The accuracy of two- and three-way positive matrix factorization models: Applying simulated multisite data sets

Abstract

The application of three-way data sets (combined multisite data sets) for source apportionment has become common, but its influence on the performance of receptor modeling techniques has not yet been explored systematically. To study the influence of site-to-site correlations of source contributions and the spatial variability of source profiles on two- and three-way positive matrix factorization (PMF), simulated three-way data sets were constructed and modeled by different applications of PMF (PMF2 for each site individually, PMF2 for data sets combining all sites together, and PMF3 for all sites). In addition, the performance of PMF was evaluated under conditions of collinearity and different source categories at two sites. The results indicated that if the sites were contributed by sources with identical profiles, the site-to-site correlations of source contributions would not influence the PMF2, and the three-way blocks could be used by PMF2. However, the PMF2 using three-way data sets was sensitive to the spatial variability of source profiles. For the three-way model, PMF3 could perform well only when all of the sources exhibited strong site-to-site associations among all sites, and at the same time, the spatial variability of source profiles were sufficiently small. It might due to the algorithm that, for each source, PMF3 produces the same source profile and the same temporal variation in daily contributions among all sites.

Implications: The application of multisite data sets for source apportionment has become common. However, limited work investigated the accuracy of two- and three-way PMFs when using multisite data sets. If the application of PMFs using multisite data sets were not appropriate, the results would be unreasonable. The unreasonable results would supply confused information for PM control strategies. In this work, simulated multisite data sets were modeled by different applications of PMFs. The effort to assess and compare the performance of two- and three-way PMFs using multisite data sets is very limited. The findings could provide information for multisite source apportionment.

Introduction

Atmospheric particulate matter (PM) has been considered one of the major pollutants in many countries (Zheng et al., 2007; Shi et al., 2011a; Shen et al., 2012; Gu et al., 2013; Pokorna et al., 2013; Zhao et al., 2013a; Titos et al., 2014). Understanding the sources of PM is of great significance for policymakers to design effective programs and strategies for the control and reduction of PM (Zheng et al., 2011; Shen et al., 2014). Receptor models are successfully applied in different scientific fields, especially when it comes to source apportionment of atmospheric particulate matter (PM) (Russell and Brunekreef, 2009; Lowenthal et al., 2012; Zhao et al., 2013b). Receptor-based source apportionment techniques have become powerful tools to understand the sources of PM (Kelly et al., 2013). Positive matrix factorization (PMF), one of the receptor models, is a factor analysis method for the modeling of environmental data sets, constraining factor loadings and scores to nonnegative values (Paatero and Tapper, 1994; Watson and Chow, 2001; Ofosu et al., 2013). It is sanctioned by the U.S. Environmental Protection Agency (EPA) and has frequently been employed in different fields of scientific research throughout the world (Paatero, 1999; Shi et al., 2009a; Chan et al., 2011; Green et al., 2012). In most studies applying two-way PMF (PMF2), numerous ambient samples obtained from a single sampling site are introduced into the PMF model and the input data set is generally characterized by two dimensions (temporal variability and chemical species). PMF2, based on a two-way factor analysis method (Paatero and Tapper, 1994; Paatero, 2007; Chen et al., 2010), extracts source profiles and quantifies contributions based on the temporal variation of chemical species because species from the same source have the same temporal variation (Pant and Harrison, 2012).

With the increasing number of monitoring sites organized for air quality monitoring networks all over the world, three-way blocks (i.e., multisite data sets) would be produced that assess chemical species, temporal variability, and spatial variability. In several studies (Escrig et al., 2009; Beuck et al., 2011; Mooibroek et al., 2011; Larsen et al., 2012; Minguillón et al., 2012), data sets combining data from multiple sites into a single input matrix (three-way blocks) have been introduced into PMF2 to enhance the number of receptor samples (Bruinen de Bruin et al., 2006; Beuck et al., 2011) and the representativeness of estimating the source contributions of PM at a regional scale (Xie et al., 2012; Keuken et al., 2013). In this context, a three-way factor analysis model, PARAFAC (Parallel Factor Analysis), which was developed to solve problems associated with three-way blocks, shows its suitability in the source apportionment of multiple sites (Paatero, 1999; Faber et al., 2003; Abdollahi et al., 2010). PMF3 (three-way PMF) is an algorithm commonly employed to solve three-way PARAFAC (Paatero, 1997, 2007).

