J. Sens. Sens. Syst., 6, 97–105, 2017www.j-sens-sens-syst.net/6/97/2017/doi:10.5194/jsss-6-97-2017© Author(s) 2017. CC Attribution 3.0 License.

Evaluation of dynamic measurement uncertainty –an open-source software package to bridge

theory and practice

Sascha Eichstädt1, Clemens Elster1, Ian M. Smith2, and Trevor J. Esward2

1Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany2National Physical Laboratory, Teddington, UK

Correspondence to: Sascha Eichstädt (sascha.eichstaedt@ptb.de)

Received: 1 September 2016 – Revised: 5 December 2016 – Accepted: 17 January 2017 – Published: 14 February 2017

Abstract. The analysis of dynamic measurements provides numerous challenges that significantly limit theuse of existing calibration facilities and mathematical methodologies. For instance, dynamic measurement anal-ysis requires the application of methods from digital signal processing, system and control theory, and mul-tivariate statistics. The design of digital filters and the corresponding evaluation of measurement uncertaintyfor high-dimensional measurands are particularly challenging. Several international research projects involv-ing national metrology institutes (NMIs), academia and industry have developed mathematical, statistical andtechnical methodologies for the treatment of dynamic measurements at NMI level. The aim of the Europeanresearch project 14SIP08 is the development of guidelines and software to extend the applicability of thosemethodologies to a wider range of users. This paper outlines the required activities towards a traceability chainfor dynamic measurements from NMIs to industrial applications. A key aspect is the development and provisionof a new open-source software package. The software is freely available, open for non-commercial distribution,and contains the most important data analysis tools for dynamic measurements.

1 Introduction

The analysis of dynamic measurements, i.e. measurementswhere at least one of the quantities of interest is time-dependent, is becoming increasingly important in metrologyand industry. Dynamic measurements are encountered in awide range of application areas, covering, for instance, singlesensors to complex sensor networks, and measured quantitieschanging on scales from picoseconds up to several minutes.Examples of dynamic measurements of mechanical quanti-ties can be found in, for instance, (Link et al., 2005) for accel-eration, in (Schlegel et al., 2012) and (Kobusch et al., 2015)for force, in (Klaus et al., 2015) for torque and in (Gardner,1981), (Matthews et al., 2014) and (Wilkens and Koch, 2004)for pressure. A more general overview on the current state ofdynamic measurements in industrial applications is given in(Schäfer, 2015). Dynamic measurement of electrical quanti-ties is covered, for instance, by (Younan et al., 1991), (Haleet al., 2009) and (Humphreys et al., 2015).

Despite the widespread occurrence of dynamic measure-ments, there is a lack of guidelines and standards for theirtreatment, application and analysis. For static measurements,i.e. measurements where no quantity of interest is time-dependent, the Guide to the Expression of Uncertainty inMeasurement (GUM) and its supplements (BIPM et al.,2008a, b, 2011) are widely considered as quasi-standards re-garding the evaluation of uncertainty. These documents haveled to the development of many software packages of vary-ing complexity, which provide easy-to-use implementationsof the GUM framework. This, together with the availabilityof international standards with uncertainty evaluation basedon the GUM framework, has led to an acceptance and ap-plication of metrologically validated uncertainty treatment.Moreover, it provides the foundation of traceability for staticmeasurements.

In contrast, the situation for dynamic measurements ismore complicated. Currently, there is a lack of harmonizedvocabulary, mathematical and statistical modelling, and mea-

Published by Copernicus Publications on behalf of the AMA Association for Sensor Technology.

98 S. Eichstädt et al.: Dynamic uncertainty

surement analysis, as outlined in (Eichstädt et al., 2016),(Ruhm, 2016) and others.

