Leveraging NMF to Investigate Air Quality in Central Taiwan

Jui-Fang Chang; Shu-Chuan Chen; Yintzer Shih

REVIEW 4 major objections 5 minor 14 references

Leveraging NMF to Investigate Air Quality in Central Taiwan

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Factoring a decade of hourly readings from 14 stations, the paper concludes that NO2 and PM2.5 in central Taiwan are mostly local, SO2 mostly imported, and PM10 split nearly evenly.

desk verdict Table 3 is internally inconsistent and unvalidated; the paper's central claim is not supported by its own analysis. read the letter →

arxiv 2411.13315 v1 pith:DHJNOQUV submitted 2024-11-20 math.NA cs.NA

Shu-Chuan Chen , Jui-Fang Chang , Yintzer Shih This is my paper · ORCID

classification math.NAcs.NA MSC 15A2365F55

keywords non-negativematrixfactorizationairpollutiondatavisualizationsourceapportionmentdomesticvstransboundarywind-roseanalysiscentralTaiwanPM2.5andNO2

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish where central Taiwan's air pollution comes from: which pollutants are generated locally and which are carried in from outside the island. The authors factor a decade of hourly concentration readings from 14 monitoring stations into a small number of nonnegative source components, then label each component domestic or transboundary according to the wind speeds and directions at which it peaks. Their central claim is that NO2 and PM2.5 are predominantly domestic, SO2 is predominantly transboundary, and PM10 is split almost evenly. A sympathetic reader would care because source attribution determines who is responsible for cleaning the air in a region that hosts Taiwan's largest thermal power plant, a politically charged emitter.

What carries the argument

The load-bearing object is the NMF factorization $A \approx WH$, with $W$ the time profiles and $H$ the spatial loadings of $k$ source components, computed by multiplicative update rules that keep every entry non-negative; non-negativity is what lets each component be read as an additive, physically meaningful source profile. Around it sits the cophenetic-correlation rule for choosing $k$ and, decisively, the wind-rose inspection that assigns each component to 'domestic' or 'transboundary' by the wind speed at which its pollution load concentrates. The wind-speed signature is what converts a statistical factorization into a geographic attribution.

What would settle it

Recompute the source-apportionment table with a single explicit wind-speed cutoff — for example the 5.5 m/s boundary of the paper's own wind classification, or 6 m/s — applied uniformly to every NMF component, and rerun the validation using the same component counts as the main analysis: if NO2's domestic share falls below half, or if the validation procedure does not reproduce the main proportions when applied to the full 2008–2017 record, the central claim fails.

Watch

Extended reading notes

Core claim

The discovery, stated on the paper's own terms, is a source-apportionment table: over 2008–2017 in central Taiwan, NO2 is 66.36% domestic and 33.64% transboundary, PM2.5 is 64.65% domestic, SO2 is 45.83% domestic (54.17% transboundary), and PM10 is 50.95% domestic (49.05% transboundary). These numbers come from decomposing each pollutant's $87{,}672 \times 14$ hourly matrix into spatial and temporal factors with non-negative matrix factorization, choosing the number of factors by cophenetic correlation, and then reading each factor's wind rose: factors whose pollution concentrates at gentle wind speeds around 3.5–4.8 m/s are classified as domestic, while factors driven by monsoon winds of 7–12 m/s are classified as transboundary. The paper further claims that this classification scheme is validated by comparing a 2010 re-run of the procedure, with larger numbers of factors, against an independently estimated 2010 domestic-ratio benchmark, finding agreement within about six percentage points for NO2, SO2, and O3.

Load-bearing premise

The load-bearing premise is that a pollutant component's wind speed alone, read visually from a wind rose, cleanly separates 'domestic' from 'transboundary' sources: components peaking near 4 m/s are called domestic and those reaching roughly 7 to 12 m/s are called transboundary, with no quantitative threshold, error margin, or physical transport model behind the split; changing that cutoff directly changes every percentage in the results table.