Although the application of the three-way data set has become common, some associated problems have not yet been explored directly and systematically. Many studies have investigated single-site source apportionment in terms of the collinearity of source profiles and the source-to-source correlation of contributions (Watson et al., 2008; Shi et al., 2009b; Habre et al., 2011). However, for three-way source apportionment, two unique problems must be faced. First, for one source category, the profiles may vary from site to site in the real atmospheric system (Kim et al., 2005), but the PMF2 and PMF3 using the three-way data set produce only one extracted profile among different sites, inconsistent with the usual condition. Second, an assumption of PMF3 is that the source emission patterns should be the same for all sites (Zeng and Hopke, 1992). However, in the real world, the contributions of one source category at different sites may exhibit different temporal variation (Hwang et al., 2008). Thus, the site-to-site correlations of daily contributions for each source may also influence the solution of the PMF when applying three-way blocks. However, the impact of the aforementioned correlations of the daily source contributions and the spatial variability of source profiles on three-way source apportionment has not been clearly understood, and the problems just mentioned tend to be ignored by environmental scientists when they use multisite data sets. Due to the presence of these two problems, we should know whether the three-way data set used in the PMF is appropriate; if not, the results of the PMF should be rejected (Öztürk et al., 2012). Importantly, the spurious results in such cases would provide inconsistent information to policymakers attempting to devise PM control strategies.

Thus, for the purpose of studying different performances of PMF when running with three-way data sets, simulated multisite data sets were constructed and modeled by different applications of two- and three-way PMF. Our goal was to evaluate how the two problems already mentioned would impact the performance of two- and three-way PMF according to their respective algorithms. In addition, the performance of PMF was explored when collinearity and different source categories at two sites occurred. Although previous works have used synthetic data sets to assess the accuracy of PMF results (Brinkman et al., 2006; Henry and Christensen, 2010; Habre et al., 2011), few studies have focused on the difference between two- and three-way PMF. To our knowledge, there is very limited work that has conducted such an assessment for three-way data sets and to compare the performance of two- and three-way PMF using multisite data sets. The findings in our work could provide valuable information for the application of two- and three-way PMF when using multisite data set.

Methodology

PMF2 and PMF3

The PMF2 attempts to apportion the source profile matrix F and the source contribution matrix G based on observation matrix X at the receptor site (Paatero and Tapper, 1994; Paatero, 2007):

1

where E is the matrix of residuals.

The three-way PMF algorithm (PMF3) can be best described in the form of the following matrix:

2

where X is also the concentration matrix at the receptor site, A relates to the source contributions, B relates to the source profiles, and C is the fractional contribution of each site. PMF3 was first presented by Paatero (1997), but the application of PMF3 has been relatively infrequent. Detailed descriptions of PMF2 and PMF3 are available in Paatero (2007). The principles of two- and three-way PMF and the parameters used in this work are supplied in the Supplemental Materials.

Synthetic data-set development

The site-to-site relationship of both source contributions and source profiles might influence the performance of two- and three-way PMF. Understanding the appropriateness of applying these PMF methods in different cases is imperative. Thus, two types of synthetic data sets were developed in this section: (1) data sets used to analyze site-to-site correlations of daily source contributions between different sites and (2) data sets used to investigate the influence of spatial variability in source profiles. Considering the feasibility of experiments and following prior studies (Javitz et al., 1988; Habre et al., 2011), the collinearity of source profiles and the source-to-source correlations between source contributions at a single site were not taken into account in this section, and the results of related tests are provided in the Supplemental Materials.

