Mode-Shape Expansion Using Physics-Constrained Gaussian Process Regression

Farid Ghahari

arxiv: 2605.23101 · v1 · pith:3WX67ERLnew · submitted 2026-05-21 · 🧮 math.NA · cs.NA· stat.ML

Mode-Shape Expansion Using Physics-Constrained Gaussian Process Regression

Farid Ghahari This is my paper

Pith reviewed 2026-05-25 05:10 UTC · model grok-4.3

classification 🧮 math.NA cs.NAstat.ML

keywords mode shape expansionGaussian process regressionphysics-constrained learningmass orthogonalitystructural dynamicssparse sensingmodal analysis

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The pith

A mass-orthogonality penalty added to Gaussian process training produces physically consistent mode shapes from sparse sensor data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how standard Gaussian process regression often yields mode shapes that break physical rules such as mass orthogonality when sensors are few. It derives a constrained single-output Gaussian process that keeps separate modal kernels but couples their hyperparameter search through an added penalty term on mass orthogonality. The marginal likelihood and its gradients are adjusted to include this term. Numerical checks on a multi-degree-of-freedom structure confirm that the resulting expansions stay faithful to the measured points yet satisfy the orthogonality condition. This matters for structural monitoring because full-field mode shapes are needed for damage detection and response prediction.

Core claim

The paper establishes that a Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework, formed by augmenting the marginal likelihood with a mass-orthogonality penalty, produces mode-shape expansions that remain consistent with sparse measurements while satisfying the physical constraint, as verified on a multi-degree-of-freedom structure.

What carries the argument

The CONS-SOGP framework that couples independent modal kernels through a mass-orthogonality penalty term during marginal-likelihood optimization.

If this is right

Expanded mode shapes satisfy mass orthogonality directly from the training step rather than through later correction.
Predictions remain accurate at the sensor locations while obeying the physical constraint across the full field.
Uncertainty estimates accompany the constrained expansions without separate post-processing.
The same penalty coupling applies across different multi-degree-of-freedom systems without structure-specific retuning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The penalty approach could be tested on experimental data sets that include measurement noise to check robustness beyond the numerical verification.
Similar penalty terms might be added for other physical invariants such as energy or boundary conditions in related regression tasks.
The method suggests a route for embedding multiple orthogonality or normalization constraints into Gaussian process models for inverse problems in dynamics.

Load-bearing premise

The mass-orthogonality penalty, when included in the optimization, yields reconstructions that stay consistent with the data without creating new fitting artifacts or requiring case-by-case tuning.

What would settle it

A numerical or experimental case in which the CONS-SOGP expansions violate mass orthogonality on the measured degrees of freedom or produce larger errors on held-out full-field data than an unconstrained Gaussian process.

read the original abstract

This paper addresses the challenge of reconstructing full-field structural mode shapes from sparse sensor data. While Gaussian Process Regression (GPR) offers a robust non-parametric framework for spatial interpolation and uncertainty quantification, standard formulations often yield physically inconsistent mode-shape reconstructions under sparse sensing conditions. A Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework is derived that utilizes independent modal kernels while coupling the optimization via a mass-orthogonality penalty. The paper presents derivations for the marginal likelihood, hyperparameter gradients, and penalty coupling. Numerical verification on a multi-degree-of-freedom structure demonstrates that the proposed method overcomes existing limitations in GP-based prediction, providing more accurate and reliable expanded mode shapes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper couples independent modal GPs via a mass-orthogonality penalty in the marginal likelihood, but provides no trade-off analysis for the penalty weight.

read the letter

The main thing here is a CONS-SOGP setup that keeps separate single-output GPs per mode and adds a mass-orthogonality penalty only during hyperparameter optimization. That specific coupling step is the technical increment beyond prior GPR work on mode shapes. They derive the marginal likelihood, the gradients, and how the penalty term enters the objective, then run a numerical check on a multi-degree-of-freedom structure that claims more accurate expanded shapes under sparse sensing. Those derivations and the choice to stay with single-output kernels while enforcing the constraint are the parts that look like actual work. The numerical example on a real structure is also a reasonable check rather than pure theory. The soft spot is exactly the one the stress-test flags: the penalty coefficient is a fixed hyperparameter chosen once per case, with no reported study of how its magnitude moves the optimum away from the data-only solution or whether the resulting predictive variances still track measurement noise. The abstract asserts that the method overcomes existing GP limitations, but without error metrics, baseline comparisons, or any sensitivity plot on the penalty, that claim cannot be judged. If the penalty dominates, the reconstructions could satisfy orthogonality while fitting the sensors worse than plain GPR, and nothing in the summary rules that out. This is a methods paper aimed at vibration engineers who already use GPs for modal expansion and want a way to bake in the mass-orthogonality constraint without switching to full multi-output models. A reader in that niche could extract the formulation and try it, but only after checking the full derivations and adding their own trade-off tests. It deserves peer review because the core idea is stated clearly and the math is presented, even though the validation section needs more substance to stand on its own.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework for expanding structural mode shapes from sparse sensor measurements. Standard GPR marginal likelihood is augmented by a mass-orthogonality penalty term; derivations are given for the objective, hyperparameter gradients, and penalty coupling. Numerical results on a multi-degree-of-freedom structure are presented to show improved accuracy and physical consistency over unconstrained GPR.

