She usually approaches me to be pet and when I do pet her she will purr and rub her head against my hand. Then If I have been petting her for a while she will stare off in space and hiss at nothing while still purring. Sometimes right after she will start rubbing her head really hard against my hand and purr twice as loud. Why does she do this? She is up to date on all her vaccinations according to her tag and never shows any signs of aggression or fear aside from the hiss.

Maybe another cat is approaching to be pet as well and she is being possessive.

Haha, that would seem quite bizarre wouldn't it. However remember that cats senses are much better than ours. It is possible that the cat is hissing at something (another cat, dog, etc) nearby which you can't see. My cat does something similar to this, only she makes more of a 'hunting' hiss when she sees birds. Wouldn't be too worried. She seems calm and good tempered to be approaching you like that and not showing any signs of aggression.

Cat owner for 10 years - personal experience.