Sensitivity evaluation of measurement uncertainty contributions of spectral data for calculated integral quantities

Abstract

Integrating spectral data (spectral responsivities of photometers or spectral distributions of light sources) to calculate integrated quantities such as tristimulus values is straightforward at first sight. However, estimating the measurement uncertainty of these integrated quantities is challenging. When calculating integrated photometric quantities, some uncertainty contributions from the spectral data transfer to the final results, some ‘cancel out’, some ‘average out’ and others increase or decrease their weight by correlation. The spectral data are usually assumed to be uncorrelated when deriving other quantities by integration, which is typically not justified. A method called the framework approach, applying orthogonal basis functions and Monte Carlo simulations, is introduced. This approach shows that neglecting partial spectral correlations may lead to a significant underestimation of the measurement uncertainty of integrated quantities. Furthermore, this paper shows how information about spectral error correlation structures can be used to obtain better estimations of the measurement uncertainty.

1. Introduction

Integration of spectral data is central to calculating photometric and colorimetric quantities, as well as quantities that are used to assess non-visual effects and photobiological safety. Calculating any of these quantities from spectral data involves determining the value of the measurand by numerical integrals (weighted sums) of source spectral distributions across the spectral range of the weighting functions.

This paper considers spectral integrals of the form:

$$X=\underset{{\lambda }_{\min }}{\overset{{\lambda }_{\max }}{\mathrm{ʃ}}}x\left(\lambda \right)S\left(\lambda \right)d\lambda$$

(1)

where x(λ) is a defined weighting function, S(λ) is the spectral distribution of a source or the spectral responsivity of a detector and the integral is calculated over the wavelength range λ_min ≤ λ ≤ λ_max. A similar spectral integral may be involved when using filtered detectors (including practical photometers). In those cases, the spectral response function, taking the place of $x\left(\lambda \right)$ , also has associated uncertainties.

While these calculations are straightforward and generally calculated using a trapezium rule, estimating the associated measurement uncertainty becomes complex and challenging, as described in the CIE198-SP2 ¹ report. In particular, where there is a significant error correlation between measured spectral data at different wavelengths, the combined uncertainty associated with the integrated quantity is affected, as demonstrated by Schmähling et al. ² To describe the effects that can be caused by correlations, one must first understand what ‘correlated’ means in this context. This paper also provides some assistance in this regard.

A general measurement uncertainty assessment, whether based on the law of propagation of uncertainties in the legacy GUM ³ or Monte Carlo (MC) methods introduced in the JCGM 101 ⁴ document, starts with a measurement model written as an equation. It is common for measurement models to be multi-stage, as described in the JCGM 106 ⁵ document. This means that some of the model’s input quantities are based on previous (earlier stage) measurement models.

For photometric and colorimetric quantities as well as quantities that are used to assess non-visual effects and photobiological safety, the measurement model will be in the form of Equation (1) and this is therefore what will be discussed in Section 2. The measurement model can be considered multi-stage in that the spectral quantity itself will be the measurand of a calibration process to determine that spectrum and will have its own measurement model.

In this work, we propagate uncertainties with complex error correlation structures through the measurement model using a method of orthogonal basis functions introduced by Kärhäet al. ⁶ To describe correlations, a framework approach for the estimation of dependencies is developed. This framework approach allows a better insight into the origin of those significant contributions to the measurement uncertainty of an integrated quantity, which come from correlations in the spectral distribution data.

With the framework approach, we introduce an evaluation method; its application (simply to show how it works) is presented based on an example in this paper.

The main contribution of this work is the estimation of sensitivity coefficients. Here, we use MC methods to propagate such error correlation structures, both individually – to evaluate the sensitivity of the integrated quantity to each source of uncertainty – and combined to provide a combined uncertainty for the integrated quantity. In this method, we follow the kind of sensitivity analysis proposed by the JCGM 101, Annex B, ⁴ which is called ‘one factor at a time’ by Razavi and Gupta. ⁷

Furthermore, this work considers partially correlated contributions associated with both wavelength and the spectral distribution amplitude, alongside with fully correlated and uncorrelated errors.

In spectral integrals, sources of uncertainty that lead to fully correlated errors are simple to propagate (and cancel out if the integrated quantity is normalised). Sources of uncertainty that lead to fully uncorrelated (independent) errors reduce significantly by the effective weighted averaging of the integral. Therefore, the most important sources of uncertainty are those that provide partial error correlation. In general, however, the form of that partial error correlation is unknown. For the description of partial error correlation, this paper builds on a method originally presented by Kärhäet al., ⁶ and extended by Vaskuri et al. ⁸ and Maham et al. ⁹ The basis function method uses a set of basis functions to model possible, (realistic) error correlation forms and uses MC methods to explore different possible correlation structures. This paper presents the final modelling equations for the basis function method and examples.

The implementation of the work presented here can be found in the open-source Python package 19nrm02, ¹⁰ which uses the LuxPy Python package by Smet. ¹¹ Additional examples of the application of the approach, including animations in the form of mp4 files, are presented in Supplemental Material for this paper (available online).

2. Methods

Integrals of the form of Equation (1) are treated as summations of a spectral quantity over a set of wavelengths λ_i:

$$X=\sum _{i=0}^{N_{\lambda }-2}x\left({\lambda }_i\right)·S_i·({\lambda }_{i+1}-{\lambda }_i)$$

(2)

It is useful to describe the spectral data with vector notation. In this paper, vectors are represented by bold italic symbols, and scalars are represented by thin italic symbols. For spectral data with $N_{\lambda }$ elements (e.g. from 360 nm to 830 nm in 5 nm steps, $N_{\lambda }$  = 95), two vectors are used:

• Amplitude vector: $\mathit{S}=[S_0,S_1,S_2,\dots ,S_{N_{\lambda }-1}]$

• Wavelength vector: $\mathbf{\lambda }=[{\lambda }_0,{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{N_{\lambda }-1}]$

where the random values of the elements in the vectors, generated during the MC simulation (MCS), can originate from wavelength uncertainty, signal uncertainty or both. Both vectors are used to calculate the spectrally integrated quantity of interest. Initially, to better understand their individual influences, they are simulated independently.

