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Diagnostics & Validation

Verify regression assumptions and validate model quality.

Regression Assumptions​

1. Linearity​

Is the relationship linear?

Check: Plot actual vs. predicted

  • Linear pattern ✅
  • Curved pattern ❌ → Use polynomial terms or Ridge

2. Independence​

Are observations independent?

Check: Is data time-series?

  • No time dependency ✅
  • Time-series ❌ → Use RLS

3. Homoscedasticity​

Is variance constant across x?

SELECT
    y_actual,
    y_predicted,
    residual,
    leverage,
    cooks_distance
FROM anofox_statistics_residual_diagnostics(
    actual_values,
    predicted_values
);

Check: Residual plot

  • Random scatter ✅
  • Funneling pattern ❌ → Use WLS

4. Normality​

Are errors normally distributed?

SELECT * FROM anofox_statistics_normality_test(
    residuals,
    0.05
);

Jarque-Bera Test:

  • p > 0.05 → Normal ✅
  • p < 0.05 → Not normal ❌

5. No Multicollinearity​

Are predictors independent?

SELECT
    variable,
    vif
FROM anofox_statistics_vif(
    ARRAY['x1', 'x2', 'x3']
);

VIF Interpretation:

  • VIF < 5 → OK ✅
  • VIF 5-10 → Caution ⚠️
  • VIF > 10 → Multicollinearity ❌ → Use Ridge

Residual Diagnostics​

SELECT
    y_actual,
    y_predicted,
    residual,
    leverage,
    cooks_distance,
    is_outlier
FROM anofox_statistics_residual_diagnostics(
    actual_values,
    predicted_values,
    outlier_threshold := 3.0
);

Cook's Distance​

Measures how much removing each observation changes coefficients

  • D < 0.5 → Normal observation
  • D > 1.0 → Influential outlier → Consider removing

Leverage​

Measures distance of x from center

  • High leverage + large residual = Problem

Model Selection Criteria​

AIC (Akaike Information Criterion)​

AIC = 2p + n×ln(RSS/n)

Penalizes model complexity

BIC (Bayesian Information Criterion)​

BIC = p×ln(n) + n×ln(RSS/n)

Stronger penalty than AIC

Rule​

Lower AIC/BIC = Better model

SELECT
    model_name,
    r_squared,
    aic,
    bic
FROM model_comparison
ORDER BY aic;  -- Choose lowest AIC model

Diagnostic Workflow​

1. Fit Model
   ↓
2. Check Residuals
   - Normal? (Jarque-Bera test)
   - Homoscedastic? (plot residuals)
   - Any outliers? (Cook's distance)
   ↓
3. Check Multicollinearity
   - VIF < 10?
   ↓
4. If Issues Found
   - Outliers → Use WLS or robust regression
   - Multicollinearity → Use Ridge
   - Non-normal → Check data quality
   ↓
5. Validate on Test Data
   - Compare predictions to holdout

Next Steps​

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