Principal Component Analysis (PCA) is a statistical technique used to reduce the complexity of a dataset by finding the underlying patterns and structures in the data.
In PCA, the data is transformed into a new coordinate system where the first axis, or principal component, captures the most variation in the data. Subsequent axes capture progressively less variation, while being orthogonal (perpendicular) to the previous axes. This process is repeated until all the variation in the data is accounted for or until the desired number of components is reached.
The resulting principal components can then be used for data visualization, dimensionality reduction, or other analysis tasks. PCA is commonly used in fields such as data science, machine learning, and finance, where large datasets with many variables are encountered.
PCA can be used for compression by reducing the dimensionality of a dataset while retaining as much of its variance or information as possible. By retaining only the most important principal components, which capture the majority of the variability in the data, PCA can effectively compress the dataset.
For example, suppose you have a dataset with 100 variables, but you only need to retain the most important information from the dataset. You can apply PCA to the dataset, which will result in a smaller set of principal components that capture the most important patterns and structures in the data. By keeping only these principal components and discarding the less important ones, you can effectively reduce the dimensionality of the data and compress it.
Note that the amount of compression achieved depends on the number of principal components retained. The fewer principal components retained, the higher the compression ratio, but at the cost of losing some information. Therefore, the number of principal components to retain should be chosen carefully to balance between compression and information loss.
Towards Data Science - "A One-Stop Shop for Principal Component Analysis": https://towardsdatascience.com/a-one-stop-shop-for-principal-component-analysis-5582fb7e0a9c
StatQuest - "Principal Component Analysis (PCA) clearly explained (2015)": https://www.youtube.com/watch?v=_UVHneBUBW0
Machine Learning Mastery - "A Gentle Introduction to Principal Component Analysis for Machine Learning": https://machinelearningmastery.com/principal-components-analysis-for-dimensionality-reduction-in-python/
Principal Component Analysis - Wikipedia: https://en.wikipedia.org/wiki/Principal_component_analysis
Principal Component Analysis - Stanford University: https://web.stanford.edu/class/cs246/slides/05-dim_red.pdf