Editorial extensions

If this is right

For NO2 and PM2.5, the policy lever is local: vehicle, industrial, and power-plant controls inside central Taiwan would address the domestic majority of these pollutants.
For SO2, local abatement alone would leave more than half the burden in place, so reducing SO2 requires cooperation with sources across the Taiwan Strait.
For PM10, domestic and transboundary measures carry roughly equal weight, so neither a purely local nor a purely regional strategy is sufficient.
The SO2 peaks recorded at the station downwind of the Taichung thermal power plant are attributed to transboundary flow, implying the plant's local SO2 fingerprint is, by this method, small relative to imported SO2.
If the method is right, the 2010 benchmark agreement (within about six percentage points for NO2, SO2, and O3) indicates the same pipeline can be reused for other pollutants and other years without redesigning the analysis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct test the paper leaves unwritten: refit the classification with one explicit wind-speed cutoff (for instance the 5.5 m/s 'gentle breeze' boundary of its own wind classification) applied uniformly, and watch whether NO2 and PM2.5 stay majority domestic; the percentages in the table move with the cutoff.
The validation section examines 2010 only and uses different numbers of components than the main runs; comparing each separate year of the decade, or re-running the 2010 validation at the main component counts, would show whether the domestic-majority claim is stable or an artifact of the chosen factor count.
Plugging the same factors into back-trajectory or chemical-transport modeling would let each NMF component be named by its actual airmass origin, turning the visual wind-speed heuristic into a falsifiable source attribution.
If the domestic/transboundary proportions survive such tests, they give both Taiwan's local governments and cross-strait negotiators a concrete basis for allocating abatement costs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The central result, Table 3, doesn't hold up. The NO2 row contradicts the paper's own classifications, and the validation section checks a different analysis than the one reported. This is not a matter of taste; the numbers don't cohere.

What's genuinely useful: the paper compiles ten years of hourly data from 14 stations in central Taiwan and applies NMF to separate pollution features. That dataset is real, and the visualizations are thorough. The idea of labeling NMF components by wind speed/direction is a routine extension of existing practice, but it's applied honestly here. The comparison to Chen et al. is a reasonable external anchor.

The soft spots are load-bearing. The wind-speed classification—components peaking at ~4 m/s called domestic, those reaching 7–12 m/s transboundary—has no thresholds, error bars, or sensitivity analysis. A different cutoff directly changes every percentage in Table 3. More seriously, the validation doesn't validate the main result. The 2010 validation runs use different k values (k=8 for NO2, k=10 for SO2 and O3) and include O3, which isn't in the main analysis. For SO2, the main result is 45.83% domestic (k=5), but the validation gives 26.92% (k=10), close to Chen et al.'s 27%. The paper claims agreement within ±6%, but the gap to the headline number is ~19 points. That gap is never discussed. And there's an internal arithmetic error: in the NO2 main analysis, summing the reported shares of components the text labels domestic gives 49.8%, not the 66.36% in Table 3. You only get 66.36% by counting one component the text explicitly calls transboundary as domestic. These are not cosmetic issues.

Who this is for: researchers working on Taiwan air quality or on source apportionment with matrix factorization might find the regional numbers worth a second look, but only after the errors are fixed. As it stands, the paper is a cautionary example of validation drifting from the headline analysis.

Recommendation: reject in current form. But I'd send it to peer review rather than desk reject, because the dataset and question are real and the errors are fixable in revision. If the authors correct the arithmetic, run sensitivity on the wind-speed thresholds, and validate with the same k and years as the main analysis, this could become a decent regional case study.

Referee Report

4 major / 5 minor

Summary. The paper applies non-negative matrix factorization (NMF) to hourly air-quality data for SO2, NO2, PM10, and PM2.5 from 14 monitoring stations in central Taiwan over 2008–2017 (87,672 hourly records). For each pollutant the data matrix is decomposed into k components; each component is then classified as 'domestic' or 'transboundary' based on wind-rose patterns, and the component shares are aggregated into Table 3, which reports domestic/transboundary proportions of 66.36%/33.64% for NO2, 45.83%/54.17% for SO2, 50.95%/49.05% for PM10, and 64.65%/35.35% for PM2.5. The paper concludes that NO2 and PM2.5 are domestically dominated, SO2 is substantially transboundary, and PM10 is nearly balanced. Section 6 re-runs the analysis on 2010 data with different k values (k=8 for NO2, k=10 for O3 and SO2) and compares the resulting domestic ratios against Chen et al. (2010), reporting agreement within a claimed ±6% margin.