Soil dust (SDUST), ammonium nitrate (AMNITR), and light-duty gasoline vehicles (LDGV) were selected as three ambient particle sources and their source profiles are referred to Marmur et al. (2005) and Shi et al. (2011b). The condition index (CI) and the variance-decomposition proposition (VDP) of these sources were introduced to investigate collinearity (Belsley et al., 2005). As listed in Table S1, only one CI value was greater than 2; however, for each CI, only one VDP exceeded 0.5. Thus, no near collinearity problem is detected in the source profile matrix used in this section (Shi et al., 2009b). The source profiles and CI values are listed in Table S1 in detail. Furthermore, the source-to-source correlations between contributions at each site were set to zero. Two sites, SI and SII, were investigated. At each site, the receptor concentration matrix was defined as the following:

3

where G_m×p was the source contribution matrix (m was the number of daily samples and was 300 in this study; p was the number of sources, which was 3 for each site); F_p×n was the source profile matrix (n was the number of species, which was 22 here); and X_m×n was the concentration matrix. The receptor concentrations were randomly perturbed to simulate analytical and measurement errors. Following Javitz et al. (1988), the coefficient of variation (CV), defined as the standard deviation of the species concentration divided by its mean, was used to perturb receptor concentrations. A CV of 10% was used in all of the concentration matrices, and random uncertainties were generated from a log-normal distribution. The contributions of each source at both sites were simulated from a normal distribution with the set mean and standard deviations, which are summarized in Table S1. For the method used to construct simulated data sets here, refer to Habre et al. (2011).

To investigate the site-to-site correlations of daily source contributions, and in consideration of both the representative and the feasibility, seven scenarios were investigated: (1) for each source, the site-to-site correlations of daily contributions between two sites were set to zero (case 1); (2) one source exhibited correlations between two sites, and the correlations were set to 0.3, 0.6,and 0.9 (cases 2–4) and (3) three sources exhibited site-to-site correlations between two sites, and the correlations were set to 0.3, 0.6, and 0.9 (cases 5–7). The correlations were selected based on Habre et al. (2011). The detailed R matrixes (the matrixes of site-to-site correlation coefficients) for each case are listed in Table S2 (Supplementary Materials). The R matrix was denoted as the following:

4

where CORREL is the symbol of correlation coefficients; g_ij^SI was the simulated contribution of the jth source to the ith sample at site SI; and g_ij^SII was the simulated contribution of the jth source to the ith sample at site SII. For each scenario, 10 simulated data sets were generated, resulting in a total of 70 data sets.

To investigate whether the spatial variability of source profiles influences PMF, synthetic data sets of another type were developed. Based on the seven scenarios already described, one scenario was chosen for this study. The same method for perturbation was used as described previously. Different CVs for source profiles were introduced to change the spatial variability of source profiles. The CVs were set to 10%, 20%, 30%, 40%, and 50% to increase the spatial variability of source profiles. For each CV, 10 data sets were generated, producing a total of 50 data sets.

Data analysis

In this study, three applications of PMF were investigated: (1) PMF2 was individually applied to the data sets of each site (called PMF2-I); (2) PMF2 was applied on the data sets that combined the two sites together (PMF2-T); and (3) PMF3 was applied to the data sets that consisted of the two sites (PMF3).

Two indicators—the correlation coefficients (S and P) between the simulated (true) and estimated daily contributions and the average absolute error (AAE)—were used to evaluate the results of the PMF. Two kinds of correlation coefficients were calculated by SPSS 16.0 and were denoted as follows:

5

6

where S is the Spearman correlation coefficient and P is the Pearson correlation coefficient. To investigate a comprehensive condition, the S and P of each site used in the following discussion are defined as the sum of correlation coefficients for all of the sources and the total concentrations (TC) of PM (sum of contributions from the three sources) at one site.

The AAE was defined as follows (Javitz et al., 1988):

7

where n was the number of samples (300 in this work); E_i was the estimated contribution of a source at one site; and T_i was the simulated contribution (namely, true contribution) that corresponded to E_i.

