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Nearest neighbor search : Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time.
Nearest neighbor search : There are numerous variants of the NNS problem and the two most well-known are the k-nearest neighbor search and the ε-approximate nearest neighbor search.
Nearest neighbor search : Shasha, Dennis (2004). High Performance Discovery in Time Series. Berlin: Springer. ISBN 978-0-387-00857-8.
Nearest neighbor search : Nearest Neighbors and Similarity Search – a website dedicated to educational materials, software, literature, researchers, open problems and events related to NN searching. Maintained by Yury Lifshits Similarity Search Wiki – a collection of links, people, ideas, keywords, papers, slides, code and data sets on nearest neighbours
Linear discriminant analysis : Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis of variance (ANOVA) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable, whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is, as they also explain a categorical variable by the values of continuous independent variables. These other methods are preferable in applications where it is not reasonable to assume that the independent variables are normally distributed, which is a fundamental assumption of the LDA method. LDA is also closely related to principal component analysis (PCA) and factor analysis in that they both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA, in contrast, does not take into account any difference in class, and factor analysis builds the feature combinations based on differences rather than similarities. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made. LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is discriminant correspondence analysis. Discriminant analysis is used when groups are known a priori (unlike in cluster analysis). Each case must have a score on one or more quantitative predictor measures, and a score on a group measure. In simple terms, discriminant function analysis is classification - the act of distributing things into groups, classes or categories of the same type.
Linear discriminant analysis : The original dichotomous discriminant analysis was developed by Sir Ronald Fisher in 1936. It is different from an ANOVA or MANOVA, which is used to predict one (ANOVA) or multiple (MANOVA) continuous dependent variables by one or more independent categorical variables. Discriminant function analysis is useful in determining whether a set of variables is effective in predicting category membership.
Linear discriminant analysis : Consider a set of observations x → (also called features, attributes, variables or measurements) for each sample of an object or event with known class y . This set of samples is called the training set in a supervised learning context. The classification problem is then to find a good predictor for the class y of any sample of the same distribution (not necessarily from the training set) given only an observation x → .: 338 LDA approaches the problem by assuming that the conditional probability density functions p ( x → \| y = 0 ) \|y=0) and p ( x → \| y = 1 ) \|y=1) are both the normal distribution with mean and covariance parameters ( μ → 0 , Σ 0 ) _,\Sigma _\right) and ( μ → 1 , Σ 1 ) _,\Sigma _\right) , respectively. Under this assumption, the Bayes-optimal solution is to predict points as being from the second class if the log of the likelihood ratios is bigger than some threshold T, so that: 1 2 ( x → − μ → 0 ) T Σ 0 − 1 ( x → − μ → 0 ) + 1 2 ln ⁡ \| Σ 0 \| − 1 2 ( x → − μ → 1 ) T Σ 1 − 1 ( x → − μ → 1 ) − 1 2 ln ⁡ \| Σ 1 \| > T (-_)^ \Sigma _^(-_)+\ln \|\Sigma _\|-(-_)^ \Sigma _^(-_)-\ln \|\Sigma _\|\ >\ T Without any further assumptions, the resulting classifier is referred to as quadratic discriminant analysis (QDA). LDA instead makes the additional simplifying homoscedasticity assumption (i.e. that the class covariances are identical, so Σ 0 = Σ 1 = Σ =\Sigma _=\Sigma ) and that the covariances have full rank. In this case, several terms cancel: x → T Σ 0 − 1 x → = x → T Σ 1 − 1 x → ^ \Sigma _^=^ \Sigma _^ x → T Σ i − 1 μ → i = μ → i T Σ i − 1 x → ^ ^_=_^ ^ because Σ i is Hermitian and the above decision criterion becomes a threshold on the dot product w → T x → > c ^ >c for some threshold constant c, where w → = Σ − 1 ( μ → 1 − μ → 0 ) =\Sigma ^(_-_) c = 1 2 w → T ( μ → 1 + μ → 0 ) \,^ (_+_) This means that the criterion of an input x → being in a class y is purely a function of this linear combination of the known observations. It is often useful to see this conclusion in geometrical terms: the criterion of an input x → being in a class y is purely a function of projection of multidimensional-space point x → onto vector w → (thus, we only consider its direction). In other words, the observation belongs to y if corresponding x → is located on a certain side of a hyperplane perpendicular to w → . The location of the plane is defined by the threshold c .
Linear discriminant analysis : The assumptions of discriminant analysis are the same as those for MANOVA. The analysis is quite sensitive to outliers and the size of the smallest group must be larger than the number of predictor variables. Multivariate normality: Independent variables are normal for each level of the grouping variable. Homogeneity of variance/covariance (homoscedasticity): Variances among group variables are the same across levels of predictors. Can be tested with Box's M statistic. It has been suggested, however, that linear discriminant analysis be used when covariances are equal, and that quadratic discriminant analysis may be used when covariances are not equal. Independence: Participants are assumed to be randomly sampled, and a participant's score on one variable is assumed to be independent of scores on that variable for all other participants. It has been suggested that discriminant analysis is relatively robust to slight violations of these assumptions, and it has also been shown that discriminant analysis may still be reliable when using dichotomous variables (where multivariate normality is often violated).
Linear discriminant analysis : Discriminant analysis works by creating one or more linear combinations of predictors, creating a new latent variable for each function. These functions are called discriminant functions. The number of functions possible is either N g − 1 -1 where N g = number of groups, or p (the number of predictors), whichever is smaller. The first function created maximizes the differences between groups on that function. The second function maximizes differences on that function, but also must not be correlated with the previous function. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions. Given group j , with R j _ sets of sample space, there is a discriminant rule such that if x ∈ R j _ , then x ∈ j . Discriminant analysis then, finds “good” regions of R j _ to minimize classification error, therefore leading to a high percent correct classified in the classification table. Each function is given a discriminant score to determine how well it predicts group placement. Structure Correlation Coefficients: The correlation between each predictor and the discriminant score of each function. This is a zero-order correlation (i.e., not corrected for the other predictors). Standardized Coefficients: Each predictor's weight in the linear combination that is the discriminant function. Like in a regression equation, these coefficients are partial (i.e., corrected for the other predictors). Indicates the unique contribution of each predictor in predicting group assignment. Functions at Group Centroids: Mean discriminant scores for each grouping variable are given for each function. The farther apart the means are, the less error there will be in classification.
Linear discriminant analysis : Maximum likelihood: Assigns x to the group that maximizes population (group) density. Bayes Discriminant Rule: Assigns x to the group that maximizes π i f i ( x ) f_(x) , where πi represents the prior probability of that classification, and f i ( x ) (x) represents the population density. Fisher's linear discriminant rule: Maximizes the ratio between SSbetween and SSwithin, and finds a linear combination of the predictors to predict group.
Linear discriminant analysis : An eigenvalue in discriminant analysis is the characteristic root of each function. It is an indication of how well that function differentiates the groups, where the larger the eigenvalue, the better the function differentiates. This however, should be interpreted with caution, as eigenvalues have no upper limit. The eigenvalue can be viewed as a ratio of SSbetween and SSwithin as in ANOVA when the dependent variable is the discriminant function, and the groups are the levels of the IV. This means that the largest eigenvalue is associated with the first function, the second largest with the second, etc..
Linear discriminant analysis : Some suggest the use of eigenvalues as effect size measures, however, this is generally not supported. Instead, the canonical correlation is the preferred measure of effect size. It is similar to the eigenvalue, but is the square root of the ratio of SSbetween and SStotal. It is the correlation between groups and the function. Another popular measure of effect size is the percent of variance for each function. This is calculated by: (λx/Σλi) X 100 where λx is the eigenvalue for the function and Σλi is the sum of all eigenvalues. This tells us how strong the prediction is for that particular function compared to the others. Percent correctly classified can also be analyzed as an effect size. The kappa value can describe this while correcting for chance agreement.Kappa normalizes across all categorizes rather than biased by a significantly good or poorly performing classes.
Linear discriminant analysis : Canonical discriminant analysis (CDA) finds axes (k − 1 canonical coordinates, k being the number of classes) that best separate the categories. These linear functions are uncorrelated and define, in effect, an optimal k − 1 space through the n-dimensional cloud of data that best separates (the projections in that space of) the k groups. See “Multiclass LDA” for details below. Because LDA uses canonical variates, it was initially often referred as the "method of canonical variates" or canonical variates analysis (CVA).
