A receptor model is evidence about patterns, not a list of culprits
How to translate aerosol positive matrix factorization is a question about evidence, not just vocabulary. A table of particulate matter chemical measurements becomes much more useful when a reader can distinguish what a receptor model directly estimates from what a scientist interprets afterward. PMF, positive matrix factorization, helps explain covariance across samples; it does not directly measure smokestack emissions or identify a particular company. This guide translates source factors, residuals, uncertainty-weighted fitting, rotational ambiguity and diagnostics into claims that a careful reader can defend.
Imagine receiving an aerosol source-apportionment report with four coloured factors, percentages and attractive labels. The real task is to reconstruct the inferential chain. Which species were actually measured? What uncertainty entered the optimisation? What does the plotted quantity count? How stable are the chosen factors when weak species, random starts, rotation and unusual episodes are reconsidered? Answering those questions prevents a diagram from turning into a stronger assertion than the observations support.
We begin with a simple model. Let X contain measured species concentrations for sample times and chemical species. PMF approximates X as G multiplied by F, plus residual error. G represents how strongly each factor contributes to each sample; F describes the chemical pattern attached to each factor. Neither factor numbering nor model colours carry inherent physical meaning. Scientists assign likely source-type names by comparing the fitted patterns with chemical knowledge and independent environmental information.
Consider a teaching dataset with ten daily filters and six measured chemical species. The fact that nitrate rises with ammonium is consistent with a shared chemical process, but covariance alone cannot say who emitted the precursor gases. Similarly, carbon and zinc changing together could reflect direct common emissions, meteorology that concentrates multiple sources, or sampling problems. The model is a disciplined tool for separating patterns, not a substitute for reasoning about chemistry and measurement.
A responsible translation of a PMF graph therefore works in layers. First report directly observed mass concentrations and their analytical definitions. Then identify mathematically estimated contributions and the quality of the reconstruction. Next explain why a source label is plausible and what other labels may fit. Finally disclose instability, uncertainty and model alternatives before drawing conclusions about actions. This is the same discipline useful in reading vocabulary, scientific English and any diagram: define the entity, denominator, mechanism and limits before interpreting its headline.
1. The question PMF answers
An ambient filter can contain sulfate, nitrate, carbon, metals and dust at once; chemistry alone does not assign the mixture to emitters. PMF seeks recurring covariance patterns across many samples. Interpret each model factor as a mathematical profile before giving it a source label. A factor resembling vehicle exhaust is evidence of a source type, not proof of a specific road.
Check independent emissions, wind and time-of-day information before claiming which activity generated the profile. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
2. Concentrations versus fingerprints
In a matrix X, each row represents one sample and each column a measured species. The factorisation X≈GF separates sample-varying contributions G from relatively stable profiles F. Rows must use compatible species definitions and a consistent analytical basis. If elemental carbon is micrograms per cubic metre while metals are nanograms, document scaling and uncertainty.
A visually strong factor can simply reflect the magnitude of a column, not its environmental importance. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
3. What the residual means
A predicted zinc concentration of 12 ng/m³ compared with 15 ng/m³ observed leaves a residual of plus 3 ng/m³ when residual is defined as observed minus predicted. That sign tells us the model underpredicts this observation. Compare the difference with measurement uncertainty rather than calling every nonzero residual failure.
State the sign convention: software outputs sometimes define residual in the opposite direction. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
4. Weighted disagreement
If a sulfate residual equals 0.8 µg/m³ and assigned uncertainty is 0.4 µg/m³, the scaled residual is plus two; its squared contribution to weighted Q is four. An identical 0.8 µg/m³ error with 1.6 µg/m³ uncertainty gives scaled residual 0.5 and contribution 0.25.
A low Q is not evidence of truth if the uncertainty values have been inflated until all deviations appear harmless. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
5. Nonnegative solutions
PMF generally constrains factor contributions and profiles to be nonnegative because source mass contributions are represented as additive quantities. The constraint makes many chemical mixtures physically interpretable but it cannot repair a negative concentration generated by aggressive blank subtraction.