Dynamic metrology is a very active field of development,and various approaches to the evaluation and propagationof uncertainty can be found in the literature. For instance,on-line evaluation of uncertainty in the application of fi-nite impulse response (FIR) filters is addressed by (Elsterand Link, 2008), and infinite impulse response (IIR) filtersby (Link and Elster, 2009); efficient Monte Carlo meth-ods for uncertainty propagation is presented in (Eichstädtet al., 2012), the efficient reporting of high-dimensional co-variance matrices is addressed by (Humphreys et al., 2015),regularized deconvolution in the frequency domain is con-sidered by (Hale et al., 2009) and (Dienstfrey and Hale,2014), and propagation of uncertainty in the application ofthe discrete Fourier transform (DFT) is addressed by (Eich-städt and Wilkens, 2016). Moreover, the European MetrologyResearch Programme (EMRP) projects IND09, “Traceabledynamic measurement of mechanical quantities1”, (2011–2014) and IND16, “Metrology for ultra-fast electronics andhigh-speed communications2”, (2011–2014) laid the founda-tions for primary dynamic calibration of force, torque andpressure sensors, as well as bridge amplifiers and ultra-fastelectronic devices. However, application of the methods de-veloped within IND09 and IND16 is still mostly limitedto national metrology institutes (NMIs). Consequently, themain goal of the European Metrology Programme for In-novation and Research (EMPIR) project 14SIP08 Standardsand software to maximize end user uptake of NMI calibra-tions of dynamic force, torque and pressure sensors3 (2015–2018) is to bridge the gap between the analysis of dynamicmeasurements at NMI-level and that within industry. There-fore, NMIs PTB (Physikalisch-Technische Bundesanstalt,Germany) and NPL (National Physical Laboratory, UK), to-gether with international companies HBM GmbH and Rolls-Royce Ltd., aim to develop practical guidelines, tutorials,training material and software. In this contribution we out-line the challenges to be tackled by the analysis of dynamicmeasurements, indicate recent publications on the state ofthe art at NMI level, and give an introduction to the pub-licly available open-source software package PyDynamic4

being developed within 14SIP08. The newly developed soft-ware allows for the first time an off-the-shelf application ofNMI-level data analysis and measurement uncertainty eval-uation methods. As mentioned above, this is a pre-requisitefor achieving a wide acceptance and application of dynamicmeasurement analysis. Moreover, a wide acceptance and val-idation of this software is ensured by a clear documentationas part of the source code and in terms of documented ex-amples together with a transparent and open system for the

1https://www.ptb.de/emrp/ind09.html2https://www.ptb.de/emrp/ultrafast.html3http://mathmet.org/projects/14SIP084https://github.com/eichstaedtPTB/PyDynamic

Figure 1. Basic outline of a dynamic measurement with analogue-to-digital conversion (ADC) of the measured system output beforeprocessing.

documentation of the software development using a publicrepository. Finally, the deployment through the establishedplatform PyPi5 allows for an easy installation with the sim-ple command pip install PyDynamic.

2 Development of standards for dynamicmeasurements

Harmonization and standardization are underpinning mostof today’s metrology and industrial areas where compara-bility, conformity and quality assurance play an importantrole. The Guide to the Expression of Uncertainty in Mea-surement (GUM) (BIPM et al., 2008a) and its supplements(BIPM et al., 2008b, 2011) represent a well-established foun-dation of an uncertainty framework that can be applied to alarge variety of application areas. It is based on a clear def-inition of the measurand, i.e. the quantity of interest, and amathematical model for its evaluation. An important aspectof metrology research for dynamic measurements is thus thedevelopment of a framework that allows the adaption of theGUM methodology for dynamic metrology. Here we give acomprehensive overview for the challenges to be addressedby such adaptions and provide questions and tasks to be con-sidered in future standardization activities.

For most applications in dynamic metrology, the analysisof dynamic measurements follows the basic workflow illus-trated in Fig. 1.

