Constraining the factor analytical solutions obtained from multiple-year receptor modeling of ambient PM2.5 data from five speciation sites in Ontario, Canada

• A new method for imposing mathematical constraints on multi-site PMF is discussed.
• The concept of softness in factor solutions is discussed.
• An iterative method that uses the measure of softness to select the individual constraints to impose is detailed.

Rotational ambiguity in factor analyses leads to solutions that are not always consistent with reality. The inherent non-negativity constraints in positive matrix factorization (PMF) help to prevent factor solutions from becoming overly unrealistic, but they are not sufficient to prevent unwanted rotations that could manifest in factors that should have similar compositions varying across multiple sites. The Canadian National Air Pollution Surveillance (NAPS) network operates five fine particulate matter (PM2.5) speciation sites in Ontario. Data from these sites from 2005 to 2010 were subjected to PMF to obtain factors representing sources of particulate matter. Eight factors were found to be common across these sites. These factors had profiles that varied greatly from one site to the other, suggesting that the PMF solutions were impacted by some rotational ambiguity. New features in the EPA PMF V5 program allow the use of a priori information to impose mathematical constraints that guide the evolution of the factor solutions. These constraints reduce the rotational space. In situations where major emissions sources are known and located in the neighborhood of receptors, or emissions inventories and literature source profiles exist, it is easy to use these profiles to force the factor solutions to conform to the expected signatures. In our case, reported source profiles were neither available nor applicable due to the large spatial span of potential sources and receptor sites. This work describes how such constraints can be generated and used in these complex situations. The fundamental principle explored in this work is the concept of ‘stiffness’ of PMF solutions to identify the desirable non-rotating factors.