DOCSLIB.ORG

Optimal Principal Component Analysis of STEM XEDS Spectrum Images Pavel Potapov1,2* and Axel Lubk2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Potapov and Lubk Adv Struct Chem Imag (2019) 5:4 https://doi.org/10.1186/s40679-019-0066-0 RESEARCH Open Access Optimal principal component analysis of STEM XEDS spectrum images Pavel Potapov1,2* and Axel Lubk2 Abstract STEM XEDS spectrum images can be drastically denoised by application of the principal component analysis (PCA). This paper looks inside the PCA workfow step by step on an example of a complex semiconductor structure con- sisting of a number of diferent phases. Typical problems distorting the principal components decomposition are highlighted and solutions for the successful PCA are described. Particular attention is paid to the optimal truncation of principal components in the course of reconstructing denoised data. A novel accurate and robust method, which overperforms the existing truncation methods is suggested for the frst time and described in details. Keywords: PCA, Spectrum image, Reconstruction, Denoising, STEM, XEDS, EDS, EDX Background components [3–9]. In general terms, PCA reduces the Scanning transmission electron microscopy (STEM) dimensionality of a large dataset by projecting it into an delivers images of nanostructures at high spatial resolu- orthogonal basic of lower dimension. It can be shown tion matching that of broad beam transmission electron that among all possible linear projections, PCA ensures microscopy (TEM). Additionally, modern STEM instru- the smallest Euclidean diference between the initial and ments are typically equipped with electron energy-loss projected datasets or, in other words, provides the mini- spectrometers (EELS) and/or X-rays energy-dispersive mal least squares errors when approximating data with a spectroscopy (XEDS, sometimes abbreviated as EDS smaller number of variables [10]. Due to that, PCA has or EDX) detectors, which allows to turn images into found a lot of applications in imaging science for data spectrum-images, i.e. pixelated images, where each pixel compression, denoising and pattern recognition (see for represents an EEL or XED spectrum. In particular, the example [11–18]) including applications to STEM XEDS recent progress in STEM instrumentation and large col- spectrum-imaging [19–24]. lection-angle silicon drift detectors (SDD) [1, 2] made A starting point for the PCA treatment is the con- D possible a fast acquisition of large STEM XEDS spec- version of a dataset into a matrix , where spectra are trum-images consisting of 10–1000 million data points. placed on the matrix rows and each row represents an Tese huge datasets show typically some correlations in individual STEM probe position (pixel). Assume for m n D the data distribution, which might be retrieved by appli- defniteness that the × matrix consists of m pix- cation of statistical analysis and then utilized for improv- els and n energy channels. Although STEM pixels may ing data quality. be originally arranged in 1D (linescan), 2D (datacube) or Te simplest and probably the most popular multivari- in a confguration with higher dimensions, they can be ate statistical technique is a principal component analysis always recasted into the 1D train as the neighborhood (PCA) that expresses available data in terms of orthog- among pixels does not play any role in the PCA treat- onal linearly uncorrelated variables called principal ment. PCA is based on the assumption that there are certain correlations among spectra constituting the data D matrix . Tese correlations appear because the data *Correspondence: [email protected] variations are governed by a limited number of the latent 1 Department of Physics, Technical University of Dresden, Dresden, factors, for example by the presence of chemical phases Germany Full list of author information is available at the end of the article with the fxed composition. Te spectral signatures of © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Potapov and Lubk Adv Struct Chem Imag (2019) 5:4 Page 2 of 21 Fig. 1 Schematic showing how data matrix D is decomposed into loading matrix P and score matrix T in PCA. A given column of T and a row of PT form a principal component. The components are sorted such as the variances of the data points in columns of T decreases from left to right. The principal components that will be retained after the truncation are flled green D latent factors might be, however, not apparent as they Te data matrix expressed by (1) can be subjected are masked by noise. In this consideration, PCA relates to dimensionality reduction or, in other words, trun- closely to the factor analysis [7] although principal com- cation of components. Such dimensionality reduction ponents generally do not coincide with the latent factors might serve various purposes, for instance, it can be a but represent rather their