Relevance feature selection of modal frequency-ambient condition pattern recognition in structural health assessment for reinforced concrete buildings

Abstract

Modal frequency is an important indicator for structural health assessment. Previous studies have shown that this indicator is substantially affected by the fluctuation of ambient conditions, such as temperature and humidity. Therefore, recognizing the pattern between modal frequency and ambient conditions is necessary for reliable long-term structural health assessment. In this article, a novel machine-learning algorithm is proposed to automatically select relevance features in modal frequency-ambient condition pattern recognition based on structural dynamic response and ambient condition measurement. In contrast to the traditional feature selection approaches by examining a large number of combinations of extracted features, the proposed algorithm conducts continuous relevance feature selection by introducing a sophisticated hyperparameterization on the weight parameter vector controlling the relevancy of different features in the prediction model. The proposed algorithm is then utilized for structural health assessment for a reinforced concrete building based on 1-year daily measurements. It turns out that the optimal model class including the relevance features for each vibrational mode is capable to capture the pattern between the corresponding modal frequency and the ambient conditions.

Introduction

The goal of structural health monitoring (SHM) is to assess the health status of a structure based on structural responses and ambient conditions measurement. ¹ Modal frequency, which is related to structural stiffness, is an important indicator in SHM. Previous studies have shown that this indicator is substantially affected by the fluctuation of ambient conditions, such as temperature and relative humidity.^2–6 In order to depict the relationship between modal parameters and ambient conditions, a number of long-term monitoring systems were operated for different types of structures, including bridge,^7–9 reinforced concrete buildings,^10–12 and other types.^13–15 In these studies, it has been shown that the fluctuation of the structural dynamical properties was significant due to the variation of ambient conditions, and the modal frequencies exhibited strong correlation with temperature and relative humidity. If the modal frequency-ambient condition pattern is not appropriately captured, there will be substantial bias in the structural health assessment results. Toward the goal for reliability enhancement of structural health assessment under changing ambient conditions, pattern recognition and machine-learning approaches have received tremendous attention.^16,17 A number of machine-learning algorithms have been proposed, including principal component analysis, ¹⁸ support vector machine, ¹⁹ nonlinear principal component analysis, ²⁰ support vector machine multi-class clustering algorithm, ²¹ and kernel-based algorithm. ²²

Due to the fact that structural health assessment exhibits significant level of uncertainty,^23–26 a number of system identification approaches have been proposed.^27–32 In particular, the Bayesian inference–based approach has attracted special attention as it provides a rigorous solution for uncertainty quantification and it is applicable for different problems in pattern recognition and system identification,^33–35 such as modal/model updating,^36–38 robust signal processing, and sensor configuration.^39–43 In this article, a novel machine-learning approach is proposed to automatically select the relevance features in modal frequency-ambient condition pattern recognition based on structural dynamic response and ambient condition measurement.

In the traditional feature selection approaches,^44,45 a set of model class candidates is first constructed, and each candidate contains a subset of the components of the full extracted feature vector. Model plausibility evaluation is implemented for each candidate, and it is obvious that the total number of candidates for evaluation is huge although the number of the components of the extracted feature vector is moderate. For instance, when the number of components of the full extracted feature vector is 10, 2¹⁰-1 candidates in total are required for consideration given that at least one feature of the full extracted feature vector should be included in a candidate. Therefore, optimal model class determination in the traditional approaches requires large computational effort. In contrast, the proposed approach is capable of conducting continuous relevance feature selection, which is done by introducing a sophisticated hyperparameterization on the weight parameter vector controlling the relevancy of different features in the prediction model. The proposed algorithm is then utilized for structural health assessment for a reinforced concrete building based on 1-year daily measurements.

The structure of this article is outlined as follows. Dataset and feature extraction are first presented. Then, the approach for relevance feature selection is developed. Finally, the proposed approach is utilized for modal frequency-ambient condition pattern recognition for the monitored structure based on measurement.

Dataset and feature extraction

Description of dataset

The structure considered in this study is a 22-story reinforced concrete residential building, namely, the East Asia Hall. Figure 1 shows the side view and its layout plan. The floor plan of the building is in asymmetric L-shape, with the building height and the length of the two spans being 64.70, 51.90, and 61.75 m, respectively.

Figure 1.

(a) Side view and (b) layout of the East Asia Hall.

