Principal component analysis (PCA) is a type of factor analysis used to transform a set of correlated variables into a set of uncorrelated variables known as principal components (PCs). PCA is typically used to reduce the dimensionality of a large set of variables while retaining as much of the information from the data set as possible.
PCA is an unsupervised statistical technique used to examine the interrelationships among the set of variables. It is a multivariate technique commonly used for exploratory data analysis to discover patterns and directions of variability in data sets with many variables.
PCA begins by decomposing a matrix of data that has been standardized to have a mean of zero and a standard deviation of one into two smaller matrices: a component matrix and a component loadings matrix. The component matrix consists of the eigenvectors of the covariance matrix of the original data, ordered by eigenvalue. Each component is associated with a component loading or weight, which is the correlation of the component with the original variables.
The component loadings matrix is the matrix of correlations between the component and the original variables. This matrix is used to determine the importance of each component in explaining the variability of the data set.
The principal component analysis technique is used to reduce the dimensionality of a data set while preserving as much of the original data as possible. If, after performing PCA, some of the components have an eigenvalue of zero or near zero, then these components can be discarded and the data set will maintain much of its original structure, but with fewer dimensions.
The use of PCA requires that the data set has many variables with a moderate to high degree of correlation. Where the variables are strongly correlated, PCA can be used to identify the underlying structure of the data. In addition, PCA can be used to identify sources of multivariate outliers such as population points that do not fit the overall pattern.
PCA is also useful for identifying areas of decreased variability due to sampling. For example, if data sets are collected by region, then PCA can be used to determine if one region has significantly fewer data points than the others.
PCA can be used to identify natural clusters or groups of data points. By plotting the data on a three-dimensional graph, it is possible to visually identify trends or patterns.
Finally, PCA can be used to reduce the dimensionality of a data set prior to applying a predictive analytics technique such as decision trees or neural networks. With fewer variables, the predictive model can be trained more quickly and will require fewer data points for validation purposes.
In summary, principal component analysis is a multivariate exploratory data analysis technique that can be used to reduce the dimensionality of a large data set while preserving as much of the information as possible. It can also be used to identify natural clusters or trends in the data, as well as to identify sources of multivariate outliers or sampling issues. Finally, PCA can also be used prior to applying a predictive model in order to reduce the number of input variables and simplify the model training process.
