# Gaussian Process Regression for Active Sensing Probabilistic Structural
Health Monitoring: Experimental Assessment Across Multiple Damage and Loading
Scenarios

Ahmad Amer  Intelligent Structural Systems Laboratory (ISSL)
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute, Troy, NY, USA
Email<EMAIL_ADDRESS>Fotis Kopsaftopoulos111Corresponding author.
Intelligent Structural Systems Laboratory (ISSL)
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute, Troy, NY, USA
Email<EMAIL_ADDRESS>
###### Abstract

In the near future, Structural Health Monitoring (SHM) technologies for
aircraft will be capable of overcoming the drawbacks in the current
maintenance and life-cycle management paradigms, namely: cost, increased
downtime, less-than-optimal safety management paradigm and the limited
applicability of fully-autonomous operations. In the context of SHM for
aircraft structures, one of the most challenging tasks is structural damage
quantification. Currently-utilized quantification techniques face accuracy
and/or robustness issues when it comes to the varying operating and
environmental conditions involved in day-to-day operations. In addition, the
damage/no-damage paradigm of current industrial frameworks does not offer much
information to maintainers on the ground for proper decision-making. In this
study, a novel structural damage quantification framework is proposed based on
the widely-used Damage Indices (DIs) and Gaussian Process Regression Models
(GPRMs) in order to overcome the aforementioned shortcomings. The proposed
framework takes a simple approach to the damage quantification problem by
using DI values for training, and provides confidence bounds on the quantified
states using a novel state prediction technique. The novelty in this approach
to state quantification lies in calculating the probability of an incoming
test DI point originating from a specific state, which allows for probability-
educated decision-making. In addition, the proposed methods are shown to
quantify multiple structural states simultaneously from incoming DI test
points. This framework is applied to three test cases: a Carbon Fiber-
Reinforced Plastic (CFRP) coupon with attached weights as simulated damage, an
aluminum coupon with a notch, and an aluminum coupon with attached weights as
simulated damage under varying loading states. The novel state prediction
method presented herein is applied to single-state quantification in the first
two test cases, as well as the third one assuming the loading state is known.
Finally, the proposed method is applied to the third test case assuming
neither the damage size nor the load is known in order to predict both
simultaneously from incoming DI test points. In applying this framework, two
forms of GPRMs (standard and variational heteroscedastic) are used in order to
critically assess their performances with respect to the three test cases.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Reference Damage Indices
    2. 2.2 Standard Gaussian Process Regression Models (SGPRMs)
      1. 2.2.1 Formulation
      2. 2.2.2 Training
      3. 2.2.3 Prediction
    3. 2.3 Variational Heteroscedastic Gaussian Process Regression Models (VHGPRMs)
      1. 2.3.1 Formulation
      2. 2.3.2 Training
      3. 2.3.3 Prediction
    4. 2.4 Damage State Quantification
      1. 2.4.1 Notes on two-state prediction using DI-trained GPRMs
  3. 3 Test Case I: Al Coupon with Varying Notch Sizes
    1. 3.1 Test Setup
    2. 3.2 Results & Discussion
  4. 4 Test Case II: CFRP Coupon with Simulated Damage
    1. 4.1 Test Setup
    2. 4.2 Results & Discussion
  5. 5 Test Case III: Al Coupon Under Varying Simulated Damage & Load States
    1. 5.1 Test Setup
    2. 5.2 Results & Discussion
      1. 5.2.1 The case of a known loading state: Single-input GPRMs
      2. 5.2.2 Multi-input GPRMs
  6. 6 Conclusion
  7. A Additional results

##  1 Introduction

In the context of engineering structures, structural safety, maintenance and
life-cycle management processes are a major factor in sustainability [1, 2].
In particular, the aerospace industry is one that depends heavily on schedule-
based procedures in order to sustain proper life cycle management, ensure
safety, and improve performance [3, 4, 5]. Most of such procedures include
some form of Non-destructive Evaluation (NDE) techniques, in which aircraft
need to be inspected on a regular basis on the ground before operations can be
resumed regardless of structural state [5, 6, 3]. This framework, although
very effective in the sustainability efforts of the aerospace industry,
suffers from a number of drawbacks; the most economically-prominent of which
are cost, increased downtime, less-than-optimal safety management paradigm
(damage can occur and grow between scheduled procedures) and the limited
applicability of fully-autonomous operations [3, 2, 6, 7]. As such, research
endeavors in the past 40 years have been directed towards developing
sustainability efforts that can be applied online (limiting downtime and
increasing safety) and in an automated fashion (limiting the need for costly
man hours, and also allowing for autonomous operation) within the frameworks
of Structural Health Monitoring (SHM) [8]. Because of the complexity of
aircraft operations, manifested in multiple operational cycles, and, within
each cycle, the varying operational and environmental conditions, the
aerospace industry poses as a very rich arena for development of SHM
techniques [6].

By convention, a complete SHM system is one that can implement all 4 levels of
SHM with high accuracy and robustness, namely: damage detection, localization,
quantification and remaining-useful-life estimation [9, 10, 11, 12]. When it
comes to SHM in aerospace structures today, the most common metric used for
interrogating structural health is composed of one or more Damage Indices
(DIs). These DIs compare specific features of the sensor response signals for
the unknown structural state to those features coming from the healthy
structure [13, 14]. The features can be the amplitude, energy or phase of the
signals, and they have been extensively used in the literature for damage
detection [15, 16, 13, 11, 14, 17] and damage quantification [18, 19, 20, 21].
From an industrial point of view, one of the major advantages of DIs is the
simplicity of application and interpretation of results, which facilitates
decision-making by maintainers on the ground [15]. When it comes to active-
sensing SHM in particular, however, the currently-utilized DIs face a number
of challenges, the most prominent of which is their deterministic nature, i.e.
they do not account for operational and environmental uncertainties [22, 23,
24, 25]. This is particularly important with the growing need for online
damage detection and quantification owing to the urban air mobility
“revolution” [26], where the SHM metric being implemented must be robust
enough to the different sources of uncertainty pertaining to flight
operations, whilst sensitive enough to detect and quantify damage accurately
[27]. In addition, the damage/healthy paradigm of DI frameworks does not
provide much useful information to maintainers, since it depends on user
expertise [28, 29], and is thus subject to error, especially under the varying
environmental and operational conditions natural to aircraft operations.

Accordingly, many researchers sought to developing active-sensing, guided-wave
SHM metrics that are more robust and accurate. Within the field of damage
quantification in particular, which is more involved than damage detection
[22], researchers used conventional/non-conventional DIs (for instance, see
[18, 19, 20, 21]), advanced signal processing techniques (see [30, 31, 32]),
advanced statistical inference/modelling techniques (see [33, 34, 35]), and
analytical guided-wave models (see for instance [36, 37, 38]) in order to
quantify damage within an active-sensing, guided-wave framework. Other
techniques involved the use of image processing of guided-wave maps (such as
in [39]). Similar to the case of damage detection [40], all of these
techniques either suffer from an incomplete treatment of the challenges
mentioned above, which face damage detection and quantification alike, or
increased model complexity. The following paragraphs briefly outline some of
the more important studies in the literature dealing with damage
quantification.

Being a relatively-simple metric to calculate, many researchers sought for
non-conventional DIs in order to obtain a robust and accurate damage
quantification metric. Wang and coworkers [41] compared 4 parametric models
which were linear combinations of different signal features (such as time of
flight and amplitude) in quantifying the size of a crack on an aircraft
riveted lap joint. Reynolds and Chattopadhyay [42] modelled the system under
interrogation using a number of transfer functions in order to decouple the
effect of structural anomalies and that of the sensor interaction with the
structure on the received signals. Then, they proposed a DI based on the
attenuation of signal intensity with damage, relating that DI to the transfer
functions of the system in order to corner down the effect of damage on the
received signals and thus quantify damage. Observing that DIs based on guided
waves alone might not meet the required level of accuracy in aerospace
applications, Vanniamparambil et al. [43] proposed a data fusion technique, in
which information from acoustic emission, guided waves, and digital image
correlation is used to indicate crack growth in an Al 2024 Alloy.

Because of the drawbacks of DIs, many researchers turned their attention to
advanced signal processing techniques in order to quantify damage in an
active-sensing framework. A common approach in this family of techniques
involved the use of wavelet transforms, measuring the change in wavelet
energies as indication of damage size [44, 45, 46]. Other techniques involved
the use of Hilbert Huang transforms [31]. While all of these techniques indeed
show superiority to approaches utilizing DIs, they still lack the proper
identification of quantification confidence bounds, and are thus not suitable
for stochastic systems without further development.

In order to tackle this problem, some researchers utilized advanced modelling
techniques where signal features are fitted using different probability
distributions in order to properly extract confidence intervals on the
quantification decision [33, 47]. For instance, Yang et al. [35, 47] used
Bayesian updating for crack size quantification in 6 Al plates having notches
with increasing lengths within an active-sensing approach. Normalized
amplitude and phase changes in the first-arrival wave mode (S0) in the signals
were used as the parameters in a linear model that predicted damage size.
Having initially calculated the parameter values from the experiments, Monte
Carlo simulations were then used to fit the parameters to a probability
distribution to be able to proceed with Bayesian updating of the notch size
model. Similar strategies have been demonstrated for estimating delamination
propagation in composites [34]. Other endeavors included the use of Gaussian
mixture models, [28], neural networks [48], or Principal Component Analysis
[49]. In general, these techniques lead to an accurate and robust damage
quantification process (same can be said for detection,) and many of them fall
under the umbrella of statistical/probabilistic SHM. However, these approaches
either involve complicated steps (such as the case of Gaussian mixture models
[28]), or require many data sets for model training and building processes
(such as the case of the matching pursuit decomposition [32] or Bayesian
updating [35]). Thus, there still lies the need for developing damage
quantification techniques that can overcome such shortcomings of currently-
used approaches. These challenges can be summarized as follows:

  * •

The lack of robustness of DI-based quantification techniques under varying
operational and/or environmental conditions.

