Taking drift-diffusion analysis from the study of turbulent flows to the study of particulate matter smog and air pollutants dynamics

L. Ingsrisawang; T.D. Frank; T. Varapongpisan

arxiv: 1907.01477 · v1 · pith:EB3ZSEKYnew · submitted 2019-06-27 · ⚛️ physics.ao-ph

Taking drift-diffusion analysis from the study of turbulent flows to the study of particulate matter smog and air pollutants dynamics

T. Varapongpisan , L. Ingsrisawang , T.D. Frank This is my paper

Pith reviewed 2026-05-25 14:03 UTC · model grok-4.3

classification ⚛️ physics.ao-ph

keywords drift-diffusion analysisair pollutantsparticulate matterozonenitrogen dioxidetime-dependent modelsChiangmai

0 comments

The pith

Drift-diffusion analysis of pollutant data from Chiangmai shows that the underlying equations for particulate matter, ozone and nitrogen dioxide must include explicit time dependence.

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

The paper adapts a method originally used for turbulent flows to extract the form of the governing equations from measured time series of three air pollutants. All three analysis variants reach the same conclusion: the annual concentration peaks are reproduced only when the parameters of the evolution equations are allowed to change periodically with the seasons. This supplies evidence that the physical-chemical models for these pollutants are non-autonomous. A reader would care because the finding implies that standard autonomous differential equations are insufficient for describing smog and pollutant dynamics in this setting.

Core claim

Three variants of drift-diffusion analysis applied to Chiangmai pollutant records all require that the parameters of the physical-chemical evolution equations vary periodically throughout the year in order to account for the observed annual peaks in the first half of the year.

What carries the argument

Drift-diffusion analysis (kernel-density, binning, and linear-approximation variants) that extracts drift and diffusion coefficients from concentration time series.

If this is right

The physical-chemical evolution equations for the three pollutants are explicitly time-dependent.
Seasonal variation must be built into the model parameters rather than treated as external forcing only.
Standard autonomous dynamical models are ruled out for these pollutant systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the time dependence is confirmed, forecasting models that assume constant coefficients will systematically miss the timing of peak pollution episodes.
Similar analysis on other cities could test whether the periodic-parameter requirement is general or specific to Chiangmai's meteorology.

Load-bearing premise

The method developed for turbulent flows can be applied unchanged to identify the structure of the governing equations for atmospheric pollutant concentrations.

What would settle it

A set of pollutant time series in which the same three analysis variants produce time-independent drift and diffusion coefficients yet still reproduce the annual peaks.

Figures

Figures reproduced from arXiv: 1907.01477 by L. Ingsrisawang, T.D. Frank, T. Varapongpisan.

**Figure 1.** Figure 1: Extreme value pollutant concentrations (solid lines) measured in Chiangmai, North Thailand, over the five years (i.e., 60 months) period from January 2010 to December 2014. Panels A, B, C show PM10, O3, and NO2, respectively. Dashed lines show model fits obtained from the linear regression model equation (6) in the deterministic case. 24001-3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Drift functions f (Y, m) for Y = PM10 extreme value concentrations determined for January, February, March, and April (panels A, B, C, and D) by means of three approximations: binning method (stair-step graphs), kernel density estimation method (solid smooth nonlinear lines), and linear regression model (solid straight lines). Dashed lines represent diagonals. functions f (Y, m) thus obtained for PM10 for … view at source ↗

