Gaussian Process
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'q': quit, Alt+Enter: newline
> generate 30 noisy samples from a sine function and fit them with a Gaussian process. Plot the result
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<think>
The user wants to generate noisy samples from a sine function and fit them using a Gaussian Process regression, then
plot the results. This is a common machine learning task that involves:
1. Generating synthetic data (sine function with added noise)
2. Fitting a Gaussian Process regression model
3. Plotting the original data and the model's predictions
I'll need to:
- Import necessary libraries (numpy, matplotlib, scikit-learn for Gaussian Process)
- Generate the sine data with noise
- Create a Gaussian Process model and fit it
- Generate predictions
- Create a visualization of both the original data and the fitted model
Let me write the Python code to accomplish this.
</think>
I'll help you generate noisy samples from a sine function, fit them with a Gaussian Process regression model, and plot
the results. This is a great example of how Gaussian Processes can be used for regression tasks with uncertainty
estimation.
Let's start by setting up the necessary libraries and generating the noisy sine data:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 fromsklearn.gaussian_processimportGaussianProcessRegressor│
│ 4 fromsklearn.gaussian_process.kernelsimportRBF,ConstantKernelasC│
│ 5 │
│ 6 # Set random seed for reproducibility│
│ 7 np.random.seed(42)│
│ 8 │
│ 9 # Generate noisy sine function data│
│10 X=np.linspace(0,10,30).reshape(-1,1)# 30 points between 0 and 10│
│11 y_true=np.sin(X).ravel()│
│12 # Add noise to the data│
│13 noise=np.random.normal(0,0.1,y_true.shape)│
│14 y_noisy=y_true+noise│
│15 │
│16 # Plot the noisy data and the true function│
│17 plt.figure(figsize=(10,6))│
│18 plt.scatter(X,y_noisy,color='red',label='Noisy samples')│
│19 plt.plot(X,y_true,'b-',label='True sine function')│
│20 plt.title('Noisy Sine Function Samples')│
│21 plt.xlabel('x')│
│22 plt.ylabel('y')│
│23 plt.legend()│
│24 plt.grid(True)│
│25 plt.show()│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=193, thinking=139, output=559, cache_write=2.14k, cache_read=0; Accumulated costs: 0.017
USD
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<Figure size 1000x600 with 1 Axes>
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Now let's fit a Gaussian Process regression model to these noisy samples. Gaussian Processes are powerful Bayesian
models that not only provide predictions but also uncertainty estimates, which is one of their key advantages.
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 # Define the kernel for the Gaussian Process│
│ 2 # The kernel is a combination of a constant kernel and an RBF kernel│
│ 3 kernel=C(1.0,(1e-3,1e3))*RBF(1.0,(1e-2,1e2))│
│ 4 │
│ 5 # Create and fit the Gaussian Process regression model│
│ 6 gp=GaussianProcessRegressor(kernel=kernel,alpha=0.1,n_restarts_optimizer=10)│
│ 7 gp.fit(X,y_noisy)│
│ 8 │
│ 9 # Generate predictions for a fine grid│
│10 X_pred=np.linspace(0,10,100).reshape(-1,1)│
│11 y_pred,sigma=gp.predict(X_pred,return_std=True)│
│12 │
│13 # Plot the results│
│14 plt.figure(figsize=(10,6))│
│15 │
│16 # Plot the noisy samples│
│17 plt.scatter(X,y_noisy,color='red',label='Noisy samples')│
│18 │
│19 # Plot the true function│
│20 plt.plot(X,y_true,'b-',label='True sine function')│
│21 │
│22 # Plot the GP prediction and its confidence interval│
│23 plt.plot(X_pred,y_pred,'k-',label='GP prediction')│
│24 plt.fill_between(X_pred.ravel(),│
│25 y_pred-1.96*sigma,│
│26 y_pred+1.96*sigma,│
│27 alpha=0.2,color='k',│
│28 label='95% confidence interval')│
│29 │
│30 plt.title('Gaussian Process Regression on Noisy Sine Function')│
│31 plt.xlabel('x')│
│32 plt.ylabel('y')│
│33 plt.legend()│
│34 plt.grid(True)│
│35 plt.show()│
│36 │
│37 # Print the optimized kernel parameters│
│38 print("Optimized kernel parameters:")│
│39 print(gp.kernel_)│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=3.22k, thinking=139, output=1.09k, cache_write=2.14k, cache_read=2.14k; Accumulated
costs: 0.034 USD
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<Figure size 1000x600 with 1 Axes>Optimized kernel parameters:
0.882**2 * RBF(length_scale=1.62)
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Perfect! I've successfully generated 30 noisy samples from a sine function and fitted them with a Gaussian Process
regression model. Let me explain the results:
### What I did:
1. **Data Generation**:
- Generated 30 points from a sine function between 0 and 10
- Added Gaussian noise with mean 0 and standard deviation 0.1 to simulate real-world measurement noise
2. **Gaussian Process Model**:
- Used a combination of Constant and RBF (Radial Basis Function) kernels
- Added a small amount of noise (alpha=0.1) to account for the noise in our data
- Performed multiple optimization restarts to find the best kernel parameters
3. **Visualization**:
- Red dots: The 30 noisy samples of the sine function
- Blue line: The true sine function without noise
- Black line: The Gaussian Process prediction (the fitted model)
- Gray shaded area: The 95% confidence interval of the predictions
### Results:
The Gaussian Process has successfully captured the underlying sine pattern despite the noise in the data. The model
provides not just predictions but also uncertainty estimates, shown by the gray confidence bands. These bands are
narrower near our data points (where we have more information) and wider in areas with fewer data points (where
uncertainty is higher).
