Missing data is a common problem in research studies that involve human subjects and technologies for data collection. They can reduce statistical power of a study and lead to erroneous conclusions [6]. To potentially mitigate this issue, the sample size for data collection is typically increased. However, this is not always possible due to research design, limitations in budget and human resources. It is not always feasible to regenerate the data by repeating the experiment, as it can be costly and time-consuming. There are existing methods of handling missing data in literature such as simply omitting observations with the missing data [7], regression imputation [8], mean substitution [9], replacing the missing data with the last observed values [10], and estimating the missing data using conditional distribution of the other variables [11]. While useful, these methods may not be optimal for tackling data missing in WMSD risk assessment. Simply omitting observations with missing data typically decreases the sample size and may adversely affect the statistical power [7]. Regression imputation predicts a missing value from other variables, but it barely adds any new information except increasing the sample size and compromising the standard error [8]. If there is a great inequality in the proportion of missing values of different variables, mean substitution may lead to inconsistent biases and underestimation of errors [9]. Replacing missing data with the last observed values assumes that there will be no changes in the outcome [10]. Using conditional distribution of the other variables becomes inappropriate when a large amount of data is missing, as in this situation, the relationship among the variables needs to be properly computed [7,11].

Moreover, in real life scenarios, a phenomenon can be described and characterized with a set of factors. Each of these factors is referred to as the ‘dimension’ of that phenomenon [12]. When a research problem includes multiple dimensions, it is considered ‘multidimensional’. This is often the case in WMSD risk studies, in which risk exposure data can be collected in terms of different measurements, often referred to as risk indicators, and working conditions for the purpose of in-depth understanding of risks. The working conditions may include various work settings and postures, and the risk indicator measurements may include kinematics, kinetics and electromyography (EMG) measurements. Quite often, multiple risk indicators can provide a more detailed insight into the worker’s risk-inducing behavior, compared to what a single risk indicator can provide as these multiple risk indicators may be correlated. In WMSD risk studies that involves analyzing the effects of working conditions on risk indicators, the working conditions and the risk indicators can be termed as a ‘dimension’ of the risk phenomenon. This type of data can be considered as high-dimensional data. When data is missing in multiple high-dimensional risk-related datasets, the existing methods will not be suitable as the interrelation among different risk indicators and potential risk factors across multiple datasets may not be well captured. Fuzzy modeling has been used in multidimensional missing data imputation [13]. However, for high-dimensional datasets where capturing the latent relationship among various dimensions is essential during imputation, the performance of Fuzzy algorithms degrades [14,15]. Besides, the performance of Fuzzy models in dealing with multiple datasets is yet to be tested. In contrast, data fusion can be an efficient method in this regard that integrates information based on the interrelation among the factors and risk indicators in such a way that it can produce a more complete representation of the measurements of the risk indicators, compared to that of an incomplete dataset [16].

In the construction industry, data fusion methods have been used for automated identification, location estimation, dislocation detection of construction materials in jobsites [17,18], automated progress tracking of construction projects [19], structural health monitoring [20], and damage identification of civil structures [21]. In construction ergonomics, a computationally efficient approach using data fusion was developed to recognize construction workers’ awkward postures [22]. Fusion of data from continuous remote monitoring of construction workers’ location and physiological status was used to identify safe and unsafe behaviors of construction workers [23]. Fusion of spatio-temporal and workers’ thoracic posture data has been done for understanding the worker’s activity type for productivity assessment [24]. A position and posture data fusion method was proposed for evaluation of construction workers’ behavioral risks [25]. However, no study has been done that uses data fusion of incomplete risk-related datasets for assessing WMSD risks.