Within this paper, where the elements of the vectors are processed individually, they are written as thin italic symbols with an index (e.g. λ_i) or in functional form (e.g. S(i) or S(λ)). The symbol $S_i$ in Equation (2) is used instead of $S\left({\lambda }_i\right)$ to indicate that the i-th element of the signal vector should be used without any interpolation, whereas the $x\left({\lambda }_i\right)$ means that we have to interpolate the x-function to the specific wavelength ${\lambda }_i$ .

The elements of these two vectors are determined experimentally, and the values can be affected by three types of uncertainty that create fluctuations in them:

• Effects that are uncorrelated from one element to another (random effects, noise).

• Effects that are fully correlated from one element to another (systematic effects, bias).

• Effects that are partially correlated, where ‘partially correlated’ can be interpreted as an infinite variety of different correlation states varying from fully uncorrelated to fully correlated.

In this paper, MCS was used to propagate uncertainties. MCS is a mathematical technique^12,13 that enables a quantitative analysis of a measurement result. The core idea behind an MCS is to represent the propagation of probability density functions (PDFs) for the input quantities (here the measured values of the spectral quantity at different wavelengths) through the measurement model (here the integral given in Equation (1)).

This involves the following steps:

• Random sampling: For each input quantity (here the spectral quantity at each wavelength), a set of representative errors are generated by generating random numbers drawn from the distributions.

• Repeated trials: In an MCS, a large number ( $N_{\mathrm{MC}}$ ) of trial runs, also known as simulations, using random inputs are performed. These trials are used to generate the PDFs of different outcomes of the model.

• Aggregation of results: After running simulations many times, the results are aggregated and analysed to estimate the statistical properties of the system being modelled, such as averages, variances, probabilities of different outcomes, correlations, etc.

2.1 Overview

A general overview of the framework approach is presented in Figure 1. The different parts of the figure will be explained in detail in this section.

Figure 1

Overview of the framework approach. See Section 2.1 for definitions of the symbols used

Figure 1 shows the calculation during two separate MCSs (MCS1 and MCS2) running each $N_{\mathrm{MC}}$ times (the number of MC trials or runs). The first part of the structure, shown in MCS1, starts with a given spectral distribution based on the outcome of a real or simulated measurement, a physical modelling (e.g. the spectral radiance of a blackbody) or a spectral distribution given in a Standard (e.g. CIE Standard Illuminant A, ¹⁴ CIE reference spectrum L41, ¹⁵ etc.) at a nominal wavelength scale $\mathbf{\lambda }$ and at this point without any uncertainty information. The nominal wavelength scale is a set of true wavelength values, used as reference values for the calculation.

In the next step, a general model, explained in Section 2.2, uses this spectral distribution to add the measurement uncertainty by generating different fluctuations around the input spectral distribution. This is done by separately modelling the wavelength and amplitude scales with different types of simulated uncertainty contributions. The uncertainty contributions are modelled as correlated, uncorrelated and partially correlated contributions. At the end, this results in a set of $N_{\mathrm{MC}}$ combinations of wavelength and signal vectors ${\mathit{M}}_{i,\mathrm{SD}}=({\mathbf{\lambda }}_i,{\mathit{S}}_i)$ .

Besides the correlated and uncorrelated uncertainty contributions, generating partially correlated uncertainty contributions using the basis function technique, as described in Section 2.3, is essential for the evaluation presented here.

In this paper, the data are simulated and do not consider the physical background of a particular measurement setup or device under test to ensure that the application does not cover just one method or describe a specific measurement technique. Possible connections of the model parameter to physical models or real measurements are given in Section 2.4.

Finally, using the $N_{\mathrm{MC}}$ generated combinations of wavelength and signal vectors ${\mathit{M}}_{i,\mathrm{SD}}=({\mathbf{\lambda }}_i,{\mathit{S}}_i)$ , integrated quantities (see Section 3) can be calculated for every of the generated ${\mathit{M}}_{i,\mathrm{SD}}$ values. This results in $N_{\mathrm{MC}}$ integrated quantities (e.g. relative luminance values $Y_{\mathrm{rel}}$ , as shown for the MCS1 example in Figure 1), and a PDF including mean, standard deviation and expanded uncertainty can be calculated from a statistical analysis of this set of output quantities.

Furthermore, the $N_{\mathrm{MC}}$ generated combinations of wavelength and signal vectors ${\mathit{M}}_{i,\mathrm{SD}}=({\mathbf{\lambda }}_i,{\mathit{S}}_i)$ can be used to summarise the data using a nominal wavelength scale $\mathbf{\lambda }$ , a mean spectral distribution $\overline{{\mathit{S}}_{\mathrm{f}}}$ , the associated standard deviation $\mathit{\sigma }\left({\mathit{S}}_{\mathrm{f}}\right),$ and a correlation matrix ${\mathit{P}}_{\mathrm{f}}$ . This set of summarised data is called the ‘compressed information’ hereafter. The methods used here are explained in Section 3.3. These data can be used to transfer the outcome of the first MCS (MCS1) to other users or to visualise the data.

In parallel to the MCS1 using the framework approach, we can use an MCS2, as shown in the lower part of Figure 1, to calculate a set of integral quantities based on a multivariate normal distribution method. This can be done using either real measurement results, for example, provided by the Physikalisch-Technische Bundesanstalt (PTB) as shown in Section 3.1 or with compressed information from MCS1 (see Section 3.2). With this method, we generate the spectral distribution data from multivariate normal distribution sampling, to again get combinations of wavelength and spectral information $\mathit{M}{\text{'}}_{i,\mathrm{SD}}=(\mathbf{\lambda },{\mathit{S}}_i)$ , but in this case with a nominal wavelength scale only. These are used for calculation and evaluation of integral data in the same manner as for MCS1, yielding compressed information $\overline{{\mathit{S}}_{\mathrm{m}}}$ , the associated standard deviation, $\mathit{\sigma }\left({\mathit{S}}_{\mathrm{m}}\right),$ and a correlation matrix, ${\mathit{P}}_{\mathrm{m}}$ . In Section 3.3, we use chromaticity coordinates as an example to show the correlation analysis for the results using this approach and to validate the data processing from MCS1.

