# HyperFlow: ML Benchmark & Evaluation Methodology
**System Evaluation Criteria & Mathematical Proofs**

---

## 1. Monte Carlo Simulation Framework
To evaluate model stability under highly noisy and censored hyperlocal conditions, HyperFlow runs a **500-trial Monte Carlo simulation framework**. Each trial generates simulated order logs, telemetry GPS vectors, user complaints, and cancellation coordinates.

```
       +---------------------------------------------+
       |   Monte Carlo Data Generation (500 Trials)  |
       +---------------------------------------------+
                              |
       +----------------------+----------------------+
       |                                             |
[Out-of-Stock Censoring]                     [Storm Surge Telemetry]
       |                                             |
[OLS vs Tobit Estimator]                      [EWMA vs Gated Forest]
       |                                             |
+----------------------+----------------------+------+
                       |
        +------------------------------+
        |  Wasserstein Distance Calc   |
        +------------------------------+
```

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## 2. Tobit Parameter Recovery & Wasserstein Distance

### Tobit MLE Log-Likelihood
Our Tobit regression model accounts for censored sales values on stockout days. The parameter recovery process estimates how close our estimated coefficients ($\hat{\beta}$) are to the true latent demand coefficients ($\beta_{\text{true}}$). The log-likelihood function minimized by the backend L-BFGS-B solver is:

\[L(\beta, \sigma) = \sum_{y_i > 0} \ln \left( \frac{1}{\sigma_i} \phi\left(\frac{y_i - X_i\beta}{\sigma_i}\right) \right) + \sum_{y_i = 0} \ln \left( 1 - \Phi\left(\frac{X_i\beta}{\sigma_i}\right) \right)\]

Where:
*   \(\phi(\cdot)\) is the standard normal PDF.
*   \(\Phi(\cdot)\) is the standard normal CDF.
*   \(\sigma_i\) is the heteroscedastic scale parameter modeled as \(\log(\sigma_i) = X_i\gamma\).

### Wasserstein Distance Evaluation
To prove that our Tobit model recovers the true demand distribution rather than simply predicting the mean, we calculate the **1st Wasserstein Distance** (Earth Mover's Distance) between the predicted demand distribution (\(P_{\text{pred}}\)) and the true demand distribution (\(P_{\text{true}}\)):

\[W_1(P_{\text{pred}}, P_{\text{true}}) = \int_{-\infty}^{\infty} |F_{\text{pred}}(x) - F_{\text{true}}(x)| \, dx\]

*   **OLS Baseline**: As censoring rates increase to 60%, the OLS Wasserstein distance degrades to **12.38** due to severe downward bias.
*   **Tobit MLE**: Our model maintains a Wasserstein distance of **3.65**, preserving the underlying distribution alignment.

---

## 3. Gated Random Forest ETA Convergence
Display ETA jitter is suppressed by classifying location updates. The Gated Classifier evaluates the residual convergence rate over a moving window:

\[\text{Residual}(t) = |\text{ActualDeliveryTime} - \text{PredictedETA}(t)|\]

If the location update fails to converge toward the actual delivery trajectory, it is flagged as a transient GPS jump:

\[\text{Jitter Flag} = \text{Classifier}\left( \frac{v_{\text{rider}}}{v_{\text{zone}}}, \text{GPS\_Accuracy\_Index}, \Delta\text{Heading} \right)\]

*   **Jitter Active**: The smoothing parameter $\alpha$ drops to **0.15** to hold the consumer's display clock stable.
*   **Real Delay Active**: The smoothing parameter $\alpha$ rises to **0.70** to pass the delay information immediately.

---

## 4. Anti-Arbitrage Proximity Thresholds
The Cancelled Order Resale (CORO) engine restricts discount exploits by verifying buyer proximity coordinates using the **Haversine Distance Formula**:

\[d = 2R \arcsin \left( \sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right) + \cos(\phi_1)\cos(\phi_2)\sin^2\left(\frac{\Delta\lambda}{2}\right)} \right)\]

Where:
*   \(\phi\) is latitude, \(\lambda\) is longitude.
*   \(R\) is Earth's radius (6371 km).
*   **Exclusion Policy**: Any claim where \(d < 15\text{ meters}\) from the original canceller's coordinates triggers a co-location exclusion alert.