Principal component analysis information

Machine learning and data mining
Part of a series on
Paradigms Supervised learning Unsupervised learning Online learning Batch learning Meta-learning Semi-supervised learning Self-supervised learning Reinforcement learning Curriculum learning Rule-based learning Quantum machine learning
Problems Classification Generative modeling Regression Clustering Dimensionality reduction Density estimation Anomaly detection Data cleaning AutoML Association rules Semantic analysis Structured prediction Feature engineering Feature learning Learning to rank Grammar induction Ontology learning Multimodal learning
Supervised learning (classification • regression) Apprenticeship learning Decision trees Ensembles Bagging Boosting Random forest k-NN Linear regression Naive Bayes Artificial neural networks Logistic regression Perceptron Relevance vector machine (RVM) Support vector machine (SVM)
Clustering BIRCH CURE Hierarchical k-means Fuzzy Expectation–maximization (EM) DBSCAN OPTICS Mean shift
Dimensionality reduction Factor analysis CCA ICA LDA NMF PCA PGD t-SNE SDL
Structured prediction Graphical models Bayes net Conditional random field Hidden Markov
Anomaly detection RANSAC k-NN Local outlier factor Isolation forest
Artificial neural network Autoencoder Cognitive computing Deep learning DeepDream Feedforward neural network Recurrent neural network LSTM GRU ESN reservoir computing Restricted Boltzmann machine GAN Diffusion model SOM Convolutional neural network U-Net Transformer Vision Mamba Spiking neural network Memtransistor Electrochemical RAM (ECRAM)
Reinforcement learning Q-learning SARSA Temporal difference (TD) Multi-agent Self-play
Learning with humans Active learning Crowdsourcing Human-in-the-loop RLHF
Model diagnostics Coefficient of determination Confusion matrix Learning curve ROC curve
Mathematical foundations Kernel machines Bias–variance tradeoff Computational learning theory Empirical risk minimization Occam learning PAC learning Statistical learning VC theory
Machine-learning venues ECML PKDD NeurIPS ICML ICLR IJCAI ML JMLR
Related articles Glossary of artificial intelligence List of datasets for machine-learning research List of datasets in computer vision and image processing Outline of machine learning
v t e

Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing.

The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified.

The principal components of a collection of points in a real coordinate space are a sequence of $p$ unit vectors, where the $i$ -th vector is the direction of a line that best fits the data while being orthogonal to the first $i-1$ vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points.^[1]

PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.

Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science.

^ Jolliffe, Ian T.; Cadima, Jorge (2016-04-13). "Principal component analysis: a review and recent developments". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 374 (2065): 20150202. Bibcode:2016RSPTA.37450202J. doi:10.1098/rsta.2015.0202. PMC 4792409. PMID 26953178.

Principal component analysis information

and 25 Related for: Principal component analysis information

Principal component analysis

Functional principal component analysis

Kernel principal component analysis

Robust principal component analysis

Multilinear principal component analysis

Principal component regression

Component analysis

Factor analysis

Independent component analysis

Multiple correspondence analysis

Analysis

Nonlinear dimensionality reduction

Dimensionality reduction

Eigenvalues and eigenvectors

Autoencoder

Multilinear subspace learning

Correspondence analysis

Sparse PCA

Functional data analysis

Feature learning

Linear discriminant analysis

Scree plot

Multivariate statistics

Parallel analysis

Eigenface