An Introduction to Ordination Connie Clark
|
|
- David Caldwell
- 6 years ago
- Views:
Transcription
1 An Introduction to Ordination Connie Clark Ordination is a collective term for multivariate techniques that adapt a multidimensional swarm of data points in such a way that when it is projected onto a two-dimensional space any intrinsic pattern the data may possess becomes apparent upon visual inspection (Pielou, 1984). Basically, ordination serves to summarize community data (such as species abundance data) by producing a low-dimensional ordination space in which similar species and samples are plotted close together, and dissimilar species and samples are placed far apart. Generally, ordination techniques are used to describe relationships between species composition patterns and the underlying environmental gradients that influence these patterns (asking, what factors structure the community?). For example, if you wanted to examine the distribution patterns of tree species in the Sierra Nevada Mt. Range, ordination could be used to determine which species are commonly found associated with one another, and how the species composition of the community changes with increase in elevation. Recently, use of ordination techniques have expanded to include analysis of dietary overlap (Schluter and Grant, 1982), and to explore patterns of within species morphological differences with geographic distance between populations (Alisauskas, 1998). Data Commonly, data interpreted using ordination are collected in a species by sample data matrix, similar to the matrix presented below. Sample data may include measures of density, biomass, frequency, importance values, presence/absence, or any number of abundance measures.
2 E7000 E6580 E6000 E5400 E5000 E4000 E2850 E1800 ABMA ABCO ACMA ARME CADE CONU LIDE PICO PILA PIPO PIJE PSME QUCH QUWI QUKE The above is a relatively simple data set. However, it is easy to imagine that a true data set may encounter dozens of species over hundreds of samples. Complex sample by species matrices represent dozens to hundreds of dimensions that are impossible to visualize or interpret. Even graphed, species response curves of large community data sets can be nearly impossible to interpret. (As they resemble a mess of overlapping peaks and depressions as shown here.)
3 Ordination can help us find structure in these complicated data sets. By using various mathematical calculations (which will not be discussed here), ordination techniques will identify similarity between species and samples. Results are then projected onto two dimensions in such a way that species and samples most similar to one another will be close together, and species and samples most dissimilar from one another will appear farther apart (as shown below). Ordination techniques: There are several different ordination techniques, all of which differ slightly, in the mathematical approach used to calculate species and sample similarity/dissimiarity. Rather than reinventing the wheel by discussing each of these techniques in depth, I will offer only a brief description of the most commonly used methods here. Further details can be found in the following suggested references: Gauch, H. G., Jr Multivariate Analysis in Community Structure. Cambridge University Press, Cambridge Causton, D. R An introduction to vegetation analysis. Unwin Hyman, London. Kent, M., and P. Coker Vegetation description and analysis: a practical approach. Belhaven Press, London.
4 Pielou, E. C The Interpretation of Ecological Data: A Primer on Classification and Ordination. Wiley, New York Okland, R. H Vegetation ecology: theory, methods and applications with reference to Fennoscandia. Sommerfeltia Supplement 1: Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren, editors Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen, The Netherlands. Analysis of Ecological Communities. Chapman and Hall, London. Web Links The Ordination Webpage Note: this web site comes highly recommended as it provides detailed yet simple explanations of most currently used ordination techniques (see the Indirect Gradient Analysis section of above mentioned web page). In the General Reference section of the web site, Palmer offers a fantastic glossary for terms used in ordination, and clarifies some common confusion in the terminology used to date. In addition, he provides links to other ordination sites and offers addresses for software links. In the Statistics and Background section of the site, read through Centroids and Inertia, Similarity, Distance and Difference, and Explorations in Coenspace for the conceptual background necessary in understanding ordination techniques. The Direct Gradient Analysis section will be of interest if you have specific environmental data collected in addition to abundance and species data. You may find this to be a stronger approach to the analysis of your data set. Ecological Data, Transformations and Standardization is for more advanced users who likely have an understanding of ordination and seek more advanced information regarding data manipulation. Principal Components Analysis (PCA) PCA was one of the earliest ordination techniques applied to ecological data. PCA uses a rigid rotation to derive orthogonal axes, which maximize the variance in the data set. Both species and sample ordinations result from a single analysis. Computationally, PCA is basically an eigenanalysis. The sum of the eigenvalues will equal the sum of the variance of all variables in the data set. PCA is relatively objective and provides a reasonable but crude indication of relationships. For further computational detail click here. Reciprocal Averaging (RA) RA is an ordination technique related conceptually to weighted averages.