Treat negative analytical values according to the stated data protocol; do not silently set all negatives to zero. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
6. Species-specific uncertainty
A trace metal measured near the detection limit can have uncertainty comparable to its reported concentration, whereas a major ion may have much tighter relative precision. The fit should not grant the near-detection signal the same authority as a precisely determined major species just because both have a numeric entry.
Document uncertainty construction, field blanks, instrument precision and any added modeling uncertainty separately. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
7. Signal-to-noise screening
A species whose between-sample signal barely exceeds uncertainty may contribute more noise than information about source patterns. Screening is a diagnostic, not a contest to eliminate every weak signal. Certain rare tracers may carry source specificity despite low concentration.
Test whether removing a weak variable changes source identity; report both the stability and the resulting loss of chemical information. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
8. Weak species settings
Many implementations allow a variable to be marked weak, raising its effective uncertainty so it influences the fitting objective less. A weak designation should be justified by noise, missingness or unreliable chemical behavior; it is not a device for making the residual charts look neat.
Keep a log of which variables were weakened and rerun the model with plausible alternatives. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
9. Bad species exclusions
A laboratory flag may make one species unsuited to factorization: for example a tracer with almost every value below quantification. Excluding it from the fit need not mean hiding it from scientific interpretation. External correlation with modeled factor contributions may still be informative.
Distinguish variables used to estimate factors from independent variables used to test labels. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
10. Below-detection results
If reporting limit is 0.3 ng/m³, a result below it does not mean exactly zero and cannot be treated as a precise value at 0.3. Some PMF workflows substitute a fraction of the limit and assign enlarged uncertainty, but the chosen convention is an analysis assumption.
Explain how results below the limit were encoded and test whether the solution depends on that convention. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
11. Missing values
A missing nitrate sample is absence of measurement, not chemical evidence that nitrate was absent from the atmosphere. Imputation or treatment as a highly uncertain placeholder requires a documented method consistent with the software version.
Inspect missingness by season, instrument outage and pollution event: structured gaps can distort factor profiles. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
12. Synchronising samples
Carbon collected over twenty-four hours cannot always be matched to a two-hour metal measurement without careful temporal aggregation. Create each sample row from measurements that refer to comparable time windows, collection cutoffs and site positions.
A model can discover sampling-schedule artifacts instead of source covariance when windows do not align. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
13. Total mass as a variable
Adding measured PM2.5 mass as a very strong fitting variable can force modeled factors to reproduce total mass even if component chemistry is incomplete. EPA guidance discusses making total mass a weak variable in suitable workflows rather than allowing it to dominate by sheer leverage.
Distinguish chemical mass reconstruction from gravimetric mass; unexplained water or organic material may remain. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
14. Source contribution units
If contributions are normalized and calibrated to total PM mass, a factor contribution can be expressed in µg/m³; without that calibration a plotted factor score may be arbitrary. A profile drawn as percent of species variation answers a different question from a source contribution to total PM2.5.
Never compare percentages with absolute mass without identifying numerator, denominator and normalization convention. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
15. Explaining factor profiles
A profile enriched in silicon, aluminium, calcium and iron may suggest mineral dust, but construction dust and resuspended road material share constituents. A source name is a hypothesis supported by a fingerprint, not something the algorithm prints as a measured ground truth.
Compare ratios, local dust mineralogy, wind episodes and relevant independent indicators. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
16. Traffic interpretation
Elemental carbon, certain metals and rush-hour timing may support a traffic-related factor, yet diesel machinery away from roads can produce overlapping signals. If weekday peaks track commuting hours, show the sampling time resolution; daily integrated filters can blur a true morning pattern.