  * •

The need for user expertise for interpreting DI results.

  * •

The inability of most of the other approaches to provide quantification
confidence intervals for proper decision-making without sacrificing simplicity
in defining/calculating the damage quantification metric.

One promising approach, which also falls under probabilistic SHM techniques,
is Gaussian Process Models (GPMs). GPMs, being probabilistic machine learning
models, have recently seen interest within the vibration-based SHM community
[50, 51, 52, 53, 54]. This interest originates from the fact that GPMs model
uncertainty in the available data regardless of whether the uncertainty
sources are known or not. Furthermore, GPMs can extract confidence bounds on
state predictions [55]. Most importantly, with the availability of data under
different conditions, the effect of any condition can be conveniently modelled
using GPMs. Consequently, the authors have recently utilized GP Regression
Models (GPRMs) [23, 56] and GP Classification Models (GPCMs) [57] for the task
of probabilistic damage quantification in active-sensing, guided-wave SHM. In
addition, GPRMs were also used by the authors to predict damage size and load
simultaneously either by themselves [27, 58], or with the assistance of
physics-based models [58].

The aim of the present study is to expand on previous work using data-based
GPRMs for damage quantification within active-sensing, guided-wave SHM in
order to critically assess the usefulness of such models for different types
of coupons/damages/states. DIs are utilized in order to train GPRMs for
predicting damage sizes for a Carbon fiber-reinforced composite (CFRP) with
simulated damage and an Al coupon with a notch, as well as damage size and/or
load state for another Al coupon with simulated damage. In addition, this
study assesses two types of GPRMs: standard homoscedastic GPRMs (SGPRMs), and
the Variational Heteroscedastic GPRMs (VHGPRMs) presented by Lazaro-Gredilla
and Titsias [59] in order to compare the performances of both approaches with
respect to accuracy and robustness. The novelty in this work can be summarized
through the following points:

  * •

The application of SGPRMs and VHGPRMs for damage quantification in active-
sensing, guided-wave SHM.

  * •

The use of conventional DIs in training accurate and robust GPRMs, which
allows for proper modelling of uncertainties using a well-known, widely-used
SHM metric.

  * •

The accurate simultaneous prediction of damage size and load states within an
active-sensing framework.

  * •

The extraction of confidence bounds over damage size and/or load prediction
for probability-educated decision-making, which contrasts with the
damage/healthy paradigm of conventional DIs.

The paper is organized as follows: Section 2 presents the overall background
of the methods used herein, including the utilized state-of-the-art DI
formulae (Section 2.1), the SGPRM (Section 2.2) and VHGPRM (Section 2.3)
formulations, and the state prediction framework used in this study (Section
2.4). Then, the following sections present the application of the proposed
framework to the three coupons: the Al coupon with a notch (Section 3), the
CFRP coupon with simulated damage (Section 4), and the Al coupon with
simulated damage under different loading conditions (Section 5). Finally,
Section 6 draws the most important conclusions and presents the areas of
development within the proposed framework.

##  2 Background

###  2.1 Reference Damage Indices

In this study, the proposed methods were compared with two formulations for
damage indices. The first one was the basic Root-Mean-Square Deviation (RMSD),
which reads:

$DI=\sqrt{\frac{\sum_{t=1}^{N}{(y_{0}[t]-y_{u}[t])^{2}}}{N}}$ (1)

In the above equation, $y_{0}[t]$ and $y_{u}[t]$ are the signals coming from
the healthy and unknown states of the system, indexed with normalized discrete
time $t$ ($t=1,2,3,\ldots,N$ where $N$ here is 2500, which includes roughly
the first $100\mu s$ of the signals). The second DI used herein was formulated
by Janapati et al. [11]. This DI was selected in particular due to its high
sensitivity to damage size, and low sensitivity to changes in the material
properties of the different components of the sensor network (coupon, sensors,
and adhesive layers). The formulation is as follows:

$Y_{u}^{n}[t]=\frac{y_{u}[t]}{\sqrt{\sum_{t=1}^{N}{y^{2}_{u}[t]}}},\quad
Y_{0}^{n}[t]=\frac{\sum_{t=1}^{N}{(y_{0}[t]\cdot
Y_{u}^{n}[t])}}{y_{0}[t]\cdot\sum_{t=1}^{N}{y_{0}^{2}[t]}},\quad
DI=\sum_{t=1}^{N}{(Y^{n}_{u}[t]-Y^{n}_{0}[t])}$ (2)

such that $Y^{n}_{0}[t]$ and $Y^{n}_{u}[t]$ are normalized baseline (healthy)
and unknown (test/inspection) signals, respectively.

###  2.2 Standard Gaussian Process Regression Models (SGPRMs)

####  2.2.1 Formulation

In this section, a brief introduction of standard GPRMs will be provided. For
a full treatment, the reader is directed to reference [60], given a training
data set $\mathcal{D}$ containing $n$ inputs-observation pairs
{$\mathbf{x}_{i}\in\mathbb{R}^{D},y_{i}\in\mathbb{R},\ i=1,2,3,\ldots,n$}, a
standard (homoscedastic) GPRM can be formulated as follows:

$y=f(\mathbf{x})+\epsilon$ (3)

where, in a Bayesian setting, a GP prior with mean $m(\mathbf{x})$ and
covariance $k(\mathbf{x},\mathbf{x^{\prime}})$ is placed on the latent
function $f(\mathbf{x})$, and an independent, identically-distributed
($\mathit{iid}$), zero-mean Gaussian prior with variance $\sigma_{n}^{2}$ is
placed on the noise term $\epsilon$, that is:

$\begin{array}[]{lr}f(\mathbf{x})\sim\mathcal{GP}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x^{\prime}})),&\,\epsilon\sim
iid\,\mathcal{N}(0,\sigma_{n}^{2})\end{array}$ (4)

As is common in the GPRM literature, $m(\mathbf{x})$ is set to zero, and the
squared exponential covariance function (kernel) is used for the latent
function GP, owing to its ability to monotonically decrease as input values go
farther from each other, which allows for similar latent function values for
close input points, and vice versa:

$k(\mathbf{x},\mathbf{x^{\prime}})=\sigma_{0}^{2}\exp(-\frac{1}{2}(\mathbf{x}-\mathbf{x^{\prime}})^{T}\Lambdaup^{-1}(\mathbf{x}-\mathbf{x^{\prime}}))$
(5)

In equation 5, $\sigma_{0}^{2}$ is the output variance, and $\Lambdaup^{-1}$
is the inverse of a diagonal matrix of the characteristic input length scales
corresponding to each dimension ($D$ i.e each covariate) in the input data.
For a single-input dimension (i.e. $D=1$), the entries along the diagonal of
$\Lambdaup^{-1}$ will be identical; otherwise, there will be a separate input
length scale for every covariate in the training input data.

####  2.2.2 Training

Training of the GPRM involves optimizing the hyperparameters
($\theta\equiv{\sigma_{0}^{2},\Lambdaup,\sigma_{n}^{2}}$), which is typically
done via Type II Maximum Likelihood [60, Chapter 5, pp. 109], whereas the
marginal likelihood (evidence) of the training observations is maximized (or
its negative log is minimized for reasons related to computational stability).
That is, the following expression is minimized with respect to $\theta$:

$\displaystyle-\log
p(\mathbf{y}|X,\theta)=-\log\mathcal{N}(\mathbf{y}|\mathbf{0},K_{XX}+\sigma_{n}^{2}\mathbb{I})$
(6a)
$\displaystyle=-\frac{1}{2}\mathbf{y}^{T}(K_{XX}+\sigma_{n}^{2}\mathbb{I})^{-1}\mathbf{y}-\frac{1}{2}\log|K_{XX}+\sigma_{n}^{2}\mathbb{I}|-\frac{n}{2}\log
2\pi$ (6b)

In the expression above, $K_{AB}$ denotes $K(A,B)$ (covariance matrix), and
$\mathbb{I}$ the identity matrix.

####  2.2.3 Prediction

Prediction can be done by assuming joint Gaussian distribution between the
training observations ($\mathbf{y}$), and a test observation ($y_{\ast}$ \- to
be predicted) at the set of test inputs ($\mathbf{x}_{\ast}$) as follows:

$\left[\begin{array}[]{c}\mathbf{y}\\\
y_{\ast}\end{array}\right]=\mathcal{N}\left[\mathbf{0},\begin{array}[]{cc}K_{XX}+\sigma_{n}^{2}\mathbb{I}&\mathbf{k}_{X\mathbf{x}_{\ast}}\\\
\mathbf{k}_{\mathbf{x}_{\ast}X}&k_{\mathbf{x}_{\ast}\mathbf{x}_{\ast}}+\sigma_{n}^{2}\mathbb{I}\end{array}\right]$
(7)

where $\mathbf{k}_{X\mathbf{x}_{\ast}}$ is the vector of covariances between
$X$ and $\mathbf{x}_{\ast}$. By invoking the properties of multivariate
Gaussian distributions [61], the predictive distribution over $y_{\ast}$ can
be defined as follows:

$\displaystyle
p(y_{\ast}|\mathbf{x}_{\ast},X,\mathbf{y})=\mathcal{N}(\mathbb{E}\\{y_{\ast}\\},\mathbb{V}\\{y_{\ast}\\})$
(8a)
$\displaystyle\mathbb{E}\\{y_{\ast}\\}=\mathbf{k}_{\mathbf{x}_{\ast}X}(K_{XX}+\sigma_{n}^{2}\mathbb{I})^{-1}\mathbf{y}$
(8b)
$\displaystyle\mathbb{V}\\{y_{\ast}\\}=k_{\mathbf{x}_{\ast}\mathbf{x}_{\ast}}-\mathbf{k}_{\mathbf{x}_{\ast}X}(K_{XX}+\sigma_{n}^{2}\mathbb{I})^{-1}\mathbf{k}_{X\mathbf{x}_{\ast}}+\sigma_{n}^{2}$
(8c)

such that $\mathbb{E}\\{y_{\ast}\\}$ and $\mathbb{V}\\{y_{\ast}\\}$ are the
predictive mean and variance, respectively, at the set of test inputs.