linear combinations [25]. frst step for more complicated multivariate statistical Principal components can be found, for example, treatment like unmixing data and extraction of latent through the diagonalization of the covariance matrix factors. In the simplest case, dimensionality reduction DDT or by applying the singular value decomposition can be utilized for removing the major part of noise D (SVD) directly to . SVD decomposes data as: from data, i.e. for its denoising. T Te following questions are at the heart of the D UV (1) U= V method. How much the dimensionality of a given data- where and are left and right hand singular vector set can be reduced? How many components must be matrices and is a diagonal matrix with singular values D retained to reproduce adequately the data variation and of on the diagonal. For the purpose of PCA formula (1) how many of them may be truncated to reduce noise? can be rewritten as: Tis paper attempts to address these crucial questions T D TP (2) on the example of typical XEDS spectrum-images P= V obtained in modern STEM instruments. where is an n n loading matrix describing prin- = ×T U At a frst glance, the reasonable number of retained cipal components and is an m n score matrix = × components should be equal to the known (or showing the contribution of components into the dataset. expected) number of latent factors behind the data var- Figure 1 illustrates the principal component decomposi- iations. It will be, however, shown that the situation is tion in the graphical form. Te columns of the loading P PT more complicated and the number of meaningful com- matrix (rows in ) represent spectra of principal com- ponents might strongly difer from the number of latent ponents expressed in the original energy channels. Typi- P factors—typically, there are less components than fac- cally, the columns of are normalized to unity, such that tors. Te reason for this deviation is unavoidable cor- all the scaling information is moved into the score matrix T ruption of data with noise. It is important to sort the principal components in the To explore the topic most comprehensively, we con- order of their signifcance. In PCA, the components are sidered an object with a very large number of latent ranked according their variance, i.e. the variance of the T factors and analyzed its experimental XEDS spectrum data along the corresponding column of . image. In parallel, we generated a twin synthetic object that mimicked the real one in all its essential features. An advantage of the synthetic data is the possibility to Potapov and Lubk Adv Struct Chem Imag (2019) 5:4 Page 3 of 21 Fig. 2 Objects employed for evaluation of PCA in the present paper: a shows the mean image of the experimentally characterized CMOS device and b represent a twin synthetic object generated to reproduce the key features of the real object. The twin object was designed to mimic the mixture of the layers composing the real object but not necessarily their exact geometry. A colored legend identifes all the constituent phases labeled according the notations in Table 1. The simulations of spectrum-images were performed in the noise-free and noisy variants. Correspondingly (c) and (d) show the mean images of the noise-free and noisy datasets exclude noise in simulations and, therefore, compare background for truncation of principal components and the noisy data with the noise-free reference. discuss the existing practical truncation methods. A PCA is often considered as a fxed procedure, where lit- novel method for automatic determination of the opti- tle can be altered or tuned. In reality, there is a number mal number of components is introduced in "Anisotropy of hidden issues hampering the treatment and leading method for truncation of principal components" section. to dubious results. Better understanding of the poten- At the end of "Truncation of principal components and tial issues might help to design the optimal treatment reconstruction" section, the results of the spectrum- fow improving the efciency and avoiding artifacts. Te image reconstruction are shown and the denoising ability systematic comparison between the experimental and of PCA is demonstrated. synthetic data sets on the one hand and between the syn- thetic noisy set and the noise-free reference on the other Results and discussion hand, allowed us to identify the typical obstacles in the Multi‑component object for spectrum‑imaging treatment fow and fnd the solutions for the optimal A modern CMOS device was chosen as an object for principal component decomposition and reconstruction XEDS spectrum imaging. Figure 2a shows a so-called of the denoised data. mean image of the device, i.e. the spectrum imaging sig- Below, it will be demonstrated that certain pre-treat- nal integrated over all available energy channels.
Recommended publications
  • Foundations of Bayesian Learning from Synthetic Data