Results and Discussion

Site-to-site correlations of source contributions

To investigate the site-to-site correlations of daily source contributions, 70 simulated data sets were computed using PMF2-I, PMF2-T, and PMF3. All simulated cases converged in each method of PMF. The mean estimated contributions for each source, the total concentration (TC) of PM (sum of contributions from the tree sources), and the standard deviations for the mean contributions of the 70 data sets estimated by the three PMF applications at both sites are summarized in Table S3 and Figure S1. The results of TC were consistent with the true mean TC, and the standard deviations were relatively small for all the PMF, ranging from 0.49 to 1.73. However, for the individual sources at different sites, the results of PMF2-I, PMF2-T, and PMF3 were different. The results using PMF3 were rather divergent and had high standard deviations. The results of PMF2-I and PMF2-T were consistent with the simulated data, and the standard deviations were low. These results indicated that the three-way data set could be used by PMF2 when the sites were affected by sources with identical profiles.

To evaluate whether the PMF methods could function well in cases with different site-to-site correlations of daily source contributions and to investigate the influence of individual cases on PMF2-I, PMF2-T, and PMF3, we calculated the Pearson and Spearman correlation coefficients and the AAEs between estimated contributions obtained by different PMF methods and corresponding true daily contributions. The P and S correlation coefficients of each site are shown in Figure 1 and Figure S2, respectively. Figure 1 shows that in all cases, the Pearson correlation coefficients of PMF2-I and PMF2-T approximately equaled 4, indicating that the estimated contributions of PMF2-I and PMF2-T were consistent with the true contributions, and both methods performed well in all cases. However, the results of PMF3 were obviously unstable. In most cases, the estimated contributions of PMF3 diverged from the true values. Notably, the PMF3 performed well in case 7, where the site-to-site correlations of daily contributions for each source between two sites were 0.9. From the case 5 to case 7, the results showed a trend toward convergence. The Spearman correlation coefficients showed consistent results.

Figure 1. The P (Pearson correlation between true and estimated daily contributions) of each site (defined as the sum of P for all the sources and TC at one site).

Furthermore, a three-dimensional heat map of AAEs is shown in Figure 2 to further demonstrate the performance of PMF for each source in the different cases. The results of this heat map were in good agreement with results of correlation coefficients. According to the heat map, it can be found that the performances of PMF2-I and PMF2-T were stable for each source at both sites for all of the site-to-site correlations. The AAEs of PMF3 are worth noting: Significant variability can be observed. PMF3 performed relatively better in cases 6 and 7, while poor results were derived in cases 1–5. These results might suggest that the site-to-site correlations of source contributions would have a strong influence on the performance of PMF3, wherein PMF3 could perform well only when all sources exhibited strong site-to-site associations between both of the investigated sites. In addition, in case 4, relatively low AAEs between the estimated and true contributions of SDUST^SI and SDUST^SII were observed, which might resulted from high site-to-site correlations that equaled 0.9 for daily contributions of SDUST at the two sites (Table S2).

Figure 2. Three-dimensional heat map of AAEs (%) between true and estimated daily contributions with different site-to-site correlations of daily source contributions by PMF2-I, PMF2-T, and PMF3.

These results can be explained by the different principles underlying PMF2 and PMF3. According to eq 2, PMF3 extracts only one Matrix A (related to source contributions) for all sites; that is, for each source, the site-to-site correlations of contributions should be unity. Therefore, PMF3 could perform well only if all sources exhibited strong site-to-site correlations among all investigated sites. In contrast, spatial variation was not considered in the solution of PMF2 ( eq 1). Thus, site-to-site correlations of source contributions do not influence the results of PMF2.

Spatial variability of source profiles

As discussed with Figure 1 and Figure 2 earlier, the site-to-site correlations of daily source contributions strongly influenced the performance of PMF3, but only weakly influenced PMF2-I and PMF2-T. In addition, the spatial variability of source profiles at different sites might also affect the performance of the PMF methods. To study the influence of spatial variability of source profiles, PMF2-I, PMF2-T, and PMF3 were calculated using different CVs of source profiles, based on case 7 where all of the PMF methods provided consistent results.