To integrate multidimensional data, it is essential to learn the relationships between those dimensions to understand the multidimensional nature of a phenomenon [12]. There are currently several existing approaches for analyzing multiple datasets. Zheng et al. [26] summarized the methods of fusing multiple datasets into three categories. The first category is stage-based fusion methods, where different datasets are used at different stages of data mining tasks [27]. The second category is deep learning-based fusion methods that use a neural network to extract original features from multiple datasets and learn a new representation of those features for classification and prediction purposes [28]. The third category is semantic meaning-based fusion methods that identify the association between different features across multiple datasets and carry the semantic meaning, i.e., the latent relationship between the features during fusion [26]. Similarity-based data fusion is one type of the semantic meaning-based fusion methods, in which similarity between the features are measured from multiple datasets. These similarities can leverage the correlation among the features collectively and help obtain any missing information of a feature based on the information available for another feature. Based on the similarities, different datasets can be fused [26]. This study is interested in fusion of risk-related datasets containing different features that represent WMSD risk factors in construction and are inherently related to each other. Understanding the latent relationship between different features across multiple datasets plays a vital role in proper fusion. Also, learning the similarities between those features is useful to impute the missing values. As a result, the similarity-based data fusion methods lend themselves well to tackling the fusion problem in this study.

Tensor decomposition is one type of similarity-based data fusion methods [29]. Tensors are generalizations of matrices to higher dimensions, and are powerful to model multidimensional data [30]. Tensors are multidimensional arrays of numerical values that are widely used in different applications in data analysis and machine learning, including filling in missing values [31], anomaly detection [32], object profiling [33], discovering patterns [34], and predicting evolution [35]. Matrix factorization was previously used in filling out missing data [36]; however, it only works with two-dimensional data and may not be applicable to high-dimensional data like those used in WMSD risk studies. Tensor decomposition is a useful tool to deal with high-dimensional datasets. It accurately extracts the correlations among various dimensions from different datasets and learns the latent structures and collaborative relationships among the dimensions to approximate the pattern of the data [37].

With increases in dimensions of the datasets, the number of the elements in a tensor also increases. This increase in elements makes the dataset difficult to deal with in terms of computational and memory requirements. By decomposing the high-dimensional data presented in form of a tensor, the problems associated with high-dimensionality can be alleviated or even removed [38]. Also, tensor decomposition can be used as an efficient tool for missing value prediction [39,40]. Though tensor decomposition has been abundantly applied in areas such as scientific computing, signal processing, and social media data analysis, its use in construction industry-related datasets is limited. One application of tensor decomposition with construction datasets was concerned with awkward posture recognition in which worker’s motion data was presented as high order tensor [22]. However, this technique has yet to be established as an alternative method of treating imperfect experimental data to provide meaningful and accurate risk assessment results in general WMSD risk assessment studies.

The two most widely used tensor decomposition algorithms are canonical polyadic decomposition (CPD) and Tucker decomposition (TD). CPD is generally advised for use with latent parameter estimation, while the TD is suggested in use of subspace estimation, compression, and dimensionality reduction [5]. As the objective of this study is to fuse two incomplete datasets based on the latent correlations among all dimensions of those datasets, CPD is more suitable than TD in fulfilling the purpose of this study. Moreover, CPD is very effective in capturing the interactions among various dimensions of a high-dimensional datasets and therefore, it can effectively be used for imputing missing data [31]. Considering many other tensor decomposition techniques are based on CPD and TD and CPD is more suitable for this study, the following subsection will only discuss CPD for tensor decomposition.

For WMSD risk assessment in construction, the effects of various work-related factors on specific risk indicators are often measured. Therefore, a typical WMSD risk-related dataset contains information such as work settings/experimental conditions and specific risk indicators. Often, a WMSD risk may not be characterized with a single type of risk indicator and in that case, multiple types of WMSD risk indicators are measured, e.g., kinematics (flexion, abduction/adduction, internal/external rotation), muscle activity data recorded by EMG, and kinetic data recorded by force plates, and can be available in multiple datasets.

