In this paper, we describe the data by X∈Rn×m, where the rows i∈[n]={1,…,n} correspond to measurements taken at consecutive times ti. The columns j∈[m]={1,…,m} correspond to the different dimensions of the measurement. Our objective is to find a lower dimensional non-negative decomposition X≈GF with p factors such that G∈R≥0n×p and F∈R≥0p×m, where p≪min⁡(n,m). In other words, the objective is to present the data as a multiplication of two much smaller matrices. The rows of Fi⋅ contain the time-independent components of the decomposition, which we call factor profiles. Factor profiles are defined to sum to unity in order to facilitate comparisons between different datasets and models. The columns of G⋅i contain the time dependency of each of the rows of F; we will call these the factor time series. Simply put, the factor profiles represent the concentration and time-independent chemical composition of the sources, and the factor time series describes the time-dependent concentration of the sources. Note that the ordering of these profiles is arbitrary for the overall solution.

We define a Bayesian probabilistic model that captures our prior assumptions of the process that generated the measurements. The only observed variables in our model are the data matrix X∈Rn×m and the uncertainty estimate σ∈R≥0n×m. Together they define the probability distribution of the observed concentration of each m/z at any point in time with the data matrix being the average and the uncertainty matrix the standard deviation of the distribution, here the Gaussian distribution. In addition to these observed variables, there are several latent variables. These include the matrices G and F mentioned above, vectors α∈R≥0p and β∈R≥0p which determine the auto-correlation behavior of the model, as well the “noise-free data matrix” Z∈R≥0n×m. We define the probabilistic model as 1Z=GF2Xij∼Normallocation=Zij,scale=σijfor alli∈[n]andj∈[m]3Fi⋅∼Dirichlet()for alli∈[p]4Gi+1,k∼Cauchylocation=Gik,scale=αkΔti+βkfor allk∈[p]andi∈[n-1], where Normal corresponds to the normal probability distribution with a given mean and standard deviation, and Dirichlet to the Dirichlet distribution parameterized by a unit vector 1m. The model specification implies that the components of Fi⋅ can have values from [0,1] with equal likelihood, but all rows must sum to unity. Cauchy is the Cauchy probability distribution, with the width depending on the time difference between the ith and (i+1)th observation (Δti=ti+1-ti). Essentially our model describes the data as a non-negative matrix decomposition (NMF) with a lag-1 auto-correlation term and a Gaussian error term for the reconstruction of X.

We chose the Cauchy distribution for the auto-correlation term because the long tails make large jumps between the ith and (i+1)th observation more probable than, e.g., a Gaussian distribution. Other choices are possible, but the experiments in this paper suggest the Cauchy distribution works as an approximation for real data. Choosing the distribution shape also implicitly influences the weight of the G auto-correlation. The αkΔti+βk determines the scale of the Cauchy distribution. The α terms allow the model to deal with time steps of different lengths and missing data, since it forms a simple linear model for the scale. It has a minimum width βk and increases linearly as Δt increases (with αk). Thus, for arbitrarily large time steps the Cauchy approaches a uniform distribution. In physical terms this means that at short time steps we expect the values of G to stay close to the previous value, and at large time steps larger deviations have higher probability. It is possible to use other formulations for the width, which would be appropriate if one wishes to include a more complex and computationally intensive description of auto-correlation. It is also possible to consider more than lag-1 auto-correlation.

We use Stan to compile and run the probabilistic models. Stan solves the probabilistic inference problem with a Markov chain Monte Carlo (MCMC) method. Instead of obtaining a single solution for the latent variables, e.g., by finding the model with the highest likelihood, we get an empirical distribution of possible solutions from which we can infer, e.g., confidence intervals. Stan takes our model and observed variables as input and outputs samples from the posterior distribution of the latent variables. We run multiple MCMC chains, starting from different initial conditions and usually extract a few thousand posterior samples per chain.

The standard way to initialize the model in Stan is by randomly sampling from the prior distributions. However, our model has many parameters with fairly strict distributions. Consequently, we found this starting point to be poor, sometimes causing Stan to markedly slow down, or even fail. Hence, we initialize the model with a point solution. We utilize Stan's capability to find a single maximum a posteriori (MAP) point solution for the parameters, which we use as the initialization. Note, however, that the solutions typically have several local optima, in which case the point solution is only one such local optimum.