    Foundations of Bayesian Learning from Synthetic Data

    Foundations of Bayesian Learning from Synthetic Data Harrison Wilde Jack Jewson Sebastian Vollmer Chris Holmes Department of Statistics Barcelona GSE Department of Statistics, Department of Statistics University of Warwick Universitat Pompeu Fabra Mathematics Institute University of Oxford; University of Warwick The Alan Turing Institute Abstract (Dwork et al., 2006), to define working bounds on the probability that an adversary may identify whether a particular observation is present in a dataset, given There is significant growth and interest in the that they have access to all other observations in the use of synthetic data as an enabler for machine dataset. DP’s formulation is context-dependent across learning in environments where the release of the literature; here we amalgamate definitions regard- real data is restricted due to privacy or avail- ing adjacent datasets from Dwork et al. (2014); Dwork ability constraints. Despite a large number of and Lei (2009): methods for synthetic data generation, there are comparatively few results on the statisti- Definition 1 (("; δ)-differential privacy) A ran- cal properties of models learnt on synthetic domised function or algorithm K is said to be data, and fewer still for situations where a ("; δ)-differentially private if for all pairs of adjacent, researcher wishes to augment real data with equally-sized datasets D and D0 that differ in one another party’s synthesised data. We use observation and all S ⊆ Range(K), a Bayesian paradigm to characterise the up- dating of model parameters when learning Pr[K(D) 2 S] ≤ e" × Pr [K (D0) 2 S] + δ (1) in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate con- Current state-of-the-art approaches involve the privati- sideration of the synthetic data generating sation of generative modelling architectures such as process and learning task at hand.
  • Synthpop: Bespoke Creation of Synthetic Data in R

    Synthpop: Bespoke Creation of Synthetic Data in R

    synthpop: Bespoke Creation of Synthetic Data in R Beata Nowok Gillian M Raab Chris Dibben University of Edinburgh University of Edinburgh University of Edinburgh Abstract In many contexts, confidentiality constraints severely restrict access to unique and valuable microdata. Synthetic data which mimic the original observed data and preserve the relationships between variables but do not contain any disclosive records are one possible solution to this problem. The synthpop package for R, introduced in this paper, provides routines to generate synthetic versions of original data sets. We describe the methodology and its consequences for the data characteristics. We illustrate the package features using a survey data example. Keywords: synthetic data, disclosure control, CART, R, UK Longitudinal Studies. This introduction to the R package synthpop is a slightly amended version of Nowok B, Raab GM, Dibben C (2016). synthpop: Bespoke Creation of Synthetic Data in R. Journal of Statistical Software, 74(11), 1-26. doi:10.18637/jss.v074.i11. URL https://www.jstatsoft. org/article/view/v074i11. 1. Introduction and background 1.1. Synthetic data for disclosure control National statistical agencies and other institutions gather large amounts of information about individuals and organisations. Such data can be used to understand population processes so as to inform policy and planning. The cost of such data can be considerable, both for the collectors and the subjects who provide their data. Because of confidentiality constraints and guarantees issued to data subjects the full access to such data is often restricted to the staff of the collection agencies. Traditionally, data collectors have used anonymisation along with simple perturbation methods such as aggregation, recoding, record-swapping, suppression of sensitive values or adding random noise to prevent the identification of data subjects.
  • Choosing Principal Components: a New Graphical Method Based on Bayesian Model Selection

    Choosing Principal Components: a New Graphical Method Based on Bayesian Model Selection

