Time-series Analysis of Environmental Systems

Zhulu LinDept of Crop and Soil Sciences

University of GeorgiaAthens, Georgia

October 12, 2005

Outline

Objectives of time-series analysis

Time-series preparation

Time-series analysis methods for

(nonlinear) environmental systems

Objectives of time-series analysis

Description of a system’s behavior

Explanation of the mechanisms underlying these behaviors

Prediction of the system’s future status (output) under given perturbation (input)

Control of the controllable variables (input) to meet (avoid) the desired (feared) state of a system

Description ▬ trend and seasonal variation

0 2 4 6 8 10 12 14 16 18 20450

500

550

600

650

700

750

800

850

900

950

Date

Flow

rate

(m3 /h

)

Influent to Athens WWTP with trend

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Date

PO

4 (mg/

L)

Outliers in orthophosphate observations

Description ▬ Data gaps and outliers

?

Description ▬ outliers ?

0 5 10 15 20 25 30 35 400

1

2

3

4

5

PO

4 (mg/

L)

PO4 observations w/ and w/o outliers removed by eyes

0 5 10 15 20 25 30 35 400

0.5

1

1.5

Date

PO

4 (mg/

L)

Explanation (or system identification)

Environmental SystemOutputInput

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

Date

Chl

orop

hyll-

a ( µ

g/L)

Chlorophyll-a observations from July 1st to August 5th, 2000

(Chlorophyll-a)

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5Orthophosphate-P observations from July 1st to August 5th, 2000

Date

PO

4 (mg/

L)

(Nutrients)

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

200

Date

PAR (K

J/m

2 /h)

Photosynthetically Active Radiation from July 1st to August 5th, 2000

(Solar radiation)

Prediction(or forecasting)

Environmental System(Whitehall Pond)

OutputInput

(Chlorophyll-a)(Nutrients)

Model

0 5 10 15 20 25 30 35-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Date

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5Orthophosphate-P observations from July 1st to August 5th, 2000

Date

PO

4 (mg/

L)

Control

Environmental System(Whitehall Pond)

Input Output

(Nutrients) (Chlorophyll-a)

Model OutputInput

(Chlorophyll-a)(Nutrients)

Outline

Objectives of time-series analysis

Time-series preparation

Time-series analysis methods for

(nonlinear) environmental systems

Time-series preparation

Data plotting

Interpolation

Smoothing

Signal extraction

Plotting Data

…Anyone who tries to analyze a time series, without

plotting it first, is asking for trouble. Not only will a

graph show up trend and seasonal variation, but it also

enables one to look for ‘wild’ observations or outliers

which do not appear to be consistent with the rest of the

data. …

⎯ C. Chatfield (1984)

05/23 07/11 08/30 10/190

500

1000

1500P

O4-P

( µg ⋅

L-1)

05/23 07/11 08/30 10/190

100

200

300

Chl

orop

hyll a

( µg ⋅

L-1

)

05/23 07/11 08/30 10/19

5

10

15

Date (May 23 ~ October 16, 2000)

DO

(mg ⋅

L-1

)(a)

(b)

(c)

Orthophosphate

Chlorophyll a

Dissolved oxygen

0 5 10 15 20 25 30 35 402

4

6

8

10

12

14

16

Date

Dis

solv

ed O

xyge

n (m

g/L)

DO observations from July 1st to August 5th, 2000

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

Date

Chl

orop

hyll-

a ( µ

g/L)

Chlorophyll-a observations from July 1st to August 5th, 2000

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5Orthophosphate-P observations from July 1st to August 5th, 2000

Date

PO

4 (mg/

L)

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

200

Date

PAR (K

J/m

2 /h)

Photosynthetically Active Radiation from July 1st to August 5th, 2000

0 5 10 15 20 25 30 35 4025

26

27

28

29

30

31

32

33

34

35

36

Date

Tem

pera

ture

(o C)

Temperature observations from July 1st to August 5th, 2000

Interpolation

0 2 4 6 8 10 12 14 16 18 20450

500

550

600

650

700

750

800

850

900

950

Date

Flow

rate

(m3 /h

)

Interpolation of influent time series

Interpolation and Smoothing

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5Interpolat ion and s m ooth ing of P O 4 t im e s eries

PO

4 (mg/

L)

Signal Extraction

0

5

10

15Trend

DO

(mg/

L)

-5

0

5Diurnal harmonics

0 5 10 15 20 25 30 35 40-0.5

0

0.5Semi-diurnal Harmonics

Date

DO

(mg/

L)D

O (m

g/L)