Before running our model, we normalize the data such that the mean of the data (X) is 1. The error estimate is scaled with the same scaling factor. The equations to do this are: fnorm=∑i,jXij/(n×m)Xij*=Xij/fnormσij*=σij/fnorm, where fnorm is a scalar normalization factor, Xij* and σij* are the scaled data and error, respectively, which we use as model inputs. This normalization is done so that we can use a consistent scale for priors and posteriors, making the modeling easier, and the denormalization is performed to return the results to familiar units. The normalization is optional, a user can also choose to use non-scaled values.

Stan outputs posterior samples from the two matrices F and G, representing the time-independent chemical composition of the factors and their time-dependent concentration. Since all the magnitude information is in G, G needs to be renormalized by simply multiplying with the normalization factor. The rows of F are constrained to sum to unity, and are, thus, directly comparable to mass spectrometric references, which are normalized similarly .

The first metric to check is if the model explains the data well (reconstruction performance). If X is not reconstructed appropriately, the solution is not acceptable. This can mean either that the data cannot be factorized this way (the model assumptions are wrong) or that the solver failed to find a solution. In such cases, the number of iterations should be increased, or solver parameters must be adjusted, such as the number of warm-up samples and parameters influencing the step size.

Even at moderate data sizes, assessing whether the original data falls within the model confidence bounds for every variable individually is not practical. Therefore, we summarized this information by computing the model residual (difference between the model input and output, mean of all samples) normalized with the uncertainty of the model output (standard deviation of all samples); essentially observing whether the original data are inside the sample standard deviation. 6Sij=Xij-EXsamples/σXsamples Data in S should be centered at 0 and have a standard deviation below 1, which means the data are often less than 1 model standard deviation away from the model mean. In addition, we also use a common evaluation metric in PMF analyses (Qm/Qexp), where Qm is defined as (, notation adapted): 7Qm=∑i,j(X-Zi,jσi,j)2. Essentially Qm describes the sum of squared model residuals normalized to the input error. We use the same reduction as where Qexp is approximated as data size and denote Qm/Qexp as Qm*.

For synthetic data – with a known ground truth – it is possible to assess how well the methods resolve the actual components in addition to the reconstruction performance. We call this evaluation factorization performance. We compare the median solutions with the corresponding actual components by calculating the average distance, Pearson and Spearman (nonlinear) correlations. Optimal matching between median solutions and true components is obtained using the Hungarian algorithm . The approach is similar to the sorting above but with the true components defining the order. Direct comparison with true components is only possible in cases where the number of true components matches the number of modeled components. Otherwise, the model must combine multiple components or create additional ones.

We use PMF, specifically the multilinear engine 2 (ME-2) controlled by the user interface SoFi , as a baseline comparison. PMF solves the decomposition in Eq. () by minimizing the sum of the squared residuals normalized with the input error – see Eq. () (object function) – given the boundary condition that all values must be positive . Since SoFi finds local optima, we ran it with different random seeds to get multiple solutions for all comparisons. As the runs have varying starting points, they often lead to different local optima, especially in cases with high rotational ambiguity. Thus, PMF provides a collection of local minima, while BAMF tries to sample the posterior distribution of the model, including plausible answers that are not minima. For simplicity, we will refer to the group of PMF solution sets as “samples”, even though PMF is not a sampler.

A priori information in the form of known rows of factor profiles or of known columns of factor time series can be added to the model to reduce the rotational ambiguity. By adding this external information, the user can reduce the space PMF searches for the optimized solution, reducing the rotational ambiguity of the solution. Using external data to run PMF is usually referred to as “constraining the solution”, and external information is used as constraints. Here we used two approaches, (a) entirely unconstrained PMF runs and (b) constrained runs using external source profiles. We rely on the commonly used a-value approach to constrain the PMF runs. In the a-value approach, the user inputs one or more factor profiles or factor time series and defines a relative tolerated deviation from the anchor (termed “a value”) . Constraint strengths are not directly comparable between the hard cut-off approach used in PMF and the softer Gaussian error term approach used in BAMF-C. For the best possible comparability, we first ran BAMF-C (constraint strength 0.001) and determined the equivalent a value by taking the maximum deviation from the anchor value on a constrained component in F (a value of 18 %). See Appendix  for the profiles used to make this comparison.