2.2 Model for uncertainty contributions in the framework approach

The framework approach uses additive (subscript ‘a’) and multiplicative (subscript ‘m’) model parameters linked to the values as uncertainty components. The uncertainty contributions are implemented during the simulation as follows:

• Uncorrelated (subscript ‘uc’): Every vector element represents a different realisation of a random process in each of the $N_{\mathrm{MC}}$ MC trials. A vector with random elements is drawn in each trial. Variations of the vector elements are independent.

• Correlated (subscript ‘c’): All vector element variations behave similarly in any given MC trial. A random number is drawn for every trial only once for the complete vector.

• Basis function (subscript ‘b’): Partially correlated variations of spectral data points are simulated by the variations of the vector elements with a periodic spectral dependence. The different vectors for the trials are calculated according to the basis function technique introduced by Kärhäet al. ⁶ with Fourier basis functions and by Vaskuri et al. ⁸ with Chebyshev basis functions. See Section 2.3 for details.

2.2.1 Amplitude vector

The model used to calculate the random numbers (subscript ‘r’) for the signal or amplitude scale is shown in Equation (3).

$${\mathit{S}}_{\mathrm{r}}={\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}*S+{\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{c}}+{\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{uc}}+{\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{b}}$$

(3)

where ${\mathit{S}}_{\mathrm{r}}$ is the random variable for the amplitude scale in the MCS (vector); $\mathit{S}$ is the nominal value of the amplitude (vector, normalised to $\max \left(\mathit{S}\right)=1$ ); ${\mathit{k}}_{S,\mathrm{m}-\mathrm{b}}$ is the multiplicative uncertainty component modelled with the basis function technique (vector, systematic and random); $*\mathrm{denotes}$ element-wise multiplication of the vector elements; ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{c}}$ is the additive fully correlated uncertainty component (vector, systematic); ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{uc}}$ is the additive uncorrelated uncertainty component (vector, random); ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{b}}$ is the additive partially correlated uncertainty component modelled with the basis function technique (vector, systematic).

2.2.2 Wavelength vector

The model to generate the random numbers for the wavelength scale is described in Equation (4).

$${\mathbf{\lambda }}_{\mathrm{r}}={\mathit{F}}_{\mathrm{d}}({\mathit{k}}_{\mathrm{\lambda },\mathrm{m}\mathrm{-}\mathrm{b}}*{\mathit{F}}_{\mathrm{n}}\left(\mathbf{\lambda }\right))+{\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{c}}+{\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{uc}}+{\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{b}}$$

(4)

where ${\mathbf{\lambda }}_{\mathrm{r}}$ is the random variable for the wavelength scale in the MCS (vector); $\mathbf{\lambda }$ is the nominal wavelength (vector); ${\mathit{k}}_{\lambda ,\mathrm{m}\mathrm{-}\mathrm{b}}$ is the multiplicative uncertainty component modelled with the basis function technique (vector, systematic and random); ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{c}}$ is the additive fully correlated uncertainty component (vector, systematic); ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{uc}}$ is the additive uncorrelated random uncertainty or noise component (vector, random); ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{b}}$  is the additive partially correlated uncertainty component modelled with the basis function technique (vector, systematic).

The functions $\mathbf{\lambda }\mathbf{\text{'}}={\mathit{F}}_{\mathrm{n}}\left(\mathbf{\lambda }\right)$ and $\mathbf{\lambda }={\mathit{F}}_{\mathrm{d}}(\mathbf{\lambda }\text{'})$ , shown in Equation (5), are used to normalise/de-normalise the wavelength scale to handle the multiplicative factor for the model in a standard way for the wavelength range $[{\lambda }_{\min },{\lambda }_{\max }]$ .

$$\begin{array}{c}\multicolumn{1}{c}{\mathbf{\lambda }\text{'}={\mathit{F}}_{\mathrm{n}}\left(\mathbf{\lambda }\right)=\frac{\mathbf{\lambda }-{\mathbf{\lambda }}_{\min }}{{\lambda }_{\max }-{\lambda }_{\min }}}\\ \multicolumn{1}{c}{\mathbf{\lambda }={\mathit{F}}_{\mathrm{d}}(\mathbf{\lambda }\text{'})=({\lambda }_{\max }-{\lambda }_{\min })\mathbf{\lambda }\prime +{\mathbf{\lambda }}_{\min }}\end{array}$$

(5)

where ${\mathbf{\lambda }}_{\min }$ and ${\mathbf{\lambda }}_{\max }$ are vectors containing ${\lambda }_{\min }$ or ${\lambda }_{\max }$ , respectively, in every vector element.

2.2.3 Combination of amplitude and wavelength scales

Pairs of wavelength vectors ${\mathbf{\lambda }}_{\mathrm{r}}$ and amplitude vectors ${\mathit{S}}_{\mathrm{r}}$ are necessary to define the spectral distribution in a matrix ${\mathit{M}}_{\mathrm{r},\mathrm{SD}}$ , as demonstrated in Equation (6).

$${\mathit{M}}_{\mathrm{r},\mathrm{SD}}=({\mathbf{\lambda }}_{\mathrm{r}},{\mathit{S}}_{\mathrm{r}})$$

(6)

2.2.4 Parameters for random numbers

For simplicity, the random numbers used for simulating the uncorrelated and correlated contributions introduced above are drawn from normal distributions with mean value $\mu ,$ and standard deviation $\sigma$ (symbol: $\mathcal{N}(\mu ;\sigma )$ ).

• Additive components: $\mathcal{N}(\mu =0;\sigma )$

• Multiplicative components: $\mathcal{N}(\mu =1;\sigma )$

The standard deviation parameter $\sigma$ is selected in a way that one can adapt the result of the simulation conveniently to the situation in a real measurement setup:

• The uncertainty of the wavelength scale parameters is, for example, 1 nm for the additive components ( $u_{\lambda ,\mathrm{a}}$ ) and, for example, 1% for the factor in the normalised wavelength scale part ( $u_{\lambda ,\mathrm{m}}$ ).

• The uncertainty of the amplitude scale parameters ( $u_S$ ) is, for example, 1%. The same parameter can also be used for the additive components using normalised data.