Label the result traffic-related unless independent information warrants a narrower vehicle category. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
17. Biomass combustion interpretation
Potassium and organic markers can support biomass-burning assignments, but potassium also appears in soil, sea spray and industrial sources. A water-soluble potassium fraction alone is not a definitive wood-smoke signature; levoglucosan can add evidence but can degrade or have multiple origins.
Seek seasonality, fire records, carbon patterns and chemical ratios before naming a burning source. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
18. Sea-salt factor
Sodium and chloride covariation may suggest marine aerosol, but chloride can be depleted by atmospheric reactions with acids. The absence of a perfect seawater chloride-to-sodium ratio does not automatically disprove marine influence.
Evaluate non-sea-salt fractions on consistent mass or molar bases and consider aged sea-salt chemistry. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
19. Secondary sulfate factor
Sulfate and ammonium appearing together frequently point toward regional secondary aerosol formation rather than a single exhaust stack. Secondary material forms after gas precursors are emitted, so a factor contribution at the receptor should not be attributed casually to one facility.
Bring in trajectories, oxidation environment, and precursor inventories with transparent uncertainty. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
20. Secondary nitrate factor
Ammonium nitrate partitions between gas and aerosol with temperature and humidity, creating seasonal patterns unrelated to emissions alone. If winter profiles show large nitrate, consider thermal stability and filter sampling artifacts before attributing the change to winter vehicle activity.
Compare gas-phase nitric acid and ammonia where available, and describe volatility limits. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
21. Crustal calcium versus cement
Strong calcium with aluminium and silicon might represent dust; unusual calcium-rich profiles with distinct local activities might implicate cement handling. Two labels can be compatible with the same fitted composition when source fingerprints overlap.
Use external site information or special tracers rather than forcing a confident label from one major element. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
22. Industrial metal factors
Nickel, vanadium, zinc, lead or copper can appear in industrially influenced profiles, but their mixtures depend on fuel, process and material history. A high nickel-to-vanadium ratio can be suggestive in context but is not a universal fingerprint across all regions.
Show uncertainty intervals for low-concentration metals before discussing highly specific plant sources. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
23. Marine shipping clues
Vanadium and nickel in some fuel-combustion mixtures may prompt a shipping hypothesis near ports; changing fuel standards and shore emissions complicate it. Distance to a harbour alone cannot establish direction of transport or the operating fuel used by a vessel.
Use wind sectors, emissions inventories and concurrent ship activity only when the temporal granularity supports the inference. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
24. Wind direction as validation
If factor scores rise during winds from a particular sector, that pattern may support a geographically clustered source; it can also result from stagnation. Stratify by wind speed and mixing conditions so high concentrations during calm nights are not mistaken for a narrow directional plume.
Treat polar plots as a consistency check, not an independent chemical identification method. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
25. Boundary-layer confounding
A shallow nocturnal boundary layer can raise concentrations of several unrelated primary pollutants simultaneously. PMF might merge sources that always co-vary under those meteorological conditions or split them if chemistry differs enough.
Test hour-of-day and boundary-layer relationships before using a merged factor as a single emission category. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
26. Seasonal factors
The chemical ratio of source material may change across seasons due to photochemistry, fuel use and atmospheric processing. A single time-invariant factor profile is a modeling simplification; an apparent seasonal source can instead reflect changing chemistry of one source.
Compare seasonal fits and a full-year fit without pretending their rotated factor identities must match one-to-one. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
27. Outlier leverage
One spectacular pollution episode can determine a rare metal-rich factor because its concentrations dominate the variance of that species. Check outlier sampling validity first; legitimate extreme events should not automatically be deleted, but their leverage should be evaluated.
Run fits with and without the extreme row and record which conclusions survive. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
28. Number of factors
Two factors may combine road and biomass patterns; ten may split noise into implausibly fine components. Neither solution is automatically optimal. Use Q changes, interpretability, residuals, stability, external source evidence and uncertainty diagnostics jointly.
