# Kalman Filter Process Noise - Model Where It Vanished

I am trying to use the Kalman filter (the scalar version) to estimate the steady state of a set measurements which is a random process. I have used a constant dynamic model as the state equation,

$$x(t+1) = x(t) + w(t)$$

Where $$w(t) \sim N(0, Q)$$.

The measurement equation

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

Where $$v(t) \sim N (0, R)$$.

The measurement noise $$v(t)$$ is taken as the uncertainty of the instrument and R is a known value with a assumed Gaussain distribution. The process error, $$w(t) = x(t+1) - x(t)$$ has high values as a constant variance (Calculated using the total error variance) or a time varying (Taking last 2, 5 readings as the set f sample). However, the state reaches the steady state when $$Q$$ is very small or $$Q = 0$$.

I have seen several other theoretical examples uses $$Q = 0$$ and reaches the steady state. But I am working with actual data. Also I have seen some other posts explaining the similar behavior. Can some explain this behavior with reasons, please?

• you are using a constant dynamic model but is the true signal actually an unknown constant ? – Fat32 Jul 24 '19 at 22:05
• Yes, trying to use this steady state estimation algirithm (Kalman) to estimate the unknown constant values of the steady state. – user44324 Jul 25 '19 at 0:53

The process noise $$w(t)$$, which is typically assumed to be a zero mean, white Gaussian noise with (power) variance $$\sigma^2$$, is used to account for any mismatches between the assumed dynamic model of the states and the actual truth.
If you do not set your $$\sigma^2$$ to zero (adding non-zero process noise to a constant dynamic model) then your state estimation will never reach steady-state, but wander around it; as you are telling the Kalman filter that the true signal is not a constant...
• Actually what happens is the Kalman Filter tries to estimate the mean of this "Random Walk" in the case $Q$ isn't vanished. – Royi Jul 28 '19 at 2:22