Kalman Filter prediction using different time step

Question

Typically Kalman Filter or any other time series forecasting methods use a single step prediction - update step.

For eg: Let us say I have sensor data collected at every 1ms.

Let z denote measurement and x denote true state.

i.e at t = 100ms I have $z_0, z_1, z_2, ... z_{100}$.

Now typically in the prediction step we predict $x_{101}$ and in the next timestep, we update the state parameters when we have a new measurement $z_{101}$.

But what if i need to predict $x_{110}$ at t=100ms?

My initial idea was to use 10ms as the timestep.

at t = 100ms, we have $z_0, z_{10}, z_{20},...z_{100}$. We can now predict $x_{110}$. But this is essentially wasting so much sensor data.

Is there a better way to approach this problem in general?

Answer 1

In the state space model $$ z_t=Ax_t+\epsilon_t\\ x_t=Bx_{t-1}+\nu_t $$ where the errors are independent and separately identically distributed the usual one-step prediction is $$ \hat{z}_{t+1|t}=E(z_{t+1}|z_t)=E(Ax_{t+1}+\epsilon_{t+1}|z_t)=AE(x_{t+1}|z_t)=A\hat{x}_{t+1|t} $$ where $\hat{x}_{t+1|t}=E(x_{t+1}|z_t)=BE(x_t|z_t)=B\hat{x}_{t|t}$. If we want the $t+h$ prediction, you simply do the same thing and write it in terms of the filtered estimate $\hat{x}_{t|t}$: $$ \hat{z}_{t+h|t}=E(z_{t+h}|z_t)=AE(x_{t+h}|z_t)=A\hat{x}_{t+h|t}=AB^h\hat{x}_{t|t}. $$

Thus, if you want a prediction for $E(x_{110}|z_{100})$ you'd use $\hat{z}_{110|100}=AB^{10}\hat{x}_{100|100}$.