AI Email Blueprint

AI Email Blueprint — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Secure coding

  Secure coding is the practice of developing computer software in such a way that guards against the accidental introduction of security vulnerabilities. Defects, bugs and logic flaws are consistently the primary cause of commonly exploited software vulnerabilities. Through the analysis of thousands of reported vulnerabilities, security professionals have discovered that most vulnerabilities stem from a relatively small number of common software programming errors. By identifying the insecure coding practices that lead to these errors and educating developers on secure alternatives, organizations can take proactive steps to help significantly reduce or eliminate vulnerabilities in software before deployment. Some scholars have suggested that in order to effectively confront threats related to cybersecurity, proper security should be coded or "baked in" to the systems. With security being designed into the software, this ensures that there will be protection against insider attacks and reduces the threat to application security. Implementing secure coding practices is part of the secure by design approach to security engineering. == Buffer-overflow prevention == Buffer overflows, a common software security vulnerability, happen when a process tries to store data beyond a fixed-length buffer. For example, if there are 8 slots to store items in, there will be a problem if there is an attempt to store 9 items. In computer memory the overflowed data may overwrite data in the next location which can result in a security vulnerability (stack smashing) or program termination (segmentation fault). An example of a C program prone to a buffer overflow is If the user input is larger than the destination buffer, a buffer overflow will occur. To fix this unsafe program, use strncpy to prevent a possible buffer overflow. Another secure alternative is to dynamically allocate memory on the heap using malloc. In the above code snippet, the program attempts to copy the contents of src into dst, while also checking the return value of malloc() to ensure that enough memory was able to be allocated for the destination buffer. == Format-string attack prevention == A Format String Attack is when a malicious user supplies specific inputs that will eventually be entered as an argument to a function that performs formatting, such as printf(). The attack involves the adversary reading from or writing to the stack. The C printf function writes output to stdout. If the parameter of the printf function is not properly formatted, several security bugs can be introduced. Below is a program that is vulnerable to a format string attack. A malicious argument passed to the program could be "%s%s%s%s%s%s%s", which can crash the program from improper memory reads. == Integer-overflow prevention == Integer overflow occurs when an arithmetic operation results in an integer too large to be represented within the available space. A program which does not properly check for integer overflow introduces potential software bugs and exploits. Below is a function in C++ which attempts to confirm that the sum of x and y is less than or equal to a defined value MAX: The problem with the code is it does not check for integer overflow on the addition operation. If the sum of x and y is greater than the maximum possible value of an unsigned int, the addition operation will overflow and perhaps result in a value less than or equal to MAX, even though the sum of x and y is greater than MAX. Below is a function which checks for overflow by confirming the sum is greater than or equal to both x and y. If the sum did overflow, the sum would be less than x or less than y. == Path traversal prevention == Path traversal is a vulnerability whereby paths provided from an untrusted source are interpreted in such a way that unauthorised file access is possible. For example, consider a script that fetches an article by taking a filename, which is then read by the script and parsed. Such a script might use the following hypothetical URL to retrieve an article about dog food: https://www.example.net/cgi-bin/article.sh?name=dogfood.html If the script has no input checking, instead trusting that the filename is always valid, a malicious user could forge a URL to retrieve configuration files from the web server: https://www.example.net/cgi-bin/article.sh?name=../../../../../etc/passwd Depending on the script, this may expose the /etc/passwd file, which on Unix-like systems contains (among others) user IDs, their login names, home directory paths and shells. (See SQL injection for a similar attack.) == Regulatory drivers == Secure coding practices are increasingly mandated by regulatory frameworks governing the development and maintenance of software systems that process sensitive data. The Health Insurance Portability and Accountability Act (HIPAA) Security Rule requires covered entities to protect the integrity of protected health information through technical safeguards under 45 CFR 164.312(c)(1) and to implement mechanisms to authenticate electronic protected health information under 45 CFR 164.312(c)(2). The Payment Card Industry Data Security Standard (PCI DSS) version 4.0 Requirement 6.2 mandates that custom software is developed securely, including training developers in secure coding techniques (6.2.2), reviewing custom code for vulnerabilities before release (6.2.3), and addressing common software attacks in development practices (6.2.4).

