DOCSLIB.ORG

Loss Functions for Clustering in Multi-Instance Learning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Loss Functions for Clustering in Multi-Instance Learning Loss Functions for Clustering in Multi-instance Learning Marek Dědič1;2, Tomáš Pevný3, Lukáš Bajer2, Martin Holeňa4 1 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, Czech Republic 2 Cisco Systems, Inc., Karlovo náměstı́ 10, Prague, Czech Republic 3 Faculty of Electrical Engineering, Czech Technical University in Prague, Karlovo náměstí 13, Prague, Czech Republic 4 Institute of Computer Science, Czech Academy of Sciences, Pod vodárenskou věží 2, Prague, Czech Republic Abstract: Multi-instance learning belongs to one of re- their work significantly by deciding about whole clusters cently fast developing areas of machine learning. It is a su- of servers instead of each one individually. pervised learning method and this paper reports research A key prerequisite for the application of multi-instance into its unsupervised counterpart, multi-instance cluster- learning to clustering is that loss functions for clustering, ing. Whereas traditional clustering clusters points, multi- originally proposed for points, are adapted for bags. In this instance clustering clusters bags, i.e. multisets of points or paper, such an adaptation is outlined for three sophisticated of other kinds of objects. The paper focuses on the problem loss functions: contrastive predictive coding, triplet loss of loss functions for clustering. Three sophisticated loss and magnet loss. functions used for clustering of points, contrastive predic- Of the three proposed methods, triplet loss and magnet tive coding, triplet loss and magnet loss, are elaborated for loss utilize some labels as part of the training process and multi-instance clustering. Finally, they are compared on 18 so are not truly unsupervised. As such, the authors expect benchmark datasets, as well as on a real-world dataset. them to outperform the method based on contrastive pre- dictive coding, which on the other hand is the most promis- ing and innovative approach from the theoretical point of 1 Introduction view. In the next section, basic properties of multi-instance Multi-instance learning (MIL) belongs to recently fast de- learning and clustering are briefly introduced. The adapta- veloping areas of machine learning. Though it is a super- tion of the three considered loss functions is explained in vised learning method, this paper addresses its application Section 3. A comprehensive comparison of them is then to unsupervised learning – clustering. Whereas traditional presented in Section 4. clustering is one of points, multi-instance clustering clus- ters multisets of points, also known as bags. Such a group- ing of the points into bags is considered a property of the 2 Multi-instance Learning and Clustering problem at hand and therefore sourced from the input data. While there have been some previous attempts at multi- Multi-instance learning (MIL) was first described by [7]. instance clustering, they use pairwise relations between all In its original form, it was developed and used for su- points in all bags, quickly becoming unwieldy and compu- pervised learning. Some prior art also exists for unsu- tationally infeasible. Our work uses multi-instance learn- pervised learning, such as [5, 31], however, it uses pair- ing as a general toolkit for learning representations of bags wise instance distances as a basis for clustering, which and then clusters the bags using those representations. This doesn’t properly utilize the inherent structure of the data builds on previous works by the authors and enables clus- and quickly becomes computationally infeasible. tering of arbitrary data-structures by expressing them as The MIL paradigm is a type of representation learning hierarchies of multi-instance problems and then using the on data which has some internal structure. Therefore, it internal structure to better solve the problems at hand. views a sample as a bag (i.e. a multiset) of an arbitrary Multi-instance clustering is evaluated in Section 4 in number of objects. The basic elements of MIL are sam- the application domain of computer and network security. ples from a space X and their corresponding labels from a While this is only one of many possible applications of space Y (a space of classes). Compared to usual supervised MIL due to its general expressive power for structured data, learning, MIL replaces individual instances with bags of it is the domain of choice for the authors. Here, MIL is instances from the space X such that every instance in X used to represent user activity as a bag of network con- belongs to at least one bag from the bag-space B. nections for each user in a fixed time window. For clus- [7] provides an example of a