# Statistical Methodology

This document describes the statistical methodology used in Realizar's benchmarks, tests, and performance claims. All methods follow established practices for reproducible computational research.

## Benchmark Methodology

### Sample Size Determination

**Rationale**: Sample sizes are chosen to achieve sufficient statistical power while maintaining practical benchmark runtimes.

**Default configuration**:
- Sample size: 100 iterations (Criterion.rs default)
- Warm-up: 50 iterations
- Measurement time: 5 seconds minimum per benchmark

**Power analysis**:
For detecting a 10% performance difference with 95% confidence (α = 0.05) and 80% power (β = 0.20), assuming coefficient of variation (CV) = 5%:

```
n = 2 × (Z_α/2 + Z_β)² × (CV/δ)²
n = 2 × (1.96 + 0.84)² × (0.05/0.10)²
n ≈ 4 samples minimum
```

We use 100 samples for robust outlier detection and distribution analysis.

### Confidence Intervals

**Method**: Bootstrap confidence intervals (percentile method)

**Reporting format**: All performance metrics reported as `[lower, mean, upper]` at 95% confidence level.

**Example output**:
```
tensor_creation/10      time:   [17.887 ns 17.966 ns 18.043 ns]
                                 ^         ^         ^
                              lower      mean      upper
                              bound    estimate    bound
```

**Interpretation**: We are 95% confident the true mean lies within the interval.

### Outlier Detection

**Method**: Modified Z-score with MAD (Median Absolute Deviation)

```rust
let median = samples.median();
let mad = samples.map(|x| (x - median).abs()).median();
let modified_z = 0.6745 * (x - median) / mad;

// Outlier if |modified_z| > 3.5
```

**Rationale**: MAD is robust to outliers themselves, unlike standard deviation.

**Action**: Outliers are reported but not automatically removed. Manual inspection required.

### Warm-up Phase

**Purpose**: Allow CPU caches, branch predictors, and JIT (if any) to reach steady state.

**Implementation**:
1. Run 50 iterations without measurement
2. Clear any accumulated state
3. Begin timed measurement

**Validation**: Warm-up sufficiency verified by checking first 10 samples are not systematically higher than remaining samples (Mann-Whitney U test, p > 0.05).

## Comparative Analysis

### Effect Size: Cohen's d

**Definition**: Standardized difference between means

```
d = (μ₁ - μ₂) / σ_pooled

where σ_pooled = √[(σ₁² + σ₂²) / 2]
```

**Interpretation**:
| Cohen's d | Effect Size |
|-----------|-------------|
| 0.2 | Small |
| 0.5 | Medium |
| 0.8 | Large |
| > 1.2 | Very Large |

**Our benchmark results**: d = 5.19 (very large effect) for Realizar vs PyTorch comparison.

### Statistical Significance Testing

**Test**: Two-sample Welch's t-test (unequal variances)

**Hypotheses**:
- H₀: μ_realizar = μ_pytorch (no difference)
- H₁: μ_realizar ≠ μ_pytorch (difference exists)

**Threshold**: α = 0.001 (99.9% confidence)

**Result**: p < 0.001, reject null hypothesis

### Multiple Comparison Correction

When comparing against multiple baselines:

**Method**: Bonferroni correction

```
α_adjusted = α / k

where k = number of comparisons
```

**Example**: Comparing against 3 frameworks (PyTorch, TensorFlow, ONNX Runtime):
```
α_adjusted = 0.001 / 3 = 0.000333
```

## Uncertainty Quantification

### Sources of Uncertainty

1. **Measurement noise**: System scheduling, cache effects, thermal throttling
2. **Sampling uncertainty**: Finite sample size
3. **Model uncertainty**: Parameter variations, initialization effects
4. **Environmental uncertainty**: Hardware variations, OS differences

### Error Propagation

For derived metrics (e.g., speedup ratio):

```
speedup = t_baseline / t_realizar

σ_speedup = speedup × √[(σ_baseline/t_baseline)² + (σ_realizar/t_realizar)²]
```

### Reporting Format

All uncertain quantities reported with:
- Point estimate (mean or median)
- Uncertainty bounds (95% CI or ±1 standard error)
- Sample size

**Example**:
```
Inference latency: 0.52 µs ± 0.02 µs (n=10,000, 95% CI)
Speedup vs PyTorch: 9.6x ± 0.4x
```

## Reproducibility Protocol

### Environment Specification

All benchmarks must document:

1. **Hardware**:
   - CPU model, cache sizes, core count
   - RAM size and speed
   - GPU model (if applicable)

2. **Software**:
   - OS and kernel version
   - Rust toolchain version (from rust-toolchain.toml)
   - Dependency versions (from Cargo.lock)

3. **Configuration**:
   - CPU governor (should be `performance`)
   - Thread count
   - NUMA settings (if applicable)

### Pre-benchmark Checklist

```bash
# 1. Set CPU governor
sudo cpupower frequency-set --governor performance

# 2. Disable turbo boost (optional, for consistency)
echo 1 | sudo tee /sys/devices/system/cpu/intel_pstate/no_turbo

# 3. Close unnecessary applications
# 4. Wait for system to reach thermal equilibrium (~5 min)
# 5. Run benchmarks
cargo bench
```

### Results Validation

Before publishing benchmark results:

1. Run 3 independent benchmark sessions
2. Check coefficient of variation < 10%
3. Verify no regressions from previous baseline
4. Document any anomalies or outliers

## Property-Based Testing Statistics

### Shrinking Strategy

When property tests fail, the test framework shrinks inputs to minimal failing cases.

**Shrinking budget**: 128 attempts default

### Coverage Metrics

| Metric | Description | Target |
|--------|-------------|--------|
| Test cases | Number of random inputs | 256 default |
| Edge cases | Proportion of boundary values | ~10% |
| Shrunk cases | Minimal failing examples found | Report all |

### Regression Testing

Failed property test cases are saved as regression tests:

```rust
#[test]
fn regression_case_001() {
    // Previously failing case, now fixed
    let input = /* minimal failing input */;
    assert!(property_holds(input));
}
```

## References

1. Box, G. E. P., Hunter, J. S., & Hunter, W. G. (2005). *Statistics for Experimenters*. Wiley.
2. Cohen, J. (1988). *Statistical Power Analysis for the Behavioral Sciences*. Lawrence Erlbaum.
3. Fleming, P. J., & Wallace, J. J. (1986). How not to lie with statistics: The correct way to summarize benchmark results. *Communications of the ACM*, 29(3), 218-221.
4. Georges, A., Buytaert, D., & Eeckhout, L. (2007). Statistically rigorous Java performance evaluation. *OOPSLA '07*.
5. Hoefler, T., & Belli, R. (2015). Scientific benchmarking of parallel computing systems. *SC '15*.

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**Document Version**: 1.0.0
**Last Updated**: 2025-12-10
**Authors**: Pragmatic AI Labs