Linear discriminant analysis : The terms Fisher's linear discriminant and LDA are often used interchangeably, although Fisher's original article actually describes a slightly different discriminant, which does not make some of the assumptions of LDA such as normally distributed classes or equal class covariances. Suppose two classes of observations have means μ → 0 , μ → 1 _,_ and covariances Σ 0 , Σ 1 ,\Sigma _ . Then the linear combination of features w → T x → ^ will have means w → T μ → i ^ _ and variances w → T Σ i w → ^ \Sigma _ for i = 0 , 1 . Fisher defined the separation between these two distributions to be the ratio of the variance between the classes to the variance within the classes: S = σ between 2 σ within 2 = ( w → ⋅ μ → 1 − w → ⋅ μ → 0 ) 2 w → T Σ 1 w → + w → T Σ 0 w → = ( w → ⋅ ( μ → 1 − μ → 0 ) ) 2 w → T ( Σ 0 + Σ 1 ) w → ^^=\cdot _-\cdot _)^^ \Sigma _+^ \Sigma _=\cdot (_-_))^^ (\Sigma _+\Sigma _) This measure is, in some sense, a measure of the signal-to-noise ratio for the class labelling. It can be shown that the maximum separation occurs when w → ∝ ( Σ 0 + Σ 1 ) − 1 ( μ → 1 − μ → 0 ) \propto (\Sigma _+\Sigma _)^(_-_) When the assumptions of LDA are satisfied, the above equation is equivalent to LDA. Be sure to note that the vector w → is the normal to the discriminant hyperplane. As an example, in a two dimensional problem, the line that best divides the two groups is perpendicular to w → . Generally, the data points to be discriminated are projected onto w → ; then the threshold that best separates the data is chosen from analysis of the one-dimensional distribution. There is no general rule for the threshold. However, if projections of points from both classes exhibit approximately the same distributions, a good choice would be the hyperplane between projections of the two means, w → ⋅ μ → 0 \cdot _ and w → ⋅ μ → 1 \cdot _ . In this case the parameter c in threshold condition w → ⋅ x → > c \cdot >c can be found explicitly: c = w → ⋅ 1 2 ( μ → 0 + μ → 1 ) = 1 2 μ → 1 T Σ 1 − 1 μ → 1 − 1 2 μ → 0 T Σ 0 − 1 μ → 0 \cdot (_+_)=_^ \Sigma _^_-_^ \Sigma _^_ . Otsu's method is related to Fisher's linear discriminant, and was created to binarize the histogram of pixels in a grayscale image by optimally picking the black/white threshold that minimizes intra-class variance and maximizes inter-class variance within/between grayscales assigned to black and white pixel classes.
Linear discriminant analysis : In the case where there are more than two classes, the analysis used in the derivation of the Fisher discriminant can be extended to find a subspace which appears to contain all of the class variability. This generalization is due to C. R. Rao. Suppose that each of C classes has a mean μ i and the same covariance Σ . Then the scatter between class variability may be defined by the sample covariance of the class means Σ b = 1 C ∑ i = 1 C ( μ i − μ ) ( μ i − μ ) T =\sum _^(\mu _-\mu )(\mu _-\mu )^ where μ is the mean of the class means. The class separation in a direction w → in this case will be given by S = w → T Σ b w → w → T Σ w → ^ \Sigma _^ \Sigma This means that when w → is an eigenvector of Σ − 1 Σ b \Sigma _ the separation will be equal to the corresponding eigenvalue. If Σ − 1 Σ b \Sigma _ is diagonalizable, the variability between features will be contained in the subspace spanned by the eigenvectors corresponding to the C − 1 largest eigenvalues (since Σ b is of rank C − 1 at most). These eigenvectors are primarily used in feature reduction, as in PCA. The eigenvectors corresponding to the smaller eigenvalues will tend to be very sensitive to the exact choice of training data, and it is often necessary to use regularisation as described in the next section. If classification is required, instead of dimension reduction, there are a number of alternative techniques available. For instance, the classes may be partitioned, and a standard Fisher discriminant or LDA used to classify each partition. A common example of this is "one against the rest" where the points from one class are put in one group, and everything else in the other, and then LDA applied. This will result in C classifiers, whose results are combined. Another common method is pairwise classification, where a new classifier is created for each pair of classes (giving C(C − 1)/2 classifiers in total), with the individual classifiers combined to produce a final classification.
Linear discriminant analysis : The typical implementation of the LDA technique requires that all the samples are available in advance. However, there are situations where the entire data set is not available and the input data are observed as a stream. In this case, it is desirable for the LDA feature extraction to have the ability to update the computed LDA features by observing the new samples without running the algorithm on the whole data set. For example, in many real-time applications such as mobile robotics or on-line face recognition, it is important to update the extracted LDA features as soon as new observations are available. An LDA feature extraction technique that can update the LDA features by simply observing new samples is an incremental LDA algorithm, and this idea has been extensively studied over the last two decades. Chatterjee and Roychowdhury proposed an incremental self-organized LDA algorithm for updating the LDA features. In other work, Demir and Ozmehmet proposed online local learning algorithms for updating LDA features incrementally using error-correcting and the Hebbian learning rules. Later, Aliyari et al. derived fast incremental algorithms to update the LDA features by observing the new samples.
Linear discriminant analysis : In practice, the class means and covariances are not known. They can, however, be estimated from the training set. Either the maximum likelihood estimate or the maximum a posteriori estimate may be used in place of the exact value in the above equations. Although the estimates of the covariance may be considered optimal in some sense, this does not mean that the resulting discriminant obtained by substituting these values is optimal in any sense, even if the assumption of normally distributed classes is correct. Another complication in applying LDA and Fisher's discriminant to real data occurs when the number of measurements of each sample (i.e., the dimensionality of each data vector) exceeds the number of samples in each class. In this case, the covariance estimates do not have full rank, and so cannot be inverted. There are a number of ways to deal with this. One is to use a pseudo inverse instead of the usual matrix inverse in the above formulae. However, better numeric stability may be achieved by first projecting the problem onto the subspace spanned by Σ b . Another strategy to deal with small sample size is to use a shrinkage estimator of the covariance matrix, which can be expressed mathematically as Σ = ( 1 − λ ) Σ + λ I where I is the identity matrix, and λ is the shrinkage intensity or regularisation parameter. This leads to the framework of regularized discriminant analysis or shrinkage discriminant analysis. Also, in many practical cases linear discriminants are not suitable. LDA and Fisher's discriminant can be extended for use in non-linear classification via the kernel trick. Here, the original observations are effectively mapped into a higher dimensional non-linear space. Linear classification in this non-linear space is then equivalent to non-linear classification in the original space. The most commonly used example of this is the kernel Fisher discriminant. LDA can be generalized to multiple discriminant analysis, where c becomes a categorical variable with N possible states, instead of only two. Analogously, if the class-conditional densities p ( x → ∣ c = i ) \mid c=i) are normal with shared covariances, the sufficient statistic for P ( c ∣ x → ) ) are the values of N projections, which are the subspace spanned by the N means, affine projected by the inverse covariance matrix. These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed by treating the means as the samples, and the denominator is the shared covariance matrix. See “Multiclass LDA” above for details.
Linear discriminant analysis : In addition to the examples given below, LDA is applied in positioning and product management.
Linear discriminant analysis : Discriminant function analysis is very similar to logistic regression, and both can be used to answer the same research questions. Logistic regression does not have as many assumptions and restrictions as discriminant analysis. However, when discriminant analysis’ assumptions are met, it is more powerful than logistic regression. Unlike logistic regression, discriminant analysis can be used with small sample sizes. It has been shown that when sample sizes are equal, and homogeneity of variance/covariance holds, discriminant analysis is more accurate. Despite all these advantages, logistic regression has none-the-less become the common choice, since the assumptions of discriminant analysis are rarely met.
Linear discriminant analysis : Geometric anomalies in higher dimensions lead to the well-known curse of dimensionality. Nevertheless, proper utilization of concentration of measure phenomena can make computation easier. An important case of these blessing of dimensionality phenomena was highlighted by Donoho and Tanner: if a sample is essentially high-dimensional then each point can be separated from the rest of the sample by linear inequality, with high probability, even for exponentially large samples. These linear inequalities can be selected in the standard (Fisher's) form of the linear discriminant for a rich family of probability distribution. In particular, such theorems are proven for log-concave distributions including multidimensional normal distribution (the proof is based on the concentration inequalities for log-concave measures) and for product measures on a multidimensional cube (this is proven using Talagrand's concentration inequality for product probability spaces). Data separability by classical linear discriminants simplifies the problem of error correction for artificial intelligence systems in high dimension.
Linear discriminant analysis : Data mining Decision tree learning Factor analysis Kernel Fisher discriminant analysis Logit (for logistic regression) Linear regression Multiple discriminant analysis Multidimensional scaling Pattern recognition Preference regression Quadratic classifier Statistical classification
Linear discriminant analysis : Duda, R. O.; Hart, P. E.; Stork, D. H. (2000). Pattern Classification (2nd ed.). Wiley Interscience. ISBN 978-0-471-05669-0. MR 1802993. Hilbe, J. M. (2009). Logistic Regression Models. Chapman & Hall/CRC Press. ISBN 978-1-4200-7575-5. Mika, S.; et al. (1999). "Fisher discriminant analysis with kernels". Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No.98TH8468). pp. 41–48. CiteSeerX 10.1.1.35.9904. doi:10.1109/NNSP.1999.788121. ISBN 978-0-7803-5673-3. S2CID 8473401. McFarland, H. Richard; Donald, St. P. Richards (2001). "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case". Journal of Multivariate Analysis. 77 (1): 21–53. doi:10.1006/jmva.2000.1924. McFarland, H. Richard; Donald, St. P. Richards (2002). "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. II. The Heterogeneous Case". Journal of Multivariate Analysis. 82 (2): 299–330. doi:10.1006/jmva.2001.2034. Haghighat, M.; Abdel-Mottaleb, M.; Alhalabi, W. (2016). "Discriminant Correlation Analysis: Real-Time Feature Level Fusion for Multimodal Biometric Recognition". IEEE Transactions on Information Forensics and Security. 11 (9): 1984–1996. doi:10.1109/TIFS.2016.2569061. S2CID 15624506.