###  2.3 Variational Heteroscedastic Gaussian Process Regression Models
(VHGPRMs)

####  2.3.1 Formulation

One of the inherent drawbacks of using standard (homoscedastic) GPRMs is the
assumption of a fixed noise variance throughout the input space, which, in
many real-life applications, is impractical. Thus, a number of modifications
have been put forward to allow for the noise variance to vary with the input
(that is, an input-dependent noise variance) [55, 61]. This is to say that the
GPRM formulation in equation 3 would become:

$y=f(\mathbf{x})+\epsilon(\mathbf{x})$ (9)

with the noise prior defined as

$\epsilon\sim\mathcal{N}(0,r(\mathbf{x}))$ (10)

One of the most common strategies is to treat this input-dependent noise as a
GP itself, which was first put forward by Goldberg et al. [62], that is

$\displaystyle r(\mathbf{x})=\exp(g(\mathbf{x}))$ (11a) $\displaystyle
g(\mathbf{x})\sim\mathcal{GP}(\mu_{0},k_{g}(\mathbf{x},\mathbf{x}^{\prime}))$
(11b)

such that $\mu_{0}$ and $k_{g}(\mathbf{x},\mathbf{x}^{\prime})$ are the mean
and covariance for the GP prior on the noise variance function, respectively,
where an exponential function is used in order to ensure that the noise
variance stays positive [59]. It is worth noting that the subscript $g$ was
introduced to differentiate between the covariance function of the noise GP
and that of the noise-free process. At this point, it is useful to introduce
shorthand notations for the covariance functions as follows:

$\displaystyle K_{j}(X,X)\equiv{K_{j}}$ $\displaystyle
K_{j}(\mathbf{x}_{\ast}X)\equiv{K_{j\ast}}$ $\displaystyle
K_{j}(\mathbf{x}_{\ast}\mathbf{x}_{\ast})\equiv{K_{j\ast\ast}}$

Such that $j$ can be $f$ or $g$. Although the formulation in equation 11a
provides a better treatment of data with hetersocedastic noise, the added
complexity results in making the marginal likelihood and the predictive
distribution over unknown observations not analytically-tractable. One of the
proposed approaches to approximate them was put forward by Lazaro-Gredilla and
Titsias [59], which is based on variational approximations. Briefly, their
approach is based on approximating $p(f,g|D)$ by $q(f)q(g)$ via the
minimization of the Kullback-Leibler divergence between them, where the $q(f)$
and $q(g)$ are the variational probability densities (arbitrary density
functions) over sets of $f$ and $g$, respectively. The resulting marginal
variational (MV) bound ($M$) becomes:

$M(\mathbf{\mu},\Sigmaup)=\log\mathcal{N}(\mathbf{y}|\mathbf{0},K_{f}+R)-\frac{1}{4}\operatorname{tr}(\Sigmaup)-\operatorname{KL}(\mathcal{N}(g|\mathbf{\mu},\Sigmaup)||\mathcal{N}(g|\mu_{0}\mathbf{1},K_{g}))$
(12)

In equation 12, the mean $\mathbf{\mu}$ and covariance $\Sigmaup$ come from
the restricting $q(g)$ to be $\mathcal{N}(g|\mathbf{\mu},\Sigmaup)$, the
$\operatorname{KL}(\cdot)$ term is the Kullback Leibler divergence between the
GP prior on $g$ and the aforementioned restriction on $q(g)$,
$\operatorname{tr}(\cdot)$ denotes the trace of the enclosed matrix,
$\mathbf{1}$ indicates a vector of ones, and $R$ is a diagonal matrix with
elements:

$R_{ii}=\exp(\mathbf{\mu}_{i}-\frac{\Sigmaup_{ii}}{2})$ (13)

where $i$ is as defined before.

####  2.3.2 Training

In equation 12, the number of free parameters to be determined becomes
n+n(n+1)/2, which would make the training process much more computationally
exhaustive. Thus, Lazaro-Gredilla and Titsias [59] proposed a
reparametrization of $\mathbf{\mu}$ and $\Sigmaup$ at the maxima of the
marginal variational bound into the following:

$\mathbf{\mu}=K_{g}(\Lambdaup-\frac{1}{2}\mathbb{I})\mathbf{1}+\mu_{0}\mathbf{1},\,\,\,\Sigmaup^{-1}=K_{g}^{-1}+\Lambdaup$
(14)

such that $\Lambdaup$ is a positive semi-definite diagonal matrix of the
variational parameters (to be determined through optimization).

####  2.3.3 Prediction

Based on the formulation presented by Lazaro-Gredilla and Titsias [59], the
predictive distribution for a new test point $y_{\ast}$ can be simplified to:

$\displaystyle q(y_{\ast})=\int\int
p(y_{\ast}|f_{\ast},g_{\ast})q(f_{\ast})q(g_{\ast})df_{\ast}dg_{\ast}$
$\displaystyle=\int\mathcal{N}(y_{\ast}|a_{\ast},c_{\ast}^{2}+\exp(g_{\ast}))\mathcal{N}(g_{\ast}|\mu_{\ast},\sigma_{\ast}^{2})dg_{\ast}$
(15)

with:

$\displaystyle a_{\ast}=\mathbf{k}_{f\ast}(K_{f}+R)^{-1}\mathbf{y}$ (16a)
$\displaystyle
c_{\ast}^{2}=k_{f\ast\ast}-\mathbf{k}_{f\ast}^{T}(K_{f}+R)^{-1}\mathbf{k}_{f\ast}$
(16b)
$\displaystyle\mu_{\ast}=\mathbf{k}_{g\ast}^{T}(\Lambdaup-\frac{1}{2}\mathbb{I})\mathbf{1}+\mu_{0}$
(16c)
$\displaystyle\sigma_{\ast}^{2}=k_{g\ast\ast}-\mathbf{k}_{g\ast}^{T}(K_{g}+\Lambdaup^{-1})^{-1}\mathbf{k}_{g\ast}$
(16d)

Although the integration above does not have an analytical solution, the first
two moments of the predictive distribution can be calculated analytically as
follows:

$\displaystyle\mathbb{E}\\{y_{\ast}|\mathbf{x}_{\ast},\mathcal{D}\\}=a_{\ast}$
(17a)
$\displaystyle\mathbb{V}\\{y_{\ast}|\mathbf{x}_{\ast},\mathcal{D}\\}=c_{\ast}^{2}+\exp(\mu_{\ast}+\frac{\sigma_{\ast}^{2}}{2})$
(17b)

Thus, under the assumption of a Gaussian predictive distribution over the
unknown test observation $y_{\ast}$, which is not necessarily true, the
variance of the test observation can be predicted along with the test
observation itself, which allows for a more flexible model compared to
standard GPRMs [61].

###  2.4 Damage State Quantification

Figure 1: A schematic flow diagram showing the steps taken in this study to
calculate the state prediction probability.

In many practical cases, such as the case in this study, the test observations
at which prediction should take place are actually the GPRM targets, not the
inputs i.e. in this study, a test DI ($y_{\ast}$ \- target) would be
available, and the damage size and/or load state ($\mathbf{x_{\ast}}$ \-
inputs) would be estimated by the GPRM. In this work, the proposed method of
estimating the inputs from the test target values is based on the probability
that a test target value $y_{\ast}$ belongs to a specific damage size and/or
load state $\mathbf{x}$. Figure 1 schematically outlines the steps to
calculate the state (input) prediction probability in this case. As shown,
this probability ($P(\mathbf{x_{\ast}}=\mathbf{x})$) can be estimated from the
Cumulative Distribution Function of the targets as follows:

$P(\mathbf{x}_{\ast}=\mathbf{x})=F_{y}(b;{E}\\{y\\},\mathbb{V}\\{y\\})-F_{X}(a;{E}\\{y\\},\mathbb{V}\\{y\\})$
(18)

such that,

$\displaystyle
F_{y}(s;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\sum_{t=-\infty}^{s}{e^{\frac{-(t-\mu)^{2}}{2\sigma^{2}}}}$
(19a) $\displaystyle a=y_{\ast}-2\sqrt{\mathbb{V}\\{y_{closest}\\}}$ (19b)
$\displaystyle b=y_{\ast}+2\sqrt{\mathbb{V}\\{y_{closest}\\}}$ (19c)

In the above equations, $y_{closest}$ is the closest training DI value to the
value of the incoming test DI point ($y_{\ast}$), and
$\mathbb{V}\\{y_{closest}\\}$ is the corresponding GPRM predictive variance.
Also, $\mathbb{E}\\{y\\}$ and $\mathbb{V}\\{y\\}$ are the GPRM predictive mean
and variance, respectively, at the training input $\mathbf{x}$. Finally,
$F_{y}(\cdot)$ is the value of the Cumulative Distribution Function of the
targets at the enclosed point and distribution. This probability is calculated
for every set of inputs (states) in the training data. As shown in this
expression, information from the uncertainty in the GPRM (predictive variance)
corresponding to the closest training DI value to the test one is leveraged in
order to properly estimate the probability of damage size and/or load state.
In addition, this framework can be easily extended to VHGPRMs by replacing the
the SGPRM predictive means and variances by VHGPRM ones. Although this
framework works for both single-state (damage size only) and two-state (load
and damage size) predictions, the readers are directed to Section 2.4.1 for
more information on the latter. Finally, it is worth noting that, although the
present treatment of the available DI data dictates the use of the state
prediction framework presented in this section, it might also be of interest
in using more elaborate multi-output, noisy-input GPRMs in which the DI values
are the inputs and the states are the outputs. In such a case, GPRM prediction
methodologies, such as those presented in Sections 2.2.3 but suited to multi-
output, noisy-input GPRMs, can be used for state quantification directly.

