Statistics Toolbox

Principal Components Analysis

One of the difficulties inherent in multivariate statistics is the problem of visualizing multidimensionality. In MATLAB, the plot command displays a graph of the relationship between two variables. The plot3 and surf commands display different three-dimensional views. When there are more than three variables, it stretches the imagination to visualize their relationships.

Fortunately, in data sets with many variables, groups of variables often move together. One reason for this is that more than one variable may be measuring the same driving principle governing the behavior of the system. In many systems there are only a few such driving forces. But an abundance of instrumentation allows us to measure dozens of system variables. When this happens, we can take advantage of this redundancy of information. We can simplify our problem by replacing a group of variables with a single new variable.

Principal components analysis is a quantitatively rigorous method for achieving this simplification. The method generates a new set of variables, called principal components. Each principal component is a linear combination of the original variables. All the principal components are orthogonal to each other so there is no redundant information. The principal components as a whole form an orthogonal basis for the space of the data.

There are an infinite number of ways to construct an orthogonal basis for several columns of data. What is so special about the principal component basis?

The first principal component is a single axis in space. When you project each observation on that axis, the resulting values form a new variable. And the variance of this variable is the maximum among all possible choices of the first axis.

The second principal component is another axis in space, perpendicular to the first. Projecting the observations on this axis generates another new variable. The variance of this variable is the maximum among all possible choices of this second axis.

The full set of principal components is as large as the original set of variables. But it is commonplace for the sum of the variances of the first few principal components to exceed 80% of the total variance of the original data. By examining plots of these few new variables, researchers often develop a deeper understanding of the driving forces that generated the original data.

The function princomp is used to find the principal components. The following sections provide an example and explain the four outputs of princomp:

• Example: Principal Components Analysis
• The Principal Components (First Output)
• The Component Scores (Second Output)
• The Component Variances (Third Output)
• Hotelling's T² (Fourth Output)

Multivariate Statistics

Example: Principal Components Analysis