Significance. If the reported proportions were robust, the paper would provide useful quantitative input for central Taiwan's air-quality policy, separating domestic from transboundary contributions. The study has several strengths: NMF is a reasonable tool for nonnegative compositional data; the validation against independent 2010 estimates from Chen et al. is a good idea; and including O3 as a sanity check (even though O3 is not part of the headline results) shows an attempt at external grounding. However, the paper ships no code, no data-availability statement, no error bars, and no sensitivity analysis for the classification rule. More importantly, the headline percentages in Table 3 are not backed by the manuscript's own component classifications, and the validation in Section 6 uses a re-configured version of the method rather than the main analysis. If the internal inconsistency and validation gap were fixed, the qualitative finding—especially for SO2 and PM10—could still be defensible, but the quantitative claims in Table 3 are not currently supported.

major comments (4)

[Section 5, Table 3] The NO2 row of Table 3 is internally inconsistent with the component classifications given in Section 5. The text labels NMF1, NMF2, and NMF4 as domestic (16.79%, 16.58%, 16.43%) and NMF3, NMF5, and NMF6 as transboundary (16.58%, 16.68%, 16.56%). Summing the domestic shares gives 49.80%, not 66.36%; the table's 66.36% is recovered only by counting NMF6—explicitly called transboundary in the text—as domestic. This is a load-bearing arithmetic error because the abstract's statement that NO2 is 'primarily influenced by local sources' hinges on this row.
[Section 6 vs Table 3] The validation in Section 6 does not actually validate Table 3. The main analysis uses k=6 for NO2, k=5 for SO2, k=4 for PM10, and k=3 for PM2.5 on the full 2008–2017 record, while Section 6 uses k=8 for NO2 and k=10 for O3 and SO2 on 2010 data only. For SO2, Table 3 reports 45.83% domestic for 2008–2017, but the 2010 k=10 run yields 26.92% domestic—an 18.9-percentage-point difference that is far outside the '±6% margin' claimed in Section 7. The paper never discusses this discrepancy; the conclusion's claim that the method is effective is therefore not supported for the headline results.
[Section 5, classification rule] The domestic/transboundary attribution is made by a visual inspection of wind roses without any quantitative decision rule or sensitivity analysis. For example, in the NO2 analysis, NMF1 is called domestic because high pollution is concentrated at 4 m/s, and NMF2 is domestic at 3.5 m/s, but NMF6—with high pollution concentrated at 3.9 m/s—is classified as transboundary with no stated meteorological justification. Changing the implicit threshold by a few m/s would directly alter every percentage in Table 3. A sensitivity analysis, an objective classifier based on wind direction and speed distributions, or at least a clear statement of the applied rule with error margins is needed.
[Section 4.1 and Section 5] The k-selection text is ambiguous and inconsistent with the displayed numbers of components. Section 4.1 describes Figures 5(a)–(d) as 'k=6 and k=5' and 'k=4 and k=3' without stating which pollutant each panel refers to, and the text does not clearly explain which k is chosen for each pollutant. The Results section then uses six components for NO2 (although Figure 6's caption says 'five components'), five for SO2, four for PM10, and three for PM2.5. The link between the cophenetic-correlation panels and the chosen k must be stated explicitly, and the figure captions should match the number of components actually used.

minor comments (5)

[Section 1] The roadmap in the introduction says 'Section 2 outlines the methods, Section 3 describes data sources, Section 4 presents the analysis results, followed by a validation of the proposed method in Section 5. Section 6 concludes the study,' but the actual structure is Section 4 (analysis), Section 5 (results), Section 6 (validation), Section 7 (conclusion). Please correct the roadmap.
[Table 2] The seasonal percentages for SO2 in Table 2 sum to 99.99% rather than 100%; check the rounding and report consistent totals.
[Figures 13 and 16] The PM2.5 analysis in Section 5 refers to 'Fig. 13 reveals distinct seasonal variations,' but Figure 13 is the PM10 day-to-year variation plot. The PM2.5 day-to-year variation is shown in Figure 16.
[Section 2, Eq. (4)–(6)] In the derivation of the multiplicative update rules, the notation '(W H H^T)' should be written as '((W H) H^T)' to avoid ambiguity about the order of multiplication; this is a presentation issue only.
[References] Reference [14] is cited as 'Chen et al.' but the entry lists the journal as 'Science of the Total Environment, Atmospheric Environment'—two journal names—and should be corrected; also consider adding a data-availability or code-availability statement, since the public EPA data link is given but the analysis code is not.

Circularity Check

2 steps flagged · score 6.0 of 10

The 2010 validation is fitted by choosing k after seeing the benchmark, and the domestic/transboundary labels are built into the wind-speed heuristic, so Table 3 and Table 4 do not independently support the headline source attribution.

fitted input called prediction [Section 6, 'Analysis on SO2' and Table 4]
"We have chosen not to select k = 3 or k = 5 in order to preserve the integrity of the data, opting instead for k = 10, as illustrated in Fig. 26. ... In conclusion, for the year 2010, domestic SO2 pollution represented 26.92%, while transboundary pollution accounted for 73.08%."