The monitoring period is 1 year between 2 May 2008 and 1 May 2009. During the monitoring period, the structure was subjected to four typhoons of 15 days. In order to focus on the pattern between the structural properties and ambient conditions under normal weather conditions, the 15-day typhoon attacking period is excluded and the rest of 350 datasets are utilized in the analysis. Two accelerometers, operated based on standard exploration geophone mass-spring systems with 50 V/g sensitivity, were installed on the 18th floor with two orthogonal directions (directions 1 and 2) shown in Figure 1(b).

In order to minimize the spatial temperature difference of the monitored structure, acceleration was measured at 11:00 p.m. every day. Based on the 10-min acceleration time history of each day, the Bayesian spectral density approach^46–48 is applied to identify the modal frequencies of the building. The Bayesian spectral density approach is a well-developed probabilistic approach for modal identification.^46–48 The modal identification is performed by utilizing the spectral density obtained from the structural response as the data to estimate the uncertain modal parameters. The efficiency and flexibility of the Bayesian spectral density approach on modal identification had been demonstrated through successful applications with in-field measurements.^11,12,38 Herein, each set of the 10-min acceleration measurement is partitioned into four segments with an equal time duration and then the averaged spectrum is obtained for modal identification. Since the squared modal frequencies of first three modes are considered, the range $[0,4]{\mathrm{s}}^{-2}$ that covers the concerned squared modal frequencies is used in the modal identification.

Figure 2 shows the identified squared modal frequencies of the first three modes ( ${\omega }_1^2,{\omega }_2^2,{\omega }_3^2$ ) and measurement of the temperature (T) and the relative humidity (H). The ranges of temperature and relative humidity are $13\circ \mathrm{C}\le T\le 35.0\circ \mathrm{C}$ and $32.0\%\le H\le 95.0\%$ , respectively. It can be seen that high temperature associates with high relative humidity, while low temperature associates with scattering relative humidity, reflecting the weather pattern influenced by monsoon and subtropical climate.

Figure 2.

Identified squared modal frequencies of the first three modes ( ${\omega }_1^2,{\omega }_2^2,{\omega }_3^2$ ) and measurement of the temperature (T) and the relative humidity (H).

Feature extraction

Extracted features are constructed based on the previous research of the relationship between modal frequency and ambient conditions. Watson and Rajapakse ⁴⁹ correlated structural properties with temperature based on a polynomial function; Xia et al. ⁵ proposed a linear relationship between modal frequencies and temperature along with humidity; Rincón et al. ⁵⁰ attempted to investigate the influence of structure due to the changing of relative humidity; Yuen and Kuok^11,12 proposed a second-order polynomial function for modal frequencies with respect to both temperature along with relative humidity; Moser and Moaveni ⁵¹ and Moaveni and Behmanesh ⁵² utilized high-order (up to fourth) polynomials for correlating modal frequencies and temperature only. In summary, most of the existing works considered low-order polynomials for modal frequencies with respect to temperature only or temperature and relatively humidity; on the other hand, very few works considered high-order polynomials (up to fourth) for frequencies with respect to temperature only, neglecting the cross combinations of temperature and relatively humidity. In this study, in order to comprehensively take different possible patterns into consideration, the extracted features include not only higher order functions of the temperature and the relative humidity but also the cross combinations of them. Finally, the feature vector x containing totally 15 components is given as follows

$$\mathit{x}={[1,T_n,H_n,T_n^2,T_n{H}_n,H_n^2,T_n^3,T_n^2{H}_n,T_n{H}_n^2,H_n^3,T_n^4,T_n^3{H}_n,T_n^2{H}_n^2,T_n{H}_n^3,H_n^4]}^{\mathrm{T}}$$

(1)

where the normalized temperature $T_n=T/T_{\max }$ and the normalized relative humidity $H_n=H/H_{\max }$ are obtained by considering the rescale of T and H by the maximum value $T_{\max }=\max (T)$ and $H_{\max }=\max (H)$ , respectively. Figure 3 shows the correlation coefficients between the squared modal frequencies and different feature components for the three modes. It is observed that all the modal frequencies possess positive correlation with each component of the feature vector. Toward the goal for reliability enhancement of the prediction model charactering modal frequency-ambient condition pattern, it is aimed to select the suitable relevance features based on the available dataset. In next section, a novel Bayesian inference–based learning approach is proposed. In contrast to the traditional approach requiring selecting relevance features by examining a large number of combinations of extracted features,^44,45 the proposed approach conducts continuous relevance feature selection by introducing a sophisticated hyperparameterization on the weight parameter vector controlling the relevancy of different features in the prediction model.