**Figure 3.** Figure 3: Model parameters A, B, and g as functions of month m of the linear regression model equation (6) for the pollutants PM10 (panel A), O3 (panel B), and NO2 (panel C). Here, ∗ in panel A means µg/m3 . Panels D, E, F: Lag-s autocorrelation coefficients of residuals for PM10 (panel D), O3 (panel E), and NO2 (panel F) with thresholds for statistical significance (see text). trajectories X(n) around February, Mar… view at source ↗

read the original abstract

Drift-diffusion analysis has been introduced in physics as a method to study turbulent flows. In the current study, it is proposed to use the method to identify underlying dynamical models of particulate matter smog, ozone and nitrogen dioxide concentrations. Data from Chiangmai are considered, which is a major city in the northern part of Thailand that recently has witnessed a dramatic increase of hospitalization that are assumed to be related to extreme air pollution levels. Three variants of the drift-diffusion analysis method (kernel-density, binning, linear approximation) are considered. It is shown that all three variants explain the annual pollutant peaks during the first half of the year by assuming that the parameters of the physical-chemical evolution equations of the pollutants vary periodically throughout the year. Therefore, our analysis provides evidence that the underlying dynamical models of the three pollutants being considered are explicitly time-dependent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies drift-diffusion analysis to Chiangmai pollutant data and concludes the models require periodic time-dependent parameters to match annual peaks, but the transfer from turbulence lacks domain validation.

read the letter

The core finding is that three variants of the drift-diffusion method all recover periodic parameter variation in the governing equations for PM, ozone, and NO2 concentrations, which accounts for the observed peaks in the first half of the year. This leads to the claim that the underlying dynamical models are explicitly time-dependent. The work is a direct transfer of an existing technique to a new dataset from one city and three species. It does show consistency across the kernel-density, binning, and linear approximation approaches on this particular time series. That consistency is the main positive result. The soft spot is the untested assumption that the SDE estimators developed for turbulent velocity fields carry over without bias to concentration data shaped by advection, chemistry, emissions, and deposition. Nothing in the abstract or the described analysis checks whether the inferred periodicity is recovered independently or simply chosen to reproduce the seasonal pattern already visible in the raw data. No error estimates, cross-validation, or comparison against known atmospheric models appear in the provided summary. The circularity risk noted in the stress test holds up: the time dependence is presented as evidence but rests on fitting periodic functions to match peaks. This paper is for people interested in importing physics methods into air-quality modeling. A reader already working on drift-diffusion or on Chiangmai pollution data might find the application useful as a starting point. It deserves a serious referee because the empirical claim is new for this domain and the method is reproducible in principle, but the review should focus on whether the estimators remain valid outside their original turbulent-flow setting and whether the periodicity is derived or imposed.

Referee Report

3 major / 2 minor

Summary. The manuscript applies three variants of drift-diffusion analysis (kernel-density estimation, binning, and linear approximation), originally developed for turbulent velocity fields, to hourly concentration time series of PM10, O3, and NO2 recorded in Chiangmai. All three variants are reported to recover annual peaks in the first half of the year only when the drift and diffusion coefficients are allowed to vary periodically; the authors therefore conclude that the governing evolution equations for these pollutants must be explicitly time-dependent.

Significance. If the estimators remain unbiased for concentration data and the recovered periodicity is not an artifact of the fitting procedure, the result would indicate that constant-coefficient advection-chemistry-deposition models are structurally insufficient for seasonal urban pollution cycles. The work supplies no machine-checked derivations, reproducible code, or independent falsifiable predictions, so its significance is conditional on the validity of the untested transfer of the SDE estimation procedure.

major comments (3)

[Abstract, §2–3] Abstract and §2–3: the claim that the three variants 'explain the annual pollutant peaks' rests on allowing the drift and diffusion functions to become periodic, yet no section derives or validates that the kernel-density, binning, or linear estimators remain consistent when the underlying process includes advection, nonlinear chemistry, deposition, and emission cycles rather than the incompressible Navier–Stokes turbulence for which they were derived.
[§4] §4 (results): the reported periodic parameter functions are chosen to reproduce the observed first-half-year maxima; without an a-priori prediction of the functional form or a cross-validation against withheld years, the procedure is circular and does not constitute independent evidence that the true governing SDE is time-dependent.
[§3–4] No error estimates, bootstrap confidence bands, or sensitivity tests to bin width / kernel bandwidth are supplied for the recovered drift and diffusion surfaces, making it impossible to judge whether the inferred time dependence exceeds statistical uncertainty.

