### e. Perform a Prediction (Kriging)

1. **Break Down the Problem**
- We need to perform Kriging prediction on a spatial grid.
- The grid covers the area defined by the coordinates $[0, 2] \times [0, 2]$.
- Ensure to generate predictions over the entire grid.

2. **Relevant Concepts**
- **Kriging** is a statistical method used for interpolation based on the spatial correlation structure of the data.
- We will set up a grid, define the spatial correlation (variogram), and apply the Kriging algorithm.
- Key components:
- Define sample data points and their respective value (if provided).
- Choose a variogram model (e.g., Gaussian, Exponential).
- Perform Kriging based on the defined model.

3. **Analysis and Detail**
- Define the grid:
- Let $x$ and $y$ range from 0 to 2, with a resolution suitable for plotting (e.g., increments of 0.1).
- Create a mesh grid.
- Collect sample data points (e.g., from observations or synthetically generated).
- Choose a variogram model.
- Calculate the Kriging predictions using the defined grid and variogram.
- For implementation, you can use Python libraries such as `pyinterp`, `scikit-learn`, or `pyKriging`.

Here is a basic structure in Python using `pyKriging`:

```python
import numpy as np
import matplotlib.pyplot as plt
from pyKriging import kriging, kriging as krig

# Sample data
x_data = np.array([0.1, 0.2, 1.0, 1.5, 1.7]) # Example x coordinates
y_data = np.array([0.1, 0.4, 1.2, 1.5, 1.8]) # Example y coordinates
z_data = np.array([1.0, 2.0, 1.5, 1.2, 3.0]) # Example observations

# Creating a grid
grid_x, grid_y = np.mgrid[0:2:100j, 0:2:100j] # 100x100 grid

# Fit the Kriging model
model = kriging(x_data, y_data, z_data, "gaussian")

# Make predictions
z_pred, _ = model.predict(np.c_[grid_x.ravel(), grid_y.ravel()])

# Reshape and Plot results
z_pred = z_pred.reshape(grid_x.shape)
plt.contourf(grid_x, grid_y, z_pred, levels=20)
plt.colorbar()
plt.title('Kriging Prediction')
plt.xlabel('X axis')
plt.ylabel('Y axis')
plt.show()
```

4. **Verify and Summarize**
- Ensure the validity of the predictions by checking if they are within the bounds of the original data.
- The plot should illustrate the predictive variance and how the sampled points influence the prediction across the grid.

### Final Answer
After running the above code, a contour plot will be generated showing the Kriging predictions for the area $[0, 2] \times [0, 2]$. The predicted surface illustrates how values vary across the specified region based on the input data points. The colors on the plot indicate higher and lower predicted values, with interpolation reflecting the underlying spatial correlation defined by the chosen variogram model.

### f. Explain the Obtained Plot
The obtained plot represents the spatial distribution of predictions made via the Kriging method. Each color level indicates a range of predicted values across the area defined by $[0, 2] \times [0, 2]$. Darker areas may represent higher densities or values, while lighter colors indicate lower intended predictions. The smooth transitions between colors are a hallmark of Kriging, as it provides a spatially continuous estimation influenced by the nearby sample points. The predictions are influenced by both the values of the sampled data and their spatial arrangement, showcasing the strength of Kriging in capturing geographic patterns. Additionally, areas with more sample data points will have more precise predictions compared to sparser areas where the prediction is more uncertain. This plot can assist in understanding spatial phenomena and can be extremely useful for decision-making processes in fields like environmental science, geology, and resource management.

Question

### e. Perform a Prediction (Kriging)

1. **Break Down the Problem**
   - We need to perform Kriging prediction on a spatial grid.
   - The grid covers the area defined by the coordinates $[0, 2] \times [0, 2]$.
   - Ensure to generate predictions over the entire grid.

2. **Relevant Concepts**
   - **Kriging** is a statistical method used for interpolation based on the spatial correlation structure of the data.
   - We will set up a grid, define the spatial correlation (variogram), and apply the Kriging algorithm.
   - Key components:
     - Define sample data points and their respective value (if provided).
     - Choose a variogram model (e.g., Gaussian, Exponential).
     - Perform Kriging based on the defined model.

3. **Analysis and Detail**
   - Define the grid:
     - Let $x$ and $y$ range from 0 to 2, with a resolution suitable for plotting (e.g., increments of 0.1).
     - Create a mesh grid.
   - Collect sample data points (e.g., from observations or synthetically generated).
   - Choose a variogram model.
   - Calculate the Kriging predictions using the defined grid and variogram.
   - For implementation, you can use Python libraries such as `pyinterp`, `scikit-learn`, or `pyKriging`.

Here is a basic structure in Python using `pyKriging`:

```python
   import numpy as np
   import matplotlib.pyplot as plt
   from pyKriging import kriging, kriging as krig

# Sample data
   x_data = np.array([0.1, 0.2, 1.0, 1.5, 1.7])  # Example x coordinates
   y_data = np.array([0.1, 0.4, 1.2, 1.5, 1.8])  # Example y coordinates
   z_data = np.array([1.0, 2.0, 1.5, 1.2, 3.0])  # Example observations

# Creating a grid
   grid_x, grid_y = np.mgrid[0:2:100j, 0:2:100j]  # 100x100 grid

# Fit the Kriging model
   model = kriging(x_data, y_data, z_data, "gaussian")

# Make predictions
   z_pred, _ = model.predict(np.c_[grid_x.ravel(), grid_y.ravel()])

# Reshape and Plot results
   z_pred = z_pred.reshape(grid_x.shape)
   plt.contourf(grid_x, grid_y, z_pred, levels=20)
   plt.colorbar()
   plt.title('Kriging Prediction')
   plt.xlabel('X axis')
   plt.ylabel('Y axis')
   plt.show()
   ```

4. **Verify and Summarize**
   - Ensure the validity of the predictions by checking if they are within the bounds of the original data.
   - The plot should illustrate the predictive variance and how the sampled points influence the prediction across the grid.

### Final Answer
After running the above code, a contour plot will be generated showing the Kriging predictions for the area $[0, 2] \times [0, 2]$. The predicted surface illustrates how values vary across the specified region based on the input data points. The colors on the plot indicate higher and lower predicted values, with interpolation reflecting the underlying spatial correlation defined by the chosen variogram model.

### f. Explain the Obtained Plot
The obtained plot represents the spatial distribution of predictions made via the Kriging method. Each color level indicates a range of predicted values across the area defined by $[0, 2] \times [0, 2]$. Darker areas may represent higher densities or values, while lighter colors indicate lower intended predictions. The smooth transitions between colors are a hallmark of Kriging, as it provides a spatially continuous estimation influenced by the nearby sample points. The predictions are influenced by both the values of the sampled data and their spatial arrangement, showcasing the strength of Kriging in capturing geographic patterns. Additionally, areas with more sample data points will have more precise predictions compared to sparser areas where the prediction is more uncertain. This plot can assist in understanding spatial phenomena and can be extremely useful for decision-making processes in fields like environmental science, geology, and resource management.

Knowee AI · Accepted Answer