<|im_start|>user
How do I do a principal component analysis of data in a NumPy array, without using libraries other than NumPy itself? I need the principal components and the fraction of the total variance explained by each component.<|im_end|>
<|im_start|>assistant
Principal Component Analysis (PCA) is a technique used for dimensionality reduction and feature extraction. Here's a step-by-step guide to perform PCA using only NumPy:

1. Import NumPy library:
```python
import numpy as np
```

2. Create or load your data as a NumPy array. Make sure your data has zero mean for each feature/column. If not, subtract the mean from each column:
```python
# Example data, replace with your actual data
data = np.array([[1, 2], [3, 4], [5, 6]])

# Subtract the mean of each column
data_centered = data - np.mean(data, axis=0)
```

3. Calculate the covariance matrix of the centered data:
```python
cov_matrix = np.cov(data_centered.T)
```

4. Compute the eigenvalues and eigenvectors of the covariance matrix:
```python
eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)
```

5. Sort the eigenvalues and corresponding eigenvectors in descending order:
```python
sorted_indices = np.argsort(eigenvalues)[::-1]
sorted_eigenvalues = eigenvalues[sorted_indices]
sorted_eigenvectors = eigenvectors[:, sorted_indices]
```

6. Calculate the principal components (PCs) by projecting the centered data onto the sorted eigenvectors:
```python
principal_components = data_centered.dot(sorted_eigenvectors)
```

7. Compute the fraction of the total variance explained by each principal component:
```python
explained_variance_ratio = sorted_eigenvalues / np.sum(sorted_eigenvalues)
```

Now you have the principal components in the `principal_components` variable and the explained variance ratios in the `explained_variance_ratio` variable.<|im_end|>
