DOCSLIB.ORG

Simulation Comparison Between an Outlier Resistant Model-Based Finite Population Estimator and Design-Based Estimators Under Contamination

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simulation Comparison Between an Outlier Resistant Model-Based Finite Population Estimator and Design-Based Estimators Under Contamination Simulation Comparison Between an Outlier Resistant Model-Based Finite Population Estimator and Design-Based Estimators under Contamination Bogong Li U.S. Bureau of Labor Statistics There are two approaches to finite population esti- we propose. Limited simulation result shows that mation. One assumes the finite population elements this estimator is as efficient as other model-based es- are fixed quantities. The randomness associated timators when there is no significant violation from with estimators comes from the random selection of model assumptions and is more efficient than least samples. Each sampled element has a weight deter- squares types of estimators when there are contam- mined by the sample design, see Cochran (1977) [2]. inating symmetrical outlying observations. Most Bureau of Labor Statistics surveys rely on this theory, for example, the Occupational Employment Statistics program which was discussed in depth in 1. Oulier-Insensitive Linear Regres- Li (2002) [10] about methods used in small area es- sion timation. The other approach assumes the popu- In the past several decades, research on influence lation elements are a random draw from a larger of outlying observations on linear model estimates population or “super-population”, in a way similar yielded classes of estimators with controllable lev- to taking random samples from a random variable. els of reaction to outlying observations. We briefly The random variable has its own mean and vari- summarize some of them here, in particularly the ance structure, usually expressed in terms of linear ones that are implemented in popular statistical soft- models. Finite population summaries of the vari- ware packages such as R, S-plus and SAS. These are able under investigation, such as mean, total are the M-estimator, least median of squares estimator estimated through fitting sample data to the linear (LMS), the least trimmed means estimator (LTS). model. The design of the sample selection is relevant Also included are other methods that are applicable only to the selection of a suitable linear model un- to more complex linear model structures, the Mal- der the “super-population”, but is irrelevant to how lows’, Schweppe’s methods, and the broader class of the estimates are derived from the model. The first the Generalized M-estimator (GM). approach is known as “design-based”, the second “model-based”. S¨arndal, Swensson and Wretman (1992) [14] discusses this distinction in greater de- • Huber M-Estimator for Regression: Estimator ˆ tail. Table 1 lists some familiar design-based estima- βn for the regression parameter β (location) is tors and their model-based equivalents. A broader defined by review on this connection is in Li (2001) [9]. Since n least squares linear estimators are very vulnerable X 0 ˆ min ρ (yi − xiβn)/σ to outlying observations, that is, a small variation ˆ βn i=1 in outlying observation produces larger variation in the estimate than non-outlying observations, we de- or sire a model-based estimator that is less responsive n X 0 ˆ to outliers while being efficient. This is especially ψ (yi − xiβ)/σ xi = 0 necessary for survey data since processing error oc- i=1 cur prominently in surveys. This study aims to for some symmetric function ρ : → + and investigate the statistical quality of such a model- R R for a fixed σ. ψ(t) is the derivative of ρ(t). based, outlier insensitive finite population estimator Examples of ρ(t) are probability density func- 2 The views expressed in this article are those of the au- tion f(t), t , |t| etc. defined within certain dis- thor and do not constitute the policy of the Bureau of Labor tance from 0. Optimal βˆ is selected to mini- Statistics. The author can be contacted at: U.S. Bureau of mize asymptotic variance while allowing as high Labor Statistics, Postal Square Building, Suite 1950, 2 Mas- sachusetts