Principal Component Analysis 편집하기

'''Principal Component Analysis (PCA)''' is a statistical technique used for dimensionality reduction by transforming a dataset into a new coordinate system. The transformation emphasizes the directions (principal components) that maximize the variance in the data, helping to reduce the number of features while preserving essential information.
==Key Concepts==
*'''Principal Components:''' New orthogonal axes computed as linear combinations of the original features. The first principal component captures the maximum variance, followed by subsequent components with decreasing variance.
*'''Explained Variance:''' The proportion of total variance captured by each principal component.
*'''Orthogonality:''' Principal components are mutually perpendicular, ensuring no redundancy.
==Steps in PCA==
#'''Standardize the Data:''' Center the data by subtracting the mean of each feature and scale it (if necessary).
#'''Compute the Covariance Matrix:''' Calculate the covariance matrix of the dataset to understand relationships between features.
#'''Calculate Eigenvectors and Eigenvalues:''' Find the eigenvectors and eigenvalues of the covariance matrix to determine the principal components and their variance contribution.
#'''Select Principal Components:''' Retain the top k principal components that explain the majority of the variance.
#'''Transform the Data:''' Project the original data onto the new feature space defined by the selected principal components.
==Applications of PCA==
PCA is widely used in various fields for the following purposes:
*'''Dimensionality Reduction:''' Reducing the number of features in datasets for efficient processing.
*'''Noise Reduction:''' Removing irrelevant or noisy dimensions to improve data quality.
*'''Data Visualization:''' Visualizing high-dimensional data in 2D or 3D for better interpretability.
*'''Feature Extraction:''' Creating new features that summarize the original dataset effectively.
*'''Anomaly Detection:''' Highlighting deviations by focusing on key patterns in data.
===Example===
Performing PCA using Python's scikit-learn library:<syntaxhighlight lang="python">
from sklearn.decomposition import PCA
import numpy as np

# Example dataset
data = np.array([[2.5, 2.4],
                 [0.5, 0.7],
                 [2.2, 2.9],
                 [1.9, 2.2],
                 [3.1, 3.0]])

# Apply PCA to reduce dimensions to 1
pca = PCA(n_components=1)
reduced_data = pca.fit_transform(data)

print("Reduced Data:", reduced_data)
print("Explained Variance Ratio:", pca.explained_variance_ratio_)
</syntaxhighlight>
==Advantages==
*'''Dimensionality Reduction:''' Simplifies complex datasets while preserving essential information.
*'''Noise Reduction:''' Eliminates redundant features, improving model accuracy.
*'''Efficient Data Representation:''' Reduces computation time and storage requirements.
==Limitations==
*'''Loss of Interpretability:''' Transformed features (principal components) are linear combinations of original features, making them harder to interpret.
*'''Assumption of Linearity:''' PCA assumes that the data's variance is best captured in a linear manner, which may not hold for all datasets.
*'''Sensitive to Scaling:''' PCA performance can be affected if the data is not properly standardized.
==Relation to SVD==
PCA is closely related to [[Singular Value Decomposition (SVD)]]. In PCA:
*Principal components are derived from the eigenvectors of the covariance matrix, which correspond to the left singular vectors in SVD.
*Eigenvalues correspond to the squared singular values from SVD.
==Related Concepts and See Also==
*[[Singular Value Decomposition]]
*[[Dimensionality Reduction]]
*[[Latent Semantic Analysis]]
*[[Feature Extraction]]
*[[Explained Variance]]
*[[Data Preprocessing]]
*[[Machine Learning]]
[[분류:Data Science]]