Algorithm/Analysis that utilises incremental information?
It looks like you are looking for time series based Machine Learning (or Sequential Machine Learning as Emre pointed out). 

This means that you have an entire dataset available to you beforehand. You train on this dataset and then use it to predict next day's weather given the weather of the last n days. Please note that you do not learn incrementally here, i.e. you do not learn from the new data that is rolling in every day. The learning is limited to the dataset that you had beforehand. 

Widely used algorithms for this kind of learning are Conditional Random Fields(CRF), Hidden Markov Models etc. Statistical methods like Autoregressive Models can also be used sometimes. A good starting point would be CRFs. 

If on the other hand you want to learn incrementally, read about online learning.