First, we generated a synthetic ToF-ACSM OA mass spectral dataset mimicking a polluted megacity environment affected by multiple OA sources. The synthetic datasets used here are based on observations from Beijing, as it is a relatively well-studied environment. In our case, the modeled sources are traffic exhaust (HOA); cooking (COA); biomass burning (BBOA); coal combustion (CCOA); and secondary OA (OOA). In addition, we also constructed more simple datasets generated with fewer factors (two factors: HOA + OOA; three factors: HOA+COA+OOA; and four factors: HOA+COA+BBOA+OOA). F were chemical fingerprints from the literature . G was created as a mix of Gaussian and Cauchy (BBOA Gaussian, others Cauchy), biased, positive random walks, with added typical diurnal concentration cycles corresponding to the matching OA sources (based on F). A mix of different random walk distributions was chosen to test whether the model approximation of Cauchy auto-correlation works with them. The random walk aims to introduce variability such as one would get from varying transport and mixing. The diurnal cycle was simply summed to the random walk to produce the time series. The overall order of magnitude and diurnal concentration variability of the components (OA sources) were estimated based on previous literature on OA sources in Beijing . For each number of factors, we constructed 10 different datasets amounting to a total of 40 datasets. For reference, one 5-component dataset in its component form is shown in Fig. . CCOA and BBOA are very similar in F and G (see Appendix ), making it a challenging dataset. The time series of CCOA and BBOA are also similar in magnitude and have similar diurnal behavior. While the short-term auto-correlation is high, the random walks of the megacity dataset are not as highly auto-correlated as the PM data in Fig. . The added diurnal peaks can be seen as the periodic peaks in the correlograms in Fig. c and f. See Appendix  for an example of the m/z dependence of the measurement errors on this dataset.

As another test, we used chemical transport model data from representing approximately 2 weeks of simulated measurements in Zurich, Switzerland. Zurich represents a typical European city with low pollution levels. In this case, the G time series come from the transport model and F is taken from the literature (HOA and BBOA from an ambient analysis presented by , biological SOA (SOAbio) from an ambient analysis presented by , anthropogenic SOA, SOAanthro, is represented by laboratory Diesel generator SOA presented by ).

This dataset differs from those in Sect.  in two important ways. Firstly, the concentrations are lower since the environment is less polluted, which affects the error estimation as larger relative measurement errors, median 1.0 % of data magnitude for this dataset and median 0.6 % for one of the datasets described in Sect. . Secondly, the sources exhibit high correlation in the time series (Appendix  shows that correlations in G are higher than in the dataset described in Sect. ), possibly indicating that meteorological conditions and transport of pollutants are important drivers of the concentration of the components. From a source apportionment analysis perspective, this simulates the worst-case situation with the data having poor separability in G. The components of this dataset compared to the megacity data can be seen in Fig. . The auto-correlation behavior of the two datasets is very similar, indicating that our fully synthetic data behave as realistically as the transport model.

In the first experiment, we assess the performance of BAMF, BAMF-0, and PMF on synthetic data mimicking the conditions in a polluted megacity described in Sect. . First, we assess the reconstruction of the input by the different models. In addition to minimizing the residuals, BAMF also includes a penalty for deviations from auto-correlation in G. Due to this and BAMF being a sampled model instead of an optimizer as described in Sect. , PMF would be expected to give answers with lower absolute measurement error-weighted residuals compared to BAMF. In other words, PMF is expected to have a better reconstruction performance.

Figure  shows the median solution reconstruction across the 10 different datasets as the number of components increases. All models reconstruct the input data well within the error estimate. The similar reconstruction metrics for BAMF and BAMF-0 suggest that the inclusion of the auto-correlation term does not substantially deteriorate the reconstruction accuracy. On the other hand, PMF has, as expected, marginally lower absolute measurement error-weighted residuals than BAMF and BAMF-0. However, they are below unity and within the error estimate, judging by the normalized residuals. Therefore, PMF most likely finds solutions that fit the noise in the data better. All other reconstruction metrics (S mean, S standard deviation, Qm*) are comparable for all models.

