Seasonality analysis

The term seasonality means a periodical repeating fluctuation of pollutant concentration, such as decreasing value in some part of the year or culminating in another. It is well observable in PAHs, which are bound to the combustion processes and therefore exhibit nearly annual fluctuation. The whole thing is to find a time period, of these changes, comparing individual values by one of the method. Autoregression method uses the lag approach, shifting the time series repeatedly by one time period between measurements and computing the highest correlation of the time series with itself.

Autocorrelation

A basic concept of repeating pattern search is the autocorrelation, which is (on equidistant time series) defined as a correlation of the time series with itself, shifted in time by a period called time lag. It is considered as an autocorrelation function (ACF) of the time lag, which starts at 0 and continues by integer multiples of the period between two consequent measurements and could be estimated as:

The higher the value of ACF is for the chosen time lag, the stronger is a tendency to repeat similar values periodically after the period of this time. This is also used for an estimation of autoregression coefficients.

Autoregression

The principle of autoregression expects the time series of the form:

where ε_t denotes the influence of random processes in the form of white noise. Thus the constants φ_i characterize the shape of the time series and could serve as an input for computation of a frequency decomposition of the time series. Thee are several ways how to compute the φ_i constants; we used the most usual set of Yule-Walker equations in our R example.

Spectrum

Assuming that every time series can be decomposed to a set of individual frequencies, we can estimate spectral characteristics representing the time series spectrum. It can be computed from the estimated autoregression coefficients or using the Fourier transform as well.

The higher spectral density of individual frequency is, the higher tendency to repeat with this period the time series has. That means if there are obvious peaks on the line inside the periodogram, the time series probably repeats with these periods (note that most of the pollutants have significant peak in about 1-year time period).

There is a build-in function spectrum in the elementary R function pool, which serves for estimation of spectral density of time series, using autoregression or fast Fourier transform (FFT) to fit the time series points. We use the option spec.ar of the function, which provides suitable periodogram, reliable enough for the visual assessment whether there is some significant periodicity in the time series.

The code for the periodogram of fluorene is simple:

x<-spec.ar(flu,order=20)$freq

y<-spec.ar(flu,order=20)$spec

Now, the x and y coordinates of the points are computed, but it is necessary to invert the x axis (standard plot draws frequency instead of period) and recalculate the period of 28 days to 1-day sampling. The number 730 is set as 2-year period, which is a range of the axis:

yy<-y[which(x>28/730)]

xx<-28/x[which(x>28/730)]

plot(xx,yy,xlim=c(0,730),cex=0,xlab="Time (days)",ylab=”Spectral density”)

lines(xx,yy,lwd=4,col="darkviolet")

You can check periodograms of all 30 compounds included in our example dataset and – as in other examples – also insert your own time series with the defined period (in days – note, that the number do not have to be an integer). By a pair of sliders you can easily expand and contract the x-axis of the periodogram to find the accurate values of peaks and meaningful parts of the curve.

The autocorrelation plot is drawn even easier way using the acf function. In default setting blue borders of a confidence interval are also plotted, which are suppressed by ci=”” argument. Resulting values of autocorrelation are stored in the autocor variable, which can be used for drawing the connecting polyline.

autocor<-acf(flu,ci="",lag.max=length(flu),xaxt="n")$acf

lines(c(0:(length(flu)-1)),autocor,lwd=3,col="forestgreen")

For the correlation plot we need even only the elementary plot function and manual shifting the concentration values vector to itself by chosen time lag:

x<-sort(c(1:(length(flu)-lag)))

y<-sort(c((lag+1):length(flu)))

plot(flu[x],flu[y])