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  Bootstrap aggregating

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

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  Boosting (machine learning)

  In machine learning (ML), boosting is an ensemble learning method that combines a set of less accurate models (called "weak learners") to create a single, highly accurate model (a "strong learner"). Unlike other ensemble methods that build models in parallel (such as bagging), boosting algorithms build models sequentially. Each new model in the sequence is trained to correct the errors made by its predecessors. This iterative process allows the overall model to improve its accuracy, particularly by reducing bias. Boosting is a popular and effective technique used in supervised learning for both classification and regression tasks. The theoretical foundation for boosting came from a question posed by Kearns and Valiant (1988, 1989): "Can a set of weak learners create a single strong learner?" A weak learner is defined as a classifier that performs only slightly better than random guessing, whereas a strong learner is a classifier that is highly correlated with the true classification. Robert Schapire's affirmative answer to this question in a 1990 paper led to the development of practical boosting algorithms. The first such algorithm was developed by Schapire, with Freund and Schapire later developing AdaBoost, which remains a foundational example of boosting. == Algorithms == While boosting is not algorithmically constrained, most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. When they are added, they are weighted in a way that is related to the weak learners' accuracy. After a weak learner is added, the data weights are readjusted, known as "re-weighting". Misclassified input data gain a higher weight and examples that are classified correctly lose weight. Thus, future weak learners focus more on the examples that previous weak learners misclassified. There are many boosting algorithms. The original ones, proposed by Robert Schapire (a recursive majority gate formulation), and Yoav Freund (boost by majority), were not adaptive and could not take full advantage of the weak learners. Schapire and Freund then developed AdaBoost, an adaptive boosting algorithm that won the prestigious Gödel Prize. Only algorithms that are provable boosting algorithms in the probably approximately correct learning formulation can accurately be called boosting algorithms. Other algorithms that are similar in spirit to boosting algorithms are sometimes called "leveraging algorithms", although they are also sometimes incorrectly called boosting algorithms. The main variation between many boosting algorithms is their method of weighting training data points and hypotheses. AdaBoost is very popular and the most significant historically as it was the first algorithm that could adapt to the weak learners. It is often the basis of introductory coverage of boosting in university machine learning courses. There are many more recent algorithms such as LPBoost, TotalBoost, BrownBoost, xgboost, MadaBoost, LogitBoost, CatBoost and others. Many boosting algorithms fit into the AnyBoost framework, which shows that boosting performs gradient descent in a function space using a convex cost function. == Object categorization in computer vision == Given images containing various known objects in the world, a classifier can be learned from them to automatically classify the objects in future images. Simple classifiers built based on some image feature of the object tend to be weak in categorization performance. Using boosting methods for object categorization is a way to unify the weak classifiers in a special way to boost the overall ability of categorization. === Problem of object categorization === Object categorization is a typical task of computer vision that involves determining whether or not an image contains some specific category of object. The idea is closely related with recognition, identification, and detection. Appearance based object categorization typically contains feature extraction, learning a classifier, and applying the classifier to new examples. There are many ways to represent a category of objects, e.g. from shape analysis, bag of words models, or local descriptors such as SIFT, etc. Examples of supervised classifiers are Naive Bayes classifiers, support vector machines, mixtures of Gaussians, and neural networks. However, research has shown that object categories and their locations in images can be discovered in an unsupervised manner as well. === Status quo for object categorization === The recognition of object categories in images is a challenging problem in computer vision, especially when the number of categories is large. This is due to high intra class variability and the need for generalization across variations of objects within the same category. Objects within one category may look quite different. Even the same object may appear unalike under different viewpoint, scale, and illumination. Background clutter and partial occlusion add difficulties to recognition as well. Humans are able to recognize thousands of object types, whereas most of the existing object recognition systems are trained to recognize only a few, e.g. human faces, cars, simple objects, etc. Research has been very active on dealing with more categories and enabling incremental additions of new categories, and although the general problem remains unsolved, several multi-category objects detectors (for up to hundreds or thousands of categories) have been developed. One means is by feature sharing and boosting. === Boosting for binary categorization === AdaBoost can be used for face detection as an example of binary categorization. The two categories are faces versus background. The general algorithm is as follows: Form a large set of simple features Initialize weights for training images For T rounds Normalize the weights For available features from the set, train a classifier using a single feature and evaluate the training error Choose the classifier with the lowest error Update the weights of the training images: increase if classified wrongly by this classifier, decrease if correctly Form the final strong classifier as the linear combination of the T classifiers (coefficient larger if training error is small) After boosting, a classifier constructed from 200 features could yield a 95% detection rate under a 10 − 5 {\displaystyle 10^{-5}} false positive rate. Another application of boosting for binary categorization is a system that detects pedestrians using patterns of motion and appearance. This work is the first to combine both motion information and appearance information as features to detect a walking person. It takes a similar approach to the Viola-Jones object detection framework. === Boosting for multi-class categorization === Compared with binary categorization, multi-class categorization looks for common features that can be shared across the categories at the same time. They turn to be more generic edge like features. During learning, the detectors for each category can be trained jointly. Compared with training separately, it generalizes better, needs less training data, and requires fewer features to achieve the same performance. The main flow of the algorithm is similar to the binary case. What is different is that a measure of the joint training error shall be defined in advance. During each iteration the algorithm chooses a classifier of a single feature (features that can be shared by more categories shall be encouraged). This can be done via converting multi-class classification into a binary one (a set of categories versus the rest), or by introducing a penalty error from the categories that do not have the feature of the classifier. In the paper "Sharing visual features for multiclass and multiview object detection", A. Torralba et al. used GentleBoost for boosting and showed that when training data is limited, learning via sharing features does a much better job than no sharing, given same boosting rounds. Also, for a given performance level, the total number of features required (and therefore the run time cost of the classifier) for the feature sharing detectors, is observed to scale approximately logarithmically with the number of class, i.e., slower than linear growth in the non-sharing case. Similar results are shown in the paper "Incremental learning of object detectors using a visual shape alphabet", yet the authors used AdaBoost for boosting. == Convex vs. non-convex boosting algorithms == Boosting algorithms can be based on convex or non-convex optimization algorithms. Convex algorithms, such as AdaBoost and LogitBoost, can be "defeated" by random noise such that they can't learn basic and learnable combinations of weak hypotheses. This limitation was pointed out by Long & Servedio in 2008. However, by 2009, multiple authors demonstrated that boosting algorithms based on non-convex optimization, such as BrownBoost, can learn from nois

