Komponenttianalyysi

Komponenttianalyysi, or component analysis, is a statistical method used to reduce the dimensionality of a dataset while retaining as much of the original variance as possible. It transforms a set of potentially correlated variables into a smaller set of uncorrelated variables, known as principal components. The first principal component captures the largest possible variance, the second captures the next largest variance orthogonal to the first, and so on.

The primary goal of component analysis is to simplify complex data, making it easier to interpret and analyze. This is achieved by identifying the underlying structure of the data. It is widely applied in fields such as psychology for personality trait analysis, finance for risk management, and image processing for feature extraction.

The process involves calculating the covariance matrix of the data and then finding its eigenvectors and eigenvalues. The eigenvectors represent the directions of maximum variance, and the corresponding eigenvalues indicate the amount of variance along those directions. By selecting the eigenvectors with the largest eigenvalues, a lower-dimensional representation of the data can be constructed.

While powerful, component analysis assumes that the principal components are linear combinations of the original variables and that variance is a good indicator of importance. It can be sensitive to the scale of the variables, often requiring standardization beforehand. Variants like independent component analysis (ICA) aim to find components that are statistically independent rather than just uncorrelated, which can be useful in different applications.