Figure 3.

Correlation coefficients between the squared modal frequency and each feature component for the three modes: (a) upper: mode 1; (b) middle: mode 2; and (c) lower: mode 3.

Relevance feature selection

Consider the mth vibrational mode of the structure. The modal frequency-ambient condition pattern given the feature vector with $N_b$ ( $N_b=15$ ) components for the dataset with N ( $N=350$ ) points is given as follows

$${\mathbf{Y}}_m=\mathbf{X}{\mathbf{b}}_m+{\mathbf{\epsilon }}_{\mathbf{m}}$$

(2)

where ${\mathbf{Y}}_m=[{\omega }_{m,1}^2,\dots ,{\omega }_{m,N}^2{]}^T\in {\mathbb{R}}^N$ is the mth mode output vector; $\mathbf{X}=[{\mathbf{X}}_1,\dots ,{\mathbf{X}}_{N_b}]\in {\mathbb{R}}^{N\times N_b}$ is the design/input matrix characterized by the feature vector of equation (1); ${\mathbf{b}}_m\in {\mathbb{R}}^{N_b}$ is the mth mode weight parameter vector to be trained; and ${\mathit{\epsilon }}_m\in {\mathbb{R}}^N$ is the mth mode residual vector following Gaussian distribution $\mathcal{N}({\mathbf{\epsilon }}_m|0,{\sigma }_m^2{\mathbf{I}}_N)$ with the corresponding mth mode prediction-error variance ${\sigma }_m^2$ . Then, the likelihood function for ${\mathrm{D}}_m=\{\mathbf{X},{\mathbf{Y}}_m\}$ of the mth mode is given as follows ⁴⁸

$$p({\mathrm{D}}_m|{\mathbf{b}}_m,{\sigma }_m^2)=\mathcal{N}({\mathbf{Y}}_m|\mathbf{X}{\mathbf{b}}_m,{\sigma }_m^2{\mathbf{I}}_N)$$

(3)

In order to conduct continuous relevance feature selection for the mth mode, a sophisticated hyperparameterization on the weight parameter vector ${\mathbf{b}}_m$ is introduced. The automatic relevance determination (ARD) prior is adopted for hyperparameterization,^53–55 which is defined as a zero-mean Gaussian distribution

$$p({\mathbf{b}}_m|{\mathbf{\alpha }}_m)=\mathcal{N}({\mathbf{b}}_m|0,{\mathbf{A}}_m^{-1}({\mathbf{\alpha }}_m))$$

(4)

where ${\mathbf{A}}_m({\mathbf{\alpha }}_m)=\mathrm{diag}\{{\alpha }_{m,1},{\alpha }_{m,2},\dots ,{\alpha }_{m,N_b}\}$ is the precision matrix (inverse of the covariance matrix) and ${\mathbf{\alpha }}_m=[{\alpha }_{m,1},{\alpha }_{m,2},\dots ,{\alpha }_{m,N_b}{]}^{\mathbf{T}}$ is the hyperparameter vector parameterizing the prior probability density function (PDF) of the weight parameter vector ${\mathbf{b}}_m$ .

Using Bayes’ theorem, the posterior PDF for ${\mathbf{b}}_m$ is given by ⁵⁶

$$\begin{array}{cc}\multicolumn{1}{c}{p({\mathbf{b}}_m|{\mathrm{D}}_m,{\mathbf{\alpha }}_m,{\sigma }_m^2)}& \propto p({\mathrm{D}}_m|{\mathbf{b}}_m,{\sigma }_m^2)p({\mathbf{b}}_m|{\mathbf{\alpha }}_m)\hfill \\ =\mathcal{N}({\mathbf{b}}_m|{\hat{\mathbf{b}}}_m,{\hat{\mathbf{\Sigma }}}_m)\hfill \end{array}$$

(5)

where

$${\hat{\mathbf{b}}}_m={\sigma }_m^{-2}{\hat{\mathbf{\Sigma }}}_m{\mathbf{X}}^{\mathrm{T}}{\mathbf{Y}}_m$$

(6)

$${\hat{\mathbf{\Sigma }}}_m={({\sigma }_m^{-2}{\mathbf{X}}^T\mathbf{X}+{\mathbf{A}}_m)}^{-1}$$

(7)

