Environmental Data Analysis with - PowerPoint Presentation

Slide1

Environmental Data Analysis with MatLab

Lecture 17:

Covariance and AutocorrelationSlide2

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

SYLLABUSSlide3

purpose of the lecture

apply the idea of covariance

to time seriesSlide4

Part 1

correlations between random variablesSlide5

Atlantic Rock DatasetScatter plot of TiO2

and Na

2OTiO2Na2OSlide6

TiO

2

Na2Od1d2scatter plotidealization as a p.d.f.Slide7

d

1

d2positive correlationd1d2d1d2negative correlationuncorrelatedtypes of correlationsSlide8

the covariance matrix

C =

σ12σ22σ1,2σ32…σ

1,3

σ

2,3

σ

1,2

σ

2,3

σ

1,3

…

…

…

…

…

…Slide9

recall the covariance matrix

C =

σ12σ22σ1,2σ32…σ

1,3

σ

2,3

σ

1,2

σ

2,3

σ

1,3

…

…

…

…

…

…

covariance of

d

1

and

d

2Slide10

estimating covariance from dataSlide11

d

i

djDdiDdj

bin, s

divide

(

d

i

,

d

j

)

plane into binsSlide12

d

i

djDdiDdj

bin, s with N

s

data

estimate total probability in bin from the data:

P

s

= p(

d

i

,

d

j

)

Δ

d

i

Δ

d

j

≈ N

s

/NSlide13
Slide14

approximate integral with sum and use p(

d

i, dj) Δdi Δdj≈Ns/NSlide15

approximate integral with sum and use p(

d

i, dj) Δdi Δdj≈Ns/Nshrink bins so no more than one data point in each binSlide16

“sample” covarianceSlide17

normalize to range ±1

“sample” correlation coefficientSlide18

Rij

-1

perfect negative correlation0no correlation1perfect positive correlation Slide19

SiO

2

TiO2Al2O2FeOMgOCaONa2OK2OSiO2TiO2Al2O2FeOMgOCaONa2O

K

2

O

|

R

ij

|

of the Atlantic Rock DatasetSlide20

SiO

2

TiO2Al2O2FeOMgOCaONa2OK2OSiO2TiO2Al2O2FeOMgOCaONa2O

K

2

O

|

R

ij

|

of the Atlantic Rock Dataset

0.73Slide21

Atlantic Rock DatasetScatter plot of TiO2

and Na

2OTiO2Na2OSlide22

Part 2

correlations between samples within a time seriesSlide23

A) time series,

d(t)

time t, daysd(t), cfsNeuse River HydrographSlide24

high degree of short-term correlation

what ever the river was doing yesterday, its probably doing today, too

because water takes time to drain awaySlide25

A) time series,

d(t)

time t, daysd(t), cfsNeuse River HydrographSlide26

low degree of intermediate-term correlation

what ever the river was doing last month, today it could be doing something completely different

because storms are so unpredictableSlide27

A) time series,

d(t)

time t, daysd(t), cfsNeuse River HydrographSlide28

moderate degree of long-term correlation

what ever the river was doing this time last year, its probably doing today, too

because seasons repeatSlide29

A) time series,

d(t)

time t, daysd(t), cfsNeuse River HydrographSlide30

1 day

3 days

30 daysSlide31

Let’s assume different samples in time series are random variables and calculate their covarianceSlide32

usual formula for the covariance

assuming time series has zero meanSlide33

now assume that the time series is stationary

(statistical properties don’t vary with time)

so that covariance depends only ontime lag between samplesSlide34

time series of length N

time lag of

(k-1)ΔtSlide35

1

using the same approximation for the sample covariance as beforeSlide36

1

using the same approximation for the sample covariance as before

autocorrelationat lag (k-1)Δt Slide37

autocorrelation in MatLabSlide38

Autocorrelation on Neuse River HydrographSlide39

Autocorrelation on Neuse River Hydrograph

symmetric about zero

a point in a time series correlates equally well with another in the future and another in the pastSlide40

Autocorrelation on Neuse River Hydrograph

peak at zero lag

a point in time series is perfectly correlated with itselfSlide41

Autocorrelation on Neuse River Hydrograph

falls off rapidly in the first few days

lags of a few days are highly correlated because the river drains the land over the course of a few daysSlide42

Autocorrelation on Neuse River Hydrograph

negative correlation at lag of 182 days

points separated by a half year are negatively correlatedSlide43

Autocorrelation on Neuse River Hydrograph

positive correlation at lag of 360 days

points separated by a year are positively correlatedSlide44

A)

B)

Autocorrelation on Neuse River Hydrographrepeating patternthe pattern of rainfall approximately repeats annuallySlide45

autocorrelation similar to convolutionSlide46

autocorrelation similar to convolution

note difference in signSlide47

Important Relation #1autocorrelation is the convolution of a time series with its time-reversed selfSlide48
Slide49

integral form of autocorrelationSlide50

integral form of autocorrelation

change of variables,

t’=-tSlide51

integral form of autocorrelation

change of variables,

t’=-twrite as convolutionSlide52

Important Relationship #2Fourier Transform of an autocorrelation

is proportional to the

Power Spectral Density of time seriesSlide53

so

sinceSlide54

so

sinceSlide55

so

since

Fourier Transform of a convolution is product of the transformsSlide56

so

since

Fourier Transform of a convolution is product of the transformsFourier Transform integralSlide57

so

since

Fourier Transform of a convolution is product of the transformsFourier Transform integraltransform of variables, t’=-tSlide58

so

since

Fourier Transform of a convolution is product of the transformsFourier Transform integraltransform of variables, t’=-tsymmetry properties of Fourier TransformSlide59

Summary

time

lag

0

frequency

0

rapidly fluctuating

time series

narrow autocorrelation function

wide spectrumSlide60

Summary

time

lag

0

frequency

0

slowly fluctuating

time series

wide autocorrelation function

narrow spectrum