Somewhat random, with some logic. Here's the "Fares 101" level of my understanding, I'm sure others will be along shortly to throw rocks. Some may tell you that it's no better than a random number generator, but be assured there are Real People, looking at Real Data that we're not privy to making adjustments as time goes on, and those people probably aren't complete fools. There's logic there, even if it's sometimes maybe poor logic or invisible logic.
In the Good Old Days, nearly every train opened 11 months out at low bucket, which meant the early bird truly got the worm. As Amtrak's revenue management gained some rudimentary sophistication, they realized that "hey, this train for the day before Thanksgiving is almost certainly going to sell out, so why are we selling tickets at low bucket?" and fares would open 11 months out at a higher level.
As time passed, this became more and more widespread, and at this point it's more like "open high, if it sells slowly we can lower them and sell tickets 5-6 months out". Sometimes that happens, and sometimes because of demand the prices just stay high and they successfully sell the train out at higher fare levels (particularly in this day and age of reduced capacity).
As far as days of the week, I'm not sure that there's enough of a "rule" there to count on. Variability for the specific train and the specific dates probably overshadows any specific rule on dates.
The general algorithm for "how can I increase my odds of traveling Amtrak on the cheap(ish)" is to look at these charts (or a scattering of fares for the endpoints you're looking at if you're not riding end to end) to get a sense of what the range of prices for your particular journey are. Start watching fares as early as you can for the time period when you're traveling. Buy tickets when they're available at a price you're willing o pay. Continue watching fares. If they go down, call Amtrak and ask that the fare be adjusted to the lower rate on offer at the current time. If they don't go down, live in the satisfaction that you probably didn't miss out on a cheaper fare and enjoy your trip!