Proceedings of ITC/ISHW2007

Physical data assessment for energy confinement scalinglaws
Heiko DREIER1) , Roland PREUSS1) , Andreas DINKLAGE1) and Volker DOSE2)
1)

Max–Planck-Institut für Plasmaphysik, EURATOM Association, Teilinstitut Greifswald, D-17489 Greifswald
2)
Max–Planck-Institut für Plasmaphysik, EURATOM Association, D-85784 Garching
The principle of data adaptive planning (DAP) is applied to estimate the information gained bya single datum from a given set of data about the parameters of interest. The information gain isthereby quantified by an information measure. By using DAP one is capable to express the importanceof a single datum. A W7-AS confinement data set was analysed with respect to different energy
confinement scaling laws, the importance of the data points is assessed with respect to the respective
scaling law. The physical question (i.e. the scaling law constraints) thereby shows significant impact
on the expected information gain of single data points.
Keywords: energy confinement, scaling laws, Connor-Taylor model, Bayesian experimental design, data
adaptive planning

1 Introduction
Fusion experiments like stellarators today are complex and also expensive devices. To gain a maximumof output by acceptable effort and costs a purposive
design and planning of such experiments is necessary.
For this, it is also important to assess the available
data from previous measurements with respect to theexperimental goals.
During the past years, data bases like the International Stellarator Confinement Data Base (ISCDB) [1]
containing the experimental results of different fusion
machines have grown and are widely used as reference
and for inter-machine comparison (see, e.g., [2, 3]).
Current work also includes the assignment of specificphysical models to the data (or sets of data), using
model comparison techniques [4, 5].
Given a set of measured data, it is useful to estimate the information gain of a single data point, e.g.,
to identify the most informative datum, with respect
to a certain physical model describing the experimental situation. It seems reasonable that in this casethe underlying model should have a significant influence on that evaluation. Also, the impact of the otherdata has to be taken into account. A possible method
for the validation of a given data set, basing on theconcept of Bayesian experimental design, is presented
in this paper. With this method, the assessment ofdata from fusion experiments with respect to different
physical models is possible.

author’s e-mail: heiko.dreier@ipp.mpg.de

1.1 Scaling laws
Because the detailed dependencies between plasma
parameters (electron density n, heating power P ),machine parameters (minor radius a, magnetic fieldstrength B) and quantities describing the energy confinement (confined energy W , energy confinement
time τE ) are not completely known, semi-empiricalscaling laws are used for comparison of different experiments. Also, the design of future experiments ispossible by extrapolation of the scaling relation.
A typical approach is given by a power scaling
law, e.g. the International Stellarator Scaling 2004
(ISS04 [3]), which was found by studies of data fromdifferent stellarator experiments:

τEISS04=0.134 · a2.28 · R0.64−0.61· Ptot
· n0.54 · B 0.84 · ι0.41
2/3 .

A different approach was introduced by Connor and Taylor [6]: Here, the basic assumption isthat scaling invariance properties of the particular
plasma model lead to constraints of scaling exponents.
Plasma models are assumed to be derived from somebasic equations like:• the Fokker-Planck equation describing collisions,• the Maxwell equations describing the influence ofβ,• and the continuity equation, momentum equationand the energy equation for the MHD descriptionof the plasma.
Following this ansatz, the Connor-Taylor scaling
model for the confined energy can be rephrased forProceedings of ITC/ISHW2007
volume averaged quantities:W∝na4 B 2

P
na4 B 3α1  3 4 α2 α3a B
1nna2(1)Here, the αi are the scaling parameters. Dependingon the physical plasma model, these parameters may
be dependent of each other, or even zero. E.g., for thecase of a collisional low-β plasma one obtains α3 = 0,
whereas α1 and α2 have to be estimated.
The scaling laws can be linearised by taking thelogarithm. Then the linearised power law reads
ln W = αc + αa ln a + αP ln P + αn ln n + αB ln B.

