# Model estimation¶

The estimation of the measurand in the analysis of dynamic measurements typically corresponds to a deconvolution problem. Therefore, a digital filter can be designed whose input is the measured system output signal and whose output is an estimate of the measurand. The package Model estimation implements methods for the design of such filters given an array of frequency response values or the reciprocal of frequency response values with associated uncertainties for the measurement system.

The package Model estimation also contains a function for the identification of transfer function models.

The package consists of the following modules:

## Fitting filters to frequency response or reciprocal¶

The module PyDynamic.model_estimation.fit_filter contains several functions to carry out a least-squares fit to a given complex frequency response and the design of digital deconvolution filters by least-squares fitting to the reciprocal of a given frequency response each with associated uncertainties.

This module contains the following functions:

PyDynamic.model_estimation.fit_filter.LSFIR(H, N, tau, f, Fs, Wt=None)[source]

Least-squares fit of a digital FIR filter to a given frequency response.

Parameters
• H ((complex) frequency response values of shape (M,)) –

• N (FIR filter order) –

• tau (delay of filter) –

• f (frequencies of shape (M,)) –

• Fs (sampling frequency of digital filter) –

• Wt ((optional) vector of weights of shape (M,) or shape (M,M)) –

Returns

Return type

filter coefficients bFIR (ndarray) of shape (N+1,)

PyDynamic.model_estimation.fit_filter.LSIIR(Hvals, Nb, Na, f, Fs, tau=0, justFit=False)[source]

Least-squares IIR filter fit to a given frequency response.

This method uses Gauss-Newton non-linear optimization and pole mapping for filter stabilization

Parameters
• Hvals (numpy array of (complex) frequency response values of shape (M,)) –

• Nb (integer numerator polynomial order) –

• Na (integer denominator polynomial order) –

• f (numpy array of frequencies at which Hvals is given of shape) –

• (M

• )

• Fs (sampling frequency) –

• tau (integer initial estimate of time delay) –

• justFit (boolean, when true then no stabilization is carried out) –

Returns

• b,a (IIR filter coefficients as numpy arrays)

• tau (filter time delay in samples)

References

PyDynamic.model_estimation.fit_filter.invLSFIR(H, N, tau, f, Fs, Wt=None)[source]

Least-squares fit of a digital FIR filter to the reciprocal of a given frequency response.

Parameters
• H (np.ndarray of shape (M,) and dtype complex) – frequency response values

• N (int) – FIR filter order

• tau (float) – delay of filter

• f (np.ndarray of shape (M,)) – frequencies

• Fs (float) – sampling frequency of digital filter

• Wt (np.ndarray of shape (M,) - optional) – vector of weights

Returns

bFIR – filter coefficients

Return type

np.ndarray of shape (N,)

References

PyDynamic.model_estimation.fit_filter.invLSFIR_unc(H, UH, N, tau, f, Fs, wt=None, verbose=True, trunc_svd_tol=None)[source]

Design of FIR filter as fit to reciprocal of frequency response values with uncertainty

Least-squares fit of a digital FIR filter to the reciprocal of a frequency response for which associated uncertainties are given for its real and imaginary part. Uncertainties are propagated using a truncated svd and linear matrix propagation.

Parameters
• H (np.ndarray of shape (M,)) – frequency response values

• UH (np.ndarray of shape (2M,2M)) – uncertainties associated with the real and imaginary part

• N (int) – FIR filter order

• tau (float) – delay of filter

• f (np.ndarray of shape (M,)) – frequencies

• Fs (float) – sampling frequency of digital filter

• wt (np.ndarray of shape (2M,) - optional) – array of weights for a weighted least-squares method

• verbose (bool, optional) – whether to print statements to the command line

• trunc_svd_tol (float) – lower bound for singular values to be considered for pseudo-inverse

Returns

• b (np.ndarray of shape (N+1,)) – filter coefficients of shape

• Ub (np.ndarray of shape (N+1,N+1)) – uncertainties associated with b

References

PyDynamic.model_estimation.fit_filter.invLSFIR_uncMC(H, UH, N, tau, f, Fs, verbose=True)[source]

Design of FIR filter as fit to reciprocal of frequency response values with uncertainty

Least-squares fit of a FIR filter to the reciprocal of a frequency response for which associated uncertainties are given for its real and imaginary parts. Uncertainties are propagated using a Monte Carlo method. This method may help in cases where the weighting matrix or the Jacobian are ill-conditioned, resulting in false uncertainties associated with the filter coefficients.