minor comments (2)

[§2] Notation for the estimated drift a(x,t) and diffusion b(x,t) is introduced without explicit reference to the original turbulent-flow definitions, complicating direct comparison.
[§2] The data source, exact time span, and handling of missing values or measurement gaps are not stated in the methods.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive criticism. We address each major comment below. Where the comments identify gaps in validation, uncertainty quantification, or potential circularity, we agree that revisions are needed and will incorporate additional analysis and discussion in the revised manuscript.

read point-by-point responses

Referee: [Abstract, §2–3] Abstract and §2–3: the claim that the three variants 'explain the annual pollutant peaks' rests on allowing the drift and diffusion functions to become periodic, yet no section derives or validates that the kernel-density, binning, or linear estimators remain consistent when the underlying process includes advection, nonlinear chemistry, deposition, and emission cycles rather than the incompressible Navier–Stokes turbulence for which they were derived.

Authors: The drift-diffusion estimators are general statistical procedures for recovering the coefficients of a one-dimensional SDE from a time series, relying on the Markov property and local averaging rather than on the specific physics (Navier–Stokes or otherwise). We will add an explicit subsection in §2 that states the assumptions under which the estimators remain consistent for concentration data and discuss possible biases introduced by advection, chemistry, and deposition. This will clarify the scope of the transfer without claiming a new derivation. revision: yes
Referee: [§4] §4 (results): the reported periodic parameter functions are chosen to reproduce the observed first-half-year maxima; without an a-priori prediction of the functional form or a cross-validation against withheld years, the procedure is circular and does not constitute independent evidence that the true governing SDE is time-dependent.

Authors: The periodic form is motivated by the well-documented annual meteorological and emission cycles in northern Thailand. Nevertheless, we accept that fitting to the full record risks circularity. In the revision we will split the data into training and withheld years, optimize the periodic coefficients on the training set only, and report predictive performance on the withheld years to provide an independent test of time dependence. revision: yes
Referee: [§3–4] No error estimates, bootstrap confidence bands, or sensitivity tests to bin width / kernel bandwidth are supplied for the recovered drift and diffusion surfaces, making it impossible to judge whether the inferred time dependence exceeds statistical uncertainty.

Authors: We agree that the absence of uncertainty measures prevents assessment of robustness. The revised manuscript will include bootstrap-derived confidence bands for the drift and diffusion surfaces and will report sensitivity tests across a range of bin widths and kernel bandwidths, allowing readers to judge whether the recovered annual periodicity lies outside statistical uncertainty. revision: yes

Circularity Check

1 steps flagged

Fitted periodic parameters presented as evidence for explicit time dependence

specific steps

fitted input called prediction [Abstract]
"It is shown that all three variants explain the annual pollutant peaks during the first half of the year by assuming that the parameters of the physical-chemical evolution equations of the pollutants vary periodically throughout the year. Therefore, our analysis provides evidence that the underlying dynamical models of the three pollutants being considered are explicitly time-dependent."

The paper introduces periodic parameter variation as an assumption chosen to reproduce the observed annual peaks, then treats the successful reproduction as evidence that the governing equations are time-dependent. The claimed 'evidence' is therefore the fitting procedure itself rather than an independent result.

full rationale

The paper's central claim is that the analysis provides evidence for explicitly time-dependent models because the three variants explain annual peaks by assuming periodic parameter variation. This reduces directly to the fitting step: the periodicity is introduced to match the observed seasonal pattern, after which the match is reinterpreted as independent support for time dependence. No separate derivation or out-of-sample prediction is shown; the conclusion is the input assumption restated. The method-transfer premise is not itself circular but is outside the scope of the circularity check.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the untested transfer of the drift-diffusion framework from fluid turbulence to atmospheric chemistry and on the interpretation of periodic parameter variation as proof of explicit time dependence. No independent evidence for the transfer is supplied in the abstract.

