Environmental time series

1. Environmental time series Вонр. проф. д-р Александар Маркоски Технички факултет – Битола 2008 год. Enviromatics 2008 - Environmental time series

2. Introduction • Solving environmental problems or management tasks it is often necessary toanalyse cycling (or periodic) processes. Cycling processes in ecology are natural.Such processes are caused mostly by natural external driving forces butalso by natural and man-made internal driving forces. They lay out different timeand frequency behaviour of ecological processes. • Mathematical equations haveto describe either the time behaviour of an indicator (function of time t) or thefrequency behaviour (function of frequency ω or cycles per time unit). Therefore,a description will be presented in the time domain in the frequency domain. • In next figure some examples of cycling processes with different periods and frequenciesare presented. Enviromatics 2008 - Environmental time series

3. Example: Cycling water quality indicators Enviromatics 2008 - Environmental time series

4. Environmental processes • The mathematical representation of environmental processes by trend functionsis unsuitable to express high frequent changes of signals. Mostly, cycling (orperiodic) ecological processes are caused by natural external driving forces. • Onthe other hand, aperiodic environmental processes are mainly influenced byartificial (man-made) external driving forces. Another distinction can be made bythe ability to reproduce a time-varying process. • In the case of correct reproductionand forecast of a process it is called a deterministic one. • Otherwise it iscalled a non-deterministic or stochastic (random) process. Enviromatics 2008 - Environmental time series

5. Fourier analysis • A Fourier polynomial is based on the assumption that a time series containsimportant deterministic cycles of known period. It provides a mean of approximatingperiodic functions by sums of sine and cosine functions, shifted andscaled. A Fourier polynomial is an approximation which represents the minimummean squared deviation of a cycling process. It is described by with - ∞ ≤ i ≤ + ∞, ω0 = 2π/T0 – frequency of the basic cycle, T0 – period of cycle. Enviromatics 2008 - Environmental time series

6. Example 1: Fourier analysis of global radiation Enviromatics 2008 - Environmental time series

7. Results • As can be seen, the fixed frequency follows the natural behaviour of the ecologicalprocess in principle only. • Comparing the process maxima a shift betweennatural system behaviour and its approximation can be seen. The Fourierpolynomial based on fixed frequencies does not regard the natural yearly differences. • On the other hand, Fourier analysis can be used to estimate the influence of basic frequencies on the total variance of an ecological process. • Next table contains results of a Fourier analysis of physical, chemical and biological waterquality indicators for drinking water reservoirs. Enviromatics 2008 - Environmental time series

8. Example 2: Fourier analysis of water quality indicators of reservoirs Enviromatics 2008 - Environmental time series

9. Example 2 cont. • Best results will be getting by physical water quality indicators. The yearly cyclesof water temperature of the investigated reservoirs are described by thefollowing equations. • Reservoir Saidenbach TEMP(t) = 12.0 + 1.458⋅cos((6π/180)t) – 4.462⋅sin((6π/180)t • Reservoir Neunzehnhain TEMP(t) = 11.9 + 0.693⋅cos((6π/180)t) + 4.415⋅sin((6π/180)t • Reservoir Klicava TEMP(t) = 11.1 - 6.650⋅cos((9π/180)t) - 7.820⋅sin((9π/180)t) • Reservoir Slapy EMP(t) = 12.0 - 7.073⋅cos((10π/180)t) - 6.684⋅sin((10π/180)t) Enviromatics 2008 - Environmental time series

10. Stationary processes • Because of time lags between input and output processes stationary processeswill then be reached if all transient processes are decayed. • Therefore, somestatistical characteristics of signals should only be grasped. If statistical characteristicswill not change in time, then these processes are called stationaryprocesses. • Process averages and dispersions will not change in time. Therefore, stationary random processes can be investigated on different time intervalsbetween - ∞ < t < + ∞. Enviromatics 2008 - Environmental time series

11. Statistical characteristics of stationary random processes • Statistical characteristics of stationary random processes can be expressedby • Probability density function p(x) of signals X(t), • Auto-correlation function Φxx(τ), • Spectral power density function Sxx(ω) • An time varying process is expressed by a stochastic signal X(t). For each timestroke tn one measured value Xn(t) will be get. The further development of theprocess can be predicted only for a short time interval. Describing the processby an analytical (deterministic) function f(t) then the time behaviour can be predictedcompletely. Enviromatics 2008 - Environmental time series