    Choosing principal components: a new graphical method based on Bayesian model selection Philipp Auer Daniel Gervini1 University of Ulm University of Wisconsin–Milwaukee October 31, 2007 1Corresponding author. Email: [email protected]. Address: P.O. Box 413, Milwaukee, WI 53201, USA. Abstract This article approaches the problem of selecting significant principal components from a Bayesian model selection perspective. The resulting Bayes rule provides a simple graphical technique that can be used instead of (or together with) the popular scree plot to determine the number of significant components to retain. We study the theoretical properties of the new method and show, by examples and simulation, that it provides more clear-cut answers than the scree plot in many interesting situations. Key Words: Dimension reduction; scree plot; factor analysis; singular value decomposition. MSC: Primary 62H25; secondary 62-09. 1 Introduction Multivariate datasets usually contain redundant information, due to high correla- tions among the variables. Sometimes they also contain irrelevant information; that is, sources of variability that can be considered random noise for practical purposes. To obtain uncorrelated, significant variables, several methods have been proposed over the years. Undoubtedly, the most popular is Principal Component Analysis (PCA), introduced by Hotelling (1933); for a comprehensive account of the methodology and its applications, see Jolliffe (2002). In many practical situations the first few components accumulate most of the variability, so it is common practice to discard the remaining components, thus re- ducing the dimensionality of the data. This has many advantages both from an inferential and from a descriptive point of view, since the leading components are often interpretable indices within the context of the problem and they are statis- tically more stable than subsequent components, which tend to be more unstruc- tured and harder to estimate.
  • Synthetic Data Generation with Probabilistic Bayesian Networks

    Synthetic Data Generation with Probabilistic Bayesian Networks

    bioRxiv preprint doi: https://doi.org/10.1101/2020.06.14.151084; this version posted September 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Synthetic data generation with probabilistic Bayesian Networks Grigoriy Gogoshin∗1a, Sergio Branciamore1b, and Andrei S. Rodin∗1c ∗Corresponding authors 1Department of Computational and Quantitative Medicine, Beckman Research Institute, and Diabetes and Metabolism Research Institute, City of Hope National Medical Center, 1500 East Duarte Road, Duarte, CA 91010 USA [email protected] [email protected] [email protected] Abstract Bayesian Network (BN) modeling is a prominent and increasingly popular computational sys- tems biology method. It aims to construct probabilistic networks from the large heterogeneous biological datasets that reflect the underlying networks of biological relationships. Currently, a va- riety of strategies exist for evaluating BN methodology performance, ranging from utilizing artificial benchmark datasets and models, to specialized biological benchmark datasets, to simulation stud- ies that generate synthetic data from predefined network models. The latter is arguably the most comprehensive approach; however, existing implementations are typically limited by their reliance on the SEM (structural equation modeling) framework, which includes many explicit and implicit assumptions that may be unrealistic in a typical biological data analysis scenario. In this study, we develop an alternative, purely probabilistic, simulation framework that more appropriately fits with real biological data and biological network models. In conjunction, we also expand on our current understanding of the theoretical notions of causality and dependence / conditional independence in BNs and the Markov Blankets within.
  • Fidelity and Privacy of Synthetic Medical Data Review of Methods and Experimental Results

    Fidelity and Privacy of Synthetic Medical Data Review of Methods and Experimental Results

    Fidelity and Privacy of Synthetic Medical Data Review of Methods and Experimental Results June 2021 Ofer Mendelevitch, Michael D. Lesh SM MD FACC, Keywords: synthetic data; statistical fidelity; safety; privacy; data access; data sharing; open data; metrics; EMR; EHR; clinical trials; review of methods; de-identification; re-identification; deep learning; generative models 1 Syntegra © - Fidelity and Privacy of Synthetic Medical Data Table of Contents Table of Contents 2 Abstract 3 1. Introduction 3 1.1 De-Identification and Re-Identification 3 1.2 Synthetic Data 4 2. Synthetic Data in Medicine 5 2.1 The Syntegra Synthetic Data Engine and Medical Mind 6 3. Statistical Fidelity Validation 6 3.1 Record Distance Metric 6 3.2 Visualize and Compare Datasets 7 3.3 Population Statistics 8 3.4 Single Variable (Marginal) Distributions 9 3.5 Pairwise Correlation 10 3.6 Multivariate Metrics 11 3.6.1 Predictive Model Performance 11 3.6.2 Survival Analysis 12 3.6.3 Discriminator AUC 13 3.7 Clinical Consistency Assessment 13 4. Privacy Validation 13 4.1 Disclosure Metrics 13 4.1.1 Membership Inference Test 14 4.1.2 File Membership Hypothesis Test 16 4.1.3 Attribute Inference Test 16 4.2 Copy Protection Metrics 17 4.2.1 Distance to Closest Record - DCR 17 4.2.2 Exposure 18 5. Experimental Results 19 5.1 Datasets 19 5.2 Results 20 5.2.1 Statistical Fidelity 20 5.2.2 Privacy 30 6. Discussion and Analysis 34 6.1 DIG dataset results analysis 34 6.2 NIS dataset results analysis 35 6.3 TEXAS dataset results analysis 35 6.4 BREAST results analysis 36 7.
  • Federated Principal Component Analysis

