Scaling by Standard Deviation
After removing the mean, the next common step in standardization is to scale the series by its standard deviation. This process yields a series with unit variance, which is beneficial when comparing multiple time series with very different variances.
Why Scale by the Standard Deviation?
- Equal Weighting: Time series with large variance can overshadow those with smaller variance in correlation analyses. Scaling by standard deviation puts all series on a level playing field.
- Normalization for Models: Many modeling techniques benefit from inputs that have similar scales, improving convergence in optimization-based models and enhancing interpretability.
How to Scale by Standard Deviation?
- Calculate the Standard Deviation:
std_value = np.std(y_mean_removed). - Divide the Series:
y_standardized = y_mean_removed / std_value.
Example (Pseudo-Code):
y_mean_removed = [-0.6, -0.1, -1.1, 1.4, 0.4] std_value = np.std(y_mean_removed) # e.g., about 0.8 y_standardized = y_mean_removed / std_value # Now y_standardized will have a standard deviation close to 1.
This step ensures that the amplitude of fluctuations is normalized. After scaling by the standard deviation, your time series will have zero mean and unit variance, making correlation coefficients and other comparative statistics more meaningful.