## Principal Components Analysis

### Background

When multivariate data are collected it is common to find some
correlated variables. One implication of these correlations is that there will
be redundancy in the information provided by the variables. In the extreme case
of two perfectly correlated variables (*x & y*) one is redundant.
Knowing the value of *x* leaves *y* with no freedom and *vice versa*.
Principal Components Analysis (PCA) exploits the redundancy in multivariate
data, enabling us to:

- pick out patterns (relationships) in the variables;
- reduce the dimensionality of a data set without a significant loss of information.

PCA is one of a family of related ordination or projection techniques that includes Factor Analysis and Principal Co-ordinates Analysis.

Note: it is Principal ( 'first in rank or importance' Concise
Oxford Dictionary) *not Principle *('a fundamental truth or law as the
basis of reasoning or action', Concise Oxford Dictionary).

### Ten Important Concepts

correlation |

variance |

covariance |

variability |

matrix |

eigen value |

eigen vector |

ordination |

standardization |

linear combination |

### Description and examples

It is important that you work through the following in the specified order. It is also important that you thoroughly understand the content before moving onto the next section.

A. Some background |

B. Matrix methods (very brief!) |

C. Eigen methods (must understand this) |

D. A graphical explanation of PCA |

E. Sample analyses |

F. Self assessment exercise 1 |

G. Self assessment exercise 2 |

### Summary

PCA and FA are two similar methods, indeed under certain circumstances (no rotation and number of factors = number of variables) they produce identical results (albeit with some rescaling of the eigen vectors).

Both methods are based on an eigen analysis of either a correlation or a covariance matrix. If a correlation matrix is used the variables are standardised