Time Series Analysis

Session II

Outline• spectral analysis• FFT• complex numbers• periodogram• power spectrum• windowing• coherence

Spectral Analysis• psd(signal,512,sf,[],256,’mean’)• what are the parameter?

0 5 10 15 20-140

-120

-100

-80

-60

-40

-20

0

20

Frequency

Pow

er S

pect

rum

Mag

nitu

de (d

B)

Fourier transform – the idea• fitting a function with sinusoids• transformation of time series to frequency domain

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

2 functions1 functions

Fourier transform – the idea• fitting a function with sinusoids• transformation of time series to frequency domain

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

10 functions3 functions

FFT- general remarks• Fast Fourier Transform• optimised algorithm• 2^N number of samples (128,256,512,1024)

FFT in matlab• Example: sunspot data• Y = fft(sunspot);• Y(1)=[]; %just the sum

1700 1750 1800 1850 1900 1950 20000

20

40

60

80

100

120

140

160

180

200Sunspot Data

0 50 100 150 200 250 300-4000

-2000

0

2000

4000

6000

8000

10000

12000

14000FFT

FFT in matlab• symmetric• remove negative frequencies• n=length(Y);• Y=Y(1:floor(n/2));• plot(abs(Y)); %periodogram

0 50 100 150 200 250 300-4000

-3000

-2000

-1000

0

1000

2000

3000

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

7

Pow

er

Period (Years/Cycle)

complex numbers

Real

Imaginary

0 10 20 30 40 50 60 70-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

amplitude: length of the arrowphase:

φ

φ

Spectral analysis with FFT• Exercise:

• type “edit fftdemo” and do the steps in the script

Problems with periodogram• Leakage• Accuracy

Leakage

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

0 50 100 150 200 250 300-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

TSFFT

Leakage• occurs for non-periodic signals• non-periodic signals violate FFT assumptions• problem for real-world signals• solution: windowing

Windowing• purpose: make signal periodic (zero at beginning and

end)

0 50 100 150 200 250 300-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

windowed signal

Windowed sinusoid

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

FFT of windowed signal

Windowing• windowing does not eliminate leakage but reduces it• actually: it changes the shape of leakage• wintool: to display and analyze windows• standard window is Hanning window• hw=hanning(256);

Accuracy of Periodogram• Question: how does variance of power estimate

decrease with increasing number of samples?• Answer: No at all!• additional information is used to compute power at finer

frequency resolution

Welchs Method

• Welchs Method: – divide time series in segments of equal length (typically 2^N

samples)– apply window– compute fft– average power spectra

Overlap• we loose information!• solution: overlap• typically: half window length

0 50 100 150 200 250 300-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

windowed signal

psd function• psd(signal,512,sf,[],256,’mean’)

• [p,f]=psd(signal,512,sf,[],256,’mean’);• p: power spectrum• f: frequency vector

segment

hanning window

overlap

frequency resolution• depends on sampling frequency and length of FFT

segment• sf=256; segment: 256 => 1 Hz resolution• sf=256; segment: 512 => 0.5 Hz resolution

Multitaper• reducing the variance further

• sinusoid (140, 150 Hz) +noise

0 50 100 150 200 250 300 350 400 450 500-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Thompson Multitaper Power Spectral Density Estimate

0 50 100 150 200 250 300 350 400 450 500-60

-50

-40

-30

-20

-10

0

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Periodogram Power Spectral Density Estimate

0 50 100 150 200 250 300 350 400 450 500-25

-20

-15

-10

-5

0

5

10

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Welch Power Spectral Density Estimate

Exercise• compute power spectrum of EMG with fft only• compute power spectrum with psd function, determine

tremor frequency with given accuracy• change length of FFT segment and type of window and

observe the effect on power spectrum

Coherence – general• Correlation in the frequency domain• normalized between 0-1 (1: complete dependence)• preferred phase difference

Coherence

?

Coherencecohere(signal1,signal2,512,sf,[],256,’mean’)

Confidence interval• analytic (Halliday et al, 1995)• numeric (use random permutation of time series and

compute coherence a large number of times)

Simulation• x=rand(1,10000);• [b,a]=butter(4,2*[10 15]/1000);• xf=filtfilt(b,a,x);• y=xf+0.5*rand(1,10000);• z=xf+0.5*rand(1,10000);• cohere(y,z,1024,1000,[],…

512,'mean')

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency

Coh

eren

ce F

unct

ion

Est

imat

e

Exercise• compute power spectra of EMG and MEG using psd

look for common frequencies• compute coherence between EMG and MEG with

different frequency resolution