The Concept of Integrated Data Analysis of Complementary Experiments - - PowerPoint PPT Presentation

The Concept of Integrated Data Analysis

• f Complementary Experiments
• R. Fischer,

, A. Din inklag age

Max-Planck-Institut für Plasmaphysik, Garching & Greifswald EURATOM Association Saratoga Springs, July 8-13, 2007

MaxEnt 2007

Motivation: Nuclear Fusion

• Different measurement techniques for the same quantities → complementary data
• Coherent combination of measurements from different diagnostics

is a major step in the analysis of experimental data from nuclear fusion devices

• Goal:
• replace combination of results from individual diagnostics
• with

combination of measured data → one-step analysis of pooled data

Outliers, inconsistent data, signal-background separation

Motivation: Single Diagnostics

Challeng nges for data anal alysis is for fus usion

• ns devices:

Reliable diagnostics

(data consistency)

⇒ Systematic and unified error analysis

• consider all

ll sta tatistic ical and systematic errors

• general agreeme

ment when we ta talk about t “error”, , “uncerta tainty ty”, “reli liabili lity”, , “signif ific icance”, “evidence”, …

Parameter correlations

(e.g. TS: Te correlated with ne)

⇒ non-Gaussian distributions in data and p parameters

• error propagati

tion? → generaliz izati tion of Gaussia ian error prop.

⇒ Robust estimation

• mixture mo

modelin ing

mapping, equilibrium calc., transport calc. ⇒ Combination with modeling Profiles and Gradients ⇒ nonparametric function estimation

• fl

flexible le AND reli liable

... an and we want to combi bine/ e/link nk di diagn gnostics ⇒ Validate theoretical models Model complexity ⇒ Model comparison

• e.g

.g. number of spectral l lines

Motivation: Multiple Diagnostics

Multi-tasking tools

(e.g. CXRS/BES or TS/IF)

⇒ Analysis of SETS of diagnostics for synergistic effects

(complementary diagnostics)

Transient effects

(e.g. W/Be/C deposit ition/erosio ion on mirrors, , degradati tion of glas fi fibers, etc.)

⇒ Combination of data for automatic in-situ calibration Combined evaluation

(“super fit”: TS, ECE, LiB, etc.)

⇒ Error reduction by combination of diagnostics

(combination of f data ta NOT result lts)

Diagnostics interdependencies

(e.g. TS, Zeff, CXRS, BES, etc.)

⇒ “One-step” analysis by combination of diagnostics

(complex error propagation!) !)

Consistent diagnostics

(global data consis istency)

⇒ Exploit redundant information

(provide in informatio ion to resolv lve data in inconsistencies)

... comb mbin ine/li link dia iagnostic ics:

Conventional vs. Integrated Data Analysis

Thomson Scattering data analysis ne(x),Te(x) ECE data analysis Te(x) mapping ρ(x) mapping ρ(x) linked result ne(ρ),Te(ρ) ne(ρ), Te(ρ), ... mapping ρ(x) → ne(x), Te(x) DTS(ne(x)),Te(x)) Thomson Scattering data dTS DECE(ne(x)),Te(x)) ECE data dECE result: p(ne(ρ),Te(ρ) | dTS,dECE) addl. information, constraints, model params, ...

con

• nven

ention

• nal

IDA (Bayesian proba bability theo eory)

estimates: ne(ρ) ± Δne(ρ), Te(ρ) ± ΔTe(ρ)

...