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  Multiclass classification

  In machine learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three or more classes (classifying instances into one of two classes is called binary classification). For example, deciding on whether an image is showing a banana, peach, orange, or an apple is a multiclass classification problem, with four possible classes (banana, peach, orange, apple), while deciding on whether an image contains an apple or not is a binary classification problem (with the two possible classes being: apple, no apple). While many classification algorithms (e.g., decision trees, k-NN, neural networks and multinomial logistic regression) naturally permit the use of more than two classes, some are by nature binary algorithms (e.g., classical binary support vector machine) and require decomposition strategies such as one-vs-all, one-vs-one, or ECOC to solve multiclass problems. Multiclass classification should not be confused with multi-label classification, where multiple labels are to be predicted for each instance (e.g., predicting that an image contains both an apple and an orange, in the previous example). == Better-than-random multiclass models == From the confusion matrix of a multiclass model, we can determine whether a model does better than chance. Let K ≥ 3 {\displaystyle K\geq 3} be the number of classes, O {\displaystyle {\mathcal {O}}} a set of observations, y ^ : O → { 1 , . . . , K } {\displaystyle {\hat {y}}:{\mathcal {O}}\to \{1,...,K\}} a model of the target variable y : O → { 1 , . . . , K } {\displaystyle y:{\mathcal {O}}\to \{1,...,K\}} and n i , j {\displaystyle n_{i,j}} be the number of observations in the set { y = i } ∩ { y ^ = j } {\displaystyle \{y=i\}\cap \{{\hat {y}}=j\}} . We note n i . = ∑ j n i , j {\displaystyle n_{i.}=\sum _{j}n_{i,j}} , n . j = ∑ i n i , j {\displaystyle n_{.j}=\sum _{i}n_{i,j}} , n = ∑ j n . j = ∑ i n i . {\displaystyle n=\sum _{j}n_{.j}=\sum _{i}n_{i.}} , λ i = n i . n {\displaystyle \lambda _{i}={\frac {n_{i.}}{n}}} and μ j = n . j n {\displaystyle \mu _{j}={\frac {n_{.j}}{n}}} . It is assumed that the confusion matrix ( n i , j ) i , j {\displaystyle (n_{i,j})_{i,j}} contains at least one non-zero entry in each row, that is λ i > 0 {\displaystyle \lambda _{i}>0} for any i {\displaystyle i} . Finally we call "normalized confusion matrix" the matrix of conditional probabilities ( P ( y ^ = j ∣ y = i ) ) i , j = ( n i , j n i . ) i , j {\displaystyle (\mathbb {P} ({\hat {y}}=j\mid y=i))_{i,j}=\left({\frac {n_{i,j}}{n_{i.}}}\right)_{i,j}} . === Intuitive explanation === The lift is a way of measuring the deviation from independence of two events A {\displaystyle A} and B {\displaystyle B} : L i f t ( A , B ) = P ( A ∩ B ) P ( A ) P ( B ) = P ( A ∣ B ) P ( A ) = P ( B ∣ A ) P ( B ) {\displaystyle \mathrm {Lift} (A,B)={\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (A)\mathbb {P} (B)}}={\frac {\mathbb {P} (A\mid B)}{\mathbb {P} (A)}}={\frac {\mathbb {P} (B\mid A)}{\mathbb {P} (B)}}} We have L i f t ( A , B ) > 1 {\displaystyle \mathrm {Lift} (A,B)>1} if and only if events A {\displaystyle A} and B {\displaystyle B} occur simultaneously with a greater probability than if they were independent. In other words, if one of the two events occurs, the probability of observing the other event increases. A first condition to satisfy is to have L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} . And the quality of a model (better or worse than chance) does not change if we over- or undersample the dataset, that is if we multiply each row R i {\displaystyle R_{i}} of the confusion matrix by a constant c i {\displaystyle c_{i}} . Thus the second condition is that the necessary and sufficient conditions for doing better than chance need only depend on the normalized confusion matrix. The condition on lifts can be reformulated with One versus Rest binary models : for any i {\displaystyle i} , we define the binary target variable y i {\displaystyle y_{i}} which is the indicator of event { y = i } {\displaystyle \{y=i\}} , and the binary model y ^ i {\displaystyle {\hat {y}}_{i}} of y i {\displaystyle y_{i}} which is the indicator of event { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} . Each of the y ^ i {\displaystyle {\hat {y}}_{i}} models is a "One versus Rest" model. L i f t ( y = i , y ^ = i ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)} only depends on