mirror of
https://github.com/mihakralj/QuanTAlib.git
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Miha Kralj
192 lines
7.2 KiB
Markdown
192 lines
7.2 KiB
Markdown
# Pseudo-Huber: Smooth Huber Approximation
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> *All the robustness of Huber, none of the discontinuities.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual vs Predicted (dual input) |
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| **Parameters** | `period`, `delta` (default 1.0) |
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| **Outputs** | Single series (PseudoHuber) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period` bars |
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| **PineScript** | [pseudohuber.pine](pseudohuber.pine) |
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- Pseudo-Huber Loss (also called Charbonnier Loss) is a smooth approximation to the Huber loss function.
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- **Similar:** [Huber](../huber/Huber.md), [LogCosh](../logcosh/LogCosh.md) | **Trading note:** Smooth approximation to Huber loss; continuously differentiable with tunable delta parameter.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Pseudo-Huber Loss (also called Charbonnier Loss) is a smooth approximation to the Huber loss function. Unlike Huber which has a piecewise definition with a kink at δ, Pseudo-Huber is continuously differentiable everywhere, making it ideal for gradient-based optimization.
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## Historical Context
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The Pseudo-Huber function emerged from the optimization and machine learning communities as a way to get Huber-like robustness while maintaining smooth gradients. It's also known as Charbonnier loss in image processing, where it's used for edge-preserving smoothing and optical flow estimation.
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## Architecture & Physics
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Pseudo-Huber uses the formula δ²(√(1 + (x/δ)²) - 1), which smoothly interpolates between quadratic behavior for small errors and linear behavior for large errors. The transition is gradual rather than abrupt, with no discontinuity in derivatives.
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### Properties
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* **Smooth everywhere**: Infinitely differentiable (unlike Huber's kink)
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* **Non-negative**: Always ≥ 0, with 0 for perfect prediction
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* **Robust**: Large errors grow linearly, not quadratically
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* **Tunable**: δ (delta) controls the L2-to-L1 transition point
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## Mathematical Foundation
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### 1. Pseudo-Huber Function
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For each error, compute:
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$$L_\delta(e) = \delta^2 \left(\sqrt{1 + \left(\frac{e}{\delta}\right)^2} - 1\right)$$
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Where:
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* $e = y - \hat{y}$ = prediction error
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* $\delta$ = tuning parameter (transition width)
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### 2. Asymptotic Behavior
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For small errors (|e| << δ):
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$$L_\delta(e) \approx \frac{e^2}{2}$$
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For large errors (|e| >> δ):
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$$L_\delta(e) \approx \delta|e| - \delta^2$$
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### 3. Gradient (Derivative)
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$$\frac{dL}{de} = \frac{e}{\sqrt{1 + (e/\delta)^2}}$$
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This approaches:
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* e for small errors (like L2)
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* δ·sign(e) for large errors (like L1)
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### 4. Running Update (O(1))
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QuanTAlib uses a ring buffer with running sum for O(1) updates:
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$$S_{new} = S_{old} - L_{oldest} + L_{newest}$$
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$$PseudoHuber = \frac{S_{new}}{n}$$
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## Implementation Details
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### Usage Patterns
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```csharp
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// Streaming mode - with custom delta
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var pseudoHuber = new PseudoHuber(period: 20, delta: 1.0);
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var result = pseudoHuber.Update(actualValue, predictedValue);
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// Batch mode - calculate for entire series
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var results = PseudoHuber.Calculate(actualSeries, predictedSeries, period: 20, delta: 1.0);
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// Span mode - zero-allocation for high performance
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PseudoHuber.Batch(actualSpan, predictedSpan, outputSpan, period: 20, delta: 1.0);
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```
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### Parameters
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| Parameter | Type | Default | Description |
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| :--- | :--- | :--- | :--- |
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| **period** | int | - | Lookback window for averaging (must be > 0) |
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| **delta** | double | 1.0 | Transition parameter (must be > 0) |
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### Properties