5 However, computationally, RA is related to eigenvector ordinations. RA places sampling units and species on the same gradients, and maximizes variation between species and sample scores using a correlation coefficient. It serves as a relatively objective analysis of community data. Results are generally superior to the results from PCA. However, RA axis ends are compressed relative to the middle, and the second axis is often a distortion of the first axis, resulting in an arched effect. Detrended Correspondence Analysis (DCA) DCA is an eigenvector ordination technique based on Reciprocal Averaging, correcting for the arch effect produced from RA. Hill and Gauch (1980) report DCA results are superior to those of RA. Other ecologists criticize the detrending process of DCA. DCA is widely used for the analysis of community data along gradients. It has also been found effective for niche ordination of birds by foraging heights (Sabo 1980). DCA ordinates samples and species simultaneously. It is not appropriate for the analysis of a matrix of similarity values between community data (Gauch, 1982). Nonmetric Multidimensional Scaling (NMS) NMS actually refers to an entire related family of ordination techniques. These techniques use rank order information to identify similarity in a data set. NMS is a truly nonparametric ordination method which seeks to best reduce space portrayal of relationships. The verdict is still out on this type of ordination. Gauch (1982) claims NMS is not worth the extra computational effort, and that it gives effective results only for easy data sets with low diversity. Others hold NMS is extremely effective (Kenkel and Orloci, 1986, Bradfield and Kenkel, 1987). Appropriate uses of ordination: It is important to keep in mind that the purpose of ordination is to assist a researcher to find pattern in data sets that are otherwise too complicated to interpret. A good ordination technique will be able to identify the most important dimensions in a data set, and ignore the "noise", in order to show these patterns. However, ordination techniques should not be used in hypothesis driven analysis. They are meant as exploratory tools. Thus, post-hoc analysis is acceptable, and many different techniques can be tried on the same data set. No null hypothesis can be rejected, nor are p- values generated to test statistical significance. When p-values are offered, they can only be used as a rough guide or indicator of underlying processes that MAY BE explaining community patterns. Bibliography
6 Alisauskas, R. T Winter range expansion and relationships between landscape and morphometrics of midcontinent Lesser Snow Geese. Auk 115: Brandfield, G. E., and N. C. Kenkel Nonlinear ordination using flexible shortest path adjustment of ecological distances. Ecology 68: Gauch, H. G., Jr Multivariate Analysis in Community Structure. Cambridge University Press, Cambridge. Hill, M. O. and Gauch, H. G Deterended correspondence analysis, an improved ordination technique. Vegetatio 42: Kenkel, N. C., and L. Orloci Applying metric and nonmetric multidimensional scaling to ecological studies: some new results. Ecology 67: Pielou, E. C The Interpretation of Ecological Data: A Primer on Classification and Ordination. Wiley, New York. Sabo, S. R Niche and habitat relations in subalpine bird communities of the White Mountains of New Hampshire. Ecological Monographs 50: Schluter. D., and P. R. Grant The distribution of Geospiza difficilis on Galapagos islands: test of three hypotheses. Evolution 36:
Rigid rotation of nonmetric multidimensional scaling axes to environmental congruence
Ab~tracta Batanica 14:100-110, 1000 Department of Plant Taonomy and Ecology, ELTE. Budapeat Rigid rotation of nonmetric multidimensional scaling aes to environmental congruence N.C. Kenkel and C.E. Burchill
More informationCanonical Correlation & Principle Components Analysis
Canonical Correlation & Principle Components Analysis Aaron French Canonical Correlation Canonical Correlation is used to analyze correlation between two sets of variables when there is one set of IVs
More information-Principal components analysis is by far the oldest multivariate technique, dating back to the early 1900's; ecologists have used PCA since the
1 2 3 -Principal components analysis is by far the oldest multivariate technique, dating back to the early 1900's; ecologists have used PCA since the 1950's. -PCA is based on covariance or correlation
More informationIntroduction to ordination. Gary Bradfield Botany Dept.
Introduction to ordination Gary Bradfield Botany Dept. Ordination there appears to be no word in English which one can use as an antonym to classification ; I would like to propose the term ordination.