the events { y = i } {\displaystyle \{y=i\}} and { y ^ = i } {\displaystyle \{{\hat {y}}=i\}} , so merging or not merging the other classes doesn't change its value. We therefore have L i f t ( y = i , y ^ = i ) = L i f t ( y i = 1 , y ^ i = 1 ) {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)=\mathrm {Lift} (y_{i}=1,{\hat {y}}_{i}=1)} and the first condition is that all binary One versus Rest models are better than chance. ==== Example ==== If K = 2 {\displaystyle K=2} and 2 is the class of interest , the normalized confusion matrix is ( s p e c i f i c i t y 1 − s p e c i f i c i t y 1 − s e n s i t i v i t y s e n s i t i v i t y ) {\displaystyle {\begin{pmatrix}\mathrm {specificity} &1-\mathrm {specificity} \\1-\mathrm {sensitivity} &\mathrm {sensitivity} \end{pmatrix}}} and we have L i f t ( y = 1 , y ^ = 1 ) − 1 = P ( y = y ^ = 1 ) λ 1 μ 1 − 1 = n 1 , 1 n n 1. n .1 − 1 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)-1={\frac {\mathbb {P} (y={\hat {y}}=1)}{\lambda _{1}\mu _{1}}}-1={\frac {n_{1,1}n}{n_{1.}n_{.1}}}-1} = n 1 , 1 ( n 1 , 1 + n 1 , 2 + n 2 , 1 + n 2 , 2 ) − ( n 1 , 1 + n 1 , 2 ) ( n 1 , 1 + n 2 , 1 ) n 1. n .1 = n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 n 1. n .1 {\displaystyle ={\frac {n_{1,1}(n_{1,1}+n_{1,2}+n_{2,1}+n_{2,2})-(n_{1,1}+n_{1,2})(n_{1,1}+n_{2,1})}{n_{1.}n_{.1}}}={\frac {n_{1,1}n_{2,2}-n_{1,2}n_{2,1}}{n_{1.}n_{.1}}}} . Thus L i f t ( y = 1 , y ^ = 1 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=1,{\hat {y}}=1)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Similarly, by swapping the roles of 1 and 2, we find that L i f t ( y = 2 , y ^ = 2 ) ≥ 1 ⟺ n 1 , 1 n 2 , 2 − n 1 , 2 n 2 , 1 ≥ 0 {\displaystyle \mathrm {Lift} (y=2,{\hat {y}}=2)\geq 1\iff n_{1,1}n_{2,2}-n_{1,2}n_{2,1}\geq 0} . Dividing by n 1. n 2. {\displaystyle n_{1.}n_{2.}} we find that the necessary and sufficient condition on the normalized confusion matrix is s e n s i t i v i t y s p e c i f i c i t y − ( 1 − s e n s i t i v i t y ) ( 1 − s p e c i f i c i t y ) ≥ 0 ⟺ s e n s i t i v i t y + s p e c i f i c i t y − 1 ≥ 0 ⟺ J ≥ 0 {\displaystyle \mathrm {sensitivity} \ \mathrm {specificity} -(1-\mathrm {sensitivity} )(1-\mathrm {specificity} )\geq 0\iff \mathrm {sensitivity} +\mathrm {specificity} -1\geq 0\iff J\geq 0} . This brings us back to the classical binary condition: Youden's J must be positive (or zero for random models). === Random models === A random model is a model that is independent of the target variable. This property is easily reformulated with the confusion matrix. This proposition shows that the model y ^ {\displaystyle {\hat {y}}} of y {\displaystyle y} is uninformative if and only if there are two families of numbers ( α i ) i {\displaystyle (\alpha _{i})_{i}} and ( β j ) j {\displaystyle (\beta _{j})_{j}} such that P ( { y = i } ∩ { y ^ = j } ) = α i β j {\displaystyle \mathbb {P} (\{y=i\}\cap \{{\hat {y}}=j\})=\alpha _{i}\beta _{j}} for any i {\displaystyle i} and j {\displaystyle j} . === Multiclass likelihood ratios and diagnostic odds ratios === We define generalized likelihood ratios calculated from the normalized confusion matrix: for any i {\displaystyle i} and j ≠ i {\displaystyle j\not =i} , let L R i , j = P ( y ^ = j ∣ y = j ) P ( y ^ = j ∣ y = i ) {\displaystyle \mathrm {LR} _{i,j}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)}}} . When K = 2 {\displaystyle K=2} , if 2 is the class of interest,, we find the classical likelihood ratios L R 1 , 2 = L R + {\displaystyle \mathrm {LR} _{1,2}=\mathrm {LR} _{+}} and L R 2 , 1 = 1 L R − {\displaystyle \mathrm {LR} _{2,1}={\frac {1}{\mathrm {LR} _{-}}}} . Multiclass diagnostic odds ratios can also be defined using the formula D O R i , j = D O R j , i = L R i , j L R j , i = n i , i n j , j n i , j n j , i = P ( y ^ = j ∣ y = j ) / P ( y ^ = i ∣ y = j ) P ( y ^ = j ∣ y = i ) / P ( y ^ = i ∣ y = i ) {\displaystyle \mathrm {DOR} _{i,j}=\mathrm {DOR} _{j,i}=\mathrm {LR} _{i,j}\mathrm {LR} _{j,i}={\frac {n_{i,i}n_{j,j}}{n_{i,j}n_{j,i}}}={\frac {\mathbb {P} ({\hat {y}}=j\mid y=j)/\mathbb {P} ({\hat {y}}=i\mid y=j)}{\mathbb {P} ({\hat {y}}=j\mid y=i)/\mathbb {P} ({\hat {y}}=i\mid y=i)}}} We saw above that a better-than-chance model (or a random model) must verify L i f t ( y = i , y ^ = i ) ≥ 1 {\displaystyle \mathrm {Lift} (y=i,{\hat {y}}=i)\geq 1} for any i {\displaystyle i} and λ i {\displaystyle \lambda _{i}} . According to the previous corollary, likelihood ratios are thus greater