The external benchmark being validated is Chen et al.'s 27% domestic SO2. The main analysis used k=5 and reported 45.83% domestic (Table 3), but the validation explicitly rejects k=5 and adopts k=10, which yields 26.92% domestic, essentially the benchmark value. Because k fixes the NMF decomposition and therefore which components are subsequently labeled domestic or transboundary, selecting k after seeing the target makes the Table 4 agreement a fitted result rather than an independent confirmation. The validation does not test the k=5 headline SO2 figure.
renaming known result [Section 5 (Results) and Table 3]
"NMF1 is notably influenced by westerly winds, with high pollution levels concentrated at a wind speed of 4 m/s, as shown in Fig. 8. Based on this, we classify NMF1 as domestic pollution, contributing 16.79% of all features. ... NMF3 ... wind speeds reaching up to 12 m/s ... classify NMF3 as transboundary pollution, accounting for 16.58% of all features."

The source categories are operationalized by a wind-speed rule: components peaking near 4 m/s are called domestic, while components reaching 7-12 m/s are called transboundary. Table 3 is then simply the sum of the shares of components carrying these labels. The conclusion that NO2 and PM2.5 are predominantly domestic while SO2 is transboundary is therefore a restatement of the low-versus-high wind-speed pattern imposed on the NMF components, not an independently measured source attribution. The external Chen et al. anchor is applied in Section 6 to a different k configuration, so it does not independently calibrate the Section 5 labels.

full rationale

The paper's NMF mathematics is standard and not circular, and no load-bearing self-citation chain is present. The circularity lies in the validation and labeling chain. In Section 6, the validation runs use different k values than the main analysis (NO2 k=8 instead of 6, SO2 k=10 instead of 5) and, for SO2, the authors state they deliberately avoided k=3 and k=5 before choosing k=10, which produces a domestic ratio almost identical to Chen et al.'s 27%. This is a fitted parameter choice presented as independent validation. Additionally, the main Table 3 percentages are the direct arithmetic of the Section 5 wind-speed classification rule, so the headline finding is a renaming of the wind-speed pattern rather than a source-attribution result calibrated by an external check. There is also a non-circular correctness problem: the NO2 row of Table 3 (66.36% domestic) cannot be reproduced from the Section 5 labels, where NMF1, NMF2, and NMF4 are domestic and NMF3, NMF5, and NMF6 are transboundary, which sums to 49.8% domestic; the printed figure is recovered only by counting the transboundary-labeled NMF6 as domestic. This inconsistency does not itself make the argument circular, but it further means Table 3 is not a reliable output of the stated derivation chain. Overall, the validation step reduces to post-hoc parameter selection and the headline attribution reduces to the labeling heuristic, so the circularity score is moderate-to-high.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The analysis relies on hand-chosen k values, visually determined wind-speed thresholds, and the physical assumption that wind speed cleanly separates local from imported pollution. These are the main free choices that shape every reported percentage.

free parameters (3)

number of components k = 5 (NO2), 5 (SO2), 4 (PM10), 3 (PM2.5) in main analysis; 8, 10, 10 in validation
Chosen by inspecting cophenetic correlation plots; the visual selection is not reproducible from the text alone.
wind speed classification thresholds = not specified numerically; examples include 4 m/s for domestic and 12 m/s for transboundary
The line between domestic and transboundary wind speeds is never stated as a rule; it is inferred from windrose plots.
component labeling (domestic vs transboundary) = manual assignment per component
Each NMF component is assigned a label by the authors using wind direction and seasonality; no independent criterion is applied.

assumptions (3)

domain assumption Each NMF component corresponds to a single pollution source regime (domestic or transboundary)
The analysis treats every factor as wholly local or wholly foreign; mixed sources are not allowed.
domain assumption Wind speed is a sufficient proxy for transport distance
The classification rule equates high wind speed with transboundary origin, without accounting for atmospheric chemistry or pollutant residence time.
standard math NMF with non-negative data gives meaningful, stable factors
Standard NMF property; stability and uniqueness over random initializations are not tested in this paper.