####  2.4.1 Notes on two-state prediction using DI-trained GPRMs

As aforementioned, the trained models in this study are used for predicting
damage size at a known load, or predicting both damage size and load state
simultaneously. Since GPRMs are trained herein using DI values, which depend
heavily on the reference signal(s) being used for their calculations, in order
to train GPRMs to predict both load state and damage size accurately, the
models must be trained with two classes of reference signals. The first class
comprises the signals coming from the healthy structure at various loading
states, which would result in DI values that show a uniform evolution with
damage size, but not necessarily with load since the reference for each
loading state (and damage size) would be the healthy signal at that specific
load, not at the unloaded state. This can be evidently seen in Figure 2 panels
a and b, which show indicative DI plots from the third test case in this study
(the Al coupon with simulated damage) when the DI’s are calculated using
reference signals from the aforementioned class of signals (class 1). The
second class of reference signals contains the signals coming form the
unloaded case at various damage sizes. This latter class would result in DI
values that evolve nicely with load states, but fail to properly follow the
effects of damage evolution since the DI values at different damage sizes
would be referenced to the unloaded case at the corresponding damage size, not
the healthy case. Figure 2 panels c and d show the evolution of the DIs
calculated using this class of signals (class 2). Thus, in order to obtain
accurate predictions of both damage size and loading state, GPRMs need to be
trained with DI values referenced to both classes of reference signals.

abcd Figure 2: Indicative DI plots from path 1-6 in the third test case in this study (Al coupon with simulated damage) showing the evolution of DIs calculated with respect to damage size and load using both classes of reference signals: (a) evolution of DI values calculated using class 1 reference signals with damage size at a load of 5 kN; (b) evolution of DI values calculated using class 1 reference signals with load in the healthy state; (c) evolution of DI values calculated using class 2 reference signals with damage size at a load of 5 kN; (d) evolution of DI values calculated using class 2 reference signals with load in the healthy state. The red dots indicate the means of the DI values at every state. Table 1: The reference signals encompassed by each of the two classes of reference signals used in this study for calculating the DI values for the purposes of training multi-input GPRMs in the third test case presented herein. Class | Reference Signal State | Switch Covariate
---|---|---
| Healthy @0 kN |
1 | Healthy @5 kN | 1
| Healthy @10 kN |
| Healthy @15 kN |
| Healthy @0 kN |
| 1 weight @0 kN |
2 | 2 weights @0 kN | 2
| 3 weights @0 kN |
| 4 weights @0 kN |

Table 1 presents the set of reference signals used in calculating the DI
values in each class. Also shown in Table 1 is the value of the so-called
“switching covariate”, which is simply a third covariate in the input space of
the trained GPRMs ($\mathbf{x}_{i}\in\mathbb{R}^{3}$, damage size and load
being the first two covariates) that acquires a value of 1 or 2 to identify
each of the two different classes of DI reference signals in the training
space. That is, the training inputs and outputs can be defined as follows:

$X\in\\{\mathbf{x}_{1}^{1}\;\mathbf{x}_{2}^{1}\;\ldots\;\mathbf{x}_{n}^{1}\;\mathbf{x}_{1}^{2}\;\mathbf{x}_{2}^{2}\;\ldots\;\mathbf{x}_{n}^{2}\\}^{T},\mathbf{y}\in\\{y_{1}^{1}\;y_{2}^{1}\;\ldots\;y_{n}^{1}\;y_{1}^{2}\;y_{2}^{2}\;\ldots\;y_{n}^{2}\\}^{T}$
(20)

where the superscript denotes the class of reference signals to which each
target (DI value) is referenced. Note that for the inputs, this superscript
only denotes the value of the third covariate (the switch covariate) and it
does not affect the values of the first two covariates (damage size and load).
It is worth noting here that it is also possible to reformulate this problem
using a multi-output GPRM approach with a 2-dimensional input space. However,
this 3-covariate framework was chosen herein for convenience.

Figure 3 schematically outlines the two-state prediction process adopted here.
As shown, with properly trained GPRMs, two-state predictions are made in a
two-step process. In the first step of this prediction process, the DI values
of the incoming test signal is first calculated with respect to the reference
signals in class 1 (healthy signals at various loading states). Each of these
test DI values (5 values in this study) is fed into the prediction process
outlined in Figure 1, and a probability is calculated for the damage size and
load, repeating this process for each of the DI values calculated with respect
to class 1 reference signals, and choosing the pair of states with highest
probability. It is important to note that, in this step, the switching
covariate is set to 1 in order to only use the predictive means and variances
related to that part of the trained GPRM. This step generally leads to an
accurate damage size prediction, but a poor load prediction, again owing to
the poor evolution of class 1 DIs with loads. Thus, only damage size
prediction is accepted in this step. In the second step of this prediction
process, a new test DI value of the incoming test signal is calculated, this
time with respect to the reference signal from the class 2 signals that
corresponds to the damage size predicted in the first step. Then, knowing the
damage size, this test DI value is fed into the prediction process outlined in
Figure 1 with the switching covariate set to 2 in order to calculate the
values of the CDF using the proper predictive moments from class 2 DIs. This
process leads to an accurate load size prediction, since it is implemented at
the known damage size predicted in the first step.

Figure 3: A schematic showing the 2-step prediction process followed in
simultaneously predicting damage size and load in the Al coupon with simulated
damage.

##  3 Test Case I: Al Coupon with Varying Notch Sizes

###  3.1 Test Setup

The first coupon used in this study was a 6061 Aluminum $152.4\times 254$ mm
($6\times 10$ in) coupon ($2.36$ mm/$0.093$ in thick) (McMaster Carr) with a
12-mm (0.5-in) diameter hole in the middle, as shown in Figure 4. Six PZT
(Lead Zirconate Titanate) SMART Layers type PZT-5A (with thickness $0.2$
mm/$0.008$ in and diameter $3.175$ mm/$1/8$ in; Acellent Technologies, Inc)
were attached to the CFRP coupon using Hysol EA 9394 adhesive. In order to
simulate damage, up to six three-gram weights were sequentially attached to
the surface of the plate using tacky tape. Damage was simulated by cutting a
notch starting from the middle hole using an end-mill and a $0.8128$-mm
($0.032$-in) hand saw, with varying lengths between $2$ and $20$ mm, in $2$-mm
increments.

In order to interrogate the coupon, each sensor was actuated, in a consecutive
manner, using 5-peak tone bursts (5-cycle Hamming-filtered sine waves) with 90
V peak-to-peak amplitude and various center frequencies. 20 response signals
per structural case were collected at each sensor at a sampling rate of 24 MHz
using a ScanGenie III data acquisition system (Acellent Technologies, Inc).
Preliminary analysis for the best separation between the first two wave
packets in various response signal paths wa done, and a center frequency of
250 kHz was chosen for the analysis presented herein. For the GPRMs, the
normalized mean squared error (NMSE) presented in ref. [59] and the residual
sum of squares divided by the sample sum of squares (RSS/SSS) were used for
assessing the trained models, and all values are reported herein with respect
to the validation (test) data that was not used in the training phase. All
analysis was done in Matlab.222Matlab version R2020a, GPRM training and
prediction: the different functions within the GPML package available at
http://www.gaussianprocess.org/gpml/code/matlab/doc/index.html. VHGPRM
training and prediction: http://www.tsc.uc3m.es/ miguel.

Figure 4: The notched Al coupon used in this study shown here with a 20-mm
notch (largest damage size). The arrows show the signal paths presented
herein.

###  3.2 Results & Discussion

Figure 5 panels a and b show the signals from the actuator-sensor paths 3-6
and 6-1, respectively, under varying notch sizes (see Figure 4 for sensor
numbering). Figure 5 panels c and d show the corresponding DI evolution from
both DI formulations used in this study. As shown in the latter two panels,
aside from the difference in amplitude, both DI formulations show similar
trends with respect to damage size. Also, it is worth noting that the
saturation phenomenon (indicated by red frames in the panels) seems to occur
under different damage sizes for both paths. As will be shown later, as these
DI values are used in training GPRMs, damage size predictions in the areas of
saturation might be challenging since the models would not be able to
differentiate between the different damage sizes having similar DI values.
Finally, because path 6-1 intersects the notch, the DI evolution using both
formulations in this path is more uniform with notch size compared to that for
path 3-6.

abcd Figure 5: Notched Al coupon: indicative signals and DI plots: (a) signals
from path 3-6; (b) signals from path 6-1; (c) DI values from path 3-6; (d) DI
values from path 6-1.

SGPRMs and VHGPRMs were trained for both paths in order to model the evolution
of both DI formulations with notch size, whiles modelling the sources of
uncertainty in the data. Tables 2 and 3 show model training and testing
information for both DI formulations. briefly, about $50\%$ of the available
DI data points for each formulation were used to train the models, while the
remaining $50\%$ was left for testing the models. As shown in the tables, the
difference in model performance between SGPRM and VHGPRM is minimal, although
the VHGPRM training process takes well over double the time required for
training the SGPRM for the reasons previously noted in Section 2.3.