The measurand is thus the sequence Y=(Y [1],Y [2], . . .Y [N ])T of discrete time values and, conse-quently, its uncertainty is a covariance matrix of dimensionN ×N , with N typically larger than 1000. In some ap-plications, certain parameters are to be derived from thissequence. Such single parameter values can, for instance, bepositive and negative peak values, an integral over a certaintime interval, the rise time of a step or the frequency of anoscillation. However, the propagation of uncertainties tosuch parameters often require knowledge of the uncertaintyover a certain time interval. The typical example is thecalculation of an integral which also requires correlationsbetween different time instants to be accounted for. Thus,we here consider as measurand the whole sequence Y.

Although the top-level workflow for dynamic measure-ments is the same as that for static measurements, the in-

5https://pypi.python.org

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S. Eichstädt et al.: Dynamic uncertainty 99

dividual low-level steps differ significantly for the followingreasons:

1. The measurement system input is a function of time;

2. The measuring device is a dynamic system (often linearand time-invariant);

3. The estimation task requires the solution of an ill-posedinverse problem;

4. The measurand is a high-dimensional multivariatequantity.

In contrast, (1) in static measurements all quantities arestatic, i.e. time dependence can be neglected; (2) the mea-surement system is usually not dynamic, but can be describedby algebraic equations; (3) the model for the evaluation ofa static measurand is usually not an ill-posed inverse prob-lem, but an algebraic equation; and (4) measurands in staticmeasurements are typically univariate or of small dimension(< 10) whereas dynamic measurands may contain severalthousand values. Altogether, the distinction between staticand dynamic measurements is that, for the latter, the fre-quency content of the involved time series in relation to thefrequency-dependent (i.e. dynamic) behaviour of the measur-ing device has to be taken into account. For instance, whenthe bandwidth of the dynamic measurand exceeds that of themeasuring system employed, significant time-dependent er-rors are to be expected in the output of the measuring system.For the compensation and correction of such errors, method-ologies from static measurement analysis are not sufficient.

These differences pose various challenges for metrologyand require the development of a new metrological vocabu-lary, the adaptation of methods from signal processing andsystem theory for metrological purposes, and the harmoniza-tion of regularization methods regarding the correspondingevaluation of uncertainty. In particular:

1. Quantities whose values are continuous functions oftime would require the translation of the GUM uncer-tainty framework to the treatment of stochastic pro-cesses as described in (Eichstädt, 2012). In order toavoid the corresponding mathematical complexities, adiscretized measurand may be considered instead. Thatis, in the workflow in Fig. 1 the discrete-time sequenceY [n] ≡ Y (tn) is considered as the measurand in orderto enable treatment within the GUM uncertainty frame-work.

2. Estimation of the measurand requires knowledge of thedynamic behaviour of the measurement system. Hence,a calibration has to identify and quantify the frequency-dependent characteristics of the complete measurementchain. As a consequence, measurement principles fromthe static case do not transfer to the dynamic case.

3. The estimation task is a mathematically ill-posed in-verse problem, which requires some kind of regulariza-tion to obtain valuable results. Although many regular-ization methods can be found in the literature, the ma-jority of the available approaches are heuristic and theirapplication to metrology is an ongoing topic of research.In particular, evaluation of the uncertainty contributionof the regularization is a challenging task, because it in-corporates prior knowledge about the measurand in theestimation process. This approach is common, for in-stance, in Bayesian statistical inference, but is not yetconsidered within the GUM uncertainty framework.

4. The reporting and dissemination of a dynamic measure-ment result cannot be carried out in the same way asfor static measurements, due to the high dimensional-ity of the measurand. Typically, a dynamic measurandis a time series of dimension greater than 1000 withits uncertainty being a corresponding covariance matrix.Conventional reporting for static measurements are of-ten based on a printed report with a table for the uncer-tainty budget. This kind of reporting is thus infeasiblefor dynamic measurands.

