{"title":"Decision Tree Classifier","authors":"","doi":"10.1017/9781108564175.024","DOIUrl":"https://doi.org/10.1017/9781108564175.024","url":null,"abstract":"in a d-dimensional space with categorical or numeric attributes X j (with j = 1,. .. , d). Further, let Y denote the (categorical) class attribute, with y i denoting the class label of point x i. We assume that there are k distinct classes, so that y i ∈ {c 1 , c 2 , · · · , c k }. The goal of decision tree classification is to build a partition-based model that predicts the classˆy i for each point x i. A decision tree uses axis-parallel hyperplanes to recursively partition the data space, so that the resulting set of partitions are as \" pure \" as possible in terms of the class labels. A decision tree consists of internal nodes that represent the decisions or split points, and leaf nodes that represent regions of the data space labeled with the majority class. A region D ⊆ D is characterized by the subset of data points that belong to that partition. Figure 19.1 illustrates the recursive hyperplane partitioning, and the corresponding decision tree. Purity The purity of a region is defined as the fraction of points with the majority where n i is the number of points x with class labels y = c i , and n is the total number of points, in the region D. Axis-Parallel Hyperplanes Recall that a hyperplane h(x) is given as h(x): w T x + b = 0 (19.2) where w ∈ R d is the weight vector that is normal to the hyperplane, and b is the offset of the hyperplane from the origin. Since a decision tree considers only axis-parallel hyperplanes, the weight vector w is restricted a priori to the set of the standard DRAFT @ 2012-10-14 21:52. Please do not distribute. Feedback is Welcome. Note that this book shall be available for purchase from Cambridge University Press and other standard distribution channels, that no unauthorized distribution shall be allowed, and that the reader may make one copy only for personal on-screen use.","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131045480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linear Discriminant Analysis","authors":"Hong Zhou","doi":"10.1017/9781108564175.025","DOIUrl":"https://doi.org/10.1017/9781108564175.025","url":null,"abstract":"Linear discriminant analysis (LDA) and the related Fisher's linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterize or separate 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 also closely related to principal component analysis (PCA) and factor analysis in that 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 on the other hand 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. The objective of LDA is to perform dimensionality reduction while preserving as much of the class discriminatory information as possible Assume we have a set of D-dimensional samples {x(1, x(2, …, x(N}, N1 of which belong to class ω1, and N2 to class ω2. We seek to obtain a scalar y by projecting the samples x onto a line Y = w T x Of all the possible lines we would like to select the one that maximizes the separability of the scalars. This is illustrated for the two-dimensional case in the following figures In order to find a good projection vector, we need to define a measure of separation between the projections The mean vector of each class in x and y feature space is We could then choose the distance between the projected means as our objective Function","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130977565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sequence Mining","authors":"Kevin Duh","doi":"10.1017/9781108564175.013","DOIUrl":"https://doi.org/10.1017/9781108564175.013","url":null,"abstract":"Many real world applications, such as in bioinformatics, web mining, text mining and so on, have to deal with sequential/temporal data. Sequence mining helps to discover frequent sequential patterns across time or positions in a given data set. Let Σ denote an alphabet, defined as a finite set of characters or symbols, and let |Σ| denote its cardinality. A sequence is defined as an ordered list of symbols, and is written as s = s 1 s 2. .. s k , where s i ∈ Σ is a symbol at position i. Here |s| = k denotes the length of the sequence. A sequence with length k is also called a k-sequence. Let Σ be the set of all possible sequences (or strings) that can be constructed using the symbols in Σ, including the empty string ∅ (which has length zero). A sequence s = s 1 s 2. .. s ns is a called subsequence of another sequence r = and the order of symbols is preserved, i.e., if j 1 < j 2 < · · · < j ns. In other words, the sequence s is embedded in r, but there might be intervening gaps between consecutive elements of s in the embedding. If s ⊆ r, we also say that r contains s. s is called a substring or consecutive subsequence of r provided s 1 s 2. .. s ns = r j r j+1. .. r j+ns−1. In other words for substrings we do not allow any gaps between the elements of s in the embedding.","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132091384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral and Graph Clustering","authors":"","doi":"10.1017/9781108564175.020","DOIUrl":"https://doi.org/10.1017/9781108564175.020","url":null,"abstract":"","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"108 39","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113944882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Neural Networks","authors":"G. Garson","doi":"10.1017/9781108564175.031","DOIUrl":"https://doi.org/10.1017/9781108564175.031","url":null,"abstract":"Artificial neural networks have emerged from the studies of how brain performs. The human brain consists of many millions of individual processing elements, called neurons that are highly interconnected. Information from the outputs of the neurons, in the form of electric pulses is received by the cells at connections called synapses. The synapses connect to the cell inputs, or dendrites and the single output the neuron appears at the axon. An electric pulse is sent down the axon when the total input stimuli for all of the dendrites exceed a certain threshold. Artificial neural networks are made up of simplified individual models of the biological neuron that are connected together to form a network. Information is stored in the network in the form of weights or different connections strengths associated with synapses in the artificial neuron models.","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127172450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Probabilistic Classification","authors":"S. Lall, S. Boyd","doi":"10.1017/9781108564175.023","DOIUrl":"https://doi.org/10.1017/9781108564175.023","url":null,"abstract":"Let the training dataset D consist of n points x i in a d-dimensional space, and let y i denote the class for each point, with y i ∈ {c 1 , c 2 , · · · , c k }. The Bayes classifier makes uses of the Bayes theorem to predict the class for a new test instance, x. It estimates the posterior probability P (c i |x) for each class c i , and chooses the class that has the largest probability. That is, the predicted labeî y for x is given asˆy = arg max i {P (c i |x)} (21.1) The Bayes theorem allows us to invert the posterior probability in terms of the likelihood and prior probability, as follows","PeriodicalId":381018,"journal":{"name":"Data Mining and Machine Learning","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121498886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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