These parameters are not used to generate a realistic measurement uncertainty. Rather, they are used to generate easy-to-use sensitivities that can be used in a linear manner for small measurement uncertainties.

Example: If we use 1 nm for the value of the model parameter for the additive fully correlated wavelength uncertainty component ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{c}}$ and we get an uncertainty of 0.0005 for the chromaticity coordinate x, we can easily calculate that we get an uncertainty contribution of 0.0001 if the real value of the model parameter is 0.2 nm. This means we have made an easy-to-use uncertainty budget that can be used to provide sensitivity information.

2.3 Basis function technique

The basis function technique is explained in detail in Kärhäet al. ⁶ for Fourier basis functions and in Vaskuri et al. ⁸ for Chebyshev basis functions. The implementation can be found in the MC Toolbox of the open-source Python package 19nrm02 ¹⁰ in the file FourierNoise.py. The basis function approach is used inside the MCS1 and also uses random parameters. First of all, a set of basis functions $b_k\left(\mathrm{\lambda }\right)$ must be generated, where $k$ is the order and $b_k\left(\mathrm{\lambda }\right)$ is used as a general notation to represent either Fourier basis functions $f_k\left(\mathrm{\lambda }\right)$ , which are generated using Equation (7), or Chebyshev basis functions $c_k\left(\mathrm{\lambda }\right)$ , which are generated using Equation (8).

Fourier basis functions

$$\begin{array}{c}\multicolumn{1}{c}{f_0\left(\lambda \right)=1}\\ \multicolumn{1}{c}{f_k\left(\lambda \right)=\sqrt{2}\sin (2\mathrm{\pi }k\frac{\lambda -{\lambda }_{\min }}{{\lambda }_{\max }-{\lambda }_{\min }}+{\varphi }_k)}\end{array}$$

(7)

Chebyshev basis functions

$$\begin{array}{c}\multicolumn{1}{c}{g_0\left(\lambda \right)=T_0\left(\lambda \right)=1}\\ \multicolumn{1}{c}{T_k\left(\lambda \right)=\cos \left[k\arccos \left(\frac{2\lambda -{\lambda }_{\min }-{\lambda }_{\max }}{{\lambda }_{\max }-{\lambda }_{\min }}\right)\right]}\\ \multicolumn{1}{c}{g_k\left(\mathrm{\lambda }\right)=T_k\left(\mathrm{\lambda }\right)/{\sigma }_k}\\ \multicolumn{1}{c}{c_k\left(\mathrm{\lambda }\right)=g_{2k-1}\left(\mathrm{\lambda }\right)\cos {\varphi }_k+g_{2k}\left(\mathrm{\lambda }\right)\sin {\varphi }_k}\end{array}$$

(8)

where $k$ is the order of the current basis function, $k\in [0\dots N_{\mathrm{B}}-1]$ ; $N_{\mathrm{B}}$ is the number of basis functions (model parameter), $N_{\mathrm{B}}\in [0\dots N_{\lambda }/2-1]$ ; ${\varphi }_k$ is a uniformly distributed random number in the range of $[0,2\mathrm{\pi }]$ ; $[{\lambda }_{\min };{\lambda }_{\max }]$ is the wavelength range; ${\mathrm{\sigma }}_k$ is the standard deviation of $T_k\left(\lambda \right)$ .

Based on the generated basis functions $b_k\left(\lambda \right)$ (depending on the method $b_k\left(\lambda \right)=f_k\left(\lambda \right)$ or $b_k\left(\lambda \right)=c_k\left(\lambda \right)$ ) and introducing a random number $Y_k$ , which has a standard normal distribution, $Y_k~\mathcal{N}(0;1)$ , one calculates the normalised weighting factors ${\gamma }_k$ with ${\gamma }_k=Y_k/\sqrt{\sum _{j=0}^{N_{\mathrm{B}}-1}{Y}_j^2}$ .

Depending on the Fourier or Chebyshev functions used, different types of basis functions are calculated. A deviation function $\delta \left(\lambda \right)$ , describing the local deviations over the scale, is then calculated according to Equation (9):

$$\delta \left(\lambda \right)=\sum _{k=0}^{N_{\mathrm{B}}-1}{\gamma }_k{b}_k\left(\lambda \right)$$

(9)

The direct influence of a specific basis function order $k=N_{\mathrm{B}}$ can be investigated using the deviation function ${\delta }_s\left(\lambda \right)$ , describing the local deviations over the scale using a single basis function only:

$${\delta }_{\mathrm{s}}\left(\lambda \right)={\gamma }_{N_{\mathrm{B}}}{b}_{N_{\mathrm{B}}}\left(\lambda \right)$$

(10)

A comparison with correlations in real datasets in Maham et al. ⁹ suggests that the summation of the results of the $N_{\mathrm{B}}$ orthogonal functions should not be equally weighted, but weighted with the reciprocal of the order of the basis functions. Applying this new approach with the one over f weighting according to Ikonen et al. ¹⁶ and Maham et al., ⁹ the basis function ${\delta }_{1\mathrm{f}}\left(\lambda \right)$ can be calculated according to Equation (11):

$${\delta }_{1\mathrm{f}}(\lambda )=\sin {\varphi }_{1\mathrm{f}}+\frac{\cos {\varphi }_{1\mathrm{f}}}{\sum _{k=1}^{N_{\mathrm{B}}-1}1/k^2}\sum _{k=1}^{N_{\mathrm{B}}-1}\frac{b_k(\lambda )}{k}$$

(11)

where ${\delta }_{1\mathrm{f}}\left(\lambda \right)$ is the function describing the local deviations over the scale using the one over f weighting; ${\varphi }_{1\mathrm{f}}$ is a uniformly distributed random number in the range of $[0,2\mathrm{\pi }]$ .

In every trial of the MCS, the random numbers ${\gamma }_k,{\varphi }_k\mathrm{and}{\varphi }_{1\mathrm{f}}$ are generated from their distributions.