The smallest Q does not choose the scientifically correct factor count by itself. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
29. Random initial solutions
Nonnegative optimization can converge on different local solutions from different starts, especially in noisy or correlated mixtures. Repeated runs with different seeds test whether a promising factor pattern is stable rather than a convenient lucky result.
Report number of runs, solution spread and why a selected solution was retained. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
30. Rotational ambiguity
Several combinations of source profiles and contributions can reconstruct nearly the same concentration matrix. Rotational tools may explore alternative solutions, but a dramatic-looking cleaner profile need not correspond to a uniquely true physical source.
Use external tracers and rotation diagnostics; describe non-uniqueness as a real limitation, not an inconvenience. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
31. Bootstrap stability
Resampling the data can show whether factors recur when the input sample set changes. A factor that maps inconsistently across bootstrap runs may not justify a confident, narrow source label.
Report unmapped factors and mapping rates instead of quoting only the most stable features. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
32. Displacement diagnostics
Displacement moves individual solution elements to probe how much a fit can change with a limited increase in the objective. Large shifts without clear Q penalty can reveal poorly constrained rotations or factor mixing.
Describe the search space and software settings so readers do not mistake one displacement run for full uncertainty propagation. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
33. Residual time series
If residuals cluster on smoky nights, a factor may be missing a combustion pattern or poorly modeling a changing profile. Plot residuals against date, pollutant concentration and meteorology rather than reporting only the global average residual.
A near-zero average can hide a large positive bias on one set of days and a negative bias on another. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
34. Observed-versus-predicted species
An overall regression line near one does not ensure all high-pollution samples are reconstructed accurately. Inspect each influential species and distinguish slope, intercept, extreme value behavior and uncertainty bands.
A good fit for sulfate cannot compensate for a systematic misfit in a source-specific metal. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
35. Cross-validation by withheld periods
A model fitted entirely to one season may seem stable because it never faces weather or source conditions outside that season. Withhold sensible time blocks or compare independent sites where possible; do not randomly intermix temporally correlated samples and call that independence.
A factor label deserves greater trust when independent evidence supports it beyond the data used to fit it. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
36. Profiles versus external inventories
An emissions inventory lists estimates at sources, whereas PMF detects chemical mixtures after transport and reaction. Differences can arise from chemical transformation, sampling footprint, source-profile overlap or inventory uncertainty.
Use inventory agreement as triangulation, not as proof that PMF has directly measured an emissions rate. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
37. Apportionment is not exposure
A contribution of 4 µg/m³ at a central monitor does not equal an individual person’s inhaled dose or indoor exposure. Exposure depends on time spent in places, infiltration, particle size and personal activity.
Keep receptor-source apportionment separate from health risk statements unless a suitable exposure analysis is supplied. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
38. Comparing two sites
Two sites with independently fitted models may assign similar factor names to profiles that are normalized differently. Compare underlying tracer composition, time aggregation and model constraints before subtracting traffic-related factor contributions.
A shared label is not a guarantee that the two mathematical factors represent identical source mixtures. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
39. Shares versus absolute reductions
If a source share rises from 25% to 40% while total PM falls from 20 to 10 µg/m³, its absolute contribution falls from 5 to 4 µg/m³. The share increased but the concentration attributable to the factor decreased. Both statements are true.
For policy comparisons, calculate absolute µg/m³ and relative percent before declaring a source got worse. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
40. Factor naming confidence
A label such as ‘industry’ hides huge differences between steelmaking, metal finishing and chemical processing. Describe a specific fingerprint and provide graduated confidence, such as ‘metal-rich factor consistent with industrial influence’.
Differentiate what is chemically observed, what is statistically inferred and what is supported by external records. The scientifically useful translation also says what the plot cannot establish. A factor can be reproduced mathematically without establishing a unique emitter or a causal health effect. This is why independent chemical and temporal evidence belongs beside the figure rather than in an appendix nobody reads.
