Estimate a scalar ARMA process

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 , OpenTURNS fits a model to the data by estimating the coefficients and the variance of the white noise.

If the User specifies a range of orders , where and , We find the best model that fits to the data and estimates the corresponding coefficients.

We proceed as follows:

• the object WhittleFactory is created with either a specified order or a range . By default, the Welch estimator (object Welch) is used with its default parameters.

• for each order , 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 and BIC of the model , 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 criterion.

• in the case of a range , 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:

with the noise defined as:

import openturns as ot
import matplotlib.pyplot as plt
ot.RandomGenerator.SetSeed(0)
ot.Log.Show(ot.Log.NONE)

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)

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 Hann filtering window
# The Welch estimator splits the time series in four blocs without overlap
myFilteringWindow = ot.Hann()
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

Out:

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=Hann blockNumber = 4 overlap = 0
AICc= 772.0387560411838 AIC= 771.0814910839188 BIC= 824.677883406151

ARMA(X_{0,t} - 0.189305 X_{0,t-1} + 0.424708 X_{0,t-2} + 0.204215 X_{0,t-3} + 0.0584364 X_{0,t-4} = E_{0,t} - 0.168415 E_{0,t-1} + 0.452162 E_{0,t-2}, E_t ~ Normal(mu = 0, sigma = 0.406733))

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

Out:

AICc= 4443.580404874942 AIC= 4443.35276259852 BIC= 4516.35727597643

ARMA(X_{0,t} + 0.419883 X_{0,t-1} + 0.176036 X_{0,t-2} = E_{0,t} + 0.422707 E_{0,t-1} + 0.183076 E_{0,t-2} - 0.19104 E_{0,t-3} - 0.110271 E_{0,t-4}, E_t ~ Normal(mu = 0, sigma = 0.409622))

Results exploitation

# Get the white noise of the (best) estimated arma
arma_range.getWhiteNoise()

WhiteNoise(Normal(mu = 0, sigma = 0.409622))