This Python code implements a Kalman filter to track the value of x based on noisy measurements over time. It initializes parameters like the process variance Q and measurement variance R, then performs the Kalman filter equations in a loop to estimate xhat and its error covariance P at each time step. It plots the noisy measurements, estimated value over time, and estimated error convergence.

1. 1
Kalman Filter Code in Python 3.7:
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (8, 8)
# intial parameters
n_iter = 50 #no of points to be observed
sz = (n_iter) # size of array
x = -0.37727 # truth value
z = np.random.normal(x,0.1,size=sz) # observations (normal about x,
sigma=0.1)
Q = 1e-5 # process variance
# allocate space for arrays
xhat=np.zeros(sz) # a posteri estimate of x
P=np.zeros(sz) # Error Co-variance
xhatminus=np.zeros(sz) # Estimate of x
Pminus=np.zeros(sz) # Error estimate
K=np.zeros(sz) # Kalman gain factor
R = 0.1**2 # estimate of measurement variance
# intial guesses
xhat[0] = 0.0
P[0] = 1.0
for k in range(1,n_iter):
# time update
xhatminus[k] = xhat[k-1]
Pminus[k] = P[k-1]+Q
# measurement update
K[k] = Pminus[k]/( Pminus[k]+R )
xhat[k] = xhatminus[k]+K[k]*(z[k]-xhatminus[k])
P[k] = (1-K[k])*Pminus[k]
plt.figure()
plt.plot(z,'k+',label='Noisy Measurements')
plt.plot(xhat,'b-',label='Estimated Position')
plt.axhline(x,color='g',label='Measured Value')

2. 2
plt.legend()
plt.title('Time vs Position', fontweight='bold')
plt.xlabel('Time')
plt.ylabel('Position')
plt.figure()
valid_iter = range(1,n_iter) # Pminus not valid at step 0
plt.plot(valid_iter,Pminus[valid_iter],label='Error Estimate')
plt.title('Estimated error vs. iteration step', fontweight='bold')
plt.xlabel('Iteration')
plt.ylabel('Error')
plt.setp(plt.gca(),'ylim',[0,.01])
plt.show()

3. 3
Results:
Fig 01: Kalman filter results
Fig 02: Error in prediction.