Data reconstruction is essential to get within the error limits. However, source apportionment aims to accurately and precisely resolve the actual components in G and F, i.e., factorization performance. The solution of each model to the example dataset is shown in Fig. . For this example, we observe that all models resolve all five components. However, BAMF has a better factorization performance both in F and G than the models not accounting for auto-correlation (BAMF-0, PMF) in Table . Similarly, for the diurnal cycles in the example in Fig. c, the diurnal concentration of OA sources, as identified by BAMF, closely resembles the ground truth components. While all models capture the time behavior of the diurnals, the absolute magnitude has a bias, with BAMF having a substantially smaller bias than the other models for four out of five components (Table ).

All models slightly underestimate OOA, which results in overestimating the other components (Table ). For the final component, CCOA, PMF has less bias, but it correlates worse with the truth. BAMF-0 and PMF significantly underestimate OOA, which results in an overestimation of other components to get the reconstruction correct. Some factors have a more pronounced cyclical temporal behavior, as witnessed by high auto-correlation at higher temporal lag (e.g., Fig. c CCOA, BBOA, COA). Based on Table , some of the factors with pronounced cyclical temporal behavior are better represented by BAMF than PMF (e.g., BBOA, COA), while this is not necessarily the case for others (e.g., CCOA).

When considering all 10 synthetic datasets with five components mimicking a polluted megacity, BAMF consistently produces factors closer in magnitude to the truth and which correlate better with the actual factors than the other models (Fig. ). BAMF is also better correlated with the time behavior of the components, while one of the components (CCOA, strongly correlated with BBOA in terms of G and F) is difficult for all models. We hypothesize BAMF-0 shows better spread in Fig. , due to being constrained by the underestimation of OOA and having to include those peaks in the spectra. Figure a shows that BAMF can over- and underestimate both BBOA and CCOA depending on the dataset. Appendix  shows the inverse relationship between CCOA and BBOA mass concentration biases and how the profile reconstruction affects the CCOA mass concentration bias for the BAMF model. Overall, using auto-correlation in source apportionment markedly improves the quality of the resolved factors while keeping the overall reconstruction metrics similar.

In a second exercise, we assessed the performance of BAMF, BAMF-0, and PMF on a synthetic dataset mimicking the conditions in a typical European city (Sect. ). In contrast to the fully synthetic dataset in Sect. , here, the true components G are OA source components computed by an air quality model . This provides G time series close to the atmosphere while still knowing the ground truth. While the three models show somewhat different components (Fig. ), the reconstruction metrics indicate that all models have acceptable solutions (Table ). In fact, the metrics also show that the European dataset is reconstructed almost within the error limits with even only three components, i.e., one component less than is present in the synthetic dataset (HOA, BBOA, SOAanthro, SOAbio). This could explain why there is significant freedom in acceptable four-component solutions and variation between them.

All models show signs of mixing between the components, most likely due to the correlation of the true G components (time behavior is very similar) as well as similar chemical signatures F. PMF mixes the SOA components while BAMF mixes the POA components. However, it is worth noting that BAMF has a significant bias on several components in G as seen in Table , but otherwise reflects their time behavior well. Given the recovered POA / SOA ratio, BAMF most likely mixes HOA and BBOA explaining that HOA is overestimated while BBOA is underestimated. PMF, on the other hand, produces two almost identical components (both in F and G) for two of the four components, and PMF cannot thus distinguish the components present in the dataset. In Fig. d BAMF-0 and BAMF also seem to overestimate higher lag auto-correlation of BBOA in a similar fashion but have a higher bias. It should be noted that even BAMF only considers lag-1 auto-correlation in the model. For HOA, BAMF and PMF do not match the anti-correlated part between lags 5 and 20 and PMF cannot match the behavior of SOA bio. Overall, all models are challenged by the European dataset, with BAMF having the most consistent performance.