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  ALL-IN-1

  ALL-IN-1 was an office automation product developed and sold by Digital Equipment Corporation in the 1980s. It was one of the first purchasable off the shelf electronic mail products. It was later known as Office Server V3.2 for OpenVMS Alpha and OpenVMS VAX systems before being discontinued. == Overview == ALL-IN-1 was advertised as an office automation system including functionality in Electronic Messaging, Word Processing and Time Management. It offered an application development platform and customization capabilities that ranged from scripting to code-level integration. ALL-IN-1 was designed and developed by Skip Walter, John Churin and Marty Skinner from Digital Equipment Corporation who began work in 1977. Sheila Chance was hired as the software engineering manager in 1981. The first version of the software, called CP/OSS, the Charlotte Package of Office System Services, named after the location of the developers, was released in May 1982. In 1983, the product was renamed ALL-IN-1 and the Charlotte group continued to develop versions 1.1 through 1.3. Digital then made the decision to move most of the development activity to its central engineering facility in Reading, United Kingdom, where a group there took responsibility for the product from version 2.0 (released in field test in 1984 and to customers in 1985) onward. The Charlotte group continued to work on the Time Management subsystem until version 2.3 and other contributions were made from groups based in Sophia Antipolis, France (System for Customization Management and the integration with VAX Notes), Reading (Message Router and MAILbus), and Nashua, New Hampshire (FMS). ALL-IN-1 V3.0 introduced shared file cabinets and the File Cabinet Server (FCS) to lay the foundation for an eventual integration with TeamLinks, Digital's PC office client. Previous integrations with PCs included PC ALL-IN-1, a DOS-based product introduced in 1989 that never proved popular with customers. Bob Wyman was the first product manager. He oversaw the growth of the product culminating in over $2 billion per year in revenue and market leadership in the proprietary office automation sector. Other consultants from Digital Equipment Corporation involved include Frank Nicodem, Donald Vickers and Tony Redmond.

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  Probabilistic latent semantic analysis

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

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  Apache Giraph

  Apache Giraph is an Apache project to perform graph processing on big data. Giraph utilizes Apache Hadoop's MapReduce implementation to process graphs. Facebook used Giraph with some performance improvements to analyze one trillion edges using 200 machines in 4 minutes. Giraph is based on a paper published by Google about its own graph processing system called Pregel. It can be compared to other Big Graph processing libraries such as Cassovary. As of September 2023, it is no longer actively developed.

