Complete Linear Regression | Python | Interpretation | Scikit-learn | Statsmodels
Link to colab notebook: https://colab.research.google.com/dri...
1. A systematic approach to creating a random dataset with a known relationship (y = 2x + 5 + noise)
2. Python code using both scikit-learn and statsmodels for the regression analysis
3. Detailed explanation of all output terms including:
Coefficients (slope and intercept)
Coefficient (slope): This tells us that for each unit increase in X, y increases by approximately 2 units. The coefficient matches our true value of 2 in the data generation process.
Intercept: This is the expected value of y when X equals zero. In our case, it's approximately 5, matching our true value of 5.
R-squared and adjusted R-squared
R-squared (R²): This measures how well the model explains the variation in the data. An R² of 0.976 means our model explains about 97.6% of the variation in y. The remaining 2.4% is due to random noise or other factors not included in our model.
Adjusted R-squared: Similar to R², but penalizes adding unnecessary predictors. Since we only have one predictor, these values are very close.
F-statistic and its p-value
F-statistic: Tests the overall significance of the regression model. A high value (4065) suggests the model is statistically significant.
Prob (F-statistic): The p-value associated with the F-statistic. A very small value (1.35e-81) indicates that our model is statistically significant at any reasonable significance level.
t-statistics and p-values for individual coefficients & Confidence intervals
coef: The estimated coefficient value (slope for x1, intercept for const)
std err: Standard error of the coefficient estimates. Smaller values indicate more precise estimates.
t: The t-statistic testing whether the coefficient is significantly different from zero.
P greater than |t|: The p-value associated with the t-statistic. Small values (less than 0.05) indicate statistically significant coefficients.
[0.025 0.975]: The 95% confidence interval for the coefficient. If this interval doesn't contain zero, the coefficient is statistically significant at the 5% level.
Diagnostic statistics like Durbin-Watson, Jarque-Bera, etc.
Omnibus & Jarque-Bera (JB): Tests for normality of residuals. Non-significant p-values suggest residuals are normally distributed.
Durbin-Watson: Tests for autocorrelation in residuals. Values near 2 suggest no autocorrelation.
Skew & Kurtosis: Measures of the distribution shape of residuals. Ideally, skew should be close to 0 and kurtosis close to 3 for normal distribution.
4. Residual analysis to verify model assumptions
The residual analysis helps us check our model assumptions:
The histogram should be roughly bell-shaped (normal distribution)
The residuals vs fitted plot should show no pattern (homoscedasticity)
The Q-Q plot should follow a straight line (normality of residuals)
PS: Ignore background noise
0:00 Introduction
1:25 Importing libraries
2:50 Creating a random dataset
6:30 Visualise dataset
9:45 Running model - Scikit-learn
15:33 Running model - statsmodels
17:28 Visualise result
19:53 Interpretation
31:25 Residual analysis
36:20 Closing notes
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