Estimate a stationary covariance function¶
The objective here is to estimate a stationary covariance model from data.
The library builds an estimation of the stationary covariance function on a ProcessSample or TimeSeries using the previous algorithm implemented in the StationaryCovarianceModelFactory class. The result consists in a UserDefinedStationaryCovarianceModel which is easy to manipulate.
Such an object is composed of a time grid and a collection of square matrices of dimension d. corresponds to the number of time steps of the final time grid on which the covariance is estimated. When estimated from a time series , the UserDefinedStationaryCovarianceModel may have a time grid different from the initial time grid of the time series.
from __future__ import print_function import openturns as ot
# Create some 1-d normal process data with an Exponential covariance model # Dimension parameter dim = 1 # Create the time grid t0 = 0.0 N = 300 t1 = 20.0 dt = (t1 - t0) / N tgrid = ot.RegularGrid(t0, dt, N) # Create the covariance model amplitude = [1.0] * dim scale = [1.0] * dim covmodel = ot.ExponentialModel(scale, amplitude) # Create a stationary Normal process with that covariance model process = ot.GaussianProcess(covmodel, tgrid) # Create a time series and a sample of time series tseries = process.getRealization() sample = process.getSample(1000)
# Build a factory of stationary covariance function covarianceFactory = ot.StationaryCovarianceModelFactory() # Set the spectral factory algorithm segmentNumber = 5 spectralFactory = ot.WelchFactory(ot.Hanning(), segmentNumber) covarianceFactory.setSpectralModelFactory(spectralFactory) # Check the current spectral factory print(covarianceFactory.getSpectralModelFactory())
class=WelchFactory window = class=FilteringWindows implementation=class=Hanning blockNumber = 5 overlap = 0.5
# Case 1 : Estimation on a ProcessSample # The spectral model factory computes the spectral density function # without using the block and overlap arguments of the Welch factories estimatedModel_PS = covarianceFactory.build(sample) # Case 2 : Estimation on a TimeSeries # The spectral model factory compute the spectral density function using # the block and overlap arguments of spectral model factories estimatedModel_TS = covarianceFactory.build(tseries) # Evaluate the covariance function at each time step # Care : if estimated from a time series, the time grid has changed for i in range(N): tau = tgrid.getValue(i) cov = estimatedModel_PS(tau)
# Drawing... sampleValueEstimated = ot.Sample(N, 1) sampleValueModel = ot.Sample(N, 1) for i in range(N): t = tgrid.getValue(i) for j in range(i - 1): s = tgrid.getValue(j) estimatedValue = estimatedModel_PS(t, s) modelValue = covmodel(t, s) if j == 0: sampleValueEstimated[i, 0] = estimatedValue[0, 0] sampleValueModel[i, 0] = modelValue[0, 0] sampleT = tgrid.getVertices() graph = ot.Graph('Covariance estimation', 'time', 'Covariance value C(0,t)', True) curveEstimated = ot.Curve(sampleT, sampleValueEstimated, 'Estimated model') graph.add(curveEstimated) curveModel = ot.Curve(sampleT, sampleValueModel, 'Exact model') curveModel.setColor('red') graph.add(curveModel) graph.setLegendPosition('topright') graph