Slides for Time Series Data and FFT

Slides for Time Series Data and FFT
Time Series Data and Fourier
Transforms
Jason Bailey
Quick Summary
• Look Time Series Data
• See data in Time domain (time series) and
Frequency domain (using Fourier Transform)
• Application: Filter data/Extract pattern with
Fourier Transform
• FFT - Fast Fourier Transform
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14-Apr
Date
21-Apr
What is Time Series Data
• A sequence of data points
• Typically at successive points in time spaced at
uniform time intervals
• Used:
• statistics, signal processing, pattern
recognition, finance, weather forecasting,
earthquake prediction, control engineering
and communications engineering
http://en.wikipedia.org/wiki/Time_series
What if we want to extract
a pattern
from time series data?
Visitors to a Learning Site
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Original Data
lunchtime rush
unidentified
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Time
https://gist.github.com/espeecat/5438953
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A sine wave or sinusoid
y(t) = A sin(2πft +ф)
Sometimes 2πf
written as ω
Cosine too
Much better to see it in a graph
• Use a tool like Matlab
– A programmable calculator with good graph/chart
abilities
• Other tools are available and much cheaper
An example of a sinusoid and FFT
plot of y=2*sin(2*pi*f*t) f=1Hz
Amplitude
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Time (seconds)
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Abs Amp
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Frequency (Hz)
https://gist.github.com/espeecat/5439069
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The Fourier Transform (FFT)
• Based on Fourier Series - represent periodic
time series data as a sum of sinusoidal
components (sine and cosine)
• (Fast) Fourier Transform [FFT] – represent time
series in the frequency domain (frequency and
power)
• The Inverse (Fast) Fourier Transform [IFFT] is the
reverse of the FFT
• Like graphic equaliser on music player
Combining Sinusoids
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Time (s)
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Time (s)
Looking at the Fourier Transforms
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Frequency (Hz)
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Applications of Fourier Transform
• Shazam – “finger printing” using Fourier
Transforms
• Images – The Gabor Transform for facial
recognition?
• Filtering data/ Extracting patterns
• Sound processing – discarding sound
• System Identification
The (Fast) Fourier Transform
• Discrete-time Fourier Transform –assumes
sampled data and limited length
• Implementations available in lots of programming
languages e.g. http://www.fftw.org/
• Python numpy.fft
Filtering Time Series Data
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Frequency Hz.
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filtered data
original data
Amplitude
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Time (s)
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Original data and filtered data
Comparison of original and filtered data
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filtered data
original data
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Amplitude
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Time (s)
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Thank you
Alternative to Matlab
• SciLab – https://www.scilab.org/
• Octave http://www.gnu.org/software/octave/
• R - http://www.r-project.org/
• Programming language & graph library
Twitter @espeecat www.jasonbailey.net
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Discrete-Time Fourier Transform
• ω = 2πƒ –angular
frequency
• Euler Formula used
but this represents
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