Significance. If the penalty-augmented optimization reliably balances data fidelity against the orthogonality constraint without introducing fitting artifacts or requiring case-by-case tuning, the method would offer a practical advance for modal analysis under limited instrumentation. The numerical verification on a multi-DOF structure provides initial evidence of benefit, but the absence of trade-off studies weakens the claim that reconstructions remain consistent with measurements while satisfying the physical constraint.

major comments (2)

[§3] §3: The derivation of the coupled gradients for the CONS-SOGP objective (standard marginal likelihood plus mass-orthogonality penalty) is presented, but the formulation treats the penalty coefficient as a fixed hyperparameter chosen once per example. No analysis is given of how its magnitude shifts the location of the optimum relative to the pure data-driven solution or whether the resulting predictive variance still reflects measurement noise.
[Numerical verification section] Numerical verification section: The claim that the method overcomes existing limitations and yields more accurate expanded mode shapes rests on results for a multi-DOF structure, yet the manuscript reports neither quantitative error metrics against baselines nor sensitivity of the reconstructions to the penalty weight. This leaves the central assertion that the reconstructions remain consistent with the measured data without new artifacts unverified.

minor comments (2)

[Abstract] The abstract states that 'independent modal kernels' are used; the main text should explicitly contrast this choice with multi-output GP formulations and justify why single-output kernels plus the penalty are preferred.
Notation for the penalty weight and the mass matrix used in the orthogonality term should be introduced once and used consistently in all equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [§3] §3: The derivation of the coupled gradients for the CONS-SOGP objective (standard marginal likelihood plus mass-orthogonality penalty) is presented, but the formulation treats the penalty coefficient as a fixed hyperparameter chosen once per example. No analysis is given of how its magnitude shifts the location of the optimum relative to the pure data-driven solution or whether the resulting predictive variance still reflects measurement noise.

Authors: We agree that the manuscript presents the penalty coefficient as a fixed hyperparameter without an accompanying sensitivity analysis. The value was selected in the numerical example to achieve a suitable balance between the marginal likelihood and the penalty term. We will add a short analysis (including a brief discussion or supplementary figure) showing the effect of varying the coefficient on the location of the optimum and on the predictive variance in the revised manuscript. revision: yes
Referee: [Numerical verification section] Numerical verification section: The claim that the method overcomes existing limitations and yields more accurate expanded mode shapes rests on results for a multi-DOF structure, yet the manuscript reports neither quantitative error metrics against baselines nor sensitivity of the reconstructions to the penalty weight. This leaves the central assertion that the reconstructions remain consistent with the measured data without new artifacts unverified.

Authors: The current numerical section provides visual comparisons demonstrating improved physical consistency relative to standard GPR. We acknowledge that quantitative error metrics (e.g., RMSE or similar against reference solutions) and explicit sensitivity results with respect to the penalty weight are not reported. We will revise the section to include these quantitative comparisons against baselines and a sensitivity study on the penalty weight to substantiate the claims of data consistency and absence of artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adds independent penalty term

full rationale

The CONS-SOGP framework augments standard GPR marginal likelihood with an explicit mass-orthogonality penalty whose coefficient is treated as a tunable hyperparameter. Section 3 derives the coupled gradients from first principles without reducing any quantity to a fitted input by construction. Numerical verification on an MDOF structure compares against standard GPR baselines using external error metrics. No self-citation chain, self-definitional loop, or renamed known result is present in the provided derivation steps. The central claim therefore rests on an independent modeling choice rather than tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard GPR assumptions plus the domain-specific choice to enforce mass orthogonality via a penalty rather than hard constraints or multi-output kernels.

