Kernel Ridge Regression
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'q': quit, Alt+Enter: newline
> generate 30 noisy samples from a sine function and fit them with Kernel Ridge regression. Plot the result
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<think>
The user wants to generate noisy samples from a sine function and fit them using Kernel Ridge regression. I'll use
Python libraries to accomplish this task:
1. Generate 30 samples from a sine function with added noise
2. Fit a Kernel Ridge regression model
3. Plot the original data and the fitted model
I'll need the following libraries:
- NumPy for generating the data
- Scikit-learn for Kernel Ridge regression
- Matplotlib for plotting
</think>
# Sine Function with Kernel Ridge Regression
I'll generate 30 noisy samples from a sine function and then fit them using Kernel Ridge regression. Let's break this
down into steps:
1. Generate data points from a sine function with added noise
2. Apply Kernel Ridge regression to fit the data
3. Plot the original noisy data and the fitted curve
Let's start by generating our data and fitting the model:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 fromsklearn.kernel_ridgeimportKernelRidge│
│ 4 fromsklearn.model_selectionimportGridSearchCV│
│ 5 │
│ 6 # Set random seed for reproducibility│
│ 7 np.random.seed(42)│
│ 8 │
│ 9 # Generate 30 points along x-axis from 0 to 2π│
│10 X=np.sort(np.random.uniform(0,2*np.pi,30))│
│11 # Generate sine values with noise│
│12 y=np.sin(X)+np.random.normal(0,0.3,30)│
│13 │
│14 # Reshape X for sklearn (requires 2D array)│
│15 X_reshaped=X.reshape(-1,1)│
│16 │
│17 # Create and fit Kernel Ridge model│
│18 kr=KernelRidge(kernel='rbf',alpha=0.1,gamma=10)│
│19 kr.fit(X_reshaped,y)│
│20 │
│21 # Create a fine grid for prediction│
│22 X_test=np.linspace(0,2*np.pi,1000).reshape(-1,1)│
│23 y_pred=kr.predict(X_test)│
│24 │
│25 # Plot the results│
│26 plt.figure(figsize=(10,6))│
│27 plt.scatter(X,y,color='red',label='Noisy samples')│
│28 plt.plot(X_test,np.sin(X_test),color='green',label='True sine function')│
│29 plt.plot(X_test,y_pred,color='blue',label='Kernel Ridge regression')│
│30 plt.xlabel('x')│
│31 plt.ylabel('y')│
│32 plt.title('Kernel Ridge Regression on Noisy Sine Function')│
│33 plt.legend()│
│34 plt.grid(True)│
│35 plt.show()│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=193, thinking=93, output=657, cache_write=2.14k, cache_read=0; Accumulated costs: 0.018
USD
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Execution result ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
<Figure size 1000x600 with 1 Axes>
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
The plot shows three key elements:
1. **Red dots**: These are our 30 noisy samples generated from the sine function with random Gaussian noise (with
standard deviation of 0.3).
2. **Green line**: This represents the true sine function without any noise, serving as a reference.
3. **Blue line**: This is the Kernel Ridge regression model's fit of the noisy data.
The Kernel Ridge regression model uses a Radial Basis Function (RBF) kernel to capture the non-linear pattern of the
sine wave. Despite the considerable noise in the data points, the model does a good job following the underlying sine
pattern.
Let's enhance our approach by using grid search to find the optimal hyperparameters for the Kernel Ridge model:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 fromsklearn.model_selectionimportGridSearchCV│
│ 2 │
│ 3 # Define parameter grid│
│ 4 param_grid={│
│ 5 'alpha':[0.001,0.01,0.1,1.0],│
│ 6 'gamma':[0.1,1.0,10.0,100.0]│
│ 7 }│
│ 8 │
│ 9 # Create grid search with cross-validation│
│10 grid=GridSearchCV(KernelRidge(kernel='rbf'),param_grid,cv=5,scoring='neg_mean_squared_error')│
│11 grid.fit(X_reshaped,y)│
│12 │
│13 print(f"Best parameters: {grid.best_params_}")│
│14 print(f"Best score: {-grid.best_score_:.4f} (MSE)")│
│15 │
│16 # Fit the model with best parameters│
│17 kr_best=grid.best_estimator_│
│18 y_pred_best=kr_best.predict(X_test)│
│19 │
│20 # Plot the results with optimal hyperparameters│
│21 plt.figure(figsize=(10,6))│
│22 plt.scatter(X,y,color='red',label='Noisy samples')│
│23 plt.plot(X_test,np.sin(X_test),color='green',label='True sine function')│
│24 plt.plot(X_test,y_pred_best,color='blue',label='Kernel Ridge (optimal params)')│
│25 plt.xlabel('x')│
│26 plt.ylabel('y')│
│27 plt.title('Optimized Kernel Ridge Regression on Noisy Sine Function')│
│28 plt.legend()│
│29 plt.grid(True)│
│30 plt.show()│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=3.32k, thinking=93, output=1.24k, cache_write=2.14k, cache_read=2.14k; Accumulated costs:
0.037 USD
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Best parameters: {'alpha': 0.1, 'gamma': 1.0}
Best score: 0.1363 (MSE)
<Figure size 1000x600 with 1 Axes>
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
## Improved Model with Optimized Hyperparameters
I performed a grid search with cross-validation to find the optimal hyperparameters for our Kernel Ridge regression
model. The results show:
- **Best hyperparameters**: alpha = 0.1, gamma = 1.0
- **Best Mean Squared Error (MSE)**: 0.1363
This means our optimized model achieves a balance between:
- Fitting the data well (controlled by gamma parameter, which defines the kernel width)
- Avoiding overfitting (controlled by alpha parameter, which penalizes complexity)
The second plot shows:
1. **Red dots**: The original 30 noisy samples
2. **Green line**: The true sine function
3. **Blue line**: The Kernel Ridge regression fit with optimized hyperparameters
The optimized model produces a smooth curve that follows the sine pattern while not being overly influenced by the noise
in our samples. This demonstrates how Kernel Ridge regression can effectively capture non-linear patterns in noisy data.
Is there anything specific about the model or the visualization you'd like me to explain or modify?
Accumulated token usage: input=7.20k, thinking=93, output=1.50k, cache_write=2.14k, cache_read=4.27k; Accumulated costs:
0.052 USD
╭─ Produced images ────────────────────────────────────────────────────────────────────────────────────────────────────╮
│workspace/images/default/5e6891f7.png│
│workspace/images/default/177566d8.png│
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'q': quit, Alt+Enter: newline
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