============================================================
REGRESSION FORECASTING ANALYSIS

STEP 1: DATA EXPLORATION

Observations: 8 weeks
Mean demand: 59.00 units
Std deviation: 10.03 units
Range: 45 - 72 units
Missing values: 0

Weekly Data:
Week Demand
1 45
2 52
3 48
4 61
5 58
6 67
7 72
8 69

Correlation (Week vs Demand): 0.9363

STEP 2: DATA QUALITY VALIDATION

Sample size check (n >= 10): FAIL (n = 8)
WARNING: Only 8 observations. Minimum recommended is 10-12.
Results will have wider confidence intervals and lower reliability.
Variance check: PASS
Missing values check: PASS

Outlier Detection (IQR method):
No outliers detected

STEP 3: REGRESSION MODEL FIT

R² = 0.8767 (explains 87.7% of variance)
Adjusted R² = 0.8561
F-statistic: 42.6449 (p = 0.0006)

Coefficients:
Intercept: 41.75 units
Slope: 3.83 units/week (p = 0.0006)

Interpretation:
Demand increases by 3.83 units per week on average.
Statistical significance: YES (p < 0.05)

STEP 4: ASSUMPTION VALIDATION

1. Linearity: OK (visual inspection via residual plot)
2. Homoscedasticity: OK (Breusch-Pagan p = 0.7621)
3. Normality: OK (Shapiro-Wilk p = 0.2778)
4. Independence: CAUTION (Durbin-Watson = 3.188)

NOTE: Time series data may show autocorrelation. Monitor if adding more data.

STEP 6: FORECASTS

Week 9 Forecast:
Point estimate: 76.2 units
95% Confidence Interval: [69.0, 83.5]
95% Prediction Interval: [64.4, 88.1]

Week 10 Forecast:
Point estimate: 80.1 units
95% Confidence Interval: [71.5, 88.6]
95% Prediction Interval: [67.4, 92.7]

EXTRAPOLATION WARNING:
Training data range: Weeks 1-8
Forecasting: Weeks 9-10
Predictions beyond training data carry higher uncertainty.

============================================================
SUMMARY AND RECOMMENDATIONS

MODEL QUALITY ASSESSMENT:

• R² = 0.8767: Model explains 87.7% of demand variation
  → STRONG fit
• Statistical significance: p = 0.0006
  → Relationship is statistically significant
• Trend: 3.83 units/week increase

ASSUMPTION STATUS:

• 2/3 core assumptions validated
  → Some violations detected, but with small samples these tests may be oversensitive

SMALL SAMPLE SIZE CONCERNS:

• Current sample: 8 observations
• Recommended minimum: 10-12 observations
• IMPACT:
  → Wider confidence/prediction intervals
  → Assumption tests less reliable
  → Model coefficients less stable
  → Higher risk of overfitting
• MITIGATION:
  → Use prediction intervals (not just point estimates)
  → Apply conservative planning buffers
  → Update model as new data arrives

LINEAR MODEL APPROPRIATENESS:

• Correlation: 0.9363 (STRONG linear relationship)
• Residual pattern: Random scatter (good)
• Linear AIC: 45.78
• Quadratic AIC: 47.60
  → Linear model is appropriate (lower AIC)

BUSINESS RECOMMENDATIONS:

1. FORECASTS FOR PLANNING:
   Week 9: Use ~76 units
   (range: 64-88)
   Week 10: Use ~80 units
   (range: 67-93)

2. UNCERTAINTY MANAGEMENT:

   • Prediction uncertainty: ±31% relative range
   • Recommendation: Build in 15-20% safety stock buffer
   • Monitor actual vs forecast to refine model

3. DATA COLLECTION PRIORITY:

   • CRITICAL: Collect 2-4 more weeks of data ASAP
   • Target: Reach n=12-15 for reliable inference
   • Refit model with each new data point
   • Watch for trend changes or seasonality

4. MODEL LIMITATIONS:

   • CORRELATION ≠ CAUSATION: Model shows trend, not root causes
   • Assumes linear growth continues unchanged
   • Small sample = high sensitivity to new data points
   • Cannot detect seasonality or structural breaks with n=8

5. ALTERNATIVE APPROACH (Given Small Sample):

   • Simple moving average (last 4 weeks): 66.5 units
   • Regression forecast (week 9): 76.2 units
   • Difference: 9.7 units
   • Methods diverge → Use range between them for planning

============================================================
CONCLUSION

The linear regression model shows a statistically significant upward trend
of 3.83 units/week with R² = 0.8767. However, the small sample
size (n=8) means forecasts carry substantial uncertainty. Use the prediction
intervals for risk management, not just point estimates. Prioritize collecting
2-4 more weeks of data to improve model reliability. For immediate planning,
apply a 15-20% buffer to the point forecasts.