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| Property | Type | Description |
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| :--- | :--- | :--- |
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| **Last** | TValue | Most recent Pseudo-Huber value |
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| **IsHot** | bool | True when buffer is full |
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| **Delta** | double | Current delta parameter |
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| **Name** | string | Indicator name (e.g., "PseudoHuber(20,1.000)") |
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| **WarmupPeriod** | int | Number of periods before valid output |
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## Performance Profile
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### Operation Count (Streaming Mode)
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O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Error computation (subtract, abs/square/log) | 1-3 | ~3-8 cy | ~5-15 cy |
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| Running accumulator update (EMA or sum) | 1 | ~4 cy | ~4 cy |
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| **Total** | **2-4** | — | **~9-19 cycles** |
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Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| Element-wise error computation | Yes | Independent per bar; fully vectorizable with `Vector<double>` |
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| Reduction (sum/mean) | Yes | Parallel reduction; AVX2 gives 4x speedup |
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| Log/exp components | Partial | Transcendental ops; polynomial approx for SIMD |
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Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~15 ns/bar | O(1) update, sqrt computation |
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| **Allocations** | 0 | Uses pre-allocated ring buffer |
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| **Complexity** | O(1) | Constant time per update |
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| **Accuracy** | 10/10 | Exact calculation |
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| **Timeliness** | 9/10 | No lag beyond the period |
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| **Smoothness** | 10/10 | Infinitely differentiable |
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## Comparison with Huber
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| Aspect | Huber | Pseudo-Huber |
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| :--- | :--- | :--- |
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| **Small errors** | e²/2 | ≈ e²/2 |
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| **Large errors** | δ\|e\| - δ²/2 | ≈ δ\|e\| - δ² |
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| **At e = δ** | Kink (C¹) | Smooth (C^∞) |
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| **Gradient** | Discontinuous 2nd derivative | Continuous all derivatives |
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| **Computation** | Conditional logic | Single formula |
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| **Optimization** | Can cause issues | Smooth convergence |
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### Numerical Comparison
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| Error (e) | Huber (δ=1) | Pseudo-Huber (δ=1) |
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| :--- | :--- | :--- |
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| **0.0** | 0.000 | 0.000 |
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| **0.5** | 0.125 | 0.118 |
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| **1.0** | 0.500 | 0.414 |
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| **2.0** | 1.500 | 1.236 |
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| **10.0** | 9.500 | 9.049 |
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Pseudo-Huber produces slightly smaller values but follows the same qualitative behavior.
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## Choosing δ
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| δ Value | Behavior | Use Case |
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| :--- | :--- | :--- |
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| **0.1** | Quickly linear | Aggressive outlier handling |
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| **1.0** | Balanced | Standard choice |
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| **10.0** | Mostly quadratic | Near-MSE behavior |
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| **100.0** | Almost pure L2 | When outliers are rare |
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## Common Use Cases
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1. **Neural Network Training**: Smooth loss for gradient descent
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2. **Computer Vision**: Optical flow, stereo matching
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3. **Robust Regression**: When smoothness matters for optimization
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4. **Image Processing**: Edge-preserving filtering (Charbonnier)
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## Edge Cases
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* **Perfect Predictions**: Returns exactly 0
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* **NaN Handling**: Uses last valid value substitution
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* **Single Input**: Not supported (requires two series)
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* **δ = 0**: Invalid (division by zero)
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* **Large Errors**: Numerically stable (no overflow)
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## Related Indicators
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* [Huber](../huber/Huber.md) - Huber Loss (piecewise, with kink)
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* [LogCosh](../logcosh/LogCosh.md) - Log-Cosh Loss (different smooth approximation)
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* [MAE](../mae/Mae.md) - Mean Absolute Error (pure L1)
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* [MSE](../mse/Mse.md) - Mean Squared Error (pure L2) |