Summary  
This chapter demonstrates how to standardize features and apply principal component analysis (PCA) using scikit-learn to reduce the dimensionality of a dataset.

General domain of usage  
Machine learning

Perform **PCA** on a real dataset using `scikit-learn`. Use the **Iris dataset**, a classic in machine learning, and follow these steps:

- Load the data;
- Prepare it for analysis;
- Standardize features;
- Apply `PCA` to reduce its dimensionality.

This process demonstrates how to implement dimensionality reduction in practical scenarios.

import numpy as np
import pandas as pd
from sklearn.datasets import load_iris
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA

# Load the Iris dataset
data = load_iris()
X = data.data
feature_names = data.feature_names

# Standardize features (important for PCA)
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# Apply PCA to reduce to 2 components
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X_scaled)

print("Original shape:", X.shape)
print("Transformed shape:", X_pca.shape)
# Each row in X_pca is a sample projected onto the first two principal components

The code above performs **PCA** on the Iris dataset by following several key steps:

#### 1. Loading the Data
The Iris dataset is loaded using `load_iris()` from `scikit-learn`. This dataset contains 150 samples of iris flowers, each described by four features: sepal length, sepal width, petal length, petal width.

#### 2. Standardizing Features
**Standardization** ensures each feature has mean `0` and variance `1`:

```python
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)
```

This step is essential because **PCA** is sensitive to the variance of each feature. Without standardization, features with larger scales would dominate the principal components, leading to misleading results.

### 3. Applying PCA
`PCA(n_components=2)` reduces the dataset from four dimensions to two:

```python
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X_scaled)
```

**Principal components** are new axes that capture the directions of maximum variance in the data. Each sample is projected onto these axes, resulting in a compact representation that retains as much information as possible.

### 4. Interpreting PCA Output
You can check how much variance each principal component explains:

```python
print(pca.explained_variance_ratio_)
```

This outputs an array, such as `[0.7277, 0.2303]`, meaning the first component explains about 73% of the variance and the second about 23%. Together, they capture most of the information from the original data.

Which statement is correct about performing PCA on the Iris dataset as shown in the example?

A comprehensive intermediate course guiding learners through the motivation, mathematical foundations, and practical implementation of Principal Component Analysis (PCA) for dimensionality reduction in data science and machine learning.

Explore the motivation, challenges, and benefits of reducing data dimensions in machine learning and data science.

Delve into the mathematical concepts that underpin PCA, including variance, covariance, and eigenvectors.

Apply PCA to real datasets using Python, interpret the results, visualize explained variance and component loadings, and compare model performance before and after PCA.

Performing PCA on a Real Dataset

1. Loading the Data

2. Standardizing Features

3. Applying PCA

4. Interpreting PCA Output