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Cite this review

Pith. "Pith review of Leveraging NMF to Investigate Air Quality in Central Taiwan." pith.science (2026). https://pith.science/paper/DHJNOQUV

@misc{pith2026241113315,
  author       = {Pith},
  title        = {Pith review of: Leveraging NMF to Investigate Air Quality in Central Taiwan},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHJNOQUV}},
  note         = {Machine review of arXiv:2411.13315}
}

abstract

This study investigates air pollution in central Taiwan, focusing on key pollutants, including SO$_2$, NO$_2$, PM$_{10}$, and PM$_{2.5}$. We use non-negative matrix factorization (NMF) to reduce data dimensionality, followed by wind direction analysis and speed to trace pollution sources. Our findings indicate that PM$_{2.5}$ and NO$_2$ levels are primarily influenced by local sources, while SO$_2$ levels are more affected by transboundary factors. For PM$_{10}$, contributions from domestic and transboundary sources are nearly equal.

Figures

Figures reproduced from arXiv: 2411.13315 by the authors.

**Figure 1.** Central Taiwan Terrain Map: Stars indicate Industrial Air Quality Monitoring Stations; squares indicate Background Air Quality Monitoring Stations; and circles indicate General Air Quality Monitoring Stations [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Visualization for hourly average concentration 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Visualization for monthly average concentration [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (26 more)

**Figure 4.** Figure 4: The flow chart for proposed data analysis 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: NMF for cophenetic correlation [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: The percentage of NO2: five components of H 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Day-to-year variation of NO2: five components of W [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: NMF for NO2 pollution Windrose (the wind direction, wind speed, and pollution concentration form a pollution Windrose) 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: The percentage of SO2: five components of H [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: Day-to-year variation of SO2: five components of W 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: NMF for SO2 pollution Windrose (the wind direction, wind speed, and pollution concentration) [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: The percentage of PM10: four components of H 16 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

**Figure 13.** Figure 13: Day-to-year variation of PM10: four components of W [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: NMF for PM10 pollution Windrose (the wind direction, wind speed, and pollution concentration) 17 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

**Figure 15.** Figure 15: The percentage of PM2.5: three components of H [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]

**Figure 16.** Figure 16: Day-to-year variation of PM2.5: three components of W 18 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

**Figure 17.** Figure 17: NMF for PM2.5 pollution Windrose (the wind direction, wind speed, and pollution concentration) [PITH_FULL_IMAGE:figures/full_fig_p019_17.png]

**Figure 18.** Figure 18: The percentage of NO2: eight components of H 19 [PITH_FULL_IMAGE:figures/full_fig_p019_18.png]

**Figure 19.** Figure 19: Day-to-year variation of NO2: eight components of W [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]

**Figure 20.** Figure 20: NMF1-NMF5 for NO2 pollution Windrose (the wind direction, wind speed and pollution concentration) 20 [PITH_FULL_IMAGE:figures/full_fig_p020_20.png]

**Figure 21.** Figure 21: NMF6-NMF8 for NO2 pollution Windrose (the wind direction, wind speed and pollution concentration) [PITH_FULL_IMAGE:figures/full_fig_p021_21.png]

**Figure 22.** Figure 22: The percentage of O3: ten components of H 21 [PITH_FULL_IMAGE:figures/full_fig_p021_22.png]

**Figure 23.** Figure 23: Day-to-year variation of O3: ten components of W [PITH_FULL_IMAGE:figures/full_fig_p022_23.png]

**Figure 24.** Figure 24: NMF1-NMF6 for O3 pollution Windrose (the wind direction, wind speed and pollution concentration) 22 [PITH_FULL_IMAGE:figures/full_fig_p022_24.png]

**Figure 25.** Figure 25: NMF7-NMF10 for O3 pollution Windrose (the wind direction, wind speed and pollution concentration) [PITH_FULL_IMAGE:figures/full_fig_p023_25.png]

**Figure 26.** Figure 26: The percentage of SO2: ten components of H 23 [PITH_FULL_IMAGE:figures/full_fig_p023_26.png]

**Figure 27.** Figure 27: Day-to-year variation of SO2: ten components of W [PITH_FULL_IMAGE:figures/full_fig_p024_27.png]

**Figure 28.** Figure 28: NMF1-NMF5 for SO2 pollution Windrose (the wind direction, wind speed and pollution concentration) 24 [PITH_FULL_IMAGE:figures/full_fig_p024_28.png]

**Figure 29.** Figure 29: NMF6-NMF10 for SO2 pollution Windrose (the wind direction, wind speed and pollution concentration) 25 [PITH_FULL_IMAGE:figures/full_fig_p025_29.png]

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