Verification of the applicability of the Gaussian mixture modelling for damage identification in reinforced concrete structures using acoustic emission testing

Abstract

This article presents an experimental study on verification of the applicability of Gaussian mixture modelling (GMM) algorithm of acoustic emissions for damage identification in reinforced concrete (RC) structures. Eight RC-flanged beam specimens with different properties were tested subjected to flexural loading. An incremental cyclic load was applied on RC-flanged beam specimens till failure, and simultaneously, the released acoustic emissions (AE) were recorded. It may be required to study crack classification in RC structures, because crack classification studies are useful to predict the structural performance and subsequently to implement the appropriate structural rehabilitation methods. AE belonging to tensile cracking and shear cracking can be studied by a probabilistic approach. It was observed that the line separating the AE clusters belonging to tensile and shear cracks was shifting towards a higher rise angle as the specimen is reaching collapse stage. This observation indicates dominance of shear cracks near the collapse stage. At the loading cycle where yielding occurred in the test specimen obtained by using GMM algorithm for AE, the load cycle entered into heavy damage zone is almost same as per NDIS-2421 damage assessment chart. The results obtained by both GMM algorithms for AE and NDIS-2421 criterion to evaluate the damage in the RC-flanged beams were compared and discussed.

Appendix A

The Gaussian mixture modelling (a multivariate probabilistic analysis) allows the user to sort large quantity of data into different clusters (sub-population or first mixture and second mixture) using the expectation (derivation of posterior probability or mixture weights)—maximization algorithm (we compute µ, ∑, and $\pi_{k} )$. The GMM (or the linear superposition of Gaussians) is given in Eq. (A1) [23]:

$$ p\left( x \right) = \mathop \sum \limits_{k = 1}^{K} \pi_{k} N \left (x|\mu_{k, } \mathop \sum \nolimits_{k} \right), $$

(A1)

where K is the number of Gaussians (or number clusters, tensile, or shear cluster) and k = 1,…K, $N(x|\mu_{k, } \mathop \sum \nolimits_{k} )$ is the normal multivariate Gaussian distribution for class K, and $\pi_{k}$ is the mixing coefficient (or weightage) for kth Gaussian distribution. A D-variate Gaussian distribution function is given in Eq. (A2):

$$ N\left( {x |\mu_{k, } \mathop \sum \nolimits_{k} } \right) = \frac{1}{{(2\pi )^{D/2} |\mathop \sum \nolimits |^{1/2} }}e^{{\frac{ - 1}{2}\left[ {(x - \mu )^{T} \mathop \sum \nolimits^{{\begin{array}{*{20}c} { - 1} \\ {} \\ \end{array} }} (x - \mu )} \right]}} , $$

(A2)

where $\mu_{k }$ is the vector form of mean for the kth Gaussian and $\mathop \sum \nolimits_{k}$ is the covariance matrix for the kth Gaussian. The mixing coefficient (or weightage) satisfies the constraint 0 ≤ $\pi_{k}$  ≤ 1 and is given in Eq. (A3):

$$ \mathop \sum \limits_{k = 1}^{K} \pi_{k} = 1. $$

(A3)

For a univariate Gaussian distribution, the probability density function [F(x)] is given in Eq. (A4):

$$ F\left( x \right) = G\left( {x|\mu ,\sigma } \right) = \frac{1}{{\sqrt {2\pi \sigma^{2} } }}e^{{\frac{ - 1}{2}\left( {\frac{x - \mu }{\sigma }} \right)^{2} }} , $$

(A4)

where − ∞ ≤ x ≤ ∞, µ is the mean, and $\sigma$ is the standard deviation of the data [30, 31]. Standard deviation (σ) can be thought of measuring how far the data values lie from the mean (µ). Assuming that the variables are independent with probability density function’s $N\left( {\mu_{1} , \sigma_{1}^{2} } \right)$, $N\left( {\mu_{2} , \sigma_{2}^{2} } \right)$, …. $N\left( {\mu_{P} , \sigma_{P}^{2} } \right)$, respectively. The joint densities are given by Eq. (A5). The variance and standard deviation describes how spread out the data is. If the data all lie close to the mean, then the standard deviation is small, while if the data are spread out over large range of values, σ is large:

$$ F \left( {x_{1} , x_{2} , \ldots x_{P} } \right) = f \left( {x_{1} } \right)* f\left( {x_{2} } \right)* \ldots f(x_{P} ). $$