For real-world source apportionment analyses, the true number of components, i.e., sources, to be resolved via matrix factorization is unknown yet crucial. Despite the importance, accurately determining and specifying the correct number of modeled components is not trivial (see, e.g., ). Typical strategies rely on reconstructing X within the measurement error, the absence of structure in the measurement error-weighted residuals and the environmental interpretability of the resolved components. Here, we assess the behavior of the BAMF, BAMF-0, and PMF models as the number of components are changed on the four-component chemical transport model dataset from Sect. . The model runs were performed both with an underspecified setting using three components and overspecified setting using five components (Figs.  and ). While underspecified, all models extract an SOAanthro component and merge the remaining three components into two. At the same time, BAMF extracts a component similar to SOAbio, while BAMF-0 and PMF extract components similar to HOA and BBOA but lack SOAbio.

For the overspecified models (five instead of four components), the results differ (Fig. ). While the models without the auto-correlation assumption (PMF, BAMF-0) split the true components into multiple sub-components (mostly the POA components, HOA and BBOA), BAMF produces an extra component that is easily identifiable as unnecessary in addition to the four components resolved with four factors. This unnecessary component is characterized by an extremely high auto-correlation and low magnitude, as seen in Fig. b, c, and d. For this component the time series is almost constant, and the composition is flat with large uncertainties. The extra component does not affect the factorization performance of the other components of BAMF substantially. At the same time, BAMF-0 and PMF have a reduced factorization performance with too many components (Fig. , Table ). In general use, one would prefer the model to indicate the limits of the factorization as BAMF does, instead of producing duplicate components. It should be noted that the behavior of BAMF can be changed by adding new model terms, such as constraints on F. For example, the model minimizes the constrained component when it is run equally overspecified (five instead of four components) but with a priori information on F as can be seen in Appendix .

As highlighted in Sect. , all models are challenged by the European dataset. Imperfect matrix factorization results are likewise often observed when using PMF for real-world chemical datasets (see, e.g., ). Often, information is available that could help resolve the sources, such as chemical fingerprints of specific components associated with different sources. In current practice, previously observed F profiles are often used as boundary conditions in source apportionment analyses. This approach has significant uncertainty in the general case because true F is unknown. Still, in our specific test case, the a priori information is precisely correct – the known information on F of the true components.

We tested the performance of the models when using a priori information on F using three different approaches on the European dataset (from Sect. ):

Full constraint. For BAMF-C and PMF, the two POA components (HOA and BBOA) were fully (for all m/zs) constrained with a roughly 18 % deviation allowed from the anchor (see Sect.  and for the determination of this).

The reconstruction and factorization performance of the different models are compared in Table . The fully constrained BAMF model (BAMF-C) performs substantially better in extracting BBOA both in F and G compared to BAMF. In fact, the extracted components are very similar to the true components with very similar temporal behavior and reduced biases in G (Fig. ). Fully constrained PMF performs marginally better than unconstrained PMF but still mixes SOAbio with SOAanthro (Fig. ). This can be expected since the a priori information is applied to HOA and BBOA, not to the SOA components. Figure d shows that PMF-C captures HOA time behavior very well, while BAMF still overestimates some auto-correlation, even though the bias in Fig. a is significantly reduced. PMF-C also has some solutions that start to approach correct SOA bio, with solutions falling between unconstrained PMF and BAMF-C. For BAMF-C and BAMF SOA bio is virtually unchanged (Figs. a, d and a, d). The incompletely constrained BAMF model performs slightly worse than the fully constrained BAMF model. Partially constrained BAMF performs worse but is still on par with the fully constrained PMF (Table ). The partially constrained BAMF model reduces bias in BBOA but starts mixing SOA bio with the other components (Fig. , Table ). Using a priori information on F improves the factorization performance of both PMF and especially BAMF, with more information leading to solutions closer to the ground truth. This is especially helpful when the additional information is on the components the model mixes when unconstrained. Such prior information is powerful, e.g., constrained PMF does better than unconstrained BAMF for all components except SOA bio. For this dataset and these constraints, BAMF-C fully constrained has the best factorization performance, fully constrained PMF and partially constrained BAMF-C perform slightly worse but about equally good, and unconstrained models fail to resolve one or more components correctly regardless of the model. A comparison of the results of fully constrained PMF and unconstrained BAMF can be seen in Appendix .