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →

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  Adobe After Effects

  Adobe After Effects is a digital effects, motion graphics, and compositing application developed by Adobe Inc.; it is used for animation and in the post-production process of film making, video games and television production. Among other things, After Effects can be used for keying, tracking, compositing, and animation. It also functions as a very basic non-linear editor, audio editor, and media transcoder. In 2019, the program won an Academy Award for scientific and technical achievement. == History == After Effects was originally created by David Herbstman, David Simons, Daniel Wilk, David M. Cotter, and Russell Belfer at the Company of Science and Art in Providence, Rhode Island. The first two versions of the software, 1.0 (January 1993) and 1.1, were released there by the company. CoSA with After Effects was acquired by Aldus Corporation in July 1993, which in turn was acquired by Adobe in 1994. Adobe acquired PageMaker as well. Adobe's first new release of After Effects was version 3.0. == Third-party integrations == After Effects functionality can be extended through a variety of third-party integrations. The most common integrations are: plug-ins, scripts, and extensions. === Plug-ins === Plug-ins are predominantly written in C or C++ and extend the functionality of After Effects, allowing for more advanced features such as particle systems, physics engines, 3D effects, and the ability to bridge the gap between After Effects and another. === Scripts === After Effects Scripts are a series of commands written in both JavaScript and the ExtendScript language. After Effects Scripts, unlike plug-ins, can only access the core functionality of After Effects. Scripts are often developed to automate repetitive tasks, to simplify complex After Effects features, or to perform complex calculations that would otherwise take a long time to complete. Scripts can also use some functionality not directly exposed through the graphical user interface. === Extensions === After Effects Extensions offer the ability to extend After Effects functionality through modern web development technologies like HTML5, and Node.js, without the need for C++. After Effects Extensions make use of Adobe's Common Extensibility Platform or CEP Panels, which means they can be built to interact with other Adobe CC apps.

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  Witness set

  In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 ⁡ | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).

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  Mating pool

  Mating pool is a concept used in evolutionary algorithms and means a population of parents for the next population. The mating pool is formed by candidate solutions that the selection operators deem to have the highest fitness in the current population. Solutions that are included in the mating pool are referred to as parents. Individual solutions can be repeatedly included in the mating pool, with individuals of higher fitness values having a higher chance of being included multiple times. Crossover operators are then applied to the parents, resulting in recombination of genes recognized as superior. Lastly, random changes in the genes are introduced through mutation operators, increasing the genetic variation in the gene pool. Those two operators improve the chance of creating new, superior solutions. A new generation of solutions is thereby created, the children, who will constitute the next population. Depending on the selection method, the total number of parents in the mating pool can be different to the size of the initial population, resulting in a new population that’s smaller. To continue the algorithm with an equally sized population, random individuals from the old populations can be chosen and added to the new population. At this point, the fitness value of the new solutions is evaluated. If the termination conditions are fulfilled, processes come to an end. Otherwise, they are repeated. The repetition of the steps result in candidate solutions that evolve towards the most optimal solution over time. The genes will become increasingly uniform towards the most optimal gene, a process called convergence. If 95% of the population share the same version of a gene, the gene has converged. When all the individual fitness values have reached the value of the best individual, i.e. all the genes have converged, population convergence is achieved. == Mating pool creation == Several methods can be applied to create a mating pool. All of these processes involve the selective breeding of a particular number of individuals within a population. There are multiple criteria that can be employed to determine which individuals make it into the mating pool and which are left behind. The selection methods can be split into three general types: fitness proportionate selection, ordinal based selection and threshold based selection. === Fitness proportionate selection === In the case of fitness proportionate selection, random individuals are selected to enter the pool. However, the ones with a higher level of fitness are more likely to be picked and therefore have a greater chance of passing on their features to the next generation. One of the techniques used in this type of parental selection is the roulette wheel selection. This approach divides a hypothetical circular wheel into different slots, the size of which is equal to the fitness values of each potential candidate. Afterwards, the wheel is rotated and a fixed point determines which individual gets picked. The greater the fitness value of an individual, the higher the probability of being chosen as a parent by the random spin of the wheel. Alternatively, stochastic universal sampling can be implemented. This selection method is also based on the rotation of a spinning wheel. However, in this case there is more than one fixed point and as a result all of the mating pool members will be selected simultaneously. === Ordinal based selection === The ordinal based selection methods include the tournament and ranking selection. Tournament selection involves the random selection of individuals of a population and the subsequent comparison of their fitness levels. The winners of these “tournaments” are the ones with the highest values and will be put into the mating pool as parents. In ranking selection all the individuals are sorted based on their fitness values. Then, the selection of the parents is made according to the rank of the candidates. Every individual has a chance of being chosen, but higher ranked ones are favored === Threshold based selection === The last type of selection method is referred to as the threshold based method. This includes the truncation selection method, which sorts individuals based on their phenotypic values on a specific trait and later selects the proportion of them that are within a certain threshold as parents.