(A5)

Solving Eq. (A5), we get

$$ F \left( {x_{1} , x_{2} , \ldots x_{P} } \right) = \frac{1}{{\left( {2\pi } \right)^{{{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} (\sigma_{1}^{2} \sigma_{2}^{2} \ldots \sigma_{P}^{2} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}{\text{e}}^{{\frac{ - 1}{2}\left[ {\left( {\frac{{x_{1} - \mu_{1} }}{{\sigma_{1} }}} \right)^{2} + \left( {\frac{{x_{2} - \mu_{2} }}{{\sigma_{2} }}} \right)^{2} + \ldots \left( {\frac{{x_{P} - \mu_{P} }}{{\sigma_{P} }}} \right)^{2} } \right]}} . $$

(A6)

1.1 Implementing GMM algorithm to study crack classification in RC structures

In the present study, the first mixture is the population (or AE hits) which belongs to tensile cracks and the second mixture is the population (or AE hits) which belongs to shear cracks. To use GMM algorithm to classify crack mode in RC test specimens under flexural loading, the input data is a 2-D vector, i.e. RA and AF, are considered:

$$ X = \left\{ {x_{1} = \left( {{\text{RA}}_{1} , {\text{AF}}_{1} } \right), x_{2} = \left( {{\text{RA}}_{2} ,{\text{AF}}_{2} } \right), \ldots ,x_{T} = \left( {{\text{RA}}_{T} , {\text{AF}}_{T} } \right)} \right\}. $$

(A7)

In addition, the hidden classes are I = {1, 2}, which represent the tensile and shear mode, respectively. In the present experimental data, there are only two clusters (or Gaussians). Therefore, from Eq. (5):

$$ F \left( {x_{1} , x_{2} } \right) = f \left( {x_{1} } \right)* f\left( {x_{2} } \right), $$

(A8)

$$ F \left( {x_{1} ,x_{2} } \right) = \frac{1}{{\left( {2\pi } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} (\sigma_{1}^{2} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}{\text{e}}^{{\frac{ - 1}{2}\left[ {\left( {\frac{{x_{1} - \mu_{1} }}{{\sigma_{1} }}} \right)^{2} } \right]}} *\frac{1}{{\left( {2\pi } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} (\sigma_{2}^{2} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}{\text{e}}^{{\frac{ - 1}{2}\left[ {\left( {\frac{{x_{2} - \mu_{2} }}{{\sigma_{1} }}} \right)^{2} } \right]}} , $$

(A9)

$$ F \left( {x_{1} , x_{2} } \right) = \frac{1}{{\left( {2\pi } \right)^{{{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} (\sigma_{1}^{2} \sigma_{2}^{2} )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}{\text{e}}^{{\frac{ - 1}{2}\left[ {\left( {\frac{{x_{1} - \mu_{1} }}{{\sigma_{1} }}} \right)^{2} + \left( {\frac{{x_{2} - \mu_{2} }}{{\sigma_{2} }}} \right)^{2} } \right]}} , $$

(A10)

where $\sigma_{1}^{2}$ is the variance of tensile clusters, $\sigma_{2 }^{2}$ is the variance of shear clusters, and $\sigma_{12} = \sigma_{21}$ = 0. Since tensile and shear clusters are independent, the exponent part of Eq. (A10) represents an ellipse. To find the constant part of Eq. (A10), calculate determinant of covariance matrix.

Since it is assumed that the variables are independent. $\sigma_{1P}$, $\sigma_{2P}$ … = 0.

