A short answer: it is not possible build a long term (longer than ~ 1 week or so) complete statistical model for all classes to predict the weather.

Indeed, pressure, temperature, wind and precipitation levels, topographical features should be included in the numerical weather models (NWM). Also, a nonlinear coupling of the ocean to the atmosphere must be included in the model. This coupling is used in global forecast models such as ocean events El Nino and La Nina. These events correspond to an oscillation of the ocean-atmosphere system. The NWM is not an exact description of the atmosphere. Many physical processes and small-scale weather elements such as boundary layers, heat and mass exchange, cloud cover, etc. are parametrized, e.g., instead of each individual cloud some parameters are being used such as average cloudiness to represent the cloud cover in a computational cell. Without such simplification, even a supercomputer cannot complete the weather calculation in a reasonable amount of time. NWM is usually run on an evenly spaced 3-dimensional grid, so initial data are needed for all grid points. However observational data are not available for great portions of the ocean and some land areas, therefore some interpolation is needed for such missing initial data points. Because of all these issues, such as nonlinear nature of the models and the initial data that cannot be precisely measured, a single weather forecast becomes increasingly inaccurate for 3 or 4 days ahead. In an attempt to push this limit, ensemble forecasts are generated, i.e. NWM are run a number of times with slightly different initial conditions to represent the uncertainties in the observation data and parametrizations. A measure of forecast uncertainty is the standard deviation of the ensemble member NWMs.

The bottom line: no reliable weather forecast is possible beyond a limit of about 1 week or so, and even this limit is often not achieved. See, for example, the NOAA site:

www.emc.ncep.noaa.gov/gmb/ens/fcsts/ensframe.html

for animated plots for North America that contain 40 ensemble members and two control splits with the Global Ensemble Forecast System.