Integrated Data Analysis

1) 1) 1) 1) Bayesian Modell lling Bayesian Modell lling

• f Individual Diagnostics
• f Individual Diagnostics

⇒ Physical model: quantity of interest ↔ data ⇒ Statistical model: IDA requires a system ematic and formaliz ized error analysis of all uncertaintie ies involved in each diag agnostic to allow for a comparable and relia iable le error analy lysis of different diagnostics. An elaborate error analysis is a MUST for the next step! 2) 2) 2) 2) Bayesian Integration: Li Linkage of Diagnostics Models ls Bayesian Integration: Li Linkage of Diagnostics Models ls ⇒ Combine statistical models of individual diagnostics ⇒ Additional information (physical constraints, modeling, ...) (identification a and quantification of errors: e extensive workload for diagnostician)

Recipe

• Identify sources of uncertainties and

quantify with probability distributions (pdf)

• Statistical (measured data, calibration data

ta, , .. ...) .) → likelihood pdf

• Systematic (mis-alignment, tr

transmis issivity ty, mir irror reflectiv ivity, , .. ...) .)

→ pdf on hyperparameter

• Model parameters (rate coeff.,

, ...) .) → pdf on model parameter

• Simplified model assumptions (ECE, plas

asma ma equilib ibri rium calc lc., ., etc tc)

• Combine probability pdfs according to Bayes theorem
• Marginalize (integrate) nuisance parameters

(systematic effects, uncerta tain model parameter, , etc.) .)

Generalization of Gaussian error propagation laws

• Result: marginal probability distributions
• f quantities of interest

Sequential or One-step Data Analysis?

Set of data {d_i} (from subsequent measurements or different

t dia iagnostics) → parameter Θ

One-step analysis:

p∣ d∝∏i pd i∣ p

Bayesian theorem Sequential analysis:

p∣d1∝ pd1∣ p

posterior using first data

p∣d1,d 2∝ pd 2∣d1, p∣d1

use old posterior as new prior

 p∣ d∝ pd N∣d N −1 ,,d1, pd 2∣d1,× pd1∣× p ⋮

product rule For independent data:

 p∣ d∝ pd N∣ pd 2∣× pd1∣× p ∝∏i pd i∣ p

Sequential ≡ one-step data analysis! Provided we use full probability distributions!

Integrated Data Analysis

⊗ ⊗ =

Thomson Scattering Soft-X-ray Interferometer Operation Int ntegrated Result

Using interdependencies: Combination of results from a set of diagnostics (W7-AS)

Probabilistic framework

• R. Fischer, A

A. . Dinkl klage, E. Pasch, P PPCF 2 2002, PPCF 2003

Integrated Data Analysis: Interdependencies

Electron density 30% reduced error

Using synergism: Combination of results from a set of diagnostics

→ synergism by exploiting full probabilistic correlation structure

⊗

Thomson Scattering Soft-X-ray

e

dT

=

∫

Profile and profile gradient

• Exponential splines

nonparametric function estimation → flexible profile reconstruction hybrid

id: polygon ↔ exp. spline ↔ splin ine

→ smooth and rapid function changes → integration over spline knot amplitudes and positions, number of knots, and

• exp. spline hyperparameter

→ robust gradient reconstruction more fl

flexible le than parametric functio ion estima mati tion (tanh)

less fl

flexible le than pointw twis ise reconstructi tion (opti timal knot t numbers)

reduced curvatu

ture regula lariz izati tion

→ uncertainties on profile gradients

W7-AS #54285

• V. Dose, R. Fischer A

AIP 2005

• R. F

Fischer et a al., to be published

Profile and profile gradient → derived quantities Ti Te ne Er Qei

Summary: IDA

✔ probability distributions: describes all kind of uncertainties ✔ multiply probability distributions, marginalization of nuisance parameters ✔ generalization of Gaussian error propagation

➢ Probabilistic modeling of individual diagnostics ➢ Probabilistic combination of different diagnostics ➢ Topics

✔ systematic and unified error analysis is a must for comparison of diagnostics ✔ error propagation beyond single diagnostics ✔ more reliable results by larger (meta-) data set (interdependencies, synergism) ✔ redundant information → provide information to resolve data inconsistencies ✔ robustness in a harsh environment ✔ flexibility vs. reliability: profile and profile gradients, model comparison ✔ robust estimation (outliers, inconsistent data, signal – background separation) ✔ IDA and Bayesian Experimental Design