Ave. NE. Washington, DC 20212-0001, or E-Mail: a breakdown points as it could be possible, see li [email protected]. Huber (1981) [7] for more details. Estimators Finite Population Tˆ Model Equivalent 2 Expansion estimator NY¯s Yi = µ + i, i ∼ (0, σ ) ˆ 2 Linear regression estimator N Y¯s + bls(¯x − x¯s) Yi = α + xiβ + i, i ∼ (0, σ ) 1 ¯ 2 2 Ratio estimator NYsx/¯ x¯s Yi = xiβ + xi i, i ∼ (0, σ ) P ¯ 2 Stratified expansion estimator h NhYhs Yhi = µh + hi, hi ∼ (0, σh) 1 P ¯ 2 2 Stratified ratio estimator h NhYhsx¯h/x¯hs Yhi = xhiβ + xhihi, hi ∼ (0, σh) P 0 Simple Proportion Estimator s Yi/n Yi ∼ ber (pi), logit(pi) = xiβ Table 1: Finite population estimators and their model-based equivalents • Least Median of Squares Regression (LMS): Re- the method we used to obtain outlier-insensitive gression parameters βˆ are implicitly defined by model parameter estimates, n o 0 0 0 ˆ 2 n −1/2 ˆo −1/2 ˆ −1 min Med (yi − xiβ) , i = 1, 2, . , n , min W ψ y − Xβ U V βˆ βˆ, Vˆ n o where only the median of the squared resid- U−1/2 W−1/2ψ y − Xβˆ , (1) uals are relevant in defining the best βˆ, see Rousseeuw and Leroy (1987) [13] for details. see Krasker and Welsch (1982) [8]. Solutions for β and V need numerical algorithm similar to • Least Trimmed Squares Regression(LTS): Sim- that for ML and REML, see Harville (1977) [5] ilar to least squares estimators, but only the and Searle, Casella and McCulloch (1992) [15] smallest h residuals are taken to minimize for methods to obtain ML and REML. summed squared residuals, h h 2. Outlier-Resistant Best Linear Un- X 0 ˆ 2 X 2 min (yi − xiβ)i:n = min (r )i:n biased Estimation ˆ ˆ β i=1 β i=1 Linear model with mixed-effects is gradually popu- 2 2 , where (r )1:n ≤ · · · ≤ (r )h:n ≤ · · · ≤ larizing with survey data usage, for example Battese, 2 (r )n:n are the ordered squared residuals, see Hater and Fuller (1987) [1]. We propose a super- Rousseeuw and Leroy (1987) [13] for more de- population structure incorporates mixed-effects: tails on this. Y1 X1 Z1 0 ··· 0 ν1 1 Y2 X2 0 Z2 ··· 0 ν2 2 • Mallows’ Method: A weight w(xi) is part of . = . β+ . . + . , . . . . . the estimating equation in Huber M-estimator . . . ··· . . . Y X 0 0 ··· Z ν which limits the influence of outlying x rows, g N×1 g g g g n where ( iid X 0 ˆ(M) ν ∼ (0, σ2) w(xi)ψ (yi − xiβ )/σ xi = 0, i ν iid i=1 2 i ∼ (0, σ I) see Mallows (1975) [11]. and Yi and xi are population and auxiliary infor- mation vectors. νi is random (can be constant) • Schweppe’s Method: Weights w(xi) depending stratum-specific effects. i is independent random on xi are as well attached to scale the residual error vector. If we rearrange the Y by observed (s) in response to outlying influential observation and non-observed (r) population elements: in x: n Ys Xs Zs 0 X 0 ˆ(S) Y = = β + ν + , w(xi)ψ (yi − xiβ )/(σw(xi)) xi = 0 Yr Xr Zr i=1 and , see Hill (1982) [6]. V V Var(Y) = ss sr , then V V • Generalized M-estimator for Mixed Models: Es- rs rr timating equations written in matrix form to the best linear unbiased predictor (BLUP) of accommodating more complex covariance struc- 0 0 0 ture and weight matrices V, W and U. This is T = lN×1Y = lsYs + lrYr which could be the mean, total or contrasts, is where τˆ is the bounded influence estimator (with IFτˆ,F < ∞) of the vector of all unknown parameters 0 0 ˆ ˆ ˆ −1 ˆ Te = lsys + lr XsβLS + VrsVss (ys − XsβLS) . in the model, (2) 0 0 τ = (β , v11, . , vij, . , vNN ) . Valliant (2000) [17] provides detailed derivation and proofs of unbiasedness and variance minimization of then τˆ is asymptotically normally distributed, Te. We propose an outlier resistant RBLUP, based Maronna and Yohai (1981) [12] discusses its asymp- on above formulation: totic mean and variance; η is a finite projection from τˆ to Te(R). Given such projection exists, by the (R) 0 0 ˆ (R) Chain Rule defined on Gˆateaux differentials, the in- Te =l ys + l WsXsβ s r fluence function of RBLUP is (R) ˆ (R) ˆ (R)−1 ˆ 0 + Vrs Vss ψ ys − Xsβ (3) IFRBLUP,F = η IFτˆ,F . 