Sequential Monte Carlo - r

Question

I was given this model, and to get the probability I am supposed to simulate the data.
x_1 ∼N(0, 102)
x_t =0.5 ∗ (x_t−1) + 25 · (x_t−1)/(1 + (x_t-1)^2) + 8 · cos(1.2 ∗ (t − 1)) + εt
, t = 2, 3, ..
y_t =(x_t)^2/25 + ηt, t = 1, 2, 3, ...
Where εT and ηt follows normal distribution.
I tried to inverse the function but I cannot do it because of the fact that I have no idea if my Xs will be positive or negative. I understood that I should use a sequential monte carlo, but I can't figure out how to find the functions of the algorithm. What is f and g, and how can we decide x(t-1) if it is equally likely to be positive or negative because of the x squared?
Algorithm:
1 Sample X1 ∼ g1(·). Let w1 = u1 = f1(x1)/g1(x1). Set t = 2
2 Sample Xt|xt−1 ∼ gt(xt|xt−1).
3 Append xt to x1:t−1, obtaining xt
4 Let ut = ft(xt|xt−1)/gt(xt|xt−1)
5 Let wt = wt−1ut , the importance weight for x1:t
6 Increment t and return to step 2

Answer 1

With a time-series model like yours, essentially the only way to compute the probability distribution of x or y is to run multiple simulations of the model, with randomly drawn values of x_0, eps_t, eta_t, and then construct histograms by aggregating the samples across all the runs. In very special cases (e.g. damped Brownian motion) it may be possible to calculate the resulting probability distributions algebraically, but I don't think there's any chance of that for your model.
In Python (I'm afraid I'm not fluent enough in R), you can simulate the time-series by something like this:
import math, random
def simSequence(steps, eps=0.1, eta=0.1):
x = random.normalvariate(0, 102)
ySamples = []
for t in range(steps):
y = (x ** 2) / 25 + random.normalvariate(0, eta)
ySamples.append(y)
x = (0.5 * x + 25 * x / (1 + x ** 2)
+ 8 * math.cos(1.2 * t) + random.normalvariate(0, eps))
return ySamples
(This replaces your t=1..n with t=0..(n-1).)
You could then generate a plot of a few examples of the y time-series:
import matplotlib.pyplot as plt
nSteps = 100
for run in range(5):
history = simSequence(nSteps)
plt.plot(range(nSteps), history)
plt.show()
to get something like:
If you then want to compute the probability distribution of y at different times, you could generate a matrix whose columns represent realizations of y_t at a common value of time and compute histograms at selected values of t:
import numpy
runs = numpy.array([ simSequence(nSteps) for run in range(10000) ])
plt.hist(runs[:,5], bins=25, label='t=5', alpha=0.5, normed=True)
plt.hist(runs[:,10], bins=25, label='t=10', alpha=0.5, normed=True)
plt.legend(loc='best')
plt.show()
which gives: