Z-score Normalization

Understanding Z-score normalization

Z-score normalization (also called standardization) converts data points into a standard scale where:

• The mean becomes 0
• The standard deviation becomes 1
• Values represent the number of standard deviations from the mean

The formula for Z-score normalization is:

z = (x - μ) / σ

where:
x = original value
μ = mean of the population
σ = standard deviation of the population

Applications in time-series analysis

Z-score normalization is particularly valuable for:

1. Anomaly Detection: Identifying unusual patterns by flagging data points with extreme Z-scores
2. Cross-series Comparison: Enabling meaningful comparisons between different time series
3. Feature Scaling: Preparing data for machine learning models that are sensitive to input scales

Implementation in financial markets

In financial applications, Z-score normalization helps in:

# Example of rolling Z-score calculation
def rolling_zscore(prices, window):
    returns = prices.pct_change()
    mean = returns.rolling(window=window).mean()
    std = returns.rolling(window=window).std()
    return (returns - mean) / std

For time-series databases like QuestDB, you can implement Z-score calculations using window functions:

WITH returns AS (
    SELECT 
        timestamp,
        (price - LAG(price) OVER(ORDER BY timestamp)) / LAG(price) OVER(ORDER BY timestamp) AS return
    FROM trades
)
SELECT 
    timestamp,
    return,
    (return - avg(return) OVER w) / stddev(return) OVER w AS zscore
FROM returns
WINDOW w AS (ORDER BY timestamp ROWS BETWEEN 20 PRECEDING AND CURRENT ROW);

Common use cases

1. Statistical Arbitrage: Identifying mean reversion opportunities in pairs trading strategies
2. Risk Management: Standardizing risk metrics across different instruments
3. Performance Analysis: Comparing returns across different market regimes

Considerations and limitations

When applying Z-score normalization, consider:

1. Window Selection: The choice of lookback period affects sensitivity
2. Non-normal Distributions: Z-scores assume normally distributed data
3. Outlier Impact: Extreme values can skew the normalization

Real-world example

Here's a practical example showing how Z-score normalization helps detect unusual price movements:

This approach is commonly used in anomaly detection systems for financial market surveillance and risk management.

Best practices

1. Data Quality: Clean outliers before normalization
2. Window Size: Choose based on the underlying data frequency
3. Monitoring: Regularly validate normalization parameters
4. Documentation: Track normalization parameters for reproducibility

The effectiveness of Z-score normalization makes it a fundamental tool in quantitative analysis and time-series processing, particularly when working with high-frequency financial data or multiple data streams that need standardization.