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

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