QuanTAlib/lib/numerics/sqrttrans/Sqrttrans.md

SQRTTRANS: Square Root Transform

"The square root is nature's variance-stabilizing trick—halving the exponent space while preserving monotonicity. When price volatility scales with level, sqrt compresses the noise."

The Square Root (SQRT) transformer applies \sqrt{x} to each value in a time series. This variance-stabilizing transformation compresses ranges where volatility scales with magnitude, making it useful for heteroscedastic data where standard deviation increases with price level.

Mathematical Foundation

Core Formula

\text{SQRT}_t = \sqrt{x_t}

where:

• x_t is the input value at time t
• x_t \geq 0 (domain restriction)

Key Properties

Property	Formula	Description
Domain	x \geq 0	Only non-negative inputs valid
Range	y \geq 0	Output always non-negative
Product Rule	\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}	Factors separate under sqrt
Quotient Rule	\sqrt{a/b} = \sqrt{a} / \sqrt{b}	Division becomes ratio of roots
Power Relation	\sqrt{x} = x^{0.5}	Half-power equivalence
Inverse	(\sqrt{x})^2 = x	Squaring reverses sqrt
Identity	\sqrt{0} = 0, \sqrt{1} = 1	Fixed points

Derivative

\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}

The derivative approaches infinity as x \to 0^+, meaning small changes near zero produce large output changes.

Financial Applications

Variance Stabilization

For data where standard deviation scales with the mean (Poisson-like behavior), sqrt transformation normalizes variance:

\text{Var}(\sqrt{X}) \approx \text{constant}

This enables statistical techniques that assume homoscedasticity.

Volatility Scaling

When volatility is proportional to price level:

\sigma_{price} \propto P \implies \sigma_{\sqrt{P}} \approx \text{constant}

The sqrt transformation can normalize volatility for cross-asset comparison.

Distance Metrics

Euclidean distance in feature space:

d = \sqrt{\sum_i (x_i - y_i)^2}

Risk Metrics

Volatility from variance:

\sigma = \sqrt{\text{Var}(R)}

Implementation Details

Negative Input Handling

Mathematical \sqrt{x} is undefined for x < 0. This implementation:

• Returns last valid value for negative inputs
• Returns last valid value for NaN/Infinity
• Starts with lastValid = 0.0 (since sqrt(0) = 0)

Precision Characteristics

Input Range	Relative Precision
x > 0	Full 15-16 digits
x = 0	Exact (returns 0)
x < 0	Substituted with last valid

Streaming Characteristics

Metric	Value
Warmup Period	0
Memory	O(1)
Complexity	O(1) per update

Performance Profile

Operation Count (Scalar)

Operation	Count	Notes
SQRT	1	Hardware instruction (FSQRT)
CMP	1	Domain check
Total	~15-20 cycles	Platform dependent

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	IEEE 754 compliant
Timeliness	10/10	Zero lag
Smoothness	N/A	Transform preserves input characteristics

Usage Examples

Basic Usage

// Create SQRT transformer
var sqrt = new Sqrttrans();

// Transform a value
var price = new TValue(DateTime.UtcNow, 100.0);
var result = sqrt.Update(price);  // 10.0

Variance Stabilization

var prices = new TSeries();
// ... populate with price data

// Apply sqrt transform for variance stabilization
var sqrtPrices = Sqrttrans.Calculate(prices);

// Now compute statistics on transformed data
var stdDev = StdDev.Calculate(sqrtPrices, 20);

Batch Processing

var source = new double[] { 1, 4, 9, 16, 25 };
var output = new double[source.Length];

Sqrttrans.Calculate(source, output);
// output: { 1, 2, 3, 4, 5 }

Chained with Square

// Round-trip: sqrt(x^2) = |x|
var values = bars.Close;
var squared = values.Select(v => new TValue(v.Time, v.Value * v.Value)).ToTSeries();
var recovered = Sqrttrans.Calculate(squared);
// recovered ≈ abs(original)

Common Pitfalls

1. Negative Input: Prices are always positive, but derived values (returns, differences) can be negative. Sqrt is undefined for negatives—this implementation returns last valid value.

2. Zero Amplification: Near zero, small changes in input cause large changes in sqrt output. \sqrt{0.01} = 0.1 but $\sqrt{0.0001} = 0.01$—a 100x input change yields only 10x output change.

3. Reversal Requires Squaring: To undo sqrt, square the result. Unlike log/exp which are inverses, sqrt/square are only one-way inverses for non-negative values.

4. Variance Stabilization Assumption: Sqrt is optimal when variance scales linearly with mean. For other heteroscedasticity patterns, log or Box-Cox may be more appropriate.

5. Magnitude Compression: Sqrt compresses large values more than small ones. \sqrt{10000} = 100 but \sqrt{100} = 10. This can distort technical analysis patterns that depend on absolute price levels.

Validation

Test	Status
Math.Sqrt Parity	✅
Perfect Squares (0,1,4,9,16,25,100)	✅
Irrational Results (√2, √3, √5)	✅
Inverse of Square	✅
Product Rule	✅
Quotient Rule	✅
Power Relationship (x^0.5)	✅
Small Values (1e-10 to 1e-2)	✅
Large Values (1e10 to 1e100)	✅

References

• Box, G.E.P., & Cox, D.R. (1964). "An Analysis of Transformations." Journal of the Royal Statistical Society, Series B, 26(2), 211-252.
• Tukey, J.W. (1977). Exploratory Data Analysis. Addison-Wesley. (Variance-stabilizing transformations)
• IEEE 754-2019. Standard for Floating-Point Arithmetic. (sqrt specification)