Vinyl Playback Optimization

Summary

This experiment investigates vinyl playback optimization. Box-Behnken design to maximize audio fidelity and minimize surface noise by tuning tracking force, anti-skate, and cartridge alignment.

The design varies 3 factors: tracking force g (g), ranging from 1.2 to 2.2, anti skate g (g), ranging from 0.5 to 2.0, and overhang mm (mm), ranging from 14 to 18. The goal is to optimize 2 responses: fidelity score (pts) (maximize) and surface noise (dB) (minimize). Fixed conditions held constant across all runs include turntable = belt_drive, cartridge type = moving_magnet.

A Box-Behnken design was chosen because it efficiently fits quadratic models with 3 continuous factors while avoiding extreme corner combinations — requiring only 15 runs instead of the 8 needed for a full factorial at two levels.

Quadratic response surface models were fitted to capture potential curvature and factor interactions. The RSM contour plots below visualize how pairs of factors jointly affect each response.

Key Findings

For fidelity score, the most influential factors were anti skate g (62.3%), tracking force g (22.0%), overhang mm (15.7%). The best observed value was 7.3 (at tracking force g = 1.7, anti skate g = 2, overhang mm = 18).

For surface noise, the most influential factors were overhang mm (37.8%), anti skate g (33.8%), tracking force g (28.4%). The best observed value was -57.0 (at tracking force g = 1.2, anti skate g = 0.5, overhang mm = 16).

Recommended Next Steps

Run confirmation experiments at the predicted optimal settings to validate the model.
Consider whether any fixed factors should be varied in a future study.

Experimental Setup

Factors

Factor	Low	High	Unit
`tracking_force_g`	1.2	2.2	g
`anti_skate_g`	0.5	2.0	g
`overhang_mm`	14	18	mm

Fixed: turntable = belt_drive, cartridge_type = moving_magnet

Responses

Response	Direction	Unit
`fidelity_score`	↑ maximize	pts
`surface_noise`	↓ minimize	dB

Configuration

use_cases/159_vinyl_playback/config.json

{
  "metadata": {
    "name": "Vinyl Playback Optimization",
    "description": "Box-Behnken design to maximize audio fidelity and minimize surface noise by tuning tracking force, anti-skate, and cartridge alignment"
  },
  "factors": [
    {
      "name": "tracking_force_g",
      "levels": [
        "1.2",
        "2.2"
      ],
      "type": "continuous",
      "unit": "g"
    },
    {
      "name": "anti_skate_g",
      "levels": [
        "0.5",
        "2.0"
      ],
      "type": "continuous",
      "unit": "g"
    },
    {
      "name": "overhang_mm",
      "levels": [
        "14",
        "18"
      ],
      "type": "continuous",
      "unit": "mm"
    }
  ],
  "fixed_factors": {
    "turntable": "belt_drive",
    "cartridge_type": "moving_magnet"
  },
  "responses": [
    {
      "name": "fidelity_score",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "surface_noise",
      "optimize": "minimize",
      "unit": "dB"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/159_vinyl_playback/sim.sh"
  }
}

Experimental Matrix

The Box-Behnken Design produces 15 runs. Each row is one experiment with specific factor settings.

Run	`tracking_force_g`	`anti_skate_g`	`overhang_mm`
1	1.7	0.5	14
2	1.7	1.25	16
3	2.2	1.25	18
4	2.2	1.25	14
5	1.7	1.25	16
6	1.7	1.25	16
7	1.2	1.25	18
8	2.2	0.5	16
9	1.7	0.5	18
10	2.2	2	16
11	1.2	1.25	14
12	1.7	2	18
13	1.2	0.5	16
14	1.2	2	16
15	1.7	2	14

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/159_vinyl_playback/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/159_vinyl_playback/config.json \
    --output use_cases/159_vinyl_playback/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/159_vinyl_playback/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/159_vinyl_playback/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/159_vinyl_playback/config.json

6

Multi-objective optimization

With 2 competing responses, use --multi to find the best compromise via Derringer–Suich desirability.

Terminal

$ doe optimize --config use_cases/159_vinyl_playback/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/159_vinyl_playback/config.json \
    --output use_cases/159_vinyl_playback/results/report.html

Features Exercised

Feature	Value
Design type	`box_behnken`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	2 (fidelity_score ↑, surface_noise ↓)
Total runs	15

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: fidelity_score

Top factors: anti_skate_g (62.3%), tracking_force_g (22.0%), overhang_mm (15.7%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
tracking_force_g	2	0.4172	0.2086	0.288	0.7569
anti_skate_g	2	3.2533	1.6266	2.249	0.1679
overhang_mm	2	0.2775	0.1388	0.192	0.8291
Lack	of	Fit	6	5.2947	0.8824
Pure	Error	2	1.4467
Error	8	6.7413	0.7233
Total	14	10.6893	0.7635

