Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs) Arnaud Czaja (SPAT Data analysis lecture Nov. 2011)

Slides:

Advertisements

Similar presentations

Eigen Decomposition and Singular Value Decomposition

Advertisements

1 To centre or not to centre …or perhaps do it twice Ian Jolliffe Universities of Reading, Southampton, Aberdeen

Krishna Rajan Data Dimensionality Reduction: Introduction to Principal Component Analysis Case Study: Multivariate Analysis of Chemistry-Property data.

Principal Component Analysis for SPAT PG course Joanna D. Haigh.

INITIAL CONDITION: 31 May 2014 Extended range prediction during 2014 season Extended range prediction during 2014 season Using CFS based Grand Ensemble.

Eigen Decomposition and Singular Value Decomposition

Reanalysis of the Iceland-Faroe inflow Bogi Hansen, Karin Margretha H. Larsen, Hjálmar Hátún and Svein Østerhus NACLIM CT2, Hamburg, January 2015.

Covariance Matrix Applications

Tensors and Component Analysis Musawir Ali. Tensor: Generalization of an n-dimensional array Vector: order-1 tensor Matrix: order-2 tensor Order-3 tensor.

Principal Component Analysis (PCA) for Clustering Gene Expression Data K. Y. Yeung and W. L. Ruzzo.

1er. Escuela Red ProTIC - Tandil, de Abril, 2006 Principal component analysis (PCA) is a technique that is useful for the compression and classification.

Principal Component Analysis CMPUT 466/551 Nilanjan Ray.

Principal Component Analysis

An introduction to Principal Component Analysis (PCA)

Principal Component Analysis

Slide 1 EE3J2 Data Mining EE3J2 Data Mining Lecture 9(b) Principal Components Analysis Martin Russell.

1 Interannual Variability in Stratospheric Ozone Xun Jiang Advisor: Yuk L. Yung Department of Environmental Science and Engineering California Institute.

EOF ANALYSIS Means of examining variations in beach profiles in a compact fashion Describes data variability in terms of orthogonal functions or statistical.

Principal Components Analysis (PCA) 273A Intro Machine Learning.

What is EOF analysis? EOF = Empirical Orthogonal Function Method of finding structures (or patterns) that explain maximum variance in (e.g.) 2D (space-time)

Principal Component Analysis Principles and Application.

Principal Component Analysis. Consider a collection of points.

Techniques for studying correlation and covariance structure

Assessment and Quantification of HF Radar Uncertainty Fearghal O’Donncha Sean McKenna Emanuele Ragnoli Teresa UpdykeHugh Roarty.

Principal Component Analysis. Philosophy of PCA Introduced by Pearson (1901) and Hotelling (1933) to describe the variation in a set of multivariate data.

Principal Component Analysis (PCA) for Clustering Gene Expression Data K. Y. Yeung and W. L. Ruzzo.

Principle Component Analysis Presented by: Sabbir Ahmed Roll: FH-227.

Principal Components Analysis (PCA). a technique for finding patterns in data of high dimension.

Presented By Wanchen Lu 2/25/2013

Feature extraction 1.Introduction 2.T-test 3.Signal Noise Ratio (SNR) 4.Linear Correlation Coefficient (LCC) 5.Principle component analysis (PCA) 6.Linear.

Eigen Decomposition Based on the slides by Mani Thomas Modified and extended by Longin Jan Latecki.

What is it? Principal Component Analysis (PCA) is a standard tool in multivariate analysis for examining multidimensional data To reveal patterns between.

Canonical Correlation Analysis and Related Techniques Simon Mason International Research Institute for Climate Prediction The Earth Institute of Columbia.

Techniques for studying correlation and covariance structure Principal Components Analysis (PCA) Factor Analysis.

Math 5364/66 Notes Principal Components and Factor Analysis in SAS Jesse Crawford Department of Mathematics Tarleton State University.

Chapter 7 Multivariate techniques with text Parallel embedded system design lab 이청용.

Central limit theorem revisited

AIRS Radiance and Geophysical Products: Methodology and Validation Mitch Goldberg, Larry McMillin NOAA/NESDIS Walter Wolf, Lihang Zhou, Yanni Qu and M.

