# 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.pine)                       |

- Covariance measures the joint variability of two random variables.
- **Similar:** [Correlation](../correlation/Correlation.md), [Beta](../beta/Beta.md) | **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

```csharp
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;