0 (R) Since both IFτˆ,F and η are finite, IFRBLUP,F is nec- where βˆ , Vˆ (R) are any bounded influence esti- essarily bounded. mators from the mixed-effects linear model; Ws are weight functions depend on Xs, ψ is the Huber ψ- 3. Simulation Study ˆ (R) ˆ (R) function. In practice, β , V are generalized We study the statistical quality of proposed RBLUP M-estimators given in section 1. Strictly speaking through a simulation. First we produce an artificial RBLUP is neither “best” in the sense of minimiz- popoulation according to the model: ing variance nor unbiased with respect to the model among all linear unbiased estimators. The name Y1 Y2 β1 given simply indicates its predecessor. = (11000 x) Y3 β2 Influence of RBLUP is necessarily bounded. By Y4 definition, influence function of a statistical func- 150 ··· 0 1 tional T (F ) ∈ k is the first derivative of T (F ): R . ν1 . 1 . . 150 . + . + , T (F) − T (F ) ∂T (F) . . IFT,F (z) := lim = , . 1250 ν . →0 ∂ 4 →0 0 ··· 1550 1000 1 where F := (1−)F +δ , ∈ , with δ the point ( iid z R z ν ∼ (0, σ2) mass at z.
Load more

Recommended publications
  • A Review on Outlier/Anomaly Detection in Time Series Data
    A review on outlier/anomaly detection in time series data ANE BLÁZQUEZ-GARCÍA and ANGEL CONDE, Ikerlan Technology Research Centre, Basque Research and Technology Alliance (BRTA), Spain USUE MORI, Intelligent Systems Group (ISG), Department of Computer Science and Artificial Intelligence, University of the Basque Country (UPV/EHU), Spain JOSE A. LOZANO, Intelligent Systems Group (ISG), Department of Computer Science and Artificial Intelligence, University of the Basque Country (UPV/EHU), Spain and Basque Center for Applied Mathematics (BCAM), Spain Recent advances in technology have brought major breakthroughs in data collection, enabling a large amount of data to be gathered over time and thus generating time series. Mining this data has become an important task for researchers and practitioners in the past few years, including the detection of outliers or anomalies that may represent errors or events of interest. This review aims to provide a structured and comprehensive state-of-the-art on outlier detection techniques in the context of time series. To this end, a taxonomy is presented based on the main aspects that characterize an outlier detection technique. Additional Key Words and Phrases: Outlier detection, anomaly detection, time series, data mining, taxonomy, software 1 INTRODUCTION Recent advances in technology allow us to collect a large amount of data over time in diverse research areas. Observations that have been recorded in an orderly fashion and which are correlated in time constitute a time series. Time series data mining aims to extract all meaningful knowledge from this data, and several mining tasks (e.g., classification, clustering, forecasting, and outlier detection) have been considered in the literature [Esling and Agon 2012; Fu 2011; Ratanamahatana et al.
    [Show full text]
  • Outlier Identification.Pdf
    STATGRAPHICS – Rev. 7/6/2009 Outlier Identification Summary The Outlier Identification procedure is designed to help determine whether or not a sample of n numeric observations contains outliers. By “outlier”, we mean an observation that does not come from the same distribution as the rest of the sample. Both graphical methods and formal statistical tests are included. The procedure will also save a column back to the datasheet identifying the outliers in a form that can be used in the Select field on other data input dialog boxes. Sample StatFolio: outlier.sgp Sample Data: The file bodytemp.sgd file contains data describing the body temperature of a sample of n = 130 people. It was obtained from the Journal of Statistical Education Data Archive (www.amstat.org/publications/jse/jse_data_archive.html) and originally appeared in the Journal of the American Medical Association. The first 20 rows of the file are shown below. Temperature Gender Heart Rate 98.4 Male 84 98.4 Male 82 98.2 Female 65 97.8 Female 71 98 Male 78 97.9 Male 72 99 Female 79 98.5 Male 68 98.8 Female 64 98 Male 67 97.4 Male 78 98.8 Male 78 99.5 Male 75 98 Female 73 100.8 Female 77 97.1 Male 75 98 Male 71 98.7 Female 72 98.9 Male 80 99 Male 75 2009 by StatPoint Technologies, Inc. Outlier Identification - 1 STATGRAPHICS – Rev. 7/6/2009 Data Input The data to be analyzed consist of a single numeric column containing n = 2 or more observations.
