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.

Property	Value
Category	Numeric
Inputs	Source (close)
Parameters	None
Outputs	Single series (SQRTTRANS)
Output range	Varies (see docs)
Warmup	0 bars
PineScript	sqrttrans.pine

• The Square Root (SQRT) transformer applies \sqrt{x} to each value in a time series.
• No configurable parameters; computation is stateless per bar.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

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)