214. Fourier

A frequency-domain representation of a time series. Decompose the series into a sum of sines and cosines at different frequencies. Useful for finding dominant cycles that aren’t obvious in the time-domain view.

214.0.1. Discrete Fourier Transform (DFT)

For a series of length :

The output is a complex number. Its magnitude measures how much the frequency (cycles per time unit) is present in the signal. The phase tells you where the cycle peaks.

The inverse DFT recovers the original series:

214.0.2. Frequencies and periods

For a series sampled at one observation per unit time, the DFT covers frequencies for :

• : DC component (mean of the series)
• : one cycle over the entire window — period
• : two cycles over the window — period
• …
• : Nyquist frequency — period 2 (highest detectable cycle)
• : redundant (mirror of lower frequencies; this is aliasing)

So you only look at the first frequencies for a real-valued series. The period corresponding to bin is .

214.0.3. FFT — the fast version

The Fast Fourier Transform (Cooley-Tukey 1965) computes the DFT in rather than the naive . It’s the standard implementation in every numerical library (NumPy fft.fft, MATLAB fft, etc.).

Use FFT in practice; conceptually it computes exactly the DFT above.

214.0.4. What you can do with the Fourier transform

214.0.4.1. 1. Find dominant cycles

Plot (the power spectrum) vs frequency. Peaks indicate dominant cycles. Example uses:

• Found a peak at in 144 months of data → period 12 → annual cycle.
• Found a peak at in 365 days → period 365 → annual cycle (in daily data).
• Multiple peaks → multiple seasonalities (e.g., daily data has both weekly and yearly peaks).

This is the core of [periodogram.typ](periodogram.typ).

214.0.4.2. 2. Filter (frequency-domain)

Remove specific frequencies:

• Low-pass filter: keep low frequencies, drop high → smooths the series.
• High-pass filter: keep high, drop low → removes trend, isolates noise/cycles.
• Band-pass filter: keep a specific range → isolate one cycle.

Procedure: FFT → zero out unwanted frequency bins → inverse FFT.

214.0.4.3. 3. Forecast with Fourier features

Add sin(2 pi k t \/ m) and cos(2 pi k t \/ m) as regressors in a linear forecasting model, where is the period. A handful of harmonics () captures most of a smooth seasonal pattern.

This is how Prophet (Facebook) and many modern forecasting libraries handle seasonality without a fixed seasonal index.

214.0.4.4. 4. Spectral density estimation

Smoothed version of the periodogram → see [periodogram.typ](periodogram.typ).

214.0.5. Limitations

• Stationarity assumption: standard FFT assumes the series is stationary. For series with changing frequency content (e.g., a chirp), use short-time Fourier transform (STFT) or wavelets.
• Boundary effects: a finite series treated as periodic introduces artifacts at the edges. Apply a windowing function (Hann, Hamming) before FFT to reduce these.
• Resolution: frequency resolution is — you can only distinguish cycles at multiples of this. Low frequencies need long series.

214.0.6. Connection to seasonality

If your series has a known seasonal period , you’d expect a peak in the spectrum at . Conversely, unknown periodicities are discovered via the spectrum — perhaps weekly + monthly + yearly all show up in the same series. Fourier analysis is the natural tool for this.

import numpy as np
Y = np.fft.fft(y)
power = np.abs(Y[:N//2])**2  # only first half (positive frequencies)
freq = np.arange(N//2) / N    # cycles per month
period = 1 / np.where(freq > 0, freq, np.inf)  # months per cycle