Linear discriminant analysis : Discriminant Correlation Analysis (DCA) of the Haghighat article (see above) ALGLIB contains open-source LDA implementation in C# / C++ / Pascal / VBA. LDA in Python- LDA implementation in Python LDA tutorial using MS Excel Biomedical statistics. Discriminant analysis StatQuest: Linear Discriminant Analysis (LDA) clearly explained on YouTube Course notes, Discriminant function analysis by G. David Garson, NC State University Discriminant analysis tutorial in Microsoft Excel by Kardi Teknomo Course notes, Discriminant function analysis by David W. Stockburger, Missouri State University Archived 2016-03-03 at the Wayback Machine Discriminant function analysis (DA) by John Poulsen and Aaron French, San Francisco State University
One-class classification : In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, tries to identify objects of a specific class amongst all objects, by primarily learning from a training set containing only the objects of that class, although there exist variants of one-class classifiers where counter-examples are used to further refine the classification boundary. This is different from and more difficult than the traditional classification problem, which tries to distinguish between two or more classes with the training set containing objects from all the classes. Examples include the monitoring of helicopter gearboxes, motor failure prediction, or the operational status of a nuclear plant as 'normal': In this scenario, there are few, if any, examples of catastrophic system states; only the statistics of normal operation are known. While many of the above approaches focus on the case of removing a small number of outliers or anomalies, one can also learn the other extreme, where the single class covers a small coherent subset of the data, using an information bottleneck approach.
One-class classification : The term one-class classification (OCC) was coined by Moya & Hush (1996) and many applications can be found in scientific literature, for example outlier detection, anomaly detection, novelty detection. A feature of OCC is that it uses only sample points from the assigned class, so that a representative sampling is not strictly required for non-target classes.
One-class classification : SVM based one-class classification (OCC) relies on identifying the smallest hypersphere (with radius r, and center c) consisting of all the data points. This method is called Support Vector Data Description (SVDD). Formally, the problem can be defined in the following constrained optimization form, min r , c r 2 subject to, \| \| Φ ( x i ) − c \| \| 2 ≤ r 2 ∀ i = 1 , 2 , . . . , n r^\|\|\Phi (x_)-c\|\|^\leq r^\;\;\forall i=1,2,...,n However, the above formulation is highly restrictive, and is sensitive to the presence of outliers. Therefore, a flexible formulation, that allow for the presence of outliers is formulated as shown below, min r , c , ζ r 2 + 1 ν n ∑ i = 1 n ζ i r^+\sum _^\zeta _ subject to, \| \| Φ ( x i ) − c \| \| 2 ≤ r 2 + ζ i ∀ i = 1 , 2 , . . . , n \|\|\Phi (x_)-c\|\|^\leq r^+\zeta _\;\;\forall i=1,2,...,n From the Karush–Kuhn–Tucker conditions for optimality, we get c = ∑ i = 1 n α i Φ ( x i ) , ^\alpha _\Phi (x_), where the α i 's are the solution to the following optimization problem: max α ∑ i = 1 n α i κ ( x i , x i ) − ∑ i , j = 1 n α i α j κ ( x i , x j ) \sum _^\alpha _\kappa (x_,x_)-\sum _^\alpha _\alpha _\kappa (x_,x_) subject to, ∑ i = 1 n α i = 1 and 0 ≤ α i ≤ 1 ν n for all i = 1 , 2 , . . . , n . ^\alpha _=10\leq \alpha _\leq i=1,2,...,n. The introduction of kernel function provide additional flexibility to the One-class SVM (OSVM) algorithm.
One-class classification : Several approaches have been proposed to solve one-class classification (OCC). The approaches can be distinguished into three main categories, density estimation, boundary methods, and reconstruction methods.
One-class classification : Multiclass classification Anomaly detection Supervised learning == References ==
Operational taxonomic unit : An operational taxonomic unit (OTU) is an operational definition used to classify groups of closely related individuals. The term was originally introduced in 1963 by Robert R. Sokal and Peter H. A. Sneath in the context of numerical taxonomy, where an "operational taxonomic unit" is simply the group of organisms currently being studied. In this sense, an OTU is a pragmatic definition to group individuals by similarity, equivalent to but not necessarily in line with classical Linnaean taxonomy or modern evolutionary taxonomy. Nowadays, however, the term "OTU" is commonly used in a different context and refers to clusters of (uncultivated or unknown) organisms, grouped by DNA sequence similarity of a specific taxonomic marker gene (originally coined as mOTU; molecular OTU). In other words, OTUs are pragmatic proxies for "species" (microbial or metazoan) at different taxonomic levels, in the absence of traditional systems of biological classification as are available for macroscopic organisms. For several years, OTUs have been the most commonly used units of diversity, especially when analysing small subunit 16S (for prokaryotes) or 18S rRNA (for eukaryotes) marker gene sequence datasets. Sequences can be clustered according to their similarity to one another, and operational taxonomic units are defined based on the similarity threshold (usually 97% similarity; however also 100% similarity is common, also known as single variants) set by the researcher. It remains debatable how well this commonly-used method recapitulates true microbial species phylogeny or ecology. Although OTUs can be calculated differently when using different algorithms or thresholds, research by Schmidt et al. (2014) demonstrated that microbial OTUs were generally ecologically consistent across habitats and several OTU clustering approaches. The number of OTUs defined may be inflated due to errors in DNA sequencing.
Operational taxonomic unit : There are three main approaches to clustering OTUs: De novo, for which the clustering is based on similarities between sequencing reads. Closed-reference, for which the clustering is performed against a reference database of sequences. Open-reference, where clustering is first performed against a reference database of sequences, then any remaining sequences that could not be mapped to the reference are clustered de novo.
Operational taxonomic unit : Hierarchical clustering algorithms (HCA): uclust & cd-hit & ESPRIT Bayesian clustering: CROP
Operational taxonomic unit : Phylotype Amplicon sequence variant
Operational taxonomic unit : Chen, W.; Zhang, C. K.; Cheng, Y.; Zhang, S.; Zhao, H. (2013). "A comparison of methods for clustering 16S rRNA sequences into OTUs.". PLOS ONE. 8 (8): e70837. Bibcode:2013PLoSO...870837C. doi:10.1371/journal.pone.0070837. PMC 3742672. PMID 23967117.
Optimal discriminant analysis and classification tree analysis : Optimal Discriminant Analysis (ODA) and the related classification tree analysis (CTA) are exact statistical methods that maximize predictive accuracy. For any specific sample and exploratory or confirmatory hypothesis, optimal discriminant analysis (ODA) identifies the statistical model that yields maximum predictive accuracy, assesses the exact Type I error rate, and evaluates potential cross-generalizability. Optimal discriminant analysis may be applied to > 0 dimensions, with the one-dimensional case being referred to as UniODA and the multidimensional case being referred to as MultiODA. Optimal discriminant analysis is an alternative to ANOVA (analysis of variance) and regression analysis.
Optimal discriminant analysis and classification tree analysis : Data mining Decision tree Factor analysis Linear classifier Logit (for logistic regression) Machine learning Multidimensional scaling Perceptron Preference regression Quadratic classifier Statistics
Optimal discriminant analysis and classification tree analysis : Yarnold, Paul R.; Soltysik, Robert C. (2004). Optimal Data Analysis. American Psychological Association. ISBN 978-1-55798-981-9. Archived from the original on 2008-11-23. Retrieved 2009-09-11. Fisher, R. A. (1936). "The Use of Multiple Measurements in Taxonomic Problems". Annals of Eugenics. 7 (2): 179–188. doi:10.1111/j.1469-1809.1936.tb02137.x. hdl:2440/15227. Martinez, A. M.; Kak, A. C. (2001). "PCA versus LDA" (PDF). IEEE Transactions on Pattern Analysis and Machine Intelligence. 23 (2): 228–233. doi:10.1109/34.908974. Mika, S.; et al. (1999). "Fisher discriminant analysis with kernels". Neural Networks for Signal Processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop (Cat. No.98TH8468). pp. 41–48. CiteSeerX 10.1.1.35.9904. doi:10.1109/NNSP.1999.788121. ISBN 978-0-7803-5673-3. S2CID 8473401.: CS1 maint: date and year (link)
Optimal discriminant analysis and classification tree analysis : LDA tutorial using MS Excel IMSL discriminant analysis function DSCRM, which has many useful mathematical definitions.