Table 2: Summary of GPRM∗ information† for the notched Al coupon based on the RMSD DI formulation. Signal | NMSE | RSS/SSS (%) | Training Time (s) | Prediction Time (s)
---|---|---|---|---
Path | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM
3-6 | 8.87E-4 | 8.88E-4 | 0.026 | 0.026 | 8.8 | 21.5 | 0.1577 | 0.1751
6-1 | 7.780E-4 | 7.781E-4 | 0.012 | 0.012 | 8.8 | 23.7 | 0.1630 | 0.1753
∗22.5% (990 points) of the data was used for training each model. |  |
†Numbers approximated to the last quoted decimal place, and times estimated
based on an Intel Core i3 laptop
with 4 Gb of RAM. |
Table 3: Summary of GPRM∗ information† for the notched Al coupon based on the DI formulation from [11] Signal | NMSE | RSS/SSS (%) | Training Time (s) | Prediction Time (s)
---|---|---|---|---
Path | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM
3-6 | 0.0011 | 0.0011 | 0.032 | 0.032 | 8.9 | 22.9 | 0.1704 | 0.1734
6-1 | 8.035E-4 | 8.033E-4 | 0.024 | 0.024 | 8.9 | 23.6 | 0.1670 | 0.1822
∗22.5% (990 points) of the data was used for training each model. |  |
†Numbers approximated to the last quoted decimal place, and times estimated
based on an Intel Core i3 laptop
with 4 Gb of RAM. |

Figure 6 shows the predictive means and confidence intervals for SGPRMs and
VHGPRMs trained using both DI data sets for path 3-6. Figure 7 shows the
corresponding plots for path 6-1. As shown in both figures, the GPRMs closely
follow the evolution of the DI in all cases and conveniently model the
uncertainty in the data sets through the underlying assumptions in the models.
In addition, both SGPRMs and VHGPRMs perform just as well, with a slight
difference in the evolution of the confidence intervals in the VHGPRM plot
(for instance, see Figure 7b towards the lower notch sizes). This similarity
between both types of models stems from the lack of changing variance with
damage size in the data sets recorded from this Al coupon.

abcd Figure 6: Notched Al coupon: GPRM predictive mean and variance for path
3-6: (a) SGPRM results based on the RMSD DI; (b) VHGPRM results based on the
RMSD DI; (c) SGPRM results based on the DI formulation from [11]; (d) VHGPRM
results based on the DI formulation from [11]. abcd Figure 7: Notched Al
coupon: GPRM predictive mean and variance for path 6-1: (a) SGPRM results
based on the RMSD DI; (b) VHGPRM results based on the RMSD DI; (c) SGPRM
results based on the DI formulation from [11]; (d) VHGPRM results based on the
DI formulation from [11].

In order to use the trained models for damage quantification, the DI data
points not used in training were considered test points coming off the coupon
with an unknown damage size. Then, following the damage size prediction
methodology outlined in Section 2.4, the probability of each DI test point
coming from each damage state in the coupon was calculated. Figure 8 shows
prediction probabilities from both model types using the RMSD DI for path 3-6
for 4 indicative test points. The shown points were selected from the healthy
case, as well as three damage cases where the original RMSD DI values (framed
area in Figure 5c) were very close just to examine whether the models have the
capability to differentiate between the different cases based on the training
data, and whether there exist any differences in quantification performance
between SGPRMs and VHGPRMs for this coupon. As shown, although both models
fail to properly predict the damage size of $20$ mm, the VHGPRM outperforms
the SGPRM in quantifying the damage size of $12$ mm. In addition, both models
can accurately predict the damage size of $16$ mm, as well as the healthy
case. Furthermore, both models can also accurately predict almost all other
damage sizes (not shown here), as will be indicated later on when analyzing
the summary prediction results.

abcd Figure 8: Notched Al coupon: damage size prediction results for path 3-6
based on the RMSD DI: (a) prediction probabilities for the healthy case; (b)
prediction probabilities for a notch size of 12 mm; (c) prediction
probabilities for a notch size of 16 mm; (d) prediction probabilities for a
notch size of 20 mm. Dashed vertical lines indicate the maximum probability
corresponding to each model.

Figure 9 shows the corresponding prediction probabilities from the GPRMs
trained using the DI formulation presented in [11]. As shown, both models
perform accurately in all selected test points except for the damage size of
$12$ mm, where the VHGPRM again outperforms the SGPRM in damage
quantification. The reason why the models trained using data sets from the
second DI formulation perform better under a damage size of $20$ mm can be
attributed to the way this DI formulation responds to that damage size, as can
be seen in Figure 5c and Figure 6 panels c and d. As shown there, unlike the
RMSD DI, the DI values from the second formulation clearly has a value
relatively far from DI values at other notch sizes, thus making it easy for
the GPRMs to accurately quantify this damage size. It is important to note
here that the predictions shown in Figures 8 and 9 are only predictions at 8
indicative DI points, and are not necessarily representing the prediction
capability of each model across all $1000+$ test points in each DI data set.

abcd Figure 9: Notched Al coupon: damage size prediction results for path 3-6
based on the DI formulation from [11]: (a) prediction probabilities for the
healthy case; (b) prediction probabilities for a notch size of 12 mm; (c)
prediction probabilities for a notch size of 16 mm; (d) prediction
probabilities for a notch size of 20 mm. Dashed vertical lines indicate the
maximum probability corresponding to each model.

In order to present prediction results from all DI test points, summary box-
plots were created presenting the models’ predictions (point of maximum
probability under each damage size) against the true damage size corresponding
to each test DI point. Figure 10 panels a and b show the summary results from
the RMSD DI-trained models, while panels c and d show the corresponding
results for the second DI formulation. In these plots, each box contains a red
horizonal line indicating the median, with the top and bottom edges of the box
indicating the $75^{th}$ and $25^{th}$ percentiles, respectively. the small
red crosses indicate outliers in the data. Closely examining these plots, a
few interesting observations come up. The first one is that the VHGPRM based
on the RMSD DI values (Figure 10b) outperforms the corresponding SGPRM (Figure
10a) in the cases of $12$ and $14$ mm, and shows a narrower prediction window
in the case of $16$ mm. Conversely, the SGPRM shows a median closer to the
true notch size in the case of an $18$-mm notch. Another observation
pertaining to the GPRMs based on the DI formulation from [11] is that the
VHGPRM again shows a more accurate prediction at the notch size of $12$ mm, as
indicated by the median predictions in Figure 10 panels c and d. finally, it
seems that the SGPRM generally outperforms the VHGPRM in predicting damage
sizes above $12$ mm, where either most of the predictions are confined around
the true damage size (notch size of $16$ mm), or are spread out with the
median prediction very close to the true notch size (notch size of $14$ mm),
or the range of predictions it self (between the $25^{th}$ and $75^{th}$
percentiles) actually encompasses the true damage size (notch size of $18$
mm).

abcd Figure 10: Notched Al coupon: true/predicted damage size boxplots for
path 3-6: (a) SGPRM state prediction based on the RMSD DI; (b) VHGPRM state
prediction based on the RMSD DI; (c) SGPRM state prediction based on the DI
formulation from [11]; (d) VHGPRM state prediction based on the DI formulation
from [11].

Figure 11 shows 4 indicative predictions by the GPRMs trained using the RMSD
DI data set from path 6-1. Next to the healthy case, three other points were
selected based on their proximity in DI value again to test the model
capability of differentiating between close test points coming from different
states of the component. Coincidentally, because the development of the RMSD
DI in the case of path 6-1 in this Al coupon is more or less monotonic, the
three points were selected based on the prediction results of the GPRMs
trained with the second DI formulation used in this study. As shown in all
panels of Figure 11, both models accurately predict damage size, with a slight
shift in the predictions of both models for the case of a $14$-mm notch.
Examining the corresponding results from the GPRMs trained with the other DI
formulation (Figure 12), one can observe that both models again perform well
except in the case of the selected DI test point at a notch size of $6$ mm,
where the SGPRM outperforms the VHGPRM in prediction. All in all, as expected,
the RMSD DI-trained GPRMs come on top compared to the GPRMs trained with the
other DI formulation due to the nice evolution of the DI values with notch
size. As with path 3-6, however, conclusions can only be withdrawn after
examining the summary prediction results from each model.

abcd Figure 11: Notched Al coupon: damage size prediction results for path 6-1
based on the RMSD DI: (a) prediction probabilities for the healthy case; (b)
prediction probabilities for a notch size of $6$ mm; (c) prediction
probabilities for a notch size of $10$ mm; (d) prediction probabilities for a
notch size of $14$ mm. Dashed vertical lines indicate the maximum probability
corresponding to each model. abcd Figure 12: Notched Al coupon: damage size
prediction results for path 6-1 based on the DI formulation in [11]: (a)
prediction probabilities for the healthy case; (b) prediction probabilities
for a notch size of $6$ mm; (c) prediction probabilities for a notch size of
$10$ mm; (d) prediction probabilities for a notch size of $14$ mm. Dashed
vertical lines indicate the maximum probability corresponding to each model.

Figure 13 panels a and b present the summary prediction results for the rmsd
DI-trained GPRMs for the standard and heteroscedastic models, respectively. As
shown, overall, VHGPRMs show sharper predictions compared to SGPRMs. This
phenomenon would generally result in sharper prediction probabilities, which
has been observed for some test DI points (not shown here for brevity).
Nonetheless, both models provide accurate predictions for each notch size. For
the models trained using the second DI formulation (Figure 13 panels c and d),
it can be observed that both model structures have trouble quantifying notch
size between $6$ mm and $14$ mm, which can be expected given the saturation
occurring in that range of notch sizes in the training DI values. This being
said, a closer examination of the box-plots shows that the trained VHGPRM
seems to give better damage size predictions at a notch size of $10$ and $12$
mm compared to the SGPRM, while the latter gives predictions that barely
encompass the true notch sizes. In contrast, the broad box-plots shown in
Figure 13c at notch sizes of $6$, $8$, and $14$ mm do cover the true damage
size, while the VHGPRM predictions completely miss the true state.

abcd Figure 13: Notched Al coupon: true/predicted damage size box-plots for
path 6-1: (a) SGPRM state prediction based on the RMSD DI; (b) VHGPRM state
prediction based on the RMSD DI; (c) SGPRM state prediction based on the DI
formulation from [11]; (d) VHGPRM state prediction based on the DI formulation
from [11].

From the analysis of both paths presented herein, the following can concluded:

  * •

GPRMs can accurately model both formulations of the DIs without the need for
user experience. this can be seen in the relatively-small training and testing
times required for GPRM training and validation processes.

  * •

Damage state prediction probabilities, extracted from model-based confidence
bounds, give a richer representation of the state of the system, compared to
the damage/no-damage paradigm of DIs.