    Federated Principal Component Analysis

    Federated Principal Component Analysis Andreas Grammenos1,3∗ Rodrigo Mendoza-Smith2 Jon Crowcroft1,3 Cecilia Mascolo1 1Computer Lab, University of Cambridge 2Quine Technologies 3Alan Turing Institute Abstract We present a federated, asynchronous, and (ε, δ)-differentially private algorithm for PCA in the memory-limited setting. Our algorithm incrementally computes local model updates using a streaming procedure and adaptively estimates its r leading principal components when only O(dr) memory is available with d being the dimensionality of the data. We guarantee differential privacy via an input-perturbation scheme in which the covariance matrix of a dataset X ∈ Rd×n is perturbed with a non-symmetric random Gaussian matrix with variance in d 2 O n log d , thus improving upon the state-of-the-art. Furthermore, contrary to previous federated or distributed algorithms for PCA, our algorithm is also invariant to permutations in the incoming data, which provides robustness against straggler or failed nodes. Numerical simulations show that, while using limited- memory, our algorithm exhibits performance that closely matches or outperforms traditional non-federated algorithms, and in the absence of communication latency, it exhibits attractive horizontal scalability. 1 Introduction In recent years, the advent of edge computing in smartphones, IoT and cryptocurrencies has induced a paradigm shift in distributed model training and large-scale data analysis. Under this new paradigm, data is generated by commodity devices with hardware limitations and severe restrictions on data- sharing and communication, which makes the centralisation of the data extremely difficult. This has brought new computational challenges since algorithms do not only have to deal with the sheer volume of data generated by networks of devices, but also leverage the algorithm’s voracity, accuracy, and complexity with constraints on hardware capacity, data access, and device-device communication.
  • Principal Component Analysis (PCA) As a Statistical Tool for Identifying Key Indicators of Nuclear Power Plant Cable Insulation

    Principal Component Analysis (PCA) As a Statistical Tool for Identifying Key Indicators of Nuclear Power Plant Cable Insulation

    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation Chamila Chandima De Silva Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Materials Science and Engineering Commons, Mechanics of Materials Commons, and the Statistics and Probability Commons Recommended Citation De Silva, Chamila Chandima, "Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation" (2017). Graduate Theses and Dissertations. 16120. https://lib.dr.iastate.edu/etd/16120 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation by Chamila C. De Silva A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Materials Science and Engineering Program of Study Committee: Nicola Bowler, Major Professor Richard LeSar Steve Martin The student author and the program of study committee are solely responsible for the content of this thesis. The Graduate College will ensure this thesis is globally accessible and will not permit alterations after a degree is conferred.
  • Synthetic Data Sharing and Estimation of Viable Dynamic Treatment Regimes with Observational Data

    Synthetic Data Sharing and Estimation of Viable Dynamic Treatment Regimes with Observational Data