    [Show full text]
  • A Machine Learning Approach to Outlier Detection and Imputation of Missing Data1
    Ninth IFC Conference on “Are post-crisis statistical initiatives completed?” Basel, 30-31 August 2018 A machine learning approach to outlier detection and imputation of missing data1 Nicola Benatti, European Central Bank 1 This paper was prepared for the meeting. The views expressed are those of the authors and do not necessarily reflect the views of the BIS, the IFC or the central banks and other institutions represented at the meeting. A machine learning approach to outlier detection and imputation of missing data Nicola Benatti In the era of ready-to-go analysis of high-dimensional datasets, data quality is essential for economists to guarantee robust results. Traditional techniques for outlier detection tend to exclude the tails of distributions and ignore the data generation processes of specific datasets. At the same time, multiple imputation of missing values is traditionally an iterative process based on linear estimations, implying the use of simplified data generation models. In this paper I propose the use of common machine learning algorithms (i.e. boosted trees, cross validation and cluster analysis) to determine the data generation models of a firm-level dataset in order to detect outliers and impute missing values. Keywords: machine learning, outlier detection, imputation, firm data JEL classification: C81, C55, C53, D22 Contents A machine learning approach to outlier detection and imputation of missing data ... 1 Introduction ..............................................................................................................................................
    [Show full text]
  • Effects of Outliers with an Outlier
    • The mean is a good measure to use to describe data that are close in value. • The median more accurately describes data Effects of Outliers with an outlier. • The mode is a good measure to use when you have categorical data; for example, if each student records his or her favorite color, the color (a category) listed most often is the mode of the data. In the data set below, the value 12 is much less than Additional Example 2: Exploring the Effects of the other values in the set. An extreme value such as Outliers on Measures of Central Tendency this is called an outlier. The data shows Sara’s scores for the last 5 35, 38, 27, 12, 30, 41, 31, 35 math tests: 88, 90, 55, 94, and 89. Identify the outlier in the data set. Then determine how the outlier affects the mean, median, and x mode of the data. x x xx x x x 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 55, 88, 89, 90, 94 outlier 55 1 Additional Example 2 Continued Additional Example 2 Continued With the Outlier Without the Outlier 55, 88, 89, 90, 94 55, 88, 89, 90, 94 outlier 55 mean: median: mode: mean: median: mode: 55+88+89+90+94 = 416 55, 88, 89, 90, 94 88+89+90+94 = 361 88, 89,+ 90, 94 416 5 = 83.2 361 4 = 90.25 2 = 89.5 The mean is 83.2. The median is 89. There is no mode.
    [Show full text]
  • How Significant Is a Boxplot Outlier?
    Journal of Statistics Education, Volume 19, Number 2(2011) How Significant Is A Boxplot Outlier? Robert Dawson Saint Mary’s University Journal of Statistics Education Volume 19, Number 2(2011), www.amstat.org/publications/jse/v19n2/dawson.pdf Copyright © 2011 by Robert Dawson all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor. Key Words: Boxplot; Outlier; Significance; Spreadsheet; Simulation. Abstract It is common to consider Tukey’s schematic (“full”) boxplot as an informal test for the existence of outliers. While the procedure is useful, it should be used with caution, as at least 30% of samples from a normally-distributed population of any size will be flagged as containing an outlier, while for small samples (N<10) even extreme outliers indicate little. This fact is most easily seen using a simulation, which ideally students should perform for themselves. The majority of students who learn about boxplots are not familiar with the tools (such as R) that upper-level students might use for such a simulation. This article shows how, with appropriate guidance, a class can use a spreadsheet such as Excel to explore this issue. 1. Introduction Exploratory data analysis suddenly got a lot cooler on February 4, 2009, when a boxplot was featured in Randall Munroe's popular Webcomic xkcd. 1 Journal of Statistics Education, Volume 19, Number 2(2011) Figure 1 “Boyfriend” (Feb. 4 2009, http://xkcd.com/539/) Used by permission of Randall Munroe But is Megan justified in claiming “statistical significance”? We shall explore this intriguing question using Microsoft Excel.