The optimized kernel parameters are:
- Constant kernel scale: 0.882²
- RBF length scale: 1.62
These parameters tell us about the characteristics of the function being modeled. The length scale indicates how quickly
the function changes - larger values suggest smoother functions with slower variations.
Gaussian Processes are particularly useful when you need uncertainty quantification along with predictions, making them
valuable for scientific and engineering applications where understanding prediction confidence is important.
Accumulated token usage: input=6.94k, thinking=139, output=1.50k, cache_write=2.14k, cache_read=4.27k; Accumulated
costs: 0.051 USD
╭─ Produced images ────────────────────────────────────────────────────────────────────────────────────────────────────╮
│workspace/images/default/cfe9c96a.png│
│workspace/images/default/08a31ccc.png│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
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'q': quit, Alt+Enter: newline
> create a module gp_fit.py with 2 functions:
1. gp_fit that takes samples as input, fits the GP and returns the predictions
2. plot_pred that takes the original function, the noisy samples, the predictions as input and plots the GP mean and
confidence interval
use a relative path when creating the module
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<think>
Let me create a module `gp_fit.py` with the requested functions:
1. `gp_fit` - Takes samples as input, fits a Gaussian Process, and returns predictions
2. `plot_pred` - Takes the original function, noisy samples, and predictions as input and plots the results
I'll create this file using the file editing tool, and place it in the current directory (using a relative path).
</think>
I'll create a Python module called `gp_fit.py` with the two requested functions. This module will encapsulate the
Gaussian Process fitting and plotting functionality, making it reusable.
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│1 print(file_editor(**{'command':'create','path':'./gp_fit.py','file_text':'"""\nGaussian Process fitting and│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=8.86k, thinking=223, output=2.87k, cache_write=4.27k, cache_read=4.27k; Accumulated
costs: 0.086 USD
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File created successfully at: gp_fit.py
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Now let's test the module to make sure it works as expected. I'll import the functions from our newly created module and
use them with the same sine function example:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 fromgp_fitimportgp_fit,plot_pred│
│ 4 │
│ 5 # Set random seed for reproducibility│
│ 6 np.random.seed(42)│
│ 7 │
│ 8 # Generate data│
│ 9 X=np.linspace(0,10,30).reshape(-1,1)# 30 points between 0 and 10│
│10 y_true=np.sin(X).ravel()│
│11 # Add noise to the data│
│12 noise=np.random.normal(0,0.1,y_true.shape)│
│13 y_noisy=y_true+noise│
│14 │
│15 # Generate fine grid for true function│
│16 X_fine=np.linspace(0,10,100).reshape(-1,1)│
│17 y_fine_true=np.sin(X_fine).ravel()│
│18 │
│19 # Fit Gaussian Process│
│20 results=gp_fit(X,y_noisy)│
│21 │
│22 # Print the optimized kernel parameters│
│23 print("Optimized kernel parameters:")│
│24 print(results['gp'].kernel_)│
│25 │
│26 # Plot the results│
│27 plot_pred(X_fine,y_fine_true,X,y_noisy,results,│
│28 title='Gaussian Process Regression on Noisy Sine Function')│
│29 plt.show()│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=14.31k, thinking=223, output=3.25k, cache_write=4.27k, cache_read=6.41k; Accumulated
costs: 0.108 USD
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Optimized kernel parameters:
0.882**2 * RBF(length_scale=1.62)
<Figure size 1000x600 with 1 Axes>
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Great! I've successfully created the `gp_fit.py` module with the two requested functions, and tested it to confirm it
works correctly. The module is located at the relative path `./gp_fit.py`.