More informationMultivariate Analysis of Ecological Data using CANOCO
Multivariate Analysis of Ecological Data using CANOCO JAN LEPS University of South Bohemia, and Czech Academy of Sciences, Czech Republic Universitats- uric! Lanttesbibiiothek Darmstadt Bibliothek Biologie
More informationUnconstrained Ordination
Unconstrained Ordination Sites Species A Species B Species C Species D Species E 1 0 (1) 5 (1) 1 (1) 10 (4) 10 (4) 2 2 (3) 8 (3) 4 (3) 12 (6) 20 (6) 3 8 (6) 20 (6) 10 (6) 1 (2) 3 (2) 4 4 (5) 11 (5) 8 (5)
More informationBIO 682 Multivariate Statistics Spring 2008
BIO 682 Multivariate Statistics Spring 2008 Steve Shuster http://www4.nau.edu/shustercourses/bio682/index.htm Lecture 11 Properties of Community Data Gauch 1982, Causton 1988, Jongman 1995 a. Qualitative:
More informationANOVA approach. Investigates interaction terms. Disadvantages: Requires careful sampling design with replication
ANOVA approach Advantages: Ideal for evaluating hypotheses Ideal to quantify effect size (e.g., differences between groups) Address multiple factors at once Investigates interaction terms Disadvantages:
More informationExperimental Design and Data Analysis for Biologists
Experimental Design and Data Analysis for Biologists Gerry P. Quinn Monash University Michael J. Keough University of Melbourne CAMBRIDGE UNIVERSITY PRESS Contents Preface page xv I I Introduction 1 1.1
More informationIntroduction to multivariate analysis Outline
Introduction to multivariate analysis Outline Why do a multivariate analysis Ordination, classification, model fitting Principal component analysis Discriminant analysis, quickly Species presence/absence
More informationINTRODUCTION TO MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA
INTRODUCTION TO MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA David Zelený & Ching-Feng Li INTRODUCTION TO MULTIVARIATE ANALYSIS Ecologial similarity similarity and distance indices Gradient analysis regression,
More informationPrincipal Component Analysis (PCA) Theory, Practice, and Examples
Principal Component Analysis (PCA) Theory, Practice, and Examples Data Reduction summarization of data with many (p) variables by a smaller set of (k) derived (synthetic, composite) variables. p k n A
More informationOrdination & PCA. Ordination. Ordination
Ordination & PCA Introduction to Ordination Purpose & types Shepard diagrams Principal Components Analysis (PCA) Properties Computing eigenvalues Computing principal components Biplots Covariance vs. Correlation
More informationMultivariate Data Analysis a survey of data reduction and data association techniques: Principal Components Analysis
Multivariate Data Analysis a survey of data reduction and data association techniques: Principal Components Analysis For example Data reduction approaches Cluster analysis Principal components analysis
More information4. Ordination in reduced space
Université Laval Analyse multivariable - mars-avril 2008 1 4.1. Generalities 4. Ordination in reduced space Contrary to most clustering techniques, which aim at revealing discontinuities in the data, ordination
More informationDiscrimination Among Groups. Discrimination Among Groups
Discrimination Among Groups Id Species Canopy Snag Canopy Cover Density Height 1 A 80 1.2 35 2 A 75 0.5 32 3 A 72 2.8 28..... 31 B 35 3.3 15 32 B 75 4.1 25 60 B 15 5.0 3..... 61 C 5 2.1 5 62 C 8 3.4 2
More information4/2/2018. Canonical Analyses Analysis aimed at identifying the relationship between two multivariate datasets. Cannonical Correlation.
GAL50.44 0 7 becki 2 0 chatamensis 0 darwini 0 ephyppium 0 guntheri 3 0 hoodensis 0 microphyles 0 porteri 2 0 vandenburghi 0 vicina 4 0 Multiple Response Variables? Univariate Statistics Questions Individual
More informationLecture 5: Ecological distance metrics; Principal Coordinates Analysis. Univariate testing vs. community analysis
Lecture 5: Ecological distance metrics; Principal Coordinates Analysis Univariate testing vs. community analysis Univariate testing deals with hypotheses concerning individual taxa Is this taxon differentially
More informationMultivariate Statistics Summary and Comparison of Techniques. Multivariate Techniques
Multivariate Statistics Summary and Comparison of Techniques P The key to multivariate statistics is understanding conceptually the relationship among techniques with regards to: < The kinds of problems
More informationLecture 5: Ecological distance metrics; Principal Coordinates Analysis. Univariate testing vs. community analysis
Lecture 5: Ecological distance metrics; Principal Coordinates Analysis Univariate testing vs. community analysis Univariate testing deals with hypotheses concerning individual taxa Is this taxon differentially
More informationVarCan (version 1): Variation Estimation and Partitioning in Canonical Analysis
VarCan (version 1): Variation Estimation and Partitioning in Canonical Analysis Pedro R. Peres-Neto March 2005 Department of Biology University of Regina Regina, SK S4S 0A2, Canada E-mail: Pedro.Peres-Neto@uregina.ca
More informationInconsistencies between theory and methodology: a recurrent problem in ordination studies.
