2.5) Closure issues in the Earth Sciences

Closure is a major issue in the analysis of Earth Science data. Unfortunately it is also generally ignored
or greatly underestimated. Closure occurs when a set of data is normalized or forced to a common sum. 
Two examples are explored here to determine the effect of closure on the correlation coefficient.

A. The following dataset lists the thicknesses of two stratigraphic marker units (A and B) in a number 
of cores from an oil field in Alberta. On the right, these have been normalized to a percentage.
What effect (if any) does this have on the correlation coefficient (Pearson and Spearman)?

location	A (m)	B(m)	A(%)	B(%)
1	15	15	50	50
2	18	25.5	41.4	58.6
3	21	33	38.9	61.1
4	22.5	42	34.9	65.1
5	24	51	32	68
6	27	60	31	69

B. By far the most common occurrence of closure in the Earth Sciences is found in major element
composition data for bulk rocks and minerals. These values are reported as oxide weight percentages
with totals summing to 100%. To determine the severity of the impact on correlation coefficients we
will perform a simple Monte Carlo simulation.

Generate 100 hypothetical sample compositions for four major element components (say SiO2, Al2O3, 
CaO and MgO) at a relative concentration ratio of 10, 5, 2, 1 and normalize these compositions to 100%

This can be done by:
- generate values using the random number generator of your spreadsheet program and multiply by the
  ratio: e.g. 10*@RAND, 5*@RAND, etc for Quattro and RANDBETWEEN(1,10), 0.5*RANDBETWEEN(1,10)  
  for excel
- sum the values in the fifth column 
- normalize the values to this total and multiply by 100%

Now calculate the correlation coefficients between the different columns (original and normalised)

Recalculate your spreadsheet by pressing F9 in Quattro or saving in Excel and export the results of 
10 of these iterations to a separate spreadsheet (paste as values). Evaluate your results.