# Evaluation of uncertainties¶

The evaluation of uncertainties is a fundamental part of the measurement analysis in metrology. The analysis of dynamic measurements typically involves methods from signal processing, such as digital filtering, the discrete Fourier transform (DFT), or simple tasks like interpolation. For most of these tasks, methods are readily available, for instance, as part of scipy.signal. This module of PyDynamic provides the corresponding methods for the evaluation of uncertainties.

The package consists of the following modules:

## Uncertainty evaluation for the DFT¶

The PyDynamic.uncertainty.propagate_DFT module implements methods for the propagation of uncertainties in the application of the DFT, inverse DFT, deconvolution and multiplication in the frequency domain, transformation from amplitude and phase to real and imaginary parts and vice versa.

The corresponding scientific publications is
S. Eichstädt und V. Wilkens GUM2DFT — a software tool for uncertainty evaluation of transient signals in the frequency domain. Measurement Science and Technology, 27(5), 055001, 2016. [DOI: 10.1088/0957-0233/27/5/055001]

This module contains the following functions:

PyDynamic.uncertainty.propagate_DFT.GUM_DFT(x, Ux, N=None, window=None, CxCos=None, CxSin=None, returnC=False, mask=None)[source]

Calculation of the DFT with propagation of uncertainty

Calculation of the DFT of the time domain signal x and propagation of the squared uncertainty Ux associated with the time domain sequence x to the real and imaginary parts of the DFT of x.

Parameters: x (numpy.ndarray of shape (M,)) – vector of time domain signal values Ux (numpy.ndarray) – covariance matrix associated with x, shape (M,M) or vector of squared standard uncertainties, shape (M,) or noise variance as float N (int, optional) – length of time domain signal for DFT; N>=len(x) window (numpy.ndarray, optional of shape (M,)) – vector of the time domain window values CxCos (numpy.ndarray, optional) – cosine part of sensitivity matrix CxSin (numpy.ndarray, optional) – sine part of sensitivity matrix returnC (bool, optional) – if true, return sensitivity matrix blocks for later use mask (ndarray of dtype bool) – calculate DFT values and uncertainties only at those frequencies where mask is True F (numpy.ndarray) – vector of complex valued DFT values or of its real and imaginary parts UF (numpy.ndarray) – covariance matrix associated with real and imaginary part of F

References

PyDynamic.uncertainty.propagate_DFT.GUM_iDFT(F, UF, Nx=None, Cc=None, Cs=None, returnC=False)[source]

GUM propagation of the squared uncertainty UF associated with the DFT values F through the inverse DFT

The matrix UF is assumed to be for real and imaginary part with blocks: UF = [[u(R,R), u(R,I)],[u(I,R),u(I,I)]] and real and imaginary part obtained from calling rfft (DFT for real-valued signal)

Parameters: F (np.ndarray of shape (2M,)) – vector of real and imaginary parts of a DFT result UF (np.ndarray of shape (2M,2M)) – covariance matrix associated with real and imaginary parts of F Nx (int, optional) – number of samples of iDFT result Cc (np.ndarray, optional) – cosine part of sensitivities (without scaling factor 1/N) Cs (np.ndarray, optional) – sine part of sensitivities (without scaling factor 1/N) returnC (if true, return sensitivity matrix blocks (without scaling factor 1/N)) – x (np.ndarray) – vector of time domain signal values Ux (np.ndarray) – covariance matrix associated with x

References

PyDynamic.uncertainty.propagate_DFT.GUM_DFTfreq(N, dt=1)[source]

Return the Discrete Fourier Transform sample frequencies

Parameters: N (int) – window length dt (float) – sample spacing (inverse of sampling rate) f – Array of length n//2 + 1 containing the sample frequencies ndarray
PyDynamic.uncertainty.propagate_DFT.DFT_transferfunction(X, Y, UX, UY)[source]

Calculation of the transfer function H = Y/X in the frequency domain

Calculate the transfer function with X being the Fourier transform of the system’s input signal and Y that of the output signal.

Parameters: X (np.ndarray) – real and imaginary parts of the system’s input signal Y (np.ndarray) – real and imaginary parts of the system’s output signal UX (np.ndarray) – covariance matrix associated with X UY (np.ndarray) – covariance matrix associated with Y H (np.ndarray) – real and imaginary parts of the system’s frequency response UH (np.ndarray) – covariance matrix associated with H

This function only calls DFT_deconv.

PyDynamic.uncertainty.propagate_DFT.DFT_deconv(H, Y, UH, UY)[source]

Deconvolution in the frequency domain

GUM propagation of uncertainties for the deconvolution X = Y/H with Y and H being the Fourier transform of the measured signal and of the system’s impulse response, respectively. This function returns the covariance matrix as a tuple of blocks if too large for complete storage in memory.

Parameters: H (np.ndarray of shape (2M,)) – real and imaginary parts of frequency response values (N an even integer) Y (np.ndarray of shape (2M,)) – real and imaginary parts of DFT values UH (np.ndarray of shape (2M,2M)) – covariance matrix associated with H UY (np.ndarray of shape (2M,2M)) – covariance matrix associated with Y X (np.ndarray of shape (2M,)) – real and imaginary parts of DFT values of deconv result UX (np.ndarray of shape (2M,2M)) – covariance matrix associated with real and imaginary part of X

References

PyDynamic.uncertainty.propagate_DFT.DFT_multiply(Y, F, UY, UF=None)[source]

Multiplication in the frequency domain

GUM uncertainty propagation for multiplication in the frequency domain, where the second factor F may have an associated uncertainty. This method can be used, for instance, for the application of a low-pass filter in the frequency domain or the application of deconvolution as a multiplication with an inverse of known uncertainty.

Parameters: Y (np.ndarray of shape (2M,)) – real and imaginary parts of the first factor F (np.ndarray of shape (2M,)) – real and imaginary parts of the second factor UY (np.ndarray either shape (2M,) or shape (2M,2M)) – covariance matrix or squared uncertainty associated with Y UF (np.ndarray of shape (2M,2M)) – covariance matrix associated with F (optional), default is None YF (np.ndarray of shape (2M,)) – the product of Y and F UYF (np.ndarray of shape (2M,2M)) – the uncertainty associated with YF
PyDynamic.uncertainty.propagate_DFT.AmpPhase2DFT(A, P, UAP, keep_sparse=False)[source]

Transformation from magnitude and phase to real and imaginary parts

Calculate the vector F=[real,imag] and propagate the covariance matrix UAP associated with [A, P]

Parameters: A (np.ndarray of shape (N,)) – vector of magnitude values P (np.ndarray of shape (N,)) – vector of phase values (in radians) UAP (np.ndarray of shape (2N,2N)) – covariance matrix associated with (A,P) or vector of squared standard uncertainties [u^2(A),u^2(P)] keep_sparse (bool, optional) – whether to transform sparse matrix to numpy array or not F (np.ndarray) – vector of real and imaginary parts of DFT result UF (np.ndarray) – covariance matrix associated with F
PyDynamic.uncertainty.propagate_DFT.DFT2AmpPhase(F, UF, keep_sparse=False, tol=1.0, return_type='separate')[source]

Transformation from real and imaginary parts to magnitude and phase

Calculate the matrix U_AP = [[U1,U2],[U2^T,U3]] associated with magnitude and phase of the vector F=[real,imag] with associated covariance matrix U_F=[[URR,URI],[URI^T,UII]]

Parameters: F (np.ndarray of shape (2M,)) – vector of real and imaginary parts of a DFT result UF (np.ndarray of shape (2M,2M)) – covariance matrix associated with F keep_sparse (bool, optional) – if true then UAP will be sparse if UF is one-dimensional tol (float, optional) – lower bound for A/uF below which a warning will be issued concerning unreliable results return_type (str, optional) – If “separate” then magnitude and phase are returned as separate arrays. Otherwise the array [A, P] is returned

If return_type is separate:

Returns: A (np.ndarray) – vector of magnitude values P (np.ndarray) – vector of phase values in radians, in the range [-pi, pi] UAP (np.ndarray) – covariance matrix associated with (A,P)

Otherwise:

Returns: AP (np.ndarray) – vector of magnitude and phase values UAP (np.ndarray) – covariance matrix associated with AP
PyDynamic.uncertainty.propagate_DFT.AmpPhase2Time(A, P, UAP)[source]