  * •

Both variations of GPRMs used herein perform well in the damage quantification
task in this simple Al coupon. This can be observed in the summary box-plots.

  * •

Neither the SGPRM nor the VHGPRM completely outperforms the other model in
damage size quantification in the coupon presented herein.

  * •

The prediction performance of DI-trained GPRMs greatly depends on the
evolution/trend of the DI values with damage size. This was evident in the
case of path 6-1, where the RMSD DI-trained models performed almost perfectly
in damage size quantification compared to the models trained with the other DI
formulation, just because of the way the RMSD DI trend evolves with damage
size.

For the sake of brevity, the analysis results for the remaining coupons
presented in this study would be based on the DI formulation from [11], and
this formulation would be denoted hereon as the DI.

##  4 Test Case II: CFRP Coupon with Simulated Damage

###  4.1 Test Setup

The second coupon used in this study was a $152.4\times 254$ mm ($6\times 10$
in) CFRP plate ($2.36$ mm/$0.093$ in thickness; ACP Composites; $0/90^{o}$
unidirectional layup with Carbon fiber prepreg). As in the case of the Al
coupon, the CFRP coupon was similarly fitted with six SMART Layers type PZT-5A
using Hysol EA 9394 adhesive. In order to simulate damage, up to six three-
gram weights were sequentially attached to the surface of the plate using
tacky tape. Figure 14 shows the CFRP coupon with 6 weights attached. Data
acquisition and analysis was done in the same manner as in the Al coupon.

Figure 14: The CFRP coupon used in this study shown here with 6 weights as
simulated damage (largest damage size). The arrows show the signal paths
presented herein.

###  4.2 Results & Discussion

abcd Figure 15: CFRP coupon: indicative signals and DI plots for the CFRP
coupon: (a) signals from path 4-1; (b) signals from path 4-3; (c) DI values
from path 4-1; (d) DI values from path 4-3. The red lines indicate the means
of the DI values at every state.

Figure 15 panels a and b show the signals received at sensors 1 and 3 in the
CFRP coupon (see Figure 14 for sensor numbering), respectively, when sensor 4
was actuated under different levels of simulated damage. Note that the abrupt
amplitude increase in panel b is due to the pre-processing of the signals
where a set number of samples at the beginning of each signal was zeroed out
to completely remove the effects of cross-talk. Figure 15 panels c and d show
the corresponding DI evolution. The red dashed line indicates the $95\%$
confidence bound for the healthy case, as calculated from the standard
deviation of the experimental baseline DI values. As shown, using this DI
formulation, there seems to be substantial overlap between the DI values under
multiple damage sizes, as well as a non-constant variance in the values with
increasing damage size. These observations can be attributed to the
nonlinearity exhibited by composites. Indeed, these paths in the CFRP coupon
were specifically selected because of these two phenomena, which may prove
interesting when quantifying damage size using GPRMs.

Table 4: Summary of GPRM∗ information† for the CFRP coupon based on the DI formulation from [11] Signal | NMSE | RSS/SSS (%) | Training Time (s) | Prediction Time (s)
---|---|---|---|---
Path | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM
4-1 | 0.0426 | 0.0425 | 2.036 | 2.035 | 4.31 | 9.82 | 0.0671 | 0.0996
4-3 | 0.0611 | 0.0611 | 2.62 | 2.62 | 3.87 | 19.72 | 0.1006 | 0.0965
∗22.5% (990 points) of the data was used for training each model. |  |
†Numbers approximated to the last quoted decimal place, and times estimated
based on an Intel Core i3 laptop
with 4 Gb of RAM. |
abcd Figure 16: CFRP coupon: GPRM predictive mean and variance: (a) SGPRM
results for path 4-1; (b) VHGPRM results for path 4-1; (c) SGPRM results for
path 4-3; (d) VHGPRM results for path 4-3.

Similar to the case of the Al coupon, about $50\%$ of the available DI data
points were used for training SGPRMs and VHGPRMs. Table 4 presents summary
information about the trained models. Figure 16 panels a and b show the
predictive means and confidence bounds of the trained SGPRM and VHGPRM,
respectively, for path 4-1 under multiple damage cases. Figure 16 panels c and
d show the corresponding plots for path 4-3. As shown, both sets of models
follow the evolution of the DI quite well, as can also be inferred from the
low model criteria shown in Table 4. In addition, the VHGPRM in both paths
exhibits a growing confidence interval, which follows the increasing variance
in the DI values with damage size (number of attached weights). This widening
in the confidence bounds should allow this model to estimate prediction
probabilities more accurately compared to the trained SGPRM, especially since
the predictive variance of the trained models is used in the prediction
process, as described in Section 2.4.

In order to initially examine how the trained models perform in damage
quantification, the prediction probabilities of a few indicative DI test
points not used in the training process were plotted. Figure 17 shows such
probabilities for the healthy (panel a), as well as 3 damaged cases. Although
both model types accurately quantify damage size in all of the presented
cases, it is worth noting that the VHGPRM prediction probability also closely
follows the evolution of the variance; the probability is narrow towards the
smaller damage sizes and broad towards the larger ones. Ont the other hand,
the SGPRM prediction probabilities, owing to the SGPRM’s constant noise term
that is trained for the larger variance in the data, are broad towards the
lower damage sizes, whereas they should be narrow because of the decreased
variance in the DI values at these damage states. This shows the importance of
utilizing heteroscedastic models, such as the VHGPRM, in order to properly
estimate the prediction probability that accurately fits the variance in the
data.

abcd Figure 17: CFRP coupon: GPRM damage size prediction results for path 4-1:
(a) prediction probabilities for the healthy case; (b) prediction
probabilities for the case of 2 weights; (c) prediction probabilities for the
case of 4 weights; (d) prediction probabilities for the case of 6 weights.
abcd Figure 18: CFRP coupon: GPRM damage size prediction results for path 4-1:
(a) prediction probabilities for the healthy case; (b) prediction
probabilities for the case of 2 weights; (c) prediction probabilities for the
case of 4 weights; (d) prediction probabilities for the case of 6 weights.

Similarly, Figure 18 shows some indicative prediction probabilities from the
trained models for path 4-3 in the CFRP coupon. Again, the adaptation of the
VHGPRM-based prediction probability to the variance in the data can be clearly
seen. In addition, both models show a slight deviation from the true damage
size in the case of 2 and 6 weights. This deviation originates from the high
overlap in the data sets between different damage sizes as can be clearly seen
in Figure 15d. Examining the summary results for this coupon (Figure 19), one
can observe that both models, overall, perform well in damage quantification,
with the bulk of predictions per each damage size (the boxes) situated around
the true damage size. The abundance of outliers in the box-plots originates
from the large variation in DI values for each damage size. Again, these
results indicate that GPRMs can accurately predict damage size in composites
within an active-sensing, guided-wave SHM framework. In addition, the results
presented in this section highlight the advantage of heteroscedastic GPRMs in
accurately estimating the damage quantification probability, which should
prove beneficial in real-life applications where the variation in the data can
be large.

abcd Figure 19: CFRP coupon: true/predicted damage size box-plots: (a) SGPRM
state prediction for path 4-1; (b) VHGPRM state prediction for path 4-1; (c)
SGPRM state prediction for path 4-3; (d) VHGPRM state prediction for path 4-3.

##  5 Test Case III: Al Coupon Under Varying Simulated Damage & Load States

###  5.1 Test Setup

The third test case in this study was a $152.4\times 304.8$ mm ($6\times 12$
in) 6061 Aluminum coupon ($2.36$ mm/$0.093$ in thick). In a similar fashion to
the two previous coupons, 6 PZT-5A sensors were attached to the plate.
However, instead of leaving the adhesive to cure in ambient pressure for 7
days as with the two previous test cases, the adhesive was cured under vacuum
for 24 hrs at room temperature. The plate was then installed onto a tensile
testing machine (Instron, Inc). This allowed for the application of 4 static
loading conditions consecutively: 0, 5, 10, and 15 kN. 1-4 three-gram weights
were attached onto the surface of the plate during each loading state in the
manner shown in Figure 20. Data acquisition and analysis was done in the same
manner as in the other two coupons, with the difference that 20
signals/sensor/damage/loading condition (a total of 2400 signal data sets)
were recorded in this test case.

Figure 20: The third test case used in this study with four 3-gm weights
simulating damage (largest damage size) shown here with the testing machine’s
grips.

###  5.2 Results & Discussion

abcd Figure 21: Al coupon with simulated damage: indicative signals and DI
plots for path 1-6: (a) signals at unloaded condition; (b) signals with 2
weights attached; (c) DI values at unloaded condition; (d) DI values with 2
weights attached. The red lines indicate the means of the DI values at every
state.

Figure 21a shows the signal from path 1-6 in the third coupon in this study at
the unloaded conditions under varying number of attached weights. Figure 21b
on the other hand shows the evolution of the signals with respect to load at a
damage state of two attached weights. Figure 21 panels c and d show their
corresponding DI plots, respectively. As shown, the DI formulation presented
in [11] closely follows the evolution of damage size and load in this path.
Also, a slight change in the variance of the DI values across multiple damage
states or loads can be observed, with little-to-no overlap between te values
in either DI plots. Examining a different path, Figure 22a shows the signals
from path 3-6 under varying damage states at a load of 15 kN, while Figure 22b
presents the signals from the same path at different loads under a damage
state of 4 attached weights. Figure 22 panels c and d again show their
respective DI plots. As shown in Figure 22c, there exists a substantial
overlap between the DI values, as well as a clearly-changing variance across
the different damage states. On the other hand, Figure 22d shows an almost
linear evolution of the DI with loads, with no overlap at all, and an almost
constant variance across the different loading states at a damage size of 4
attached weights. For a presentation of the full array of DI plots at all
damage sizes and loading states, the readers are directed to Figures A.1-A.4
in the Appendix.

abcd Figure 22: Al coupon with simulated damage: indicative signals and DI
plots for path 3-6: (a) signals at 15 kN; (b) signals with 4 attached weights;
(c) DI values at 15 kN; (d) DI values with 4 attached weights. The red lines
indicate the means of the DI values at every state.

