Using Power Spectra to Analyze Time-Series Structure

Summary

This article explains how the discrete Fourier transform (DFT) represents a time series as frequency components and how a power spectrum summarizes the energy at each frequency. It outlines the fast Fourier transform as a more efficient way to compute the transform, then describes preprocessing: centering and applying a Welch window to reduce boundary effects and spectral leakage. The article also introduces Savitzky–Golay smoothing to make a spectrum with many narrow peaks easier to interpret.

The examples are implemented in an MQL5 class that calculates power, cumulative power, and cumulative spectrum deviation, with plots for analysis. The stated applications include examining seasonality, supporting autoregressive model order selection, deciding whether seasonal differencing may be needed, and assessing prediction models. The document provides implementation detail and example scripts for autoregressive, seasonal, and white-noise series, but the supplied text does not include their plotted results or a quantitative evaluation. It cautions that windowing can alter the data and that maximum-entropy spectra may reveal narrow features but can also produce spurious peaks, so they should be compared with DFT results.

Key ideas

• The DFT expresses a finite time series as periodic sine and cosine components.
• A power spectrum shows how signal energy is distributed across frequencies.
• Centering and windowing can reduce boundary artifacts, while windowing may also distort the original signal.
• Smoothing can make dense spectra easier to read but should not replace understanding the underlying frequency components.
• Power spectra can help investigate seasonality, autoregressive structure, and forecast-model behavior.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.