(SSM) refers to a class of probabilistic graphical model (Koller and Friedman, 2009) that describes the probabilistic dependence between the latent state variable and the observed measurement. The state or the measurement can be either continuous or discrete. The term “” originated in 1960s in the area of control engineering (Kalman, 1960). SSM provides a general framework for analyzing deterministic and that are measured or observed through a
is the set of all possible states of a ; each state of the system corresponds to a unique point in the state space. For example, the state of an idealized pendulum is uniquely defined by its angle and angular velocity, so the state space is the set of all possible pairs (angle, velocity). A dynamical system is a rule for time evolution on a state space.
The most well studied SSM is the Kalman filter, which defines an optimal for inferring linear Gaussian systems.
Like hidden markov models, in SSM we have ,
- Observation Equation
- State Equation.
where yt is observations at time t. Zt, Tt and Rt are system matrices. Alpha is latent state variable. et is normally distributed with 0 mean and Ht standard deviation (sd) and nt is also normally distributed with 0 mean and Qt sd.
Example of Gaussian State space model ( using KFAS package in R)-
Providing Initial values of parameters-
deaths <- window(alcohol[,2], end =2007)
population <- window(alcohol[,6], end = 2007)
## defining all system matrices
Zt <- matrix(c(1,0),1,2) #matrix(data = NA, nrow = 1, ncol = 1)
Tt <- matrix(c(1,0,1,1),2,2)
Rt <- matrix(c(1,0), 2,1)
Qt <- matrix(NA)
a1 <- matrix(c(1,0),2,1)
P1 <- matrix(0,2,2)
P1inf <- diag(2)
|red-forecasted & black-actual values|