# Robust analysis of compositional data

Matthias Templ
GEOSTAT 2016

### Acknowledgement

• Karel Hron and Peter Filzmoser for a long-term and fruitful cooperation

• Karel Hron for providing his slides on compositional data analysis (used in some parts of the presentation)

### Geostatistics is applied in (Wikipedia, 08.09.2016)

• petroleum geology, hydrogeology, hydrology
• meteorology, oceanography
• geochemistry, geometallurgy
• geography
• forestry, environmental control, landscape ecology
• soil science, agriculture

### CoDa are present in all topics

• petroleum geology, hydrogeology, hydrology
• meteorology, oceanography
• geochemistry, geometallurgy
• geography
• forestry, environmental control, landscape ecology
• soil science, agriculture

Example compositional spatial and temporal data: Proportions of land use/land types or forest fragmentation proportions in each grid cell with potential (covariates may include elevation range, road length, population, median household income, and housing levels).

### What you will see/learn?

• What are compositional data?
• Real space vs the simplex, representation in Coordinates
• Examples
• Applications in multivariate statistics using geochemical data
• Why to use robust methods?
• The R package robCompositions

### What are compositional data?

• $$D$$-part vectors, describing quantitatively the parts of some whole, which carry exclusively relative information between the parts (Aitchison, 1986; Pawlowsky-Glahn et al., 2015)
• Typical units of measurement: percentages, mg/kg, mg/l
• Examples: geochemical data - proportions of minerals in a rock; concentations of fenolical acids in wine (mg/l); household expenditures on various commodities (foodstuff, housing, clothing), forest fragmentation proportions, etc.
• Compositional data consist of multivariate observations with positive values that sum up to a constant. Examples are proportional data or percentages, for which the values sum up row-wise to 1 or 100.

### Still compositional data?

• One or more variables of multivariate data are not available or has not been measured?
• When rounding errors leads to violate the prescribed constraint?
• Or what happens if the sum is not constant at all, but very different for each compositional observations?

The answer: it (always) depends on the analysis goals

## Compositional data are treated as multivariate data where relative rather than absolute information is relevant for the analysis.

• Absolute information: refers to the original raw data, in their concrete units such as counts, monetary units, temperature, precipitation, etc.
• Relative information: refers to a relative data representation, like proportions or percentages such as concentration of chemical elements in parts per million (ppm) or mg/kg, share of family income to gross household income, percentage of votes for a political party, daylight per day, etc..

### Generally,

• relative information is analyzed by considering (log-)ratios between the variables.
• representation of data in orthonormal/orthogonal coordinates
• analysis on orthonormal/orthogonal coordinates and backtransformation to the original space

A NO GO:
statistical analysis of compositional data using standard statistical methods with the assumption of Euclidean geometry in real space is just wrong but typically applied in practice.

### Example GEMAS data, univariate case

Absolute and relative concentrations of Phosphor (P) for samples extracted by X-ray fluorescence (XRF) from agricultural soils in Europe.