Using Fourier Methods to Approximate Time Series with Trigonometric Terms
Summary
The document asks how to fit a time series with a fixed sum of sine and cosine terms when both coefficients and periodicities are to be chosen. It identifies a difficulty with trying many possible frequencies in trigonometric regression and asks whether an approximation method can select them optimally. The responses point first to Fourier decomposition, which represents a sequence through frequency components; one suggested denoising approach is to remove higher-frequency components.
A second response recommends the discrete cosine transform as a standard method for expressing finite sequences as sums of cosines. These are practical starting points for frequency-based approximation, but the discussion does not derive an optimization procedure for arbitrary, independently chosen periodicities or demonstrate that either transform finds a globally best fit under a specified error measure. It also provides no data, comparison, or empirical results, so model choice and validation remain open.
Key ideas
- Fourier methods decompose a time series into frequency components.
- Removing high-frequency components is suggested as a way to reduce noise.
- The discrete cosine transform represents finite sequences as sums of cosines.
- The responses recommend standard transforms as starting points rather than giving a general optimizer for arbitrary periods.
- The document contains no empirical comparison or specified error criterion.
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Full text
# Approximating a function with trignometric polynomials # Approximating a function with trignometric polynomials Let’s say I have a function, which is a time series of data points, I am trying to find a polynomial of fixed sine's and cosines that bests approximate the data points. I know Chebyshev Approximation is popular tool to approximate a dataset function with a polynomial. I am trying to approximate the function with a trigonometric polynomial that finds the optimal coefficients and periodicities for each cosine/sine term that best minimizes the error. I have done this before using trigonometric regressions to find a best fit for a time series but my problems with trig regressions is there are infitinitly many periodicities to test. Is there an approximation method that can be used to approximating the best coefficient on each cosine/sine and the optimal periodicity that best fits the data? Thank you! ## Answer by Lucas Morin (score 4) https://quant.stackexchange.com/a/8266 Working on trigonometric polynomial decomposition, the first step is to take a big look at Fourier transformation. It is very powerfull, well documented and probably well implemented on your favorite language. It will give you the decomposition of your time series. You can remove highest frequencies, which correspond to noise, to have a good estimation. ## Answer by Karol J. Piczak (score 3) https://quant.stackexchange.com/a/8270 Building upon +Imorin answer, you should have a look specifically at discrete cosine transforms. It's a standard approach when trying to express finite sequences as a sum of cosines. I would start from there, especially as it's implemented in every common language (R, Matlab, Python for starters). Only then evaluate if you need more.
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