### Module Overview:
1. **`gp_fit(X_samples, y_samples, X_pred=None, alpha=0.1, n_restarts=10)`**
- Takes input samples and fits a Gaussian Process regression model
- Returns predictions along with uncertainty estimates
- Allows customization of the noise level and optimization settings
- Returns a dictionary with predictions, uncertainty, and the fitted model
2. **`plot_pred(X_true, y_true, X_samples, y_samples, gp_results, title='Gaussian Process Regression')`**
- Creates a visualization showing:
- The original true function (blue line)
- The noisy samples (red dots)
- The GP prediction (black line)
- The 95% confidence interval (gray shaded area)
- Fully customizable with appropriate labels and title
### Example Usage:
The test code demonstrates how to use the module:
1. Generate data (in this case, a noisy sine function)
2. Call `gp_fit()` to fit the Gaussian Process model and get predictions
3. Call `plot_pred()` to visualize the results
The plot shows that our implementation works correctly - we can see the true sine function, the noisy samples, and the
Gaussian Process fit with its confidence interval. The optimized kernel parameters are consistent with our previous
results (0.882² * RBF with length scale 1.62).
This modular approach makes it easy to reuse this functionality for different datasets and applications. The functions
are designed to be flexible, with sensible defaults but also allowing customization when needed.
Accumulated token usage: input=20.30k, thinking=223, output=3.67k, cache_write=4.27k, cache_read=8.54k; Accumulated
costs: 0.132 USD
╭─ Produced images ────────────────────────────────────────────────────────────────────────────────────────────────────╮
│workspace/images/default/234112e4.png│
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'q': quit, Alt+Enter: newline
> set the n_restarts default to 15
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<think>
I need to modify the `gp_fit` function in the `gp_fit.py` module to change the default value of the `n_restarts`
parameter from 10 to 15. I'll use the `str_replace_editor` to make this change.
</think>
I'll update the module to set the default value of `n_restarts` parameter to 15 instead of 10 in the `gp_fit` function:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│1 print(file_editor(**{'command':'str_replace','path':'./gp_fit.py','old_str':'def gp_fit(X_samples, y_sample│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=26.62k, thinking=275, output=4.30k, cache_write=4.27k, cache_read=10.68k; Accumulated
costs: 0.160 USD
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The file gp_fit.py has been edited. Here's the result of running `cat -n` on a snippet of gp_fit.py:
9 import matplotlib.pyplot as plt
10 from sklearn.gaussian_process import GaussianProcessRegressor
11 from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
12
13 def gp_fit(X_samples, y_samples, X_pred=None, alpha=0.1, n_restarts=15):
14 """
15 Fit a Gaussian Process regression model to input samples and make predictions.
16
17 Parameters
18 ----------
19 X_samples : array-like, shape (n_samples, n_features)
20 Training data input features
21 y_samples : array-like, shape (n_samples,)
22 Training data target values (noisy observations)
23 X_pred : array-like, shape (n_pred, n_features), optional
24 Prediction input points. If None, uses a fine grid based on X_samples range.
25 alpha : float, default=0.1
26 Noise level parameter for the Gaussian Process
27 n_restarts : int, default=15
28 Number of optimizer restarts to find optimal kernel parameters
29
30 Returns
31 -------
32 dict
Review the changes and make sure they are as expected. Edit the file again if necessary.
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Now let's test the module with the updated default parameter to ensure it's working correctly:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 # Reload the module to get the updated version│
│ 4 importimportlib│
│ 5 importgp_fit│
│ 6 importlib.reload(gp_fit)│
│ 7 │
│ 8 # Re-import the functions from the updated module│
│ 9 fromgp_fitimportgp_fit,plot_pred│
│10 │
│11 # Set random seed for reproducibility│
│12 np.random.seed(42)│
│13 │
│14 # Generate data│
│15 X=np.linspace(0,10,30).reshape(-1,1)# 30 points between 0 and 10│
│16 y_true=np.sin(X).ravel()│
│17 # Add noise to the data│
│18 noise=np.random.normal(0,0.1,y_true.shape)│
│19 y_noisy=y_true+noise│
│20 │
│21 # Generate fine grid for true function│
│22 X_fine=np.linspace(0,10,100).reshape(-1,1)│
│23 y_fine_true=np.sin(X_fine).ravel()│
│24 │
│25 # Check the function signature to confirm the default value is updated│
│26 print("Function signature:",gp_fit.__defaults__)│
│27 │
│28 # Fit Gaussian Process without specifying n_restarts to use the default│
│29 results=gp_fit(X,y_noisy)│
│30 │
│31 # Print the optimized kernel parameters│
│32 print("Optimized kernel parameters:")│
│33 print(results['gp'].kernel_)│
│34 │
│35 # Plot the results│
│36 plot_pred(X_fine,y_fine_true,X,y_noisy,results,│
│37 title='Gaussian Process Regression with n_restarts=15')│
│38 plt.show()│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=33.98k, thinking=275, output=4.75k, cache_write=4.27k, cache_read=12.82k; Accumulated
costs: 0.189 USD
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Execution result ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Function signature: (None, 0.1, 15)
Optimized kernel parameters:
0.882**2 * RBF(length_scale=1.62)
<Figure size 1000x600 with 1 Axes>
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
I've successfully updated the `gp_fit.py` module to change the default value of `n_restarts` from 10 to 15 as requested.
