Changed the piecewise regression example using bambi

README.md

@@ -125,60 +125,95 @@ After this, we can evaluate the model as before.
 
 
 ### Piecewise Regression
+Let's walk through an example of performing a piecewise regression using Bambi.
+
+```python
+
+# Create example data
+np.random.seed(42)
+x = np.linspace(0, 10, 100)
+y = np.where(x < 5, 2*x + np.random.normal(0, 1, 100), -3*x + 20 + np.random.normal(0, 1, 100))
+data = pd.DataFrame({'x': x, 'y': y})
+
+# Plot the data
+plt.scatter(data['x'], data['y'])
+plt.xlabel('x')
+plt.ylabel('y')
+plt.show()
+
+# Define the model with a spline term for 'x'
+model = bmb.Model('y ~ 0 + x + (x > 5) + (x - 5) * (x > 5)', data)
+
+# Fit the model
+results = model.fit(draws=2000, cores=2)
+
+# Summarize the results
+print(results.summary())
+
+# Predict and plot the fitted values
+x_pred = np.linspace(0, 10, 100)
+data_pred = pd.DataFrame({'x': x_pred})
+y_pred = model.predict(idata=results, data=data_pred).posterior_predictive.mean(axis=0)
+
+plt.scatter(data['x'], data['y'], label='Data')
+plt.plot(x_pred, y_pred, color='red', label='Fitted Piecewise Regression')
+plt.xlabel('x')
+plt.ylabel('y')
+plt.legend()
+plt.show()
+```
+
+This should give you a piecewise regression model fitted to your data using Bambi. The plot will show the original data points and the fitted piecewise regression line.
+
+# Potential
+
+we'll demonstrate the concept of potential in a probabilistic model using a likelihood function. In this case, we'll use a Gaussian distribution (Normal distribution) to represent the likelihood and add a potential function to constrain the model.
+
+```python
+def likelihood(x, mu, sigma):
+    """
+    Gaussian likelihood function
+    """
+    return (1 / (np.sqrt(2 * np.pi) * sigma)) * np.exp(-0.5 * ((x - mu) / sigma) ** 2)
+
+def potential(x):
+    """
+    Potential function to constrain the model
+    """
+    # Example potential: Quadratic potential centered at x = 2
+    return np.exp(-0.5 * (x - 2) ** 2)
+
+def posterior(x, mu, sigma):
+    """
+    Posterior function combining likelihood and potential
+    """
+    return likelihood(x, mu, sigma) * potential(x)
+
+# Define parameters
+mu = 0
+sigma = 1
+x_values = np.linspace(-5, 5, 100)
+
+# Calculate likelihood, potential, and posterior
+likelihood_values = likelihood(x_values, mu, sigma)
+potential_values = potential(x_values)
+posterior_values = posterior(x_values, mu, sigma)
+
+# Plot the functions
+plt.figure(figsize=(10, 6))
+plt.plot(x_values, likelihood_values, label='Likelihood', linestyle='--')
+plt.plot(x_values, potential_values, label='Potential', linestyle=':')
+plt.plot(x_values, posterior_values, label='Posterior')
+plt.xlabel('x')
+plt.ylabel('Probability Density')
+plt.title('Likelihood, Potential, and Posterior')
+plt.legend()
+plt.show()
+```
+
+This example visually demonstrates how adding a potential function can constrain the model and influence the resulting distribution.
 
-We'll use synthetic data to demonstrate the piecewise regression.
 
-import numpy as np
-import pandas as pd
-import statsmodels.formula.api as smf
-
-# Generate synthetic data
-np.random.seed(0)
-x = np.linspace(0, 30, 100)
-y = 3 + 0.5 * x + 1.5 * truncate(x, 10) + 2.0 * truncate(x, 20) + np.random.normal(0, 1, 100)
-
-# Create a DataFrame
-df = pd.DataFrame({'x': x, 'response': y})
-
-# Define the truncate function
-def truncate(x, l):
-    return (x - l) * (x >= l)
-
-# Fit the piecewise regression model
-formula = "response ~ x + truncate(x, 10) + truncate(x, 20)"
-model = smf.ols(formula, data=df).fit()
-
-# Display the model summary
-print(model.summary())
-
-The output will include the summary of the fitted regression model, showing coefficients for each term:
-
-                            OLS Regression Results
-==============================================================================
-Dep. Variable:               response   R-squared:                       0.963
-Model:                            OLS   Adj. R-squared:                  0.962
-Method:                 Least Squares   F-statistic:                     838.8
-Date:                Thu, 23 Jan 2025   Prob (F-statistic):           7.62e-71
-Time:                        22:42:00   Log-Likelihood:                -136.22
-No. Observations:                 100   AIC:                             280.4
-Df Residuals:                      96   BIC:                             290.9
-Df Model:                           3
-Covariance Type:            nonrobust
-==============================================================================
-                 coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------
-Intercept      3.1241      0.150     20.842      0.000       2.826       3.422
-x              0.4921      0.014     34.274      0.000       0.465       0.519
-truncate(10)   1.4969      0.024     61.910      0.000       1.449       1.544
-truncate(20)   1.9999      0.018    110.199      0.000       1.964       2.036
-==============================================================================
-Omnibus:                        0.385   Durbin-Watson:                   2.227
-Prob(Omnibus):                  0.825   Jarque-Bera (JB):                0.509
-Skew:                          -0.046   Prob(JB):                        0.775
-Kurtosis:                       2.669   Cond. No.                         47.4
-==============================================================================
-
-This summary shows the coefficients for the intercept, x, truncate(x, 10), and truncate(x, 20) terms, along with their statistical significance.
 
 ### More