There are a lot of supplemental parameters that might be important for specific purposes. 

Snowflake — three exes L, a, b, tolerance

Brand owners can use Snowflake to define color boundaries in a more detailed way. The first step - generating PDF should be evaluated on a properly calibrated and profiled Display.  The next step - which requires prints - means in practice a perfect proofer being able to print with high accuracy.

For example, for paper, there are a couple of common parameters:

• Glossiness (where measurement conditions are specified). Please note that for substrate a dedicated method has been developed - TAPPI T 480 where 75° angle and special geometry are critical.
  
  
• Whiteness (several formulas to select from, where CIE15 is the most popular)

Whenever we are working with different kinds of lighting, various raw materials, inks, and paints A key parameter might be :

• Color Inconstancy Index
  
  
• Metamerism Index

For the CC Nano users, ChromaChecker developed couple of additional parameters that can become a part of the tolerance definition.