free parameters (2)

GP kernel hyperparameters
Standard in any GPR; optimized via marginal likelihood.
mass-orthogonality penalty weight
Controls the strength of coupling between modes; must be chosen or tuned.

axioms (2)

domain assumption Mode shapes satisfy mass orthogonality
Invoked as the physical constraint that the penalty enforces.
domain assumption Independent modal kernels remain appropriate when coupled only through the penalty
The framework treats modes separately except for the added term.

pith-pipeline@v0.9.0 · 5639 in / 1331 out tokens · 20988 ms · 2026-05-25T05:10:20.010835+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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[2]

L., & Jennings, P

Beck, J. L., & Jennings, P. C. 1980. Structural identification using linear models and earthquake records. Earthquake enginee ring & structural dynamics, 8(2), 145-160

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R., & Doebling, S

Farrar, C. R., & Doebling, S. W. 1997. An overview of modal-based damage identification methods

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Sohn, H. 2007. Effects of environmental and operational variability on structural health monitoring. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365(1851), 539-560

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Vidal, F., Navarro, M., Aranda, C., & Enomoto, T. 2014. Changes in dynamic characteristics of Lorca RC buildings from pre-and post- earthquake ambient vibration data. Bulletin of Earthquake Engineering, 12(5), 2095-2110

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Ceravolo, R., Matta, E., Quattrone, A., & Zanotti Fragonara, L. 2017. Amplitude dependence of equivalent modal parameters in monitored buildings during earthquake swarms. Earthquake Engineering & Structural Dynamics, 46(14), 2399-2417

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work page 2022
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F., Nateghi, F., & Taciroglu, E

Abazarsa, F., Ghahari, S. F., Nateghi, F., & Taciroglu, E. 2013. Response‐only modal identification of structures using limit ed sensors. Structural Control and Health Monitoring, 20(6), 987-1006

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Yang, Y., & Nagarajaiah, S. 2013. Output-only modal identification with limited sensors using sparse component analysis. Journal of Sound and Vibration, 332(19), 4741-4765

work page 2013
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F., Abazarsa, F., Ghannad, M

Ghahari, S. F., Abazarsa, F., Ghannad, M. A., Celebi, M., & Taciroglu, E. 2014. Blind modal identification of structures from spatially sparse seismic response signals. Structural Control and Health Monitoring, 21(5), 649-674

work page 2014
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Tarpø, M., Nabuco, B., Georgakis, C., & Brincker, R. 2020. Expansion of experimental mode shape from operational modal analys is and virtual sensing for fatigue analysis using the modal expansion method. International journal of fatigue, 130, 105280

work page 2020
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Guyan, R. J. 1965. Reduction of stiffness and mass matrices. AIAA journal, 3(2), 380-380

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O’Callahan, J. C. 1989. System equivalent reduction expansion process. In Proc. of the 7th Inter. Modal Analysis Conf., 1989

work page 1989
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Smith, S., & Beattie, C. 1990. Simultaneous expansion and orthogonalization of measured modes for structure identification. In Dynamics Specialists Conference (p. 1218)

work page 1990
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Levine-West, M., Milman, M., & Kissil, A. 1996. Mode shape expansion techniques for prediction -experimental evaluation. AIAA journal, 34(4), 821-829

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Krige, D. G., 1951. A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of Southern African Institute of Mining and Metallurgy, 52(6)

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Tamhidi, A., Kuehn, N., Ghahari, S.F., Rodgers, A. J., Kohler, M. D., Taciroglu, E., and Bozorgnia, Y., 2022. Conditioned simulation of ground‐motion time series at uninstrumented sites using Gaussian process regression. Bulletin of the Seismological Society of America, 112(1), 331–347. Seismological Society of America

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Ghahari, F., Swensen, D., Haddadi, H., & Taciroglu, E. 2024. A hybrid model -data method for seismic response reconstruction of instrumented buildings. Earthquake Spectra, 40(2), 1235-1268

work page 2024
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Ghahari, F., Swensen, D., & Haddadi, H. 2025. Optimal Sensor Placement in Buildings: Stationary Excitation. Sensors, 25(24), 7470

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Ghahari, F., Swensen, D., & Haddadi, H. 2026. Optimal Sensor Placement in Buildings: Earthquake Excitation. Sensors, 26(8), 2383

work page 2026
[22]