multi-instance problem tering, this enables detection of compromised users based where each bag represents a key chain with some keys (in- on their complex behaviours. Clustering opens a new win- stances). To solve the problem of finding which key opens dow of opportunity here by e.g. grouping servers with sim- a particular lock, a “proxy” MIL problem is presented – de- ilar behaviour together, allowing a human analyst to boost termining which key chain opens that particular lock. This Copyright ©2020 for this paper by its authors. Use permitted under line of thinking leads to the pivotal definition of the label Creative Commons License Attribution 4.0 International (CC BY 4.0). of a bag as being positive iff the bag contains at least one positive instance. In later works such as [23], this interpre- The functions fI and fB are realized by a deep neural net- tation of MIL is abandoned in favor of a more general one. work using ReLU as its activation function. The aggrega- An instance is no longer viewed as having a meaning in and tion g is realized as an element-wise mean or maximum of of itself, but only in the context of its bag. The notion of all the vectors. an instance-level label is dropped because in this interpre- The structure of the data is highly exploited using the tation, the bag is the atomic unit of interest. To this end, embedded-space paradigm. The approach uses multiple the embedded-space paradigm, described in Section 2.2, is layers of MIL nested in each other – the instances of a used. bag do not necessarily need to be feature vectors, but can be bag in themselves. The HTTP traffic of a particular 2.1 A probabilistic formulation of multi-instance client is therefore represented as a bag of all second-level learning domains the client has exchanged datagrams with. Each second-level domain is then represented as a bag of indi- A probabilistic way of describing multi-instance learning vidual URLs which the client has connected to. Individual was first introduced in [23] and builds on the previous work URLs are then split into 3 parts, domain, path and query, [19]. and each part is represented as a bag of tokens, which can Let for the space X exist a measurable space (X ; A), then be broken down even further. In the end, the model X where A is a σ-algebra on X . Let P denote the set of consists of 5 nested MIL problems. all probability measures on (X ; A). A bag B is viewed as A benefit of MIL is also its easy use for explainability a random sample with replacement of a random variable and interpretability. The authors of [22] present a way to X governed by a particular probability distribution pB 2 P , extract indicators of compromise and explain the decision that is using the learned MIL model. B = fxijxi ∼ pB; i 2 f1; : : : ; mgg where m 2 N: (1) 2.3 Clustering 2.2 Embedded-space paradigm for solving Clustering is a prime example of a problem typically asso- multi-instance problems ciated with unsupervised learning. The problem at hand, however, is not one of clustering ordinary number vectors While it is possible to use several approaches to solv- in a linear space. Instead, a clustering of objects repre- ing multi-instance problems, in this work, the embedded- sented by bags is explored, as that is the problem solved for space paradigm was used. For an overview of the other datasets introduced later in Section 4. While [28] present a paradigms, see [6]. clustering of bags using a modified Hausdorff distance on In the embedded space paradigm, labels are only defined bags and [17] present a clustering using maximum mean on the level of bags. In order for these bag labels to be discrepancy, a different approach to clustering of bags is φ : B! X¯ learned, an embedding function of the form used in this work. Among the main reasons for this choice X¯ must be defined, where is a latent space, which may or is the prohibitively high computational complexity of the X may not be identical to . Using this function, each bag previously mentioned approaches on large datasets and the φ (B) 2 X¯ can be represented by an object , which makes it possibility of utilizing the representations previously intro- possible to use any off-the-shelf supervised learning algo- duced in [22]. X¯ rithm acting on . Among the simplest embedding func- An approach based on the embedded-space paradigm for tions are, e.g. element-wise minimum, maximum, and MIL was chosen. In order to utilize the structure of the mean. A more complicated embedding function may for data, a MIL model is used to represent each bag in the la- example apply a neural network to each instance of the bag tent space X¯. This presents the issue of how to train the em- and subsequently pool the instances using one of the afore- bedding function φ, because its learning in standard MIL mentioned functions.
