Multivariate Methods • Principal Components • Overview of Principal Component Analysis
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Overview of Principal Component Analysis
A principal component analysis models the variation in a set of variables in terms of a smaller number of independent linear combinations (principal components) of those variables.
If you want to see the arrangement of points across many correlated variables, you can use principal component analysis to show the most prominent directions of the high-dimensional data. Using principal component analysis reduces the dimensionality of a set of data. Principal components representation is important in visualizing multivariate data by reducing it to graphable dimensions. Principal components is a way to picture the structure of the data as completely as possible by using as few variables as possible.
For p variables, p principal components are formed as follows:
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The first principal component is the linear combination of the standardized original variables that has the greatest possible variance.
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Each subsequent principal component is the linear combination of the variables that has the greatest possible variance and is uncorrelated with all previously defined components.
Each principal component is calculated by taking a linear combination of an eigenvector of the correlation matrix (or covariance matrix or sum of squares and cross products matrix) with the variables. The eigenvalues represent the variance of each component.
The Principal Components platform allows you to conduct your analysis on the correlation matrix, the covariance matrix, or the unscaled data. You can also conduct Factor Analysis within the Principal Components platform. See the Factor Analysis chapter in the Consumer Research book for details.