Predict step — predicting the next state from the model
Under the steady-state model the value does not change, but uncertainty grows over time. Verify P⁻ = P + Q with a case study.
The Kalman filter does not use each observation as is. It first uses a model to predict "this is what it should be next," and only then mixes in the observation.
This course uses the minimal model
Here we only use a one-dimensional steady-state model. The steady-state model is a model that assumes the target does not move (its position does not change), so we set "the next time step will, for now, be the same value as the previous one."
x̂⁻ = x̂Thanks to this simplification we can focus on what really matters right now: the division of labor between prediction and update.
Uncertainty grows over time
Even if the value itself does not change, uncertainty grows the further into the future we go. We express this with the variance P and update it in the prediction step with the following formula.
P⁻ = P + QQ expresses "how easily the model tends to be off." The longer we go without new observations, the more P grows.
Check 1 — Trace the prediction step by hand
From the previous-step estimate, build the predicted value x̂⁻ and predicted variance P⁻.
Q1. With the previous estimate x̂ = 12, P = 4, and Q = 1.5, what is the predicted value x̂⁻ under the steady-state model?
Under the steady-state model x̂⁻ = x̂, so the value remains 12.
Q2. Under the same conditions, what is the predicted variance P⁻?
P⁻ = P + Q = 4 + 1.5 = 5.5. The value itself stays the same, but uncertainty grows a little.
Q3. With initial variance P₀ = 2 and Q = 0.5, if we only predict three times in a row without any observations, what is the variance?
2 + 0.5 + 0.5 + 0.5 = 3.5. Without observations, predicting alone steadily grows the uncertainty.
The size of Q changes the behavior
Check 2 — Grasp the role of Q
Compare Case A and Case B with different values of Q, then put the meaning of Q back into words.
Q1. With previous variance P = 1 and Q = 0.01, what is the predicted variance P⁻?
P⁻ = 1 + 0.01 = 1.01. Because Q is small, the uncertainty barely grows.
Q2. With the same P = 1 and Q = 4.0, what is the predicted variance P⁻?
P⁻ = 1 + 4.0 = 5.0. The larger Q is, the faster the prediction's uncertainty grows.
Q3. Which statement best captures a case where Q is very small?
The correct answer is the first choice. A small Q means we assume the model is accurate, so it is safe to lean on the prediction.
Takeaway of this chapter
Prediction is the operation "the value does not change, but uncertainty grows a little." In the next chapter we look at how much to move that prediction once an observation arrives.