⏳Intro to Time Series Unit 11 – Spectral Analysis in Time Series

Key Concepts and Definitions

• Spectral analysis examines the frequency content of a time series signal
• Frequency domain representation shows how much of each frequency is present in a signal
• Fourier transform decomposes a time series into its constituent frequencies
• Power spectral density (PSD) measures the power of a signal at different frequencies
• Periodogram provides an estimate of the spectral density of a signal
• Spectral estimation techniques (Welch's method, multitaper method) improve the accuracy of spectral estimates
• Nyquist frequency is the highest frequency that can be accurately represented in a sampled signal (half the sampling rate)
• Aliasing occurs when a signal is sampled at a rate lower than twice its highest frequency component

Time Domain vs Frequency Domain

• Time domain representation shows how a signal changes over time
  • Useful for understanding the temporal evolution of a signal
  • Examples include time series plots, autocorrelation functions, and cross-correlation functions
• Frequency domain representation shows how much of each frequency is present in a signal
  • Useful for identifying periodic components, noise, and other frequency-specific features
  • Obtained through Fourier transform or spectral estimation techniques
• Both representations provide complementary information about a signal
• Time-frequency analysis (wavelet analysis, short-time Fourier transform) combines both domains to analyze non-stationary signals

Fourier Transform Basics

• Fourier transform decomposes a signal into its constituent frequencies
• Continuous Fourier transform (CFT) applies to continuous-time signals
  • Defined as: $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
• Discrete Fourier transform (DFT) applies to discrete-time signals
  • Defined as: $X(k) = \sum_{n=0}^{N-1} x(n) e^{-j2\pi kn/N}$
• Inverse Fourier transform reconstructs the time-domain signal from its frequency components
• Fast Fourier transform (FFT) is an efficient algorithm for computing the DFT
• Fourier transform properties (linearity, scaling, shifting, convolution) facilitate signal analysis and processing

Power Spectral Density (PSD)

• PSD measures the power of a signal at different frequencies
• Defined as the Fourier transform of the autocorrelation function: $S(f) = \int_{-\infty}^{\infty} R(\tau) e^{-j2\pi f\tau} d\tau$
• Provides information about the distribution of power across frequencies
• Can identify dominant frequencies, periodic components, and noise in a signal
• Units are power per unit frequency (e.g., watts per hertz)
• Estimation methods include periodogram, Welch's method, and multitaper method

Periodogram Analysis

• Periodogram is an estimate of the spectral density of a signal
• Computed as the squared magnitude of the DFT: $P(f) = \frac{1}{N} |X(f)|^2$
• Provides a simple and intuitive way to visualize the frequency content of a signal
• Suffers from high variance and spectral leakage
  • Variance can be reduced by averaging multiple periodograms (Welch's method)
  • Spectral leakage can be mitigated using window functions (Hamming, Hann, Blackman)
• Periodogram averaging trades off frequency resolution for reduced variance

Spectral Estimation Techniques

• Aim to improve the accuracy and reliability of spectral estimates
• Welch's method divides the signal into overlapping segments, computes periodograms for each segment, and averages them
  • Reduces variance at the cost of frequency resolution
  • Overlap between segments (50-75%) helps maintain statistical stability
• Multitaper method applies multiple orthogonal window functions (tapers) to the signal and averages the resulting spectral estimates
  • Tapers are designed to minimize spectral leakage and maximize concentration of energy
  • Provides better bias-variance trade-off compared to periodogram
• Parametric methods (autoregressive, moving average, ARMA) model the signal as the output of a linear system
  • Estimate model parameters and derive the PSD from the model
  • Suitable for short and noisy data, but require model order selection

Interpreting Spectral Results

• Identify dominant frequencies and their relative power
  • Peaks in the spectrum indicate strong periodic components
  • Width of peaks relates to the stability and coherence of the oscillations
• Assess the overall shape of the spectrum
  • Flat spectrum suggests white noise (equal power at all frequencies)
  • 1/f spectrum (pink noise) has power inversely proportional to frequency
  • Narrowband spectrum has power concentrated around specific frequencies
• Compare spectra across different conditions or time segments
  • Changes in spectral content can reveal underlying dynamics or transitions
• Consider the limitations and uncertainties of spectral estimates
  • Finite data length, noise, and non-stationarity can affect the results
  • Use confidence intervals or significance tests to assess the reliability of the estimates

Applications and Case Studies

• Geophysical time series (climate data, seismic signals)
  • Identify climate oscillations (El Niño, Pacific Decadal Oscillation)
  • Detect and characterize seismic events and earth's natural frequencies
• Biomedical signals (EEG, ECG, EMG)
  • Analyze brain rhythms and their spatial distribution
  • Detect heart rate variability and abnormalities
  • Assess muscle activity and fatigue
• Speech and audio processing
  • Identify formants and pitch in speech signals
  • Detect and remove noise or interference
  • Compress and encode audio using frequency-domain techniques
• Mechanical vibrations and rotating machinery
  • Monitor the health and performance of machines based on their vibration spectra
  • Detect faults, imbalances, or wear in bearings, gears, and other components
• Financial time series (stock prices, exchange rates)
  • Identify market cycles and trends
  • Assess the impact of economic events on different frequency components
  • Develop trading strategies based on spectral features