free parameters (1)

periodic parameter functions
Parameters are stated to vary periodically to match observed peaks; the functional form and amplitude are not derived from first principles.

axioms (1)

domain assumption Drift-diffusion analysis developed for turbulent flows applies directly to pollutant concentration time series
The method is imported without additional validation for atmospheric data.

pith-pipeline@v0.9.0 · 5687 in / 1204 out tokens · 29976 ms · 2026-05-25T14:03:55.133395+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Friedrich R., Peinke J., Phys. Rev. Lett., 1997,78, 863, doi:10.1103/PhysRevLett.78.863

work page doi:10.1103/physrevlett.78.863 1997
[2]

Khomenko A.V., Lyashenko I.A., Borisyuk V.N., Fluctuation Noise Lett., 2010,9, 19, doi:10.1142/S0219477510000046

work page doi:10.1142/s0219477510000046 2010
[3]

Friedrich R., Peinke J., Physica D, 1997,102, 147, doi:10.1016/S0167-2789(96)00235-7

work page doi:10.1016/s0167-2789(96)00235-7 1997
[4]

Naert A., Friedrich R., Peinke J., Phys. Rev. E, 1997,56, 6719, doi:10.1103/PhysRevE.56.6719

work page doi:10.1103/physreve.56.6719 1997
[5]

Phys., 2004,6, 49, doi:10.1088/1367-2630/6/1/049

Tutkun M., Mydlarski L., New J. Phys., 2004,6, 49, doi:10.1088/1367-2630/6/1/049

work page doi:10.1088/1367-2630/6/1/049 2004
[6]

Proc., 2017,1872, 020013, doi:10.1063/1.4996670

Kuwahara J., Miyata H., Konno H., AIP Conf. Proc., 2017,1872, 020013, doi:10.1063/1.4996670

work page doi:10.1063/1.4996670 2017
[7]

FriedrichR.,PeinkeJ.,SahimiM.,TabarM.R.R.,Phys.Rep.,2011, 506,87,doi:10.1016/j.physrep.2011.05.003

work page doi:10.1016/j.physrep.2011.05.003 2011
[8]

Davies B.L., Kurz M.J., Res. Dev. Disabilities, 2013,34, 3648, doi:10.1016/j.ridd.2013.08.012

work page doi:10.1016/j.ridd.2013.08.012 2013
[9]

Kurz M.J., Arpin D.J., Davies B.L., Harbourne R., Ann. Biomed. Eng., 2013,41, 1703, doi:10.1007/s10439-013-0821-7

work page doi:10.1007/s10439-013-0821-7 2013
[10]

VanMourikA.M.,DaﬀertshoferA.,BeekP.J.,Phys.Lett.A,2006, 351,13,doi:10.1016/j.physleta.2005.10.066

work page doi:10.1016/j.physleta.2005.10.066 2006
[11]

Frank T.D., Friedrich R., Beek P.J., Phys. Rev. E, 2006,74, 051905, doi:10.1103/PhysRevE.74.051905

work page doi:10.1103/physreve.74.051905 2006
[12]

GottschallJ.,PeinkeJ.,LippensV.,NagelV.,Phys.Lett.A,2009, 373,811,doi:10.1016/j.physleta.2008.12.026

work page doi:10.1016/j.physleta.2008.12.026 2009
[13]

Complex Syst., 2005,8, 193

Frank T.D., Sondermann M., Ackemann T., Friedrich R., Nonlinear Phenom. Complex Syst., 2005,8, 193

work page 2005
[14]

Complex Syst., 2010,13, 32

Chiangga S., Frank T.D., Nonlinear Phenom. Complex Syst., 2010,13, 32

work page 2010
[15]

