# Feast Quality Model: Full Parameter Specification (simon note: AI generated)
## Executive Summary
This document presents two piecewise linear models for predicting feast quality based on the balance of spicy, sweet, and bulk characteristics. Both models use learned per-dish weights for each dimension, with penalties applied when totals fall outside optimal zones.
**Key Finding:** The “spicy peak” model shows marginal RMSE improvement and is **marginally significant** (p=0.034) by F-test, but **BIC favors the simpler flat model**. Given the mixed evidence, we recommend targeting the **center of the flat model’s optimal zones** for maximum robustness.
---
## Model Comparison
| Metric | All-Flat Model | Spicy Peak Model |
|--------|----------------|------------------|
| RMSE | 1.9635 | 1.9583 |
| Parameters | 64 | 67 |
| AIC | 2440.9 | 2437.9 |
| BIC | 2789.5 | 2802.8 |
**Statistical Test (F-test):**
- F-statistic: 2.892
- Degrees of freedom: (3, 1647)
- **p-value: 0.034**
**Interpretation:** The peak model’s improvement is marginally significant (p=0.034), but BIC favors the simpler flat model (delta = −13.3). This mixed evidence suggests caution in adopting the more complex model.
---
## Model 1: All-Flat (3-Piece Piecewise Linear)
### Structure
For each dimension (spicy, sweet, bulk):
- **Below optimal zone:** Linear penalty with slope
- **In optimal zone:** No penalty (flat)
- **Above optimal zone:** Linear penalty with slope
1. **Peak model has mixed evidence:** The F-test p-value of 0.034 is below 0.05, but BIC favors the flat model. The evidence is not conclusive.
2. **High degeneracy:** 119 different feasts fall within all optimal zones of the flat model. The center-targeting approach provides a principled way to select among them.
3. **Weights are continuous:** The per-dish weights are fitted continuous values. Integer or simple-fraction approximations may exist but are not explored here.
4. **Variance heterogeneity:** Residual variance increases when above optimal thresholds (especially for bulk). This is not captured in the point predictions above.
5. **Model uncertainty:** All predictions have associated uncertainty (~2.0 quality points RMSE). The differences between top feasts are often smaller than this uncertainty.
---
## Summary
**Use the center-targeted recommendation** for maximum robustness:
# Feast Quality Model: Full Parameter Specification (simon note: AI generated)
## Executive Summary
This document presents two piecewise linear models for predicting feast quality based on the balance of spicy, sweet, and bulk characteristics. Both models use learned per-dish weights for each dimension, with penalties applied when totals fall outside optimal zones.
**Key Finding:** The “spicy peak” model shows marginal RMSE improvement and is **marginally significant** (p=0.034) by F-test, but **BIC favors the simpler flat model**. Given the mixed evidence, we recommend targeting the **center of the flat model’s optimal zones** for maximum robustness.
---
## Model Comparison
| Metric | All-Flat Model | Spicy Peak Model |
|--------|----------------|------------------|
| RMSE | 1.9635 | 1.9583 |
| Parameters | 64 | 67 |
| AIC | 2440.9 | 2437.9 |
| BIC | 2789.5 | 2802.8 |
**Statistical Test (F-test):**
- F-statistic: 2.892
- Degrees of freedom: (3, 1647)
- **p-value: 0.034**
**Interpretation:** The peak model’s improvement is marginally significant (p=0.034), but BIC favors the simpler flat model (delta = −13.3). This mixed evidence suggests caution in adopting the more complex model.
