time series analysis matlab tutorial - university of...
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Time series analysis Matlab tutorial
Joachim Gross
Outline• Terminology• Sampling theorem• Plotting• Baseline correction• Detrending• Smoothing• Filtering• Decimation
Remarks• Focus on practical aspects, exercises, getting
experience (not on equations, theory)• Focus on “How to do …”• Learn some basic skills for TS analysis
• Note: Usually there is not a single perfectly correct way of doing a TS operation! => learn the limitations!
What is a time series?A sequence of measurements over time
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Terminology– Continuous TS: continuous observations– Discrete TS: observations at specific times usually equally spaced
– Deterministic TS: future values can be exactly predicted from past values
– Stochastic TS: exact prediction not possible
Objectives of TS analysis• Description• Explanation• Prediction• Control
Simple descriptive analysisSummary statistics (mean, std) is not always meaningful for
TS
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Sampling• Converting a continuous signal into a discrete time series• Reconstruction is possible if sampling frequency is greater than twice
the signal bandwidth
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Time (s)0 0.2 0.4 0.6 0.8 1
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75 Hz sampling
Sampling• Nyquist frequency: half of sampling frequency
10 Hz sampling 10 Hz reconstruction
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Sampling
8 Hz sampling
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• Aliasing: Frequencies above Nyquist frequency are reconstructed below Nyquist frequency
Sampling
8 Hz sampling
• Aliasing: Frequencies above Nyquist frequency are reconstructed below Nyquist frequency
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Frequency
Pow
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Simple operations on TS• Plotting• Removing a baseline• Removing a trend• Smoothing• Filtering• Decimation
Plotting in Matlab• For visual inspection of TS• For publications/talks
• plot• sptool
Data preprocessing I• Removing offset• ts=ts-mean(ts);
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Data preprocessing I• Removing a baseline• basel=find(t<=0);• ts=ts-mean(ts(basel));
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Data preprocessing II• Removing a trend• ts=detrend(ts);• subtracts best fitting line• detrend can be used to subtract mean: detrend(ts,’constant’)
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data 1 lineardata 2
Data preprocessing III• Smoothing• ts=filter(ones(1,30)/30,1,ts); %mean filter, moving average• uses zeros at beginning! • => baseline correction or do not use first 30 samples
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Data preprocessing III• introduces a shift! => either correct for it or • ts=filtfilt(ones(1,15)/15,1,ts); %mean filter, forward and reverse• no shift!• filter can take any smoothing kernel (gaussian, etc)
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7shifted by 15 samples filtfilt
Data preprocessing III• Smoothing• ts=medfilt1(ts,30); %median filter, takes into account the shift• uses 0 at beginning and end !
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Data preprocessing III• Smoothing• ts=sgolayfilt(ts,3,41); %Savitzky-Golay filter• fits 3rd order polynomial to frames of size 41• good at preserving high frequencies in the data
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Data preprocessing III• Smoothing• compare unsmoothed and smoothed data• check for shift• check beginning (and end) of the smoothed time series
Exercise 1
Data preprocessing IV• Filtering• FIR-Filter (finite impulse
response)• stable• high filter order• usually have linear phase(phase change is proportional to
frequency)
• IIR-Filter (infinite impulse response)
• potentially unstable• low filter order• non-linear phase distortion• computationally efficient
Data preprocessing IV• IIR-Filter:
– Butterworth– Elliptic– Chebychev Typ 1– Chebychev Typ 2– Bessel
• FIR-Filter:– fir1
Data preprocessing IV• lowpass• highpass• bandpass• bandstop
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Magnitude Response (dB)
dB is logarithmic unit0dB = factor of 13dB = factor of 210dB= factor of 10
5 Hz lowpass
Data preprocessing IV• lowpass• highpass• bandpass• bandstop
dB is logarithmic unit0dB = factor of 13dB = factor of 210dB= factor of 10
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30 Hz highpass
Data preprocessing IV• lowpass• highpass• bandpass• bandstop
dB is logarithmic unit0dB = factor of 13dB = factor of 210dB= factor of 10
2-30 Hz bandpass
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Data preprocessing IV• lowpass• highpass• bandpass• bandstop
dB is logarithmic unit0dB = factor of 13dB = factor of 210dB= factor of 10
30-40 Hz bandstop
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Magnitude Response (dB)
Simple design: FIR• [b]=fir1(4,2*4/sf); %4 Hz lowpass• [b]=fir1(4,2*4/sf,’high’); %4 Hz highpass• [b]=fir1(4,2*[4 10]/sf); %4-10 Hz bandpass• [b]=fir1(4,2*[4 10]/sf,’stop’); %4-10 Hz bandstop
• tsf=filter(b,1,ts);• tsf=filtfilt(b,1,ts); %forward and reverse
Simple design: IIR• [b,a]=butter(4,2*4/sf); %4 Hz lowpass• [b,a]=butter(4,2*4/sf,’high’); %4 Hz highpass• [b,a]=butter(4,2*[4 10]/sf); %4-10 Hz bandpass• [b,a]=butter(4,2*[4 10]/sf,’stop’); %4-10 Hz bandstop
• tsf=filter(b,a,ts);• tsf=filtfilt(b,a,ts); %forward and reverse
Simple Inspection
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Pha
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freqz(b,a,100,100);sf
number of frequencies
Complex design• fdatool
– magnitude response– phase response– impulse response– compare filters– effect of changing filter order
Filter artifacts• onset transients
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2x 10-3 1 Hz lowpass (4th order Butterworth)
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filtfiltfilter
Filter artifacts• ringing
with artifact
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without artifact
Filter artifacts• ringing
filter, 20 Hz lowpass (12th order Butterworth)
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filtfilt
Filter artifacts• beginning and end of filtered ts is distorted• filtering artifacts is dangerous• filtering may change the latency of effects!• filtering may change the phase
Suggestions• be careful with low frequencies• use low order butterworth forward and reverse (to avoid
phase distortions)• carefully check beginning and end of filtered ts• make sure you don’t have artifacts in the data• use surrogate data (filtered noise)
Data preprocessing V• Decimation• ts=decimate(ts,4);• decimate uses a lowpass filter to avoid aliasing artifacts
Exercises 2-4