Principal Coordinate Analysis

Principal Coordinate Analysis

Principal Coordinate Analysis (PCoA) is a powerful and popular multivariate analysis method that lets you analyze a proximity matrix, whether it is a dissimilarity matrix, e.g. a euclidean distance matrix, or a similarity matrix, e.g. a correlation matrix. 

XLSTAT provides a PCoA feature with several standard options that will let you represent your data efficiently and gain a deep insight on them:

• Run a PCoA on a similarity or a dissimilarity matrix
• Correct negative eigenvalues if needed using the Square root or Lingoes correction
• Filter factors by fixing a maximum number of axes to be retained or by fixing a minimum of variance explained
  
  
   

What is Principal Coordinate Analysis?

Principal Coordinate Analysis (often referred to as PCoA) is aimed at graphically representing a resemblance matrix (similarity matrix or dissimilarity matrix) between p elements (individuals, variables, objects, among others).

The algorithm can be divided into three steps:

1. Computation of a distance matrix, e.g. a euclidean distance matrix, for the p elements
2. Centering of the matrix by rows and columns
3. Eigen-decomposition of the centered distance matrix

The rescaled eigenvectors correspond to the principal coordinates (principal axes), which are synthetic variables, that can be used to display the p objects in a space with 1, 2, ... , p-1 dimensions.

As with PCA (Principal Component Analysis) eigenvalues can be interpreted in terms of percentage of total variability that is being represented in a reduced space.

Results of Principal Coordinate Analysis in XLSTAT

• Delta1 matrix: This matrix corresponds to the D1 matrix of Gower, used to compute the eigen-decomposition.
• Eigenvalues and percentage of inertia: this table displays the eigenvalues and the corresponding percentage of inertia.
• Principal coordinates: This table displays the principal coordinates of the objects that are used to create the chart where the proximities between the charts can be interpreted.
• Contributions: This table displays the contributions that help evaluate how much an object contributes to a given axis.
• Squared cosines: This table displays the contributions that help evaluate how close an object is to a given axis.

Principal Coordinate Analysis and Principal Component Analysis

PCA and Principal Coordinate Analysis are quite similar in the way that the PCA can also represent observations in a space with less dimensions, the latter being optimal in terms of carried variability. A Principal Coordinate Analysis applied to matrix of Euclidean distances between observations (calculated after standardization of the columns using the unbiased standard deviation) leads to the same results as a PCA based on the correlation matrix. The eigenvalues obtained with the Principal Coordinate Analysis are equal to (p-1) times those obtained with the PCA.

Principal Coordinate Analysis and Multidimensional Scaling

Principal Coordinate Analysis and MDS (Multidimensional Scaling) share the same goal of representing objects for which we have a proximity matrix.

MDS has two drawbacks when compared with Principal Coordinate Analysis:

• The algorithm is much more complex and performs slower.
• Axes obtained with MDS cannot be interpreted in terms of variability.

MDS has two advantages compared with Principal Coordinate Analysis:

• The algorithm allows having missing data in the proximity matrix.
• The non-metric version of MDS provides a simpler and clear way to handle matrices where only the ranking of the distances is important.

Tutorial on how to run PCoA in Excel using the XLSTAT software

Here is an example on how to run a Principal coordinate analysis (PCoA) with XLSTAT.