Several research efforts in dynamic metrology have devel-oped initial answers to some of the challenges listed above.For instance, primary dynamic calibration methods for sev-eral mechanical and electrical quantities have been devel-oped at NMIs during the last few years – see Sect. 1 andreferences therein.

Despite the availability of many publications on dynamicmetrology, the translation into international standards andguidelines is still at an early stage. Some national guide-lines, such as the draft German DKD-R 3–10 on dynamiccalibration of uni-axial testing machines, and internationalstandards, such as (ISO 16063-43:2015) on methods for thecalibration of vibration and shock transducers, directly ad-dress dynamic calibration. The majority of current standardsand guidelines, however, either refer to the lack of commonmethods and harmonized treatment and the general need forresearch or they are limited to static measurements only.

In a first step, a harmonized vocabulary has to be deter-mined. For instance, (Ruhm, 2016) proposes calling dynamicmeasurement devices “systems” and the dynamic quantities“signals”. Then, system and control theory are the founda-tion for a comprehensive vocabulary in dynamic metrology.First attempts in this regard are made within EMPIR 14SIP08by providing input to the BIPM JCGM working group 1 forthe development of the third supplement to the GUM, whichwill focus on the topic of modelling. Based on a commonvocabulary, guidelines and standards for dynamic calibrationcan be developed in a consistent way. Many standardizationbodies already have technical committees which are work-ing on such documents. The required mathematical tools,though, are often much more complicated than the corre-sponding methods in the realm of static measurements. For

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100 S. Eichstädt et al.: Dynamic uncertainty

certain examples this may result in the acceptance of largererrors and possibly unreliable uncertainties in order to ben-efit from easily applicable mathematical methods. For in-stance, the (CIE TC2-60:2013) guideline “Effect of instru-mental bandpass function and measurement interval on spec-tral quantities” advises the use of methods related to classi-cal approaches in that field despite well-known drawbacks,because of their easier application. In order to compensatefor the mathematical complexity of a superior method, PTBhad developed a software tool with a graphical user inter-face; see (Eichstädt et al., 2013). This resulted in a wideracceptance and application of the method, which would nothave been possible with the availability of the method alone6.Similar situations can be found in many other applications.For instance, the parametric dynamic calibration method inISO 16063 requires the availability of a measured frequencyresponse with associated uncertainties and the propagationof uncertainties to the estimated transducer model param-eters. The required mathematical methods are beyond thestandard toolbox of most dedicated laboratories. Another ex-ample can be found in the standard (IEC 62127-1:2007-11)for hydrophones used for the characterization of medical ul-trasound devices. There, the need of deconvolution for non-ideal hydrophones is identified. An incorporation of respec-tive mathematical procedures, though, is postponed until eas-ily applicable approaches are available. Therefore, several re-search activities are on the way in order to lay the foundationfor a revision of this standard. The availability of approach-able methods and ready-to-use software will be an importantaspect for the acceptance of the revision in practice.

3 PyDynamic – software for dynamic metrology

Many tasks in dynamic metrology involve the applicationof signal processing, for which ready-to-use implementa-tions are available in almost all major software packages.These software implementations, though, lack the corre-sponding evaluation of uncertainty. As a consequence, un-certainty evaluation is frequently undertaken using eitherrule-of-thumb methods or time-consuming simulation ap-proaches, or is neglected completely. The EMPIR project14SIP08 develops a user-friendly software environment tocarry out data processing for dynamic metrology. The soft-ware is called PyDynamic and it implements recently pub-lished mathematical and statistical methods required to carryout the workflow shown in Fig. 1. Since the methods havebeen published elsewhere, we will focus on the demonstra-tion of their simple application by using PyDynamic. There-fore, the currently implemented methods are illustrated usingthree typical tasks in the analysis of dynamic measurements.

6The software can be downloaded free of charge from the PTBwebsite.