Transformation from amplitude and phase to time domain

GUM propagation of covariance matrix UAP associated with DFT amplitude A and phase P to the result of the inverse DFT. Uncertainty UAP is assumed to be given for amplitude and phase with blocks: UAP = [[u(A,A), u(A,P)],[u(P,A),u(P,P)]]

Parameters: A (np.ndarray of shape (N,)) – vector of amplitude values P (np.ndarray of shape (N,)) – vector of phase values (in rad) UAP (np.ndarray of shape (2N,2N)) – covariance matrix associated with [A,P] x (np.ndarray) – vector of time domain values Ux (np.ndarray) – covariance matrix associated with x
PyDynamic.uncertainty.propagate_DFT.Time2AmpPhase(x, Ux)[source]

Transformation from time domain to amplitude and phase

Parameters: x (np.ndarray of shape (N,)) – time domain signal Ux (np.ndarray of shape (N,N)) – squared uncertainty associated with x A (np.ndarray) – amplitude values P (np.ndarray) – phase values UAP (np.ndarray) – covariance matrix associated with [A,P]
PyDynamic.uncertainty.propagate_DFT.Time2AmpPhase_multi(x, Ux, selector=None)[source]

Transformation from time domain to amplitude and phase

Perform transformation for a set of M signals of the same type.

Parameters: x (np.ndarray of shape (M,N)) – M time domain signals of length N Ux (np.ndarray of shape (M,)) – squared standard deviations representing noise variances of the signals x selector (np.ndarray of shape (L,), optional) – indices of amplitude and phase values that should be returned; default is 0:N-1 A (np.ndarray of shape (M,N)) – amplitude values P (np.ndarray of shape (M,N)) – phase values UAP (np.ndarray of shape (M, 3N)) – diagonals of the covariance matrices: [diag(UPP), diag(UAA), diag(UPA)]

## Uncertainty evaluation for digital filtering¶

This module contains functions for the propagation of uncertainties through the application of a digital filter using the GUM approach.

This modules contains the following functions:

Note

The Elster-Link paper for FIR filters assumes that the autocovariance is known and that noise is stationary!

PyDynamic.uncertainty.propagate_filter.FIRuncFilter(y, sigma_noise, theta, Utheta=None, shift=0, blow=None, kind='corr')[source]

Uncertainty propagation for signal y and uncertain FIR filter theta

Parameters: y (np.ndarray) – filter input signal sigma_noise (float or np.ndarray) – float: standard deviation of white noise in y 1D-array: interpretation depends on kind theta (np.ndarray) – FIR filter coefficients Utheta (np.ndarray) – covariance matrix associated with theta shift (int) – time delay of filter output signal (in samples) blow (np.ndarray) – optional FIR low-pass filter kind (string) – only meaningfull in combination with isinstance(sigma_noise, numpy.ndarray) “diag”: point-wise standard uncertainties of non-stationary white noise “corr”: single sided autocovariance of stationary (colored/corrlated) noise (default) x (np.ndarray) – FIR filter output signal ux (np.ndarray) – point-wise uncertainties associated with x

References

PyDynamic.uncertainty.propagate_filter.IIRuncFilter(x, noise, b, a, Uab)[source]

Uncertainty propagation for the signal x and the uncertain IIR filter (b,a)

Parameters: x (np.ndarray) – filter input signal noise (float) – signal noise standard deviation b (np.ndarray) – filter numerator coefficients a (np.ndarray) – filter denominator coefficients Uab (np.ndarray) – covariance matrix for (a[1:],b) y (np.ndarray) – filter output signal Uy (np.ndarray) – uncertainty associated with y

References

## Monte Carlo methods for digital filtering¶

The propagation of uncertainties via the FIR and IIR formulae alone does not enable the derivation of credible intervals, because the underlying distribution remains unknown. The GUM-S2 Monte Carlo method provides a reference method for the calculation of uncertainties for such cases.

This module implements Monte Carlo methods for the propagation of uncertainties for digital filtering.