In the last step, the wavelength (Equation (12)) or amplitude (Equation (13)) data can be modified accordingly by the basis functions:

$$\begin{array}{c}\multicolumn{1}{c}{k_{\lambda ,\mathrm{m}-\mathrm{b}}\left(\lambda \right)=(1+u_{\lambda ,\mathrm{m}}\delta \left(\lambda \right))}\\ \multicolumn{1}{c}{{\lambda }_{\mathrm{a}\mathrm{-}\mathrm{b}}\left(\lambda \right)=u_{\lambda ,\mathrm{a}}\delta \left(\lambda \right)}\end{array}$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{k_{S,\mathrm{m}\mathrm{-}\mathrm{b}}\left(\lambda \right)=(1+u_S\delta \left(\lambda \right))}\\ \multicolumn{1}{c}{S_{\mathrm{a}\mathrm{-}\mathrm{b}}\left(\lambda \right)=u_S\delta \left(\lambda \right)}\end{array}$$

(13)

where $u_{\lambda ,m}$ is the multiplicative standard uncertainty of the wavelength scale; $u_{\lambda ,a}$ is the additive standard uncertainty of the wavelength scale; $u_S$ is the standard uncertainty of the amplitude scale.

The corresponding vectors ( ${\mathit{k}}_{\lambda ,\mathrm{m}\mathrm{-}\mathrm{b}}$ , ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{b}}$ , ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ and ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{b}}$ ) are formed from the elements given in Equations (12) and (13).

2.4 Connection to physical models

The framework approach can be linked to the physical properties of measurement systems. An overview is given in Table 1 together with the uncertainty components in the framework approach.

Table 1

Connection of the uncertainty components in the framework approach to physical models, for the example of spectroradiometric measurements of a lamp

Symbol	Description	Origin in other (physical) models
${\mathit{k}}_{\mathbf{\lambda }\mathbf{,}\mathbf{m}\mathrm{-}\mathbf{b}}$	Uncertainty of the wavelength calibration factor	The measurement result obtained using several narrowband spectral lamps for the wavelength calibration is the scale factor in nm per pixel or nm per degree of the spectroradiometric system. The uncertainty of this scale factor, for example, from a polynomial regression of different wavelength measurement points, can be used to model ${\mathit{k}}_{\mathit{\lambda }\mathbf{,}\mathrm{m}\mathrm{-}\mathrm{b}}$
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{c}}$	Wavelength scale shift of the whole scale	The stability (reproducibility or repeatability) of the homing/initialisation of a monochromator can be used to model ${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{c}}$
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{uc}}$	Wavelength scale shift of single measurement points	Reproducibility or repeatability of the wavelength setting at a single wavelength position
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{b}}$	General uncertainty of the wavelength scale	A complicated relationship between different wavelength settings can be modelled by ${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{b}}$ . An example of a physical-based modelling for such a parameter can be found in White et al. 17 The basis function technique can model the described behaviour if a correction is not possible
${\mathit{k}}_{\mathit{S},\mathbf{m}\mathrm{-}\mathbf{b}}$	Uncertainty of the absolute/relative amplitude calibration factor	A sophisticated modelling of the calibration factor can be introduced, for example, ageing of the calibration reference artefact can cause correlations between the calibration factors at different wavelengths
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{c}}$	Correlated uncertainty of a global offset	Modelling the global dark signal offset
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{uc}}$	Uncorrelated uncertainty of the offset signals	Modelling the individual dark signal at every measurement position
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{b}}$	Uncertainty modelling for the offset	A time-dependent offset of a reference voltage can cause wavelength-dependent offset values in the amplitude scale

3. Comparison of real and modelled results

In Section 3.1, real measurement data, provided by the PTB, are used for the MCS2 path to compare the results with modelled data from the framework approach from the MCS1 path of the simulation as presented in Section 3.2. The reduction of the datasets from the simulation into a form that can be transferred to other users or to other simulation paths is described in Section 3.3. As an example, we calculate tristimulus values and chromaticity data as explained in Section 3.4.

3.1 Real measurement data

During the development of the framework method, the PTB provided a dataset from a real measurement of the spectral irradiance of an FEL lamp. The data consisted of a mean spectral distribution vector $\overline{{\mathit{S}}_{\mathrm{PTB}}}$ , the standard deviation $\mathit{\sigma }\left({\mathit{S}}_{\mathrm{PTB}}\right)$ (see Figure 2) and the correlation matrix ${\mathit{P}}_{\mathrm{PTB}}$ (see Figure 3), all at a nominal wavelength scale $\mathbf{\lambda }$ . The correlation matrix included the uncertainty information associated with the nominal wavelength scale.

Figure 2

Spectral distribution $\overline{{\mathit{S}}_{\mathrm{PTB}}}$ and standard deviation $\mathit{\sigma }\left({\mathit{S}}_{\mathrm{PTB}}\right)$ of a real FEL lamp measurement at PTB

Figure 3

Correlation matrix ${\mathit{P}}_{\mathrm{PTB}}$ of the real spectral distribution measurement of the FEL lamp, shown in Figure 2

The data from PTB were then used to generate a value matrix ${\mathit{M}}_{\mathrm{r},\mathrm{SD},\mathrm{PTB}}$ to describe the spectral characteristics of the lamp, by making draws from a multivariate normal distribution using the nominal wavelength $\mathbf{\lambda }$ , according to Equation (14).

$${\mathit{M}}_{\mathrm{r},\mathrm{SD},\mathrm{PTB}}=(\mathbf{\lambda },\mathcal{N}(\overline{{\mathit{S}}_{\mathrm{PTB}}},\mathit{\sigma }\left({\mathit{S}}_{\mathrm{PTB}}\right),{\mathit{P}}_{\mathrm{PTB}}))$$

(14)

3.2 Modelled data using the framework approach

In this study, we used the model as described in this section to perform a sensitivity analysis on the calculated relative luminance, chromaticity coordinates and correlated colour temperature (CCT). Separate MCSs were performed for each source of uncertainty with respect to Equations (3) and (4), with the random number parameters described in Section 2.2, so that the sensitivity of the output values to each source of uncertainty in turn could be calculated. This follows the type of sensitivity analysis proposed by Razavi and Gupta. ⁷

The spectral distribution of a blackbody at a temperature of 3077 K was used as input quantity for the simulated measurement, to make the simulation as simple as possible. The chosen blackbody temperature is the same as the CCT of the FEL lamp measured at PTB (see Section 3.1) and gives a spectral distribution very close to the relative spectral distribution of the measured lamp. Based on this theoretical blackbody, the amplitude and wavelength values were modified during the MCS as described above (MCS1). With the modified spectral distributions, one can calculate integrated output quantities and study their behaviour during the simulation. Using these simulation data one can estimate the PDFs and statistical parameters for all output quantities, including mean and standard deviation.