    Synthetic Data Sharing and Estimation of Viable Dynamic Treatment Regimes with Observational Data by Nina Zhou A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Biostatistics) in The University of Michigan 2021 Doctoral Committee: Professor Ivo D. Dinov, Co-Chair Professor Lu Wang, Co-Chair Research Associate Professor Daniel Almirall Assistant Professor Zhenke Wu Professor Chuanwu Xi Nina Zhou [email protected] ORCID iD: 0000-0002-1649-3458 c Nina Zhou 2021 ACKNOWLEDGEMENTS Firstly, I would like to express my sincere gratitude to my advisors Dr. Lu Wang and Dr. Ivo Dinov for their continuous support of my Ph.D. study and related research, for their patience, motivation, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis. Besides my advisors, I am grateful to the the rest of my thesis committee: Dr. Zhenke Wu, Dr. Daniel Almirall and Dr.Chuanwu Xi for their insightful comments and encouragements, but also for the hard question which incented me to widen my research from various perspectives. I would like to thank my fellow doctoral students for their feedback, cooperation and of course friendship. In addition, I would like to express my gratitude to Dr. Kristen Herold for proofreading all my writings. Last but not the least, I would like to thank my family: my parents and grand parents for supporting me spiritually throughout writing this thesis and my life in general. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS :::::::::::::::::::::::::: ii LIST OF FIGURES ::::::::::::::::::::::::::::::: vi LIST OF TABLES :::::::::::::::::::::::::::::::: viii LIST OF ALGORITHMS ::::::::::::::::::::::::::: x ABSTRACT ::::::::::::::::::::::::::::::::::: xi CHAPTER I.
  • Benchmarking Unsupervised Outlier Detection with Realistic Synthetic Data

    Benchmarking Unsupervised Outlier Detection with Realistic Synthetic Data

    Benchmarking Unsupervised Outlier Detection with Realistic Synthetic Data GEORG STEINBUSS, Karlsruhe Institute of Technology (KIT), Germany KLEMENS BÖHM, Karlsruhe Institute of Technology (KIT), Germany Benchmarking unsupervised outlier detection is difficult. Outliers are rare, and existing benchmark data contains outliers with various and unknown characteristics. Fully synthetic data usually consists of outliers and regular instance with clear characteristics and thus allows for a more meaningful evaluation of detection methods in principle. Nonetheless, there have only been few attempts to include synthetic data in benchmarks for outlier detection. This might be due to the imprecise notion of outliers or to the difficulty to arrive at a good coverage of different domains with synthetic data. In this work we propose a generic processfor the generation of data sets for such benchmarking. The core idea is to reconstruct regular instances from existing real-world benchmark data while generating outliers so that they exhibit insightful characteristics. This allows both for a good coverage of domains and for helpful interpretations of results. We also describe three instantiations of the generic process that generate outliers with specific characteristics, like local outliers. A benchmark with state-of-the-art detection methods confirms that our generic process is indeed practical. CCS Concepts: • Mathematics of computing ! Distribution functions; • Computing methodologies ! Machine learning algorithms; Anomaly detection. Additional Key Words and Phrases: Outlier Detection, Unsupervised, Benchmark, Synthetic Data. 1 INTRODUCTION Unsupervised outlier detection aims at finding data instances in an unlabeled data set that deviate from most other instances. Such outliers tend to be rare and usually exhibit unknown characteristics. This renders the evaluation of detection performance and hence comparisons of unsupervised outlier-detection methods difficult.
  • Exam-Srm-Sample-Questions.Pdf

