Sept 18 2025 Miika

Principal Component Analysis

• Before modeling and building predictive models, first explore the data
  • Association between species and temporal or geographic units
  • Co-variant species
• First, reconstruct climate, then start to look at the association between reconstructed climate and other variables.

The task

• DuPont et. al. 2019
• Given a fossil pollen datasetfrom the past 2 million years (marine core from close to Limpopo River mouth in South Africa that drains much of southern Africa)
• Pollen % numbers for 116 plant taxa for 220 stratigraphic units
• Looking for ecologically meaningful structure or patterns in the data

Start by exploring the data

• Examine how species co-vary over time
• Some taxa are vary big and take up a large percentage of the data
  • Peak in Podocarpus around 500 years ago
  • This corresponds with low in Asteraceae and Poaceae (sunflowers and grasses)
  • Plotting a correlation between these reveals a negative correlation

Pattern recognition is difficult with this much data

• We need to reduce the dimensionality by combining variables that show the same information to build a model
• We can use Principal Component Analysis (PCA) to notice underlying structure in the data
  • We should notice a structure that can be interpreted as the result of temperature variability
• Looking at a PCA scatterplot and showing only the species, we can see that Podocarpus is very different from the others
  • PC1 loading 0.358, meaning it has a positive correlation with PC1
  • Some other species also have some correlation with PC1
  • Asteraceae has an equally strong negative correlation
• The six taxa with the strongest positive loading to PC1 are all trees (woody plants)
• The six taxa with strongest negative loading are all herbaceaeous plants (grasses and forbs)
• This could represent temporal changes between woody and herbaceous vegetation
  • Alternation perhaps
• PC1 scores plotted vs time:
  • We use the scores instead of the loadings because these are assocaited with the observation number (which in this case is the stratigraphic unit)
  • There is a trend showing that there was possibly more herbaceous species in this region 1 million years ago which trended towards woody species, but with "up-down" cycles of about 100 years
  • These up-down cycles correspond with Milokovic cycles in orbital eccentricity and precession
  • However, it also correlates with changes in CO₂ levels.
    • An ongoing debate in African paleoecology is whether CO₂ or climate drives vegetation change in Africa

Example 2: Micro mammal fossil data

• From a cave, 25 species and their frequencies from 14 stratigraphic units and their age
• Using PCA, we can squeeze this 25-dimensional data into two dimensions
  • Angle between arrows indicates relatedness
  • Which are positively and negatively correlated
  • How stratigraphic units are related based on variable associations

Chi-square test

• Species composition in fossil or ecological sample is a cateogrical variable
• No numerical values
• We need cross-tables to analyze correlation
• In paleoecology we are often interested in analyzing these correlations
• Why not use PCA?

Cross-table or Contigency Table

• Data classified according to two categorical variables
• Values in the table are the frequency for that pair/category
• e.g. Rows are stratigraphic layer, columns are taxa
  • Cell B2 is Species 2, Layer 2 frequency
• Chi-square test of independence tests if these two variables are associated or are independent
  • e.g. Are these two layers different of similar to each other? Or can it be explained by natural variance?
  • If it is found to be statistically significant, we can say that there is an association between certain stratigraphic layers and a certain species
  • First, tests expected frequencies according to null hypothesis
  • Next, compares observed frequencies and determines if the difference is big enough to reject null hypothesis
  • In R: reports Chi-squared and p-value

Residuals

• Calculates expected frequencies from column total * row total / table total
• Filling the table with observed- expected gives "residuals"
  • In our example, we have a strong association between Layer 1 and Species 1, and negative between Layer 4 an Species 1. and some other trends.

Contributions to chi-square statistic

• These values report the effect that that each value has on the chi-square statistic overall

Correspondence Analysis (CA)

• The math is similar to a PCA, it's sometimes called a PCA for categorical data
  • although sometimes people use PCA for this too, but they probably shouldn't
• This is also a dimension-reducing technique (by reducing to a 2D represenation)
• Using the same table from our chi-square example, it shows:
  • Dimension 1 explains 86% of the variation
  • Dimension 2 explains 13% of the variation

Back to the micromammal example

• 70% of variation can be explained by this first dimension, but 20% by the second axis
• The LGM layers are especially affected by the second dimension
• The results are somewhat similar to the PCA plot

Summary

• PCA and CA are both useful for exploring data
• Not actually used for predictive models, but they can give a start to determine which variables to look at if you can find a useful interpretation or pattern in the data
• If you have some ecological knowledge, you can interpret the PCA or CA axes as being explained by a certain variable like temperature, precipitation, potential evapotranspiration, moisture, CO₂ concentration
• Both PCA and CA scatterplots can reveal clusters and outliers (similar and unique site assemblages)
• Plotting PC or Ca scores against time can reveal interesting temporal patterns