A total of 156 coal mine dust samples were collected from nine active coal mines using standardized personal sampling trains as per CFR 30, Part 70.2. Coal mine dust samplers were worn by NIOSH personnel in each active mine and configured to collect only the respirable fraction of airborne particulate matter onto pre-weighed 37 mm-diameter PVC filters secured in sealed two- or three-piece cassettes (MSA, Inc.). Coarse particulates were removed prior to reaching the filter using a 10 mm Dorr–Oliver cyclone (50% cut point at 3.5 μm aerodynamic diameter).³¹ Sampling pumps were operated at a flow rate of 2.0 L·min^–1 for a standard eight-hour shift (Zefon, Inc.).

A total of 87 samples were collected from one limestone, two sandstone, and one silver mine using sampling trains standardized for MNM mines.³² Samplers were configured identically to coal mines with the exception of a sampling pump flow rate of 1.7 L·min^–1. After collection, all cassettes were removed from samplers and sealed in plastic bags for gravimetric analysis.

Gravimetric analysis was conducted according to NIOSH method 600.³³ The difference between filter pre- and post-weight was used to determine respirable dust mass (milligram, mg). Silica masses were also determined using NIOSH method 7500 in the case of the MNM samples and the NIOSH method 7603 in the case of coal dust samples. The distribution of sample mass according to mine type is presented in Table I.

A Bruker Alpha FT-IR spectrometer operating in transmission mode was used to measure mine dust spectra on PVC filters. The filters were mounted in a stainless-steel holder in the instrument sampling compartment such that the infrared beam interrogated the center of the sample at normal incidence.³⁰ The instrument was configured to operate at 4 cm^–1 resolution and averaging 40 scans per sample. A spectrum was acquired using a room temperature pyroelectric detector in approximately one minute once the scan sequence was initiated.

Interferograms were zero-filled prior to Fourier transformation at a factor of two and apodized using the Blackman–Harris three-term function. Sample transmittance was ratioed against blank filter transmittance to adjust the measurement for filter and chamber background. The absorption spectrum was calculated as the negative decadic logarithm of this ratio. As only a single blank filter was used as background for all samples some artifacts were present in the absorption spectra, conditional upon deviations in filter thickness between the sample spectrum and reference blank. The final spectrum contained 2542 absorption channels over a wavenumber range of 4000–400 cm^–1.

Spectra from the thirteen mines were collected into a design matrix for multivariate calibration. This design matrix is organized such that the 243 rows are sample observations (N), and the 2542 columns are the input variables/predictors supplied to the calibration models (p). This N×p design matrix may be presented in matrix and vector notation as: (1) X=(x1,x2,…,xN−1,xN)Tfori=1,…,N, where X is the design matrix comprised of a collection of N vectors of length p. The ith sample (row) is regularly indicated as xi throughout this work as context dictates. Furthermore, the dependent/response variable (y) of gravimetric mass measurements is represented as an N-length vector with elements (yi) paired to the appropriate rows in the design matrix (xi). Following the partitioning of samples into training and testing sets, the subscript “c” or “t” is used to denote affiliation with the calibration and test set, respectively.

Second derivative spectra were generated from absorption spectra using a second-order Savitzky–Golay filter with a 21-point window.³⁴ Second derivative spectroscopy has proven beneficial in minimizing the impact of background absorption on multivariate calibrations, particularly those that utilize dust-laden PVC filter spectra.²³ The design matrix presented in Eq. 1 was regenerated following second derivative transformation with the number of rows and columns remaining unchanged.

A principal component analysis (PCA) was next performed on mean-centered derivative spectra to ascertain the predominant absorption variations, clusters present among spectra, and to identify potential outliers.³⁵ The principal components explaining at least 95% of the variance in the spectra were considered for the subsequent exploratory analysis. Exploratory analysis involved plotted eigenvectors against wavenumbers whereby infrared vibrations were assigned to features (minima) known to absorb in mine spectra.^36,37 Interpreting eigenvectors was aided by studying PC-score plots that were pseudo-colored using qualitative information (e.g., mine type, sampling location) about the spectra.³⁸

Samples were then grouped by site identifier and sorted by sample mass. Every third sample in each of the 13 groups (9 coal + 4 MNM sites) was placed in the test set yielding 156 and 87 samples for training/calibration and testing, respectively. Partitioning samples this way ensured that the mass distribution of the training and test samples were as similar as possible. This is particularly important as the median mass and range associated with each type of mine varied widely (Table I).

