Adding UTM parameters to URLs that advertise your publication will allow you to see if a new susbcriber came from one of those URLs.
What are some techniques to help grow the number of subscribers to your list?
How can I ask my subscribers if they are still interested in receiving my newsletter?
Create Google Analytics goals to help track subscriptions to your publication.
Integrate Facebook advertising with your publication to help grow your subscriber base.
Integrate Twitter advertising with your publication to help grow your subscriber base.
Integrate Twitter Lead Generation with your publication to allow Twitter users to subscribe directly from the promoted Tweet.