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Miha Kralj
109 lines
4.1 KiB
Markdown
109 lines
4.1 KiB
Markdown
# Variance (VAR)
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> *Volatility is the price of admission for high returns.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Statistic |
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| **Inputs** | Source (close) |
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| **Parameters** | `period`, `isPopulation` (default false) |
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| **Outputs** | Single series (Variance) |
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| **Output range** | Varies (see docs) |
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| **Warmup** | `period` bars |
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| **PineScript** | [variance.pine](variance.pine) |
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- Variance measures how far a set of numbers is spread out from their average value.
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- **Similar:** [StdDev](../stddev/StdDev.md), [MeanDev](../meandev/MeanDev.md) | **Trading note:** Rolling variance; squared deviation from mean. Foundation of portfolio risk calculations.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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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.
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## Historical Context
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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.
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## Architecture & Physics
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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.
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### O(1) Calculation
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The algorithm maintains two running sums:
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1. Sum of values ($\sum x$)
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2. Sum of squared values ($\sum x^2$)
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When a new value enters and an old value leaves:
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$$ \sum x_{new} = \sum x_{old} - x_{out} + x_{in} $$
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$$ \sum x^2_{new} = \sum x^2_{old} - x^2_{out} + x^2_{in} $$
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This avoids iterating over the entire window for each update.
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## Mathematical Foundation
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Variance ($\sigma^2$ or $s^2$) is defined as:
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### Population Variance (N)
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$$ \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} $$
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Using the computational formula:
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$$ \sigma^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{N}}{N} $$
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### Sample Variance (N-1)
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$$ s^2 = \frac{\sum_{i=1}^{N} (x_i - \bar{x})^2}{N-1} $$
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Using the computational formula:
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$$ s^2 = \frac{\sum x^2 - \frac{(\sum x)^2}{N}}{N-1} $$
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Where:
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* $N$ is the period.
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* $\mu$ or $\bar{x}$ is the mean.
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## Performance Profile
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### Operation Count (Streaming Mode)
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Variance uses Welford-style running sums of x and x^2 for exact O(1) update (no sqrt needed).
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Ring buffer add/evict | 1 | 3 cy | ~3 cy |
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| Update sum_x and sum_x2 | 2 | 2 cy | ~4 cy |
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| Compute variance via shortcut formula | 1 | 5 cy | ~5 cy |
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| NaN guard + state update | 1 | 2 cy | ~2 cy |
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| **Total** | **O(1)** | — | **~14 cy** |
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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.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | 5 ns/bar | O(1) complexity using running sums. |
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| **Allocations** | 0 | Zero-allocation in hot path. |
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| **Complexity** | O(1) | Constant time update. |
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| **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). |
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## Validation
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **Skender** | ✅ | Matches `StdDev^2` (Sample Variance). |
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| **TA-Lib** | ✅ | Matches `VAR` (Population Variance usually, check specific implementation). |
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## Usage
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```csharp
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using QuanTAlib;
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// Create a 20-period Sample Variance indicator
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var variance = new Variance(20, isPopulation: false);
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// Update with a new value
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var result = variance.Update(new TValue(DateTime.UtcNow, 100.0));
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// Access the last calculated value
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Console.WriteLine($"Variance: {variance.Last.Value}"); |