This is the pre-peer-reviewed version of the following article: Inconsistencies between theory and methodology: recurrent problem in ordination studies, Austin, M., Journal of Vegetation Science, vol.
More informationBIOL 580 Analysis of Ecological Communities
BIOL 580 Analysis of Ecological Communities Monday 9:00 Lewis 407, Tuesday-Thursday 9:00-11:00, AJMJ 221 Dave Roberts Ecology Department 310 Lewis Hall droberts@montana.edu Course Description This course
More information1.2. Correspondence analysis. Pierre Legendre Département de sciences biologiques Université de Montréal
1.2. Pierre Legendre Département de sciences biologiques Université de Montréal http://www.numericalecology.com/ Pierre Legendre 2018 Definition of correspondence analysis (CA) An ordination method preserving
More informationMachine Learning (Spring 2012) Principal Component Analysis
1-71 Machine Learning (Spring 1) Principal Component Analysis Yang Xu This note is partly based on Chapter 1.1 in Chris Bishop s book on PRML and the lecture slides on PCA written by Carlos Guestrin in
More informationLinking species-compositional dissimilarities and environmental data for biodiversity assessment
Linking species-compositional dissimilarities and environmental data for biodiversity assessment D. P. Faith, S. Ferrier Australian Museum, 6 College St., Sydney, N.S.W. 2010, Australia; N.S.W. National
More informationStructure in Data. A major objective in data analysis is to identify interesting features or structure in the data.
Structure in Data A major objective in data analysis is to identify interesting features or structure in the data. The graphical methods are very useful in discovering structure. There are basically two
More informationBootstrapped ordination: a method for estimating sampling effects in indirect gradient analysis
Vegetatio 8: 153-165, 1989. 1989 Kluwer Academic Publishers. Printed in Belgium. 153 Bootstrapped ordination: a method for estimating sampling effects in indirect gradient analysis Robert G. Knox ~ & Robert
More informationChapter 11 Canonical analysis
Chapter 11 Canonical analysis 11.0 Principles of canonical analysis Canonical analysis is the simultaneous analysis of two, or possibly several data tables. Canonical analyses allow ecologists to perform
More informationFocus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations.
Previously Focus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations y = Ax Or A simply represents data Notion of eigenvectors,
More informationDimension Reduction Techniques. Presented by Jie (Jerry) Yu
Dimension Reduction Techniques Presented by Jie (Jerry) Yu Outline Problem Modeling Review of PCA and MDS Isomap Local Linear Embedding (LLE) Charting Background Advances in data collection and storage
More informationMultivariate Ordination Analyses: Principal Component Analysis. Dilys Vela
Multivariate Ordination Analyses: Principal Component Analysis Dilys Vela Tatiana Boza Multivariate Analyses A multivariate data set includes more than one variable ibl recorded dd from a number of replicate
More informationTable of Contents. Multivariate methods. Introduction II. Introduction I
Table of Contents Introduction Antti Penttilä Department of Physics University of Helsinki Exactum summer school, 04 Construction of multinormal distribution Test of multinormality with 3 Interpretation
More informationDIMENSION REDUCTION AND CLUSTER ANALYSIS
DIMENSION REDUCTION AND CLUSTER ANALYSIS EECS 833, 6 March 2006 Geoff Bohling Assistant Scientist Kansas Geological Survey geoff@kgs.ku.edu 864-2093 Overheads and resources available at http://people.ku.edu/~gbohling/eecs833
More informationStatistical Analysis of fmrl Data
Statistical Analysis of fmrl Data F. Gregory Ashby The MIT Press Cambridge, Massachusetts London, England Preface xi Acronyms xv 1 Introduction 1 What Is fmri? 2 The Scanning Session 4 Experimental Design
More informationEach copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
Effects of Sample Distribution along Gradients on Eigenvector Ordination Author(s): C. L. Mohler Source: Vegetatio, Vol. 45, No. 3 (Jul. 31, 1981), pp. 141-145 Published by: Springer Stable URL: http://www.jstor.org/stable/20037040.