The CVs of source profiles were set to 10%, 20%, 30%, 40%, and 50%; 50 data sets were computed using PMF2-I, PMF2-T, and PMF3. The P and S correlation coefficients between estimated and simulated contributions at two sites with different CVs are exhibited in Figure S3, for PMF2-I, PMF2-T, and PMF3, respectively. Figure S3 indicates that variation in source profiles played an important role in the performance of PMF2-T and PMF3, but weakly influenced PMF2-I. The performance of PMF2-T and PMF3 declined when the CVs of source profiles were high. The mean AAEs of each source at two sites with different spatial variability of source profiles are shown in Table 1. A trend of increasing AAEs with increasing CVs was observed for PMF2-T and PMF3, while low, stable AAEs were observed for PMF2-I, which was consistent with the results of P and S correlation coefficients.

Table 1. Mean AAEs (%) of each source category at site SI and SII under different spatial variability of source profiles, estimated by PMF2-I, PMF2-T, and PMF3

		SI, for CV:					SII, for CV:
		10%	20%	30%	40%	50%	10%	20%	30%	40%	50%
PMF2-I	SDUST	4.8%	4.6%	5.1%	4.9%	4.8%	5.2%	5.0%	5.6%	5.6%	5.6%
	AMNITR	6.8%	6.5%	7.0%	6.3%	7.3%	6.8%	6.8%	7.8%	7.4%	6.4%
	LDGV	6.8%	6.2%	6.5%	6.6%	6.3%	6.7%	6.9%	9.1%	7.8%	6.6%
PMF2-T	SDUST	9.5%	19.5%	32.0%	40.3%	45.5%	12.5%	19.3%	32.9%	45.0%	50.0%
	AMNITR	12.1%	20.3%	52.8%	41.4%	62.7%	14.5%	18.8%	33.7%	44.4%	62.4%
	LDGV	12.2%	19.4%	40.1%	59.1%	51.1%	13.6%	27.6%	40.2%	52.8%	101.3%
PMF3	SDUST	16.9%	24.4%	35.8%	41.5%	40.4%	17.4%	23.3%	35.5%	46.5%	45.4%
	AMNITR	17.2%	26.9%	39.7%	40.2%	48.9%	15.0%	28.7%	44.8%	49.7%	47.1%
	LDGV	21.6%	35.6%	33.0%	48.2%	30.1%	21.9%	38.9%	40.7%	43.1%	40.0%

To further investigate the source profiles and contributions estimated by different PMF applications, a case study (two synthetic tests with CVs of 10% and 30%, respectively) was carried out. The normalized simulated (true) and estimated source profiles of SDUST for the two experiments are shown in Figure 3. For CV = 10%, the simulated source profiles showed little spatial variability for different sites. Source profiles estimated by PMF2-I, PMF2-T, and PMF3 showed similar trends with the simulated profiles, although the source profiles calculated by PMF2-T and PMF3 were similar for different sites. However, for CV = 30%, high spatial variability was observed among the true source profiles. Under this condition, the results of PMF2-I resembled the simulated profiles, while the performance of PMF2-T and PMF3 was poor, which might be because they can only estimate the same source profiles for different sites whose source profiles had high spatial variability. Figure S4 shows the true and estimated source contributions of SDUST for two sites, which were calculated using PMF, for which CV = 10% and 30% (Supplemental Materials). The performance of PMF was in agreement with the preceding discussion. For CV = 10%, the estimated contributions calculated using all PMF methods were consistent with the simulated contributions; however, differences were observed for CV = 30%.

Figure 3. Normalized simulated and estimated source profiles of SDUST for site SI and SII by PMF2-I, PMF2-T, and PMF3.