Signal Extraction

0

100

200

300Trend

-50

0

50Diurnal Harmonics

0 5 10 15 20 25 30 35 40-5

0

5Semi-diurnal Harmonics

Chl

orop

hyll-

a ( µ

g/L)

Signal Extraction

-3

-2

-1

0

1

2

3

8/15 8/16 8/17 8/18 8/19 8/20

Chla

,DO

, Tem

p

-12

-7

-2

3

8

Chla (µg/L) DO (mg/L) Temp (°C) PAR (µE/s/m2)

Signal Extraction

-6

-4

-2

0

2

4

6

8

10

8/15 8/16 8/17 8/18 8/19 8/20

-4

-3

-2

-1

0

1

2

3

4

PO4-P (µg/L) TOC/100 (µg/L)

TIN (µg/L) Chla (µg/L)

Outline

Objectives of time-series analysis

Time-series preparation

Time-series analysis methods for

environmental systems

Methods for stationary process

Time domain Frequency domain

Univariateprocess

Autocorrelation function (ACF) and Partial ACF (PACF)

Spectral analysis

Linear process Impulse response function, step

response functions or ODE’s

Frequency response function

or Transfer function

Bivariateprocess

Cross-correlation Cross-spectrum

Methods for stationary process

Not commonly used in environmental system analysis, but useful for residual assessment in environmental modeling;

An example.

ITSM

Observations = Simulation + AR(5) + noises

Methods for non-stationary process

Classical approach

Pre-processing + methods for stationary process

Where, pre-processing includes: classical decomposition, or transformation, differencing, etc

Modern approach

Recursive estimation (forward filtering and backward smoothing)

Comparing the classical and modern approaches

A simple example

Classical Modern

Comparing the classical and modern approaches

A not-so complicated example

DO time series from Whitehall pond

Classical Modern

Comparing the classical and modern approaches

A more complicated example

PO4 time series from Whitehall pond

Classical Modern

An introduction to the modern methods

Signal processing method (univariateprocess)

System identification method (multivariate process)

Dynamic Harmonic Regression (DHR) models for signal processing

DHR Model:

Examples:Interpolation of broken DO time seriesInterpolation and smoothing of PO4 time series

( ) ( ) ( ) ( ); 1,2,...,y k T k S k e k k N= + + =

{ }1 1

( ) ( ) ( ) cos( ) ( )sin( )R R

i i i i ii i

S k S k a k t b k tω ω= =

= = +∑ ∑

MATLAB

Data-based mechanistic (DBM) modeling methodology for system

identification

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

200

Date

PAR (K

J/m

2 /h)

Photosynthetically Active Radiation from July 1st to August 5th, 2000

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

Date

Chl

orop

hyll-

a ( µ

g/L)

Chlorophyll-a observations from July 1st to August 5th, 2000

Environmental System(Whitehall Pond)

OutputInput

(Solar radiation) (Chlorophyll-a)

Output = [Transfer Function] × Input

First-order transfer function (TF)

)(1

)( 11

0 δ−+

= − kuza

bky z-1 operator form

1

0

1 abG+

=)ln(

1

1aT

−−=

Time constant Steady state gain

General transfer function (TF) model

1

1

1 11

1 10 1

( , )( ) ( ) ( )( , )

( , ) 1 ( ) ( )

( , ) ( ) ( ) ( )

nn

mm

B k zy k u k kA k z

A k z a k z a k z

B k z b k b k z b k z

δ ν−

−

− − −

− − −

= − +

= + + ⋅⋅ ⋅ +

= + + ⋅ ⋅ ⋅ +

To determine:1. values of n, m, δ;2. estimates of ai(k), bi(k), i = 0, 1, 2, …, (n, m);3. nonlinearity exhibited by ai(k), bi(k).

First-order TF model with TVP’s

01

1

( )( ) ( )1

b ty t u ta z

δ−= −−

Estimation of TVP

0 50 100 150 200 250 300 350 400 450-5

0

5

10

15

20

25

Samples

TVP

, sca

led

y(k)

, mod

el

y(k)TVPmodel

)3(9523.01

)|(ˆ)( 0 −

−= ku

Nkbky

State Dependent Parameter Modeling

0 50 100 150 200 250 300-5

0

5

10

15

20

25

y(k)

TVP

)()|(ˆ0 kyNkb α=

Model output and observations

0 50 100 150 200 250 300 350 400 4500

50

100

150

200

250

300

Samples

chla

(ug/

L)