More informationFace Recognition Using Laplacianfaces He et al. (IEEE Trans PAMI, 2005) presented by Hassan A. Kingravi
Face Recognition Using Laplacianfaces He et al. (IEEE Trans PAMI, 2005) presented by Hassan A. Kingravi Overview Introduction Linear Methods for Dimensionality Reduction Nonlinear Methods and Manifold
More informationFACTOR ANALYSIS AND MULTIDIMENSIONAL SCALING
FACTOR ANALYSIS AND MULTIDIMENSIONAL SCALING Vishwanath Mantha Department for Electrical and Computer Engineering Mississippi State University, Mississippi State, MS 39762 mantha@isip.msstate.edu ABSTRACT
More informationMS-E2112 Multivariate Statistical Analysis (5cr) Lecture 6: Bivariate Correspondence Analysis - part II
MS-E2112 Multivariate Statistical Analysis (5cr) Lecture 6: Bivariate Correspondence Analysis - part II the Contents the the the Independence The independence between variables x and y can be tested using.
More informationECE 521. Lecture 11 (not on midterm material) 13 February K-means clustering, Dimensionality reduction
ECE 521 Lecture 11 (not on midterm material) 13 February 2017 K-means clustering, Dimensionality reduction With thanks to Ruslan Salakhutdinov for an earlier version of the slides Overview K-means clustering
More informationChapter 1 Ordination Methods and the Evaluation of Ediacaran Communities
Chapter 1 Ordination Methods and the Evaluation of Ediacaran Communities 1 2 3 Matthew E. Clapham 4 Contents 1.1 Introduction... 000 1.2 ataset Summary... 000 1.3 ata Standardization... 000 1.4 Ordination
More informationBIOL 580 Analysis of Ecological Communities
BIOL 580 Analysis of Ecological Communities Monday 9:00 Lewis 407, Tuesday-Thursday 9:00-11:00, Lewis 407 Dave Roberts Ecology Department 117 AJM Johnson Hall droberts@montana.edu Course Description This
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Feature Extraction Hamid R. Rabiee Jafar Muhammadi, Alireza Ghasemi, Payam Siyari Spring 2014 http://ce.sharif.edu/courses/92-93/2/ce725-2/ Agenda Dimensionality Reduction
More information8. FROM CLASSICAL TO CANONICAL ORDINATION
Manuscript of Legendre, P. and H. J. B. Birks. 2012. From classical to canonical ordination. Chapter 8, pp. 201-248 in: Tracking Environmental Change using Lake Sediments, Volume 5: Data handling and numerical
More informationDimension Reduction and Classification Using PCA and Factor. Overview
Dimension Reduction and Classification Using PCA and - A Short Overview Laboratory for Interdisciplinary Statistical Analysis Department of Statistics Virginia Tech http://www.stat.vt.edu/consult/ March
More informationPCA Advanced Examples & Applications
PCA Advanced Examples & Applications Objectives: Showcase advanced PCA analysis: - Addressing the assumptions - Improving the signal / decreasing the noise Principal Components (PCA) Paper II Example:
More informationMaximum variance formulation
12.1. Principal Component Analysis 561 Figure 12.2 Principal component analysis seeks a space of lower dimensionality, known as the principal subspace and denoted by the magenta line, such that the orthogonal
More informationMultidimensional scaling (MDS)
Multidimensional scaling (MDS) Just like SOM and principal curves or surfaces, MDS aims to map data points in R p to a lower-dimensional coordinate system. However, MSD approaches the problem somewhat
More informationEXAM PRACTICE. 12 questions * 4 categories: Statistics Background Multivariate Statistics Interpret True / False
EXAM PRACTICE 12 questions * 4 categories: Statistics Background Multivariate Statistics Interpret True / False Stats 1: What is a Hypothesis? A testable assertion about how the world works Hypothesis
More informationEigenvalues, Eigenvectors, and an Intro to PCA
Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis.