Carlos H. R. Lima - Depto. of Civil and Environmental Engineering, University of Brasilia. Brazil. Upmanu Lall - Water Center, Columbia.

EE4-62 MLCV Lecture Face Recognition – Subspace/Manifold Learning Tae-Kyun Kim 1 EE4-62 MLCV.

Christina Bonfanti University of Miami- RSMAS MPO 524.

Empirical Orthogonal Functions (EOFs) successive eigenvalues should be distinct – if not, the eigenvalues and associated patterns are noise – 1 from 2,

Principal Component Analysis Zelin Jia Shengbin Lin 10/20/2015.

Principal Component Analysis (PCA).

Principal Component Analysis and Linear Discriminant Analysis for Feature Reduction Jieping Ye Department of Computer Science and Engineering Arizona State.

3 “Products” of Principle Component Analysis

Central limit theorem revisited Throw a dice twelve times- the distribution of values is not Gaussian Dice Value Number Of Occurrences.

Principal Components Analysis ( PCA)

Central limit theorem - go to web applet. Correlation maps vs. regression maps PNA is a time series of fluctuations in 500 mb heights PNA = 0.25 *

EMPIRICAL ORTHOGONAL FUNCTIONS 2 different modes SabrinaKrista Gisselle Lauren.

PRINCIPAL COMPONENT ANALYSIS(PCA) EOFs and Principle Components; Selection Rules LECTURE 8 Supplementary Readings: Wilks, chapters 9.

Principal Component Analysis (PCA)

Principal Component Analysis

Dimensionality Reduction

Eigen Decomposition Based on the slides by Mani Thomas and book by Gilbert Strang. Modified and extended by Longin Jan Latecki.

Principal Component Analysis

Singular Value Decomposition North Atlantic SST

Principal Component Analysis (PCA)

Northern Hemispheric Flood Fingerprints

Introduction PCA (Principal Component Analysis) Characteristics:

Recitation: SVD and dimensionality reduction

SVD, PCA, AND THE NFL By: Andrew Zachary.

ALL the following plots are subject to the filtering :

Eigen Decomposition Based on the slides by Mani Thomas

Principal Component Analysis

Canonical Correlation Analysis and Related Techniques

Eigen Decomposition Based on the slides by Mani Thomas

Outline Variance Matrix of Stochastic Variables and Orthogonal Transforms Principle Component Analysis Generalized Eigenvalue Decomposition.

Presentation transcript:

Principal Component Analysis (PCA) or Empirical Orthogonal Functions (EOFs) Arnaud Czaja (SPAT Data analysis lecture Nov. 2011)

Outline Motivation Mathematical formulation (on the board) Illustration: analysis of ~100yr of sea surface temperature fluctuations in the North Atlantic How to compute EOFs Some issues regarding EOF analysis

Motivation Data compression...to “carry less luggage” Original pictures 6 EOFs 12 EOFs 24 EOFs

Motivation Data compression... to simplify with the hope of better understanding and forecasting Selten (1995) Mean Z300 (CI=100m) r.m.s Z300 (CI=10m) 20-EOF modelQG model (231 var.)

Motivation Identify “modes” empirically from data “Annular modes” in pressure data Thompson and Wallace (2000)

Some examples of calculations

Pictures Mean “picture” EOF1 EOF2EOF3

North Atlantic sea surface temperature variability (Deser and Blackmon 1993) PC2PC1 EOF2 12% EOF1 45%

How to compute EOFs Compute the covariance matrix Σ of the observation matrix X Compute its eigenvalues (variance explained) and eigenvectors (=eof) The principal component is then obtained by “projection”: pc(t) = X * eof Another (more efficient) method: singular value decomposition of X (come and see me if you are interested)

Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues) See North et al. (1982)

Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues)

Main issues with EOF analysis Sensitivity to size of dataset (“sampling” issues)

Main issues with EOF analysis Orthogonality constraint is not physical. Methods have been developed to deal with this (“rotated EOFs”) The link between EOFs and physical modes of a system is not clear

Main issues with EOF analysis Orthogonality constraint is not physical. Methods have been developed to deal with this (“rotated EOFs”) The link between EOFs and physical modes of a system is not clear Good luck if you try EOFs... Do not hesitate to come and see me!