3.1 Design of a compensation filter

Estimation of the dynamic measurand in the workflow de-picted in Fig. 1 can be undertaken through the application ofa digital compensation filter; cf. (Eichstädt et al., 2010). Tothis end, a digital finite impulse response (FIR) filter,

g(z)=M∑k=0

bkz−k, (1)

can be designed such that its frequency response,

G(e−jω/Fs )=M∑k=0

bke−jkω/Fs , (2)

approximates that of the inverse measurement system in acertain frequency interval, i.e.

H (jω)G(e−jω/Fs )e−jωτ ≈ 1, |ω| ≤ ω1, (3)

with H (jω) the frequency response of the measurement sys-tem. The time delay τ = n0Ts is introduced as a means ofaddressing the unphysical nature of the inverse system; cf.(Eichstädt et al., 2010). An example of such a filter is shownin Fig. 2.

Provided that the frequency response of the measurementsystem is available at a set of frequencies, the design of acompensation filter can be carried out by solving the lin-ear least-squares problem for the filter coefficient vector b

of length M + 1:

b = argminb(H −Db)>W−1 (H −Db) , (4)

with D the design matrix of dimension 2N × (M + 1), W achosen 2N × 2N symmetric weighting matrix, and the 2Nfrequency response values of the measurement system ex-pressed in terms of real and imaginary parts,

H = (<H (jω1), . . .,=H (jωN )) . (5)

See (Elster and Link, 2008). Typically, these values are deter-mined by dynamic calibration experiments or derived frominformation provided by the manufacturer. Thus, the valuesof H are accompanied by a statement of their uncertainty UHwhich has to be propagated to an uncertainty Ub associatedwith the filter coefficients b. Since the filter coefficient esti-mate is evaluated by means of weighted linear least-squares,the associated uncertainty is the covariance matrix,

Ub =(

D>WD)−1

D>WUHWD(

D>WD)−1

. (6)

Care must be taken to avoid numerical errors that may ariseif the design matrix is ill-conditioned. In this case, truncatedsingular value decomposition can be used to calculate a sta-ble pseudo-inverse; see (Elster and Link, 2008). The deriva-tion of a mathematical model for the evaluation of the mea-surand thus comprises (i) the provision of the frequency re-sponse values with associated uncertainties, (ii) the formula-tion of the filter estimation problem as a least-squares model,

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Figure 2. Frequency response of the measurement system of theFIR deconvolution filter and the resulting compensation as a productof the measurement system and deconvolution filter.

(iii) the numerical solution of the (weighted) least-squaresproblem and finally (iv) the propagation of uncertainties tothe estimated filter coefficients. In PyDynamic, task (i) cantypically be carried out by using the methods for workingwith the discrete Fourier transform, and tasks (ii)–(iv) arecarried out by the function LSFIR_unc:

b, Ub = LSFIR_unc(H, UH, M, n0, f, Fs)

with f the vector of frequencies at which the system’s fre-quency response is given and Fs the sampling frequency.Thus, the mathematical complexity of the filter design taskis encapsulated within one Python function call.

Figure 3 shows the result of applying an FIR deconvolu-tion filter to the output of a measurement system whose reso-nance frequency is excited by the simulated input signal. Onthe scale of Fig. 3, the FIR filter output is almost indistin-guishable from the simulated input signal.

If, instead of an FIR filter, an IIR filter is sought, the Py-Dynamic function

b, a, Uab = LSIIR_unc(H,UH,Mb,Ma,f,Fs)

which implements a Monte Carlo method for uncertaintypropagation can be used. Here, Mb and Ma denote the or-der of the numerator and denominator IIR filter part, respec-tively.

3.2 Uncertainty propagation for digital filtering

The application of digital filters is one of the most basic tasksin the processing of dynamic measurement data. A commonexample is the application of a low-pass filter for noise at-tenuation or a compensation filter for input estimation, asdescribed above. The implementation of digital filtering isstraightforward in almost all scientific software packages,whereas the propagation of uncertainty is typically neglected.

