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Trend analysis: methodology

Trend analysis: methodology. Victor Shatalov. Meteorological Synthesizing Centre East. Main topics. Trend analysis of annual averages of concentration/deposition fluxes Trend analysis of monthly averages (with seasonal variations). Trend analysis: generalities. Residue. Trend.

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Trend analysis: methodology

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  1. Trend analysis: methodology Victor Shatalov Meteorological Synthesizing Centre East

  2. Main topics • Trend analysis of annual averages of concentration/deposition fluxes • Trend analysis of monthly averages (with seasonal variations)

  3. Trend analysis: generalities Residue Trend Aim: investigation of general tendencies in time series such as: • Measured and calculated pollutant concentrations at monitoring sites • Averageconcentrations/deposition fluxes in EMEP countries … Method: trend analysis – decomposition of the considered series into regular component (trend) and random component (residue)

  4. Main steps • Detection of trend and its character: • increasing • decreasing • mixed • Identification of trend type: • linear • quadratic • exponential • other • Quantification of trend: • total reduction • annual reduction • magnitude of seasonal variations • magnitude of random component • other • Interpretation of the obtained results Presentation by Markus Wallasch, 15 TFMM meeting, April 2014

  5. Determination of trend existence B[a]P measurements: SE12 0.14 Decreasing pair 0.12 0.10 0.08 3 ng/m 0.06 Increasing pair 0.04 0.02 0.00 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Mixed trend character: In the period from 1990 to 2000 – statistically significant (at 95% level) decreasing trend In the period from 2004 to 2010 – statistically significant (at 90% level) increasing trend Z = - 4.05 Z = 1.8 Mann-Kendall test: Z = (number of increasing pairs) – (number of decreasing pairs) with normalization. Critical values: ± 1.44 at 85% level ± 1.65 at 90% level ±1.96 at 95% level Z = - 1.49 Decreasing trend at 85% significance level Typical situation for HMs and POPs

  6. Z = - 3.1 decrease Z = 3.8 increase Determination of trend type: linear trend Conc = A · Time + B + ω Calculation of A and B: regression or Sen’s slope ω – residues (random component) Residual trend exists Criterion of the choice of trend type: Mann-Kendall test should not show statistically significant trend on all sub-periods of the time series

  7. Criterion of non-linearity Cichord Non-linear trend Δi Chord Linear trend Criterion of non-linearity of the obtained trend in time: NL = max[abs(Δi /Cichord)] · 100% i Fraction of non-linear trends Heavy metals (Pb) 87% POPs (B[a]P) 62% Supposed threshold value: 10%

  8. Determination of trend type: mono-exponential trend Conc = A · exp(- Time / t) + ω, t – characteristic time Calculation of A and t: least square method Z = 3.2 Z = - 3.3 increase decrease Residual trend exists

  9. Determination of trend type: polynomial trend Calculation of A, B and C: least square method Conc = A · Time2 + B · Time + C + ω Z = 0.5 Z = -2.3 no trend decrease Residual trend exists

  10. Determination of trend type: bi-exponential trend Calculated by least square method Conc = A1 · exp(- Time /t1) + A2 · exp(- Time /t2) Ai – amplitudes, ti – characteristic times Z = 0 Z = -1.4 no trend no trend No statistically significant residual trend obtained See [Smith, 2002]

  11. Statistical significance of increasing trend Confidence interval for trend slope: [TS0 + A, TS0 + B] TS0 – slope of calculated trend [A, B] – confidence interval for slope of random component Typical situation for B[a]P: increase in the end of the period Mann-Kendall test for 2004 – 2010: • does not confirm statistically significant increasing trend • does not claim the absence of increasing trend Z = 1.8 Increase is statistically significant

  12. Non-linear trend analysis Conc = A1 · exp(- Year / t1) + A2 · exp(- Year / t2) + ω Regression model, non-linear in the parameters t1 and t2 Non-linear regression models are widely investigated, for example: • Nonlinear regression, Gordon K. Smith, in Encyclopedia on Environmetrics, ISBN 0471899976, Wiley&Sons, 2002, vol 3, pp. 1405 – 1411 • Estimating and Validating Nonlinear Regression Metamodels in Simulation, I. R. dos Santos and A. M. O. Porta Nova, Communications in Statistics, Simulation and Computation, 2007, vol. 36: pp. 123 – 137 • Nonlinear regression, G. A. F. Seber and C. J. Wild, Wiley-Interscience, 2003

