11.3 Applications of spectral analysis

Spectral analysis is a powerful tool for understanding time series data across various fields. By decomposing complex signals into their frequency components, it reveals hidden patterns, cycles, and relationships that might not be apparent in the raw data.

From economics to engineering and geophysics, spectral analysis helps identify periodic components, trends, and noise in time series. It also enables the study of relationships between multiple time series, providing valuable insights for forecasting, control, and optimization in real-world applications.

Applications of Spectral Analysis

Application of spectral analysis techniques

• Apply spectral analysis techniques to real-world time series data from various fields
  • Economics: Analyze business cycles (GDP fluctuations), stock market trends (S&P 500), and economic indicators (inflation rates)
  • Engineering: Study vibrations (bridge oscillations), power systems (electrical grid frequency), and control systems (aircraft stability)
  • Geophysics: Investigate seismic activity (earthquake waveforms), ocean waves (tidal patterns), and atmospheric phenomena (temperature variations)
• Steps to apply spectral analysis techniques:
  1. Collect and preprocess the time series data by removing trends, outliers, and missing values
  2. Compute the power spectral density (PSD) or periodogram using Fourier transform or Welch's method
  3. Identify significant frequencies and their corresponding powers from the PSD plot
  4. Interpret the results in the context of the specific field and application to gain insights into underlying processes

Identification of time series components

• Spectral analysis decomposes a time series into its constituent frequencies and helps identify periodic components, trends, and noise
  • Periodic components appear as peaks in the power spectrum at their corresponding frequencies (daily, weekly, or annual cycles)
  • Trends manifest as low-frequency components in the power spectrum (long-term growth or decline)
  • Noise is characterized by a continuous distribution of power across all frequencies (white noise or colored noise)
• Techniques for identifying components:
  • Periodogram: Estimate the power spectrum by computing the squared magnitude of the Fourier transform of the time series
  • Smoothing: Apply techniques like moving average or Hamming window to reduce noise and enhance significant peaks
  • Significance testing: Determine the statistical significance of peaks using methods like Fisher's g-test or the chi-squared test

Relationships between multiple time series

• Cross-spectral analysis extends spectral analysis to multiple time series and helps detect and quantify relationships between them
  • Cross-periodogram: Compute the cross-power spectrum between two time series to identify common frequencies
  • Coherence: Measure the linear relationship between two time series as a function of frequency
    • Coherence ranges from 0 (no relationship) to 1 (perfect linear relationship) and helps assess the strength of the relationship
  • Phase: Quantify the lead-lag relationship between two time series at each frequency to determine which series leads or lags the other
• Applications of cross-spectral analysis:
  • Identifying common periodic components or trends across multiple time series (stock prices and economic indicators)
  • Detecting and quantifying causal relationships or feedback loops between variables (temperature and CO2 levels)
  • Studying the propagation of signals or disturbances through a system (seismic waves through different layers of the Earth)

Interpretation of spectral analysis results

• Interpreting the power spectrum:
  • Dominant frequencies indicate the presence of periodic components or cycles (annual sales patterns or tidal cycles)
  • The power at each frequency represents the relative importance or strength of that component (amplitude of seasonal variations)
  • Broad peaks suggest the presence of quasi-periodic behavior or noise (irregular business cycles or background seismic noise)
• Relating spectral results to the underlying processes:
  • Economic processes: Business cycles (3-5 years), seasonal patterns (quarterly or monthly), or long-term trends (decades)
  • Physical processes: Natural frequencies (bridge resonance), resonance (aircraft wing flutter), or energy transfer between components (coupled oscillators)
  • Geophysical processes: Tidal cycles (diurnal and semidiurnal), climate oscillations (El Niño), or seismic wave propagation (P-waves and S-waves)
• Combining spectral analysis with domain knowledge:
  • Use the insights gained from spectral analysis to refine models or theories (economic models incorporating cyclical components)
  • Identify potential causal relationships or feedback mechanisms (climate variables influencing each other)
  • Develop strategies for forecasting (predicting future sales based on seasonal patterns), control (damping vibrations in structures), or optimization (designing filters to remove noise) based on the spectral properties of the system