    [Show full text]
  • Lecture 14 Testing for Kurtosis
    9/8/2016 CHE384, From Data to Decisions: Measurement, Kurtosis Uncertainty, Analysis, and Modeling • For any distribution, the kurtosis (sometimes Lecture 14 called the excess kurtosis) is defined as Testing for Kurtosis 3 (old notation = ) • For a unimodal, symmetric distribution, Chris A. Mack – a positive kurtosis means “heavy tails” and a more Adjunct Associate Professor peaked center compared to a normal distribution – a negative kurtosis means “light tails” and a more spread center compared to a normal distribution http://www.lithoguru.com/scientist/statistics/ © Chris Mack, 2016Data to Decisions 1 © Chris Mack, 2016Data to Decisions 2 Kurtosis Examples One Impact of Excess Kurtosis • For the Student’s t • For a normal distribution, the sample distribution, the variance will have an expected value of s2, excess kurtosis is and a variance of 6 2 4 1 for DF > 4 ( for DF ≤ 4 the kurtosis is infinite) • For a distribution with excess kurtosis • For a uniform 2 1 1 distribution, 1 2 © Chris Mack, 2016Data to Decisions 3 © Chris Mack, 2016Data to Decisions 4 Sample Kurtosis Sample Kurtosis • For a sample of size n, the sample kurtosis is • An unbiased estimator of the sample excess 1 kurtosis is ∑ ̅ 1 3 3 1 6 1 2 3 ∑ ̅ Standard Error: • For large n, the sampling distribution of 1 24 2 1 approaches Normal with mean 0 and variance 2 1 of 24/n 3 5 • For small samples, this estimator is biased D. N. Joanes and C. A. Gill, “Comparing Measures of Sample Skewness and Kurtosis”, The Statistician, 47(1),183–189 (1998).
    [Show full text]
  • A SYSTEMATIC REVIEW of OUTLIERS DETECTION TECHNIQUES in MEDICAL DATA Preliminary Study
    A SYSTEMATIC REVIEW OF OUTLIERS DETECTION TECHNIQUES IN MEDICAL DATA Preliminary Study Juliano Gaspar1,2, Emanuel Catumbela1,2, Bernardo Marques1,2 and Alberto Freitas1,2 1Department of Biostatistics and Medical Informatics, Faculty of Medicine, University of Porto, Porto, Portugal 2CINTESIS - Center for Research in Health Technologies and Information Systems University of Porto, Porto, Portugal Keywords: Outliers detection, Data mining, Medical data. Abstract: Background: Patient medical records contain many entries relating to patient conditions, treatments and lab results. Generally involve multiple types of data and produces a large amount of information. These databases can provide important information for clinical decision and to support the management of the hospital. Medical databases have some specificities not often found in others non-medical databases. In this context, outlier detection techniques can be used to detect abnormal patterns in health records (for instance, problems in data quality) and this contributing to better data and better knowledge in the process of decision making. Aim: This systematic review intention to provide a better comprehension about the techniques used to detect outliers in healthcare data, for creates automatisms for those methods in the order to facilitate the access to information with quality in healthcare. Methods: The literature was systematically reviewed to identify articles mentioning outlier detection techniques or anomalies in medical data. Four distinct bibliographic databases were searched: Medline, ISI, IEEE and EBSCO. Results: From 4071 distinct papers selected, 80 were included after applying inclusion and exclusion criteria. According to the medical specialty 32% of the techniques are intended for oncology and 37% of them using patient data.