    Exam-Srm-Sample-Questions.Pdf

    SOCIETY OF ACTUARIES EXAM SRM - STATISTICS FOR RISK MODELING EXAM SRM SAMPLE QUESTIONS AND SOLUTIONS These questions and solutions are representative of the types of questions that might be asked of candidates sitting for Exam SRM. These questions are intended to represent the depth of understanding required of candidates. The distribution of questions by topic is not intended to represent the distribution of questions on future exams. April 2019 update: Question 2, answer III was changed. While the statement was true, it was not directly supported by the required readings. July 2019 update: Questions 29-32 were added. September 2019 update: Questions 33-44 were added. December 2019 update: Question 40 item I was modified. January 2020 update: Question 41 item I was modified. November 2020 update: Questions 6 and 14 were modified and Questions 45-48 were added. February 2021 update: Questions 49-53 were added. July 2021 update: Questions 54-55 were added. Copyright 2021 by the Society of Actuaries SRM-02-18 1 QUESTIONS 1. You are given the following four pairs of observations: x1=−==−=( 1,0), xx 23 (1,1), (2, 1), and x 4(5,10). A hierarchical clustering algorithm is used with complete linkage and Euclidean distance. Calculate the intercluster dissimilarity between {,xx12 } and {}x4 . (A) 2.2 (B) 3.2 (C) 9.9 (D) 10.8 (E) 11.7 SRM-02-18 2 2. Determine which of the following statements is/are true. I. The number of clusters must be pre-specified for both K-means and hierarchical clustering. II. The K-means clustering algorithm is less sensitive to the presence of outliers than the hierarchical clustering algorithm.
  • Dynamic Factor Models Catherine Doz, Peter Fuleky

    Dynamic Factor Models Catherine Doz, Peter Fuleky

    Dynamic Factor Models Catherine Doz, Peter Fuleky To cite this version: Catherine Doz, Peter Fuleky. Dynamic Factor Models. 2019. halshs-02262202 HAL Id: halshs-02262202 https://halshs.archives-ouvertes.fr/halshs-02262202 Preprint submitted on 2 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. WORKING PAPER N° 2019 – 45 Dynamic Factor Models Catherine Doz Peter Fuleky JEL Codes: C32, C38, C53, C55 Keywords : dynamic factor models; big data; two-step estimation; time domain; frequency domain; structural breaks DYNAMIC FACTOR MODELS ∗ Catherine Doz1,2 and Peter Fuleky3 1Paris School of Economics 2Université Paris 1 Panthéon-Sorbonne 3University of Hawaii at Manoa July 2019 ∗Prepared for Macroeconomic Forecasting in the Era of Big Data Theory and Practice (Peter Fuleky ed), Springer. 1 Chapter 2 Dynamic Factor Models Catherine Doz and Peter Fuleky Abstract Dynamic factor models are parsimonious representations of relationships among time series variables. With the surge in data availability, they have proven to be indispensable in macroeconomic forecasting. This chapter surveys the evolution of these models from their pre-big-data origins to the large-scale models of recent years. We review the associated estimation theory, forecasting approaches, and several extensions of the basic framework.
  • Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE

    Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE

    IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 5, MAY 2007 1361 Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE Abstract—One-dimensional Generalized Autoregressive Con- Markov noise. It is also claimed that objects in imagery create a ditional Heteroscedasticity (GARCH) model is widely used for response over several scales in a multiresolution representation modeling financial time series. Extending the GARCH model to of an image, and therefore, the wavelet transform can serve as multiple dimensions yields a novel clutter model which is capable of taking into account important characteristics of a wavelet-based a means for computing a feature set for input to a detector. In multiscale feature space, namely heavy-tailed distributions and [17], a multiscale wavelet representation is utilized to capture innovations clustering as well as spatial and scale correlations. We periodical patterns of various period lengths, which often ap- show that the multidimensional GARCH model generalizes the pear in natural clutter images. In [12], the orientation and scale casual Gauss Markov random field (GMRF) model, and we de- selectivity of the wavelet transform are related to the biological velop a multiscale matched subspace detector (MSD) for detecting anomalies in GARCH clutter. Experimental results demonstrate mechanisms of the human visual system and are utilized to that by using a multiscale MSD under GARCH clutter modeling, enhance mammographic features. rather than GMRF clutter modeling, a reduced false-alarm rate Statistical models for clutter and anomalies are usually can be achieved without compromising the detection rate. related to the Gaussian distribution due to its mathematical Index Terms—Anomaly detection, Gaussian Markov tractability.