The posterior mean vector ${\hat{\mathbf{b}}}_m$ and covariance matrix ${\hat{\mathbf{\Sigma }}}_m$ , as well as the relevancy of each feature in the design matrix, continuously depends on the hyperparameter vector ${\mathbf{\alpha }}_m$ . This conclusion can be validated by considering two extreme cases for the value of the ith component of the hyperparameter vector ${\alpha }_{m,i}$ . If ${\alpha }_{m,i}\to 0$ , then the prior PDF $p(b_{m,i}|{\alpha }_{m,i})$ is Gaussian distribution with very small precision (very large variance), corresponding to the case that the posterior PDF is dominated by the likelihood function so $b_{m,i}\to b_{m,i}^{\mathrm{mle}}$ where $b_{m,i}^{\mathrm{mle}}$ is the maximum likelihood estimate of $b_{m,i}$ , and its associated feature ${\mathbf{X}}_i$ is retained in the model for modal frequency-ambient condition pattern of the mth mode in equation (2). On the other hand, if ${\alpha }_{m,i}\to \infty$ , then the prior PDF $p(b_{m,i}|{\alpha }_{m,i})$ is Gaussian distribution with very high precision (i.e. very small variance), corresponding to the case that the posterior PDF is dominated by the prior PDF so $b_{m,i}\to 0$ and its associated feature ${\mathbf{X}}_i$ is irrelevant to the model in equation (2). In the remaining of this section, aiming for searching the optimal model class with the relevance features for each vibrational mode of the structure, an efficient strategy is proposed for optimization of the hyperparameter vector ${\mathbf{\alpha }}_m$ along with the prediction-error variance ${\sigma }_m^2$ .

Optimization of hyperparameter vector and prediction-error variance

According to Bayes’ theorem, the posterior PDF of the hyperparameter vector ${\mathbf{\alpha }}_m$ and the prediction-error variance ${\sigma }_m^2$ is given by $p({\mathbf{\alpha }}_m,{\sigma }_m^2|{\mathrm{D}}_m)\propto p({\mathrm{D}}_m|{\mathbf{\alpha }}_m,{\sigma }_m^2)p({\mathbf{\alpha }}_m,{\sigma }_m^2)$ , where $p({\mathrm{D}}_m|{\mathbf{\alpha }}_m,{\sigma }_m^2)$ is the evidence conditional on ${\mathbf{\alpha }}_m$ and ${\sigma }_m^2$ , and $p({\mathbf{\alpha }}_m,{\sigma }_m^2)$ is the prior PDF of ${\mathbf{\alpha }}_m$ and ${\sigma }_m^2$ . As the maximum a posteriori estimation can be well approximated by the maximum evidence estimation, the optimal values for ${\mathbf{\alpha }}_m$ and ${\sigma }_m^2$ can be determined by considering the following optimization ⁵⁷

$$[{\hat{\mathbf{\alpha }}}_m,{\hat{\sigma }}_m^2]=\underset{{\mathbf{\alpha }}_m\in {\mathbb{R}}^{N_b},{\sigma }_m^2\in {\mathbb{R}}^{+}}{\arg max}{\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)$$

(8)

where ${\mathfrak{L}}_m({\alpha }_m,{\sigma }_m^2)\equiv \ln p({\mathrm{D}}_m|{\alpha }_m,{\sigma }_m^2)$ is the logarithm of the evidence of the mth vibrational mode, which can be evaluated as follows ⁵⁷

$$\begin{array}{c}\multicolumn{1}{c}{{\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)=\ln \int p({\mathrm{D}}_m|{\mathbf{\alpha }}_m,{\sigma }_m^2)p({\mathbf{b}}_m|{\mathbf{\alpha }}_m)\mathrm{d}{\mathbf{b}}_m}\\ =-\frac{1}{2}[N\ln 2\pi +\ln |{\mathit{C}}_m|+{\mathit{Y}}_m^T{\mathit{C}}_m^{-1}{\mathit{Y}}_m]\hfill \end{array}$$

(9)

with the matrix ${\mathbf{C}}_m\in {\mathbb{R}}^{N\times N}$ given by

$${\mathbf{C}}_m={\sigma }_m^2{\mathbf{I}}_N+{\mathbf{X}}^{\mathrm{T}}{\mathbf{A}}_m^{-1}\mathbf{X}$$

