Time Series Spectral Representation

0 2 4 6 8 10 12

060

0014

000

Month

Mea

n flo

w m

gal

1 2 3 4 5 6 7 8 9 10 11 12

Alafia River Monthly Mean Streamflow

Z(t) = {Z1, Z2, Z3, … Zn}

)1S()n/kt2Sinbn/kt2Cosa(Z2/n

0kkkt

Any mathematical function has a representation in terms of sin and cos functions.

Spectral Analysis• Represent a time series in terms of the

wavelengths associated with oscillations, rather than individual data values

• The spectral density function describes the distribution of these wavelengths

• Spectral analysis involves estimating the spectral density function.

• Fourier analysis involves representing a function as a sum of sin and cos terms and is the basis for spectral analysis

Why Spectral Analysis

• Yields insight regarding hidden periodicities and time scales involved

• Provides the capability for more general simulation via sampling from the spectrum

• Supports analytic representation of linear system response through its connection to the convolution integral

• Is widely used in many fields of data analysis

Frequency and Wavenumber

)2S()tSinwbtCoswa(Z2/n

0kkkkkt

n/k2wk n/kfk

Due to orthogonality of Sin and Cos functions

)3S(

2

1n...1ktCoswZ

n

2

evennif2/n,0ktCoswZn

1

an

1tkt

n

1tkt

k

)4S(tSinwZn

2b

n

1tktk

cycles/time radians/time

Frequency Limits

Lowest frequency resolved

Highest frequency resolved, corresponds to n/2 (Nyquist frequency)

General case, time interval ∆t

Nw 2/1fN

n/2w1 n/1f1

t/wN t2/1fN

tn/2w1 tn/1f1

1

k

kw

tn/k2w

AliasingDue to discretization, a sparsely sampled high frequency process may be erroneously attributed to a lower frequency

0 1 2 3 4 5 6

-1.0

-0.5

0.0

0.5

1.0

t

Example. Fourier representation of streamflow

1 2 3 4 5 650

015

0025

0035

00

k

sqrt

(a *

a +

b *

b)

0 2 4 6 8 10 12

040

0010

000

Month

Mea

n flo

w m

gal

1 3 5 7 9 11

Alafia River Monthly Mean Streamflow

Equivalent complex number representation

wsiniwcoseiw

)6S(eZn

1C

n

1t

tiwtk

k

)5S(eCZ)2/n(trunc

21n

trunck

tiwkt

k

Note: Integer truncation is used in the sum limits. For example if n=5 (odd) the limits are -2, 2. If n=6 (even) the limits are -2, 3. (Same number of fourier coefficients as data points)

n/k2wk

Complex conjugate pair

kk wn/k2w

*kk CC

2

ibaC kk

k

2

ibaC kk*

k

Fourier representation of an infinite (non periodic) discrete series

n

1t

tiwtk

keZn

1C

)2/n(trunc

2

1ntrunck

tiwkt

keCZDiscrete transform pair

n/2w1

n/2w

Lowest frequency

Highest frequency

Spacing

As

Nw

n 0w

w- on continuous wwkwk

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.8

wk

Ck

w

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.8

wk

Ck

0.0 0.5 1.0 1.5 2.0 2.5 3.00.

20.

8

wk

Ck

Fourier representation of an infinite (non periodic) discrete series

n

1t

tiwtk

keZn

1C

)2/n(trunc

2

1ntrunck

tiwkt

keCZ

n/k2wk n/2w

Discrete transform pair

2/)1n(

2/)1n(t

tiwtk

keZn

1C

)2/n(trunc

2

1ntrunck

tiwkt

keCZRe-center on [(-n-1)/2, (n-1)/2]

2

w

n

1

n

2/)1n(

2/)1n(t

tiwt

k keZ2

1

w

Cwe

w

CZ

)2/n(trunc

2

1ntrunck

tiwkt

k

w

C)w(C k

for 0w

t

iwtteZ

2

1)w(C

kw

dwe)w(CZ iwtt

Fourier transforms for discrete function on infinite domain

(aperiodic)

An infinite series of discrete data points at spacing ∆t obtained from the finite discrete case taking the limit T,n→ . The highest resolvable frequency (Nyquist) is t/w N

dwe)w(CZN

N

w

w

iwtt

n

iwtteZ

2

t)w(C

Wei section 10.5

Fourier transforms for continuous function on finite domain (periodic)

Z(t) defined on the finite domain (-T/2,T/2), taken as periodic with period length T. The lowest resolved frequency is T/2wo

k

tikwk

oeC)t(Z dte)t(ZT

1c

2/T

2/T

tikwk

o

Wei section 10.6.1

Fourier transforms for continuous function on infinite domain

(aperiodic)

