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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +7316 +10.21105/joss.07316 + +ESAT: Environmental Source Apportionment Toolkit Python +package + + + +https://orcid.org/0009-0009-4015-5270 + +Smith +Deron + + + + +https://orcid.org/0000-0002-8630-873X + +Cyterski +Michael + + + + +https://orcid.org/0000-0002-5886-7876 + +Johnston +John M + + + + +https://orcid.org/0009-0009-4679-5922 + +Wolfe +Kurt + + + + +https://orcid.org/0009-0005-2221-0433 + +Parmar +Rajbir + + + + + +United States Environmental Protection Agency, Office of +Research and Development, Center for Environmental Measurement and +Modeling + + + + +19 +11 +2024 + +9 +104 +7316 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2024 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Python +Rust +source apportionment +matrix factorization +environmental research +air quality +water quality + + + + + + Summary +

Source apportionment is an important tool in environmental science + where sample or sensor data are often the product of many, often + unknown, contributing sources. Source apportionment is used to + understand the relative contributions of air sources + (Bhandari + et al., 2022) like vehicle emissions, industrial activities, or + dust; as well as particulate matter pollution and to identify relative + contributions of point sources and non-point sources in water bodies + such as lakes, rivers, and estuaries + (Jiang + et al., 2019; + Mamun + & An, 2021). Using non-negative matrix factorization (NMF), + source apportionment models estimate potential source profiles and + contributions providing a cost-efficient method for further strategic + data collection or modeling.

+

Environmental Source Apportionment Toolkit (ESAT) is an open-source + Python package that provides a flexible and transparent workflow for + source apportionment using NMF algorithms, developed to replace the + EPA’s Positive Matrix Factorization version 5 (PMF5) application + (EPA, + 2014; + Pentti + Paatero, 1999). ESAT recreates the + source apportionment workflow of PMF5 including pre- and + post-processing analytical tools, batch modeling, uncertainty + estimations and customized constraints. ESAT + offers a simulator for generating datasets from synthetic profiles and + contributions, allowing for model output evaluation. The synthetic + profiles can be randomly generated, use a pre-defined set of profiles, + or be a combination of the two. The random synthetic contributions can + follow specified curves and value ranges. By running + ESAT using the synthetic datasets, users are + able to accurately assess ESAT’s ability to + find a solution that recreates the original synthetic profiles and + contributions.

+ + + Statement of Need +

The EPA’s PMF5, released in 2014, provides a widely-used source + apportionment modeling and analysis workflow that is no longer + supported and relies on the proprietary Multilinear Engine v2 (ME2). + ESAT has been developed as a replacement to + PMF5, and has been designed for increased flexibility, documentation + and transparency.

+

The Python API and CLI of ESAT provides a + programmatic interface that can recreate the PMF5 workflow. The matrix + factorization algorithms in ESAT have been + written in Rust for runtime optimization. The + ESAT API and CLI provides a flexible way to + create source apportionment workflows and novel research applications. + ESAT was developed for environmental research, + though it’s not limited to that domain, as matrix factorization is + used in many different fields.

+ + Algorithms +

Source apportionment algorithms use a loss function to quantify + the difference between the input data matrix (V) and the product of + a factor contribution matrix (W) and a factor profile matrix (H), + weighted by an uncertainty matrix (U) + (Pentti + Paatero & Tapper, 1994). The goal is to find factor + matrices that best reproduce the input matrix, while constraining + all, or most of, the factor elements to be non-negative. The + solution, W and H, can be used to calculate the residuals and + overall model loss. ESAT has two NMF + algorithms for updating the profile and contribution matrices: + least-squares NMF (LS-NMF) + (Wang + et al., 2006) and weighted-semi NMF (WS-NMF) + (Ding + et al., 2008; + Melo + & Wainer, 2012).

+

The loss function used in ESAT, and PMF5, + is a variation of squared-error loss, where data uncertainty is + taken into consideration (both in the loss function and in the + matrix update equations):

+

+ + Q=∑i=1n∑j=1m[Vij−∑k=1KWikHkjUij]2 + here + + V + is the input data matrix of features + (columns= + + M) + by samples (rows= + + N), + + + U + is the uncertainty matrix of the input data matrix, + + + W + is the factor contribution matrix of samples by + factors= + + K, + + + H + is the factor profile of factors by features.

+

The ESAT versions of NMF algorithms + convert the uncertainty + + U + into weights defined as + + Uw=1U2. + The update equations for LS-NMF then become:

+

+ + Ht+1=Ht∘Wt(V∘Uw)Wt((WtHt)∘Uw)

+

+ + Wt+1=Wt∘(V∘Uw)Ht+1((WtHt+1)∘Uw)Ht+1

+

while the update equations for WS-NMF:

+

+ + Wt+1,i=(HTUwidH)−1(HTUwidVi)

+

+ + Ht+1,i=Ht,i((VTUw)Wt+1)i++[Ht(Wt+1TUwW)−]i((VTUw)Wt+1)i−+[Ht(Wt+1TUwW)+]i

+

where + + W−=(|W|−W)2.0 + and + + W+=(|W|+W)2.0.

+ + + Error Estimation +

An important part of the source apportionment workflow is + quantifying potential model error. ESAT + offers the error estimation methods that were developed and made + available in PMF5 + (Brown + et al., 2015; + P. + Paatero et al., 2014).

+

The displacement method (DISP) determines the amount that a + source profile feature, a single value in the H matrix, must + increase and decrease to cause targeted changes to the loss value. + One or more features can be selected in the DISP uncertainty + analysis. The bootstrap method (BS) uses block bootstrap resampling + with replacement to create datasets with the original dimensions of + the input, where the order of the samples has been modified in + blocks of a specified size. The BS method then calculates a new + model from the bootstrap dataset, and original initialization, to + evaluate how the profiles and concentrations change as a result of + sample reordering. The bootstrap-displacement method (BS-DISP) is + the combination of the two techniques, where DISP is run for each + bootstrap model on one or more features.

+

These error estimation methods address different uncertainty + aspects: DISP targets rotational uncertainty, BS addresses random + errors and sample variability, and BS-DISP provides the most + comprehensive understanding of how the uncertainty impacts a source + apportionment solution.

+ + + + Acknowledgements +

This paper has been reviewed in accordance with EPA policy and + approved for publication. ESAT development has been funded by U.S. + EPA. Mention of any trade names, products, or services does not + convey, and should not be interpreted as conveying, official EPA + approval, endorsement, or recommendation. The views expressed in this + paper are those of the authors and do not necessarily represent the + views or policies of the U.S. EPA.

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