Common Ways to Normalize Time Series for Correlation Studies

Before conducting correlation studies on random signals or time series, it is often necessary to normalize or preprocess the data. This step helps ensure that assumptions related to correlation (such as stationarity) are more closely met, and that the effects of trends, differing scales, or seasonal patterns do not unduly influence the results. Some standard approaches include:

1. Detrending

• Linear Detrending: Remove a best-fit linear trend from the data, ensuring that long-term upward or downward drifts don't dominate the correlation.
• Higher-order Detrending: For more complex patterns, remove polynomial trends or model-based trends to isolate a stationary component.

2. Mean and Variance Normalization (Standardization)

• Mean Removal: Subtracting the sample mean from each time series yields zero-mean data, aligning both series for correlation analysis.
• Scaling by Standard Deviation: Dividing by the standard deviation ensures that each series has unit variance, preventing a high-amplitude series from dominating the correlation.

3. De-Seasonalizing

• Removing Seasonal Components: If the data exhibit periodic structures (e.g., seasonal cycles), remove these predictable patterns. Methods like seasonal differencing or decompositions can isolate more stationary residuals.

4. Differencing to Achieve Stationarity

• First-order Differencing: Replace values with their successive differences to remove slow trends and stabilize the mean.
• Higher-order Differencing: For more pronounced non-stationarity, second or third differences might be employed, albeit at the cost of interpretability.

5. Filtering or Smoothing

• Bandpass Filtering: Isolate certain frequency bands to remove low-frequency trends or high-frequency noise.
• Low-pass/High-pass Filtering: Target specific frequency components that distort correlation patterns, enhancing the signal of interest.

6. Pre-Whitening

• Model-Based Pre-Whitening : Fit an appropriate model (such as ARIMA) to transform a series into a white-noise-like process. Correlations computed on these residuals can be more meaningful.

7. Min-Max Normalization (Less Common for Correlation)

• Scaling Data Range: Adjusting the data to a fixed range (e.g., [0, 1]) can help when comparing series of vastly different magnitudes. However, for correlation analysis, standardization is usually preferred.

Practical Considerations: The choice of normalization technique depends on the data characteristics and the nature of the correlation study. In practice, multiple methods (such as detrending and standardization) are often combined to create a more stationary and comparable set of time series for correlation analysis.