Since this assessment is not based on an actual measurement system, we only generate sensitivity values rather than absolute values. This means one can state at the end, for example: The sensitivity coefficient for the chromaticity coordinate x is 0.0006 for every 1 nm correlated error in the wavelength scale.

Starting from a blackbody at 3077 K at a nominal wavelength scale from 360 nm to 830 nm with 1 nm steps, and using the model equations presented previously, we generate $N_{\mathrm{MC}}$ pairs of random wavelength ${\mathbf{\lambda }}_{\mathrm{r}}$ and amplitude ${\mathit{S}}_r$ realisations with defined parameters.

For the amplitude scale, Equation (3), the following parameters are used in the simulation:

$\mathit{S}$ blackbody at 3077 K, normalised to $\max \left(\mathit{S}\right)=1$

${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ multiplicative uncertainty component modelled with the basis function technique (used with the cumulative basis function approach and $N_{\mathrm{B}}=7$ Chebyshev basis functions) with $u_S=0.01$

${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{c}}$ additive fully correlated uncertainty component (vector, systematic), ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{c}}~\mathcal{N}(0;0.01)$

${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{uc}}$ additive uncorrelated uncertainty component (vector, random), ${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{uc}}~\mathcal{N}(0;0.01)$

${\mathit{S}}_{\mathrm{a}\mathrm{-}\mathrm{b}}$ additive partially correlated uncertainty component modelled with the basis function technique (vector, systematic), with $u_S=0.01$

For the wavelength scale, Equation (4), the following parameters are used in the simulation:

${\mathit{k}}_{\mathbf{\lambda },\mathrm{m}\mathrm{-}\mathrm{b}}$ multiplicative uncertainty component modelled with the basis function technique (used with the cumulative basis function approach and $N_{\mathrm{B}}=7$ Chebyshev basis functions, see Equation (10)) with $u_{\lambda ,\mathrm{m}}=0.01$ (description: ‘basis c 7’)

$\mathbf{\lambda }$ nominal wavelength (vector)

${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{c}}$ additive fully correlated uncertainty component (vector, systematic), ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{c}}~\mathcal{N}(0;1\mathrm{nm})$

${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{uc}}$ additive uncorrelated random uncertainty or noise (vector, random), ${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{uc}}~\mathcal{N}(0;1\mathrm{nm})$

${\mathbf{\lambda }}_{\mathrm{a}\mathrm{-}\mathrm{b}}$ additive partially correlated uncertainty component modelled with the basis function technique (vector, systematic), with $u_{\lambda ,\mathrm{a}}=1\mathrm{nm}$

3.3 Data transfer and interpolation

For data visualisation or the transfer of measurement results to other users, the matrix ${\mathit{M}}_{\mathrm{r},\mathrm{SD}}$ of spectral distributions can be reduced to a mean vector $\overline{\mathit{S}\prime }$ , a standard deviation $\mathit{\sigma }(\mathit{S}\prime )$ and a correlation matrix $\mathit{P}=\mathrm{Corr}({\mathit{S}}_{\mathrm{r}}\prime )$ . To do this, the nominal wavelength scale $\mathbf{\lambda }$ is required and not the wavelength scale ${\mathbf{\lambda }}_{\mathrm{r}}$ from the MC trials described in Section 3.2. Every pair of wavelength and spectral distribution data $({\mathbf{\lambda }}_{\mathrm{r}},{\mathit{S}}_{\mathrm{r}})$ is interpolated to the nominal wavelength $\mathbf{\lambda }$ .

$${\mathit{M}}_{\mathrm{r},\mathrm{SD}}\prime =(\mathbf{\lambda },{\mathit{S}}_{\mathrm{r}}\prime )=\mathrm{Interpolate}({\mathbf{\lambda }}_{\mathrm{r}},{\mathit{S}}_{\mathrm{r}})$$

(15)

Using a non-nominal wavelength scale usually requires interpolation for further calculations. This interpolation of the amplitude vector to regular, equidistant wavelength intervals means that the measurement uncertainties of the wavelength vector are converted to the amplitude values and influence the measurement uncertainties and correlations for the amplitude vector. Care is needed when doing the interpolation, particularly if the amplitude vector is noisy, since artificial spectral structure can be introduced if an inappropriate interpolation method is used. Guidance on interpolation is given in CIE 15:2018, Clause 7.2.3. ¹⁴ (Rule of thumb: Interpolate the smooth function and not the noisy one.)

New data pairs ${\mathit{M}}_{\mathrm{r},\mathrm{SD}}\prime \prime$ for the interpolated data can now be generated based on a multivariate normal distribution $\mathcal{N}$ according to Equation (16):

$${\mathit{M}}_{\mathrm{r},\mathrm{SD}}"=(\mathbf{\lambda },\mathcal{N}(\overline{\mathit{S}\prime },\mathit{\sigma }(\mathit{S}\prime ),\mathit{P}))$$

(16)

3.4 Calculation and evaluation of integral data

As an example, to show the effect of the different measurement uncertainties (with regard to the parameters of the models introduced above), several integrated quantities (tristimulus values, chromaticity coordinates and CCT) are calculated from the simulated spectral distributions.

According to CIE 15:2018, Clause 7 ¹⁴ the tristimulus values, X, Y, Z, are calculated with Equation (17) using the nominal wavelength scale.

$$\begin{array}{c}\multicolumn{1}{c}{X=\sum _{i=0}^{N_{\lambda }-1}\overline{x}\left({\lambda }_i\right)·S\left({\lambda }_i\right)·\mathrm{\Delta }\lambda ;}\\ \multicolumn{1}{c}{Y=\sum _{i=0}^{N_{\lambda }-1}\overline{y}\left({\lambda }_i\right)·S\left({\lambda }_i\right)·\mathrm{\Delta }\lambda ;}\\ \multicolumn{1}{c}{Z=\sum _{i=0}^{N_{\lambda }-1}\overline{z}\left({\lambda }_i\right)·S\left({\lambda }_i\right)·\mathrm{\Delta }\lambda }\end{array}$$

(17)

where $\overline{x}$ ( ${\lambda }_i$ ), $\overline{y}$ ( ${\lambda }_i$ ), $\overline{z}$ ( ${\lambda }_i$ ) are colour-matching functions at wavelength λ_i; S(λ_i) is the (relative) spectral distribution of the light source at wavelength λ_i; Δλ is the step width for the summation; N_λ is the number of elements.

