Principal Coordinate Analysis
Principal Coordinate Analysis (PCoA) is used to visualize proximity matrices. Available in Excel using the XLSTAT add-on statistical software.
What is Principal Coordinate Analysis
Principal Coordinate Analysis (often referred to as PCoA) is aimed at graphically representing a resemblance matrix between p elements (individuals, variables, objects, among others).
The algorithm can be divided into three steps:
- Computation of a distance matrix for the p elements
- Centering of the matrix by rows and columns
- Eigen-decomposition of the centered distance matrix
The rescaled eigenvectors correspond to the principal coordinates 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 of 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 that the PCA can also represents observations in a space with less dimensions, the later 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.