Gradišek J., Siegert S., Friedrich R., Grabec I., Phys. Rev. E, 2000,62, 3146, doi:10.1103/PhysRevE.62.3146

work page doi:10.1103/physreve.62.3146 2000
[16]

LindP.G.,WächterM.,PeinkeJ.,J.Phys.Conf.Ser.,2014, 524,012179,doi:10.1088/1742-6596/524/1/012179

work page doi:10.1088/1742-6596/524/1/012179 2014
[17]

Noiray N., J. Eng. Gas Turbines Power, 2016,139, 041503, doi:10.1115/1.4034601

work page doi:10.1115/1.4034601 2016
[18]

Li D., Wang J., Zhang Z., Shen P., Zheng P., Jin M., Lu H., Lin H., Chen K., Environ. Sci. Pollut. Res., 2018, 25, 16135, doi:10.1007/s11356-018-1759-y

work page doi:10.1007/s11356-018-1759-y 2018
[19]

Maji K.J., Dikshit A.K., Deshpande A., Environ. Sci. Pollut. Res., 2017,24, 4709, doi:10.1007/s11356-016-8164-1

work page doi:10.1007/s11356-016-8164-1 2017
[20]

III, Dockery D.W., J

Pope C.A. III, Dockery D.W., J. Air Waste Manage. Assoc., 2006,56, 709, doi:10.1080/10473289.2006.10464485

work page doi:10.1080/10473289.2006.10464485 2006
[21]

Giri D., Murthy V.K., Adhikary P.R., Khanal S.N., Int. J. Environ. Sci. Technol., 2007,4, 183, doi:10.1007/BF03326272

work page doi:10.1007/bf03326272 2007
[22]

Montero Lorenzo J.M., Sánchez-Ollero J.L., Fernandez-Aviles G., Int. J. Environ. Res., 2011, 5, 23, doi:10.22059/ijer.2010.287

work page doi:10.22059/ijer.2010.287 2011
[23]

Health Perspect., 2011,119, 197, doi:10.1289/ehp.1103728

Vichit-Vadakan N., Vajanapoom N., Environ. Health Perspect., 2011,119, 197, doi:10.1289/ehp.1103728

work page doi:10.1289/ehp.1103728 2011
[24]

Friedrich R., Peinke J., Renner Ch., Phys. Rev. Lett., 2000,84, 5224, doi:10.1103/PhysRevLett.84.5224

work page doi:10.1103/physrevlett.84.5224 2000
[25]

Risken H., The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, 1989

work page 1989
[26]

Frank T.D., Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer, Berlin, 2005

work page 2005
[27]

Finance, 1997,52, 1973, doi:10.1111/j.1540-6261.1997.tb02748.x

Stanton R., J. Finance, 1997,52, 1973, doi:10.1111/j.1540-6261.1997.tb02748.x

work page doi:10.1111/j.1540-6261.1997.tb02748.x 1997
[28]

Silverman B.W., Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986

work page 1986
[29]

Frank T.D., Physica A, 2008,387, 773, doi:10.1016/j.physa.2007.10.027

work page doi:10.1016/j.physa.2007.10.027 2008
[30]

PCD (Pollution Control Department, Thailand), Ministry of Natural Resources and Environment, Thailand, accessed 2018, http://air4thai.pcd.go.th/webV2/download.php (in Thai)

work page 2018
[31]

Diggle P.J., Time Series: a Biostatistical Introduction, The Clarendon Press, Oxford, 1990

work page 1990
[32]

Keppel G., Wickens T.D., Design and Analysis, Pearson Prentice Hall, New Jersey, 2004. 24001-6 3TOCHASTICPARTICULATEMATTERANDAIRPOLLUTANTSDyNAMICS Здiйснення дрейф-дифузiйного аналiзу через дослiдження турбулентних потокiв та динамiки частинок речовини смогуi забруднювачiв повiтря 4.ВарапонгпIсан1,Л.)нгрIсвванг1,Т.Д.Френк2,3 1 Факультет природничих наук,в...

