QuanTAlib/lib/statistics/covariance/Covariance.md

Covariance: Covariance

Correlation is just covariance normalized by standard deviation. But sometimes you want the raw, unadulterated relationship.

Property	Value
Category	Statistic
Inputs	Source (close)
Parameters	period, isPopulation (default false)
Outputs	Single series (Cov)
Output range	Varies (see docs)
Warmup	period bars
PineScript	covariance.pine

• Covariance measures the joint variability of two random variables.
• Similar: Correlation, Beta | Trading note: Rolling covariance; measures how two assets move together. Foundation of portfolio theory.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Covariance measures the joint variability of two random variables. It indicates the direction of the linear relationship between variables.

Architecture & Physics

Covariance is calculated using a sliding window approach. It maintains running sums of x, y, and xy to allow for O(1) updates.

• Positive Covariance: Indicates that the two variables tend to move in the same direction.
• Negative Covariance: Indicates that the two variables tend to move in opposite directions.
• Zero Covariance: Indicates that the two variables are uncorrelated.

Mathematical Foundation

1. Population Covariance

Cov(X, Y) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{n}

2. Sample Covariance

Cov(X, Y) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{n - 1}

3. Computational Formula (Running Sums)

Cov(X, Y) = \frac{\sum xy - \frac{(\sum x)(\sum y)}{n}}{n} \quad \text{(or } n-1 \text{)}

Performance Profile

Operation Count (Streaming Mode)

Covariance uses a dual-input sliding window with running cross-product sums for O(1) update.

Operation	Count	Cost (cycles)	Subtotal
Ring buffer add/evict (2 inputs)	2	3 cy	~6 cy
Update 3 running sums (Sx, Sy, Sxy)	3	2 cy	~6 cy
Compute covariance formula	1	5 cy	~5 cy
NaN guard + state update	1	2 cy	~2 cy
Total	O(1)	—	~19 cy

O(1) per update using online running sums. Periodic resync every 1000 bars prevents floating-point drift accumulation.

Metric	Score	Notes
Throughput	High	O(1) updates using running sums.
Allocations	0	No heap allocations in hot path.
Complexity	O(1)	Constant time update regardless of period.
Accuracy	High	Uses double precision; periodic resync prevents drift.

Validation

Library	Status	Notes
Manual	✅	Verified against manual calculation.
Excel	✅	Matches COVARIANCE.P and COVARIANCE.S.

Usage

using QuanTAlib;

// Create a Covariance indicator with period 20 (Sample Covariance by default)
var cov = new Covariance(20);

// Update with new values
cov.Update(price1, price2);

// Access the result
double result = cov.Last.Value;