# Principal Coordinate Analysis

Principal Coordinate Analysis (PCoA) is used to visualize proximity matrices. Available in Excel using the XLSTAT add-on statistical software.

## 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:

- Computation of a distance matrix, e.g. a euclidean 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 (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.

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