Load more

Recommended publications
  • Adaptive Wavelet Clustering for Highly Noisy Data
    Adaptive Wavelet Clustering for Highly Noisy Data Zengjian Chen Jiayi Liu Yihe Deng Department of Computer Science Department of Computer Science Department of Mathematics Huazhong University of University of Massachusetts Amherst University of California, Los Angeles Science and Technology Massachusetts, USA California, USA Wuhan, China [email protected] [email protected] [email protected] Kun He* John E. Hopcroft Department of Computer Science Department of Computer Science Huazhong University of Science and Technology Cornell University Wuhan, China Ithaca, NY, USA [email protected] [email protected] Abstract—In this paper we make progress on the unsupervised Based on the pioneering work of Sheikholeslami that applies task of mining arbitrarily shaped clusters in highly noisy datasets, wavelet transform, originally used for signal processing, on which is a task present in many real-world applications. Based spatial data clustering [12], we propose a new wavelet based on the fundamental work that first applies a wavelet transform to data clustering, we propose an adaptive clustering algorithm, algorithm called AdaWave that can adaptively and effectively denoted as AdaWave, which exhibits favorable characteristics for uncover clusters in highly noisy data. To tackle general appli- clustering. By a self-adaptive thresholding technique, AdaWave cations, we assume that the clusters in a dataset do not follow is parameter free and can handle data in various situations. any specific distribution and can be arbitrarily shaped. It is deterministic, fast in linear time, order-insensitive, shape- To show the hardness of the clustering task, we first design insensitive, robust to highly noisy data, and requires no pre- knowledge on data models.
    [Show full text]
  • Logistic Regression Trained with Different Loss Functions Discussion
    Logistic Regression Trained with Different Loss Functions Discussion CS6140 1 Notations We restrict our discussions to the binary case. 1 g(z) = 1 + e−z @g(z) g0(z) = = g(z)(1 − g(z)) @z 1 1 h (x) = g(wx) = = w −wx − P wdxd 1 + e 1 + e d P (y = 1jx; w) = hw(x) P (y = 0jx; w) = 1 − hw(x) 2 Maximum Likelihood Estimation 2.1 Goal Maximize likelihood: L(w) = p(yjX; w) m Y = p(yijxi; w) i=1 m Y yi 1−yi = (hw(xi)) (1 − hw(xi)) i=1 1 Or equivalently, maximize the log likelihood: l(w) = log L(w) m X = yi log h(xi) + (1 − yi) log(1 − h(xi)) i=1 2.2 Stochastic Gradient Descent Update Rule @ 1 1 @ j l(w) = (y − (1 − y) ) j g(wxi) @w g(wxi) 1 − g(wxi) @w 1 1 @ = (y − (1 − y) )g(wxi)(1 − g(wxi)) j wxi g(wxi) 1 − g(wxi) @w j = (y(1 − g(wxi)) − (1 − y)g(wxi))xi j = (y − hw(xi))xi j j j w := w + λ(yi − hw(xi)))xi 3 Least Squared Error Estimation 3.1 Goal Minimize sum of squared error: m 1 X L(w) = (y − h (x ))2 2 i w i i=1 3.2 Stochastic Gradient Descent Update Rule @ @h (x ) L(w) = −(y − h (x )) w i @wj i w i @wj j = −(yi − hw(xi))hw(xi)(1 − hw(xi))xi j j j w := w + λ(yi − hw(xi))hw(xi)(1 − hw(xi))xi 4 Comparison 4.1 Update Rule For maximum likelihood logistic regression: j j j w := w + λ(yi − hw(xi)))xi 2 For least squared error logistic regression: j j j w := w + λ(yi − hw(xi))hw(xi)(1 − hw(xi))xi Let f1(h) = (y − h); y 2 f0; 1g; h 2 (0; 1) f2(h) = (y − h)h(1 − h); y 2 f0; 1g; h 2 (0; 1) When y = 1, the plots of f1(h) and f2(h) are shown in figure 1.