(10)

where ${\mathbf{I}}_N$ is the $N\times N$ identity matrix. The optimization in equation (8) is $(N_b+1)$ -dimensional, which is infeasible when the number of components of the feature vector $N_b$ is large. Here, separation optimization strategy is proposed for efficiency enhancement. Consider the following decomposition for the matrix ${\mathbf{C}}_m$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{C}}_m={\sigma }_m^2{\mathbf{I}}_N+\sum _{k=1}^{N_b}{\alpha }_{m,k}^{-1}{\mathit{X}}_k{\mathit{X}}_k^T}\\ ={\mathbf{C}}_{m,-i}+{\alpha }_{m,i}^{-1}{\mathbf{X}}_i{\mathbf{X}}_i^{\mathbf{T}}\hfill \end{array}$$

(11)

where ${\mathbf{C}}_{m,-i}={\sigma }_m^2{\mathbf{I}}_N+\sum _{k\ne i}{\alpha }_{m,k}^{-1}{\mathit{X}}_k{\mathit{X}}_k^T$ represents ${\mathbf{C}}_m$ excluding the ith term in the sum ${\alpha }_{m,i}^{-1}{\mathbf{X}}_i{\mathbf{X}}_i^{\mathbf{T}}$ . Then, $|{\mathbf{C}}_m|$ and ${\mathbf{C}}_m^{-1}$ can be rewritten as follows ⁵⁸

$$\left|{\mathbf{C}}_m\right|=\left|{\mathbf{C}}_{m,-i}\right||1+{\alpha }_{m,i}^{-1}{\mathbf{X}}_i{\mathbf{C}}_{m,-i}^{-1}{\mathbf{X}}_i^{\mathbf{T}}|$$

(12)

$${\mathbf{C}}_m^{-1}={\mathbf{C}}_{m,-i}^{-1}-\frac{{\mathbf{C}}_{m,-i}^{-1}{\mathbf{X}}_i{\mathbf{X}}_i^{\mathbf{T}}{\mathbf{C}}_{m,-i}^{-1}}{{\mathbf{\alpha }}_{\mathbf{m}\mathbf{,}\mathbf{i}}+{\mathbf{X}}_i^{\mathit{T}}{\mathbf{C}}_{m,-i}^{-1}{\mathbf{X}}_i}$$

(13)

Let ${\mathbf{\alpha }}_{m,-i}$ denote the hyperparameter vector ${\mathbf{\alpha }}_m$ excluding ${\alpha }_{m,i}$ . By plugging equations (12) and (13) into equations (9) and (10), ${\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)$ can be decomposed into two parts

$${\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)={\mathfrak{L}}_m({\mathbf{\alpha }}_{m,-i},{\sigma }_m^2)+{\mathfrak{l}}_m({\alpha }_{m,i},{\sigma }_m^2)$$

(14)

The first term ${\mathfrak{L}}_m({\mathbf{\alpha }}_{m,-i},{\sigma }_m^2)$ is the log-evidence removing ${\alpha }_{m,i}$ (and thus ${\mathrm{b}}_{m,i}$ and ${\mathbf{X}}_i$ ) from the model of the mth mode

$${\mathfrak{L}}_m\left({\alpha }_{m,-i},{\sigma }_m^2\right)=-\frac{1}{2}\left[N\ln 2\pi +\ln \left|C_{m,-i}^{-1}\right|+Y_m^T{C}_{m,-i}^{-1}{Y}_m\right]$$

(15)

The second term $\mathfrak{l}({\alpha }_{m,i},{\sigma }_m^2)$ , related to ${\alpha }_{m,i}$ and ${\sigma }_m^2$ , is given by

$${\mathfrak{l}}_m\left({\alpha }_{m,i},{\sigma }_m^2\right)=\frac{1}{2}\left[\ln {\alpha }_{m,i}-\ln \left({\alpha }_{m,i}+S_{m,i}\right)+\frac{Q_{m,i}^2}{{\alpha }_{m,i}+S_{m,i}}\right]$$

(16)

with

$$S_{m,i}={\mathbf{X}}_i^{\mathbf{T}}{\mathbf{C}}_{m,-i}^{-1}{\mathbf{X}}_i$$

(17)

$$Q_{m,i}={\mathbf{X}}_i^{\mathbf{T}}{\mathbf{C}}_{m,-i}^{-1}{\mathbf{Y}}_m$$

(18)

