The diagonal elements of D store the variance of the respective principal components. The matrix V contains the coefficients for the principal components. This calculation can also be carried out using the zscore function from the Statistics Toolbox:Д¬alculating the coefficients of the principal components and their respective variances is done by finding the eigenfunctions of the sample covariance matrix: Sometimes there are good reasons for modifying or not performing this step, but I will recommend that you standardize unless you have a good reason not to. Here, 'standardization' means subtracting the sample mean from each observation, then dividing by the sample standard deviation. Most often, the first step in PCA is to standardize the data. To summarize the data, we calculate the sample mean vector and the sample standard deviation vector: We determine the size of this data set thus:

Performing PCA will be illustrated using the following data set, which consists of 3 measurements taken of a particular subject over time: This article walks through the specific mechanics of calculating the principal components of a data set in MATLAB, using either the MATLAB Statistics Toolbox, or just the base MATLAB product. PCA is completely reversible (the original data may be recovered exactly from the principal components), making it a versatile tool, useful for data reduction, noise rejection, visualization and data compression among other things. Further principal components, if there are any, exhibit decreasing variance and are uncorrelated with all other principal components. The second principal component is calculated to have the second most variance, and, importantly, is uncorrelated (in a linear sense) with the first principal component. The coefficients of the principal components are calculated so that the first principal component contains the maximum variance (which we may tentatively think of as the 'maximum information'). Each principal component is a linear transformation of the entire original data set. Given a table of two or more variables, PCA generates a new table with the same number of variables, called the principal components. One such technique is principal component analysis ('PCA'), which rotates the original data to new coordinates, making the data as 'flat' as possible. These relationships are often linear, or at least approximately so, making them amenable to common analysis techniques. Real-world data sets usually exhibit relationships among their variables.

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