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i would like to understand basic idea of state space model,generally definition of state space model says that

State space model (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

now consider following situation,suppose our model is consisted by stat equation and observation equation,where state equation is given by following formula

$X(t+1)=A*x(t)+n(t)$

where A is an n × n state-transition matrix,so it means that it describes transition places for given position?and $n(t)$ is Gaussian noise,and also observation equation is described by: The m-dimensional measurement y(t) is subject to a linear transformation of the hidden state x(t) and is further corrupted by a measurement noise process v(t)

$y(t)=B*x(t)+v(t)$

now i am confused in the following things,first equation simply says that state equation represents similar to random walk,we have transition matrix as given in the following link(https://en.wikipedia.org/wiki/Stochastic_matrix), but what does second equation shows? what is a measurement in this case?i am interested meaning of measurement in such models, i have started learning state space models a few days ago and i am still confused in several terminology,firs such terminology is measurement definition,thanks a lot

Asaf Karagila
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dato datuashvili
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  • First, in many applications, the $A$ matrix is not a transition matrix in the sense of giving probabilities of moving to a new state, rather, it is an autoregression vector. That is, it tells you how $x$ depends on past values of itself. Thus, in an economic context, output today might depend on past output and past consumption. $B$ can be similar: observed output (or rocket position etc.) can depend on some linear combination of actual output plus noise. – Trurl Jun 18 '13 at 12:34
  • thanks for replay @Trurl,for make it more clear for me,A matrix tell us how x depend on it's pas value,but observed output in this case what does represent?let say airplane position at given time is depend on it's position at previous time +noise right?but what about observed position what does it shows? it is also called measurement also yes? – dato datuashvili Jun 18 '13 at 12:39
  • @Amzoti i have updated it,sorry – dato datuashvili Jun 18 '13 at 12:42
  • On the air traffic controller's radar screen, the observed position is the actual position plus noise (roughly) but you can imagine that the radar bounce depends on other factors in $x$ such as speed, altitude, even past position (because of slight lags in the radar equipment). Or, in the context I know better, you only have an estimate of the actual unemployment rate, not the true rate, and perhaps with unemployment being high for a while, people are embarrased to admit that, so your measurement error depends on lagged values as well. Hope this helps. – Trurl Jun 18 '13 at 12:46
  • so it means that position and observation is almost same,just in case of observation we are involving some additional parameters to actual position?so it means that if we have estimated airplane position from previous time,then for estimate observation we should consider other factors for get true position?does it means that we are not trusting actual position estimation only by noise and previous position? – dato datuashvili Jun 18 '13 at 12:52
  • or take this example,let us consider airplane position fixed by radar control system,we know it's position at time let say $t=10$,for get position at time $t=11$,we need first to add previous position multiplied by some auto-regressive vector and + noise right? we get position,but now i want to estimate observation,this observation what does show me ?just it is same as previous position,but considered several other factor,like wind speed,acceleration.velocities and so on? – dato datuashvili Jun 18 '13 at 12:57
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    Basically, your fist equation is your system model, i.e. how you expect your system to move on to step 11 given step 10. $n(t)$ denotes the modeling error, e.g. errors due to numerical calculations.$$$$

    The second equation describes your system observation i.e. the output of your sensors and can be anything from velocity to torque or position. The error term $v(t)$ denotes the measurement error which stems from non ideal sensors.

     – Karl Hardr Jun 18 '13 at 13:14
  • now it became clear for me i think,it is like how i expect my system move from one position to another,while observation is how does it really move right? – dato datuashvili Jun 18 '13 at 15:12
  • Well, the way I've usually interpreted it is that $x(t)$ describes the actual position of the system, the truth, and the $x(t+1)$ equation describes how the position changes. The airplane's change of position depends on speed, direction, and so forth, but also random things like gusts of wine (the $n(t)$). But you don't know the truth, you only have an imperfect observation--and how that observation-what you see--depends on the truth, is what the $y(t)$ equation tells you. – Trurl Jun 18 '13 at 18:59
  • i see now,thank a lot of – dato datuashvili Jun 18 '13 at 19:39

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