    [Show full text]
  • Cluster Analysis, a Powerful Tool for Data Analysis in Education
    International Statistical Institute, 56th Session, 2007: Rita Vasconcelos, Mßrcia Baptista Cluster Analysis, a powerful tool for data analysis in Education Vasconcelos, Rita Universidade da Madeira, Department of Mathematics and Engeneering Caminho da Penteada 9000-390 Funchal, Portugal E-mail: [email protected] Baptista, Márcia Direcção Regional de Saúde Pública Rua das Pretas 9000 Funchal, Portugal E-mail: [email protected] 1. Introduction A database was created after an inquiry to 14-15 - year old students, which was developed with the purpose of identifying the factors that could socially and pedagogically frame the results in Mathematics. The data was collected in eight schools in Funchal (Madeira Island), and we performed a Cluster Analysis as a first multivariate statistical approach to this database. We also developed a logistic regression analysis, as the study was carried out as a contribution to explain the success/failure in Mathematics. As a final step, the responses of both statistical analysis were studied. 2. Cluster Analysis approach The questions that arise when we try to frame socially and pedagogically the results in Mathematics of 14-15 - year old students, are concerned with the types of decisive factors in those results. It is somehow underlying our objectives to classify the students according to the factors understood by us as being decisive in students’ results. This is exactly the aim of Cluster Analysis. The hierarchical solution that can be observed in the dendogram presented in the next page, suggests that we should consider the 3 following clusters, since the distances increase substantially after it: Variables in Cluster1: mother qualifications; father qualifications; student’s results in Mathematics as classified by the school teacher; student’s results in the exam of Mathematics; time spent studying.
    [Show full text]
  • Regularized Regression Under Quadratic Loss, Logistic Loss, Sigmoidal Loss, and Hinge Loss
    Regularized Regression under Quadratic Loss, Logistic Loss, Sigmoidal Loss, and Hinge Loss Here we considerthe problem of learning binary classiers. We assume a set X of possible inputs and we are interested in classifying inputs into one of two classes. For example we might be interesting in predicting whether a given persion is going to vote democratic or republican. We assume a function Φ which assigns a feature vector to each element of x — we assume that for x ∈ X we have d Φ(x) ∈ R . For 1 ≤ i ≤ d we let Φi(x) be the ith coordinate value of Φ(x). For example, for a person x we might have that Φ(x) is a vector specifying income, age, gender, years of education, and other properties. Discrete properties can be represented by binary valued fetures (indicator functions). For example, for each state of the United states we can have a component Φi(x) which is 1 if x lives in that state and 0 otherwise. We assume that we have training data consisting of labeled inputs where, for convenience, we assume that the labels are all either −1 or 1. S = hx1, yyi,..., hxT , yT i xt ∈ X yt ∈ {−1, 1} Our objective is to use the training data to construct a predictor f(x) which predicts y from x. Here we will be interested in predictors of the following form where β ∈ Rd is a parameter vector to be learned from the training data. fβ(x) = sign(β · Φ(x)) (1) We are then interested in learning a parameter vector β from the training data.
