GY663 Lisa Orme
Natasha Langton
1 May 2019
For the purpose of this assignment the cycles of the Central England Temperature record and North Atlantic Oscillation were investigated. Initially the data was investigated individually, and then a cross-spectral analysis and a wavelet analysis was conducted. The following is the code used to conduct the analysis.
Central England Temperature (CET) record.
Set working directory to where the data is stored.
setwd("~/College/Msc Climate Change/GY663/Lisa Orme/ENC__Final_GY663_Workhshop")Load the appropriate packages for the analysis.
library(bspec)##
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## samplelibrary(lomb)
library(WaveletComp)
library(zoo)## Warning: package 'zoo' was built under R version 3.5.3##
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## as.Date, as.Date.numericlibrary(quantmod)## Warning: package 'quantmod' was built under R version 3.5.3## Loading required package: xts## Warning: package 'xts' was built under R version 3.5.3## Loading required package: TTR## Warning: package 'TTR' was built under R version 3.5.3## Version 0.4-0 included new data defaults. See ?getSymbols.library(dplyr)## Warning: package 'dplyr' was built under R version 3.5.3##
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## format.pval, unitsFor the purpose of this analysis, going to look at Central England Temperature record. Need to load the csv file into R using the following code.
CET = read.csv(file = "CET.csv", header = TRUE, sep = ",")Use head and tail function to visually inspect the data. From the output it is clear that there 14 columns that contain data regarding the years and months for CET. The data ranges from 2018 to 1668.
head(CET, n = 10)## Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC YEAR
## 1 2018 5.3 2.9 4.9 9.8 13.2 16.1 19.1 16.6 13.7 10.6 8.3 6.9 10.68
## 2 2017 4.0 6.1 8.7 8.9 13.2 16.0 16.8 15.6 13.5 12.4 6.8 4.8 10.58
## 3 2016 5.4 4.9 5.8 7.5 12.5 15.2 16.9 17.0 16.0 10.9 5.6 6.0 10.34
## 4 2015 4.4 4.0 6.4 9.0 10.8 14.0 15.9 15.9 12.6 11.0 9.5 9.7 10.31
## 5 2014 5.7 6.2 7.6 10.2 12.2 15.1 17.7 14.9 15.1 12.5 8.6 5.2 10.95
## 6 2013 3.5 3.2 2.7 7.5 10.4 13.6 18.3 16.9 13.7 12.5 6.2 6.3 9.61
## 7 2012 5.4 3.8 8.3 7.2 11.7 13.5 15.5 16.6 13.0 9.7 6.8 4.8 9.72
## 8 2011 3.7 6.4 6.7 11.8 12.2 13.8 15.2 15.4 15.1 12.6 9.6 6.0 10.72
## 9 2010 1.4 2.8 6.1 8.8 10.7 15.2 17.1 15.3 13.8 10.3 5.2 -0.7 8.86
## 10 2009 3.0 4.1 7.0 10.0 12.1 14.8 16.1 16.6 14.2 11.6 8.7 3.1 10.14tail(CET, n = 10)## Year JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC YEAR
## 351 1668 5 5 5 8 10 14 16 16 14 10 6 5 9.51
## 352 1667 0 4 2 7 10 15 17 16 13 9 6 3 8.52
## 353 1666 4 5 6 8 11 15 18 17 14 11 6 3 9.86
## 354 1665 1 1 5 7 10 14 16 15 13 9 6 2 8.29
## 355 1664 4 5 5 8 11 15 16 16 13 9 6 4 9.34
## 356 1663 1 1 5 7 10 14 15 15 13 10 7 5 8.63
## 357 1662 5 6 6 8 11 15 15 15 13 11 6 3 9.52
## 358 1661 5 5 6 8 11 14 15 15 13 11 8 6 9.78
## 359 1660 0 4 6 9 11 14 15 16 13 10 6 5 9.10
## 360 1659 3 4 6 7 11 13 16 16 13 10 5 2 8.87Can use the plot function to view the temperature variation for each of the months from 1668 to 2018. However from the output it is difficult to see if there is trends or cycles from the plot.