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.
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 ≤ 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 ) )=\sigma (\theta _-\mathbf \cdot \mathbf ) 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 ) -\mathbf \cdot \mathbf )=-\mathbf \cdot \mathbf ) 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 ) ) -\mathbf \cdot \mathbf )=1-\exp(-\exp(\theta _-\mathbf \cdot \mathbf )) which gives the proportional hazards model.
Ordinal regression : In machine learning, alternatives to the latent-variable models of ordinal regression have been proposed. An early result was PRank, a variant of the perceptron algorithm that found multiple parallel hyperplanes separating the various ranks; its output is a weight vector w and a sorted vector of K−1 thresholds θ, as in the ordered logit/probit models. The prediction rule for this model is to output the smallest rank k such that wx < θk. Other methods rely on the principle of large-margin learning that also underlies support vector machines. Another approach is given by Rennie and Srebro, who, realizing that "even just evaluating the likelihood of a predictor is not straight-forward" in the ordered logit and ordered probit models, propose fitting ordinal regression models by adapting common loss functions from classification (such as the hinge loss and log loss) to the ordinal case.
Ordinal regression : ORCA (Ordinal Regression and Classification Algorithms) is an Octave/MATLAB framework including a wide set of ordinal regression methods. R packages that provide ordinal regression methods include MASS and Ordinal.
Ordinal regression : Logistic regression
Ordinal regression : Agresti, Alan (2010). Analysis of ordinal categorical data. Hoboken, N.J: Wiley. ISBN 978-0470082898. Greene, William H. (2012). Econometric Analysis (Seventh ed.). Boston: Pearson Education. pp. 824–842. ISBN 978-0-273-75356-8. Hardin, James; Hilbe, Joseph (2007). Generalized Linear Models and Extensions (2nd ed.). College Station: Stata Press. ISBN 978-1-59718-014-6.
Probit model : In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model. A probit model is a popular specification for a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. It is most often estimated using the maximum likelihood procedure, such an estimation being called a probit regression.
Probit model : Suppose a response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example, Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes the form P ( Y = 1 ∣ X ) = Φ ( X T β ) , \beta ), where P is the probability and Φ is the cumulative distribution function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood. It is possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable Y ∗ = X T β + ε , =X^\beta +\varepsilon , where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive: Y = = 1&Y^>0\\0&\end\right\=\left.1&X^ \beta +\varepsilon >0\\0&\end\right\ The use of the standard normal distribution causes no loss of generality compared with the use of a normal distribution with an arbitrary mean and standard deviation, because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount. To see that the two models are equivalent, note that P ( Y = 1 ∣ X ) = P ( Y ∗ > 0 ) = P ( X T β + ε > 0 ) = P ( ε > − X T β ) = P ( ε < X T β ) by symmetry of the normal distribution = Φ ( X T β ) P(Y=1\mid X)&=P(Y^>0)\\&=P(X^ \beta +\varepsilon >0)\\&=P(\varepsilon >-X^ \beta )\\&=P(\varepsilon <X^ \beta )&\\&=\Phi (X^ \beta )\end
Probit model : The suitability of an estimated binary model can be evaluated by counting the number of true observations equaling 1, and the number equaling zero, for which the model assigns a correct predicted classification by treating any estimated probability above 1/2 (or, below 1/2), as an assignment of a prediction of 1 (or, of 0). See Logistic regression § Model for details.
Probit model : Consider the latent variable model formulation of the probit model. When the variance of ε conditional on x is not constant but dependent on x , then the heteroscedasticity issue arises. For example, suppose y ∗ = β 0 + B 1 x 1 + ε =\beta _+B_x_+\varepsilon and ε ∣ x ∼ N ( 0 , x 1 2 ) ^) where x 1 is a continuous positive explanatory variable. Under heteroskedasticity, the probit estimator for β is usually inconsistent, and most of the tests about the coefficients are invalid. More importantly, the estimator for P ( y = 1 ∣ x ) becomes inconsistent, too. To deal with this problem, the original model needs to be transformed to be homoskedastic. For instance, in the same example, 1 [ β 0 + β 1 x 1 + ε > 0 ] +\beta _x_+\varepsilon >0] can be rewritten as 1 [ β 0 / x 1 + β 1 + ε / x 1 > 0 ] /x_+\beta _+\varepsilon /x_>0] , where ε / x 1 ∣ x ∼ N ( 0 , 1 ) \mid x\sim N(0,1) . Therefore, P ( y = 1 ∣ x ) = Φ ( β 1 + β 0 / x 1 ) +\beta _/x_) and running probit on ( 1 , 1 / x 1 ) ) generates a consistent estimator for the conditional probability P ( y = 1 ∣ x ) . When the assumption that ε is normally distributed fails to hold, then a functional form misspecification issue arises: if the model is still estimated as a probit model, the estimators of the coefficients β are inconsistent. For instance, if ε follows a logistic distribution in the true model, but the model is estimated by probit, the estimates will be generally smaller than the true value. However, the inconsistency of the coefficient estimates is practically irrelevant because the estimates for the partial effects, ∂ P ( y = 1 ∣ x ) / ∂ x i ′ , will be close to the estimates given by the true logit model. To avoid the issue of distribution misspecification, one may adopt a general distribution assumption for the error term, such that many different types of distribution can be included in the model. The cost is heavier computation and lower accuracy for the increase of the number of parameter. In most of the cases in practice where the distribution form is misspecified, the estimators for the coefficients are inconsistent, but estimators for the conditional probability and the partial effects are still very good. One can also take semi-parametric or non-parametric approaches, e.g., via local-likelihood or nonparametric quasi-likelihood methods, which avoid assumptions on a parametric form for the index function and is robust to the choice of the link function (e.g., probit or logit).
Probit model : The probit model is usually credited to Chester Bliss, who coined the term "probit" in 1934, and to John Gaddum (1933), who systematized earlier work. However, the basic model dates to the Weber–Fechner law by Gustav Fechner, published in Fechner (1860), and was repeatedly rediscovered until the 1930s; see Finney (1971, Chapter 3.6) and Aitchison & Brown (1957, Chapter 1.2). A fast method for computing maximum likelihood estimates for the probit model was proposed by Ronald Fisher as an appendix to Bliss' work in 1935.
Probit model : Generalized linear model Limited dependent variable Logit model Multinomial probit Multivariate probit models Ordered probit and ordered logit model Separation (statistics) Tobit model
Probit model : Albert, J. H.; Chib, S. (1993). "Bayesian Analysis of Binary and Polychotomous Response Data". Journal of the American Statistical Association. 88 (422): 669–679. doi:10.1080/01621459.1993.10476321. JSTOR 2290350. Amemiya, Takeshi (1985). "Qualitative Response Models". Advanced Econometrics. Oxford: Basil Blackwell. pp. 267–359. ISBN 0-631-13345-3. Gouriéroux, Christian (2000). "The Simple Dichotomy". Econometrics of Qualitative Dependent Variables. New York: Cambridge University Press. pp. 6–37. ISBN 0-521-58985-1. Liao, Tim Futing (1994). Interpreting Probability Models: Logit, Probit, and Other Generalized Linear Models. Sage. ISBN 0-8039-4999-5. McCullagh, Peter; John Nelder (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0-412-31760-5.
Probit model : Media related to Probit model at Wikimedia Commons Econometrics Lecture (topic: Probit model) on YouTube by Mark Thoma
Quadratic classifier : In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier.
Quadratic classifier : Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c Ax +\mathbf x +c In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces.
Quadratic classifier : Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 ,\mu _ and covariance matrices Σ 0 , Σ 1 ,\Sigma _ corresponding to y = 0 and y = 1 respectively. Then the likelihood ratio is given by Likelihood ratio = 2 π \| Σ 1 \| − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) 2 π \| Σ 0 \| − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t =\|^\exp \left(-(\mathbf -_)^\Sigma _^(\mathbf -_)\right)\|^\exp \left(-(\mathbf -_)^\Sigma _^(\mathbf -_)\right)<t for some threshold t . After some rearrangement, it can be shown that the resulting separating surface between the classes is a quadratic. The sample estimates of the mean vector and variance-covariance matrices will substitute the population quantities in this formula.
Quadratic classifier : While QDA is the most commonly-used method for obtaining a classifier, other methods are also possible. One such method is to create a longer measurement vector from the old one by adding all pairwise products of individual measurements. For instance, the vector [ x 1 , x 2 , x 3 ] ,\;x_,\;x_] would become [ x 1 , x 2 , x 3 , x 1 2 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 3 2 ] . ,\;x_,\;x_,\;x_^,\;x_x_,\;x_x_,\;x_^,\;x_x_,\;x_^]. Finding a quadratic classifier for the original measurements would then become the same as finding a linear classifier based on the expanded measurement vector. This observation has been used in extending neural network models; the "circular" case, which corresponds to introducing only the sum of pure quadratic terms x 1 2 + x 2 2 + x 3 2 + ⋯ ^+x_^+x_^+\cdots \; with no mixed products ( x 1 x 2 , x 1 x 3 , … x_,\;x_x_,\;\ldots \; ), has been proven to be the optimal compromise between extending the classifier's representation power and controlling the risk of overfitting (Vapnik-Chervonenkis dimension). For linear classifiers based only on dot products, these expanded measurements do not have to be actually computed, since the dot product in the higher-dimensional space is simply related to that in the original space. This is an example of the so-called kernel trick, which can be applied to linear discriminant analysis as well as the support vector machine.