    [Show full text]
  • Recommended Methods for Outlier Detection and Calculations Of
    Recommended Methods for Outlier Detection and Calculations of Tolerance Intervals and Percentiles – Application to RMP data for Mercury-, PCBs-, and PAH-contaminated Sediments Final Report May 27th, 2011 Prepared for: Regional Monitoring Program for Water Quality in San Francisco Bay, Oakland CA Prepared by: Dr. Don L. Stevens, Jr. Principal Statistician Stevens Environmental Statistics, LLC 6200 W. Starr Road Wasilla, AK 99654 Introduction This report presents results of a review of existing methods for outlier detection and for calculation of percentiles and tolerance intervals. Based on the review, a simple method of outlier detection that could be applied annually as new data becomes available is recommended along with a recommendation for a method for calculating tolerance intervals for population percentiles. The recommended methods are applied to RMP probability data collected in the San Francisco Estuary from 2002 through 2009 for Hg, PCB, and PAH concentrations. Outlier detection There is a long history in statistics of attempts to identify outliers, which are in some sense data points that are unusually high or low . Barnett and Lewis (1994) define an outlier in a set of data as “an observation (or subset of observations) which appears to be inconsistent with the remainder of that set of data. This captures the intuitive notion, but does not provide a constructive pathway. Hawkins (1980) defined outliers as data points which are generated by a different distribution than the bulk observations. Hawkins definition suggests something like using a mixture distribution to separate the data into one or more distributions, or, if outliers are few relative to the number of data points, to fit the “bulk” distribution with a robust/resistant estimation procedure, and then determine which points do not conform to the bulk distribution.
    [Show full text]
  • Identify the Outlier in the Data Set. Determine How the Outlier Affects the Mean, Median, and Mode of the Data
    11-5 Appropriate Measures 1. The number of minutes spent studying are: 60, 70, 45, 60, 80, 35, and 45. Find the measure of center that best represents the data. Justify your selection and then find the measure of center. ANSWER: The mean best represents the data. There are no extreme values. mean: 56.4 minutes median: 2. The table shows monthly rainfall in inches for five months. Identify the outlier in the data set. Determine how the outlier affects the mean, median, and mode of the data. Then tell which measure of center best describes the data with and without the outlier. Round to the nearest hundredth. Justify your selection. ANSWER: outlier: 2.43 in.; without the outlier: mean: 7.36 in., median: 7.19 in., mode: none; with the outlier: mean: 6.54 in., median: 6.83 in., mode: none; The mean rainfall best describes the data without the outlier. The median rainfall best describes the data with the outlier. 3. The table shows the average depth of several lakes. a. Identify the outlier in the data set. b. Determine how the outlier affects the mean, median, mode, and range eSolutionsof theManual data.- Powered by Cognero Page 1 c. Tell which measure of central tendency best describes the data with and without the outlier. ANSWER: a. 1,148 b. With the outlier, the mean is 216.83 ft, the median is 33.5 ft, there is no mode, and the range is 1,138. Without the outlier, the mean is 30.6 ft, the median is 24 ft, there is no mode, and the range is 52.
    [Show full text]
  • Topics in Biostatistics: Part II
    Basic Statistics for Virology Sarah J. Ratcliffe, Ph.D. Center for Clinical Epidemiology and Biostatistics U. Penn School of Medicine Email: [email protected] September 2011 CCEB Outline Summarizing the Data Hypothesis testing Example Interpreting results Resources CCEB Summarizing the Data Two basic aspects of data Centrality Variability Different measures for each Optimal measure depends on type of data being described CCEB Centrality Mean Sum of observed values divided by number of observations Most common measure of centrality Most informative when data follow normal distribution (bell-shaped curve) Median “middle” value: half of all observed values are smaller, half are larger Best centrality measure when data are skewed Mode Most frequently observed value CCEB Mean can Mislead Group 1 data: 1,1,1,2,3,3,5,8,20 Mean: 4.9 Median: 3 Mode: 1 Group 2 data: 1,1,1,2,3,3,5,8,10 Mean: 3.8 Median: 3 Mode: 1 When data sets are small, a single extreme observation will have great influence on mean, little or no influence on median In such cases, median is usually a more informative measure of centrality CCEB CCEB CCEB Variability Most commonly used measure to describe variability is standard deviation (SD) SD is a function of the squared differences of each observation from the mean If the mean is influenced by a single extreme observation, the SD will overstate the actual variability CCEB Variability SD describes variability in original data SE = SD / √n describes variability in the mean. Confidence intervals for the mean use SE e.g. Mean ± t* SE Confidence interval for predicted observation uses SD.