To describe the proposed method, consider two datasets DS1 and DS2 (denoted for Incomplete Dataset 1 and Incomplete Dataset 2 in Fig. 2). Each dataset contains two different types of WMSD risk indicators collected from several subjects, investigating various work-induced risks for a number of work settings (e.g., working technique, working posture, working condition). This is common practice of WMSD risk assessment studies in construction, where different types of risk indicators are often measured from the same group of people and the same work settings for an in-depth understanding of a specific type of risk. In this case, a data value in DS1 or DS2 represents a value of a risk indicator measured from a subject at certain work settings. For example, to measure the knee WMSD risk, two common types of risk indicators are measured – awkward knee rotations and maximum EMG of knee postural muscles. Awkward knee rotations may include flexion, abduction/adduction, and internal/external rotations. Maximum EMG of knee postural muscles may include measurements of maximum flexor and extensor muscle activations. The datasets can be represented in a tabular form as shown in Table 1.

In Table 1, each dataset contains a type of risk indicator measurements collected for several subjects at a number of work settings. The column subject contains subjects’ IDs. The columns setting₁, setting₂, …, setting_q contain arrangements of q work settings at which measurements of m risk indicators in columns ind₁¹, ind₁², …, ind₁^m are collected in DS1, and n risk indicators are collected in columns ind₂¹, ind₂², …, ind₂ⁿ in DS2. For example, the column setting₁ can represent working postures that have two arrangements – static and dynamic. The column setting₂ can represent slopes of the working site that have three arrangements – 0°, 15° and 30°. So, in the column setting₁, the values are either ‘static’ or ‘dynamic’ and in the column setting₂ the values are either ‘0°’, ‘15°’ or ‘30°’. If a study involves only these two settings, then the value of q will be 2. Similarly, the columns ind₁¹, ind₁², …, ind₁^m can contain kinematics measures (flexion, abduction/adduction, internal/external rotation) and ind₂¹, ind₂², …, ind₂ⁿ can contain EMG measures of several muscles. To be more specific with an example, in DS1, a value of any of the m risk indicator columns can be a certain measurement value of a risk indicator (e.g., flexion) captured for certain subject at certain arrangements of q work settings. Note that in both datasets, the values of columns of subject and setting₁, setting₂ …, setting_q are identical; the only difference is the measurements of the risk indicators. These datasets become incomplete when a significant amount of risk indicator measurements are not captured during data collection process.

Since the datasets contain multiple risk indicator measurements for the same combination of subjects and settings, the risk indicator measurements from both datasets are concatenated, yielding a new dataset (DS3) as illustrated in Table 2. This process is called early integration data fusion [36].

In Table 2, each risk indicator measurement in DS3 is collected for certain subject at different arrangements of q work settings. Thus, ‘subject’ and each ‘work setting’ are considered as dimensions of each risk indicator, resulting in a total of (1 + _q) dimensions. For multiple risk indicators, this method allows datasets to be represented as a ‘multi-dimensional’ or ‘multi-modal’ tensor, where each cell of the tensor contains a value of a certain risk indicator, measured for a subject at certain arrangements of q work settings. In this study, the ‘subject’, each ‘work setting’, and the entire set of ‘risk indicators’ can be considered as dimensions of the tensor which has a total of (1 + q + 1) dimensions.

To illustrate this tensor representation with a simplified example, consider only one work setting (e.g., working postures) denoted by ‘setting’. In this case, DS3 can be represented as a three-dimensional tensor (subject × risk indicators × setting) (M_{i, j, k}) ∈ ℝ^s×r×t (Fig. 3), where i, j and k correspond to dimensions ‘subject’, ‘risk indicators’ and ‘setting’ respectively; and s, r and t are their sizes where i∈{1,2, …s}, j∈{1,2, …r} and k∈{1,2, …t}. Each slice along the setting axis in Fig. 3 represents a matrix (M_{i, j}) ∈ ℝ^s×r given a setting arrangement k.

