Estimate a scalar ARMA processΒΆ

The objective here is to estimate an ARMA model from a scalar stationary time series using the Whittle estimator and a centered normal white noise.

The data can be a unique time series or several time series collected in a process sample.

If the user specifies the order (p,q), OpenTURNS fits a model ARMA(p,q) to the data by estimating the coefficients (a_1, \dots, a_p), (b_1, \dots, b_q) and the variance \sigma of the white noise.

If the User specifies a range of orders (p,q) \in Ind_p \times Ind_q, where Ind_p = [p_1, p_2] and Ind_q = [q_1, q_2], We find the best model ARMA(p,q) that fits to the data and estimates the corresponding coefficients.

We proceed as follows:

  • the object WhittleFactory is created with either a specified order (p,q) or a range Ind_p \times Ind_q. By default, the Welch estimator (object Welch) is used with its default parameters.
  • for each order (p,q), the estimation of the parameters is done by maximizing the reduced equation of the Whittle likelihood function ([lik2]), thanks to the method build of the object WhittleFactory. This method applies to a time series or a process sample. If the user wants to get the quantified criteria AIC_c, AIC and BIC of the model ARMA(p,q), he has to specify it by giving a Point of size 0 (Point()) as input parameter of the method build.
  • the output of the estimation is, in all the cases, one unique ARMA: the ARMA with the specified order or the optimal one with respect to the AIC_c criterion.
  • in the case of a range Ind_p \times Ind_q, the user can get all the estimated models thanks to the method getHistory of the object WhittleFactory. If the build has been parameterized by a Point of size 0, the user also has access to all the quantified criteria.

The synthetic data is generated using the following 1-d ARMA process:

X_{0,t} + 0.4 X_{0,t-1} + 0.3 X_{0,t-2} + 0.2 X_{0,t-3} + 0.1 X_{0,t-4} = E_{0,t} + 0.4 E_{0,t-1} + 0.3 E_{0,t-2}

with the noise E defined as:

E \sim Triangular(-1, 0, 1)

In [5]:
from __future__ import print_function
import openturns as ot
import matplotlib.pyplot as plt
ot.RandomGenerator.SetSeed(0)
In [6]:
# Create an arma process

tMin = 0.0
n = 1000
timeStep = 0.1
myTimeGrid = ot.RegularGrid(tMin, timeStep, n)

myWhiteNoise = ot.WhiteNoise(ot.Triangular(-1.0, 0.0, 1.0), myTimeGrid)
myARCoef = ot.ARMACoefficients([0.4, 0.3, 0.2, 0.1])
myMACoef = ot.ARMACoefficients([0.4, 0.3])
arma = ot.ARMA(myARCoef, myMACoef, myWhiteNoise)

tseries = ot.TimeSeries(arma.getRealization())

# Create a sample of N time series from the process
N = 100
sample = arma.getSample(N)
In [7]:
# CASE 1 : we specify a (p,q) order

# Specify the order (p,q)
p = 4
q = 2

# Create the estimator
factory = ot.WhittleFactory(p, q)
print("Default spectral model factory = ", factory.getSpectralModelFactory())

# To set the spectral model factory
# For example, set WelchFactory as SpectralModelFactory
# with the Hanning filtering window
# The Welch estimator splits the time series in four blocs without overlap
myFilteringWindow = ot.Hanning()
mySpectralFactory = ot.WelchFactory(myFilteringWindow, 4, 0)
factory.setSpectralModelFactory(mySpectralFactory)
print("New spectral model factory = ", factory.getSpectralModelFactory())

# Estimate the ARMA model from a time series
# To get the quantified AICc, AIC and BIC criteria
arma42, criterion = factory.buildWithCriteria(tseries)
AICc, AIC, BIC = criterion[0:3]
print('AICc=', AICc, 'AIC=', AIC, 'BIC=', BIC)
arma42
Default spectral model factory =  class=WelchFactory window = class=FilteringWindows implementation=class=Hamming blockNumber = 1 overlap = 0
New spectral model factory =  class=WelchFactory window = class=FilteringWindows implementation=class=Hanning blockNumber = 4 overlap = 0
AICc= 771.8917262722518 AIC= 770.9344613149868 BIC= 824.530853637219
Out[7]:

ARMA(X_{0,t} - 0.214424 X_{0,t-1} + 0.432622 X_{0,t-2} + 0.203859 X_{0,t-3} + 0.0512422 X_{0,t-4} = E_{0,t} - 0.194383 E_{0,t-1} + 0.461067 E_{0,t-2}, E_t ~ Normal(mu = 0, sigma = 0.406619))

In [8]:
# CASE 2 : we specify a range of (p,q) orders
###################################

# Range for p
pIndices = [1, 2, 4]
# Range for q = [4,5,6]
qIndices = [4, 5, 6]

# Build a Whittle factory with default SpectralModelFactory (WelchFactory)
# this time using ranges of order p and q
factory_range = ot.WhittleFactory(pIndices, qIndices)

# Estimate the arma model from a process sample
arma_range, criterion = factory_range.buildWithCriteria(sample)
AICc, AIC, BIC = criterion[0:3]
print('AICc=', AICc, 'AIC=', AIC, 'BIC=', BIC)
arma_range
AICc= 4443.4456045627585 AIC= 4443.217962286336 BIC= 4516.222475664246
Out[8]:

ARMA(X_{0,t} + 0.382771 X_{0,t-1} + 0.185752 X_{0,t-2} = E_{0,t} + 0.385312 E_{0,t-1} + 0.192682 E_{0,t-2} - 0.191497 E_{0,t-3} - 0.102842 E_{0,t-4}, E_t ~ Normal(mu = 0, sigma = 0.409595))

In [9]:
# Results exploitation

# Get the white noise of the (best) estimated arma
arma_range.getWhiteNoise()
Out[9]:

WhiteNoise(Normal(mu = 0, sigma = 0.409595))