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  Bondy's theorem

  In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. == Statement == The theorem is as follows: Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct. In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct. == Example == Consider the 4 × 4 matrix [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] {\displaystyle {\begin{bmatrix}1&1&0&1\\0&1&0&1\\0&0&1&1\\0&1&1&0\end{bmatrix}}} where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix [ 1 0 1 1 0 1 0 1 1 1 1 0 ] {\displaystyle {\begin{bmatrix}1&0&1\\1&0&1\\0&1&1\\1&1&0\end{bmatrix}}} no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix [ 1 1 1 0 1 1 0 0 1 0 1 0 ] {\displaystyle {\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\\0&1&0\end{bmatrix}}} are distinct. Another possibility would have been deleting the fourth column. == Learning theory application == From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows: Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C. This implies that every finite concept class C has its teaching dimension bounded by |C| − 1.

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  NationBuilder

  NationBuilder is a Los Angeles-based technology start-up that develops content management and customer relationship management (CRM) software. Although the company initially targeted political campaigns and nonprofit organizations, it later expanded its marketing efforts to include other people and organizations trying to build an online following, such as artists, musicians and restaurants. The software uses voter data such as names, addresses and other information, such as previous voting records in the case of political campaigns, to allow users to centralize, build and manage campaigns by integrating various communication tools like websites, newsletters, text messaging and social media channels under one platform. Among other features, the software enables users to quickly create websites, build databases through registrations, send targeted newsletters, analyse data from multiple sources and leverage micro-donations. The software's appeal towards political campaigns comes from the combination of a number of previously separate campaigning services, channels and data sources into a single platform that was presented as a facile solution for non-technical users and which enabled political campaigners to quickly deploy campaigns by convincing numerous people to donate. == History == NationBuilder was founded in 2009 in Los Angeles by Jim Gilliam and launched in 2011. In 2012 Joe Green joined NationBuilder as co-founder and president. He left that role 11 months later in February 2013. Gilliam was previously a movie-maker who co-founded Brave New Films with Robert Greenwald and had sought funding for his films through crowd-sourcing. Green, who studied organizing at Harvard and was Mark Zuckerberg's roommate, is also the co-founder of the Causes Facebook app; he left NationBuilder in 2013. Since its founding, the company has helped campaigns raise $1.2 billion. In 2012, NationBuilder announced that 1,000 subscribers have used its software to amass 2.5 million supporters and raise $12 million in campaign donations. In 2015 it has helped raise $264 million, recruit over one million volunteers and coordinate some 129,000 events. By 2016, the company said its software was used by about 40 percent of all contested elections at the state and national level in the U.S., which included 3,000 political campaigns. Using such software is easier in the U.S. than Europe, where comprehensive data protection and privacy laws are in effect since 2018. The Scottish National Party was the first political party to use NationBuilder, harvesting vast amounts of data pertaining to voter activity via websites such as Facebook and Twitter. This revelation prompted outrage over privacy concerns. Guy Herbert of the No2ID campaign called the use of such data harvesting tools by the SNP "utterly hypocritical". == Funding == Investors in NationBuilder include Chris Hughes - the Facebook co-founder, Sean Parker - first president of Facebook and co-founder of Napster and Causes, Dan Senor - the former Republican foreign-policy adviser and Ben Horowitz, co-founder of Andreessen Horowitz. In 2012, it has raised $6.3 million in funding from a number of investors. == Notable implementations == The software is reported to have played a role in some public elections in Europe, the US and New Zealand, as well as non-profit initiatives, and political parties in Australia. Notable users include Bernie Sanders, Mitch McConnell, Andrew Yang, Theresa May, Amnesty International, the NAACP and Donald Trump. === France === La République En Marche used NationBuilder in their campaign for the 2017 National Assembly. === New Zealand === NationBuilder's services are used by New Zealand political parties, including in the campaigns of both the National and Labour parties in the 2017 general election. === United Kingdom === Despite stricter data protection and privacy laws in the UK and EU, NationBuilder was used to significant impact in a number of UK elections, most notably in the 2016 campaign for withdrawal of the United Kingdom from the European Union. The company later made a public announcement that both sides in that campaign had used its software. === United States === NationBuilder was used in the Donald Trump presidential campaign to advance his election efforts and eventually win the 2016 presidential race. Jill Stein of the Green Party, Republican Rick Santorum, and independent supporters of various candidates all used NationBuilder during their 2016 runs for president. During the 2018 US election cycle, political entities paid more than $1 million for the use of NationBuilder. Among the entities paying the most were Donald J. Trump for President, Prosperity Action and the Republican Party of Tennessee.