Because PMF2-I was run independently among sites, it was not surprising to observe that spatial variability in the source files weakly influenced PMF2-I. However, eqs 1 and 2 predicted that PMF2-T and PMF3 would produce the same source profiles (matrix F in eq 1 and matrix B in eq 2), so spatial variability in the source profiles significantly influenced PMF2-T and PMF3. In conclusion, if the spatial variability of source profiles was high, PMF2-T and PMF3 could not work well, and only PMF2 for each site individually could be used reliably. Thus, the results in this study indicated that PMF2 and PMF3 would not perform well and would be inconsistent with empirical data if the spatial variability of source profiles were too large, even though the same common source categories were permitted, resolving for different sites by individual analyses.

In addition, scenarios of collinearity in the source profiles and source contributions and different source categories at two sites were explored. The results and discussion are provided in the Supplemental Materials, as shown in Tables S4–S8 and Figures S5 and S6.

Conclusion

Different applications of two- and three-way PMF showed that both the site-to-site correlations of daily source contributions and the spatial variability of source profiles would influence the performance of different applications of PMF, along with the accuracy of the estimated results. For PMF2, if the investigated sites were contributed by sources with identical profiles, it would not be influenced by the site-to-site correlations of source contributions, so PMF2 could perform well using a three-way data set that contained the same source profiles. However, PMF2 was sensitive to the spatial variability of source profiles. When the spatial variability was high, PMF2 using the three-way data set could not function well. For the three-way model, the application of PMF3 was limited by both the site-to-site correlations of daily source contributions and the spatial variability of source profiles. PMF3 could perform well only when all of the sources exhibited strong site-to-site associations among all of the investigated sites and simultaneously, the spatial variability of source profiles was small. These conclusions may provide valuable information to environmental scientists when they carry out multisite source apportionment.

Funding

This study is supported by the National Natural Science Foundation of China (21207070 and 41375132), Special Funds for Research on Public Welfare of the Ministry of Environmental Protection of China (201409003), and the Combined Laboratory of the Tianjin Meteorological Bureau.

Supplemental Material

Supplemental data for this article can be accessed at http://dx.doi.org/10.1080/10962247.2014.926300.

Additional information

Notes on contributors

Ying-Ze Tian

Ying-Ze Tian, Guo-Liang Shi, Xiao-Yu Zhou, Jiao Wang, and Yin-Chang Feng are with the State Environmental Protection Key Laboratory of Urban Ambient Air Particulate Matter Pollution Prevention and Control, College of Environmental Science and Engineering, at Nankai University, Tianjin, China.

Guo-Liang Shi

Ying-Ze Tian, Guo-Liang Shi, Xiao-Yu Zhou, Jiao Wang, and Yin-Chang Feng are with the State Environmental Protection Key Laboratory of Urban Ambient Air Particulate Matter Pollution Prevention and Control, College of Environmental Science and Engineering, at Nankai University, Tianjin, China.

Bo Han

Bo Han and Wei Wang are with the College of Software, Nankai University, Tianjin, China.

Wei Wang

Bo Han and Wei Wang are with the College of Software, Nankai University, Tianjin, China.

Xiao-Yu Zhou

Ying-Ze Tian, Guo-Liang Shi, Xiao-Yu Zhou, Jiao Wang, and Yin-Chang Feng are with the State Environmental Protection Key Laboratory of Urban Ambient Air Particulate Matter Pollution Prevention and Control, College of Environmental Science and Engineering, at Nankai University, Tianjin, China.

Jiao Wang

Ying-Ze Tian, Guo-Liang Shi, Xiao-Yu Zhou, Jiao Wang, and Yin-Chang Feng are with the State Environmental Protection Key Laboratory of Urban Ambient Air Particulate Matter Pollution Prevention and Control, College of Environmental Science and Engineering, at Nankai University, Tianjin, China.

Xiang Li

Xiang Li is with the Department of Computer Science, University of Georgia, Athens, GA, USA.

Yin-Chang Feng

Ying-Ze Tian, Guo-Liang Shi, Xiao-Yu Zhou, Jiao Wang, and Yin-Chang Feng are with the State Environmental Protection Key Laboratory of Urban Ambient Air Particulate Matter Pollution Prevention and Control, College of Environmental Science and Engineering, at Nankai University, Tianjin, China.