Rules extraction system family : The rules extraction system (RULES) family is a family of inductive learning that includes several covering algorithms. This family is used to build a predictive model based on given observation. It works based on the concept of separate-and-conquer to directly induce rules from a given training set and build its knowledge repository. Algorithms under RULES family are usually available in data mining tools, such as KEEL and WEKA, known for knowledge extraction and decision making.
Rules extraction system family : RULES family algorithms are mainly used in data mining to create a model that predicts the actions of a given input features. It goes under the umbrella of inductive learning, which is a machine learning approach. In this type of learning, the agent is usually provided with previous information to gain descriptive knowledge based on the given historical data. Thus, it is a supervised learning paradigm that works as a data analysis tool, which uses the knowledge gained through training to reach a general conclusion and identify new objects using the produced classifier. Inductive learning had been divided into two types: decision tree (DT) and covering algorithms (CA). DTs discover rules using decision tree based on the concept of divide-and-conquer, while CA directly induces rules from the training set based on the concept of separate and conquers. Although DT algorithms was well recognized in the past few decades, CA started to attract the attention due to its direct rule induction property, as emphasized by Kurgan et al. [1]. Under this type of inductive learning approach, several families have been developed and improved. RULES family [2], known as rule extraction system, is one family of covering algorithms that separate each instance or example when inducing the best rules. In this family, the resulting rules are stored in an ‘IF condition THEN conclusion’ structure. It has its own induction procedure that is used to induce the best rules and build the knowledge repository.
Rules extraction system family : To induce the best rules based on a given observation, RULES family start by selecting (separating) a seed example to build a rule, condition by condition. The rule that covers the most positive examples and the least negative examples are chosen as the best rule of the current seed example. It allows the best rule to cover some negative examples to handle the increase flexibility and reduce the overfitting problem and noisy data in the rule induction. When the coverage performance reaches a specified threshold, it marks the examples that match the induced rules without deletion. This prevents the repetition of discovering the same rule as well as preserves the coverage accuracy and the generality of new rules. After that, the algorithm is repeated to select (conquer) another seed example until all the examples are covered. Hence, only one rule can be generated at each step.
Rules extraction system family : Several versions and algorithms have been proposed in RULES family, and can be summarized as follows: RULES-1 [3] is the first version in RULES family and was proposed by prof. Pham and prof. Aksoy in 1995. RULES-2 [4] is an upgraded version of RULES-1, in which every example is studied separately. RULES-3 [5] is another version that contained all the properties of RULES-2 as well as other additional features to generates more general rules. RULES-3Plus [6] is an extended version of RULES-3 with two additional functionalities. RULES-4 [7] is the first incremental version in the RULES family. RULES-5 [8] is the first RULES version that handles continuous attributes without discretization. It was also extended to produce RULES-5+[9], which improves the performance using a new rule space representation scheme. RULES-6 [10] is a scalable version of RULES family developed as an extension of RULES-3 plus. RULES-F [11] is an extension of RULES-5 that handles not only continuous attributes but also continuous classes. A new rule space representation scheme was also integrated to produce an extended version called RULES-F+ [9]. RULES-SRI [12] is another scalable RULES algorithm, developed to improve RULES-6 scalability. Rule Extractor-1 (REX-1) [13] is an improvement of RULES-3, RULES-3 Plus, and RULES-4 to shortened the process time and produced simpler models with fewer rules. RULES-IS [14] an incremental algorithm inspired by the immune systems. RULES-3EXT [15] is an extension of RULES-3 with additional features. RULES-7 [16] is an extension of RULES-6, in which it applies specialization over one seed at a time. RULES-8 [17] is an improved version that deals with continuous attributes online. RULES-TL [18] is another scalable algorithm that was proposed to enhance the performance and speed while introducing more intelligent aspects. RULES-IT [19] is an incremental version that is built based on RULES-TL to incrementally deal with large and incomplete problems.
Rules extraction system family : Covering algorithms, in general, can be applied to any machine learning application field, as long as it supports its data type. Witten, Frank and Hall [20] identified six main fielded applications that are actively used as ML applications, including sales and marketing, judgment decisions, image screening, load forecasting, diagnosis, and web mining. RULES algorithms, in particular, were applied in different manufacturing and engineering applications [21]. RULES-3 EXT was also applied over signature verification and the algorithm performance was verified by Aksoy and Mathkour [22]. Recently, Salem and Schmickl [23] have studied the efficiency of RULEs-4 in predating agent's density.
Rules extraction system family : Decision Tree WEKA KEEL Machine learning C4.5 algorithm
Rules extraction system family : [1] L. A. Kurgan, K. J. Cios, and S. Dick, "Highly Scalable and Robust Rule Learner: Performance Evaluation and Comparison," IEEE SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, vol. 36, pp. 32–53, 2006. [2] M. S. Aksoy, "A review of rules family of algorithms," Mathematical and Computational Applications, vol. 13, pp. 51–60, 2008. [3] D. T. Pham and M. S. Aksoy, "RULES: A simple rule extraction system," Expert Systems with Applications, vol. 8, pp. 59–65, 1995. [4] D. T. Pham and M. S. Aksoy, "An algorithm for automatic rule induction," Artificial Intelligence in Engineering, vol. 8, pp. 277–282, 1993. [5] D. T. Pham and M. S. Aksoy, "A new algorithm for inductive learning," Journal of Systems Engenering, vol. 5, pp. 115–122, 1995. [6] D. T. Pham and S. S. Dimov, "The RULES-3 Plus inductive learning algorithm," in In Proceedings of the Third World Congress on Expert Systems, Seoul, Korea, 1996, pp. 917–924. [7] D. T. Pham and S. S. Dimov, "An algorithm for incremental inductive learning," Journal of Engineering Manufacture, vol. 211, pp. 239–249, 1997. [8] D. Pham, S. Bigot, and S. Dimov, "RULES-5: a rule induction algorithm for classification problems involving continuous attributes," in Institution of Mechanical Engineers, 2003, pp. 1273–1286. [9] S. Bigot, "A new rule space representation scheme for rule induction in classification and control applications," Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 2011. [10] D. T. Pham and A. A. Afify, "RULES-6: A Simple Rule Induction Algorithm for Supporting Decision Making," in 31st Annual Conference of IEEE Industrial Electronics Society (IECON '05), 2005, pp. 2184–2189. [11] D. T. Pham, S. Bigot, and S. S. Dimov, "RULES-F: A fuzzy inductive learning algorithm," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 220, pp. 1433–1447, 2006. [12] A. A. Afify and D. T. Pham, "SRI: A Scalable Rule Induction Algorithm," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 220, pp. 537–552, 2006. [13] Ö. Akgöbek, Y. S. Aydin, E. Öztemel, and M. S. Aksoy, "A new algorithm for automatic knowledge acquisition in inductive learning," Knowledge-Based Systems, vol. 19, pp. 388–395, 2006. [14] D. T. Pham and A. J. Soroka, "An Immune-network inspired rule generation algorithm (RULES-IS)," in Third Virtual International Conference on Innovative Production Machines and Systems, WhittlesDunbeath, 2007. [15] H. I. Mathkour, "RULES3-EXT Improvement on RULES-3 Induction Algorithm," Mathematical and Computational Applications, Vol. 15, No. 3, pp., 2010, vol. 15, pp. 318–324, 2010. [16] K. Shehzad, "EDISC: A Class-tailored Discretization Technique for Rule-based Classification," IEEE Transactions on Knowledge and Data Engineering, vol. 24, pp. 1435–1447, 2012. [17] D. Pham, "A novel rule induction algorithm with improved handling of continuous valued attributes," Doctor of Philosophy, School of Engineering, Cardiff University, Cardiff, 2012. [18] H. ElGibreen and M. S. Aksoy, "RULES – TL: A simple and Improved RULES Algorithm for Incomplete and Large Data," Journal of Theoretical and Applied Information Technology, vol. 47, pp. 28–40, 2013. [19] H. Elgibreen and M. Aksoy, "RULES-IT: incremental transfer learning with RULES family," Frontiers of Computer Science, vol. 8, pp. 537–562, 2014. [20] I. H. Witten, E. Frank, and M. A. Hall, Data Mining Practical Machine Learning Tools and Techniques, Third ed.: Morgan Kaufmann, 2011. [21] D. Pham and A. Afify, "Machine-learning techniques and their applications in manufacturing," Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture, vol. 219, pp. 395–412, 2005. [22] M. S. Aksoy and H. Mathkour, "Signature verification using rules 3-ext inductive learning system," International Journal of the Physical Sciences, vol. 6, pp. 4428–4434, 2011. [23] Z. Salem and T. Schmickl, "The efficiency of the RULES-4 classification learning algorithm in predicting the density of agents," Cogent Engineering, vol. 1, p. 986262, 2014.
