In the present work we discuss and extend an existing Bayesian Hierarchical Gaussian Process Model (BHGP) used to integrate data with different accuracies. The low-accuracy data are the deterministic output of a computer experiment and the high-accuracy data come from a more precise computer simulation or a physical experiment. A Gaussian process model is used to fit the low-accuracy data. Then the high-accuracy data are linked to the low-accuracy data using a flexible adjustment model where two further Gaussian processes perform scale and location adjustments. An empirical Bayesian approach is chosen and a Monte Carlo Markov Chain (MCMC) algorithm is used to approximate the predictive distribution at new input sites. The existing BHGP model is then extended in order to model the more general situation where also the low accuracy data come from a physical experiment. A measurement error term needs to be included in the model for the low-accuracy data and the MCMC prediction method is accordingly adjusted. The BHGP model is implemented in Matlab and a validation study is performed to verify the developed code and to evaluate the predictive performance of the model. The extended BHGP model is then applied to a set multi-sensor metrology data in order to model the surface of an object. The low-accuracy data are measured with an innovative optical-based Mobile Spatial Coordinate Measuring System II (MScMS-II), developed at Politecnico di Torino, Italy, and the high-resolution data are acquired with a Coordinate-Measuring Machine (CMM). Comparing the BHGP model with other existing methods allows us to conclude that significative improvements (by 11%-22%) in terms of prediction error are achieved when low-resolution and high-resolution data are combined using an appropriate adjustment model.
Nel presente lavoro si analizza e si estende un modello bayesiano gerarchico che sfrutta i processi gaussiani (BHGP) con lo scopo di integrare dati con diversa accuratezza. I dati a bassa accuratezza provengono da un esperimento computazionale deterministico e quelli ad elevata accuratezza da una simulazione numerica più precisa o da un esperimento fisico. Un processo gaussiano modella i dati a bassa accuratezza, mentre un modello flessibile di aggiustamento collega i dati molto accurati a quelli poco accurati, sfruttando due ulteriori processi gaussiani che svolgono la funzione di parametri di scala e di localizzazione. Si adotta un approccio bayesiano empirico per fare inferenza sui parametri del modello e si sfrutta un algoritmo Markov Chain Monte Carlo (MCMC) per approssimare la distribuzione a posteriori predittiva in corrispondenza di nuovi punti sperimentali. Il modello BHGP esistente viene esteso in modo da poter essere applicato al caso più generale in cui anche i dati a bassa accuratezza provengono da un esperimento fisico. Un temine di errore casuale è introdotto nel modello dei dati a bassa accuratezza e i passi dell'algoritmo MCMC devono essere corretti di conseguenza. Dopo aver implementato il modello in Matlab, si svolge uno studio di validazione per verificare la correttezza del codice e le prestazioni del modello in termini predittivi. Il modello BHGP viene infine applicato a dati di metrologia, provenienti da due distinti strumenti di misura a coordinate. I dati a bassa accuratezza sono misurati con un innovativo dispositivo portatile per la misura a coordinate su larga scala (MScMS-II) sviluppato presso il Politecnico di Torino, mentre quelli ad elevata accuratezza sono acquisiti con una macchina di misura a coordinate (CMM). Paragonando il modello BHGP ad altri modelli analizzati, si riscontra un significativo miglioramento delle prestazioni (dall'11% al 22%) in termini di errore di predizione, quando i dati multi-risoluzione sono combinati usando un opportuno modello di aggiustamento.
Bayesian Hierarchical Gaussian Process Model: an Application to Multi-Resolution Metrology Data
DOLCI, LUCIA
2009/2010
Abstract
In the present work we discuss and extend an existing Bayesian Hierarchical Gaussian Process Model (BHGP) used to integrate data with different accuracies. The low-accuracy data are the deterministic output of a computer experiment and the high-accuracy data come from a more precise computer simulation or a physical experiment. A Gaussian process model is used to fit the low-accuracy data. Then the high-accuracy data are linked to the low-accuracy data using a flexible adjustment model where two further Gaussian processes perform scale and location adjustments. An empirical Bayesian approach is chosen and a Monte Carlo Markov Chain (MCMC) algorithm is used to approximate the predictive distribution at new input sites. The existing BHGP model is then extended in order to model the more general situation where also the low accuracy data come from a physical experiment. A measurement error term needs to be included in the model for the low-accuracy data and the MCMC prediction method is accordingly adjusted. The BHGP model is implemented in Matlab and a validation study is performed to verify the developed code and to evaluate the predictive performance of the model. The extended BHGP model is then applied to a set multi-sensor metrology data in order to model the surface of an object. The low-accuracy data are measured with an innovative optical-based Mobile Spatial Coordinate Measuring System II (MScMS-II), developed at Politecnico di Torino, Italy, and the high-resolution data are acquired with a Coordinate-Measuring Machine (CMM). Comparing the BHGP model with other existing methods allows us to conclude that significative improvements (by 11%-22%) in terms of prediction error are achieved when low-resolution and high-resolution data are combined using an appropriate adjustment model.File | Dimensione | Formato | |
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2011_03_Dolci.pdf
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https://hdl.handle.net/10589/16465