Partial least squares regression was chosen as the basis to compare subsequent models as it is arguably the de facto standard for multivariate spectroscopic calibrations.³⁹ Moreover, PLS regression has proven accurate and precise in determining aerosol species collected on polymeric filters in mine dusts and atmospheric samples.^{19,21,23,38,40–43} The popularity of PLS stems from its “optimal” use of infrared absorption for calibration as well as the ease in visualizing the key relationships within and between the spectra.⁴⁴ Solution optimality follows from an orthogonal transformation of the spectroscopic predictors (i.e., columns of X), constrained to maximize the covariance between the response variable and spectra. Orthogonal transformation solves the collinearity problem in spectra-based calibrations while covariance maximization ensures that components are information rich in terms of the response.⁴⁵

Five-fold interleaved cross validation was used to select the number of PLS components (i.e., latent variables or LVs). Training samples were organized into groups (by site) and sorted to ensure that the cross-validation training and “test” sets exhibit analogous mass distributions during model selection. The number of PLS components were selected by minimizing the root mean square error of cross-validation.

Regression diagnostic plots aided in assessing heterogeneity in the sample data, the potential impact on calibration, and thus motivated the use of MoE models. First, each sample’s Q residual and Hotelling T2 value (i.e., squared Mahalanobis distance) are plotted, comparing the residual/unused spectral variance to the spectral variance used by the PLS model, respectively.⁴⁶ Critical values were calculated for both measures to flag extreme samples (α=0.05). Calculating a critical limit is straightforward for T2 values as they are assumed to follow a chi-squared distribution.⁴⁶ The Wilson–Hilferty approximation estimates the critical value for Q residuals as they follow a noncentral chi-squared distribution.^47,48

A second diagnostic plot compared Cook’s distance (D(i)) to Q residuals to visualize residual absorption in terms of a sample’s “influence” on the PLS calibration coefficients.⁴⁹ As a leave-one-out measure, the ith sample’s Cook’s distance was calculated by (i) deleting the ith case, (ii) fitting the PLS model, (iii) calculating the residuals, (iv) estimating the sum of square residuals, and (v) normalizing the result by mean square error of calibration scaled by the regression degrees of freedom (dfPLS). Degrees of freedom were estimated using the kernel PLS algorithm.⁵⁰ Two cutoff values described a sample as either highly influential using an F-distribution (D(i)≥F(0.5,dfPLS,Nc−dfPLS)) or as moderately influential using a common heuristic (D(i)>4/Nc).⁵¹

The univariate response case of the MoE model was implemented in this study and given as (2) f(yi∣xiG,xiE)=∑g=1Gτg(xiG)ϕ(yi∣θg(x˜iE))+τ0(xiG)V where f(yi∣xiG,xiE) describes the conditional probability density of yi given the input variables xi for the ith sample, τg(xiG) the mixture component density for the gth cluster dependent on the “gate” variables xiG, and ϕ(yi∣θg(x˜iE)) the gth Gaussian cluster estimated from the sample data conditioned on parameters θg and the “expert” variables x˜iE.²⁶ The tilde (∼) over x˜iE indicates the ith observation is augmented to include a model intercept. A noise component, τ0(xiG)/V, is included in each MoE model to identify outliers and uses the gate variables to guide estimation when specified. A gated noise component assumes that the gate variables influence the fitted noise probabilities and leaves Eq. 2 unmodified. The so-called “expert only” models use a non-gated noise term (τ0(xiG)→τ0) that assumes noise probabilities are constant across samples. For all models tested, the hypervolume (V) was set at five times MSHA’s PEL of respirable coal dust exposure per working shift (5·1.5 mg/m³; 30 CFR, 70.100) as we do not anticipate observing values larger than this in practice (∼6 mg/filter) nor do we expect samples this heavily loaded to obey linear absorption laws.

Gaussian (normal) clusters are parameterized as θg(x˜iE)={x˜iEγg,σg2} such that γg are the linear calibration coefficients used to predict sample mass. This takes the familiar form: (3) y^ig=x˜iEγ^g where y^ig corresponds to the ith mass prediction for the gth cluster model and γ^g the estimated calibration coefficients for the gth cluster model. Gate parameters (not shown) are estimated assuming a multinomial logistic form. Cluster variances are constrained as equal for all models (σ12=σ22=…=σG2). Cluster variances, mixture components, gate, and expert parameters are estimated by maximizing the log likelihood of f(yi∣xiG,xiE) via the expectation maximization (EM) algorithm.⁵² The EM algorithm was initialized using hierarchical clustering.