41. Sensitivity to laboratory blanks
If copper on filters partly comes from field or transport blanks, PMF may interpret batch-specific contamination as a source pattern. Investigate the field blank series and preparation batches; corrections introduce their own uncertainty.
The strongest apparent signal is not necessarily environmental if sample handling covaries with dates or technicians. Translate the numerical result with the measured unit first, followed by the statistical interpretation, and only then the environmental hypothesis. The order matters because a reader must be able to distinguish arithmetic that anyone can reproduce from source attribution that requires extra evidence.
42. Measurement uncertainty propagation
Using only instrumental precision may understate errors from flow, recovery, blank variability and transformation. A combined uncertainty model must respect units and correlations; some errors affect whole campaigns, not independent rows.
Report which components entered the PMF uncertainty matrix and which remain outside its formal intervals. An instructive test is to ask whether the interpretation would remain the same if the uncertain chemical variable were removed or its error bars widened. If the story collapses, present it as a sensitive possibility rather than a robust discovery. That caution is useful even when the plot itself appears exceptionally clean.
43. Reporting a conclusion
A defensible summary links model factors to measurable tracers, time behavior, uncertainties and alternative explanations. For example write ‘a carbon-rich factor contributed an estimated 3–5 µg/m³ on sampled days and is consistent with combustion’, not ‘traffic caused 40% of disease’.
Keep ecological interpretation, source attribution and policy causal claims at their appropriate evidential levels. Put this detail into a report by writing one sentence for the observation, one for the model calculation and one for the strongest defensible external check. It prevents a source name from swallowing the distinctions between measurement, assumption, numerical inference and real-world mechanism.
Worked translation and four independent tests
A complete reader can now translate a PMF result without treating fitted factors as named eyewitnesses. Start with the measured input table; inspect uncertainties and exclusions; understand how G and F explain the data; inspect residuals and solution stability; triangulate chemical labels; and state what remains unmeasured. When a report omits a stage, the honest response is to state the missing evidence rather than fill the gap with certainty. The most useful PMF interpretation is often an appropriately qualified one: several source-related patterns are consistent with the observed aerosol mixture, with specified confidence and limits.
Independent transfer problem one: an observed sulfate value is 5.0 µg/m³, prediction 4.4 µg/m³ and assigned uncertainty 0.3 µg/m³. Find the observed-minus-predicted residual and its scaled value. Answer: plus 0.6 µg/m³ and plus 2, contributing 4 to a squared weighted objective. Independent transfer problem two: a modelled source contribution declines from 6 to 4 µg/m³ while total PM2.5 declines from 24 to 10 µg/m³. Its share changes from 25% to 40%, even while absolute concentration decreases by 2 µg/m³; therefore a share increase is not evidence of an absolute source increase.
Independent transfer problem three: a source factor aligns with a port’s wind direction but lacks identifying chemical markers and has low bootstrap mapping stability. Which claim is stronger: ‘ships caused the factor’ or ‘a poorly constrained factor shows some port-sector association’? The second. Independent transfer problem four: nitrate and ammonium vary together during cool weather. Does the covariance prove a single exhaust source? No. Secondary aerosol chemistry and temperature-sensitive partitioning are relevant alternatives. Test these alternatives with additional measurement types and meteorology before narrowing the label.
To use this guide across disciplines, ask four reusable questions whenever a scientific graph is translated into prose. What was measured rather than assumed? What calculation transformed the inputs? What would a different but plausible assumption change? What statement survives even if the most specific source label fails? Those questions make an explanation useful without making it falsely certain.
First-party methods and further reading
Original methodological sources include the US EPA Positive Matrix Factorization overview, the EPA PMF 5.0 Fundamentals and User Guide, the EPA PMF frequently asked questions and EPA receptor modelling. Related eduKate tutorials on aerosol metals by ICP-MS and ion chromatography and charge balance explain the measurements behind many PMF columns. These sources support methodological descriptions, not the invented teaching examples in this article.