Local Weather Forecast (Self-Learning Algorithm) First of all, the question! :

Is it possible to predict local weather tendencies for the upcoming hours (let's say 2-3 hours) using only local data?
If yes, which data is needed to make such predictions and which predictions are possible?

I'm not talking about very precise predictions, just basic stuff like: "The temperature will rise/drop" or "If humidity goes up and air pressure drops, it will probably rain" etc.. you get the idea.

I was thinking of using data for temperature, air pressure, humidity, time of day and wind speed/direction.. maybe something else?

Some Background :

for a little DIY Home-Project I decided it would be an interesting idea to set up a little "weather station" in front of my home and collect all sorts of weather related data in short intervalls (e.g. temperature, air pressure, humidity, wind speed/direction, time of day etc.)

The idea is to collect data over the period of a few months (or more) and to develop an algorithm that is able to predict the upcoming weather with a certain probability.
The result should be a little program that was trained on past data and is then able to predict weather tendencies (Temperature, rain, etc..) using the data of the last couple of hours. Let's say the temperature and the air pressure drop and the humidty rises over a certain period of time, then it should tell me that it will likely rain. 

I will not bother you with the details on how to write such a self-learning algorithm, but since I'm not very well educated regarding meteorology I'm not even sure if such predictions are possible or to what extent.

I think you've understood the idea. Please give me your Ideas about it. What's possible what's not. Maybe experiences, which data I need, what predictions i could possibly make and which not and so on.

Thanks for any help!
(not an answer, too long for a comment)

Strictly speaking, the answer is yes, since the temperature generally increases during the day and falls at night (and local time is a local variable), the barometric pressure and humidity can indicate if a storm is coming (there were some really old fashioned weather prediction "clocks" that did this). Realistically, upcoming weather in your local area depends on what's nearby, since that's the weather that moves into your area. Also make sure you don't overfit your model: even a model that perfectly fits your existing data isn't necessarily predictive. Finally, there are lots of weather data available, including local weather data, and you might want to use those data as well.