MaxEnt 2007 The Concept of Integrated Data Analysis of Complementary Experiments R. Fischer, , A. Din inklag age Max-Planck-Institut für Plasmaphysik, Garching & Greifswald EURATOM Association Saratoga Springs, July 8-13, 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 combination of measured data → one-step analysis of pooled data with
Motivation: Single Diagnostics Challeng nges for data anal alysis is for fus usion ons devices: ⇒ Systematic and unified error analysis Reliable diagnostics consider all ll sta tatistic ical and systematic errors (data consistency) general agreeme ment when we ta talk about t “error”, , “uncerta tainty ty”, “reli liabili lity”, , “signif ific icance”, “evidence”, … ⇒ non-Gaussian distributions in data and p Parameter correlations parameters (e.g. TS: T e correlated with n e ) error propagati tion? → generaliz izati tion of Gaussia ian error prop. ⇒ Robust estimation Outliers, inconsistent data, signal-background separation mixture mo modelin ing mapping, equilibrium calc., ⇒ Combination with modeling transport calc. ⇒ nonparametric function estimation Profiles and Gradients fl flexible le AND reli liable ⇒ Model comparison Validate theoretical models Model complexity e.g .g. number of spectral l lines ... an and we want to combi bine/ e/link nk di diagn gnostics ⇒
Motivation: Multiple Diagnostics ... comb mbin ine/li link dia iagnostic ics: Consistent diagnostics ⇒ Exploit redundant information (global data consis istency) (provide in informatio ion to resolv lve data in inconsistencies) ⇒ Error reduction by combination of diagnostics Combined evaluation (combination of f data ta NOT result lts) (“super fit”: TS, ECE, LiB, etc.) ⇒ “One-step” analysis by combination of diagnostics Diagnostics interdependencies (e.g. TS, Z eff , CXRS, BES, etc.) (complex error propagation!) !) ⇒ Analysis of SETS of diagnostics Multi-tasking tools for synergistic effects (e.g. CXRS/BES or TS/IF) (complementary diagnostics) Transient effects ⇒ Combination of data for automatic in-situ calibration (e.g. W/Be/C deposit ition/erosio ion on mirrors, , degradati tion of glas fi fibers, etc.)
Conventional vs. Integrated Data Analysis con onven ention onal IDA (Bayesian proba bability theo eory) n e (ρ), T e (ρ), ... Thomson ECE ... Scattering mapping ρ(x) data data → n e (x), T e (x) analysis analysis D TS (n e (x)),T e (x)) D ECE (n e (x)),T e (x)) addl. n e (x),T e (x) T e (x) information, Thomson constraints, Scattering ECE model params, mapping mapping data d TS data d ECE ... ρ(x) ρ(x) linked result result: p(n e (ρ),T e (ρ) | d TS ,d ECE ) n e (ρ),T e (ρ) estimates: n e (ρ) ± Δn e (ρ), T e (ρ) ± ΔT e (ρ)
Integrated Data Analysis 1) 1) Bayesian Modell Bayesian Modell lling of Individual Diagnostics of Individual Diagnostics 1) 1) lling ⇒ 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! (identification a and quantification of errors: e extensive workload for diagnostician) 2) Bayesian Integration: Li 2) Bayesian Integration: Li Linkage of Diagnostics Models ls 2) 2) Linkage of Diagnostics Models ls ⇒ Combine statistical models of individual diagnostics ⇒ Additional information (physical constraints, modeling, ...)
Recipe Identify sources of uncertainties and quantify with probability distributions (pdf) Statistical (measured data, calibration data .) → likelihood pdf ta, , .. ...) 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 of quantities of interest
Sequential or One-step Data Analysis? Set of data { d_i } (from subsequent measurements or different iagnostics) → parameter Θ t dia One-step analysis: d ∝ ∏ i p d i ∣ p p ∣ Bayesian theorem posterior using first data Sequential analysis: p ∣ d 1 ∝ p d 1 ∣ p use old posterior as new prior p ∣ d 1, d 2 ∝ p d 2 ∣ d 1, p ∣ d 1 ⋮ p ∣ product rule d ∝ p d N ∣ d N − 1 , ,d 1, p d 2 ∣ d 1, × p d 1 ∣× p For independent data: p ∣ d ∝ p d N ∣ p d 2 ∣× p d 1 ∣× p ∝ ∏ i p d i ∣ p Sequential ≡ one-step data analysis! Provided we use full probability distributions!
Integrated Data Analysis Using interdependencies: Combination of results from a set of diagnostics (W7-AS) ⊗ ⊗ = Interferometer Int ntegrated Thomson Scattering Soft-X-ray Operation Result Probabilistic framework R. Fischer, A A. . Dinkl klage, E. Pasch, P PPCF 2 2002, PPCF 2003
Integrated Data Analysis: Interdependencies Using synergism: Combination of results from a set of diagnostics ∫ = ⊗ dT e Thomson Soft-X-ray Scattering Electron density 30% reduced error → synergism by exploiting full probabilistic correlation structure
Profile and profile gradient Exponential splines W7-AS #54285 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 V. Dose, R. Fischer A AIP 2005 R. F Fischer et a al., to be published
Profile and profile gradient T i T e n e → derived quantities E r Q ei
Summary: IDA ➢ Probabilistic modeling of individual diagnostics ✔ probability distributions: describes all kind of uncertainties ✔ multiply probability distributions, marginalization of nuisance parameters ✔ generalization of Gaussian error propagation ➢ Probabilistic combination of different diagnostics ✔ 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 ➢ Topics ✔ flexibility vs. reliability: profile and profile gradients, model comparison ✔ robust estimation (outliers, inconsistent data, signal – background separation) ✔ IDA and Bayesian Experimental Design
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