What are wavelets in time series analysis?

Wavelets are mathematical tools used in time series analysis to decompose signals into components that vary across both time and frequency. Unlike traditional methods like the Fourier transform, which only reveals frequency content, wavelets localize information in time, making them ideal for analyzing non-stationary data—where patterns (like trends or seasonality) change over time. A wavelet is a small “wave” (a localized oscillation) that can be scaled and shifted to match different parts of a signal. For example, the Haar wavelet detects abrupt changes (like spikes), while the Daubechies family smooths noise. This adaptability lets wavelets capture transient features, such as sudden shifts or short-lived anomalies, which are common in real-world time series like sensor readings or financial data.

Wavelet analysis works by breaking a time series into approximations (low-frequency components) and details (high-frequency components) at multiple resolution levels. This is done using a pair of filters: a low-pass filter for approximations and a high-pass filter for details, followed by downsampling to reduce data size. For instance, in a stock price time series, the approximation might represent the overall trend, while details capture daily volatility. Each decomposition level corresponds to a specific frequency band, allowing you to isolate features like hourly fluctuations or multi-day cycles. Libraries like PyWavelets in Python simplify this process—by applying pywt.wavedec(), you can decompose a signal and reconstruct specific components (e.g., removing noise by zeroing out high-frequency detail coefficients).

Wavelets are widely used for tasks like denoising, anomaly detection, and feature extraction. For example, in ECG signal processing, wavelets remove high-frequency noise without blurring critical peaks. In industrial settings, they detect equipment failures by identifying sudden changes in vibration sensor data. A developer might use the Haar wavelet to spot abrupt temperature spikes in IoT sensor data or apply the Morlet wavelet to analyze seasonal sales patterns. The choice of wavelet depends on the data: Haar is simple but effective for sharp transitions, while Daubechies (e.g., ‘db4’) balances smoothness and localization. By focusing on relevant coefficients, wavelets also enable efficient data compression—retaining only key features reduces storage and computation costs.