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  Loss function

  In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss. == Examples == === Regret === Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known. === Quadratic loss function === The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t {\displaystyle t} , then a quadratic loss function is λ ( x ) = C ( t − x ) 2 {\displaystyle \lambda (x)=C(t-x)^{2}\;} for some constant C {\displaystyle C} ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL). Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, log-cosh and SMAE losses are used when the data has many large outliers. === 0-1 loss function === In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = { 0 if y = y ^ 1 if y ≠ y ^ {\displaystyle L({\hat {y}},y)={\begin{cases}0&{\text{if }}y={\hat {y}}\\1&{\text{if }}y\neq {\hat {y}}\end{cases}}} In information theory, this loss function is known as Hamming distortion. == Constructing loss and objective functions == In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions. == Expected loss == In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X {\displaystyle X} . === Statistics === Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. ==== Frequentist expected loss ==== We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle \delta } and the parameter θ {\displaystyle \theta } . Here the decision rule depends on the outcome of X {\displaystyle X} . The risk function is given by: R ( θ , δ ) = E θ ⁡ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).} Here, θ {\displaystyle \theta } is a fixed but possibly unknown state of nature, X {\displaystyle X} is a vector of observations stochastically drawn from a population, E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X {\displaystyle X} , d P θ {\displaystyle \mathrm {d} P_{\theta }} is a probability measure over the event space of X {\displaystyle X} (parametrized by θ {\displaystyle \theta } ) and the integral is evaluated over the entire support of X {\displaystyle X} . ==== Bayes Risk ==== In a Bayesian approach, the expectation is calculated using the prior distribution π ∗ {\displaystyle \pi ^{}} of the parameter θ {\displaystyle \theta } : ρ ( π ∗ , a ) = ∫ Θ ∫ X L ( θ , a ( x ) ) d P ( x | θ ) d π ∗ ( θ ) = ∫ X ∫ Θ L ( θ , a ( x ) ) d π ∗ ( θ | x ) d M ( x ) {\displaystyle \rho (\pi ^{},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})} where M ( x ) {\displaystyle M(\mathbf {x} )} is known as the predictive likelihood wherein θ {\displaystyle \theta } has been "integrated out," π ∗ ( θ | x ) {\displaystyle \pi ^{}(\theta |\mathbf {x} )} is the posterior distribution, and the order of integration has been changed. One then should choose the action a ∗ {\displaystyle a^{}} which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside d x {\displaystyle \mathrm {d} x} is known as the Posterior Risk, and minimising it with respect to decision a {\displaystyle a} also minimizes the overall Bayes Risk. This optimal decision, a ∗ {\displaystyle a^{}} is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ {\displaystyle \theta } , over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ {\displaystyle \theta } . ==== Examples in statistics ==== For a scalar parameter θ {\displaystyle \theta } , a decision function whose output θ ^ {\displaystyle {\hat {\theta }}} is an estimate of θ {\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the estimate, R ( θ , θ ^ ) = E θ ⁡ [ ( θ − θ ^ ) 2 ] . {\displaystyle R(\theta ,{\hat {\thet

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  Promoter based genetic algorithm

  The promoter based genetic algorithm (PBGA) is a genetic algorithm for neuroevolution developed by F. Bellas and R.J. Duro in the Integrated Group for Engineering Research (GII) at the University of Coruña, in Spain. It evolves variable size feedforward artificial neural networks (ANN) that are encoded into sequences of genes for constructing a basic ANN unit. Each of these blocks is preceded by a gene promoter acting as an on/off switch that determines if that particular unit will be expressed or not. == PBGA basics == The basic unit in the PBGA is a neuron with all of its inbound connections as represented in the following figure: The genotype of a basic unit is a set of real valued weights followed by the parameters of the neuron and proceeded by an integer valued field that determines the promoter gene value and, consequently, the expression of the unit. By concatenating units of this type we can construct the whole network. With this encoding it is imposed that the information that is not expressed is still carried by the genotype in evolution but it is shielded from direct selective pressure, maintaining this way the diversity in the population, which has been a design premise for this algorithm. Therefore, a clear difference is established between the search space and the solution space, permitting information learned and encoded into the genotypic representation to be preserved by disabling promoter genes. == Results == The PBGA was originally presented within the field of autonomous robotics, in particular in the real time learning of environment models of the robot. It has been used inside the Multilevel Darwinist Brain (MDB) cognitive mechanism developed in the GII for real robots on-line learning. In another paper it is shown how the application of the PBGA together with an external memory that stores the successful obtained world models, is an optimal strategy for adaptation in dynamic environments. Recently, the PBGA has provided results that outperform other neuroevolutionary algorithms in non-stationary problems, where the fitness function varies in time.

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