Estimate a spectral density function

The objective of this example is to estimate the spectral density function from data, which can be a sample of time series or one time series.

The following example illustrates the case where the available data is a sample of realizations of the process, defined on the time grid , discretized every . The spectral model of the process is the Cauchy model parameterized by and .

The figure draws the graph of the real spectral model and its estimation from the sample of time series.

import openturns as ot
import openturns.viewer as otv

Generate some data

# Create the time grid
# In the context of the spectral estimate or Fourier transform use,
# we use data blocs with size of form 2^p
tMin = 0.0
tstep = 0.1
size = 2**12
tgrid = ot.RegularGrid(tMin, tstep, size)

# We fix the parameter of the Cauchy model
amplitude = [5.0]
scale = [3.0]
model = ot.CauchyModel(amplitude, scale)
process = ot.SpectralGaussianProcess(model, tgrid)

# Get a time series or a sample of time series
tseries = process.getRealization()
sample = process.getSample(1000)

Build a spectral model factory

segmentNumber = 10
overlapSize = 0.3
factory = ot.WelchFactory(ot.Hann(), segmentNumber, overlapSize)

Estimation on a TimeSeries or on a ProcessSample

estimatedModel_TS = factory.build(tseries)
estimatedModel_PS = factory.build(sample)

Change the filtering window

factory.setFilteringWindows(ot.Hamming())

Get the frequencyGrid

frequencyGrid = ot.SpectralGaussianProcess(estimatedModel_PS, tgrid).getFrequencyGrid()

# With the model, we want to compare values
# We compare values computed with theoritical values
plotSample = ot.Sample(frequencyGrid.getN(), 3)

# Loop of comparison ==> data are saved in plotSample
for k in range(frequencyGrid.getN()):
    freq = frequencyGrid.getStart() + k * frequencyGrid.getStep()
    plotSample[k, 0] = freq
    plotSample[k, 1] = abs(estimatedModel_PS(freq)[0, 0])
    plotSample[k, 2] = abs(model(freq)[0, 0])

# Some cosmetics : labels, legend position, ...
graph = ot.Graph(
    "Estimated spectral function - Validation",
    "Frequency",
    "Spectral density function",
    True,
    "upper right",
)
graph.setLogScale(ot.GraphImplementation.LOGY)

# The first curve is the estimate density as function of frequency
curve1 = ot.Curve(plotSample.getMarginal([0, 1]))
curve1.setColor("blue")
curve1.setLegend("estimate model")

# The second curve is the theoritical density as function of frequency
curve2 = ot.Curve(plotSample.getMarginal([0, 2]))
curve2.setColor("red")
curve2.setLegend("Cauchy model")

graph.add(curve1)
graph.add(curve2)
view = otv.View(graph)
otv.View.ShowAll()