This module contains the following functions:

PyDynamic.uncertainty.propagate_MonteCarlo.MC(x, Ux, b, a, Uab, runs=1000, blow=None, alow=None, return_samples=False, shift=0, verbose=True)[source]

Standard Monte Carlo method

Monte Carlo based propagation of uncertainties for a digital filter (b,a) with uncertainty matrix $$U_{\theta}$$ for $$\theta=(a_1,\ldots,a_{N_a},b_0,\ldots,b_{N_b})^T$$

Parameters: x (np.ndarray) – filter input signal Ux (float or np.ndarray) – standard deviation of signal noise (float), point-wise standard uncertainties or covariance matrix associated with x b (np.ndarray) – filter numerator coefficients a (np.ndarray) – filter denominator coefficients Uab (np.ndarray) – uncertainty matrix $$U_\theta$$ runs (int,optional) – number of Monte Carlo runs return_samples (bool, optional) – whether samples or mean and std are returned

If return_samples is False, the method returns:

Returns: y (np.ndarray) – filter output signal Uy (np.ndarray) – uncertainty associated with

Otherwise the method returns:

Returns: Y – array of Monte Carlo results np.ndarray

References

PyDynamic.uncertainty.propagate_MonteCarlo.SMC(x, noise_std, b, a, Uab=None, runs=1000, Perc=None, blow=None, alow=None, shift=0, return_samples=False, phi=None, theta=None, Delta=0.0)[source]

Sequential Monte Carlo method

Sequential Monte Carlo propagation for a digital filter (b,a) with uncertainty matrix $$U_{\theta}$$ for $$\theta=(a_1,\ldots,a_{N_a},b_0,\ldots,b_{N_b})^T$$

Parameters: x (np.ndarray) – filter input signal noise_std (float) – standard deviation of signal noise b (np.ndarray) – filter numerator coefficients a (np.ndarray) – filter denominator coefficients Uab (np.ndarray) – uncertainty matrix $$U_\theta$$ runs (int, optional) – number of Monte Carlo runs Perc (list, optional) – list of percentiles for quantile calculation blow (np.ndarray) – optional low-pass filter numerator coefficients alow (np.ndarray) – optional low-pass filter denominator coefficients shift (int) – integer for time delay of output signals return_samples (bool, otpional) – whether to return y and Uy or the matrix Y of MC results theta (phi,) – parameters for AR(MA) noise model $$\epsilon(n) = \sum_k \phi_k\epsilon(n-k) + \sum_k \theta_k w(n-k) + w(n)$$ with $$w(n)\sim N(0,noise_std^2)$$ Delta (float,optional) – upper bound on systematic error of the filter

If return_samples is False, the method returns:

Returns: y (np.ndarray) – filter output signal (Monte Carlo mean) Uy (np.ndarray) – uncertainties associated with y (Monte Carlo point-wise std) Quant (np.ndarray) – quantiles corresponding to percentiles Perc (if not None)

Otherwise the method returns:

Returns: Y – array of all Monte Carlo results np.ndarray

References

PyDynamic.uncertainty.propagate_MonteCarlo.UMC(x, b, a, Uab, runs=1000, blocksize=8, blow=1.0, alow=1.0, phi=0.0, theta=0.0, sigma=1, Delta=0.0, runs_init=100, nbins=1000, credible_interval=0.95)[source]

Batch Monte Carlo for filtering using update formulae for mean, variance and (approximated) histogram. This is a wrapper for the UMC_generic function, specialised on filters

Parameters: x (np.ndarray, shape (nx, )) – filter input signal b (np.ndarray, shape (nbb, )) – filter numerator coefficients a (np.ndarray, shape (naa, )) – filter denominator coefficients, normalization (a[0] == 1.0) is assumed Uab (np.ndarray, shape (naa + nbb - 1, )) – uncertainty matrix $$U_\theta$$ runs (int, optional) – number of Monte Carlo runs blocksize (int, optional) – how many samples should be evaluated for at a time blow (float or np.ndarray, optional) – filter coefficients of optional low pass filter alow (float or np.ndarray, optional) – filter coefficients of optional low pass filter phi (np.ndarray, optional,) – see misc.noise.ARMA noise model theta (np.ndarray, optional) – see misc.noise.ARMA noise model sigma (float, optional) – see misc.noise.ARMA noise model Delta (float, optional) – upper bound of systematic correction due to regularisation (assume uniform distribution) runs_init (int, optional) – how many samples to evaluate to form initial guess about limits nbins (int, list of int, optional) – number of bins for histogram credible_interval (float, optional) – must be in [0,1] central credible interval size

By default, phi, theta, sigma are chosen such, that N(0,1)-noise is added to the input signal.

