Comprehensive review and comprehension check
Alternate numerical calculations and conceptual questions to confirm the flow of the 1D scalar Kalman filter end to end.
The final step is comprehensive problems. Alternate numerical calculations and conceptual questions, and walk through the flow of the 1D scalar Kalman filter end to end.
How to approach the comprehensive review
Solve these in the order "first run the numbers, then judge what the settings mean." Writing intermediate steps on paper helps the hand-calculation items stick. The questions are numbered Q1〜Q12 in a single sequence.
Exercise 1 — Work through the first step end to end (Q1〜Q4)
With initial values x̂₀ = 0, P₀ = 3, Q = 1, R = 4, and the first observation z₁ = 8, follow the predict → update chain through once.
Q1. Q1: What is the predicted variance P₁⁻ = P₀ + Q?
P₁⁻ = 3 + 1 = 4.
Q2. Q2: What is the Kalman gain K₁ = P₁⁻ / (P₁⁻ + R)?
K₁ = 4 / (4 + 4) = 0.5.
Q3. Q3: What is the updated estimate x̂₁?
The difference between observation and prediction is 8 − 0 = 8, so x̂₁ = 0 + 0.5 × 8 = 4.
Q4. Q4: What is the updated variance P₁?
P₁ = (1 − 0.5) × 4 = 2.
Exercise 2 — Carry into the second step (Q5〜Q8)
Use the previous results x̂₁ = 4 and P₁ = 2 to process the second observation z₂ = 6.
Q1. Q5: What is the predicted variance P₂⁻?
P₂⁻ = 2 + 1 = 3.
Q2. Q6: What is the Kalman gain K₂?
K₂ = 3 / (3 + 4) ≈ 0.4286.
Q3. Q7: What is the updated estimate x̂₂?
The difference between observation and prediction is 6 − 4 = 2, so x̂₂ ≈ 4 + 0.4286 × 2 ≈ 4.8571.
Q4. Q8: What is the updated variance P₂?
P₂ ≈ (1 − 0.4286) × 3 ≈ 1.7143.
Exercise 3 — Judge what the settings mean (Q9〜Q12)
The final set alternates numerical calculations and conceptual questions. Make sure you can describe each setting in words.
Q1. Q9: Which combination best matches "observations are fairly good but the model is unreliable"?
When the model tends to be off, raise Q; when observations are trustworthy, lower R.
Q2. Q10: Which combination produces an estimate that is "smoother but slower to follow change"?
Small Q and large R make the filter rely on the prediction, smoothing the output but delaying responses to sudden changes.
Q3. Q11: With predicted variance P⁻ = 0.5 and observation noise R = 4, what is the Kalman gain K?
K = 0.5 / (0.5 + 4) ≈ 0.1111. The prediction's uncertainty is small and the observation is a bit shaky, so we only move slightly toward the observation.
Q4. Q12: Which statement best captures the 1D scalar Kalman filter loop?
The Kalman filter repeats "predict → look at the difference from the observation → mix with the Kalman gain → update the variance as well."
What you can do once you understand this far
K and P, not as a vague impression.