Chang, M., & Pakzad, S. N. 2014. Optimal sensor placement for modal identification of bridge systems considering number of sensing nodes. Journal of Bridge Engineering, 19(6), 04014019

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Simon, P., Goldack, A., & Narasimhan, S. 2016. Mode shape expansion for lively pedestrian bridges through kriging. Journal of Bridge Engineering, 21(6), 04016015

work page 2016
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MATLAB version 24.1.0 (R2024a)

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F., Baltay, A., Çelebi, M., Parker, G

Ghahari, S. F., Baltay, A., Çelebi, M., Parker, G. A., McGuire, J. J., & Taciroglu, E. 2022. Earthquake early warning for est imating floor shaking levels of tall buildings. Bulletin of the Seismological Society of America, 112(2), 820-849

work page 2022
[26]

F., Sargsyan, K., Çelebi, M., & Taciroglu, E

Ghahari, S. F., Sargsyan, K., Çelebi, M., & Taciroglu, E. 2022. Quantifying modeling uncertainty in simplified beam models fo r building response prediction. Structural Control and Health Monitoring, 29(11), e3078

work page 2022
[27]

A., Swensen, D., Çelebi, M., Haddadi, H., & Taciroglu, E

Ghahari, F., Sargsyan, K., Parker, G. A., Swensen, D., Çelebi, M., Haddadi, H., & Taciroglu, E. 2024. Performance-based earthquake early warning for tall buildings. Earthquake Spectra, 40(2), 1425-1451

work page 2024
[28]

Li, R., & Sudjianto, A. 2005. Analysis of computer experiments using penalized likelihood in Gaussian Kriging models. Technometrics, 47(2), 111-120. 9 APPENDIX Figure A1. MATLAB implementation of the SOGP method for mode shape expansion . function [Phi_Mean,Phi_Std,Theta] = SOGP_func(Y_phys,obs_floors,N,noise_std_phys) % Input: % Y_phys : Measured mode sh...

work page 2005

[1] [1]

Ibrahim, S. R. 1977. Random decrement technique for modal identification of structures. Journal of spacecraft and rockets, 14 (11), 696-700

work page 1977

[2] [2]

L., & Jennings, P

Beck, J. L., & Jennings, P. C. 1980. Structural identification using linear models and earthquake records. Earthquake enginee ring & structural dynamics, 8(2), 145-160

work page 1980

[3] [3]

R., & Doebling, S

Farrar, C. R., & Doebling, S. W. 1997. An overview of modal-based damage identification methods

work page 1997

[4] [4]

Sohn, H. 2007. Effects of environmental and operational variability on structural health monitoring. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365(1851), 539-560

work page 2007

[5] [5]

Vidal, F., Navarro, M., Aranda, C., & Enomoto, T. 2014. Changes in dynamic characteristics of Lorca RC buildings from pre-and post- earthquake ambient vibration data. Bulletin of Earthquake Engineering, 12(5), 2095-2110

work page 2014

[6] [6]

Ceravolo, R., Matta, E., Quattrone, A., & Zanotti Fragonara, L. 2017. Amplitude dependence of equivalent modal parameters in monitored buildings during earthquake swarms. Earthquake Engineering & Structural Dynamics, 46(14), 2399-2417

work page 2017

[7] [7]

Sivori, D., Cattari, S., & Lepidi, M. 2022. A methodological framework to relate the earthquake -induced frequency reduction to structural damage in masonry buildings. Bulletin of Earthquake Engineering, 20(9), 4603-4638

work page 2022

[8] [8]

F., Nateghi, F., & Taciroglu, E

Abazarsa, F., Ghahari, S. F., Nateghi, F., & Taciroglu, E. 2013. Response‐only modal identification of structures using limit ed sensors. Structural Control and Health Monitoring, 20(6), 987-1006

work page 2013

[9] [9]

Yang, Y., & Nagarajaiah, S. 2013. Output-only modal identification with limited sensors using sparse component analysis. Journal of Sound and Vibration, 332(19), 4741-4765

work page 2013

[10] [10]

F., Abazarsa, F., Ghannad, M

Ghahari, S. F., Abazarsa, F., Ghannad, M. A., Celebi, M., & Taciroglu, E. 2014. Blind modal identification of structures from spatially sparse seismic response signals. Structural Control and Health Monitoring, 21(5), 649-674

work page 2014

[11] [11]

Tarpø, M., Nabuco, B., Georgakis, C., & Brincker, R. 2020. Expansion of experimental mode shape from operational modal analys is and virtual sensing for fatigue analysis using the modal expansion method. International journal of fatigue, 130, 105280

work page 2020

[12] [12]