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for fidelity_score

Half-Normal Plot of Effects

Model Diagnostics

Response: surface_noise

Top factors: overhang_mm (37.8%), anti_skate_g (33.8%), tracking_force_g (28.4%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
tracking_force_g	2	1.4690	0.7345	0.184	0.8357
anti_skate_g	2	2.3262	1.1631	0.291	0.7553
overhang_mm	2	2.0048	1.0024	0.251	0.7842
Lack	of	Fit	6	27.1333	4.5222
Pure	Error	2	8.0000
Error	8	35.1333	4.0000
Total	14	40.9333	2.9238

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Normal probability plot for surface_noise

Half-Normal Plot of Effects

Model Diagnostics

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

fidelity score anti skate g vs overhang mm

fidelity score tracking force g vs anti skate g

fidelity score tracking force g vs overhang mm

surface noise anti skate g vs overhang mm

surface noise tracking force g vs anti skate g

surface noise tracking force g vs overhang mm

Multi-Objective Optimization

When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.

Overall Desirability

D = 0.9784

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`fidelity_score`	1.5	1.0000	7.68 1.0000 7.68 pts	↑
`surface_noise`	1.0	0.9470	-56.96 0.9470 -56.96 dB	↓

Recommended Settings

Factor	Value
`tracking_force_g`	1.2 g
`anti_skate_g`	2 g
`overhang_mm`	18 mm

Source: from RSM model prediction

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`surface_noise`	-56.96	-57.00	+0.04

Top 3 Runs by Desirability

Run	D	Factor Settings
#6	0.7879	tracking_force_g=1.2, anti_skate_g=1.25, overhang_mm=18
#2	0.7711	tracking_force_g=2.2, anti_skate_g=0.5, overhang_mm=16

Model Quality

Response	R²	Type
`surface_noise`	0.7333	quadratic

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.9784

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
fidelity_score               1.5       1.0000        7.68 pts   ↑
surface_noise                1.0       0.9470      -56.96 dB   ↓

Recommended settings:
  tracking_force_g = 1.2 g
  anti_skate_g = 2 g
  overhang_mm = 18 mm
  (from RSM model prediction)

Trade-off summary:
  fidelity_score: 7.68 (best observed: 7.30, sacrifice: -0.38)
  surface_noise: -56.96 (best observed: -57.00, sacrifice: +0.04)

Model quality:
  fidelity_score: R² = 0.8226 (quadratic)
  surface_noise: R² = 0.7333 (quadratic)

Top 3 observed runs by overall desirability:
  1. Run #5 (D=0.8395): tracking_force_g=2.2, anti_skate_g=1.25, overhang_mm=14
  2. Run #6 (D=0.7879): tracking_force_g=1.2, anti_skate_g=1.25, overhang_mm=18
  3. Run #2 (D=0.7711): tracking_force_g=2.2, anti_skate_g=0.5, overhang_mm=16

Full Analysis Output

doe analyze
=== Main Effects: fidelity_score ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
anti_skate_g             1.2750       0.2256            62.3%
tracking_force_g         0.4500       0.2256            22.0%
overhang_mm              0.3214       0.2256            15.7%

=== ANOVA Table: fidelity_score ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
tracking_force_g             2       0.4172       0.2086      0.288     0.7569
anti_skate_g                 2       3.2533       1.6266      2.249     0.1679
overhang_mm                  2       0.2775       0.1388      0.192     0.8291
Lack of Fit                  6       5.2947       0.8824      1.220     0.5155
Pure Error                   2       1.4467       0.7233
Error                        8       6.7413       0.7233
Total                       14      10.6893       0.7635

=== Summary Statistics: fidelity_score ===

tracking_force_g:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.2                 4     6.2250     1.0145     5.0000     7.2000
  1.7                 7     6.0571     0.7547     5.1000     7.1000
  2.2                 4     5.7750     1.1206     4.6000     7.3000

anti_skate_g:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.5                 4     5.4000     0.2828     5.0000     5.6000
  1.25                7     6.0143     0.9856     4.6000     7.2000
  2                   4     6.6750     0.6752     5.8000     7.3000

overhang_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  14                  4     5.9750     1.1266     4.6000     7.2000
  16                  7     5.9286     0.8480     5.0000     7.3000
  18                  4     6.2500     0.8737     5.4000     7.1000

=== Main Effects: surface_noise ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
overhang_mm              1.0000       0.4415            37.8%
anti_skate_g             0.8929       0.4415            33.8%
tracking_force_g         0.7500       0.4415            28.4%

=== ANOVA Table: surface_noise ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
tracking_force_g             2       1.4690       0.7345      0.184     0.8357
anti_skate_g                 2       2.3262       1.1631      0.291     0.7553
overhang_mm                  2       2.0048       1.0024      0.251     0.7842
Lack of Fit                  6      27.1333       4.5222      1.131     0.5394
Pure Error                   2       8.0000       4.0000
Error                        8      35.1333       4.0000
Total                       14      40.9333       2.9238