Figure 3. Input signal for a simulated measurement, calculated out-put signal, and estimate obtained by the application of a FIR decon-volution filter to the system output.

This statement in particular holds true when the filter coeffi-cients have associated uncertainty. However, the propagationof uncertainties is a prerequisite for the final step in the work-flow depicted in Fig. 1.

3.2.1 FIR filtering

Consider the FIR filter with coefficient vector b havingassociated uncertainty Ub, and the filter input signal x =

(x(t1), . . .,x(tN ))> with associated point-wise uncertaintiesux = (ux1 , . . .,uxN )>. Following (Elster and Link, 2008), thefilter output is obtained as

y(tn)=M∑k=0

bkx(tn−k), (7)

with uncertainty evaluated as

u2yn= b>UX(n)b+X>(n)UbX(n)+Tr(UX(n)Ub), (8)

where UX(n) denotes the covariance matrix associated withX(n)= (x(tn), . . .,x(tn−M ))> and Tr denotes the trace of amatrix. When b is a deconvolution filter, its application tothe measured system output is typically complemented witha low-pass filter for noise attenuation; see (Elster and Link,2008). Then UX(n) also contains the correlation introducedby the low-pass filter. Otherwise, the covariance matrix UX(n)contains only a diagonal with elements equal to ux . Hence,the propagation of uncertainties through an FIR filter withuncertain coefficients requires the calculation of the time-dependent covariance matrix UX(n) and the implementationof Eq. (8). InPyDynamic, this task is carried out simply by

y, uy = FIRuncFilter(x, ux, b, Ub)

The uncertainty calculated for the FIR estimation resultdepicted in Fig. 3 is shown in Fig. 4. The time dependence ofthe uncertainty associated with the measurand is typical fordynamic measurements.

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102 S. Eichstädt et al.: Dynamic uncertainty

Figure 4. Point-wise standard uncertainties associated with the out-put of the FIR deconvolution filter.

3.2.2 IIR filtering

The application of a digital filter with IIR is given mathemat-ically by

y(tn)=M∑k=0

bkx(tn−k)−M∑k=1

aky(tn−k). (9)

An example of the application of an IIR filter is given inFig. 5.

The recursive structure of the IIR filter makes an analyticcalculation of the uncertainty associated with its output dif-ficult. Therefore, (Link and Elster, 2009) considered a trans-formation of the model equation into a state-space systeminstead, yielding

z(n+ 1)=Az(n)+ qx(n), (10)

y(n)=c>z(n)+ b0x(n), (11)

with q = (0, . . .,0,1)>, c = (bM−b0aM , . . .,b1−b0a1)> and

A=

0 1 0 · · · 0 00 0 1 · · · 0 0...

......

. . ....

...

0 0 0 · · · 1 00 0 0 · · · 0 1−aM −aM−1 −aM−2 · · · −a2 −a1

.

(Link and Elster, 2009) derived an uncertainty propagationscheme based on a linearization of the state-space model withrespect to the filter coefficients and the input sequence x[n].The resulting uncertainty calculation can be carried out inparallel with the application of the IIR filter. The mathemati-cal procedure as proposed by (Link and Elster, 2009) requiresthe implementation of a state-space model with time-varyingderivatives in order to propagate the uncertainty through theIIR filter with uncertain coefficients. In PyDynamic, this taskis carried out simply by

y, uy = IIRuncFilter(x, ux, b, a, Uab)

Figure 5. Rectangular signal and corresponding output of a sixth-order IIR low-pass filter of Butterworth type.

