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

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

3. 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

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

5. 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 =
Decreasing trend at 85% significance level
Z =
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

6. 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 = 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

7. 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

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 = - 3.3
Z = 3.2
decrease
increase
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
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 , 2002, vol 3, pp
No statistically significant residual trend obtained
See [Smith, 2002]

11. 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

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 , Wiley&Sons, 2002, vol 3, pp – 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 , 2002, vol 3, pp

13. 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

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% 14

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

16. 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

17. 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

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

19. Usage of higher harmonics
Average B[a]P concentrations in Europe from 1990 to (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

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
Relative annual reductions (as above): Rmin, Rmax, Rav, Rtot 20

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
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

22. 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

23. 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

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% 24