Using fourier analysis for time series prediction
Question:
For data that is known to have seasonal, or daily patterns I’d like to use fourier analysis be used to make predictions. After running fft on time series data, I obtain coefficients. How can I use these coefficients for prediction?
I believe FFT assumes all data it receives constitute one period, then, if I simply regenerate data using ifft, I am also regenerating the continuation of my function, so can I use these values for future values?
Simply put: I run fft for t=0,1,2,..10 then using ifft on coef, can I use regenerated time series for t=11,12,..20 ?
Answers:
When you run an FFT on time series data, you transform it into the frequency domain. The coefficients multiply the terms in the series (sines and cosines or complex exponentials), each with a different frequency.
Extrapolation is always a dangerous thing, but you’re welcome to try it. You’re using past information to predict the future when you do this: “Predict tomorrow’s weather by looking at today.” Just be aware of the risks.
I’d recommend reading “Black Swan”.
It sounds like you want a combination of extrapolation and denoising.
You say you want to repeat the observed data over multiple periods. Well, then just repeat the observed data. No need for Fourier analysis.
But you also want to find “patterns”. I assume that means finding the dominant frequency components in the observed data. Then yes, take the Fourier transform, preserve the largest coefficients, and eliminate the rest.
X = scipy.fft(x)
Y = scipy.zeros(len(X))
Y[important frequencies] = X[important frequencies]
As for periodic repetition: Let z = [x, x], i.e., two periods of the signal x. Then Z[2k] = X[k] for all k in {0, 1, …, N-1}, and zeros otherwise.
Z = scipy.zeros(2*len(X))
Z[::2] = X
I’m aware that this question may be not actual for you anymore, but for others that are looking for answers I wrote a very simple example of fourier extrapolation in Python https://gist.github.com/tartakynov/83f3cd8f44208a1856ce
Before you run the script make sure that you have all dependencies installed (numpy, matplotlib). Feel free to experiment with it.
P.S. Locally Stationary Wavelet may be better than fourier extrapolation. LSW is commonly used in predicting time series. The main disadvantage of fourier extrapolation is that it just repeats your series with period N, where N – length of your time series.
you can use the library that @tartakynov posted and, to not repeat exactly the same time series in the forcast (overfitting), you can add a new parameter to the function called n_param and fix a lower bound h for the amplitudes of the frequencies.
def fourierExtrapolation(x, n_predict,n_param):
usually you will find that, in a signal, there are some frequencies that have significantly higher amplitude than others, so, if you select this frequencies you will be able to isolate the periodic nature of the signal
you can add this two lines who are determinated by certain number n_param
h=np.sort(x_freqdom)[-n_param]
x_freqdom=[ x_freqdom[i] if np.absolute(x_freqdom[i])>=h else 0 for i in range(len(x_freqdom)) ]
just adding this you will be able to forecast nice and smooth
another useful article about FFt:
forecast FFt in R
For data that is known to have seasonal, or daily patterns I’d like to use fourier analysis be used to make predictions. After running fft on time series data, I obtain coefficients. How can I use these coefficients for prediction?
I believe FFT assumes all data it receives constitute one period, then, if I simply regenerate data using ifft, I am also regenerating the continuation of my function, so can I use these values for future values?
Simply put: I run fft for t=0,1,2,..10 then using ifft on coef, can I use regenerated time series for t=11,12,..20 ?
When you run an FFT on time series data, you transform it into the frequency domain. The coefficients multiply the terms in the series (sines and cosines or complex exponentials), each with a different frequency.
Extrapolation is always a dangerous thing, but you’re welcome to try it. You’re using past information to predict the future when you do this: “Predict tomorrow’s weather by looking at today.” Just be aware of the risks.
I’d recommend reading “Black Swan”.
It sounds like you want a combination of extrapolation and denoising.
You say you want to repeat the observed data over multiple periods. Well, then just repeat the observed data. No need for Fourier analysis.
But you also want to find “patterns”. I assume that means finding the dominant frequency components in the observed data. Then yes, take the Fourier transform, preserve the largest coefficients, and eliminate the rest.
X = scipy.fft(x)
Y = scipy.zeros(len(X))
Y[important frequencies] = X[important frequencies]
As for periodic repetition: Let z = [x, x], i.e., two periods of the signal x. Then Z[2k] = X[k] for all k in {0, 1, …, N-1}, and zeros otherwise.
Z = scipy.zeros(2*len(X))
Z[::2] = X
I’m aware that this question may be not actual for you anymore, but for others that are looking for answers I wrote a very simple example of fourier extrapolation in Python https://gist.github.com/tartakynov/83f3cd8f44208a1856ce
Before you run the script make sure that you have all dependencies installed (numpy, matplotlib). Feel free to experiment with it.
P.S. Locally Stationary Wavelet may be better than fourier extrapolation. LSW is commonly used in predicting time series. The main disadvantage of fourier extrapolation is that it just repeats your series with period N, where N – length of your time series.
you can use the library that @tartakynov posted and, to not repeat exactly the same time series in the forcast (overfitting), you can add a new parameter to the function called n_param and fix a lower bound h for the amplitudes of the frequencies.
def fourierExtrapolation(x, n_predict,n_param):
usually you will find that, in a signal, there are some frequencies that have significantly higher amplitude than others, so, if you select this frequencies you will be able to isolate the periodic nature of the signal
you can add this two lines who are determinated by certain number n_param
h=np.sort(x_freqdom)[-n_param]
x_freqdom=[ x_freqdom[i] if np.absolute(x_freqdom[i])>=h else 0 for i in range(len(x_freqdom)) ]
just adding this you will be able to forecast nice and smooth
another useful article about FFt:
forecast FFt in R