  13. ΔCi For the considered example: Rmin= - 6% (growth) Rmax= 15% Rav = 6% Rtot = 69% Parameters for trend characterization: reduction/growth Total reduction per period Rtot = (Сbeg–Cend)/Cbeg=1–Cend/Cbeg Cbeg Relative annual reduction Ri = ΔCi / Ci = (1 – Ci+1 / Ci) Cend Average annual reduction Rav = 1 – (Cend / Cbeg) 1/(N-1) where N – number of years Negative values of reduction mean growth Reduction parameters Rmin = min (Ri) Rmax = max (Ri) Rav Rtot

  14. Parameters for trend characterization: random component Frand Δ Parameter: standard deviation of random component normalized by trend values Frand = σ(Δ/Ctrend) For the considered example: Frand = 11%

  15. Seasonal variations of pollution Pb concentrations measured at EMEP site DE7 from 1990 to 2008. Seasonal variations are also seen. B[a]P concentrations measured at EMEP site CZ3 from 1996 to 2010. Pronounced seasonal variations are seen. Seasonal variations are characteristic of heavy metals and (particularly) for POPs

  16. Possible approaches to description of seasonal variations Mono-exponential approximation *) Conc = A · exp(– t / t + B · cos(2p · t – φ)) or Log(Conc) = A’ – t / t + B · cos(2p · t – φ) *) Kong et al., Statistical analysis of long-term monitoring data… Environ. Sci. Techn., 10/2014 t – time t – chatracteristic times, A, B – constants, φ – phase shifts. Bi-exponential approximation Conc = A1 · exp(– t / t1) · (1 + B1 · cos(2p · t – φ1)) + A2 · exp(– t / t2) · (1 + B2 · cos(2p · t – φ2))

  17. Usage of higher harmonics Possibility to avoid negative values: usage of higher harmonics Conc = Tr1 + Tr2 , Tri = Ai·exp(– t / ti)·(1+Bi·cos(2p·t–φi)+Ci·cos(4p·t–ψi)) Measurement data at CZ3 from 1996 to 2010 Trendcalculated by bi-exponential approach.Possible artifact: negative trend values Statistical significance of second harmonic: Fisher’s test F

  18. Poor approximation for small values of concentrations Residues for one-harmonic approximation Usage of higher harmonics Average B[a]P concentrations in Europe from 1990 to 2010 (main harmonic only) Pronounced harmonic trend with doubled frequency

  19. Poor approximation for small values of concentrations Trend including two harmonics Usage of higher harmonics Average B[a]P concentrations in Europe from 1990 to 2010 (main harmonic only) Significance of second harmonic is confirmed by Fisher’s test

  20. Splitting trends to particular components Crand Cmain Full trend Cseas Relative annual reductions (as above): Rmin, Rmax, Rav, Rtot Example: average B[a]P concentrations for Germany from 1990 to 2010. Ctot Ctot = Cmain + Cseas + Crand

  21. Splitting trends to particular components Crand Cmain Full trend Cseas Average value of the annual amplitude of the normalized seasonal component Fseas Threshold value: 10% Normalization: Cseas/Cmain Example: average B[a]P concentrations for Germany from 1990 to 2010. Ctot Ctot = Cmain + Cseas + Crand Fraction of trends with essential seasonality Heavy metals (Pb) 93% POPs (B[a]P) 100%

  22. Splitting trends to particular components Crand Cmain Full trend Normalization: Crand/Cmain Cseas Standard deviation of normalized random component Frand Example: average B[a]P concentrations for Germany from 1990 to 2010. Ctot Ctot = Cmain + Cseas + Crand

  23. Phase shift as a fingerprint of source type 14 Anthropogenic Trends for PB concentrations at CZ1 Secondary 12 3 10 8 Air concentrations, ng/m 6 4 Δφ 2 0 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 1990 1991 1992 2007 2008 2009 2010 Difference Δφ of phase shift φ between Pb pollution at CZ1 location due to anthropogenic and secondary sources. Phase shift can be used to determine which source type (anthropogenic or secondary) mainly contributes to the pollution at given location (in a particular country).

  24. List of trend parameters Parameters for trend characterization: • Relative reduction over the whole period (Rtot), • Relative annual reductions of contamination: • average over the period (Rav), • maximum (Rmax), • minimum (Rmin). • Relative contribution of seasonal variability (Fseas). • Relative contribution of random component (Frand). • Phase shift of maximum values of contamination with respect to the beginning of the year (φ). Statistical tests: • Non-linearity parameter (NL) 10% • Relative contribution of seasonal variability (Fseas) 10%

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