Anomaly detection is an important problem that has been researched within diverse research areas and application domains. Many anomaly detection techniques have been specifically developed for certain application domains, while others are more generic. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection. We have grouped existing techniques into different categories based on the underlying approach adopted by each technique. For each category we have identified key assumptions, which are used by the techniques to differentiate between normal and anomalous behavior. When applying a given technique to a particular domain, these assumptions can be used as guidelines to assess the effectiveness of the technique in that domain. For each category, we provide a basic anomaly detection technique, and then show how the different existing techniques in that category are variants of the basic technique. This template provides an easier and more succinct understanding of the techniques belonging to each category. Further, for each category, we identify the advantages and disadvantages of the techniques in that category. We also provide a discussion on the computational complexity of the techniques since it is an important issue in real application domains. We hope that this survey will provide a better understanding of the different directions in which research has been done on this topic, and how techniques developed in one area can be applied in domains for which they were not intended to begin with.

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Anomaly detection: A survey

A Monte Carlo study compared 14 methods to test the statistical significance of the intervening variable effect. An intervening variable (mediator) transmits the effect of an independent variable to a dependent variable. The commonly used R. M. Baron and D. A. Kenny (1986) approach has low statistical power. Two methods based on the distribution of the product and 2 difference-in-coefficients methods have the most accurate Type I error rates and greatest statistical power except in 1 important case in which Type I error rates are too high. The best balance of Type I error and statistical power across all cases is the test of the joint significance of the two effects comprising the intervening variable effect.

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A comparison of methods to test mediation and other intervening variable effects.

Structural Equations with Latent Variables.

Mediation models are widely used, and there are many tests of the mediated effect. One of the most common questions that researchers have when planning mediation studies is, "How many subjects do I need to achieve adequate power when testing for mediation?" This article presents the necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters, to provide a guide for researchers when designing studies or applying for grants.

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Required Sample Size to Detect the Mediated Effect

List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

One of the most vexing of problems in data analysis is the determination of whether or not to discard some observations because they are inconsistent with the rest of the observations and/or the probability distribution assumed to be the underlying distribution of the data. One direction of research activity related to this problem is that of the study of robust statistical procedures (cf. Huber [1981]), primarily procedures for estimating population parameters which are insensitive to the effect of “outliers”, i.e., observations inconsistent with the assumed model of the random process generating the observations. Typically, the robust procedure involves some “trimming” or down-weighting procedure, wherein some fraction of the extreme observations are automatically eliminated or given less weight to guard against the potential effect of outliers.

Identification of Outliers

Consider the situation where X and Y are related by Y = α + βX, where α and β are unknown and where we observe X and Y with error, i.e., we observe x = X + u and y = Y + v. Assume that Eu = Ev = 0 and that the errors (u and v) are uncorrelated with the true values (X and F). We survey and comment on the solutions to the problem of obtaining consistent estimates of α and β from a sample of (x, y)'s, (1) when one makes various assumptions about properties of the errors and the true values other than those mentioned above, and (2) when one has various kinds of “additional information” which aids in constructing these consistent estimates. The problems of obtaining confidence intervals for β and of testing hypotheses about β are not discussed, though approximate variances of some of the estimates of β are given. * This paper is an outgrowth of a Master's Thesis submitted to the Department of Statistics, University of Chicago. I am indebted for helpful comments and criticisms to T. E. Harris, W. H. Kr...

The fitting of straight lines when both variables are subject to error

Consider a linear-programming problem in which the “right-hand side” is a random vector whose expected value is known and where the expected value of the objective function is to be minimized. An approximate solution is often found by replacing the “right-hand side” by its expected value and solving the resulting linear programming problem. In this paper conditions are given for the equality of the expected value of the objective function for the optimal solution and the value of the objective function for the approximate solution; bounds on these values are also given. In addition, the relation between this problem and a related problem, where one makes an observation on the “right-hand side” and solves the nonstochastic linear programming problem based on this observation, is discussed.

Inequalities for Stochastic Linear Programming Problems

The first course in Statistics typically covers inferential procedures which are valid only if a number of preconditions are satisfied by the data. This book, designed for a second course, contains a collection of statistical diagnostics and prescriptions necessary for the applied statistician so that he can deal with the realities of inference from data and not merely with the kind of classroom problems where all the data satisfy the assumptions associated with the technique being taught. The book begins with four chapters on data diagnostics, and then proceeds to discuss prescriptions for using the data, given its diagnosed characteristics. The book concludes with two chapters on techniques for making inferences from specialized data, mixing categorical and measured data, and cross-classified data.

Prescriptions for working statisticians

: Upper and lower bounds on the expectation of a convex function of a vector valued random variable are derived by examining the boundary of an appropriate multivariate moment space.

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Albert Madansky

Papers

Identification of Outliers

The fitting of straight lines when both variables are subject to error

Inequalities for Stochastic Linear Programming Problems

Prescriptions for working statisticians

Bounds on the Expectation of a Convex Function of a Multivariate Random Variable