1.2 Data adaptive planning
The approach of Bayesian experimental design (BED)
offers a mathematically consistent way for the optimisation of diagnostics as well as for the design of experiments and experimental campaigns using a physicalquestion, expressed by a set of parameters of interest,as the design criterion. It bases on the maximisationof a utility function, namely the Kullback-Leibler distance, which is a measure for the information gain bya measurement. The method was proposed by Lindley [7], a review on different applications can be foundin [8], the design of plasma diagnostics is studied in[9, 10].
The BED approach can also be utilised for theassessment of a data point. For this, the expected
information gain for this particular measurement iscalculated with respect to• the physical question of interest,• the experimental configuration,• the measurement error and error statistics and• other data available.
If existing data sets are implemented, the KullbackLeibler distance as the utility function for the assessment reads:
ZUKL (D, d, ξ) =dα p(α| D, d, ξ)p(α| D, d, ξ)· lnp(α| d)This expression is an measure for the information
gained by the new datum D about the parametersof interest, α. It is given in bit if the base-2 logarithm is used. The probability density function(PDF) p(α| D, d, ξ) is called ”posterior distribution”,describing the knowledge about α given the new datum D, the experimental configuration ξ of the newmeasurement, and the old data d. The prior function p(α| d), on the other hand, shows the knowledge

about α before the new measurement (given only theold data).
For Bayesian experimental design, the information measure has to be averaged over the expected
Rvalues for D, described by the PDF p(D|ξ, d) =e p(α|e d) p(D| α,e ξ), to cover all possible outcomedαe is the result forof the future experiment. Here, α
the parameters of interest given only the old measurements d. This leads to the Expected Utility (EU)functionZ
EU (ξ, d) =dD p(D| d, ξ) · U (D, d, ξ)ZZe p(α|e d) dD p(D| α,e ξ)=dαZdα p(α| D, d, ξ)p(α| D, d, ξ)· ln.
p(α| d)For a linear physical problem d = X ·α and D = ξ T ·α(matrix X contains the experimental configurationsof the old data) and assuming a Gaussian error statistics, the EU can be calculated analytically after somealgebra:
G
1log (1 + G) −, (2)EU (d, ξ) =2(1 + G)2withξT

XT C X

−1ξ

; Cii = 1/s2i .
(3)
σ2
The measurement error of the new datum is thereby
given with σ, the error of the old data are encodedwith s.
G=2 Results
For the study presented here, the Expected Utilityof every datum from a given set was calculated according to linearised scaling laws. For this, the respective
datum was removed from the data set and treated as”new datum”. The experimental configuration (n, P ,a and B) of this measurement then corresponds to ξ,
X and d are the configuration and the data outcome
from the other data points in the set (see eqs. (2) and(3)).
The data set itself consists of 153 ι = 1/3 dataof W7-AS taken from ISDB. In this data base, thevalues for the configuration parameters n, P , a and B
as well as for the measured confined energy are listed.
Furthermore, the measurement errors for all quantities
are given. For the data analysed here the collisional
low-β model was identified to be the most probable
one [5]; therefore, α3 = 0 was used in the ConnorTaylor scaling law. As a second model, a linearised
power law was applied.

Proceedings of ITC/ISHW2007

(a)

(b)

Fig. 1 Data set plotted against experimental and theoretical value of the confined energy. The theoretical value was calculated by a power scaling law (a)and the Connor-Taylor model for collisional low-β
plasmas (b). The EU of the respective datum iscolor-coded.

Figure 1 shows the data set plotted against theexperimental value of the confined energy, Wexp andthe theoretical value Wtheo calculated by the respective scaling law. The EU is given as color scheme.
For both analysed scaling laws the data points
with the highest expected information gain are foundat high values of W . As a difference, the most informative point with respect to the low-β model is not
the one at maximum W , which is the case for thepower law model.
The absolute scales for the EU are different for
both scaling laws. This results from the fact that twodifferent physical questions are analysed here. The setof parameters of interest, the scaling exponents, are
not identical. Therefore, the absolute values of theexpected information gain for the different problems
cannot be compared directly.

