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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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

http://en.wikipedia.org/wiki/Time_series

What if we want to extract a pattern

from time series data?

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Original Data

lunchtime rush

unidentified

https://gist.github.com/espeecat/5438953

A sine wave or sinusoid

Cosine too

y(t) = A sin(2πft +ф) Sometimes 2πf written as ω

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

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plot of y=2*sin(2*pi*f*t) f=1Hz

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https://gist.github.com/espeecat/5439069

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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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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filtered data

original data

Original data and filtered data

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Comparison of original and filtered data

filtered data

original data

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

https://www.scilab.org/http://www.gnu.org/software/octave/http://www.r-project.org/http://www.r-project.org/http://www.r-project.org/

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Discrete-Time Fourier Transform

• ω = 2πƒ –angular frequency

• Euler Formula used but this represents