    [Show full text]
  • Visualizing Outliers
    Visualizing Big Data Outliers through Distributed Aggregation Leland Wilkinson Fig. 1. Outliers revealed in a box plot [72] and letter values box plot [36]. These plots are based on 100,000 values sampled from a Gaussian (Standard Normal) distribution. By definition, the data contain no probable outliers, yet the ordinary box plot shows thousands of outliers. This example illustrates why ordinary box plots cannot be used to discover probable outliers. Abstract— Visualizing outliers in massive datasets requires statistical pre-processing in order to reduce the scale of the problem to a size amenable to rendering systems like D3, Plotly or analytic systems like R or SAS. This paper presents a new algorithm, called hdoutliers, for detecting multidimensional outliers. It is unique for a) dealing with a mixture of categorical and continuous variables, b) dealing with big-p (many columns of data), c) dealing with big-n (many rows of data), d) dealing with outliers that mask other outliers, and e) dealing consistently with unidimensional and multidimensional datasets. Unlike ad hoc methods found in many machine learning papers, hdoutliers is based on a distributional model that allows outliers to be tagged with a probability. This critical feature reduces the likelihood of false discoveries. Index Terms—Outliers, Anomalies. 1 INTRODUCTION Barnett and Lewis [6] define an outlier in a set of data as “an observa- • We demonstrate why other visual analytic tools cannot reliably tion (or subset of observations) which appears to be inconsistent with be used to detect multidimensional outliers. the remainder of that set of data.” They go on to explain that their defi- nition rests on the assumption that the data constitute a random sample • We introduce some novel applications of outlier detection and from a population and that outliers can be represented as points in a accompanying visualizations based on hdoutliers.
    [Show full text]
  • Outlier Detection and Robust Mixture Modeling Using Nonconvex Penalized Likelihood
    Journal of Statistical Planning and Inference 164 (2015) 27–38 Contents lists available at ScienceDirect Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi Outlier detection and robust mixture modeling using nonconvex penalized likelihood Chun Yu a, Kun Chen b, Weixin Yao c,∗ a School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China b Department of Statistics, University of Connecticut, Storrs, CT 06269, United States c Department of Statistics, University of California, Riverside, CA 92521, United States article info a b s t r a c t Article history: Finite mixture models are widely used in a variety of statistical applications. However, Received 29 April 2014 the classical normal mixture model with maximum likelihood estimation is prone to the Received in revised form 9 March 2015 presence of only a few severe outliers. We propose a robust mixture modeling approach Accepted 10 March 2015 using a mean-shift formulation coupled with nonconvex sparsity-inducing penalization, Available online 19 March 2015 to conduct simultaneous outlier detection and robust parameter estimation. An efficient iterative thresholding-embedded EM algorithm is developed to maximize the penalized Keywords: log-likelihood. The efficacy of our proposed approach is demonstrated via simulation EM algorithm Mixture models studies and a real application on Acidity data analysis. Outlier detection ' 2015 Elsevier B.V. All rights reserved. Penalized likelihood 1. Introduction Nowadays finite mixture distributions are increasingly important in modeling a variety of random phenomena (see Everitt and Hand, 1981; Titterington et al., 1985; McLachlan and Basford, 1988; Lindsay, 1995; Böhning, 1999). The m-component finite normal mixture distribution has probability density m I D X I 2 f .y θ/ πiφ.y µi; σi /; (1.1) iD1 T 2 where θ D .π1; µ1; σ1I ::: I πm; µm; σm/ collects all the unknown parameters, φ.·; µ, σ / denotes the density function 2 Pm D of N(µ, σ /, and πj is the proportion of jth subpopulation with jD1 πj 1.
    [Show full text]