I need an advance because I have to create a mathematical model based on Markov stochastic process to indicate where a human step will be in the space. I have to start from this:
Pr ( S(t1) = S1 | S(t0) = S0) = Pr(step length, step frequency) Pr(change length step, change step frequency)
The first probability of the product is deterministic while the second one is based on empirical estimates. I have to make this product explicit using factors such as the step length, step frequency and the weight of the person.
I could be wrong, but this sounds a lot like a classic odometry problem. It's complicated by the fact that step length (or more correctly, cumulative locations of successive steps) is a continuous markov process whose distribution is not all that clear (depends on terrain, person's physical attributes, etc.) Plus, step frequency and length are not independent of each other. I don't imagine that it's even a stationary process. Perhaps you can assume stationarity for the short term (using a Gaussian distribution for the uncertainty in frequency/length with correlation between frequency and length). It would make sense to use some type of feedback from sensors (accelerometers, gyros, GPS, etc. smartphone is a ready-made IMU!) in a filtering scheme (Kalman, particle?) for improving the estimate of step placement.
Thanks so much for the advice. However I have to explicit that product of probabilities and yes, for the second one, I can use a Gaussian distribution, but the problem for me is how to create the mathematical model starting from this expression because I can't find anything of useful on Internet. I'm at the third year of industrial engineering and this is very difficult for me because I don't have enough knowledge on the subject. If you were me, how do you make the first factor (i.e. the first probability) explicit? In other words, I would like to know which algorithms I can use to make explicit, approximately, that probability using those parameters.
You could set up a toy problem as a Monte-Carlo random walk. You need an initial x,y distribution (can be a delta function if first position is known or perhaps gaussian if there is some uncertainty). You then need an x,y transition distribution (how much randomness is there going from time step k to time step k+1, use gaussian?).
1) set initial x,y position (pick a point at random using the initial distribution)
2) sample the x,y transition distribution using a random number generator (basing the distribution on some set of rules like direction of travel, stride, etc.)
3) move to new position
4) repeat from 2
If you want to include step frequency, it gets a bit more complicated. In this case, you will have a 3-vector (x, y, w)=(stride_x, stride_y, frequency) of random variables that follow the Markov recipe. You will need to probably use some coordinate transformations to translate stride & frequency to step position. Your MArkov transition rule might not be a gaussian any more. (I would need to think about that a bit).
Try doing this using a large ensemble (many experiments at once at each time step; a "cloud" of x,y points, so to speak). You can then generate statistics over each time step and even estimate the distribution.
You will find that over time, your position will become more and more uncertain unless you have some feedback from sensor data. The "cloud" will spread out. You will need to then implement some form of filtering where you use your random-walk model to predict the next position and the sensor data to correct this initial prediction. If you do the ensemble calculation, you can use it as a basis for a particle filter. Google is your friend, although you are correct. It can be difficult to know where to look.
Try:
Sampling distributions in Matlab/Octave
Metropolis Monte-Carlo
Particle Filters
Kalman filters
EM algorithm
Fokker Planck Continuous Markov models
Forward algorithm
Hidden Markov Model
Training hidden Markov model
Minimum Mean Square Error Estimators
Thanks so much. You've been really kind. Now I try to look at what you told me and then I hope I can create this model.
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