As the ith hyperparameter ${\alpha }_{m,i}$ has been isolated from the log-evidence ${\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)$ , the optimal value of ${\alpha }_{m,i}$ can be obtained by taking the partial derivative of ${\mathfrak{l}}_m({\alpha }_{m,i},{\sigma }_m^2)$ in equation (16) with respect to ${\alpha }_{m,i}$

$$\frac{\partial {\mathfrak{l}}_m({\alpha }_{m,i},{\sigma }_m^2)}{\partial {\alpha }_{m,i}}=\frac{{\alpha }_{m,i}^{-1}{S}_{m,i}^2-(Q_{m,i}^2-S_{m,i})}{2{({\alpha }_{m,i}+S_{m,i})}^2}$$

(19)

Thus, the optimal value of the ith hyperparameter ${\hat{\alpha }}_{m,i}$ is achieved when $\partial {\mathfrak{l}}_m({\alpha }_{m,i},{\sigma }_m^2)/\partial {\alpha }_{m,i}=0$ under the constrain that ${\alpha }_{m,i}\ge 0$ (because ${\alpha }_{m,i}$ is the diagonal element of the precision matrix ${\mathbf{A}}_m$ ). Thus, there are two possible values for ${\hat{\alpha }}_{m,i}$

$${\hat{\alpha }}_{m,i}=\{\begin{array}{cc}\infty ,\hfill & \mathrm{if}{Q}_{m,i}^2\le S_{m,i}\hfill \\ \multicolumn{1}{c}{S_{m,i}^2/(Q_{m,i}^2-S_{m,i}),}& \multicolumn{1}{c}{\mathrm{if}{Q}_{m,i}^2>S_{m,i}}\end{array}$$

(20)

Based on the updated value of ${\mathbf{\alpha }}_m$ , the optimal prediction-error variance is re-estimated by considering the derivative of ${\mathfrak{L}}_m({\mathbf{\alpha }}_m,{\sigma }_m^2)$ with respect to ${\sigma }_m^2$ ^53,54

$${\hat{\sigma }}_m^2=\frac{{({\mathbf{Y}}_m-\mathbf{X}{\hat{\mathbf{b}}}_m)}^{\mathrm{T}}({\mathbf{Y}}_m-\mathbf{X}{\hat{\mathbf{b}}}_m)}{N-N_b+\sum _{i=1}^{N_b}{\hat{\alpha }}_{m,i}{\hat{\mathbf{\Sigma }}}_{m,i}}$$

(21)

where ${\hat{\mathbf{b}}}_m$ is obtained by equation (6); ${\hat{\mathbf{\Sigma }}}_m$ is obtained by equation (7) and ${\hat{\mathit{\Sigma }}}_{m,i}$ is the ith diagonal element of the posterior covariance matrix ${\hat{\mathbf{\Sigma }}}_m$ .

Summary of the proposed approach

The procedures of the proposed algorithm are shown in Figure 4 and are as follows. For pattern recognition of the mth mode, the algorithm starts from a single feature and proceeds by adding, pertaining and/or deleting features.

1. Initialize ${\sigma }_m^2$ . The default value is 20% of the sample variance of the mth mode output ${\mathbf{Y}}_m=[{\omega }_{m,1}^2,\dots ,{\omega }_{m,N}^2{]}^{\mathrm{T}}$ .

2. Select the initial feature ${\mathbf{X}}_i$ . The correlation coefficients between the output ${\mathbf{Y}}_m$ and each feature are calculated. The initial feature is selected as the one which possesses the maximum absolute correlation coefficient.

3. Initialize ${\mathbf{\alpha }}_m$ . Since only one of the initial feature ${\mathbf{X}}_i$ is included, each component of ${\mathbf{\alpha }}_{m,-i}$ is infinity and, hence, ${\mathbf{C}}_{m,-i}={\sigma }_m^2{\mathbf{I}}_N$ . The hyperparameter associated with the initial feature ${\mathbf{X}}_i$ can be estimated using equation (20)

   $${\alpha }_{m,i}=\frac{S_{m,i}^2}{Q_{m,i}^2-S_{m,i}}=\frac{{\mathbf{X}}_i^T{\mathbf{X}}_i}{{({\mathbf{X}}_i^T{\mathbf{Y}}_m)}^2/({\mathbf{X}}_i^T{\mathbf{X}}_i)-{\sigma }_m^2}$$

   (22)

4. Compute ${\hat{\mathbf{b}}}_m={\sigma }_m^{-2}{\hat{\mathbf{\Sigma }}}_m{\mathbf{X}}^{\mathrm{T}}{\mathbf{Y}}_m$ and ${\hat{\mathbf{\Sigma }}}_m=({\sigma }_m^{-2}{\mathbf{X}}^T\mathbf{X}+{\mathbf{A}}_m{)}^{-1}$ by equations (6) and (7) based on ${\mathbf{A}}_m$ and ${\sigma }_m^2$ .