plot(CET$Year, CET$JAN, type = 'l', col = 'blue', main = 'CET temperature variation 1668 to 2018', xlab = 'Years', ylab = 'Temperature', ylim = c(-4,20), xlim = c(1590,2020))
lines(CET$Year, CET$FEB, col = 'cyan')
lines(CET$Year, CET$MAR, col = 'Green3')
lines(CET$Year, CET$APR, col = 'Orange')
lines(CET$Year, CET$MAY, col = 'Purple')
lines(CET$Year, CET$JUN, col = 'Yellow')
lines(CET$Year, CET$JUL, col = 'Pink')
lines(CET$Year, CET$AUG, col = 'Dark blue')
lines(CET$Year, CET$SEP, col = 'Dark green')
lines(CET$Year, CET$OCT, col = 'Red')
lines(CET$Year, CET$NOV, col = 'violet')
lines(CET$Year, CET$Dec, col = 'olivedrab')
legend("topleft",
legend=c("January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December"),
col= c("Blue", "Cyan", "Green3", "Orange", "Purple", "Yellow", "Pink", "Dark Blue",
"Dark Green", "Red", "Violet", "Olivedrab"), cex = 0.8, lty=1,lwd=1, bty="n")
Therefore, plot the annually averaged CET records to visualise any trends.There appears to be an increasing trend in temperature from about 1970 onwards.
plot(CET$Year, CET$YEAR, type = 'l', main = 'Annual average CET from 1668 to 2018', xlab = 'Years', ylab = 'Annually averaged Temperature', col = 'Olivedrab')
Want to do seasonal analysis so therefore make new dataframes that contain Winter (DJF), Spring (MAM), Summer (JJA) and Autumn (SON)
CET.winter <- CET[c(1:360), c(1,2,3,13)]
CET.spring <- CET[c(1:360), c(1,4,5,6)]
CET.summer <- CET[c(1:360), c(1,7,8,9)]
CET.autumn <- CET[c(1:360), c(1,10,11,12)]Get the rowMeans of each of the months. This calculates the mean of each row, using square brackets allows for selection of columns, in which row means is to be conducted. The mean of the three rows is then stored in a new column named average.
CET.winter$average <- rowMeans(CET.winter[,c(2,3,4)])
CET.spring$average <- rowMeans(CET.spring[,c(2,3,4)])
CET.summer$average <- rowMeans(CET.summer[,c(2,3,4)])
CET.autumn$average <- rowMeans(CET.autumn[,c(2,3,4)])For the purpose of this analysis, we just want the annual averages for each of the seasons. Therefore remove the month columns from each of the dataframes. This can be done by using the square brackets.
CET.winter <- CET.winter[c(1:360), c(1,5)]
CET.spring <- CET.spring[c(1:360), c(1,5)]
CET.summer <- CET.summer[c(1:360), c(1,5)]
CET.autumn <- CET.autumn[c(1:360), c(1,5)]Can use the plot function to have a look at the seasonal trends in temperature.
plot(CET.winter$Year, CET.winter$average, type = 'l', xlab = 'years', ylab = 'temperature', main = 'Plot of seasonal temperature', col = 'green3', ylim = c(-2,18))
lines(CET.summer$Year, CET.summer$average, col = 'orange')
lines(CET.autumn$Year, CET.autumn$average, col = 'violet')
lines(CET.spring$Year, CET.spring$average, col = 'cyan')
legend("bottomright",
legend=c("Winter", "Spring", "Summer", "Autumn"),
col= c("Green3", "Cyan", "Orange","Violet"), cex = 0.8, lty=1,lwd=1, bty="n")
From the graph it is clear that summer has the highest temperatures, followed by Autumn. Winter has the lowest temperatures and is followed by Spring.
Detrend each of the seasonal data. This removes the linear trend from stratigraphic series.
