Fast Fourier Transforms VS. Autoregressive Moving Model Average

Recap:

Last week we talked about how Fast Fourier Transforms are used to quickly compute the Discrete Fourier Transform of a time series thereby transforming the time series into its Frequency Domain. This week we will talk about the Autoregressive Moving Model Average and how it compares to Fast Fourier Transforms.

Why FFT or ARMA in the first place?

“Perfect feature extraction would make the job of the classifier trivial, and an omnipotent classifier would not need the help of feature extraction,”

What is the Autoregressive Moving Model Average:

Given a time series of data Xt, the autoregressive moving average model (ARMA), is used as a tool for understanding and possibly predicting future values in the time series (Wikipedia 2006). The ARMA is typically applied to time series data (Wikipedia 2006).

Autoregressive model:

ϕ
1,...ϕp are the parameters and ε is the error term

Moving average model:

w
here the θ1, ..., θq are the parameters of the model and the εt, εt-1,... are the error terms

Autoregressive moving average model:

Taking the AR model and the MA model, we get the ARMA model. The notation ARMA(p, q) refers to a model with p autoregressive terms and q moving average terms (Wikipedia 2006). This model subsumes the AR and MA models,

Autoregressive vs. Fast Fourier

• The ARMA estimate may have very sharp spectral features so one cm evaluate it on a very fine mesh near to those features, and more coarsely farther away from them

• With ARMA Good spectral estimates can be obtained from short EEG segments, where as FFT assumes the function continues to infinity on both sides.

• ARMA is preferred for prediction

• In some ARMA models Coefficients can be analyzed independent of the function.

Power approximations of a 12 Hz sine wave added to a 20 Hz sine wave sampled at 200 Hz for 0.3 seconds. Top part of the figure shows the FFT approximation, and the bottom part shows the AR model of 8th order.

Disadvantages of ARMA:

• Problem of selecting the proper model order:

  • With noisy inputs and the order being too high the result is spurious peaks

  • Too low an order yields a smooth spectra

• ARMA can be an order of magnitude slower than FFT.

  • N = the size of the signal

  • M = model order or magnitude

  • if the magnitude of M is greater then logN then ARMA is slower then FFT (NlogN)

[1] M. J. Polak, "Adaptive Logic Networks in a Brain-Computer Interface System," in Computing Science. vol. Doctor of Philosophy Edmonton, Alberta: University of Alberta, 2000, p. 122.

[2] B. Hodgson ”What is an autoregressive moving average model(ARMA)?” http://cnx.org/content/m13395/latest/