Using the spectral distributions generated with the framework approach, the tristimulus values, $X_{\mathrm{r}},Y_{\mathrm{r}},Z_{\mathrm{r}}$ , and from these the chromaticity coordinates, $x_{\mathrm{r}}$ and $y_{\mathrm{r}}$ , are calculated according to Equations (18) to (21).

Note that, due to the variation of the wavelength scale in each MC trial depending on the modelling, the wavelength steps are calculated separately for each trial and therefore the integrated values obtained using Equations (18) to (20) are also calculated separately for each trial:

$$X_{\mathrm{r}}=\sum _{i=0}^{N_{\lambda }-2}\overline{x}\left({\lambda }_{\mathrm{r},i}\right)·S_{\mathrm{r},i}·({\lambda }_{\mathrm{r},i+1}-{\lambda }_{\mathrm{r},i})$$

(18)

$$Y_{\mathrm{r}}=\sum _{i=0}^{N_{\lambda }-2}\overline{y}\left({\lambda }_{\mathrm{r},i}\right)·S_{\mathrm{r},i}·({\lambda }_{\mathrm{r},i+1}-{\lambda }_{\mathrm{r},i})$$

(19)

$$Z_{\mathrm{r}}=\sum _{i=0}^{N_{\lambda }-2}\overline{z}\left({\lambda }_{\mathrm{r},i}\right)·S_{\mathrm{r},i}·({\lambda }_{\mathrm{r},i+1}-{\lambda }_{\mathrm{r},i})$$

(20)

$$x_{\mathrm{r}}=\frac{X_{\mathrm{r}}}{X_{\mathrm{r}}+Y_{\mathrm{r}}+Z_{\mathrm{r}}};y_{\mathrm{r}}=\frac{Y_{\mathrm{r}}}{X_{\mathrm{r}}+Y_{\mathrm{r}}+Z_{\mathrm{r}}}$$

(21)

The CCT is calculated from the chromaticity coordinates $(x_{\mathrm{r}},y_{\mathrm{r}})$ based on standard algorithms according to Robertson ¹⁸ and Baxter et al. ¹⁹

4. Results

All diagrams, tables and calculations in the following are based on the open-source Python package 19nrm02 ¹⁰ and implemented in the Jupyter Notebook MCSim_PM.ipynb. The calculations in the MCS were performed with $N_{\mathrm{MC}}$  = 20 000 trials.

Table 2 shows the result of the MCS1 for the described model settings and the output data relative luminance (Y), chromaticity coordinates (x, y) and CCT ( $T_{\mathrm{cp}}$ ). In every row of the table, only the parameter described in the first column is used as random. All others are set to their nominal values. For the last row, all parameters are set to their random inputs to evaluate the results of all model parameters together.

Table 2

Evaluation table for the MU calculation using the framework approach in the MCS1

Quantity	Unit	Mean	SD	Distribution	$u\left(Y\right)$	$u\left(x\right)$	$u\left(y\right)$	$u\left(T_{\mathrm{cp}}\right)$ /K
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{c}}$	nm	0.0	1.000	Normal	0.0058	0.00056	0.00026	7.8
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{uc}}$	nm	0.0	1.000	Normal	0.00048	0.0001	0.00012	1.7
${\mathit{\lambda }}_{\mathbf{a}\mathrm{-}\mathbf{b}}$	nm	0.0	1.000	Basis c 7	0.0067	0.00085	0.00078	14
${\mathit{k}}_{\mathbf{\lambda }\mathbf{,}\mathbf{m}\mathrm{-}\mathbf{b}}$	1	1.0	0.010	Normal	0.013	0.0011	0.0011	23
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{c}}$	1	0.0	0.010	Normal	0.021	0.0025	0.0018	30
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{uc}}$	1	0.0	0.010	Normal	0.0017	0.00041	0.0005	6.5
${\mathit{S}}_{\mathbf{a}\mathrm{-}\mathbf{b}}$	1	0.0	0.010	Basis c 7	0.024	0.0036	0.0036	52
${\mathit{k}}_{\mathit{S},\mathbf{m}\mathrm{-}\mathbf{b}}$	1	1.0	0.010	Basis c 7	0.0065	0.00098	0.00088	20
All					0.037	0.0048	0.0043	70

Mean values: Y = 1.000, (x, y) = (0.43155, 0.40216), $T_{\mathrm{cp}}$  = 3077 K.

Relative luminance: The additive correlated components and the additive partially correlated components (basis functions for both the wavelength and the amplitude scales) have a greater contribution to the uncertainty than the other effects.

Chromaticity coordinates: Uncorrelated contributions of the model average out and correlated additive amplitude component as well as the multiplicative basis function contribution for the wavelength scale and the additive basis function contribution for the amplitude scale are significant.

CCT: Besides all the partially correlated parameters modelled by the basis functions, the correlated additive amplitude component contributes significantly to the uncertainty.

The results of the uncertainty evaluation for the chromaticity coordinates are shown graphically in Figures 4 and 5, in terms of expanded uncertainty ellipsoids. Figure 4 shows the effect of the wavelength scale parameters, and Figure 5 shows the effect of the amplitude scale parameters. The label ‘All’ is used for the results in the last row of Table 2 containing all modelled uncertainty contributions. In addition to the data in Table 2, these plots allow the chromaticity uncertainty and the correlation to be analysed together.

Figure 4

Covariance plots for the chromaticity coordinates based on the wavelength uncertainty contributions

Figure 5

Covariance plots for the chromaticity coordinates based on the amplitude uncertainty contributions

The correlated additive components result in fully correlated chromaticity uncertainties, displayed as a line. The other contributions result in ellipsoids with different principal axis orientations and lengths. A major contribution is the partially correlated amplitude component, which is modelled as an additive basis function.