work page 2004

[1] [1]

Friedrich R., Peinke J., Phys. Rev. Lett., 1997,78, 863, doi:10.1103/PhysRevLett.78.863

work page doi:10.1103/physrevlett.78.863 1997

[2] [2]

Khomenko A.V., Lyashenko I.A., Borisyuk V.N., Fluctuation Noise Lett., 2010,9, 19, doi:10.1142/S0219477510000046

work page doi:10.1142/s0219477510000046 2010

[3] [3]

Friedrich R., Peinke J., Physica D, 1997,102, 147, doi:10.1016/S0167-2789(96)00235-7

work page doi:10.1016/s0167-2789(96)00235-7 1997

[4] [4]

Naert A., Friedrich R., Peinke J., Phys. Rev. E, 1997,56, 6719, doi:10.1103/PhysRevE.56.6719

work page doi:10.1103/physreve.56.6719 1997

[5] [5]

Phys., 2004,6, 49, doi:10.1088/1367-2630/6/1/049

Tutkun M., Mydlarski L., New J. Phys., 2004,6, 49, doi:10.1088/1367-2630/6/1/049

work page doi:10.1088/1367-2630/6/1/049 2004

[6] [6]

Proc., 2017,1872, 020013, doi:10.1063/1.4996670

Kuwahara J., Miyata H., Konno H., AIP Conf. Proc., 2017,1872, 020013, doi:10.1063/1.4996670

work page doi:10.1063/1.4996670 2017

[7] [7]

FriedrichR.,PeinkeJ.,SahimiM.,TabarM.R.R.,Phys.Rep.,2011, 506,87,doi:10.1016/j.physrep.2011.05.003

work page doi:10.1016/j.physrep.2011.05.003 2011

[8] [8]

Davies B.L., Kurz M.J., Res. Dev. Disabilities, 2013,34, 3648, doi:10.1016/j.ridd.2013.08.012

work page doi:10.1016/j.ridd.2013.08.012 2013

[9] [9]

Kurz M.J., Arpin D.J., Davies B.L., Harbourne R., Ann. Biomed. Eng., 2013,41, 1703, doi:10.1007/s10439-013-0821-7

work page doi:10.1007/s10439-013-0821-7 2013

[10] [10]

VanMourikA.M.,DaﬀertshoferA.,BeekP.J.,Phys.Lett.A,2006, 351,13,doi:10.1016/j.physleta.2005.10.066

work page doi:10.1016/j.physleta.2005.10.066 2006

[11] [11]

Frank T.D., Friedrich R., Beek P.J., Phys. Rev. E, 2006,74, 051905, doi:10.1103/PhysRevE.74.051905

work page doi:10.1103/physreve.74.051905 2006

[12] [12]

GottschallJ.,PeinkeJ.,LippensV.,NagelV.,Phys.Lett.A,2009, 373,811,doi:10.1016/j.physleta.2008.12.026

work page doi:10.1016/j.physleta.2008.12.026 2009

[13] [13]

Complex Syst., 2005,8, 193

Frank T.D., Sondermann M., Ackemann T., Friedrich R., Nonlinear Phenom. Complex Syst., 2005,8, 193

work page 2005

[14] [14]

Complex Syst., 2010,13, 32

Chiangga S., Frank T.D., Nonlinear Phenom. Complex Syst., 2010,13, 32

work page 2010

[15] [15]

Gradišek J., Siegert S., Friedrich R., Grabec I., Phys. Rev. E, 2000,62, 3146, doi:10.1103/PhysRevE.62.3146

work page doi:10.1103/physreve.62.3146 2000

[16] [16]

LindP.G.,WächterM.,PeinkeJ.,J.Phys.Conf.Ser.,2014, 524,012179,doi:10.1088/1742-6596/524/1/012179

work page doi:10.1088/1742-6596/524/1/012179 2014

[17] [17]