Guyan, R. J. 1965. Reduction of stiffness and mass matrices. AIAA journal, 3(2), 380-380

work page 1965

[13] [13]

O’Callahan, J. C. 1989. System equivalent reduction expansion process. In Proc. of the 7th Inter. Modal Analysis Conf., 1989

work page 1989

[14] [14]

Smith, S., & Beattie, C. 1990. Simultaneous expansion and orthogonalization of measured modes for structure identification. In Dynamics Specialists Conference (p. 1218)

work page 1990

[15] [15]

Levine-West, M., Milman, M., & Kissil, A. 1996. Mode shape expansion techniques for prediction -experimental evaluation. AIAA journal, 34(4), 821-829

work page 1996

[16] [16]

G., 1951

Krige, D. G., 1951. A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of Southern African Institute of Mining and Metallurgy, 52(6)

work page 1951

[17] [17]

E., and Williams, C

Rasmussen, C. E., and Williams, C. K. I., 2006. Gaussian Processes for Machine Learning. Cambridge, Massachusetts: MIT Press

work page 2006

[18] [18]

J., Kohler, M

Tamhidi, A., Kuehn, N., Ghahari, S.F., Rodgers, A. J., Kohler, M. D., Taciroglu, E., and Bozorgnia, Y., 2022. Conditioned simulation of ground‐motion time series at uninstrumented sites using Gaussian process regression. Bulletin of the Seismological Society of America, 112(1), 331–347. Seismological Society of America

work page 2022

[19] [19]

Ghahari, F., Swensen, D., Haddadi, H., & Taciroglu, E. 2024. A hybrid model -data method for seismic response reconstruction of instrumented buildings. Earthquake Spectra, 40(2), 1235-1268

work page 2024

[20] [20]

Ghahari, F., Swensen, D., & Haddadi, H. 2025. Optimal Sensor Placement in Buildings: Stationary Excitation. Sensors, 25(24), 7470

work page 2025

[21] [21]

Ghahari, F., Swensen, D., & Haddadi, H. 2026. Optimal Sensor Placement in Buildings: Earthquake Excitation. Sensors, 26(8), 2383

work page 2026

[22] [22]

Chang, M., & Pakzad, S. N. 2014. Optimal sensor placement for modal identification of bridge systems considering number of sensing nodes. Journal of Bridge Engineering, 19(6), 04014019

work page 2014

[23] [23]

Simon, P., Goldack, A., & Narasimhan, S. 2016. Mode shape expansion for lively pedestrian bridges through kriging. Journal of Bridge Engineering, 21(6), 04016015

work page 2016

[24] [24]

MATLAB version 24.1.0 (R2024a)

MATLAB 2024. MATLAB version 24.1.0 (R2024a). Natick, MA: The MathWorks Inc. 8

work page 2024

[25] [25]

F., Baltay, A., Çelebi, M., Parker, G

Ghahari, S. F., Baltay, A., Çelebi, M., Parker, G. A., McGuire, J. J., & Taciroglu, E. 2022. Earthquake early warning for est imating floor shaking levels of tall buildings. Bulletin of the Seismological Society of America, 112(2), 820-849

work page 2022

[26] [26]

F., Sargsyan, K., Çelebi, M., & Taciroglu, E

Ghahari, S. F., Sargsyan, K., Çelebi, M., & Taciroglu, E. 2022. Quantifying modeling uncertainty in simplified beam models fo r building response prediction. Structural Control and Health Monitoring, 29(11), e3078

work page 2022

[27] [27]

A., Swensen, D., Çelebi, M., Haddadi, H., & Taciroglu, E

Ghahari, F., Sargsyan, K., Parker, G. A., Swensen, D., Çelebi, M., Haddadi, H., & Taciroglu, E. 2024. Performance-based earthquake early warning for tall buildings. Earthquake Spectra, 40(2), 1425-1451

work page 2024

[28] [28]

Li, R., & Sudjianto, A. 2005. Analysis of computer experiments using penalized likelihood in Gaussian Kriging models. Technometrics, 47(2), 111-120. 9 APPENDIX Figure A1. MATLAB implementation of the SOGP method for mode shape expansion . function [Phi_Mean,Phi_Std,Theta] = SOGP_func(Y_phys,obs_floors,N,noise_std_phys) % Input: % Y_phys : Measured mode sh...

work page 2005