It is well known that uncertainty evaluation using theGUM uncertainty framework can produce unreliable resultsdue to the use of a linearization of the model function. AMonte Carlo method, as described in GUM Supplement 1(cf. (BIPM et al., 2008b)), can instead be applied. A straight-forward implementation of the Monte Carlo method, though,is infeasible due to the high dimensionality of the measur-and. Therefore, (Eichstädt et al., 2012) developed an efficientsequential implementation of the GUM Monte Carlo proce-dure specifically for measurement models that involve digitalfiltering. With the sequential implementation of the MonteCarlo simulation, the required computer memory is indepen-dent of the length of the involved signals. In (Eichstädt et al.,2012), the corresponding algorithms are provided as pseudo-code. The implementation, however, may require advancedprogramming skills in order to carry out the required steps ef-ficiently. In PyDynamic, the sequential Monte Carlo methodis used by calling

y, uy = SMC(x,noise,b,a,Uab,runs=10000)

and it also allows for the sequential calculation of point-wise coverage intervals with prescribed coverage probability.Figure 6 shows the uncertainty associated with the output ofthe low-pass filter depicted in Fig. 5 when the filter cut-offfrequency is uncertain, which consequently results in uncer-tain filter coefficients.

3.3 Uncertainty evaluation for the discrete Fouriertransform

The DFT and inverse DFT are common tools applied insignal processing, and all major scientific software pack-ages provide corresponding implementations. Uncertaintyevaluation, though, is usually neglected due to the lack ofsuitable software implementations. To this end, (Eichstädtand Wilkens, 2016) proposed efficient implementations forGUM-compliant uncertainty evaluation for the DFT, inverse

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Figure 6. Point-wise standard uncertainties associated with the IIRlow-pass filter output signal, when the filter cut-off frequency is un-certain.

DFT, multiplication in the frequency domain, deconvolu-tion in the frequency domain, and the conversion from anamplitude–phase representation of a system’s frequency re-sponse to its representation by real and imaginary parts.In PyDynamic, these methods are contained in the modulepropagate_DFT.

For instance, the propagation through the application ofthe DFT for the discrete-time signal y with associated uncer-tainty Uy is carried out by

Y, UY = GUM_DFT(y,Uy)

A deconvolution in the frequency domain to obtain an es-timate X of the DFT of the system input x from knowledgeof the system output y and the system frequency responseH (jω) with associated uncertainty UH is carried out by

X, UX = DFT_deconv(H, Y, UH, UY)

A low-pass filter for noise-attenuation can then be appliedto the result of the deconvolution by

Xl, UXl = DFT_multiply(X, UX, HL)

with HL the frequency response of the chosen low-passfilter.

The DFT domain methods in PyDynamic provide an end-to-end propagation of uncertainties in many important ap-plication areas. For instance, dynamic calibration of second-order systems based on measurement of the input and outputsignal can be carried out by using (i) GUM_DFT to propagatethe time-domain signals and their uncertainty to the Fourierdomain, (ii) DFT_deconv to calculate the frequency re-sponse of the system to be calibrated and its associated un-certainty, (iii) fit_sos to fit the parameters of a second-order system to the uncertain frequency response and calcu-late their associated uncertainties, (iv) LSFIR_unc to de-sign a corresponding FIR-type deconvolution filter with un-certain coefficients and, finally, (v) FIRuncFilter to ap-ply that filter to a measured system output and calculate an

estimate of its input and its associated uncertainty in line withthe GUM framework. Similar workflows can be outlined formany other application areas of dynamic metrology. In thisway PyDynamic lays the foundation for a wide implementa-tion of reliable NMI-level and GUM-compliant tools in theanalysis of dynamic measurements.