    [Show full text]
  • The Central Limit Theorem in Differential Privacy
    Privacy Loss Classes: The Central Limit Theorem in Differential Privacy David M. Sommer Sebastian Meiser Esfandiar Mohammadi ETH Zurich UCL ETH Zurich [email protected] [email protected] [email protected] August 12, 2020 Abstract Quantifying the privacy loss of a privacy-preserving mechanism on potentially sensitive data is a complex and well-researched topic; the de-facto standard for privacy measures are "-differential privacy (DP) and its versatile relaxation (, δ)-approximate differential privacy (ADP). Recently, novel variants of (A)DP focused on giving tighter privacy bounds under continual observation. In this paper we unify many previous works via the privacy loss distribution (PLD) of a mechanism. We show that for non-adaptive mechanisms, the privacy loss under sequential composition undergoes a convolution and will converge to a Gauss distribution (the central limit theorem for DP). We derive several relevant insights: we can now characterize mechanisms by their privacy loss class, i.e., by the Gauss distribution to which their PLD converges, which allows us to give novel ADP bounds for mechanisms based on their privacy loss class; we derive exact analytical guarantees for the approximate randomized response mechanism and an exact analytical and closed formula for the Gauss mechanism, that, given ", calculates δ, s.t., the mechanism is ("; δ)-ADP (not an over- approximating bound). 1 Contents 1 Introduction 4 1.1 Contribution . .4 2 Overview 6 2.1 Worst-case distributions . .6 2.2 The privacy loss distribution . .6 3 Related Work 7 4 Privacy Loss Space 7 4.1 Privacy Loss Variables / Distributions .
    [Show full text]
  • Cluster Analysis Y H Chan
    Basic Statistics For Doctors Singapore Med J 2005; 46(4) : 153 CME Article Biostatistics 304. Cluster analysis Y H Chan In Cluster analysis, we seek to identify the “natural” SPSS offers three separate approaches to structure of groups based on a multivariate profile, Cluster analysis, namely: TwoStep, K-Means and if it exists, which both minimises the within-group Hierarchical. We shall discuss the Hierarchical variation and maximises the between-group variation. approach first. This is chosen when we have little idea The objective is to perform data reduction into of the data structure. There are two basic hierarchical manageable bite-sizes which could be used in further clustering procedures – agglomerative or divisive. analysis or developing hypothesis concerning the Agglomerative starts with each object as a cluster nature of the data. It is exploratory, descriptive and and new clusters are combined until eventually all non-inferential. individuals are grouped into one large cluster. Divisive This technique will always create clusters, be it right proceeds in the opposite direction to agglomerative or wrong. The solutions are not unique since they methods. For n cases, there will be one-cluster to are dependent on the variables used and how cluster n-1 cluster solutions. membership is being defined. There are no essential In SPSS, go to Analyse, Classify, Hierarchical Cluster assumptions required for its use except that there must to get Template I be some regard to theoretical/conceptual rationale upon which the variables are selected. Template I. Hierarchical cluster analysis. For simplicity, we shall use 10 subjects to demonstrate how cluster analysis works.
    [Show full text]
  • Bayesian Classifiers Under a Mixture Loss Function
    Hunting for Significance: Bayesian Classifiers under a Mixture Loss Function Igar Fuki, Lawrence Brown, Xu Han, Linda Zhao February 13, 2014 Abstract Detecting significance in a high-dimensional sparse data structure has received a large amount of attention in modern statistics. In the current paper, we introduce a compound decision rule to simultaneously classify signals from noise. This procedure is a Bayes rule subject to a mixture loss function. The loss function minimizes the number of false discoveries while controlling the false non discoveries by incorporating the signal strength information. Based on our criterion, strong signals will be penalized more heavily for non discovery than weak signals. In constructing this classification rule, we assume a mixture prior for the parameter which adapts to the unknown spar- sity. This Bayes rule can be viewed as thresholding the \local fdr" (Efron 2007) by adaptive thresholds. Both parametric and nonparametric methods will be discussed. The nonparametric procedure adapts to the unknown data structure well and out- performs the parametric one. Performance of the procedure is illustrated by various simulation studies and a real data application. Keywords: High dimensional sparse inference, Bayes classification rule, Nonparametric esti- mation, False discoveries, False nondiscoveries 1 2 1 Introduction Consider a normal mean model: Zi = βi + i; i = 1; ··· ; p (1) n T where fZigi=1 are independent random variables, the random errors (1; ··· ; p) follow 2 T a multivariate normal distribution Np(0; σ Ip), and β = (β1; ··· ; βp) is a p-dimensional unknown vector. For simplicity, in model (1), we assume σ2 is known. Without loss of generality, let σ2 = 1.