CET.winter.detrend <- detrend(CET.winter)##
## ----- SUBTRACTING LINEAR TREND FROM STRATIGRAPHIC SERIES -----
## * Slope= 0.003953477
## * y-intercept= -3.516986
CET.spring.detrend <- detrend(CET.spring)##
## ----- SUBTRACTING LINEAR TREND FROM STRATIGRAPHIC SERIES -----
## * Slope= 0.002825459
## * y-intercept= 2.978171
CET.summer.detrend <- detrend(CET.summer)##
## ----- SUBTRACTING LINEAR TREND FROM STRATIGRAPHIC SERIES -----
## * Slope= 0.001107789
## * y-intercept= 13.28379
CET.autumn.detrend <- detrend(CET.autumn)##
## ----- SUBTRACTING LINEAR TREND FROM STRATIGRAPHIC SERIES -----
## * Slope= 0.003253849
## * y-intercept= 3.736503
Need to reorder the CET record data so that it starts in 1659 and ends in 2018. This will enable the creation of a timeseries dataframe.
CET.winter <- as.data.frame(CET.winter)
CET.winter <- CET.winter[order(CET.winter$Year),]
CET.spring <- as.data.frame(CET.spring)
CET.spring <- CET.spring[order(CET.spring$Year),]
CET.summer <- as.data.frame(CET.summer)
CET.summer <- CET.summer[order(CET.summer$Year),]
CET.autumn <- as.data.frame(CET.autumn)
CET.autumn <- CET.autumn[order(CET.autumn$Year),]Convert the average temperature for each month in timeseries data
CET.winter.temp <- ts(CET.winter$average, start = 1659, end = 2018)
CET.spring.temp <- ts(CET.spring$average, start = 1659, end = 2018)
CET.summer.temp <- ts(CET.summer$average, start = 1659, end = 2018)
CET.autumn.temp <- ts(CET.autumn$average, start = 1659, end = 2018)Calculate the welch method for each of the seasonal average temperatures and plot. Segment length is set to 70 so that cycles and frequencies may be identified.
CET.winter.spec1 <- welchPSD(CET.winter.temp, seglength=70)
plot(CET.winter.spec1$frequency, CET.winter.spec1$power, log='y',type="l") 
CET.spring.spec1 <- welchPSD(CET.spring.temp, seglength=70)
plot(CET.spring.spec1$frequency, CET.spring.spec1$power, log='y',type="l") 
CET.summer.spec1 <- welchPSD(CET.summer.temp, seglength=70)
plot(CET.summer.spec1$frequency, CET.summer.spec1$power, log='y',type="l") 
CET.autumn.spec1 <- welchPSD(CET.autumn.temp, seglength=70)
plot(CET.autumn.spec1$frequency, CET.autumn.spec1$power, log='y',type="l") 
To find the frequency and the cycles for Winter data, the following code was used. With a threshold of 5, the only frequency peak is equal to 0.1. The cycle identified based on a threshold of 5 has one peak of 10 years.
CET.winter.spec1$frequency[findPeaks(CET.winter.spec1$power, thresh=5)]## [1] 0.11/0.1## [1] 10To find the frequency and cycles for the spring data the threshold was set to 5. This gave one frequency peak of 0.0714. The cycle identified has one peak of 14.0056 years.
CET.spring.spec1$frequency[findPeaks(CET.spring.spec1$power, thresh=5)]## [1] 0.071428571/0.0714## [1] 14.0056To find the cycles and frequency for summer data the threshold was set to 5. This resulted in one frequency peak of 0.0571. The cycle identified had one peak of 17.5131 years.
CET.summer.spec1$frequency[findPeaks(CET.summer.spec1$power, thresh=5)]## [1] 0.057142861/0.0571## [1] 17.51313To find the frequency and cycles associated with the autumn data, the threshold was set to 5. This resulted in one frequency peak of 0.1286. The cycle identified had one peak of 7.7760 years.