Because this coupon exhibits variation in both damage size and load, it is of
interest to examine two cases in a typical state-quantification scenario. The
first case is when the applied load on the component is known, and it is of
interest to only quantify damage size. The second case, is when both the
applied load and the damage state are unknown, and both need to be
simultaneously quantified. The following sections present indicative results
from both cases.

####  5.2.1 The case of a known loading state: Single-input GPRMs

For the first case, again $50\%$ of the available DI data points were used in
training single-input SGPRMs and VHGPRMs under each loading state. Table 5
shows the model information for the trained models pertaining to the paths
presented in Figures 21 and 22. As shown in the table, again SGPRMs and
VHGPRMs exhibit almost identical model criteria (NMSE and RSS/SSS), with the
substantial difference in training time. It is important to note here that, as
presented in the case of the CFRP coupon, although the model criteria might be
the same, the performance of both model types with respect to accurate
estimation of the quantification probabilities can be different according to
the nature of the training data.

Table 5: Summary of single-input GPRM∗ information† for the Al coupon with simulated damage. Signal | Load | NMSE | RSS/SSS (%) | Training Time (s) | Prediction Time (s)
---|---|---|---|---|---
Path | (kN) | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM
1-6 | 0 | 0.0034 | 0.0034 | 0.073 | 0.073 | 16.413 | 102.982 | 0.1913 | 0.3686
5 | 0.0052 | 0.0052 | 0.1 | 0.1 | 17.294 | 86.772 | 0.1895 | 0.4157
10 | 0.0012 | 0.0012 | 0.028 | 0.028 | 17.108 | 85.548 | 0.2558 | 0.3755
15 | 0.0016 | 0.0016 | 0.036 | 0.036 | 16.561 | 108.205 | 0.2276 | 0.3589
3-6 | 0 | 0.0226 | 0.0226 | 0.97 | 0.97 | 10.348 | 23.044 | 0.1279 | 0.1126
5 | 0.105 | 0.105 | 3.16 | 3.16 | 9.037 | 22.989 | 0.0957 | 0.112
10 | 0.0473 | 0.0473 | 1.77 | 1.77 | 9.25 | 25.79 | 0.1136 | 0.1082
15 | 0.0379 | 0.0379 | 1.59 | 1.59 | 9.061 | 25.689 | 0.094 | 0.109
∗50% (1000 points) of the data was used for training each model. |  |
†Numbers approximated to the last quoted decimal place, and times estimated
based on an Intel Core i3 laptop
with 4 Gb of RAM. |  |
abcd Figure 23: Al coupon with simulated damage: GPRM predictive mean and
variance: (a) SGPRM for path 1-6 at unloaded conditions; (b) VHGPRM for path
1-6 at unloaded conditions; (c) SGPRM for path 3-6 at 15 kN; (d) VHGPRM for
path 3-6 at 15 kN.

Figure 23 panels a and b show the predictive means and confidence bounds from
the GPRMs trained using the DI values presented in Figure 21c for path 1-6.
The difference between VHGPRMs and SGPRMs can be clearly seen at a damage size
of 2 attached weights, where the VHGPRM can adapt its confidence bounds to fit
the relatively-large variance in the training DI values at that specific
damage size. The same observation can be persists in Figure 23 panels a and b,
which show the predictive means and confidence bounds from the GPRMs trained
using the DI values presented in Figure 22c for path 3-6 (a load of 15 kN). In
both paths, though, each model type can clearly follow the evolution of the DI
with damage size. Although this section is particular to the case where the
load is known and the damage size is being quantified, for comparison, GPRMs
trained at a specific damage size and unknown loading state are also presented
herein. Figure 24 shows the predictive means and confidence bounds from such
GPRMs trained using DI values for paths 1-4 (panels a and b) and 3-6 (panels c
and d) corresponding to the cases shown in Figure 21d (two attached weights)
and 22d (four attached weights) respectively.

abcd Figure 24: Al coupon with simulated damage: GPRM predictive mean and
variance: (a) SGPRM for path 1-6 with 2 weights attached; (b) VHGPRM for path
1-6 with 2 weights attached; (c) SGPRM for path 3-6 with 2 weights attached;
(d) VHGPRM for path 3-6 with 2 weights attached.

Figure 25 shows the prediction probabilities extracted from the GPRMs shown in
Figure 23 corresponding to 4 indicative test DI points not used in the
training process from both SGPRMs and VHGPRMs for path 1-6. As shown, asside
from a slight deviation in the prediction of both model types for the case of
3 attached weights, both models accurately predict damage size. In addition,
as shown in Figure 25 panels a and b, the VHGPRM more accurately estimates the
prediction probabilities as a bit narrow (panel a) or broad (panel b) given
the respective change in variance in the training DI values at the healthy
state, as well as the damage state of 2 attached weights. Examining the
results for path 3-6 (Figure 26, the superiority of VHGPRMs can be clearly
shown here with more accurate predictions in the healthy case, as well as the
case of 4 attached weights. Both models however fail to accurately quantify
the case of three attached weights (Figure 26c), while both of them fall just
short of accurately predicting the case of 2 attached weights (Figure 26b).
This is attributed again to the immense overlap between the DI values at these
two damage cases.

abcd Figure 25: Al coupon with simulated damage: damage size prediction
results for path 1-6 at the unloaded condition: (a) prediction probabilities
for the healthy case; (b) prediction probabilities for the case of 1 attached
weight; (c) prediction probabilities for the case of 3 attached weights; (d)
prediction probabilities for the case of 4 attached weights. abcd Figure 26:
Al coupon with simulated damage: damage size prediction results for path 3-6
under 15 kN: (a) prediction probabilities for the healthy case; (b) prediction
probabilities for the case of 1 attached weight; (c) prediction probabilities
for the case of 3 attached weights; (d) prediction probabilities for the case
of 4 attached weights.

In order to analyze the SGPRMs and VHGPRMs in damage quantification for the Al
coupon with simulated damage, summary box-plots were plotted for both paths
under the different loading states exhibited by the coupon in this study.
Figure 27 presents the summary results for path 1-6. Examining the different
panels across each loading state, it can be observed that both model types
perform just as well in damage size quantification. In addition, all
predictions for all test points under all loading states are accurate, except
for the 5-kN case for the damage sizes of 3 and 4 attached weights, where it
can be observed that the SGPRMs underestimate the latter damage size, while
the VHGPRMs overestimate the former.

abcdefgh Figure 27: Al coupon with simulated damage: true/predicted damage
size boxplots for path 1-6: (a) SGPRM state predictions at 0 kN; (b) VHGPRM
state predictions at 0 kN; (c) SGPRM state predictions at 5 kN; (d) VHGPRM
state predictions at 5 kN; (e) SGPRM state predictions at 10 kN; (f) VHGPRM
predictions at 10 kN; (g) SGPRM state predictions at 15 kN; (h) VHGPRM state
predictions at 15 kN.

Looking at the summary results from path 3-6 (Figure 28), panels a and b show
the results for the unloadeding condition. It can be seen that the VHGPRM
outperforms the SGPRM here with narrower prediction ranges for all damage
sizes. Panels c and d show the prediction results for the 5-kN case. As shown,
both models underestimate the case of 4 attached weights, with the SGPRM
outperforming the VHGPRM in predicting the case of 2 attached weights.
Examining the loading case of 10 kN, it can be observed that both models
perform well in damage size quantification except when 2 weights are attached
to the Al coupon, in which case the SGPRM seem to be a bit more accurate with
the median prediction around the true damage size. Finally, when 15 kN are
applied, although the VHGPRM outperforms the SGPRM in model accuracy in the
healthy, as well as the case of 4 attached weights, the SGPRM outperforms the
VHGPRM models when 1 weight is attached to the coupon. Again, in all cases,
both models seem to perform well in the task of damage size quantification in
this coupon.

abcdefgh Figure 28: Al coupon with simulated damage: true/predicted damage size boxplots for path 3-6: (a) SGPRM state predictions at 0 kN; (b) VHGPRM state predictions at 0 kN; (c) SGPRM state predictions at 5 kN; (d) VHGPRM state predictions at 5 kN; (e) SGPRM state predictions at 10 kN; (f) VHGPRM state predictions at 10 kN; (g) SGPRM state predictions at 15 kN; (h) VHGPRM state predictions at 15 kN. Table 6: Summary of multi-input GPRM∗ information† for the Al coupon with simulated damage. Signal | NMSE | RSS/SSS (%) | Training Time (s) | Prediction Time (s)
---|---|---|---|---
Path | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM | SGPRM | VHGPRM
1-6 | 0.0026 | 0.0026 | 0.12 | 0.12 | 7.3113 | 36.5709 | 0.2571 | 0.8023
3-6 | 0.0023 | 0.0023 | 0.14 | 0.15 | 8.2146 | 47.5281 | 0.2943 | 0.5018
∗10% (800 points) of the data was used for training each model. |
†Numbers approximated to the last quoted decimal place, and times estimated
based on an Intel Core i3 laptop
with 4 Gb of RAM. |

####  5.2.2 Multi-input GPRMs

abcd Figure 29: Al coupon with simulated damage: Multi-input GPRM predictive
mean and standard deviation for path 1-6: (a) SGPRM predictive mean; (b)
VHGPRM predictive mean; (c) SGPRM predictive standard deviation; (d) VHGPRM
predictive standard deviation. abcd Figure 30: Al coupon with simulated
damage: Multi-input GPRM predictive mean and standard deviation for path 3-6:
(a) SGPRM predictive mean; (b) VHGPRM predictive mean; (c) SGPRM predictive
standard deviation; (d) VHGPRM predictive standard deviation..