    [Show full text]
  • Cluster Analysis Objective: Group Data Points Into Classes of Similar Points Based on a Series of Variables
    Multivariate Fundamentals: Distance Cluster Analysis Objective: Group data points into classes of similar points based on a series of variables Useful to find the true groups that are assumed to really exist, BUT if the analysis generates unexpected groupings it could inform new relationships you might want to investigate Also useful for data reduction by finding which data points are similar and allow for subsampling of the original dataset without losing information Alfred Louis Kroeber (1876-1961) The math behind cluster analysis A B C D … A 0 1.8 0.6 3.0 Once we calculate a distance matrix between points we B 1.8 0 2.5 3.3 use that information to build a tree C 0.6 2.5 0 2.2 D 3.0 3.3 2.2 0 … Ordination – visualizes the information in the distance calculations The result of a cluster analysis is a tree or dendrogram 0.6 1.8 4 2.5 3.0 2.2 3 3.3 2 distance 1 If distances are not equal between points we A C D can draw a “hanging tree” to illustrate distances 0 B Building trees & creating groups 1. Nearest Neighbour Method – create groups by starting with the smallest distances and build branches In effect we keep asking data matrix “Which plot is my nearest neighbour?” to add branches 2. Centroid Method – creates a group based on smallest distance to group centroid rather than group member First creates a group based on small distance then uses the centroid of that group to find which additional points belong in the same group 3.
    [Show full text]
  • A Unified Bias-Variance Decomposition for Zero-One And
    A Unified Bias-Variance Decomposition for Zero-One and Squared Loss Pedro Domingos Department of Computer Science and Engineering University of Washington Seattle, Washington 98195, U.S.A. [email protected] http://www.cs.washington.edu/homes/pedrod Abstract of a bias-variance decomposition of error: while allowing a more intensive search for a single model is liable to increase The bias-variance decomposition is a very useful and variance, averaging multiple models will often (though not widely-used tool for understanding machine-learning always) reduce it. As a result of these developments, the algorithms. It was originally developed for squared loss. In recent years, several authors have proposed bias-variance decomposition of error has become a corner- decompositions for zero-one loss, but each has signif- stone of our understanding of inductive learning. icant shortcomings. In particular, all of these decompo- sitions have only an intuitive relationship to the original Although machine-learning research has been mainly squared-loss one. In this paper, we define bias and vari- concerned with classification problems, using zero-one loss ance for an arbitrary loss function, and show that the as the main evaluation criterion, the bias-variance insight resulting decomposition specializes to the standard one was borrowed from the field of regression, where squared- for the squared-loss case, and to a close relative of Kong loss is the main criterion. As a result, several authors have and Dietterich’s (1995) one for the zero-one case. The proposed bias-variance decompositions related to zero-one same decomposition also applies to variable misclassi- loss (Kong & Dietterich, 1995; Breiman, 1996b; Kohavi & fication costs.
    [Show full text]
  • Are Loss Functions All the Same?