12. Cont. • Only some statistical statements on the future developmentof the process X(t) can be given: • Prob(X(tn+1) ≤ x) ≡ P(x), • or • Prob(a < X(t) ≤ b) = ∫ p(x)dx. • The Gaussian distribution with a bell-shaped density is one of the most importantprobability density distributions where p(x) = 1/√2πσ⋅exp-(x-x*)2/2σ2. Importantexpectations are linear average: E(x) = ∫ x⋅p(x) dx and squared average:E(x2) = ∫ x2⋅p(x) dx.T • he probability density function gives an information about the probability of theprocess X(t) that the amplitude at time t lies between x and (x + Δx): • Prob(x < X(t) ≤ x + Δx) ≈ p(x)⋅Δx. Enviromatics 2008 - Environmental time series

13. Time correlation functions • No statements on changes of X(t) within time intervals Δx are made. It cannotbe seen if a process contains lower and/or higher frequencies. Therefore, multipleprobability distribution functions are necessary to describe the time varyingprocess behaviour. • The probability that X(t) at time t = t1 lies between x1 and x1+ Δx1 and at time t = t2 = t1 + τ between x2 + Δx2 (after τ time units) is approximatelygiven by • Prob(x1 < X(t1)≤ x1 + Δx1, x2 < X(t2) ≤ x2 + Δx2) ≈ p(x1, x2)⋅Δx1⋅Δx2. Enviromatics 2008 - Environmental time series

14. Cont… • The parameter τ gives information on the statistical coupling of data x(t1) andx(t1 + τ). • The auto-correlation function (ACF) gives information on the innercorrelation between data with the distance τ on the time axes: ∫∫(x(t)⋅x(t + τ)⋅p[x(t), x(t + τ)]dx(t)⋅dx(t + τ) ≡ Φxx(τ) • The cross-correlation function (CCF) gives information on the statistical correlationof two different processes X(t) and Y(t): ∫∫(x(t)⋅y(t + τ)⋅p[x(t), y(t + τ)]dx(t)⋅dy(t + τ) ≡ Φxy(τ). Enviromatics 2008 - Environmental time series

15. Frequency functions • Transforming the time correlation functions into the frequency domain one getsthe auto-power spectrum or the cross-power spectrum. The auto-power spectrumSxx(ω) of x(t) is the Fourier transform of the ACF: Sxx(ω) = Sxx(-ω) = 1/2π⋅∫Φxx(τ)⋅e-jωτ dτ. • The auto-power spectrum of an ecological process or signal is visualised by aperiodogram. It represents the dominant frequency of the process. It gives thespectrum of a stationary signal which is a distribution of the variance of the signalas a function of frequency. • The frequency components that account for thelargest share of the variance are revealed. Each peak represents the part of thevariance of the signal that is due to a cycle of a different period or length. Significantperiodicity in the signal will induce a sharp peak in a periodogram. Theauto-covariance function is the time domaincounterpart of periodogram. Enviromatics 2008 - Environmental time series

16. Example 3: Periodogram Enviromatics 2008 - Environmental time series

17. Coherency function • The cross-power spectrum Sxy(ω) of two stochastic ecological processes x(t)and y(t) is the Fourier transform of the CCF: • Sxy(ω) = 1/2π⋅∫Φxy(τ)⋅e-jωτ dτIt is a complex function. Coherency function • The coherency function Co(ω) is a measure of synchronicity of (two) signals. Itis calculated on the base of periodograms of both signals by • Coxy(ω) = |Sxy(ω)|2/Sxx(ω)⋅Syy(ω), • where |Sxy(ω)| = √Re(Sxx(ω))2 + Re(Syy(ω))2 and for the phase shift betweenboth signals ϕ(ω) = arc tan (Im(Sxy(ω))/Re(Sxy(ω))) is valid. • The limitation of CCF is considered by what is called a window function h(τ): • ~Sxy(ω) = 1/2π⋅∫Φxy(τ)⋅h(τ)⋅e-jωτ dτ = Sxy(α)⋅H(ω - α), • where H(ω) is the Fourier transform of h(τ) which distorts Sxy(ω) to ~Sxy(ω). Enviromatics 2008 - Environmental time series

18. Environmental time series The End Enviromatics 2008 - Environmental time series