Following tensor construction, CPD is applied to decompose the tensor into factor matrices. As mentioned earlier, CPD learns the latent structures and collaborative relationships among the dimensions of a tensor. Thus, each factor matrix represents the latent relationships between the corresponding dimension and all the other dimensions captured by latent factor R (previously discussed in Section 2.3.1). Before applying CPD, the latent factor R should be computed properly to ensure unique mapping of the original tensor to the decomposed tensor in the form of factor matrices. In the above example, (M_{i, j, k}) ∈ ℝ^s×r×t is associated with three dimensions – ‘subject’, ‘risk indicators’ and ‘setting’; hence, three factor matrices are obtained after decomposition, denoted as FM₁ ∈ ℝ^s×R, FM₂ ∈ ℝ^r×R, FM₃ ∈ ℝ^t×R, as shown in Fig. 4. Here, the factor matrix FM₁ obtained for the dimension ‘subject’ represents the latent relationships between the subject and the risk indictors collected at different arrangements of the setting. FM₂ is obtained for the dimension ‘risk indicators’ and represents the latent relationships between the risk indicators and the subjects at different arrangements of the setting for which those risk indicators were collected. FM₃ is obtained for the dimension ‘setting’ and represents the latent relationships between different arrangements of the setting and the risk indicators collected for the subjects at those arrangements of the setting. CPD leverages all the inter-dimensional correlations among all the dimensions of a tensor and performs data fusion based on those correlations to create a reconstructed tensor. The structures of the factor matrices and the method of computing R will be discussed further in “Implementation and results”.

After tensor decomposition, the resulting factor matrices are used to reconstruct a new fused tensor where the missing values are filled and values of the WMSD risk indicators are fused based on the inter-dimensional correlations. In the prior example, such tensor can be denoted as (M′i,j,k)∈ℝs×r×t, which is computed using the following equation: (4)M′i,j,k=∑r=1RFM1r∘FM2r∘FM3r where FM_1r, FM_2r, FM_3r are the column vectors of FM₁, FM₂ and FM₃ respectively, R is the latent factor and ‘∘’ is the vector outer product. Using the newly reconstructed tensor, values of the WMSD risk indicators for different subjects at different settings are extracted and reorganized into a tabular form, readily available for risk assessment.

The current study considered two risk-related datasets that were collected from the authors’ prior human subject laboratory experimental studies. These studies assessed work-related factors for knee WMSDs among residential construction roofers who work on sloped environments. One dataset contains calculated knee rotation (kinematics) data, representing five knee rotational angles – flexion, abduction, adduction, internal and external rotation [42]. The second data set contains EMG data, describing muscle activation of five knee postural muscles – biceps femoris, recus femoris, semitendinosus, vastus lateralis and vastus medialis [43]. Awkward knee rotations and maximum muscle activation have been identified as two quantitative risk indicators of WMSDs in low extremities [44,45]. The five knee rotation angles represent the WMSD risk associated with awkward and extreme kneeling postures, while the EMG data represent the WMSD risk associated with heightened activation of knee postural muscles that might cause knee joint overloading. Both types of data were collected from the experimental study in which 9 subjects simulated the shingle installation roofing task at three roof slopes (0°, 15° and 30°) with two kneeling postures (static and dynamic). Each participant performed the task for five trials. Although the experiment was designed and implemented carefully, both datasets initially collected from the experiment had missing values. About 2% data were missing in the knee rotation dataset and about 20% data were missing in the EMG dataset at random. In this study, observations with missing data were intentionally removed from the initial datasets for the sake of obtaining complete datasets for evaluation. In the knee rotation dataset, each data point represents the maximum angle value of a knee rotation for a certain subject at a specific slope, posture and trial. In the EMG dataset, each data point represents the maximum normalized EMG value of a knee postural muscle of a certain subject at a specific slope, posture and trial. Since only maximum measurements in both datasets were collected as risk indicators, the two datasets could be easily synchronized.

To evaluate the performance of the proposed method in dealing with missing data, this study was designed to fuse knee rotation and EMG datasets for different proportions of missing values using the CPD method. A proportion of data was randomly removed from both knee kinematics and EMG datasets with percentages of missing data as shown in Table 3. Therefore, fourteen incomplete datasets—seven for knee rotations and seven for EMG—were constructed with different portions of missing values.