Syntactic pattern recognition : Syntactic pattern recognition, or structural pattern recognition, is a form of pattern recognition in which each object can be represented by a variable-cardinality set of symbolic nominal features. This allows for representing pattern structures, taking into account more complex relationships between attributes than is possible in the case of flat, numerical feature vectors of fixed dimensionality that are used in statistical classification. Syntactic pattern recognition can be used instead of statistical pattern recognition if clear structure exists in the patterns. One way to present such structure is via strings of symbols from a formal language. In this case, the differences in the structures of the classes are encoded as different grammars. An example of this would be diagnosing heart problems with electrocardiogram (ECG) measurements. ECG waveforms can be approximated with diagonal and vertical line segments. If normal and unhealthy waveforms can be described as formal grammars, ECG signals can be classified as healthy or unhealthy by first describing them in terms of the basic line segments, and then trying to parse the descriptions according to the grammars. Another example is tessellation of tiling patterns. A second way to represent relations are graphs, where nodes are linked if corresponding subpatterns are related. An item can be assigned a certain class label if its graph representation is isomorphic with prototype graphs of that class. Typically, patterns are constructed from simpler sub-patterns in a hierarchical fashion. This helps divide the recognition task into easier subtasks of first identifying sub-patterns, and then the actual patterns. Structural methods provide descriptions of items, which may be useful in their own right. For example, syntactic pattern recognition can be used to determine what objects are present in an image. Furthermore, structural methods are strong when applied to finding a "correspondence mapping" between two images of an object. Under natural conditions, corresponding features will be in different positions and/or may be occluded in the two images, due to camera attitude and perspective, as in face recognition. A graph matching algorithm will yield the optimal correspondence.
Syntactic pattern recognition : Grammar induction String matching Hopcroft–Karp algorithm Structural information theory
Syntactic pattern recognition : Schalkoff, Robert (1992). Pattern recognition - statistical, structural and neural approaches. John Wiley & sons. ISBN 0-471-55238-0. Bunke, Horst (1993). Structural and syntactic pattern recognition, Chen, Pau & Wang (Eds.) Handbook of pattern recognition & computer vision. World Scientific. pp. 163–209. ISBN 981-02-1136-8. Flasinski, Mariusz (2019). Syntactic pattern recognition. World Scientific. ISBN 978-981-3278-46-2.
Whitening transformation : A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1. The transformation is called "whitening" because it changes the input vector into a white noise vector. Several other transformations are closely related to whitening: the decorrelation transform removes only the correlations but leaves variances intact, the standardization transform sets variances to 1 but leaves correlations intact, a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.
Whitening transformation : Suppose X is a random (column) vector with non-singular covariance matrix Σ and mean 0 . Then the transformation Y = W X with a whitening matrix W satisfying the condition W T W = Σ − 1 W=\Sigma ^ yields the whitened random vector Y with unit diagonal covariance. If X has non-zero mean μ , then whitening can be performed by Y = W ( X − μ ) . There are infinitely many possible whitening matrices W that all satisfy the above condition. Commonly used choices are W = Σ − 1 / 2 (Mahalanobis or ZCA whitening), W = L T where L is the Cholesky decomposition of Σ − 1 (Cholesky whitening), or the eigen-system of Σ (PCA whitening). Optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of X and Y . For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original X and whitened Y is produced by the whitening matrix W = P − 1 / 2 V − 1 / 2 V^ where P is the correlation matrix and V the diagonal variance matrix.
Whitening transformation : Whitening a data matrix follows the same transformation as for random variables. An empirical whitening transform is obtained by estimating the covariance (e.g. by maximum likelihood) and subsequently constructing a corresponding estimated whitening matrix (e.g. by Cholesky decomposition).
Whitening transformation : This modality is a generalization of the pre-whitening procedure extended to more general spaces where X is usually assumed to be a random function or other random objects in a Hilbert space H . One of the main issues of extending whitening to infinite dimensions is that the covariance operator has an unbounded inverse in H . Nevertheless, if one assumes that Picard condition holds for X in the range space of the covariance operator, whitening becomes possible. A whitening operator can be then defined from the factorization of the Moore–Penrose inverse of the covariance operator, which has effective mapping on Karhunen–Loève type expansions of X . The advantage of these whitening transformations is that they can be optimized according to the underlying topological properties of the data, thus producing more robust whitening representations. High-dimensional features of the data can be exploited through kernel regressors or basis function systems.
Whitening transformation : An implementation of several whitening procedures in R, including ZCA-whitening and PCA whitening but also CCA whitening, is available in the "whitening" R package published on CRAN. The R package "pfica" allows the computation of high-dimensional whitening representations using basis function systems (B-splines, Fourier basis, etc.).
Whitening transformation : Decorrelation Principal component analysis Weighted least squares Canonical correlation Mahalanobis distance (is Euclidean after W. transformation).
Whitening transformation : http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf The ZCA whitening transformation. Appendix A of Learning Multiple Layers of Features from Tiny Images by A. Krizhevsky.
Winnow (algorithm) : The winnow algorithm is a technique from machine learning for learning a linear classifier from labeled examples. It is very similar to the perceptron algorithm. However, the perceptron algorithm uses an additive weight-update scheme, while Winnow uses a multiplicative scheme that allows it to perform much better when many dimensions are irrelevant (hence its name winnow). It is a simple algorithm that scales well to high-dimensional data. During training, Winnow is shown a sequence of positive and negative examples. From these it learns a decision hyperplane that can then be used to label novel examples as positive or negative. The algorithm can also be used in the online learning setting, where the learning and the classification phase are not clearly separated.
Winnow (algorithm) : The basic algorithm, Winnow1, is as follows. The instance space is X = n ^ , that is, each instance is described as a set of Boolean-valued features. The algorithm maintains non-negative weights w i for i ∈ , which are initially set to 1, one weight for each feature. When the learner is given an example ( x 1 , … , x n ) ,\ldots ,x_) , it applies the typical prediction rule for linear classifiers: If ∑ i = 1 n w i x i > Θ ^w_x_>\Theta , then predict 1 Otherwise predict 0 Here Θ is a real number that is called the threshold. Together with the weights, the threshold defines a dividing hyperplane in the instance space. Good bounds are obtained if Θ = n / 2 (see below). For each example with which it is presented, the learner applies the following update rule: If an example is correctly classified, do nothing. If an example is predicted incorrectly and the correct result was 0, for each feature x i = 1 =1 , the corresponding weight w i is set to 0 (demotion step). ∀ x i = 1 , w i = 0 =1,w_=0 If an example is predicted incorrectly and the correct result was 1, for each feature x i = 1 =1 , the corresponding weight w i multiplied by α(promotion step). ∀ x i = 1 , w i = α w i =1,w_=\alpha w_ A typical value for α is 2. There are many variations to this basic approach. Winnow2 is similar except that in the demotion step the weights are divided by α instead of being set to 0. Balanced Winnow maintains two sets of weights, and thus two hyperplanes. This can then be generalized for multi-label classification.
Winnow (algorithm) : In certain circumstances, it can be shown that the number of mistakes Winnow makes as it learns has an upper bound that is independent of the number of instances with which it is presented. If the Winnow1 algorithm uses α > 1 and Θ ≥ 1 / α on a target function that is a k -literal monotone disjunction given by f ( x 1 , … , x n ) = x i 1 ∪ ⋯ ∪ x i k ,\ldots ,x_)=x_\cup \cdots \cup x_ , then for any sequence of instances the total number of mistakes is bounded by: α k ( log α ⁡ Θ + 1 ) + n Θ \Theta +1)+ . == References ==
Blockmodeling : Blockmodeling is a set or a coherent framework, that is used for analyzing social structure and also for setting procedure(s) for partitioning (clustering) social network's units (nodes, vertices, actors), based on specific patterns, which form a distinctive structure through interconnectivity. It is primarily used in statistics, machine learning and network science. As an empirical procedure, blockmodeling assumes that all the units in a specific network can be grouped together to such extent to which they are equivalent. Regarding equivalency, it can be structural, regular or generalized. Using blockmodeling, a network can be analyzed using newly created blockmodels, which transforms large and complex network into a smaller and more comprehensible one. At the same time, the blockmodeling is used to operationalize social roles. While some contend that the blockmodeling is just clustering methods, Bonacich and McConaghy state that "it is a theoretically grounded and algebraic approach to the analysis of the structure of relations". Blockmodeling's unique ability lies in the fact that it considers the structure not just as a set of direct relations, but also takes into account all other possible compound relations that are based on the direct ones. The principles of blockmodeling were first introduced by Francois Lorrain and Harrison C. White in 1971. Blockmodeling is considered as "an important set of network analytic tools" as it deals with delineation of role structures (the well-defined places in social structures, also known as positions) and the discerning the fundamental structure of social networks.: 2, 3 According to Batagelj, the primary "goal of blockmodeling is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily". Blockmodeling was at first used for analysis in sociometry and psychometrics, but has now spread also to other sciences.
