• Underlying structure of the data
• Think of them as latent variables that are structuring the data
• E.g. in Pollen Analysis, annual temperature cannot be measured directly in the data, but it structures the data and influences it. We can interpret the data structure as the result of temperature variability
• This is a low-dimensional representation of the data
• Widely used in Ecological Modeling

Mathematical Foundation

(Brief intro from Paleoecology lecture)

• PCA is not necessarily suitable for count or percentage data - it needs to be transformed first
• We first need to standardize the data so that each feature has a mean of zero and SD of 1
  • This sounds dramatic but doesn't affect differences between data points, just affects the scale
  • Only works for normal distribution likely?
• PCA calculates covariance or correlation matrix
  • Finds eigenvectors (directions of max variance)
  • and eigenvalues (amount of variance explained)
  • Projects this dataon to new axes
• Isolates the largest possible explainable spread in the data and draws a line through it
• Algorithm tries to find an angle for this line where the spread is the widest (capturing the largest variation in the data)

PC Scores

• Values of new variables (principal component) of each data point (observation)
• Gives the position for each data point along the new variable

PC loading

• THese are represented by the length of arrows in the chart
• Weights that indicate the degree to which this component contributes to the principal component

Proportion of Variance

• Can be shown in R and gives an estimate of how much each PC contributes to the variance

Main idea

It is reducing the dimensionality by transforming it

Example

Given in Ordination Methods for Fossil Data