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

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

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

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 Interpretation of the obtained results Presentation by Markus Wallasch, 15 TFMM meeting, April 2014 4

Determination of trend existence 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 B[a]P measurements: SE12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 ng/m 3 Decreasing pair Increasing pair Z = - 1.49 Decreasing trend at 85% significance level Z = - 4.05 Z = 1.8 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 Typical situation for HMs and POPs 5

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) Z = - 3.1 decrease Z = 3.8 increase 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

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

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 = - 3.3 Z = 3.2 decrease increase Residual trend exists

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

Determination of trend type: bi-exponential trend Conc = A1 · exp(- Time /t1) + A2 · exp(- Time /t2) Ai – amplitudes, ti – characteristic times Calculated by least square method Z = 0 Z = -1.4 no trend no trend Nonlnear regression, Gordon K. Smith, in Encyclopedia on Environmetrix, ISBN 0471899976, Wiley@Sons, 2002, vol 3, pp. 1405 - 1411 No statistically significant residual trend obtained See [Smith, 2002]

Statistical significance of increasing trend 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 TS0 – slope of calculated trend Confidence interval for trend slope: [TS0 + A, TS0 + B] [A, B] – confidence interval for slope of random component Increase is statistically significant 11

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 Nonlnear regression, Gordon K. Smith, in Encyclopedia on Environmetrix, ISBN 0471899976, Wiley@Sons, 2002, vol 3, pp. 1405 - 1411

Parameters for trend characterization: reduction/growth Total reduction per period Rtot = (Сbeg–Cend)/Cbeg=1–Cend/Cbeg Cbeg ΔCi Relative annual reduction Ri = ΔCi / Ci = (1 – Ci+1 / Ci) For the considered example: Rmin = - 6% (growth) Rmax = 15% Rav = 6% Rtot = 69% 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

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% 14

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

Possible approaches to description of seasonal variations 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)) 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 16

Usage of higher harmonics Measurement data at CZ3 from 1996 to 2010 Trend calculated by bi-exponential approach. Possible artifact: negative trend values 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)) Statistical significance of second harmonic: Fisher’s test F 17

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

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

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

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

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

Phase shift as a fingerprint of source type Trends for PB concentrations at CZ1 2 4 6 8 10 12 14 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Air concentrations, ng/m 3 Anthropogenic Secondary Δφ 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). 23

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% 24