted as gradient descent on a cost function. By adapting not only the closest feature vector but all of them with a step size decreasing with increasing distance order, compared to (online) k-means clustering a much more robust convergence of the algorithm can be achieved. The neural gas model does not delete a node and also does not create new nodes. === Comparison with SOM === Compared to self-organized map, the neural gas model does not assume that some vectors are neighbors. If two vectors happen to be close together, they would tend to move together, and if two vectors happen to be apart, they would tend to not move together. In contrast, in an SOM, if two vectors are neighbors in the underlying graph, then they will always tend to move together, no matter whether the two vectors happen to be neighbors in the Euclidean space. The name "neural gas" is because one can imagine it to be what an SOM would be like if there is no underlying graph, and all points are free to move without the bonds that bind them together. == Variants == A number of variants of the neural gas algorithm exists in the literature so as to mitigate some of its shortcomings. More notable is perhaps Bernd Fritzke's growing neural gas, but also one should mention further elaborations such as the Growing When Required network and also the incremental growing neural gas. A performance-oriented approach that avoids the risk of overfitting is the Plastic Neural gas model. === Growing neural gas === Fritzke describes the growing neural gas (GNG) as an incremental network model that learns topological relations by using a "Hebb-like learning rule", only, unlike the neural gas, it has no parameters that change over time and it is capable of continuous learning, i.e. learning on data streams. GNG has been widely used in several domains, demonstrating its capabilities for clustering data incrementally. The GNG is initialized with two randomly positioned nodes which are initially connected with a zero age edge and whose errors are set to 0. Since in the GNG input data is presented sequentially one by one, the following steps are followed at each iteration: It is calculating the errors (distances) between the two closest nodes to the current input data. The error of the winner node (only the closest one) is respectively accumulated. The winner node and its topological neighbors (connected by an edge) are moving towards the current input by different fractions of their respective errors. The age of all edges connected to the winner node are incremented. If the winner node and the second-winner are connected by an edge, such an edge is set to 0. Else, an edge is created between them. If there are edges with an age larger than a threshold, they are removed. Nodes without connections are eliminated. If the current iteration is an integer multiple of a predefined frequency-creation threshold, a new node is inserted between the node with the largest error (among all) and its topological neighbor presenting the highest error. The link between the former and the latter nodes is eliminated (their errors are decreased by a given factor) and the new node is connected to both of them. The error of the new node is initialized as the updated error of the node which had the largest error (among all). The accumulated error of all nodes is decreased by a given factor. If the stopping criterion is not met, the algorithm takes a following input. The criterion might be a given number of epochs, i.e., a pre-set number of times where all data is presented, or the reach of a maximum number of nodes. === Incremental growing neural gas === Another neural gas variant inspired by the GNG algorithm is the incremental growing neural gas (IGNG). The authors propose the main advantage of this algorithm to be "learning new data (plasticity) without degrading the previously trained network and forgetting the old input data (stability)." === Growing when required === Having a network with a growing set of nodes, like the one implemented by the GNG algorithm was seen as a great advantage, however some limitation on the learning was seen by the introduction of the parameter λ, in which the network would only be able to grow when iterations were a multiple of this parameter. The proposal to mitigate this problem was a new algorithm, the Growing When Required network (GWR), which would have the network grow more quickly, by adding nodes as quickly as possible whenever the network identified that the existing nodes would not describe the input well enough. === Plastic neural gas === The ability to only grow a network may quickly introduce overfitting; on the other hand, removing nodes on the basis of age only, as in the GNG model, does not ensure that the removed nodes are actually useless, because removal depends on a model parameter that should be carefully tuned to the "memory length" of the stream of input data. The "Plastic Neural Gas" model solves this problem by making decisions to add or remove nodes using an unsupervised version of cross-validation, which controls an equivalent notion of "generalization ability" for the unsupervised setting. While growing-only methods only cater for the incremental learning scenario, the ability to grow and shrink is suited to the more general streaming data problem. == Implementations == To find the ranking i 0 , i 1 , … , i N − 1 {\displaystyle i_{0},i_{1},\ldots ,i_{N-1}} of the feature vectors, the neural gas algorithm involves sorting, which is a procedure that does not lend itself easily to parallelization or implementation in analog hardware. However, implementations in both parallel software and analog hardware were actually designed.

T-distributed stochastic neighbor embedding

t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

Ordinal regression

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

WinFIG

WinFIG is a proprietary shareware vector graphics editor application. The file format and rendering are as close to Xfig as possible, but the program takes advantage of Windows features like clipboard, printer preview, multiple documents etc. As of 2011, WinFIG is under active development, with new features being added regularly. == History == The first release was in March 2003 and based on the Amiga program AmiFIG by the same author, which is also an Xfig compatible vector drawing application. WinFIG was not created by porting the Xfig source code to Windows. It is an independent implementation. Starting with release 4.0 WinFIG was ported from MFC to the Qt toolkit as the application framework and thereby enabling the first release of a Linux version. After Version 7.8 the Version scheme changes to years with version 2021.1. == Interface and usability == WinFIG is designed to provide a clear, efficient and convenient graphical user interface. It allows working on multiple documents using an MDI user interface and provides unlimited undo and redo of actions. == Features == === Object creation === The basic types of objects in WinFIG are: Open and closed Splines Ellipses Polylines and Polygons Texts LaTeX formatted texts Arcs Images: PNG, GIF, JPEG, EPS and more Compound objects, which are hierarchical compositions of objects Objects can have several attributes, which depend on the object type: Line width Line style Line cap style Line join style Arrows Outline color, fill color and fill pattern === Object manipulation === move copy scale rotate align add/delete points from lines or splines copy object attributes Numerical input of point coordinates === Exports === WinFIG can export into various formats: Raster formats: GIF, JPEG, PNG, PPM, XBM, XPM, PCX, TIFF, SLD Formats for printed documents: PostScript, PDF, LaTeX, HP-GL (printer control language used by Hewlett-Packard plotters), Vector graphics formats: EPS, SVG, PSTricks, TPIC, PIC, CGM, Metafont, MetaPost, EMF, Tk. === Miscellaneous === Winfig can handle smart links. A smart link is a moving connection from a source to a target object. It is established by connecting the end point of a line or spline to another object. The connecting line or spline segment follows the movements of the target object. Smart links are useful for diagrams, graphs etc. WinFIG can show a grid and provides several magnet modes for constraining editing operations to discrete coordinates. Objects can be organized in layers to control their Z-order. This is important to control overlapping of filled shapes. Object library: drawings can be stored in a special sub-folder in the program installation directory, which makes them available in the library dialog for easy reuse.

Memtransistor

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