(a)

(b)

Fig. 2 Expected Utilities of the data set with respect tothe experimental configuration parameters P andn, (a) for the power law model, (b) for the low-βmodel.

The behavior of the Expected Utility was thenanalysed with respect to the experimental configuration parameters a, n, B and P . To study the influenceof these parameters, the data set is plotted in the n-P
plane next (fig. 2). This plane is of interest because
density and heating power can be varied. Again, one
finds similarities between the different scaling laws: In
both cases, data points in regions with a low sample
rate (high n and P ), far away from the most of thedata, are very informative. In case of the power scaling law, the highest EU is given for the datum withmaximum n and P . In contrast, this is not the casefor the low-β scaling.
In order to study the impact of the further control
parameters, the data set is plotted with respect to theminor radius a and the magnetic field B (fig. 3): In
the case of the power law high values of the EU occur
in all regions of a and B, for the Connor-Taylor model
the highest values of the EU are found only at theProceedings of ITC/ISHW2007

(a)

(b)

(c)

(d)

Fig. 3 Expected Utilities of the data set with respect tothe minor radius a (upper row) and the magnetic
field strength B (lower row), for the power law
model (left column) and for the low-β model (rightcolumn), respectively.

highest values of a and B.
These findings can be explained by taking into account the dependency on a and B in the low-β model:
Both parameters are related to the α1 and α2 termswith high exponents (a4 , B 3 , see eq. (1)), so highvalues of a and B will have a strong influence on theoutcome of the model. In contrast, the parameters nand P enter the model only to the power of 1. Therefore, data points with high a and/or B will lead tohigher information gain in case of the low-β model.
This indicates that measurements at high a and B
are more important for the validation of this model.

3 Conclusion
In this work, data adaptive planning was used to calculate the expected information gain of a datum withrespect to a set of 153 ι = 1/3 data from W7-AS. The
capability of DAP to implement a physical question as
an assessment criterion in a mathematical consistent
way was demonstrated by applying the method to two
different energy confinement scaling laws: a power law
similar to the International Stellarator Scaling, andthe Connor-Taylor scaling for collisional low-β plasmas. The expected information gain from the single
data points turned out to be different for both models: Whereas for the power law the information gainof data points with values of low a and B are on a par
data with values of high a and B, for the low-β modelthe data in the range of high a and B turn out to be
more valuable. This can be explained by the depen-dence of the Connor-Taylor terms on high powers ofa (∝ a4 ) and B (∝ B 3 ). The Connor-Taylor terms
are therefore dominated by these parameters at highvalues.
These results show the influence of the physicalquestion on the Expected Utility: Different physicalmodels will lead to different values of the information
gain from a certain datum. DAP is a tool to quantify
these differences and to estimate the contribution ofindividual data to the overall result. It is a new quality
for the assessment of data going beyond qualitativearguments.
For future work, further analysis of the parameters influencing the value of the expected information
gain have to be made. In particular, the measurement
error and the error statistics may have significant impact (see [10]).
In addition, the DAP formalism can be used forthe systematic planning of future experiments andexperimental campaigns: The calculation of the expected information gain for a possible future measurement offers the possibility to estimate the mostinformative experimental condition for the next experiment.
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Physical Society Conference on Plasma Physics, Warsaw (2007)[6] J. Connor, J. Taylor, Nucl. Fusion 17, 1047-1055(1977)[7] Lindley, D. V., Bayesian Statistics - A Review, SIAM,
Philadelphia (1972)[8] K. Chaloner, I. Verdinelli, Stat. Sci. 10, 273-304 (1995)[9] R. Fischer, in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conference Proc. 735 (2004), 76-83[10] H. Dreier et al., Fusion Sci. Technol. 50, 262-267(2006)