---
## Model 1: All-Flat (3-Piece Piecewise Linear)
### Structure
For each dimension (spicy, sweet, bulk):
- **Below optimal zone:** Linear penalty with slope
- **In optimal zone:** No penalty (flat)
- **Above optimal zone:** Linear penalty with slope
```
Quality = Intercept + SpicyEffect + SweetEffect + BulkEffect
SpicyEffect = -slope_low * max(0, t1 - total_spicy) - slope_high * max(0, total_spicy—t2)
SweetEffect = -slope_low * max(0, t1 - total_sweet) - slope_high * max(0, total_sweet—t2)
BulkEffect = -slope_low * max(0, t1 - total_bulk) - slope_high * max(0, total_bulk—t2)
```
### Optimal Zones and Penalties
| Dimension | Lower Threshold (t1) | Upper Threshold (t2) | Slope Below | Slope Above |
|-----------|---------------------|---------------------|-------------|-------------|
| Spicy | 3.213 | 3.558 | 1.573 | 0.899 |
| Sweet | 3.068 | 4.461 | 1.593 | 1.273 |
| Bulk | 7.136 | 8.677 | 0.877 | 1.028 |
**Intercept:** 16.742
### Per-Dish Weights
| Dish | Spicy Weight | Sweet Weight | Bulk Weight |
|------|-------------|--------------|-------------|
| Ambrosial Applesauce | 0.000 | 1.456 | 0.000 |
| BBQ Basilisk Brisket | 0.821 | 0.831 | 1.404 |
| Chili Con Chimera | 1.304 | 0.078 | 1.502 |
| Displacer Dumplings | 0.052 | 0.075 | 0.086 |
| Ettin Eye Eclairs | 0.124 | 2.815 | 0.371 |
| Fiery Formian Fritters | 1.337 | 0.000 | 0.918 |
| Geometric Gelatinous Gateau | 0.176 | 2.078 | 0.583 |
| Honeyed Hydra Hearts | 0.000 | 1.537 | 0.851 |
| Killer Kraken Kebabs | 2.146 | 0.054 | 2.434 |
| Mighty Minotaur Meatballs | 0.069 | 0.181 | 2.233 |
| Opulent Owlbear Omelette | 0.008 | 0.016 | 1.788 |
| Pegasus Pinion Pudding | 0.124 | 0.621 | 0.786 |
| Roc Roasted Rare | 0.004 | 0.290 | 3.493 |
| Scorching Salamander Stew | 2.138 | 0.035 | 1.520 |
| Troll Tenderloin Tartare | 0.062 | 0.000 | 0.960 |
| Vicious Vampire Vindaloo | 3.015 | 0.148 | 1.570 |
| Wyvern Wing Wraps | 0.657 | 0.181 | 0.749 |
---
## Model 2: Spicy Peak (4-Piece for Spicy, 3-Piece for Sweet/Bulk)
### Structure
For spicy: 4-piece continuous function with potential peak
- Piece 1: spicy < t1 → slope s1
- Piece 2: t1 ⇐ spicy < t2 → slope s2 (if positive, quality increases)
- Piece 3: t2 ⇐ spicy < t3 → slope s3 (if negative, quality decreases = peak at t2)
- Piece 4: spicy >= t3 → slope s4
For sweet/bulk: Same 3-piece flat structure as Model 1.
### Spicy Parameters (4-Piece)
| Parameter | Value | Interpretation |
|-----------|-------|----------------|
| t1 (breakpoint 1) | 1.016 | Below this: slope s1 |
| t2 (breakpoint 2) | 2.697 | **Peak location** (if s2>0, s3<0) |
| t3 (breakpoint 3) | 5.872 | Above this: slope s4 |
| s1 (slope 1) | 2.043 | Slope for spicy < t1 |
| s2 (slope 2) | 1.856 | Slope for t1 ⇐ spicy < t2 |
| s3 (slope 3) | −1.253 | Slope for t2 ⇐ spicy < t3 |
| s4 (slope 4) | −0.268 | Slope for spicy >= t3 |
**Peak Location:** ~2.70 (where slope changes from positive to negative)
### Sweet/Bulk Parameters (3-Piece)
| Dimension | Lower Threshold | Upper Threshold | Slope Below | Slope Above |
|-----------|-----------------|-----------------|-------------|-------------|
| Sweet | 3.028 | 4.591 | 1.600 | 1.299 |
| Bulk | 6.692 | 8.486 | 0.980 | 1.126 |
**Intercept:** 13.752
### Per-Dish Weights (Peak Model)
| Dish | Spicy Weight | Sweet Weight | Bulk Weight |
|------|-------------|--------------|-------------|
| Ambrosial Applesauce | 0.000 | 1.452 | 0.000 |
| BBQ Basilisk Brisket | 0.642 | 0.871 | 1.395 |
| Chili Con Chimera | 1.020 | 0.071 | 1.427 |
| Displacer Dumplings | 0.036 | 0.062 | 0.089 |
| Ettin Eye Eclairs | 0.070 | 2.904 | 0.410 |
| Fiery Formian Fritters | 1.079 | 0.000 | 0.973 |
| Geometric Gelatinous Gateau | 0.139 | 2.104 | 0.522 |
| Honeyed Hydra Hearts | 0.000 | 1.515 | 0.745 |
| Killer Kraken Kebabs | 1.693 | 0.042 | 2.322 |
| Mighty Minotaur Meatballs | 0.000 | 0.182 | 2.228 |
| Opulent Owlbear Omelette | 0.052 | 0.010 | 1.673 |
| Pegasus Pinion Pudding | 0.125 | 0.592 | 0.653 |
| Roc Roasted Rare | 0.000 | 0.279 | 3.229 |
| Scorching Salamander Stew | 1.691 | 0.041 | 1.614 |
| Troll Tenderloin Tartare | 0.071 | 0.000 | 0.914 |
| Vicious Vampire Vindaloo | 2.361 | 0.163 | 1.545 |
| Wyvern Wing Wraps | 0.518 | 0.157 | 0.711 |
---
## Optimal Feast Recommendations
### Degeneracy Analysis
Both models have **degenerate optimal solutions** - multiple feasts achieve the same (or nearly the same) predicted quality:
| Model | Best Score | # Degenerate Solutions |
|-------|------------|------------------------|
| Flat Model | 16.74 | 135 |
| Peak Model | 16.87 | 6 |
| In All Flat Zones | 16.74 | 119 |
To break this degeneracy, we select the feast **closest to the center** of each optimal zone, providing maximum robustness against model uncertainty.