Figures 6 to 9 show selected correlation matrices of the generated spectral distributions inside the MCS1 based only on the influence of the model parameter ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ . This means, after calculating the value pairs ${\mathit{M}}_{\mathrm{r},\mathrm{SD}}=(\mathbf{\lambda },{\mathit{S}}_{\mathrm{r}})$ during the MCS1, we can summarise the data as explained in Section 3.3 and display the correlation matrix $\mathit{P}$ to get some insights about the correlation structure of the simulated data. A correlation matrix is a symmetric matrix with values between −1 and 1 and with other specific properties (e.g. all eigenvalues must be non-negative). Therefore, it is almost impossible to generate the correlation matrices directly via an MC simulation. This makes the basis function approach more valuable.

Figure 6

Correlation matrix for the spectral distribution with ${\mathit{N}}_{\mathrm{B}}=7$ Chebyshev basis functions ( ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ Model parameter only)

Figure 7

Correlation matrix for the spectral distribution with ${\mathit{N}}_{\mathrm{B}}=7$ Fourier basis functions ( ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ Model parameter only)

Figure 8

Correlation matrix for the spectral distribution with seventh base function only (Fourier basis functions, ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ Model parameter only)

Figure 9

Correlation matrix for the spectral distribution with ${\mathit{N}}_{\mathrm{B}}=7$ Fourier basis functions and one over f weighting ( ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ Model parameter only)

The nominal wavelength scale is used for the ordinate and abscissa axes in these plots. For the following examples, one can see that there is only a partial correlation generated by the basis function approach between different wavelength settings and not a full correlation as could be expected.

Figure 6 shows the correlation matrix for the spectral distribution based on the ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ modelling with $N_{\mathrm{B}}=7$ and Chebyshev basis functions. Figure 7 shows the same calculation with Fourier basis functions. Figure 8 shows a correlation matrix using Fourier basis functions as in Figure 7, but with a single basis function of order $k=7$ only. Figure 9 shows a correlation matrix using the same basis functions as in Figure 7 and a one over f weighting.

Figures 6 to 9 indicate different correlation matrices that are also structured completely differently from the measurement results of the PTB, which are shown in Figure 3. The authors cannot yet provide general interpretations of the selected correlation matrices displayed here.

4.1 Evaluation of different basis function numbers and types

Iterating the MCS over several basis function numbers ( $N_{\mathrm{B}}=0\dots N_{\lambda }/2-1$ ) and types will result in a lot of information as can be seen in Figures 10 to 13. $N_{\lambda }/2-1$ is the maximum number of basis functions possible, due to the Nyquist rule.

Figure 10

Measurement uncertainty of relative tristimulus value, u(Y_rel), as a function of the number of basis functions, N_B.

Figure 11

Measurement uncertainty of the chromaticity value, u(x), as a function of the number of basis functions, N_B

Figure 12

Measurement uncertainty of chromaticity value, u(y), as a function of the number of basis functions, N_B

Figure 13

Measurement uncertainty of the correlated colour temperature value, u(T_cp), as a function of the number of basis functions, N_B

Figures 10 to 13 show the number of basis functions on the horizontal axis (log scale) and on the vertical axis the measurement uncertainty of different quantities (tristimulus value (Y), chromaticity coordinates $(x,y)$ and CCT (T_cp) resulting from changes of the model parameter ${\mathit{k}}_{S,\mathrm{m}\mathrm{-}\mathrm{b}}$ only.

Besides identifying the basis function number that has the maximum impact on the final uncertainty in each case, three specific points are usually of particular interest and marked separately in Figures 10 to 13:

• $N_{\mathrm{B}}=0$ , the fully correlated case, is marked with the label ‘corr’.

• $N_{\mathrm{B}}=N_{\lambda }/2-1$ , the uncorrelated case, is marked with the label ‘un-corr’ and a corresponding horizontal line.

• The uncertainties evaluated from the measurement results from PTB are shown as an additional point and a corresponding horizontal line marked with the label ‘PTB’.

The different basis function techniques are displayed with different graphs and labelled with ‘f’ for the cumulative Fourier basis functions (see Equation (9)), ‘f s’ for a single Fourier basis function (see Equation (10)) and ‘c’ for cumulative Chebyshev basis functions. The one over f approach using Fourier basis functions is labelled with ‘f 1f’ (see Equation (11)).

5. Discussion

The simulations show that, as expected, uncorrelated and fully correlated contributions generally have no significant influence on chromaticity coordinates and the other evaluated quantities. The exception is the contribution from the additive, fully correlated errors in the values of the wavelengths (e.g. caused by the homing/initialising procedure of a monochromator or by the wavelength adjustment of an array spectroradiometer with a few spectral lines only), which makes significant contributions to nearly all investigated output quantities.

However, it was shown by modelling with orthogonal basis functions that partial correlations contribute significantly to the measurement uncertainty. The tristimulus functions change slowly with wavelength and therefore only the long-wave basis functions (i.e. those where $N_{\mathrm{B}}$ is small) produce effects that do not cancel each other out. Thus, these reflect the possible maximum effects of correlations. This is a possible contribution to the measurement uncertainty that needs to be considered for physical models.

Using single basis functions provides information about the most sensitive frequency for the basis function technique. It is usually observed that ignoring correlations for intermediate spectral frequencies leads to a greater underestimation of the uncertainty. It therefore makes sense to carry out this single basis function analysis and to analyse the results carefully.

6. Conclusion

The framework approach presented in this paper makes it possible to understand the main contributions of the spectral measurement uncertainties to the estimated combined measurement uncertainty and to consider their impact for specific cases. Additionally, this approach identifies possible correlations among the output quantities.

An MCS1 has shown that when evaluating quantities and uncertainties derived using spectral integrations, the fully correlated and uncorrelated errors contribute much less to the uncertainty than the partially correlated errors modelled by the basis functions. As expected, the basis functions with fewer terms (functions with low spectral frequencies representing errors with short autocorrelation length) provide a larger uncertainty in the evaluation of these spectrally integrated quantities and thus provide a reasonable estimate of the maximum uncertainty.