Prediction of rainfall using nonhomogeneous Hidden Markov Model - matlab

Question

I am new to HMM but I have gone through enough literature. I am working on a project in which I will be predicting rainfall using atmospheric parameters.
I have four observable characteristics of the atmosphere (humidity, temperature, wind, sea level height) for 10 years. I have also rainfall amount data with me.
As per I can understand, for each day a weather state will be specified on the basis of the spatial rainfall. So here goes the question. Lets suppose I have data for 100 days.
Rainfall = { 1,2,3,4... 100}. So if I want to generate weather states what should I do?
Lets suppose
temperature = { 30 to 45, some kind of distribution }
humidity = { 25 to 80 }
wind = { 60 to 100 }
sea level height = { 35 to 90 }
How to find
P(X_0) Initial parameter,
P(X_t|X_t-1) state transition matrix,
P(Y_t|X_t) dependence of observation on state
Do I need some clustering for generating states?
I am coding it in MATLAB.
You can come with your example or any source which can explain the procedure to implement in program.

Answer 1

An HMM has a discrete number of states, so your first step will be to define your states. Once you have well-defined states, come up with a numbering scheme for your states and write a function that can accept the data for a given time period, and output the state number that corresponds to that state.
Once you have a function (let's call it get_state) that maps data to a state number, you can create your state transition matrix as follows:
T = zeros(num_states);
for day = 2:num_days
s1 = get_state(data(day-1));
s2 = get_state(data(day));
T(s1,s2) = T(s1,s2) + 1;
end
The i,j-th element of the matrix T now gives you the transition counts from state i to j. You can turn this into transition probabilities as follows:
M = bsxfun(#rdivide,T+1,sum(T+1,2));
The dependence of the observation on the state is harder. You will have to figure out how you want to turn the observed data into a probability density function or probability mass function. You can have mutliple observed distributions from a single state instead of combining temperature, humidity, etc., into a single observation.
This is obviously not a full implementation, but hopefully it is enough to give you a starting point.