=== Summary Statistics: surface_noise ===

tracking_force_g:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.2                 4   -54.5000     1.7321   -56.0000   -52.0000
  1.7                 7   -54.4286     1.9881   -57.0000   -52.0000
  2.2                 4   -53.7500     1.5000   -55.0000   -52.0000

anti_skate_g:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  0.5                 4   -54.7500     2.2174   -57.0000   -52.0000
  1.25                7   -53.8571     1.5736   -56.0000   -52.0000
  2                   4   -54.5000     1.7321   -56.0000   -52.0000

overhang_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  14                  4   -54.7500     2.0616   -57.0000   -52.0000
  16                  7   -54.2857     1.7043   -56.0000   -52.0000
  18                  4   -53.7500     1.7078   -56.0000   -52.0000

Optimization Recommendations

doe optimize
=== Optimization: fidelity_score ===
Direction: maximize

Best observed run: #6
  tracking_force_g = 1.7
  anti_skate_g = 2
  overhang_mm = 18
  Value: 7.3

RSM Model (linear, R² = 0.0138, Adj R² = -0.2552):
  Coefficients:
    intercept                      +6.0267
    tracking_force_g               -0.0125
    anti_skate_g                   +0.0750
    overhang_mm                    +0.1125

RSM Model (quadratic, R² = 0.3508, Adj R² = -0.8177):
  Coefficients:
    intercept                      +5.7333
    tracking_force_g               -0.0125
    anti_skate_g                   +0.0750
    overhang_mm                    +0.1125
    tracking_force_g*anti_skate_g  -0.1750
    tracking_force_g*overhang_mm   -0.3500
    anti_skate_g*overhang_mm       +0.6750
    tracking_force_g^2             -0.1417
    anti_skate_g^2                 +0.1833
    overhang_mm^2                  +0.5083

  Curvature analysis:
    overhang_mm                    coef=+0.5083  convex (has a minimum)
    anti_skate_g                   coef=+0.1833  convex (has a minimum)
    tracking_force_g               coef=-0.1417  concave (has a maximum)

  Notable interactions:
    anti_skate_g*overhang_mm       coef=+0.6750  (synergistic)
    tracking_force_g*overhang_mm   coef=-0.3500  (antagonistic)

  Predicted optimum (from linear model, at observed points):
    tracking_force_g = 1.7
    anti_skate_g = 2
    overhang_mm = 18
    Predicted value: 6.2142

  Surface optimum (via L-BFGS-B, linear model):
    tracking_force_g = 1.2
    anti_skate_g = 2
    overhang_mm = 18
    Predicted value: 6.2267

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. overhang_mm  (effect: 0.6, contribution: 58.6%)
  2. anti_skate_g  (effect: 0.2, contribution: 22.0%)
  3. tracking_force_g  (effect: 0.2, contribution: 19.3%)

=== Optimization: surface_noise ===
Direction: minimize

Best observed run: #7
  tracking_force_g = 1.2
  anti_skate_g = 0.5
  overhang_mm = 16
  Value: -57.0

RSM Model (linear, R² = 0.0550, Adj R² = -0.2028):
  Coefficients:
    intercept                      -54.2667
    tracking_force_g               +0.3750
    anti_skate_g                   +0.0000
    overhang_mm                    +0.3750

RSM Model (quadratic, R² = 0.3546, Adj R² = -0.8070):
  Coefficients:
    intercept                      -54.6667
    tracking_force_g               +0.3750
    anti_skate_g                   +0.0000
    overhang_mm                    +0.3750
    tracking_force_g*anti_skate_g  +0.2500
    tracking_force_g*overhang_mm   +1.5000
    anti_skate_g*overhang_mm       +0.2500
    tracking_force_g^2             -0.1667
    anti_skate_g^2                 +0.0833
    overhang_mm^2                  +0.8333

  Curvature analysis:
    overhang_mm                    coef=+0.8333  convex (has a minimum)
    tracking_force_g               coef=-0.1667  concave (has a maximum)
    anti_skate_g                   coef=+0.0833  negligible curvature

  Notable interactions:
    tracking_force_g*overhang_mm   coef=+1.5000  (synergistic)

  Predicted optimum (from linear model, at observed points):
    tracking_force_g = 2.2
    anti_skate_g = 1.25
    overhang_mm = 18
    Predicted value: -53.5167

  Surface optimum (via L-BFGS-B, linear model):
    tracking_force_g = 1.2
    anti_skate_g = 1.15832
    overhang_mm = 14
    Predicted value: -55.0167

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. overhang_mm  (effect: 1.2, contribution: 60.7%)
  2. tracking_force_g  (effect: 0.8, contribution: 37.5%)
  3. anti_skate_g  (effect: 0.0, contribution: 1.8%)