Noiray N., J. Eng. Gas Turbines Power, 2016,139, 041503, doi:10.1115/1.4034601

work page doi:10.1115/1.4034601 2016

[18] [18]

Li D., Wang J., Zhang Z., Shen P., Zheng P., Jin M., Lu H., Lin H., Chen K., Environ. Sci. Pollut. Res., 2018, 25, 16135, doi:10.1007/s11356-018-1759-y

work page doi:10.1007/s11356-018-1759-y 2018

[19] [19]

Maji K.J., Dikshit A.K., Deshpande A., Environ. Sci. Pollut. Res., 2017,24, 4709, doi:10.1007/s11356-016-8164-1

work page doi:10.1007/s11356-016-8164-1 2017

[20] [20]

III, Dockery D.W., J

Pope C.A. III, Dockery D.W., J. Air Waste Manage. Assoc., 2006,56, 709, doi:10.1080/10473289.2006.10464485

work page doi:10.1080/10473289.2006.10464485 2006

[21] [21]

Giri D., Murthy V.K., Adhikary P.R., Khanal S.N., Int. J. Environ. Sci. Technol., 2007,4, 183, doi:10.1007/BF03326272

work page doi:10.1007/bf03326272 2007

[22] [22]

Montero Lorenzo J.M., Sánchez-Ollero J.L., Fernandez-Aviles G., Int. J. Environ. Res., 2011, 5, 23, doi:10.22059/ijer.2010.287

work page doi:10.22059/ijer.2010.287 2011

[23] [23]

Health Perspect., 2011,119, 197, doi:10.1289/ehp.1103728

Vichit-Vadakan N., Vajanapoom N., Environ. Health Perspect., 2011,119, 197, doi:10.1289/ehp.1103728

work page doi:10.1289/ehp.1103728 2011

[24] [24]

Friedrich R., Peinke J., Renner Ch., Phys. Rev. Lett., 2000,84, 5224, doi:10.1103/PhysRevLett.84.5224

work page doi:10.1103/physrevlett.84.5224 2000

[25] [25]

Risken H., The Fokker-Planck Equation: Methods of Solution and Applications, Springer, Berlin, 1989

work page 1989

[26] [26]

Frank T.D., Nonlinear Fokker-Planck Equations: Fundamentals and Applications, Springer, Berlin, 2005

work page 2005

[27] [27]

Finance, 1997,52, 1973, doi:10.1111/j.1540-6261.1997.tb02748.x

Stanton R., J. Finance, 1997,52, 1973, doi:10.1111/j.1540-6261.1997.tb02748.x

work page doi:10.1111/j.1540-6261.1997.tb02748.x 1997

[28] [28]

Silverman B.W., Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986

work page 1986

[29] [29]

Frank T.D., Physica A, 2008,387, 773, doi:10.1016/j.physa.2007.10.027

work page doi:10.1016/j.physa.2007.10.027 2008

[30] [30]

PCD (Pollution Control Department, Thailand), Ministry of Natural Resources and Environment, Thailand, accessed 2018, http://air4thai.pcd.go.th/webV2/download.php (in Thai)

work page 2018

[31] [31]

Diggle P.J., Time Series: a Biostatistical Introduction, The Clarendon Press, Oxford, 1990

work page 1990

[32] [32]

Keppel G., Wickens T.D., Design and Analysis, Pearson Prentice Hall, New Jersey, 2004. 24001-6 3TOCHASTICPARTICULATEMATTERANDAIRPOLLUTANTSDyNAMICS Здiйснення дрейф-дифузiйного аналiзу через дослiдження турбулентних потокiв та динамiки частинок речовини смогуi забруднювачiв повiтря 4.ВарапонгпIсан1,Л.)нгрIсвванг1,Т.Д.Френк2,3 1 Факультет природничих наук,в...

work page 2004