Steps towards an integrated data analysis Basic concepts and(2)

Aspects of the integration of different data sources are discussed with respect to improvement of reliability and significance of experimental results. Thomson scattering data have been analysed by means of Bayesian probability theory. This much improved s

measurements), to give a full joint evaluation of diagnostic data. To achieve this goal we

have formulated a framework which will allow us to extract the most reliable profile

information from different sources, including a consistent mapping on magnetic surfaces.

Moreover, the statistical models developed for the error analysis are used for diagnostic

improvement and diagnostic design. Outstanding tools for tackling these issues are given

by Bayesian probability theory.

Evaluation of Thomson scattering data by means of Bayesian probability theory

For a systematic statistical modelling of Thomson scattering, Bayesian probability theory

(BPT, [2,3]) was employed. The main advantages of BPT result from the possibility to

derive the probability density function (pdf) of the quantities of interest, to treat nuisance

parameters in a concise way and to include expert knowledge (priors). The analysis starts

from Bayes theorem, which in our case links the marginal posterior pdf P(Te,ne|d, σ, I) to

find an electron density ne and temperature Te, given the data d, its uncertainties σ and

some additional information I, to the likelihood P(d|,Te,ne,σ,I) and the so-called prior

P(Te,ne,|I):

P(Te,ne|d,σ,I)=P(d|Te,ne,σ,I)P(Te,ne|I)P(d|I)

For our purposes, the evidence P(d|I) serves for normalization only. The major objective of our investigation, which is to determine reliable uncertainty measures of the experimental

results allowing one to combine different diagnostics sources, is reflected by the marginal

posterior probability density functions of the quantities of interest. It must be emphasized

that much care and effort has been taken for a detailed investigation of the uncertainties of all parameters that enter the statistical model.

Fig. 1: Marginal probability densities for ne (left panel) and Te (right panel) at two spatial positions for a Wendelstein 7-AS plasma # 47894. Different lines depict uncertainties included in the

analysis. The dotted and the dashed lines are hidden for the Te analysis because inclusion of both

the Raman scattering errors (Ra) and the polarization factors (Pol) are negligible if compared to the errors in the Thomson data (Th) and spectral sensitivities of the polychromators (τ).

Aspects of the integration of different data sources are discussed with respect to improvement of reliability and significance of experimental results. Thomson scattering data have been analysed by means of Bayesian probability theory. This much improved s

For a systematic derivation of probability density functions from diagnostic data, the

evaluation of ne and Te profiles from Thomson scattering on Wendelstein 7-AS (spatially

resolved, 0.5 J, 10 ns pulsed Nd:YAG laser based TS at 20 Hz, discrete spectral channel

detection) employing BPT was performed. It was shown that BPT reproduced results from different analysis techniques which have been proven for validity in the parameter regimes chosen for comparison [4]. Moreover, due to systematic inclusion of all accessible

uncertainties that affect the outcome – or, in a different terminology, by the discussion of a complete statistical model of the diagnostics – substantial improvements with respect to

sensitivity could be achieved by the BPT analysis, which contributes to investigations in

parameter regimes [5] the diagnostics was originally not designed for.

BPT can be employed for assessing and improvement of diagnostic capabilities [6]. In

figure 1 the outcome of the BPT analysis – the marginal pdf for ne and Te – is shown. In

this representation the influence of different nuisance parameters can be visualized. E.g. the Te measurement is barely affected by the Raman calibration and the polarization

dependence of the detectors, but spectral sensitivities τ strongly affect the uncertainties.

The latter result is valid for ne as well, where τ even shifts the maximum of the Te pdf.

These considerations may be used to quantify the effect of any improvement related to a

given nuisance parameters. Moreover, figure 1 shows that the resulting pdfs may be non-

Gaussian, but Gaussians are an essential prerequisite for error propagation laws. A detailed discussion of the results may be found in [4].

Joint evaluation of profile data

The Bayesian framework is especially well suited for the combination of different sources

of information. The reason for this is the direct association of uncertainties not only with

observations, but also with the unknown parameters of a model. Since unknown parameters are described directly by probability distributions, relationships between the unknown

parameters of a model can be stated explicitly. Such relationships are easily formalised and visualized by the use of Bayesian graphical models [7], where nodes represent unknown

parameters or observations, and the directed edges represent probabilistic or logical

dependencies between the nodes. By applying Bayes theorem [2], the joint posterior

probability distribution of the unknown nodes given the observed nodes can then be

calculated. In figure 2 such a model is shown for an integrated solution of the problem of

combining profile measurements of electron temperature and density from two different

Thomson scattering diagnostics. The mapping on magnetic surfaces is here an integral part of the solution, since it will both depend on and influence the inferred profiles. To

demonstrate the generality of the method, an extra non-profile diagnostic signal (the total

plasma energy) has been added as an auxiliary information source.