As aforementioned, a second scenario was looked at when studying this coupon,
which is the case of simultaneously-unknown damage size and load. For that,
multi-input GPRMs were trained using about as low as $0.05\%$ of all the
available DI data sets under varying damage sizes and loads for the two paths
presented herein. Table 6 presents some important aspects regarding the
trained models. In order to visualize the trained GPRMs, the predictive means
and standard deviations of the trained SGPRM and VGPRM were plotted for path
1-6 in Figure 29. One interesting observation is that the VHGPRM predictive
mean (panel a) looks almost identical to the SGPRM mean (panel b).
Additionally, the predictive standard deviation of the VHGPRM is clearly more
adaptive to the data, as can be seen from the high deviations in the regions
where no training data was available (panel d - between weights and between
loads), and as can also be seen with the slightly higher standard deviation at
the regions where training data existed compared to the case of the SGPRM
(panel c). This observation is clearer when looking at the same plots for path
3-6 (Figure 30 panels c and d), where the VHGPRM again shows more adaptability
to the data, which is beneficial in capturing the true trends in the DI
evolution with damage size and load. Figure 31 shows the prediction
probabilities estimated by the trained multi-input models for path 1-6 at two
indicative DI test points. As shown, both models perform excellently in
simultaneously predicting both damage size and load state. These results show
that multi-input GPRMs can accurately perform multi-state quantification for
active-sensing, guided-wave SHM. Figure 32 also shows the prediction
probabilities for 2 other indicative test DI points for path 3-6. Again, both
models perform well in predicting both damage size and load.

abcd Figure 31: Al coupon with simulated damage: Multi-input GPRM prediction
results for path 1-6: (a) SGPRM prediction at 5 kN for 2 attached weights; (b)
VHGPRM prediction at 5 kN for 2 attached weights; (c) SGPRM prediction at 15
kN for 3 attached weights; (d) VHGPRM prediction at 15 kN for 3 attached
weights. abcd Figure 32: Al coupon with simulated damage: Multi-input GPRM
prediction results for path 3-6: (a) SGPRM prediction at 0 kN with 1 attached
weight; (b) VHGPRM prediction at 0 kN with 1 attached weight; (c) SGPRM
prediction at 5 kN for 4 attached weights; (d) VHGPRM prediction at 5 kN for 4
attached weights.

In addition to the indicative results presented here, summary prediction error
results were plotted In order to gain some insights into how prediction error
evolves when using multi-input GPRMs. Figure 33 panels a and b present the
evolution of the load and damage size prediction errors plotted with respect
to loading condition for SGPRM and VHGPRM of path 1-6, respectively, from all
7500+ test DI values not used in training. It is worth noting here that each
pair of facing orange and blue bars corresponding to the load and damage size
prediction errors, respectively, comes from the same test DI point i.e. there
are just over 7500 bars with either color. Panels c and d show the like but
plotted with respect to damage size. Doing an overall examination of where the
models fall short in accurately predicting damage size and load (where the
blue and yellow bars have high values) and comparing that with the DI plots in
Figures A.1 and A.2 in the Appendix, it can be observed that the GPRMs only
fall short at areas where there is significant overlap between DI values
across different states. Particularly, examining panels a and b of Figure 33,
it can be seen that damage size miss-predictions come from the 0, 5, and 15 kN
cases; examining panels c and d, it can be observed that most of these miss-
predictions are related to the damage size of 4 weights, with some related to
3 and 2 weights also. This is expected since the 4-weight DI values in path
1-6 did exhibit significant overlap with smaller damage sizes (see Figure
A.1). The same can be said about the load miss-predictions, where most of them
come form the 10 and 15-kN cases, as evident in panel a and b, and can be
correlated to the damage sizes of 3 and 4 weights as shown in panels c and d.
Again, examining the evolution of the DI values with load under multiple
damage sizes as shown in figure A.2, the roots of these load miss-predictions
can be well understood.

abcd Figure 33: Al coupon with simulated damage: Multi-input GPRM state
prediction error results for 7500+ test DI values from path 1-6. Black
horizontal lines divide regions of specific loads (panels a and b) or specific
damage sizes (panels c and d): (a) SGPRM state prediction error with respect
to load; (b) VHPRM state prediction error with respect to load; (c) SGPRM
state prediction error with respect to damage size; (d) VHGPRM state
prediction error with respect to damage size. In panels a and b (c and d),
black lines separate bars of different loads (damage sizes) and green dashed
lines separate bars of different damage sizes (loads).

Moving onto path 3-6, inspecting Figure 34, one can observe, again, a nice
agreement between the miss-predictions in damage size and load and the overlap
in DI values as evident in Figures A.3 and A.4 in the Appendix. In particular,
because this is a damage non-intersecting path, the evolution of the DI values
with damage size across different loading states is not uniform, and thus
there is significant overlap between DI values across the different damage
sizes. This explains the higher damage size miss-classifications in this path
compared to path 1-6. On the other hand, the overall miss-predictions in load
are less than with path 1-6, which is again expected given the much more
uniform evolution of the DI with load under different damage sizes for path
3-6 (see Figure A.4). Overall, both of the DI-trained SGPRM and VHGPRM perform
well in simultaneously predicting damage size and load within the capabilities
of the training DI data sets.

abcd Figure 34: Al coupon with simulated damage: Multi-input GPRM state
prediction error results for 7500+ test DI values from path 3-6. The black
horizontal lines divide regions of specific loads (panels a and b) or specific
damage sizes (panels c and d): (a) SGPRM state prediction error with respect
to load; (b) VHPRM state prediction error with respect to load; (c) SGPRM
state prediction error with respect to damage size; (d) VHGPRM state
prediction error with respect to damage size. In panels a and b (c and d),
black lines separate bars of different loads (damage sizes) and green dashed
lines separate bars of different damage sizes (loads).

From all of the results presented in this test case, a few conclusions can be
withdrawn as follows:

  * •

Single-input VHGPRMs outperform SGPRMs in accurately estimating prediction
probabilities based on the CDF. This is due to the capability of the former
models to adapt to the changing noise level in the data.

  * •

DI-trained GPRMs can accurately simultaneously predict damage size and loading
state in an active-sensing, guided-wave SHM framework.

  * •

State miss-classification originates from the evolution of the DI with the
states being quantified; if the evolution is uniform, state quantification is
accurate. Otherwise, the trained models cannot provide an accurate
quantification of the true state. Future work can target more robust SHM
metrics that show a uniform evolution with respect to the different states for
training GPRMs.

##  6 Conclusion

In this study, a novel method for damage/state quantification within the
framework of active-sensing, guided-wave SHM was proposed and applied to three
experimental test cases. The proposed method is based on damage index-trained
Gaussian Process Regression Models (DI-trained GPRMs) in which the structural
state is predicted based on the probability of an incoming test DI value from
an unknown state of the structure originating from different structural
states. The advantages of this framework lie in its simplicity, with only DI
values needed for training a robust and accurate quantification model, as well
as in the probability-based quantification process this method entertains.
Three experimental test cases were presented in this study in which the
proposed framework was applied for damage size quantification (Al coupon with
notch and CFRP coupon with simulated damage), as well as for damage size
and/or load state quantification (Al coupon with simulated damage under
different loading conditions). In addition, two types of GPRMs were
implemented within the proposed framework, namely: standard homoscedastic
(SGPRMs) and variational heteroscedastic (VHGPRMs) models. Both types of GPRMs
showed accurate damage size and/or load state quantification, with the VHGPRMs
allowing for a better representation of the evolution of the DI with damage
size and load, especially for signal paths that exhibited DIs with varying
noise across different states. Also, the limit of accurate damage size and/or
load state quantification seemed to be controlled by the training data; states
at which training DI values overlapped were indistinguishable by the GPRMs
when came to calculating the prediction probabilities. Finally, at least two
interesting points remain open for exploration and are currently being
explored by the authors. Firstly, although the proposed framework provides a
simple route to accurate and robust state quantification in active-sensing,
guided-wave SHM, it does involve multiple steps in some cases (2-state
predictions for instance). Thus, other more straightforward, yet
mathematically-involved, approaches, such as multi-output noisy-input GPRMs,
might be a better alternative to the multi-step approach presented in this
study. Secondly, because of the nature of how DI values can overlap across
multiple states, using more robust and accurate SHM metrics, such as non-
parametric and/or parametric time series representations, in training GPRMs
might allow for limiting or totally avoiding state miss-classifications.

## Acknowledgment

This work is carried out at the Rensselaer Polytechnic Institute under the
Vertical Lift Research Center of Excellence (VLRCOE) Program, grant number
W911W61120012, with Dr. Mahendra Bhagwat and Dr. William Lewis as Technical
Monitors.

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##  Appendix A Additional results

abcd Figure A.1: Al coupon with simulated damage: DI plots from path 1-6 under
multiple loading conditions: (a) 0 kN; (b) 5 kN; (c) 10 kN; (d) 15 kN. The red
dashed lines indicate the healthy $95\%$ confidence bounds, and the red
circles indicate the mean DI values at each state. abcde Figure A.2: Al coupon
with simulated damage: DI plots from path 1-6 under multiple damage
conditions: (a) Healthy; (b) One weight attached; (c) Two weights attached;
(d) Three weights attached; (e) Four weights attached. The red dashed lines
indicate the healthy $95\%$ confidence bounds, and the red circles indicate
the mean DI values at each state. abcd Figure A.3: Al coupon with simulated
damage: DI plots from path 3-6 under multiple loading conditions: (a) 0 kN;
(b) 5 kN; (c) 10 kN; (d) 15 kN. The red dashed lines indicate the healthy
$95\%$ confidence bounds, and the red circles indicate the mean DI values at
each state. abcde Figure A.4: Al coupon with simulated damage: DI plots from
path 3-6 under multiple damage conditions: (a) Healthy; (b) One weight
attached; (c) Two weights attached; (d) Three weights attached; (e) Four
weights attached. The red dashed lines indicate the healthy $95\%$ confidence
bounds, and the red circles indicate the mean DI values at each state.