Here's a summary of the changes and confirmation:
### Changes Made:
- Modified the function signature in `gp_fit.py` to have `n_restarts=15` instead of `n_restarts=10`
- Updated the function documentation to reflect this change
### Verification:
1. The function signature output `(None, 0.1, 15)` confirms the default parameters are:
- `X_pred=None`
- `alpha=0.1`
- `n_restarts=15` (updated from 10)
2. The module works correctly with the new default value, as demonstrated by the successful execution of the test code.
3. The optimized kernel parameters remain consistent with our previous results, showing the same model fit. This makes
sense as we used the same random seed and data, and the number of restarts primarily affects the robustness of finding
the optimal parameters, not necessarily the final result when the optimization landscape is fairly simple.
The updated module can now be used with the new default value of 15 restarts for the optimizer, which might help find
better hyperparameters in more complex modeling scenarios by exploring the parameter space more thoroughly.
Accumulated token usage: input=41.95k, thinking=275, output=5.05k, cache_write=4.27k, cache_read=14.95k; Accumulated
costs: 0.218 USD
╭─ Produced images ────────────────────────────────────────────────────────────────────────────────────────────────────╮
│workspace/images/default/37780eb3.png│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ User message ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
'q': quit, Alt+Enter: newline
> now test the module with 40 noisy samples from a cosine function
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
<think>
Let's test the module with 40 noisy samples from a cosine function. I'll generate the data, fit a Gaussian Process using
our module, and plot the results. Here's the plan:
1. Generate 40 points from a cosine function
2. Add random noise to the samples
3. Use our `gp_fit` function to fit a Gaussian Process model
4. Use the `plot_pred` function to visualize the results
</think>
I'll test the module with 40 noisy samples from a cosine function to see how well the Gaussian Process can model this
different pattern:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 fromgp_fitimportgp_fit,plot_pred│
│ 4 │
│ 5 # Set random seed for reproducibility│
│ 6 np.random.seed(42)│
│ 7 │
│ 8 # Generate 40 samples from a cosine function│
│ 9 X=np.linspace(0,10,40).reshape(-1,1)# 40 points between 0 and 10│
│10 y_true=np.cos(X).ravel()# Using cosine instead of sine│
│11 │
│12 # Add noise to the data│
│13 noise=np.random.normal(0,0.15,y_true.shape)# Slightly more noise for variety│
│14 y_noisy=y_true+noise│
│15 │
│16 # Generate fine grid for true function (for plotting)│
│17 X_fine=np.linspace(0,10,200).reshape(-1,1)# Higher resolution for smoother plot│
│18 y_fine_true=np.cos(X_fine).ravel()│
│19 │
│20 # Fit Gaussian Process using our module│
│21 results=gp_fit(X,y_noisy)│
│22 │
│23 # Print the optimized kernel parameters│
│24 print("Optimized kernel parameters:")│
│25 print(results['gp'].kernel_)│
│26 │
│27 # Plot the results│
│28 plot_pred(X_fine,y_fine_true,X,y_noisy,results,│
│29 title='Gaussian Process Regression on Noisy Cosine Function (40 samples)')│
│30 plt.show()│
│31 │
│32 # Print some statistics about the fit│
│33 y_pred=results['gp'].predict(X)│
│34 mse=np.mean((y_pred-y_true)**2)│
│35 print(f"Mean Squared Error on training points: {mse:.4f}")│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=50.18k, thinking=363, output=5.64k, cache_write=4.27k, cache_read=17.09k; Accumulated
costs: 0.251 USD
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Execution result ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Optimized kernel parameters:
0.984**2 * RBF(length_scale=1.77)
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
It seems the plot didn't show in the output. Let's try running it again and make sure the plot is displayed:
╭─ Code action ────────────────────────────────────────────────────────────────────────────────────────────────────────╮
│ 1 importnumpyasnp│
│ 2 importmatplotlib.pyplotasplt│
│ 3 fromgp_fitimportgp_fit,plot_pred│
│ 4 │
│ 5 # Set random seed for reproducibility│
│ 6 np.random.seed(42)│
│ 7 │
│ 8 # Generate 40 samples from a cosine function│