    Are Loss Functions All the Same? L. Rosasco∗ E. De Vito† A. Caponnetto‡ M. Piana§ A. Verri¶ September 30, 2003 Abstract In this paper we investigate the impact of choosing different loss func- tions from the viewpoint of statistical learning theory. We introduce a convexity assumption - which is met by all loss functions commonly used in the literature, and study how the bound on the estimation error changes with the loss. We also derive a general result on the minimizer of the ex- pected risk for a convex loss function in the case of classification. The main outcome of our analysis is that, for classification, the hinge loss appears to be the loss of choice. Other things being equal, the hinge loss leads to a convergence rate practically indistinguishable from the logistic loss rate and much better than the square loss rate. Furthermore, if the hypothesis space is sufficiently rich, the bounds obtained for the hinge loss are not loosened by the thresholding stage. 1 Introduction A main problem of statistical learning theory is finding necessary and sufficient condi- tions for the consistency of the Empirical Risk Minimization principle. Traditionally, the role played by the loss is marginal and the choice of which loss to use for which ∗INFM - DISI, Universit`adi Genova, Via Dodecaneso 35, 16146 Genova (I) †Dipartimento di Matematica, Universit`adi Modena, Via Campi 213/B, 41100 Modena (I), and INFN, Sezione di Genova ‡DISI, Universit`adi Genova, Via Dodecaneso 35, 16146 Genova (I) §INFM - DIMA, Universit`adi Genova, Via Dodecaneso 35, 16146 Genova (I) ¶INFM - DISI, Universit`adi Genova, Via Dodecaneso 35, 16146 Genova (I) 1 problem is usually regarded as a computational issue (Vapnik, 1995; Vapnik, 1998; Alon et al., 1993; Cristianini and Shawe Taylor, 2000).
    [Show full text]
  • Cluster Analysis: What It Is and How to Use It Alyssa Wittle and Michael Stackhouse, Covance, Inc
    PharmaSUG 2019 - Paper ST-183 Cluster Analysis: What It Is and How to Use It Alyssa Wittle and Michael Stackhouse, Covance, Inc. ABSTRACT A Cluster Analysis is a great way of looking across several related data points to find possible relationships within your data which you may not have expected. The basic approach of a cluster analysis is to do the following: transform the results of a series of related variables into a standardized value such as Z-scores, then combine these values and determine if there are trends across the data which may lend the data to divide into separate, distinct groups, or "clusters". A cluster is assigned at a subject level, to be used as a grouping variable or even as a response variable. Once these clusters have been determined and assigned, they can be used in your analysis model to observe if there is a significant difference between the results of these clusters within various parameters. For example, is a certain age group more likely to give more positive answers across all questionnaires in a study or integration? Cluster analysis can also be a good way of determining exploratory endpoints or focusing an analysis on a certain number of categories for a set of variables. This paper will instruct on approaches to a clustering analysis, how the results can be interpreted, and how clusters can be determined and analyzed using several programming methods and languages, including SAS, Python and R. Examples of clustering analyses and their interpretations will also be provided. INTRODUCTION A cluster analysis is a multivariate data exploration method gaining popularity in the industry.
    [Show full text]
  • Factors Versus Clusters
    Paper 2868-2018 Factors vs. Clusters Diana Suhr, SIR Consulting ABSTRACT Factor analysis is an exploratory statistical technique to investigate dimensions and the factor structure underlying a set of variables (items) while cluster analysis is an exploratory statistical technique to group observations (people, things, events) into clusters or groups so that the degree of association is strong between members of the same cluster and weak between members of different clusters. Factor and cluster analysis guidelines and SAS® code will be discussed as well as illustrating and discussing results for sample data analysis. Procedures shown will be PROC FACTOR, PROC CORR alpha, PROC STANDARDIZE, PROC CLUSTER, and PROC FASTCLUS. INTRODUCTION Exploratory factor analysis (EFA) investigates the possible underlying factor structure (dimensions) of a set of interrelated variables without imposing a preconceived structure on the outcome (Child, 1990). The analysis groups similar items to identify dimensions (also called factors or latent constructs). Exploratory cluster analysis (ECA) is a technique for dividing a multivariate dataset into “natural” clusters or groups. The technique involves identifying groups of individuals or objects that are similar to each other but different from individuals or objects in other groups. Cluster analysis, like factor analysis, makes no distinction between independent and dependent variables. Factor analysis reduces the number of variables by grouping them into a smaller set of factors. Cluster analysis reduces the number of observations by grouping them into a smaller set of clusters. There is no right or wrong answer to “how many factors or clusters should I keep?”. The answer depends on what you’re going to do with the factors or clusters.
    [Show full text]