Are you sure climate isn't the main factor? If I were homeless I'd go to SF over NYC due to the mild winters.
71% in SF homeless already lived in SF, 90% already lived in California: https://socketsite.com/archives/2016/02/san-franciscos-homel...
the cite link in that article is dead but refers to http://hsh.sfgov.org/wp-content/uploads/2016/06/2015-San-Fra...
75% of LA homeless already lived in LA: https://www.lahsa.org/documents?id=2059-2018-greater-los-ang...
1) It was self-reported
2) "Lived in SF" in report meant they were SF residents and this is requires just any bill with city address within 30 days (consider cases like: moved to SF with hopes of better opportunities, out on the streets in couple of months - but sf resident for a short term prior).