The incomplete rotation and EMG datasets were then used for tensor construction. As the knee rotation and EMG data were collected from the same subjects at the same work settings (i.e., roof slope, working posture, and trial), they could be combined by the early integration method [46]. In this method, the columns of these two types of risk indicators were concatenated to form a single set of length ten vectors which was referred as ‘risk indicators’ in this study. For tensor construction, the following forty-nine incomplete datasets as shown in Table 4 were considered.

An N dimensional tensor T_N ∈ℝ^{I₁×I₂×……. ×I_N} has size (T_N) = I₁ × I₂ × ……… × I_N, where, I_i is the size of its i^th dimension [31]. As the risk indicators were measured from multiple subjects at different working postures, roof slopes and multiple trials, the collected data could be represented as a five-dimensional sparse tensor T ∈ ℝ^{I₁×I₂×I₃×I₄×I₅}, where I₁ = No. of subjects (total 9), I₂ = No. of risk indicators [EMG and kinematics; total 10 (5 knee rotation angles and 5 knee muscle EMG)], I₃ = No. of trials (total 5), I₄ = No. of postures [total 2 (static and dynamic)], I₅ = No. of slopes [total 3 (0°,15°,30°)]. In short, the tensor T maps the knee rotations and EMG measurements of nine subjects, obtained from five trials, performed on three roof slopes, using two postures. Tensor T was referred to as ‘incomplete tensor’ in this study.

For a certain posture and slope, the sparse tensor T is denoted as T_{i, j} which represents the values of the risk indicators obtained from five trials performed by each subject at that roof slope and posture. In T_{i, j}, i ∈ {1, 2}, with 1 and 2 representing the static and dynamic postures respectively; j ∈ {1,2,3}, with 1, 2 and 3 representing the 0°, 15° and 30° slopes respectively. Then, the five-dimensional tensor T can be illustrated in Fig. 5.

Similarly, the original datasets collected from the previous experiments were constructed to a five-dimensional tensor X ∈ ℝ^{I₁×I₂×I₃×I₄×I₅} as the benchmark data, which was referred to as the “original tensor” in this study and was used for validation of performance of the fusion method.

Once a tensor T was constructed, the CPD tensor decomposition was applied to decompose the tensor T into five factor matrices, representing information in each dimension. The factor matrices were represented as matrices A, B, C, D, and E, providing latent information in dimensions of subjects, risk indicators, trials, postures, and slopes, respectively.

CPD factorizes the five-dimensional sparse tensor T ∈ ℝ^{I₁×I₂×I₃×I₄×I₅} as: (5)T≈∑r=1Rar∘br∘cr∘dr∘er≡〚A,B,C,D,E〛 where a_r ∈ ℝ^I₁, b_r ∈ ℝ^I₂, c_r ∈ ℝ^I₃, d_r ∈ ℝ^I₄, e_r ∈ ℝ^I₅, for r = 1, …, R. Factor matrix A = [a₁,a₂………a_R] ∈ ℝ^I₁×R, B = [b₁, b₂………b_R] ∈ ℝ^I₂×R, C= [ c₁,c₂………c_R] ∈ ℝ^I₃×R, D = [ d₁,d₂……… d_R] ∈ ℝ^I₄×R, and E= [e₁, e₂…e_R] ∈ ℝ^I₅×R are collections of all R vectors of a_r, b_r, c_r, d_r, e_r respectively.

Fig. 6 illustrates the process of the CPD tensor decomposition.

Eq. (5) can be represented as: (6)T≈∑RAr∘Br∘Cr∘Dr∘Er

Factor matrices A, B, C, D and E are computed by solving the following optimization equation: (7)arg min‖T−∑RAr∘Br∘Cr∘Dr∘Er‖

To solve Eq. (7), the alternating least squares algorithm was applied using an optimization-based program cpd_als in Tensor Toolbox from MATLAB [47].

