QuanTAlib/lib/statistics/variance/Variance.md

Variance (VAR)

Volatility is the price of admission for high returns.

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

• Variance measures how far a set of numbers is spread out from their average value.
• Similar: StdDev, MeanDev | Trading note: Rolling variance; squared deviation from mean. Foundation of portfolio risk calculations.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

Variance measures how far a set of numbers is spread out from their average value. In finance, it is a key measure of volatility and risk.

Historical Context

Variance is a fundamental concept in statistics, formalized by Ronald Fisher in 1918. In finance, it gained prominence with Modern Portfolio Theory (Markowitz, 1952), where it serves as the standard measure of risk.

Architecture & Physics

The Variance indicator uses a sliding window (RingBuffer) to maintain the last N data points. It calculates the variance using an O(1) running sum of squares algorithm, ensuring constant time complexity regardless of the period length.

O(1) Calculation

The algorithm maintains two running sums:

1. Sum of values (\sum x)
2. Sum of squared values (\sum x^2)

When a new value enters and an old value leaves:

\sum x_{new} = \sum x_{old} - x_{out} + x_{in} \sum x^2_{new} = \sum x^2_{old} - x^2_{out} + x^2_{in}

This avoids iterating over the entire window for each update.

Mathematical Foundation

Variance (\sigma^2 or s^2) is defined as:

Population Variance (N)

\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}

Using the computational formula:

\sigma^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{N}}{N}

Sample Variance (N-1)

s^2 = \frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1}

Using the computational formula:

s^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{N}}{N-1}

Where:

• N is the period.
• \mu or \bar{x} is the mean.

Performance Profile

Operation Count (Streaming Mode)

Variance uses Welford-style running sums of x and x^2 for exact O(1) update (no sqrt needed).

Operation	Count	Cost (cycles)	Subtotal
Ring buffer add/evict	1	3 cy	~3 cy
Update sum_x and sum_x2	2	2 cy	~4 cy
Compute variance via shortcut formula	1	5 cy	~5 cy
NaN guard + state update	1	2 cy	~2 cy
Total	O(1)	—	~14 cy

O(1) per update. Slightly faster than StdDev (no sqrt). Periodic resync prevents floating-point drift in long series where sum_x2 >> (sum_x)^2/N.

Metric	Score	Notes
Throughput	5 ns/bar	O(1) complexity using running sums.
Allocations	0	Zero-allocation in hot path.
Complexity	O(1)	Constant time update.
Accuracy	9	High accuracy, though running sums can accumulate floating point errors over very long periods (mitigated by periodic resync if needed, though not strictly implemented here as window is finite).

Validation

Library	Status	Notes
Skender	✅	Matches StdDev^2 (Sample Variance).
TA-Lib	✅	Matches VAR (Population Variance usually, check specific implementation).

Usage

using QuanTAlib;

// Create a 20-period Sample Variance indicator
var variance = new Variance(20, isPopulation: false);

// Update with a new value
var result = variance.Update(new TValue(DateTime.UtcNow, 100.0));

// Access the last calculated value
Console.WriteLine($"Variance: {variance.Last.Value}");