For the simple model xt = xt1 + wwt and yt = xt + vvt, where wt N(0, 1), vt N(0, 1), and

For the simple model xt = xt−1 + σwwt and yt = xt + σvvt, where wt ∼ N(0, 1), vt ∼ N(0, 1), and they are independent. Implement the following algorithms:

(A.1) SMC using the full information proposal distribution.

(A.2) Bootstrap particle filter (state equation as proposal distribution).

(A.3) Independent particle filter (observation equation as proposal distribution)

with single matching L = 1.

(A.4) Kalman filter.

Use the following variances when simulating data:

Use the following resampling schedules:

(C.1) Resample every step.

(C.2) Resample every five steps.

(C.3) No resampling.

Simulate 500 series, each of 200 observations, starting at x0 = 0 under the three variance settings, and run the four algorithms under the three resampling schedules (for SMC algorithms). Use initial distribution x0 ∼ N(0, 1).
Make comparisons.

ow (B.1) = 1 and = 1. (B.2) = 4 and o = 1. (B.3) w W = 1 and o = 4.