5. Update $S_{m,j}={\mathbf{X}}_j^{\mathbf{T}}{\mathbf{C}}_{m,-j}^{-1}{\mathbf{X}}_j$ and $Q_{m,j}={\mathbf{X}}_j^{\mathbf{T}}{\mathbf{C}}_{m,-j}^{-1}{\mathbf{Y}}_m$ by equations (17) and (18) for $j=1,\dots ,N_b$ . Select a candidate feature ${\mathbf{X}}_j$ , update its associated hyperparameter ${\alpha }_{m,j}$ as follows

6. If $Q_{m,j}^2>S_{m,j}$ and ${\alpha }_{m,j}<\infty$ , then update ${\alpha }_{m,j}=S_{m,j}^2/(Q_{m,j}^2-S_{m,j})$ (implying that ${\mathbf{X}}_j$ has been already included in the model in current iteration);

7. If $Q_{m,j}^2>S_{m,j}$ and ${\alpha }_{m,j}=\infty$ , then include ${\mathbf{X}}_j$ and update ${\alpha }_{m,j}=S_{m,j}^2/(Q_{m,j}^2-S_{m,j})$ .

Update ${\hat{\sigma }}_m^2$ by equation (21) with the updated ${\hat{\mathbf{\Sigma }}}_m$ and ${\hat{\mathbf{b}}}_m$ .

Repeat Steps 4–6 until convergence achieves so the final learning results are obtained.

Figure 4.

The procedures of the proposed algorithm.

Modal frequency-ambient condition pattern recognition

Table 1 shows the weight parameter identification results of the optimal models of the three modes, the optimal values (outside the parentheses), and the standard deviations (inside the parentheses), and the sign “–” indicates that the weight parameter is not included in the optimal model, representing that the associated feature is irrelevant. It is observed that the optimal models of the three modes possess different relevant features shown as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\omega }_1^2=1.92+0.04H_0-0.931T_0^2{H}_0}\\ \multicolumn{1}{c}{+1.014T_0^3{H}_0+0.255T_0^2{H}_0^2}\end{array}$$

(23)

$${\omega }_2^2=2.582+0.266T_0-0.187T_0^2{H}_0+0.358T_0^2{H}_0^2$$

(24)

$$\begin{array}{c}\multicolumn{1}{c}{{\omega }_3^2=3.332-0.033T_0{H}_0-0.931T_0^2{H}_0}\\ \multicolumn{1}{c}{-0.015T_0^4+0.332T_0^2{H}_0^2}\end{array}$$

(25)

Table 1.

Weight parameter identification results of the optimal models of the three modes.

Mode	$b_{m,1}$	$b_{m,2}$	$b_{m,3}$	$b_{m,4}$	$b_{m,5}$	$b_{m,6}$	$b_{m,7}$	$b_{m,8}$
m = 1	1.920 (0.015)	−	0.040 (0.037)	−	−	−	−	−0.931 (0.127)
m = 2	2.582 (0.025)	0.226 (0.069)	−	−	−	−	−	−0.187 (0.126)
m = 3	3.332 (0.013)	−	−	−	−0.033 (0.057)	−	−	−
Mode	$b_{m,9}$	$b_{m,10}$	$b_{m,11}$	$b_{m,12}$	$b_{m,13}$	$b_{m,14}$	$b_{m,15}$	${\sigma }_m^2(\times {10}^{-4})$
m = 1	−	−	−	1.014 (0.144)	0.255 (0.073)	−	−	8.68
m = 2	−	−	−	−	0.358 (0.087)	−	−	6.02
m = 3	−	−	−0.015 (0.012)	−	0.332 (0.061)	−	−	7.34

– indicates that the weight parameter is not included in the optimal model.