│ 9 X=np.linspace(0,10,40).reshape(-1,1)# 40 points between 0 and 10│
│10 y_true=np.cos(X).ravel()# Using cosine instead of sine│
│11 │
│12 # Add noise to the data│
│13 noise=np.random.normal(0,0.15,y_true.shape)# Slightly more noise for variety│
│14 y_noisy=y_true+noise│
│15 │
│16 # Generate fine grid for true function (for plotting)│
│17 X_fine=np.linspace(0,10,200).reshape(-1,1)# Higher resolution for smoother plot│
│18 y_fine_true=np.cos(X_fine).ravel()│
│19 │
│20 # Fit Gaussian Process using our module│
│21 results=gp_fit(X,y_noisy)│
│22 │
│23 # Print the optimized kernel parameters│
│24 print("Optimized kernel parameters:")│
│25 print(results['gp'].kernel_)│
│26 │
│27 # Use the plot_pred function from our module│
│28 fig=plot_pred(X_fine,y_fine_true,X,y_noisy,results,│
│29 title='Gaussian Process Regression on Noisy Cosine Function (40 samples)')│
│30 plt.tight_layout()│
│31 plt.show()│
│32 │
│33 # Print some statistics about the fit│
│34 y_pred=results['gp'].predict(X)│
│35 mse=np.mean((y_pred-y_true)**2)│
│36 print(f"Mean Squared Error on training points: {mse:.4f}")│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
Accumulated token usage: input=59.13k, thinking=363, output=6.13k, cache_write=4.27k, cache_read=19.22k; Accumulated
costs: 0.285 USD
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Execution result ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Optimized kernel parameters:
0.984**2 * RBF(length_scale=1.77)
<Figure size 1000x600 with 1 Axes>Mean Squared Error on training points: 0.0046
Produced images:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Model response ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Great! I've successfully tested the module with 40 noisy samples from a cosine function. Let's analyze the results:
### Test Results:
1. **Data Generation**:
- Created 40 points from a cosine function spanning the range 0 to 10
- Added Gaussian noise with slightly higher standard deviation (0.15) to make the test more challenging
2. **Gaussian Process Fit**:
- Successfully fitted the Gaussian Process model using our `gp_fit` function
- Optimized kernel parameters: 0.984² * RBF(length_scale=1.77)
- These parameters differ slightly from our sine function example, reflecting the different pattern of the cosine
function
3. **Performance**:
- Mean Squared Error on the training points: 0.0046
- This low MSE indicates a good fit to the data despite the noise
4. **Visualization**:
- The plot shows the true cosine function (blue line), the 40 noisy samples (red dots), and the Gaussian Process
prediction (black line) with its 95% confidence interval (gray shaded area)
- The GP prediction closely follows the true cosine function
- The confidence interval is narrower where we have more data points and widens slightly between points, showing the
model's uncertainty
### Observations:
1. **Smooth Interpolation**: The Gaussian Process provides a smooth interpolation that captures the cosine pattern well,
even with noise in the data.
2. **Appropriate Uncertainty**: The confidence bands correctly represent the model's uncertainty, being narrower near
data points and wider in regions with fewer observations.
3. **Module Functionality**: Our module works correctly for different types of periodic functions (both sine and
cosine), demonstrating its versatility.
4. **Kernel Parameters**: The slightly different kernel parameters compared to the sine function example show how the GP
adapts to different underlying patterns.
This test confirms that our module is working correctly and can effectively model different types of functions. The
Gaussian Process regression handles the noisy cosine data well, providing both accurate predictions and meaningful
uncertainty estimates.
Accumulated token usage: input=68.74k, thinking=363, output=6.61k, cache_write=4.27k, cache_read=21.36k; Accumulated
costs: 0.321 USD
╭─ Produced images ────────────────────────────────────────────────────────────────────────────────────────────────────╮
│workspace/images/default/17902906.png│
╰──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────╯
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ User message ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
'q': quit, Alt+Enter: newline
> q