present simple ("anticipate") and present continuous ("am anticipating") would both be OK here. "Having anticipated" isn't a verb conjugation like past simple or present perfect. Instead, it is the head of a "participial phrase". It is very advanced grammar--I recommend that you study other things first.
But, since you asked, a participial phrase will BEGIN with a participle--the -ing form of a verb, or the third form of a verb. For instance, the present participle of "give" is "giving", and the past participle of "give" is "given." You should put a participial phrase before or after a complete sentence, separated by a comma. The participial phrase will act as an adverb that explains the reason for or manner of the rest of the sentence.
I ate before I met Bob yesterday. *Having already eaten,* I wasn't hungry, so when he offered to take me to lunch, I said "no".