Parameters
• H (np.ndarray of shape (M,) and dtype complex) – frequency response values

• UH (np.ndarray of shape (2M,2M)) – uncertainties associated with the real and imaginary part of H

• N (int) – FIR filter order

• tau (int) – time delay of filter in samples

• f (np.ndarray of shape (M,)) – frequencies corresponding to H

• Fs (float) – sampling frequency of digital filter

• verbose (bool, optional) – whether to print statements to the command line

Returns

• b (np.ndarray of shape (N+1,)) – filter coefficients of shape

• Ub (np.ndarray of shape (N+1, N+1)) – uncertainties associated with b

References

PyDynamic.model_estimation.fit_filter.invLSIIR(Hvals, Nb, Na, f, Fs, tau, justFit=False, verbose=True)[source]

Design of a stable IIR filter as fit to reciprocal of frequency response values

Least-squares fit of a digital IIR filter to the reciprocal of a given set of frequency response values using the equation-error method and stabilization by pole mapping and introduction of a time delay.

Parameters
• Hvals (np.ndarray of shape (M,) and dtype complex) – frequency response values.

• Nb (int) – order of IIR numerator polynomial.

• Na (int) – order of IIR denominator polynomial.

• f (np.ndarray of shape (M,)) – frequencies corresponding to Hvals

• Fs (float) – sampling frequency for digital IIR filter.

• tau (float) – initial estimate of time delay for filter stabilization.

• justFit (bool) – if True then no stabilization is carried out.

• verbose (bool) – If True print some more detail about input parameters.

Returns

• b (np.ndarray) – The IIR filter numerator coefficient vector in a 1-D sequence

• a (np.ndarray) – The IIR filter denominator coefficient vector in a 1-D sequence

• tau (int) – time delay (in samples)

References

PyDynamic.model_estimation.fit_filter.invLSIIR_unc(H, UH, Nb, Na, f, Fs, tau=0)[source]

Design of stabel IIR filter as fit to reciprocal of given frequency response with uncertainty

Least-squares fit of a digital IIR filter to the reciprocal of a given set of frequency response values with given associated uncertainty. Propagation of uncertainties is carried out using the Monte Carlo method.

Parameters
• H (np.ndarray of shape (M,) and dtype complex) – frequency response values.

• UH (np.ndarray of shape (2M,2M)) – uncertainties associated with real and imaginary part of H

• Nb (int) – order of IIR numerator polynomial.

• Na (int) – order of IIR denominator polynomial.

• f (np.ndarray of shape (M,)) – frequencies corresponding to H

• Fs (float) – sampling frequency for digital IIR filter.

• tau (float) – initial estimate of time delay for filter stabilization.

Returns

• b,a (np.ndarray) – IIR filter coefficients

• tau (int) – time delay (in samples)

• Uba (np.ndarray of shape (Nb+Na+1, Nb+Na+1)) – uncertainties associated with [a[1:],b]

References

## Identification of transfer function models¶

The module PyDynamic.model_estimation.fit_transfer contains a function for the identification of transfer function models.

This module contains the following function:

PyDynamic.model_estimation.fit_transfer.fit_som(f, H, UH=None, weighting=None, MCruns=None, scaling=0.001)[source]

Fit second-order model to complex-valued frequency response

Fit second-order model (spring-damper model) with parameters $$S_0, delta$$ and $$f_0$$ to complex-valued frequency response with uncertainty associated with real and imaginary parts.

For a transformation of an uncertainty associated with amplitude and phase to one associated with real and imaginary parts, see PyDynamic.uncertainty.propagate_DFT.AmpPhase2DFT.

Parameters
• f (np.ndarray of shape (M,)) – vector of frequencies

• H (np.ndarray of shape (2M,)) – real and imaginary parts of measured frequency response values at frequencies f

• UH (np.ndarray of shape (2M,) or (2M,2M)) – uncertainties associated with real and imaginary parts When UH is one-dimensional, it is assumed to contain standard uncertainties; otherwise it is taken as covariance matrix. When UH is not specified no uncertainties assoc. with the fit are calculated.

• weighting (str or array) – Type of weighting (None, ‘diag’, ‘cov’) or array of weights ( length two times of f)

• MCruns (int, optional) – Number of Monte Carlo trials for propagation of uncertainties. When MCruns is ‘None’, matrix multiplication is used for the propagation of uncertainties. However, in some cases this can cause trouble.

• scaling (float) – scaling of least-squares design matrix for improved fit quality

Returns

• p (np.ndarray) – vector of estimated model parameters [S0, delta, f0]

• Up (np.ndarray) – covariance associated with parameter estimate