Z(t) defined on the entire interval (-,), obtained from the finite domain case by letting T→ . All frequencies are resolved.

w

iwtdwe)w(C)t(Z

t

iwtdte)t(Z2

1)w(C

The placement of 2π in this definition varies amongst references

Wei section 10.6.2

Frequency domain representation of a random process

• Z(t) and C(w) are alternative equivalent representations of the data

• If Z(t) is a random process C(w) is also random

• ACF(Z(t))≡Spectral density function(C(w))

1940 1950 1960 1970 1980 1990 2000

020

000

6000

0

Year

mga

l

Alafia River Streamflow

Spectral decomposition of any function

n

1t

tiwtk

keZn

1c

0 100 200 300 400

050

010

00

k

ck

1.04 2.087 3.134

0 10 20 30 40 50 60

-10

12

1:n1

z[1:

n1]

0 5 10 15 20

0.0

0.4

0.8

Lag

AC

F

Series z

0.0 0.1 0.2 0.3 0.4 0.5

02

46

810

Frequency

Pow

er

0.0 0.1 0.2 0.3 0.4 0.5

0.5

1.0

2.0

frequency

spec

trum

Series: zSmoothed Periodogram

bandwidth = 0.0197

Spectral representation of a stationary random process

Time Series

Smoothing

Fourier Transform

Autocorrelation

Fourier Transform

Fourier coefficients Spectral density function

Autocorrelation function

Time Domain Frequency Domain

)t(Z )w(C

Z1, Z2, Z3, … C1, C2, C3, …

0 5 10 15 20

0.0

0.4

0.8

Lag

AC

F

Series z

0.0 0.1 0.2 0.3 0.4 0.5

01

23

45

67

Frequency

Pow

er

0.0 0.1 0.2 0.3 0.4 0.5

01

23

45

67

Frequency

Pow

er

The Periodogram|Ck|2 is random because Zt is random

Power at low frequency Persistence indicated by ACF

2k

2t |C|Z

n

1

Decomposition of variance

|Ck|2

k/n

Wei section 12.1

0 10 20 30 40 50 60

-1.5

-0.5

0.5

1.5

1:n1

z[1:

n1]

0 5 10 15 20

0.0

0.4

0.8

Lag

AC

F

Series z

0 5 10 15 20

0.0

0.4

0.8

Lag

AC

F

Series z

0 10 20 30 40 50 60

-4-3

-2-1

01

2

1:n1

z[1:

n1]

r=0.9

r=0.2

Time Domain

0 10 20 30 40 50 60

-1.5

-0.5

0.5

1.5

1:n1

z[1:

n1]

0 10 20 30 40 50 60

-4-3

-2-1

01

2

1:n1

z[1:

n1]

r=0.9

r=0.2

Frequency Domain

0.0 0.1 0.2 0.3 0.4 0.5

01

23

4

Frequency

Pow

er

0.0 0.1 0.2 0.3 0.4 0.5

02

46

810

Frequency

Pow

er

The Spectral Density Function

The spectral density function is defined as

0.0 0.1 0.2 0.3 0.4 0.5

02

46

8

Frequency

Pow

er

dw)w(Se)Z,Z(Cov iwk

ktt

S(w)dw=E(|C(w)|2)

dw)w(S)Z(Var 2t

Wei section 12.2, 12.3

Problem Estimating S(w)

• Z(t) Stationary C(w) Independent

• |C(w)|2 has 2 degrees of freedom (from real and imaginary parts

• More data ∆w gets smaller, but still 2 degrees of freedom

0.0 0.1 0.2 0.3 0.4 0.5

02

46

8

Frequency

Pow

er

0.1 0.2 0.3 0.4 0.5

02

46

8

Frequency

Pow

er

n=50 n=200

S(w) estimated by smoothing the periodogram

• Balance Spectral resolution versus precision• Tapering to minimize leakage to adjacent

frequencies • Confidence bounds by 2 based on number of

degrees of freedom involved with smoothing• Multitaper methods• A lot of lore

0.0 0.1 0.2 0.3 0.4 0.5

05

1015

Frequency

Pow

er

Different degrees of smoothing

Spectral analysis gives us • Decomposition of process into dominant frequencies• Diagnosis and detection of periodicities and

repeatable patterns• Capability to, through sampling from the spectrum,

simulate a process with any S(w) and hence any Cov()

• By comparison of input and output spectra infer aspects of the process based on which frequencies are attenuated and which propagate through