Sampling: simple stable Autoregressive

I want to try out the convex hull inclusivity tester on data that is less random. One way to get such data is to sample a stable ARMA process. In fourier space, an ARMA can be expressed as a quotient of two polynomials:



P contains the poles, and Q contains the zeroes, or nills. In this post, I use only Q = 1.
The criterion for stability is that all roots of P is in the unit circle. To get a uniform sampling from the unit circle, we can sample an angle from a uniform distribution, and an absolute value from a triangle distribution (since the density must increase linearly with the absolute value). The code:

	def poles(k):
	"""
	Generates k pairs of poles in the unit circle
	Returns the resulting polynomial coefficients
	"""
	pol = np.array([1])
	for _ in range(k):
	theta = np.random.random()*np.pi
	r = np.random.random()**0.5 # to get uniform over area
	coeffs = np.array([1, -2*np.cos(theta)*r, r**2])
	pol = np.convolve(pol, coeffs) # polynomial multiplication
	return pol
	def main():
	ar = poles(5)
	ps = ArmaProcess(ar, [1])
	y = ps.generate_sample(1000)
	plt.plot(y)
	plt.show()

view raw poles.py hosted with ❤ by GitHub

This output format fits perfectly into the ArmaProcess object from statsmodels.

Random stable AR process from 5 pairs of poles.