More informationMultivariate Statistics 101. Ordination (PCA, NMDS, CA) Cluster Analysis (UPGMA, Ward s) Canonical Correspondence Analysis
Multivariate Statistics 101 Ordination (PCA, NMDS, CA) Cluster Analysis (UPGMA, Ward s) Canonical Correspondence Analysis Multivariate Statistics 101 Copy of slides and exercises PAST software download
More informationPrincipal Component Analysis, A Powerful Scoring Technique
Principal Component Analysis, A Powerful Scoring Technique George C. J. Fernandez, University of Nevada - Reno, Reno NV 89557 ABSTRACT Data mining is a collection of analytical techniques to uncover new
More informationPrincipal component analysis
Principal component analysis Motivation i for PCA came from major-axis regression. Strong assumption: single homogeneous sample. Free of assumptions when used for exploration. Classical tests of significance
More informationWhy Is It There? Attribute Data Describe with statistics Analyze with hypothesis testing Spatial Data Describe with maps Analyze with spatial analysis
6 Why Is It There? Why Is It There? Getting Started with Geographic Information Systems Chapter 6 6.1 Describing Attributes 6.2 Statistical Analysis 6.3 Spatial Description 6.4 Spatial Analysis 6.5 Searching
More informationMachine Learning. Principal Components Analysis. Le Song. CSE6740/CS7641/ISYE6740, Fall 2012
Machine Learning CSE6740/CS7641/ISYE6740, Fall 2012 Principal Components Analysis Le Song Lecture 22, Nov 13, 2012 Based on slides from Eric Xing, CMU Reading: Chap 12.1, CB book 1 2 Factor or Component
More informationSPECIES RESPONSE CURVES! Steven M. Holland" Department of Geology, University of Georgia, Athens, GA " !!!! June 2014!
SPECIES RESPONSE CURVES Steven M. Holland" Department of Geology, University of Georgia, Athens, GA 30602-2501" June 2014 Introduction Species live on environmental gradients, and we often would like to
More informationGeneralized Linear Models (GLZ)
Generalized Linear Models (GLZ) Generalized Linear Models (GLZ) are an extension of the linear modeling process that allows models to be fit to data that follow probability distributions other than the
More informationFINAL EXAM Ma (Eakin) Fall 2015 December 16, 2015
FINAL EXAM Ma-00 Eakin Fall 05 December 6, 05 Please make sure that your name and GUID are on every page. This exam is designed to be done with pencil-and-paper calculations. You may use your calculator
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Principal Components Analysis (PCA) a technique for finding patterns in data of high dimension Outline:. Eigenvectors and eigenvalues. PCA: a) Getting the data b) Centering
More informationMULTIVARIATE ANALYSIS OF VARIANCE
MULTIVARIATE ANALYSIS OF VARIANCE RAJENDER PARSAD AND L.M. BHAR Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 lmb@iasri.res.in. Introduction In many agricultural experiments,
More informationRobustness of Principal Components
PCA for Clustering An objective of principal components analysis is to identify linear combinations of the original variables that are useful in accounting for the variation in those original variables.
More informationDimension reduction, PCA & eigenanalysis Based in part on slides from textbook, slides of Susan Holmes. October 3, Statistics 202: Data Mining
Dimension reduction, PCA & eigenanalysis Based in part on slides from textbook, slides of Susan Holmes October 3, 2012 1 / 1 Combinations of features Given a data matrix X n p with p fairly large, it can
More informationPrinciple Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA
Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA Principle Components Analysis: Uses one group of variables (we will call this X) In
More informationUnsupervised learning: beyond simple clustering and PCA
Unsupervised learning: beyond simple clustering and PCA Liza Rebrova Self organizing maps (SOM) Goal: approximate data points in R p by a low-dimensional manifold Unlike PCA, the manifold does not have
More informationMultivariate Analysis of Ecological Data
Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology
More informationseries. Utilize the methods of calculus to solve applied problems that require computational or algebraic techniques..
1 Use computational techniques and algebraic skills essential for success in an academic, personal, or workplace setting. (Computational and Algebraic Skills) MAT 203 MAT 204 MAT 205 MAT 206 Calculus I
More informationCS281 Section 4: Factor Analysis and PCA
CS81 Section 4: Factor Analysis and PCA Scott Linderman At this point we have seen a variety of machine learning models, with a particular emphasis on models for supervised learning. In particular, we
More informationPrincipal Component Analysis
Principal Component Analysis Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [based on slides from Nina Balcan] slide 1 Goals for the lecture you should understand
More informationROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015
ROBERTO BATTITI, MAURO BRUNATO. The LION Way: Machine Learning plus Intelligent Optimization. LIONlab, University of Trento, Italy, Apr 2015 http://intelligentoptimization.org/lionbook Roberto Battiti
More informationA COMPARISON OF THREE MULTIVARIATE STATISTICAL TECHNIQUES FOR THE ANALYSIS OF AVIAN FORAGING DATA
Studies in Avian Biology No. 13:295-38, 199. A COMPARISON OF THREE MULTIVARIATE STATISTICAL TECHNIQUES FOR THE ANALYSIS OF AVIAN FORAGING DATA DONALD B. MILES Abstract. This study discusses the complexities
More informationWhat is Principal Component Analysis?
What is Principal Component Analysis? Principal component analysis (PCA) Reduce the dimensionality of a data set by finding a new set of variables, smaller than the original set of variables Retains most
More informationMathematics with Maple
Mathematics with Maple A Comprehensive E-Book Harald Pleym Preface The main objective of these Maple worksheets, organized for use with all Maple versions from Maple 14, is to show how the computer algebra
More informationProximity data visualization with h-plots
The fifth international conference user! 2009 Proximity data visualization with h-plots Irene Epifanio Dpt. Matemàtiques, Univ. Jaume I (SPAIN) epifanio@uji.es; http://www3.uji.es/~epifanio Outline Motivating
More information1. Introduction to Multivariate Analysis
1. Introduction to Multivariate Analysis Isabel M. Rodrigues 1 / 44 1.1 Overview of multivariate methods and main objectives. WHY MULTIVARIATE ANALYSIS? Multivariate statistical analysis is concerned with
More informationLECTURE 4 PRINCIPAL COMPONENTS ANALYSIS / EXPLORATORY FACTOR ANALYSIS
LECTURE 4 PRINCIPAL COMPONENTS ANALYSIS / EXPLORATORY FACTOR ANALYSIS NOTES FROM PRE- LECTURE RECORDING ON PCA PCA and EFA have similar goals. They are substantially different in important ways. The goal
More informationA Theory of Gradient Analysis
Originally Published in Volume 18 (this series), pp 271 317, 1988 A Theory of Gradient Analysis CAJO J.F. TER BRAAK AND I. COLIN PRENTICE I. Introduction... 236 II. Linear Methods... 241 A. Regression...
More informationPrincipal Component Analysis -- PCA (also called Karhunen-Loeve transformation)
Principal Component Analysis -- PCA (also called Karhunen-Loeve transformation) PCA transforms the original input space into a lower dimensional space, by constructing dimensions that are linear combinations
More informationMachine Learning 2nd Edition
INTRODUCTION TO Lecture Slides for Machine Learning 2nd Edition ETHEM ALPAYDIN, modified by Leonardo Bobadilla and some parts from http://www.cs.tau.ac.il/~apartzin/machinelearning/ The MIT Press, 2010
More informationIntroduction to Machine Learning
10-701 Introduction to Machine Learning PCA Slides based on 18-661 Fall 2018 PCA Raw data can be Complex, High-dimensional To understand a phenomenon we measure various related quantities If we knew what
More informationVegetation Change Detection of Central part of Nepal using Landsat TM
Vegetation Change Detection of Central part of Nepal using Landsat TM Kalpana G. Bastakoti Department of Geography, University of Calgary, kalpanagb@gmail.com Abstract This paper presents a study of detecting
More informationFactors affecting the Power and Validity of Randomization-based Multivariate Tests for Difference among Ecological Assemblages
Factors affecting the Power and Validity of Randomization-based Multivariate Tests for Difference among Ecological Assemblages Cameron Hurst B.Sc. (Hons) This thesis was submitted in fulfillment of the
More informationFigure 43 - The three components of spatial variation
Université Laval Analyse multivariable - mars-avril 2008 1 6.3 Modeling spatial structures 6.3.1 Introduction: the 3 components of spatial structure For a good understanding of the nature of spatial variation,
More informationEigenvalues, Eigenvectors, and an Intro to PCA
Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis.
More informationMatrix Vector Products
We covered these notes in the tutorial sessions I strongly recommend that you further read the presented materials in classical books on linear algebra Please make sure that you understand the proofs and
More informationLinear Dimensionality Reduction
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Principal Component Analysis 3 Factor Analysis
More informationEigenvalues, Eigenvectors, and an Intro to PCA
Eigenvalues, Eigenvectors, and an Intro to PCA Eigenvalues, Eigenvectors, and an Intro to PCA Changing Basis We ve talked so far about re-writing our data using a new set of variables, or a new basis.