Blockmodeling : A network as a system is composed of (or defined by) two different sets: one set of units (nodes, vertices, actors) and one set of links between the units. Using both sets, it is possible to create a graph, describing the structure of the network. During blockmodeling, the researcher is faced with two problems: how to partition the units (e.g., how to determine the clusters (or classes), that then form vertices in a blockmodel) and then how to determine the links in the blockmodel (and at the same time the values of these links). In the social sciences, the networks are usually social networks, composed of several individuals (units) and selected social relationships among them (links). Real-world networks can be large and complex; blockmodeling is used to simplify them into smaller structures that can be easier to interpret. Specifically, blockmodeling partitions the units into clusters and then determines the ties among the clusters. At the same time, blockmodeling can be used to explain the social roles existing in the network, as it is assumed that the created cluster of units mimics (or is closely associated with) the units' social roles. Blockmodeling can thus be defined as a set of approaches for partitioning units into clusters (also known as positions) and links into blocks, which are further defined by the newly obtained clusters. A block (also blockmodel) is defined as a submatrix, that shows interconnectivity (links) between nodes, present in the same or different clusters. Each of these positions in the cluster is defined by a set of (in)direct ties to and from other social positions. These links (connections) can be directed or undirected; there can be multiple links between the same pair of objects or they can have weights on them. If there are not any multiple links in a network, it is called a simple network.: 8 A matrix representation of a graph is composed of ordered units, in rows and columns, based on their names. The ordered units with similar patterns of links are partitioned together in the same clusters. Clusters are then arranged together so that units from the same clusters are placed next to each other, thus preserving interconnectivity. In the next step, the units (from the same clusters) are transformed into a blockmodel. With this, several blockmodels are usually formed, one being core cluster and others being cohesive; a core cluster is always connected to cohesive ones, while cohesive ones cannot be linked together. Clustering of nodes is based on the equivalence, such as structural and regular. The primary objective of the matrix form is to visually present relations between the persons included in the cluster. These ties are coded dichotomously (as present or absent), and the rows in the matrix form indicate the source of the ties, while the columns represent the destination of the ties. Equivalence can have two basic approaches: the equivalent units have the same connection pattern to the same neighbors or these units have same or similar connection pattern to different neighbors. If the units are connected to the rest of network in identical ways, then they are structurally equivalent. Units can also be regularly equivalent, when they are equivalently connected to equivalent others. With blockmodeling, it is necessary to consider the issue of results being affected by measurement errors in the initial stage of acquiring the data.
Blockmodeling : Regarding what kind of network is undergoing blockmodeling, a different approach is necessary. Networks can be one–mode or two–mode. In the former all units can be connected to any other unit and where units are of the same type, while in the latter the units are connected only to the unit(s) of a different type.: 6–10 Regarding relationships between units, they can be single–relational or multi–relational networks. Further more, the networks can be temporal or multilevel and also binary (only 0 and 1) or signed (allowing negative ties)/values (other values are possible) networks. Different approaches to blockmodeling can be grouped into two main classes: deterministic blockmodeling and stochastic blockmodeling approaches. Deterministic blockmodeling is then further divided into direct and indirect blockmodeling approaches. Among direct blockmodeling approaches are: structural equivalence and regular equivalence. Structural equivalence is a state, when units are connected to the rest of the network in an identical way(s), while regular equivalence occurs when units are equally related to equivalent others (units are not necessarily sharing neighbors, but have neighbour that are themselves similar).: 24 Indirect blockmodeling approaches, where partitioning is dealt with as a traditional cluster analysis problem (measuring (dis)similarity results in a (dis)similarity matrix), are: conventional blockmodeling, generalized blockmodeling: generalized blockmodeling of binary networks, generalized blockmodeling of valued networks and generalized homogeneity blockmodeling, prespecified blockmodeling. According to Brusco and Steinley (2011), the blockmodeling can be categorized (using a number of dimensions): deterministic or stochastic blockmodeling, one–mode or two–mode networks, signed or unsigned networks, exploratory or confirmatory blockmodeling.
Blockmodeling : Blockmodels (sometimes also block models) are structures in which: vertices (e.g., units, nodes) are assembled within a cluster, with each cluster identified as a vertex; from such vertices a graph can be constructed; combinations of all the links (ties), represented in a block as a single link between positions, while at the same time constructing one tie for each block. In a case, when there are no ties in a block, there will be no ties between the two positions that define the block. Computer programs can partition the social network according to pre-set conditions.: 333 When empirical blocks can be reasonably approximated in terms of ideal blocks, such blockmodels can be reduced to a blockimage, which is a representation of the original network, capturing its underlying 'functional anatomy'. Thus, blockmodels can "permit the data to characterize their own structure", and at the same time not seek to manifest a preconceived structure imposed by the researcher. Blockmodels can be created indirectly or directly, based on the construction of the criterion function. Indirect construction refers to a function, based on "compatible (dis)similarity measure between paris of units", while the direct construction is "a function measuring the fit of real blocks induced by a given clustering to the corresponding ideal blocks with perfect relations within each cluster and between clusters according to the considered types of connections (equivalence)".
Blockmodeling : Blockmodeling is done with specialized computer programs, dedicated to the analysis of networks or blockmodeling in particular, as: BLOCKS (Tom Snijders), CONCOR, Model (Vladimir Batagelj), Model2 (Vladimir Batagelj), Pajek (Vladimir Batagelj and Andrej Mrvar), R–package Blockmodeling (Aleš Žiberna), StOCNET (Tom Snijders),...
Blockmodeling : Stochastic block model Mathematical sociology Role assignment Multiobjective blockmodeling Blockmodeling linked networks == References ==
Vladimir Batagelj : Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks (blockmodeling).
Vladimir Batagelj : Vladimir Batagelj completed his Ph.D. at the University of Ljubljana in 1986 under the direction of Tomaž Pisanski. He stayed at the University of Ljubljana as a professor until his retirement, where he was a professor of sociology and statistics, while also being a chair of the Department of Sociology of the Faculty of Social Sciences. As visiting professor, he was taught at the University of Pittsburgh (1990-91) and at the University of Konstanz (2002). He was also a member of editorial boards of two journals: Informatica and Journal of Social Structure. His work has been cited over 11000 times. His book Exploratory Social Network Analysis with Pajek on blockmodeling, coauthored with Wouter de Nooy and Andrej Mrvar, is Batagelj's most cited work and has over 3300 citations. The book was translated into Chinese and Japanese. The revised and expanded third edition has been published by Cambridge University Press. In 1975, 11 years before completing his PhD, Batagelj published a solo paper in Communications of the ACM. Batagelj authored more than 20 textbooks in Slovenian, covering topics like TeX, combinatorics and discrete mathematics. He has also written extensively in the Slovenian popular science journal Presek. Batagelj has advised 9 Ph.D. students.
Vladimir Batagelj : Batagelj is particularly known for his work on Pajek, a freely available software for analysis and visualization of large networks. He began work on Pajek in 1996 with Andrej Mrvar, who was then his PhD student.
Vladimir Batagelj : First prizes for contributions (with Andrej Mrvar) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. In 2007 the book Generalized blockmodeling was awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association In 2007 he was awarded (together with Anuška Ferligoj) the Simmel Award by INSNA. In 2013, Vladimir Batagelj and Andrej Mrvar received the INSNA's William D. Richards Software award for their work on Pajek.
Vladimir Batagelj : Vladimir Batagelj, Social Network Analysis, Large-Scale [1]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 8245–8265. Vladimir Batagelj, Complex Networks, Visualization of [2]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 1253–1268. Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press 2005 (ISBN 0-521-60262-9). ESNA in Japanese, TDU, 2010. Patrick Doreian, Vladimir Batagelj, Anuška Ferligoj, Mark Granovetter (Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)
Vladimir Batagelj : Vladimir Batagelj publications indexed by Google Scholar ResearcherId: B-9105-2008
Blockmodeling : Blockmodeling is a set or a coherent framework, that is used for analyzing social structure and also for setting procedure(s) for partitioning (clustering) social network's units (nodes, vertices, actors), based on specific patterns, which form a distinctive structure through interconnectivity. It is primarily used in statistics, machine learning and network science. As an empirical procedure, blockmodeling assumes that all the units in a specific network can be grouped together to such extent to which they are equivalent. Regarding equivalency, it can be structural, regular or generalized. Using blockmodeling, a network can be analyzed using newly created blockmodels, which transforms large and complex network into a smaller and more comprehensible one. At the same time, the blockmodeling is used to operationalize social roles. While some contend that the blockmodeling is just clustering methods, Bonacich and McConaghy state that "it is a theoretically grounded and algebraic approach to the analysis of the structure of relations". Blockmodeling's unique ability lies in the fact that it considers the structure not just as a set of direct relations, but also takes into account all other possible compound relations that are based on the direct ones. The principles of blockmodeling were first introduced by Francois Lorrain and Harrison C. White in 1971. Blockmodeling is considered as "an important set of network analytic tools" as it deals with delineation of role structures (the well-defined places in social structures, also known as positions) and the discerning the fundamental structure of social networks.: 2, 3 According to Batagelj, the primary "goal of blockmodeling is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily". Blockmodeling was at first used for analysis in sociometry and psychometrics, but has now spread also to other sciences.