Here, $\sigma_{11}$ is the standard deviation for only tensile clusters. $\sigma_{22}$ is the standard deviation of only shear clusters; ∑ is the covariance matrix. $\sigma_{21}$ is the covariance for both tensile and shear clusters. $\sigma_{11}^{2}$ and $\sigma_{22}^{2}$ are known as variance. If the tensile and shear clusters are independent each other, then their covariance is zero:

$$ \left| {\mathop \sum \nolimits } \right|^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} = \left( {\sigma_{1}^{2} \sigma_{2}^{2} \ldots \sigma_{P}^{2} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} . $$

(A11)

For the exponential part of Eq. (A4), in matrix multiplication, square term is equal to the term multiplied by its transpose, and therefore

$$ {\text{Exponent }} = \frac{ - 1}{2}\left( {X - \mu } \right)^{T} \mathop \sum \nolimits^{ - 1} \left( {X - \mu } \right). $$

(A12)

Substituting Eqs. (A11) and (A12) into Eq. (A6), we get the general equation for a P variate Gaussian distribution:

$$ P\left( {X |\mu ,\mathop \sum \nolimits } \right) = F \left( {x_{1} , x_{2} , \ldots x_{P} } \right) = \frac{1}{{\left( {2\pi } \right)^{{{\raise0.7ex\hbox{$P$} \!\mathord{\left/ {\vphantom {P 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \left| {\mathop \sum \nolimits } \right|^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}{\text{e}}^{{\frac{ - 1}{2} [\left( {X - \mu } \right)^{T} \mathop \sum \nolimits^{ - 1} \left( {X - \mu } \right)]}} . $$

(A13)

Estimating the (∑, µ) of a distribution maximum-likelihood (ML) estimation is used.

Taking logarithm on both sides of Eq. (A13):

$$ \ln p(X|\mu , \mathop \sum \nolimits ) = - \frac{p}{2}\ln \left( {2\pi } \right) - \frac{1}{2}\ln |\mathop \sum \nolimits | - \frac{1}{2}\left( {x - \mu } \right)^{T} \mathop \sum \nolimits^{ - 1} \left( {x - \mu } \right). $$

(A14)

To maximize, take derivative of Eq. (A14) with respect to µ and equate it to zero:

$$ \frac{{\partial \left[ {\ln p(X|\mu , \mathop \sum \nolimits )} \right]}}{\partial \mu } = 0, $$

(A15)

$$ \mu_{\text{ML}} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} X_{n} . $$

(A16)

Take derivative of Eq. (A14) with respect to ∑ and equate it to zero:

$$ \frac{{\partial \left[ {\ln p(X|\mu , \mathop \sum \nolimits )} \right]}}{\partial \mathop \sum \nolimits } = 0, $$

(A17)

$$ \mathop \sum \nolimits_{\text{ML}} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} (X_{n} - \mu_{\text{ML}} )(X_{n} - \mu_{\text{ML}} )^{T} , $$

(A18)

where N is the number of data points. Similarly, for a mixture of Gaussians also called as “Linear superposition of Gaussian’s”:

$$ p\left( x \right) = \mathop \sum \limits_{k = 1}^{2} \pi_{k} N\left( {x|\mu_{k, } \mathop \sum \nolimits_{k} } \right), $$

(A19)

where K is the number of Gaussians (or clusters). When k equal to 1 indicates for tensile clusters and k equal to 2 for shear cluster, $N(x|\mu_{k, } \mathop \sum \nolimits_{k} )$ is the normal multivariate Gaussian distribution for class K; $\pi_{k}$ is the mixing coefficient (or weightage) for each Gaussian distribution. $\pi_{1}$ is the mixing coefficient (or weightage) for tensile cluster:

$$ \pi_{1} = \frac{{N_{1} }}{N}, $$

(A20)

where N₁ is the number of AE hits in tensile cluster. N is the total number of AE hits. $\pi_{2}$ is the mixing coefficient (or weightage) for shear cluster:

$$ \pi_{2} = \frac{{N_{2} }}{N}. $$

(A21)