Scores taken from the PLS calibration are used as expert variables in the MoE models (x˜iE). In this way, the MoE models reflect increasingly more complicated perturbations of the original PLS calibration dependent on: (i) The inclusion/exclusion of gating variables, (ii) the inclusion/exclusion of a noise component, (iii) the number of clusters selected, and (iv) the variance constraints placed on the clusters (σg2). In fact, as the PLS scores are used as the expert variables, it is straightforward to show that the original PLS calibration corresponds to a special case of the MoE model, i.e., one that uses a single-cluster (G=1), no gate variables, and fits no noise component to the data.

Three MoE models were fitted and tested to systematically assess the degree that gate variables and the noise component improve method accuracy. First, an “expert only” MoE model was fitted to assess whether multiple clusters (and a noise term) improve the prediction of sample mass. The second MoE model used “mine type” as a categorical gate variable with the four levels following directly from the “Mine Type” column in Table I. The “cassette type” covariate was then included as a second gate variable with the three levels given in the “Cassette Type” column in Table I. For all fitted models, the optimal number of clusters (G) was identified as the one minimizing the Bayesian information criterion.⁵³

Models are validated and compared on figures of merit calculated from test set data. Non-robust metrics include the coefficient of determination (R2) and root mean square error of prediction (RMSEP).⁵⁴ Robust measures include the median absolute error (MAE), median absolute percentage error, median bias, and median percentage bias.³⁷ The MAEs are calculated as k times the median absolute difference between the measured and predicted mass or (4) MAE=k×median(|y−y^|) where the scalar k=1.4826 makes MAE a consistent estimator of average prediction error, i.e., MAE and RMSEP are asymptotically equivalent when errors are unbiased and follow a normal distribution.⁵⁵ Median absolute percentage error (MAPE) is calculated by normalizing the term y−y^ by y in Eq. 4 with k=100 to convert to a percentage basis. Median bias and median percentage bias follow analogously from Eq. 4 by dropping the absolute value from the calculation. Again, errors are normalized and k set equal to 100 for median percentage bias.

Ninety-five percent confidence intervals are calculated for median and median percentage bias using a nonparametric bootstrap percentile approach.⁵⁶ Intervals were calculated by resampling training set residuals to generate 5000 bootstrap replicates. Confidence interval endpoint adjustments used bias correction and acceleration (i.e., BCA). Bootstrap intervals are used to formally test the two-sided null hypothesis that median biases are equal to zero. If confidence intervals around bias estimates do not contain a zero the null hypothesis is rejected at α=0.05 probability level.

Mass predictions for any MoE model are possible using two techniques when more than a single cluster is used for modelling. The first approach predicts mass in each sample using all available calibrations, providing G predictions of mass for a sample. Predictions are then multiplied by mixing probabilities (τ^g) and summed to generate a final value of sample mass. Mass predictions under this framework are given as (5) y^i=∑g=1Gτ^gy^ig With y^i indicating the final estimate of sample mass. Alternatively, sample mass may be determined using the cluster model having the largest mixing probability or max (τ^ig). We adopted the approach in Eq. 5, known as Bayesian model averaging,²⁸ as it protects against the possibility of false cluster assignment, ambiguous cluster composition, or the belief that a single “correct” cluster exists to determine sample mass under the fitted model. The latter point is particularly important as the levels of the “mine type” and “cassette type” gate variables are not strong proxies for mine geology and sampling deposition biases, respectively. We therefore expect MoE clusters to reflect information from mixed geologic sources and a model that imperfectly adjusts calibration factors for deposition differences (at best). Note that MoE outliers are still quantified for all test samples using Eq. 5; albeit, with probabilities renormalized on the G clusters (excluding the outlier cluster). Leaving outliers in the test set keeps a comparison between MoE models and the base-case PLS calibration as fair as possible as all models are compared on the same samples.

All analyses were performed using R Statistical Software (v4.1.2; R Core Team). The BCA bootstrap percentile method was coded by the authors with computation time reduced using the cluster computing functions available in the parallel package (v4.1.2). The PCA and visualization were performed using the PCA Tools package (v2.5.15). The kernel PLS algorithm was available in the PLS package (v2.8-1). MoE models were fitted using the MoEClust package (v.1.5.1). The PLS model was wrapped into the cross-validation search routine using control functions available in the caret package (v.6.0-94).