### Target Centers (Flat Model)
| Dimension | Optimal Zone | Center | Width |
|-----------|--------------|--------|-------|
| Spicy | [3.21, 3.56] | 3.386 | 0.344 |
| Sweet | [3.07, 4.46] | 3.764 | 1.393 |
| Bulk | [7.14, 8.68] | 7.907 | 1.540 |
---
## RECOMMENDED FEAST (Center-Targeted)
This feast is in all optimal zones AND closest to the center of each zone.
| Dimension | Value | Target Center | Deviation |
|-----------|-------|---------------|-----------|
| Spicy | 3.388 | 3.386 | 0.003 |
| Sweet | 3.675 | 3.764 | 0.089 |
| Bulk | 8.041 | 7.907 | 0.135 |
**Predicted Quality:** 16.74 (flat model), 16.80 (peak model)
**Normalized Distance from Center:** 0.160
**Dishes (7):**
- BBQ Basilisk Brisket
- Displacer Dumplings
- Geometric Gelatinous Gateau
- Killer Kraken Kebabs
- Opulent Owlbear Omelette
- Pegasus Pinion Pudding
- Troll Tenderloin Tartare
---
## Alternative Recommendations
### Best by Flat Model
*(135 feasts tied at this score)*
**Example:** Predicted Quality: 16.74
- Spicy: 3.388, Sweet: 3.675, Bulk: 8.041
- Dishes (7): BBQ Basilisk Brisket, Displacer Dumplings, Geometric Gelatinous Gateau, Killer Kraken Kebabs, Opulent Owlbear Omelette, Pegasus Pinion Pudding, Troll Tenderloin Tartare
### Best by Peak Model
*(6 feasts tied at this score)*
**Example:** Predicted Quality: 16.87
- Spicy: 2.695, Sweet: 3.138, Bulk: 6.864
- Dishes (5): Geometric Gelatinous Gateau, Pegasus Pinion Pudding, Roc Roasted Rare, Troll Tenderloin Tartare, Vicious Vampire Vindaloo
---
## Caveats and Limitations
1. **Peak model has mixed evidence:** The F-test p-value of 0.034 is below 0.05, but BIC favors the flat model. The evidence is not conclusive.
2. **High degeneracy:** 119 different feasts fall within all optimal zones of the flat model. The center-targeting approach provides a principled way to select among them.
3. **Weights are continuous:** The per-dish weights are fitted continuous values. Integer or simple-fraction approximations may exist but are not explored here.
4. **Variance heterogeneity:** Residual variance increases when above optimal thresholds (especially for bulk). This is not captured in the point predictions above.
5. **Model uncertainty:** All predictions have associated uncertainty (~2.0 quality points RMSE). The differences between top feasts are often smaller than this uncertainty.
---
## Summary
**Use the center-targeted recommendation** for maximum robustness:
**BBQ Basilisk Brisket, Displacer Dumplings, Geometric Gelatinous Gateau, Killer Kraken Kebabs, Opulent Owlbear Omelette, Pegasus Pinion Pudding, Troll Tenderloin Tartare**
This feast:
- Falls within all optimal zones of the flat model
- Is closest to the center of each zone (normalized distance: 0.160)
- Scores well on both flat (16.74) and peak (16.80) models