Environmental Data Analysis with - PowerPoint Presentation

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Environmental Data Analysis with MatLab

Lecture 16:

Orthogonal Functions

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Lecture 01

Using

MatLabLecture 02 Looking At DataLecture 03 Probability and Measurement Error Lecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares ProblemsLecture 09 Fourier SeriesLecture 10 Complex Fourier SeriesLecture 11 Lessons Learned from the Fourier Transform Lecture 12 Power Spectral DensityLecture 13 Filter Theory Lecture 14 Applications of Filters Lecture 15 Factor Analysis Lecture 16 Orthogonal functions Lecture 17 Covariance and AutocorrelationLecture 18 Cross-correlationLecture 19 Smoothing, Correlation and SpectraLecture 20 Coherence; Tapering and Spectral Analysis Lecture 21 InterpolationLecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-TestsLecture 24 Confidence Limits of Spectra, Bootstraps

SYLLABUS

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purpose of the lecture

further develop Factor Analysis

and introduceEmpirical Orthogonal Functions

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review of the last lecture

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example

Atlantic Rock Dataset

chemical composition for several thousand rocks

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Rocks are a mix of minerals, and …

mineral

1

mineral

2

mineral

3

rock 1

rock 2

rock 3

rock 4

rock 5

rock 6

rock 7

…minerals have a well-defined

composition

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rocks contain elements

rocks contain minerals

andminerals contain elementssimpler

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rocks contain elements

rocks contain minerals

andminerals contain elementssimplersamplessamplesfactorsfactors

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the sample matrix,

S

N samples by M elementse.g.sediment samplesrock samplesword element is used in the abstract sense and may not refer to actual chemical elements

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the factor matrix, F

P

factors by M elementse.g.sediment sourcesmineralsnote that there are P factorsa simplification if P<M

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the loading matrix, C

N

samples by P factorsspecifies the mix of factors for each sample

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the key question

how many factors are

needed to represent the samples?and what are these factors?

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singular value decomposition

the methodology for

answer these questions

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the matrix of

P mutually-perpendicular vectors, each of length M

diagonal matrix Σ,of P singular valuesthe matrix of P mutually-perpendicular vectors, each of length N

sample matrix

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the matrix of loadings,

C

. the matrix of factors, Fsince C depends on Σ,the samples contains more of the factors with large singular values than of the factors with the small singular values

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in MatLab

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singular values,

S

iiindex, iplot of M singular values, sorted by size

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singular values,

S

iiindex, idiscard, since close to zerouse it to discard near-zero singular values

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singular values,

S

iiindex, iand to determine the number P of factorsP=5

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graphical representation of factors 2 through 5

f

5f2f3f4SiO2TiO2Al2O3FeOtotalMgOCaONa2OK2OAtlantic Rock Dataset

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C

2

C3C4factor loadings C2 through C4 plotted in 3Dfactors 2 through 4 capture most of the variability of the rocks

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end of review

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Part 1: Creating Spiky Factors

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can we find “better” factors

that those returned by

svd()?

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mathematically

S

= CF = C’ F’with F’ = M F and C’ = M-1 Cwhere M is any P×P matrix with an inversemust rely on prior information to choose M

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one possible type of prior information

factors should contain mainly just a few elements

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example of minerals

Mineral

CompositionQuartzSiO2RutileTiO2AnorthiteCaAl2Si2O8FosteriteMg2SiO4

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spiky factors

factors containing mostly just a few elements

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How to quantify spikiness?

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variance as a measure of spikiness

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modification for factor analysis

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modification for factor analysis

depends on the square, so positive and negative values are treated the same

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f(1)= [1, 0, 1, 0, 1, 0]

T

is much spikier than f(2)= [1, 1, 1, 1, 1, 1]T

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f(2)=[1, 1, 1, 1, 1, 1]

T

is just as spiky as f(3)= [1, -1, 1, -1, -1, 1]T

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“varimax” procedure

find spiky factors without changing P

start with P svd() factorsrotate pairs of them in their plane by angle θto maximize the overall spikiness

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f

B

fAf’Bf’Aq

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determine θ by maximizing

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after tedious trig the solution can be shown to be

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and the new factors are

in this example A=3 and B=5

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now one repeats for every pair of factors

and then iterates the whole process several times

until the whole set of factors is as spiky as possible

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A)

B

)f5f2f3f4f’5f’2f’3f’4SiO2TiO2Al2O3FeOtotalMgO

CaO

Na

2

O

K

2

O

example: Atlantic Rock dataset

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Part 2: Empirical Orthogonal Functions

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row number in the sample matrix could be meaningful

example: samples collected at a succession of times

time

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column number in the sample matrix could be meaningful

example: concentration of the same chemical element at a sequence of positions

distance

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S = CF

becomes

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S = CF

becomes

distance dependencetime dependence

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S = CF

becomes

each loading: a temporal pattern of variability of the corresponding factoreach factor:a spatial pattern of variability of the element

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S = CF

becomes

there are P patterns and they are sorted into order of importance

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S = CF

becomes

factors now called EOF’s (empirical orthogonal functions)

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example

sea surface temperature in the Pacific Ocean

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29

S

29N124E290Elatitudelongitudeequatorial Pacific Oceansea surface temperature (black = warm)CAC Sea Surface Temperature

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the image is 30 by 84 pixels in size, or 2520 pixels total

to use

svd(), the image must be unwrapped into a vector of length 2520

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2520 positions in the equatorial Pacific ocean

399 times

“element” means temperature

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s

ingular values,

Siiindex, isingular values

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s

ingular values,

Siiindex, isingular valuesno clear cutoff for P, but the first 10 singular values are considerably larger than the rest

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using SVD to approximate data

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S=C

M

FMS=CPFPS≈CP’FP’ With M EOF’s, the data is fit exactlyWith P chosen to exclude only zero singular values, the data is fit exactlyWith P’<P, small non-zero singular values are excluded too, and the data is fit only approximately

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A) Original

B) Based on first 5 EOF’s