4 Outlook

With the availability of a harmonized vocabulary, a princi-pal and general mathematical modelling approach, togetherwith established routines for the evaluation of measurementuncertainties and the development of a traceability chain forindustrial end users of dynamic measurement, can finally beachieved. The next steps in the development of PyDynamicwill thus focus on the implementation of further mathemat-ical and statistical approaches to common tasks in dynamicmetrology. This includes, for instance, the identification ofgeneral transfer function models to frequency response datawith associated uncertainties, the propagation of the uncer-tainty associated with dynamic quantities of high dimension-ality, sub-sampling and interpolation of dynamic quantities.There is an increasing use of sensors in distributed networkswith automated data assimilation and evaluation. This re-quires common data protocols in order to enable a reliablecommunication for the sensor network. Therefore, we are de-veloping a custom data format “Signal” for PyDynamic thatallows the user to carry out standard data operations with-out the need to manually propagate the uncertainties. Thatis, “Signals” can be added, subtracted from one another us-ing standard “+” and “−” operations; digital filters with orwithout uncertain coefficients can be applied to a “Signal”;application of sub-sampling, interpolation and multiplicationwith a scalar or a vector can be carried out easily. Each “Sig-nal” has at least three properties: a time axis, signal valuesand associated uncertainties. For all operations on and with“Signals”, the propagation of uncertainties is carried withoutintervention of the user. In this way, complex programs andcalculations can be carried out without additional costs re-garding the implementation of the corresponding uncertaintyevaluation.

PyDynamic is distributed under the LGPLv3 software li-cense which allows the incorporation of PyDynamic routinesin closed source code. Together with the implemented ver-satile data analysis methods, this opens the possibility of in-telligent sensors with embedded data analysis that providesdata values with associated dynamic uncertainty. In addition,data analysis for sensor networks can then be based on PyDy-namic’s “Signal” data format and the implemented functions.Moreover, due to the employed object-oriented programmingapproach for the data structure, users can easily extend theexisting code functionality to their needs.

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5 Conclusions

Analysis of dynamic measurements is the topic of a grow-ing number of research initiatives. The majority of publica-tions in this area focus on measurements at the level of NMIs.However, dynamic measurements are routinely carried out atthe industrial level and mathematical and statistical methods,guidelines and best-practice guides, which are suitable fortypical industrial applications, are required. The prerequisitefor the development and wide acceptance of such guidancedocuments, though, is the availability of well-established andapproachable methodologies. At present, there is a signifi-cant lack of methods and advice, standard software tools andinternational standards. This lack has been acknowledgedin several publications and support is being requested by agrowing number of standardization groups. Therefore, in theEMPIR project 14SIP08, NMIs PTB (Germany) and NPL(UK), together with international companies HBM GmbHand Rolls-Royce Ltd., aim to develop practical guidelines, tu-torials, training material and software. One of the outcomesof this project is the software package PyDynamic, whichafter only one year of development already provides imple-mentations of the major tools required for the analysis of dy-namic measurements. The software development will con-tinue throughout and beyond the duration of 14SIP08. Theintention is for PyDynamic to act as ready-to-use softwarethat removes the barrier between the analysis of static anddynamic measurements, and makes dynamic measurementanalysis standard practice within both NMIs and industry. Weoutlined, for three typical-use cases in dynamic metrology,how such a software tool can enable the application of so-phisticated mathematical approaches. In many applications,the complete data analysis workflow can already be carriedout with the help of PyDynamic functions, making the prop-agation of uncertainties through that workflow a simple taskfor the user. In the future, this will be improved even moreby the provision of the custom data format “Signal” whichallows the propagation of uncertainties without the need toknow which PyDynamic function would be required for theoperation on the data. Together with the cooperation of EM-PIR 14SIP08 with JCGM WG1 and the publication of guid-ance documents, this lays the foundation for future standardsand international guidelines in dynamic metrology.

6 Code and data availability

The “data” used for this publication is simulated data, gen-erated by the code available for download at https://github.com/eichstaedtPTB/PyDynamic/tree/master/examples(Eichstädt and Smith, 2016).

Competing interests. The authors declare that they have no con-flict of interest.

Acknowledgements. This work has been carried out as part ofthe European Metrology Programme for Innovation and Research(EMPIR) project 14SIP08. The EMPIR initiative is co-fundedby the European Union’s Horizon 2020 research and innovationprogramme and the EMPIR participating states.

Edited by: K.-D. SommerReviewed by: three anonymous referees

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