Here, N₂ is the number of AE hits in shear cluster and N is the total number of AE hits recorded. Taking logarithmic-likelihood on both sides of Eq. (A14), we get

$$ \ln p(X|\mu_{k} , \mathop \sum \nolimits_{k} ,\pi_{k} ) = \mathop \sum \limits_{n = 1}^{N} \ln p\left( {x_{n} } \right) = \mathop \sum \limits_{n = 1}^{N} \ln \left[ {\mathop \sum \limits_{k = 1}^{K} \pi_{k} N\left( {x_{n} |\mu_{k} ,\mathop \sum \nolimits_{k} } \right)} \right] , $$

(A22)

where N indicates all data points (total number of AE hits recorded in the test) and n may equal to 1 and 2. Note that, inside the logarithm, there is a summation over K Gaussian’s, and outside, there is summation for N number of sample data points. The maximum-likelihood can be obtained iteratively using expectation–maximization (EM) method. EM algorithm is used to estimate a set of parameters which are the mixing coefficient ($\pi_{k}$), set of class means ($\mu_{k}$), and set of covariance ($\mathop \sum \nolimits_{k}$). EM algorithm method contains two steps: namely, estimation (or expectation) step and maximization step.

Step I: By applying Bayes’ rule for a given parameter, the expected values of a latent variable are computed:

$$ \gamma_{k} \left( x \right) = \frac{{\pi_{k} N(x|\mu_{k} ,\mathop \sum \nolimits_{k} )}}{{\mathop \sum \nolimits_{k = 1}^{K} \pi_{k} N(x|\mu_{k} ,\mathop \sum \nolimits_{k} )}}, $$

(A23)

where $\pi_{k} = \frac{{N_{k} }}{N}$, $N_{k}$ = Number of samples for kth class or it is the effective number of data points assigned to cluster k and N = total number of samples, $\gamma_{k}$ is the latent variable, $P\left( {k |X} \right)$ is the posterior probability, $p\left( k \right)$ is the class prior, and $p(x)$ is the unconditional prior.

For tensile cluster, k = 1.

$$ \gamma_{1} \left( x \right) = \frac{{\pi_{1} N(x|\mu_{1} ,\mathop \sum \nolimits_{1} )}}{{\mathop \sum \nolimits_{k = 1}^{2} \pi_{k} N(x|\mu_{k} ,\mathop \sum \nolimits_{k} )}}. $$

(A24)

For shear cluster, k = 2.

$$ \gamma_{2} \left( x \right) = \frac{{\pi_{2} N(x|\mu_{2} ,\mathop \sum \nolimits_{2} )}}{{\mathop \sum \nolimits_{k = 1}^{2} \pi_{k} N(x|\mu_{k} ,\mathop \sum \nolimits_{k} )}}, $$

(A25)

where the numerator corresponds to the kth Gaussian and the denominator is the sum of all Gaussians.

Step II: In this step, the parameters are re-estimated using the current latent variable:

$$ \mu_{j} = \frac{{\mathop \sum \nolimits_{n = 1}^{N} \gamma_{k} \left( {x_{n} } \right)x_{n} }}{{\mathop \sum \nolimits_{n = 1}^{N} \gamma_{k} \left( {x_{n} } \right)}}, $$

(A26)

where j = 1,2. In case of tensile clusters, j is equal to 1 and j is equal to 2 for shear clusters:

$$ \mathop \sum \nolimits_{j} = \frac{{\mathop \sum \nolimits_{n = 1}^{N} \gamma_{k} \left( {x_{n} } \right)(x_{n} - \mu_{k} ) (x_{n} - \mu_{k} )^{T} }}{{\mathop \sum \nolimits_{n = 1}^{N} \gamma_{k} \left( {x_{n} } \right)}} $$

(A27)

$$ \pi_{j} = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N} \gamma_{k} \left( {x_{n} } \right). $$

(A28)

Here, n = a single AE hit. N = total number of AE hits recorded

The log-likelihood is estimated using Eq. (A22) for the above obtained parameters over a number of successive iterations and checked whether it truly represents the data sample or not. This process repeats until a convergence is obtained which is governed by the following criteria; in successive iterations, the parameters (µ, ∑, π) do change or over the last few iterations; second, when the log-likelihood itself does not change over a set of iterations [3].