Blockmodeling : A network as a system is composed of (or defined by) two different sets: one set of units (nodes, vertices, actors) and one set of links between the units. Using both sets, it is possible to create a graph, describing the structure of the network. During blockmodeling, the researcher is faced with two problems: how to partition the units (e.g., how to determine the clusters (or classes), that then form vertices in a blockmodel) and then how to determine the links in the blockmodel (and at the same time the values of these links). In the social sciences, the networks are usually social networks, composed of several individuals (units) and selected social relationships among them (links). Real-world networks can be large and complex; blockmodeling is used to simplify them into smaller structures that can be easier to interpret. Specifically, blockmodeling partitions the units into clusters and then determines the ties among the clusters. At the same time, blockmodeling can be used to explain the social roles existing in the network, as it is assumed that the created cluster of units mimics (or is closely associated with) the units' social roles. Blockmodeling can thus be defined as a set of approaches for partitioning units into clusters (also known as positions) and links into blocks, which are further defined by the newly obtained clusters. A block (also blockmodel) is defined as a submatrix, that shows interconnectivity (links) between nodes, present in the same or different clusters. Each of these positions in the cluster is defined by a set of (in)direct ties to and from other social positions. These links (connections) can be directed or undirected; there can be multiple links between the same pair of objects or they can have weights on them. If there are not any multiple links in a network, it is called a simple network.: 8 A matrix representation of a graph is composed of ordered units, in rows and columns, based on their names. The ordered units with similar patterns of links are partitioned together in the same clusters. Clusters are then arranged together so that units from the same clusters are placed next to each other, thus preserving interconnectivity. In the next step, the units (from the same clusters) are transformed into a blockmodel. With this, several blockmodels are usually formed, one being core cluster and others being cohesive; a core cluster is always connected to cohesive ones, while cohesive ones cannot be linked together. Clustering of nodes is based on the equivalence, such as structural and regular. The primary objective of the matrix form is to visually present relations between the persons included in the cluster. These ties are coded dichotomously (as present or absent), and the rows in the matrix form indicate the source of the ties, while the columns represent the destination of the ties. Equivalence can have two basic approaches: the equivalent units have the same connection pattern to the same neighbors or these units have same or similar connection pattern to different neighbors. If the units are connected to the rest of network in identical ways, then they are structurally equivalent. Units can also be regularly equivalent, when they are equivalently connected to equivalent others. With blockmodeling, it is necessary to consider the issue of results being affected by measurement errors in the initial stage of acquiring the data.
Blockmodeling : Regarding what kind of network is undergoing blockmodeling, a different approach is necessary. Networks can be one–mode or two–mode. In the former all units can be connected to any other unit and where units are of the same type, while in the latter the units are connected only to the unit(s) of a different type.: 6–10 Regarding relationships between units, they can be single–relational or multi–relational networks. Further more, the networks can be temporal or multilevel and also binary (only 0 and 1) or signed (allowing negative ties)/values (other values are possible) networks. Different approaches to blockmodeling can be grouped into two main classes: deterministic blockmodeling and stochastic blockmodeling approaches. Deterministic blockmodeling is then further divided into direct and indirect blockmodeling approaches. Among direct blockmodeling approaches are: structural equivalence and regular equivalence. Structural equivalence is a state, when units are connected to the rest of the network in an identical way(s), while regular equivalence occurs when units are equally related to equivalent others (units are not necessarily sharing neighbors, but have neighbour that are themselves similar).: 24 Indirect blockmodeling approaches, where partitioning is dealt with as a traditional cluster analysis problem (measuring (dis)similarity results in a (dis)similarity matrix), are: conventional blockmodeling, generalized blockmodeling: generalized blockmodeling of binary networks, generalized blockmodeling of valued networks and generalized homogeneity blockmodeling, prespecified blockmodeling. According to Brusco and Steinley (2011), the blockmodeling can be categorized (using a number of dimensions): deterministic or stochastic blockmodeling, one–mode or two–mode networks, signed or unsigned networks, exploratory or confirmatory blockmodeling.
Blockmodeling : Blockmodels (sometimes also block models) are structures in which: vertices (e.g., units, nodes) are assembled within a cluster, with each cluster identified as a vertex; from such vertices a graph can be constructed; combinations of all the links (ties), represented in a block as a single link between positions, while at the same time constructing one tie for each block. In a case, when there are no ties in a block, there will be no ties between the two positions that define the block. Computer programs can partition the social network according to pre-set conditions.: 333 When empirical blocks can be reasonably approximated in terms of ideal blocks, such blockmodels can be reduced to a blockimage, which is a representation of the original network, capturing its underlying 'functional anatomy'. Thus, blockmodels can "permit the data to characterize their own structure", and at the same time not seek to manifest a preconceived structure imposed by the researcher. Blockmodels can be created indirectly or directly, based on the construction of the criterion function. Indirect construction refers to a function, based on "compatible (dis)similarity measure between paris of units", while the direct construction is "a function measuring the fit of real blocks induced by a given clustering to the corresponding ideal blocks with perfect relations within each cluster and between clusters according to the considered types of connections (equivalence)".
Blockmodeling : Blockmodeling is done with specialized computer programs, dedicated to the analysis of networks or blockmodeling in particular, as: BLOCKS (Tom Snijders), CONCOR, Model (Vladimir Batagelj), Model2 (Vladimir Batagelj), Pajek (Vladimir Batagelj and Andrej Mrvar), R–package Blockmodeling (Aleš Žiberna), StOCNET (Tom Snijders),...
Blockmodeling : Stochastic block model Mathematical sociology Role assignment Multiobjective blockmodeling Blockmodeling linked networks == References ==
Confirmatory blockmodeling : Confirmatory blockmodeling is a deductive approach in blockmodeling, where a blockmodel (or part of it) is prespecify before the analysis, and then the analysis is fit to this model. When only a part of analysis is prespecify (like individual cluster(s) or location of the block types), it is called partially confirmatory blockmodeling.: 26–27 This is so-called indirect approach, where the blockmodeling is done on the blockmodel fitting (e.g., a priori hypothesized blockmodel).: 215 Opposite approach to the confirmatory blockmodeling is an inductive exploratory blockmodeling.: 27
Deterministic blockmodeling : Deterministic blockmodeling is an approach in blockmodeling that does not assume a probabilistic model, and instead relies on the exact or approximate algorithms, which are used to find blockmodel(s). This approach typically minimizes some inconsistency that can occur with the ideal block structure. Such analysis is focused on clustering (grouping) of the network (or adjacency matrix) that is obtained with minimizing an objective function, which measures discrepancy from the ideal block structure. However, some indirect approaches (or methods between direct and indirect approaches, such as CONCOR) do not explicitly minimize inconsistencies or optimize some criterion function. This approach was popularized in the 1970s, due to the presence of two computer packages (CONCOR and STRUCTURE) that were used to "find a permutation of the rows and columns in the adjacency matrix leading to an approximate block structure". The opposite approach to deterministic blockmodeling is a stochastic blockmodeling approach.
Deterministic blockmodeling : Blockmodeling == References ==
Exploratory blockmodeling : Exploratory blockmodeling is an (inductive) approach (or a group of approaches) in blockmodeling regarding the specification of an ideal blockmodel.: 234 This approach, also known as hypotheses-generating, is the simplest approach, as it "merely involves the definition of the block types permitted as well as of the number of clusters." With this approach, researcher usually defines the best possible blockmodel, which then represent the base for the analysis of the whole network. This approach is usually based on:: 284 previous analyses and theoretical considerations, using stricker blockmodel and block types, where the structural equivalence is stricker than the regular equivalence and using smaller number of classes. The opposite approach is called a confirmatory blockmodeling. == References ==
Generalized blockmodeling : In generalized blockmodeling, the blockmodeling is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Generalized blockmodeling was introduced in 1994 by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj.
Generalized blockmodeling : Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the blockmodel is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using criterion function.: 16–17 Using local optimization procedure, firstly the initial clustering (with specified number of clusters is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a vertex is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as blockmodels for the future exploration of the network. Different types of generalized blockmodeling are: generalized binary blockmodeling, generalized valued blockmodeling and generalized homogeneity blockmodeling.
Generalized blockmodeling : According to Patrick Doreian, the benefits of generalized blockmodeling, are as follows: usage of explicit criterion function, compatible with a given type of equivalence, results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; with generalized blockmodeling it's possible to specify new types of blockmodels; this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; researcher can speficy which (pair of) vertices must be (not) clustered together; this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block.

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