Both the temperature and relative humidity exist in all the three optimal models, showing that these two ambient factors influence the modal frequencies of different modes and they should be considered simultaneously in modal frequencies prediction. This conclusion is consistent with that by Xia et al. ⁵ and Yuen and Kuok.^11,12 On the other hand, the optimal models for different modes possess different combinations of the two ambient factors. The reasons are given as follows. In practical monitoring of a target structure under changing ambient and operational conditions, fluctuations of modal frequencies are induced by multiple environmental factors, not only temperature and relative humidity but also other factors, such as wind load, ⁵⁹ rainfall, support reaction, ⁶⁰ and soil–structure interaction,^61,62 and the degrees of participation of different environmental factors to the modal parameters of different modes are possibly not identical. On the other hand, it is worth noting that although only the temperature and relative humidity are utilized in the study, the temperature and relative humidity are the primary effects while other factors (wind load, rainfall, support reaction, soil–structure interaction) are the secondary effects in modal frequencies prediction. This conclusion is supported by checking the normalized residual plot of the three modes shown in Figure 5 that most of the normalized residuals satisfy $\left|{\overline{\epsilon }}_m\right|\le 1.96,m=1,2,3$ , where the threshold “1.96” corresponds to that there is 95% in probability for a standard normal distributed sample falling into the interval [−1.96, 1.96]. Therefore, it can be concluded that the optimal models given in equations (23)–(25) are capable to properly capture the wandering pattern of the modal frequencies under normal weather conditions.

Figure 5.

Normalized residual plot of the three modes ( ${\overline{\epsilon }}_1,{\overline{\epsilon }}_2,\mathrm{and}{\overline{\epsilon }}_3$ are the normalized residuals of modes 1, 2, and 3, respectively).

Note that in Bayesian spectral density approach,^46–48 not only identified values but also associated uncertainties of modal frequencies are available. It is worth noting that unequal uncertainty values for modal frequencies of different days may lead to heterogeneous error pattern, requiring the development of the heterogeneous Bayesian inference and learning framework. ⁶³

Figures 6 and 7 show the squared modal frequencies versus the temperature and relative humidity of the three modes, respectively. The modal frequency–temperature curves of Figure 6 are plotted with the relative humidity evaluated at its sample mean (67.9%), and the modal frequency–relative humidity curves of Figure 7 are plotted with the temperature with its sample mean (24.7°C). The increasing trends of these curves satisfy the correlation analysis results of Figure 3 for different modes. Finally, the design modal frequency contours of the optimal values and the associated uncertainties for the three modes are given in Figure 8. The left three subplots are for the optimal values of modal frequencies given the temperature and the relative humidity for modes 1, 2, and 3, respectively, and the right three subplots are for the corresponding associated uncertainties for modes 1, 2, and 3, respectively. Note that different modes associate with different uncertainty contours because they possess different optimal models. In addition, it can be observed that the uncertainty level of the corner regions in the uncertainty contours is higher than that of the central region. This is due to the non-uniform distribution of the input measurement that most of the measured data points locate in the center region while very few locate in the corner region, especially for the top right and bottom right regions. On the other hand, the non-uniform distribution of the input measurement also reflects the ambient condition pattern of the city for the monitored structure that, high temperature is with intermediate level of relative humidity rather than high or low level of relative humidity.

Figure 6.

The squared modal frequencies versus the temperature of the three modes.

Figure 7.

The squared modal frequencies versus the relative humidity of the three modes.

Figure 8.

Design modal frequency contours of the optimal values (left three subplots) and the associated uncertainties (right three subplots) for the three modes.

Conclusion

In this article, a novel machine-learning algorithm is proposed to automatically select relevance features in modal frequency-ambient condition pattern recognition based on structural dynamic response and ambient condition measurement. In the traditional feature selection approaches, 3 × (2¹⁵-1) candidates in total are required to obtain the optimal model classes for the three modes based on the 15-component full extracted feature vector, which requires large computational effort. On the other hand, in the proposed approach, the optimal model classes for modal frequency predictions of different vibrational modes of the structure are determined by first introduction of the hyperparameter in the ARD prior for the weight parameter vector, which controls the relevancy of different features in the prediction model, then conducting hyperparameter optimization by maximum evidence estimation. During the optimization process, the irrelevance components of the weight parameter vector as well as the corresponding features are automatically pruned out while the relevance terms are retained. The proposed algorithm is utilized for structural health assessment for a reinforced concrete building based on 1-year daily measurements. It turns out that the optimal model classes of different vibrational modes of the structure are capable to capture the modal frequency-ambient condition patterns of different modes. In addition, the design modal frequency contours of the optimal values and the associated uncertainties are given, which provides an efficient tool for modal frequency prediction in future analysis.