Returns: y (np.ndarray) – filter output signal Uy (np.ndarray) – uncertainty associated with y_cred_low (np.ndarray) – lower boundary of credible interval y_cred_high (np.ndarray) – upper boundary of credible interval happr (dict) – dictionary keys: given nbin dictionary values: bin-edges val[“bin-edges”], bin-counts val[“bin-counts”]

References

• Eichstädt, Link, Harris, Elster [Eichst2012]
• ported to python in 2019-08 from matlab-version of Sascha Eichstaedt (PTB) from 2011-10-12
• copyright on updating formulae parts is by Peter Harris (NPL)
PyDynamic.uncertainty.propagate_MonteCarlo.UMC_generic(draw_samples, evaluate, runs=100, blocksize=8, runs_init=10, nbins=100, return_samples=False, n_cpu=1)[source]

Generic Batch Monte Carlo using update formulae for mean, variance and (approximated) histogram. Assumes that the input and output of evaluate are numeric vectors (but not necessarily of same dimension). If the output of evaluate is multi-dimensional, it will be flattened into 1D.

Parameters: draw_samples (function(int nDraws)) – function that draws nDraws from a given distribution / population needs to return a list of (multi dimensional) numpy.ndarrays evaluate (function(sample)) – function that evaluates a sample and returns the result needs to return a (multi dimensional) numpy.ndarray runs (int, optional) – number of Monte Carlo runs blocksize (int, optional) – how many samples should be evaluated for at a time runs_init (int, optional) – how many samples to evaluate to form initial guess about limits nbins (int, list of int, optional) – number of bins for histogram return_samples (bool, optional) – see return-value of documentation n_cpu (int, optional) – number of CPUs to use for multiprocessing, defaults to all available CPUs

Example

draw samples from multivariate normal distribution: draw_samples = lambda size: np.random.multivariate_normal(x, Ux, size)

build a function, that only accepts one argument by masking additional kwargs: evaluate = functools.partial(_UMCevaluate, nbb=b.size, x=x, Delta=Delta, phi=phi, theta=theta, sigma=sigma, blow=blow, alow=alow) evaluate = functools.partial(bigFunction, **dict_of_kwargs)

By default the method

Returns: y (np.ndarray) – mean of flattened/raveled simulation output i.e.: y = np.ravel(evaluate(sample)) Uy (np.ndarray) – covariance associated with y happr (dict) – dictionary of bin-edges and bin-counts output_shape (tuple) – shape of the unraveled simulation output can be used to reshape y and np.diag(Uy) into original shape

If return_samples is True, the method additionally returns all evaluated samples. This should only be done for testing and debugging reasons, as this removes all memory-improvements of the UMC-method.

Returns: sims – dict of samples and corresponding results of every evaluated simulation samples and results are saved in their original shape dict

References

## Uncertainty evaluation for interpolation¶

The PyDynamic.uncertainty.interpolation module implements methods for the propagation of uncertainties in the application of standard interpolation methods as provided by scipy.interpolate.interp1d.

This module contains the following function:

PyDynamic.uncertainty.interpolation.interp1d_unc(t_new: numpy.ndarray, t: numpy.ndarray, y: numpy.ndarray, uy: numpy.ndarray, kind: Optional[str] = 'linear') → Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray][source]

Interpolate arbitrary time series considering the associated uncertainties

The interpolation timestamps must lie within the range of the original timestamps. In addition, at least two of each of the original timestamps, measured values and associated measurement uncertainties are required and an equal number of each of these three.

Parameters: t_new ((M,) array_like) – The timestamps at which to evaluate the interpolated values. t ((N,) array_like) – timestamps in ascending order y ((N,) array_like) – corresponding measurement values uy ((N,) array_like) – corresponding measurement values’ uncertainties kind (str, optional) – Specifies the kind of interpolation for the measurement values as a string (‘previous’, ‘next’, ‘nearest’ or ‘linear’). t_new ((M,) array_like) – interpolation timestamps y_new ((M,) array_like) – interpolated measurement values uy_new ((M,) array_like) – interpolated measurement values’ uncertainties

References