Once the tensor was decomposed into factor matrices, these factor matrices were used to reconstruct a new tensor from its decomposed state. This reconstructed tensor is an approximate mapping of the sparse tensor T, which was referred to as ‘fused tensor’ and represented as T ′ ∈ ℝ^{I₁×I₂×I₃×I₄×I₅}, calculated by: (8)T′=∑r=1RAr∘Br∘Cr∘Dr∘Er

In this tensor, all the missing values were imputed, and the risk indicator values were reconstructed. Tensorlab (a MATLAB package for tensor computation) was used for the imputation and reconstruction [50]. Note that the factor matrices computed by Eq. (7) ensured the minimal reconstruction error during the imputation and reconstruction of the risk indicators. The reconstructed risk indicators were then extracted to a spread sheet, readily for use in WMSD risk assessment.

To allow for reliable observations, the procedures in Subsections 5.2 to 5.5 (except for construction of the five-dimensional tensor X from the original datasets) were repeated three times. In each repeat, the random removal of certain portion of data as designed earlier led to different preservation in each dataset. Three repeats ensured a two-thirds (67%) probability that the averaged results were more accurate than a single experiment while maintaining time-efficiency in implementation.

The performance of the data fusion method was evaluated by comparing the reconstructed data to the original data using two statistical measures—root mean square error (RMSE) and mean absolute error (MAE). RMSE and MAE were used to measure the standard error and average magnitude of the error of the fused tensor, respectively [51,52].

The RMSE was calculated using the following equation: (9)RMSE=1N[∑i1=1I1∑i2=1I2∑i3=1I3∑i4=1I4∑i5=1I5(Xi1,i2,i3,i4,i5−T′i1,i2,i3,i4,i5)2]

Here, X_{i1, i2, i3, i4, i5} is an element (a value of a risk indicator of a certain subject at a certain slope, posture and trial) in the original tensor X and T′_{i1, i2, i3, i4, i5} is an element in the re-constructed tensor (fused dataset). N is the total number of cells of the tensor which was computed as N = I₁ × I₂ × I₃ × I₄ × I₅.

The MAE was calculated using the following equation: (10)MAE=1N[∑i1=1I1∑i2=1I2∑i3=1I3∑i4=1I4∑i5=1I5|(Xi1,i2,i3,i4,i5−T′i1,i2,i3,i4,i5)|]

The performance of the data fusion method was also evaluated by comparing the risk assessment results obtained from tensor T′ to those obtained from tensor X collected in previous experimental studies. In this comparison, consistency of the effects of roof slope and working posture on five knee rotation angles and five knee postural muscles’ EMG between the original and fused datasets were considered as a performance metric. In previous studies, the authors explored if the work-related factors (i.e., roof slope and working posture) are associated with knee WMSDs among construction roofers [42,43]. Those studies considered the original tensor (X) for risk assessments, whereas in this study, the fused tensor (T′) was considered for assessing the similar risk to see if the risk assessment results were consistent. This way, the performance of the fusion method could be evaluated.

The consistency (Con) was computed using the following equation: (11)Con=∑i=1Pf(mi,m′i)P×100. where, P = Total number of possible effects of roof slope, working posture and their interactions on five knee rotation angles and EMG of five knee postural muscles [total 30 = 3 factors (slope, posture, slope-posture interaction) × 10 response variables (5 knee rotations and 5 EMG measurements)].

m_t denotes the effects of the factors (slope and posture) on the response variables obtained from the original tensor (X).

m′_i denotes the effects of the factors (slope and posture) on the response variables obtained from the fused tensor (T′).

All the resulting measurements of the performance metrics were averaged over the three trial repetitions.

Table 5 summarizes the results, including the RMSE, MAE, consistency (Con), number of rank-1 tensors R (latent factor) and core consistency values computed for the knee rotation and EMG datasets with different proportions of missing values.