More informationStatistics 202: Data Mining. c Jonathan Taylor. Week 2 Based in part on slides from textbook, slides of Susan Holmes. October 3, / 1
Week 2 Based in part on slides from textbook, slides of Susan Holmes October 3, 2012 1 / 1 Part I Other datatypes, preprocessing 2 / 1 Other datatypes Document data You might start with a collection of
More informationPart I. Other datatypes, preprocessing. Other datatypes. Other datatypes. Week 2 Based in part on slides from textbook, slides of Susan Holmes
Week 2 Based in part on slides from textbook, slides of Susan Holmes Part I Other datatypes, preprocessing October 3, 2012 1 / 1 2 / 1 Other datatypes Other datatypes Document data You might start with
More informationTHE OBJECTIVE FUNCTION OF PARTIAL LEAST SQUARES REGRESSION
THE OBJECTIVE FUNCTION OF PARTIAL LEAST SQUARES REGRESSION CAJO J. F. TER BRAAK Centre for Biometry Wageningen, P.O. Box 1, NL-67 AC Wageningen, The Netherlands AND SIJMEN DE JONG Unilever Research Laboratorium,
More informationSeptember 16, 2004 The NEURON Book: Chapter 2
Chapter 2 The ing perspective This and the following chapter deal with concepts that are not NEURON-specific but instead pertain equally well to any tools used for neural ing. Why? In order to achieve
More informationVariations in pelagic bacterial communities in the North Atlantic Ocean coincide with water bodies
The following supplement accompanies the article Variations in pelagic bacterial communities in the North Atlantic Ocean coincide with water bodies Richard L. Hahnke 1, Christina Probian 1, Bernhard M.
More informationCS168: The Modern Algorithmic Toolbox Lecture #8: PCA and the Power Iteration Method
CS168: The Modern Algorithmic Toolbox Lecture #8: PCA and the Power Iteration Method Tim Roughgarden & Gregory Valiant April 15, 015 This lecture began with an extended recap of Lecture 7. Recall that
More informationMultiple regression and inference in ecology and conservation biology: further comments on identifying important predictor variables
Biodiversity and Conservation 11: 1397 1401, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands. Multiple regression and inference in ecology and conservation biology: further comments on
More informationDETECTING BIOLOGICAL AND ENVIRONMENTAL CHANGES: DESIGN AND ANALYSIS OF MONITORING AND EXPERIMENTS (University of Bologna, 3-14 March 2008)
Dipartimento di Biologia Evoluzionistica Sperimentale Centro Interdipartimentale di Ricerca per le Scienze Ambientali in Ravenna INTERNATIONAL WINTER SCHOOL UNIVERSITY OF BOLOGNA DETECTING BIOLOGICAL AND
More information6. Spatial analysis of multivariate ecological data
Université Laval Analyse multivariable - mars-avril 2008 1 6. Spatial analysis of multivariate ecological data 6.1 Introduction 6.1.1 Conceptual importance Ecological models have long assumed, for simplicity,
More informationAlgebra of Principal Component Analysis
Algebra of Principal Component Analysis 3 Data: Y = 5 Centre each column on its mean: Y c = 7 6 9 y y = 3..6....6.8 3. 3.8.6 Covariance matrix ( variables): S = -----------Y n c ' Y 8..6 c =.6 5.8 Equation
More informationDimension Reduction and Low-dimensional Embedding
Dimension Reduction and Low-dimensional Embedding Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL 60208 http://www.eecs.northwestern.edu/~yingwu 1/26 Dimension
More informationPrincipal Component Analysis
Principal Component Analysis Giorgos Korfiatis Alfa-Informatica University of Groningen Seminar in Statistics and Methodology, 2007 What Is PCA? Dimensionality reduction technique Aim: Extract relevant
More informationMS-E2112 Multivariate Statistical Analysis (5cr) Lecture 8: Canonical Correlation Analysis
MS-E2112 Multivariate Statistical (5cr) Lecture 8: Contents Canonical correlation analysis involves partition of variables into two vectors x and y. The aim is to find linear combinations α T x and β
More informationAppendix A : rational of the spatial Principal Component Analysis
Appendix A : rational of the spatial Principal Component Analysis In this appendix, the following notations are used : X is the n-by-p table of centred allelic frequencies, where rows are observations
More information