CET.autumn.spec1$frequency[findPeaks(CET.autumn.spec1$power, thresh=5)]## [1] 0.12857141/0.1286## [1] 7.77605Export the detrended seasonal temperatures.
write.table(CET.winter.detrend, file = "CET.winter.detrend.csv", row.names = FALSE, col.names = TRUE, sep = ",")
write.table(CET.spring.detrend, file = "CET.spring.detrend.csv", row.names = FALSE, col.names = TRUE, sep = ",")
write.table(CET.summer.detrend, file = "CET.summer.detrend.csv", row.names = FALSE, col.names = TRUE, sep = ",")
write.table(CET.autumn.detrend, file = "CET.autumn.detrend.csv", row.names = FALSE, col.names = TRUE, sep = ",")NAO
Import the NAO data using the read.csv and read.table function.
NAO.1 = read.csv(file = "NAO.csv", header = FALSE, sep = ",")
NAO <- read.table("NAO.csv", header=FALSE, sep=",")[,2] Use and head and tail function to have a quick look at the data. The data starts in 1826 and ends in 2017.
head(NAO.1)## V1 V2
## 1 1825 0.16
## 2 1826 0.27
## 3 1827 -0.45
## 4 1828 -0.88
## 5 1829 -0.09
## 6 1830 0.88tail(NAO.1)## V1 V2
## 188 2012 -0.63
## 189 2013 0.59
## 190 2014 0.10
## 191 2015 1.16
## 192 2016 0.39
## 193 2017 0.57Plot NAO to visualise the variability. From the graph it is clear that NAO is highly variable, however there is a sharp decline in NAO around 2010.
plot(NAO.1$V1, NAO.1$V2, type = 'l', xlab = 'Years', ylab = 'NAO', main = 'NAO 1826 to 2017', col = 'darkblue')
Need to convert NAO into a timeseries for the purpose of this analysis.
NAO.ts <- ts(NAO, start=1826, end=2017)Carry out the Welch spectral analysis on the NAO data.
NAO.Spec1 <- welchPSD(NAO.ts, seglength=100)
plot(NAO.Spec1$frequency, NAO.Spec1$power, type = 'l', ylab = 'power', xlab = 'frequency', main = 'Welch Spectral Analysis NAO', col = 'violet' )
To find the frequency and cycles associated with the NAO data, the following code was used and the threshold was set to 1. This resulted in three frequency peaks of 0.09, 0.14 and 0.21.
NAO.Spec1$frequency[findPeaks(NAO.Spec1$power, thresh=1)] ## [1] 0.09 0.14 0.21To find the cycles use the following code. The cycle identified had three peaks of 11.1111 years, 7.14285 years and 4.761905 years.
1/0.09## [1] 11.111111/0.14## [1] 7.1428571/0.21## [1] 4.761905Cross Spectral Analysis of CET record and NAO variability to investigate if there is a shared cycle.
Need to use the timeseries data for both NAO variability and CET seasonal records. Going to rename each of the data with .ts so that it is easy to recognise that they are timeseries data.
CET.winter.ts <- CET.winter.temp
CET.spring.ts <- CET.spring.temp
CET.summer.ts <- CET.summer.temp
CET.autumn.ts <- CET.autumn.temp Need to combine the NAO data with the seasonal data for the CET records into one dataframe. This only keeps the records that span the same length which is 1825 to 2016.
winter.ts <- ts.intersect(CET.winter.ts, NAO.ts, dframe = TRUE)
spring.ts <- ts.intersect(CET.spring.ts, NAO.ts, dframe = TRUE)
summer.ts <- ts.intersect(CET.summer.ts, NAO.ts, dframe = TRUE)
autumn.ts <- ts.intersect(CET.autumn.ts, NAO.ts, dframe = TRUE)Use the correlation function to investigate if the two timeseries for Winter have a significant correlation.
correlation = rcorr(winter.ts$CET.winter.ts,winter.ts$NAO.ts, type="pearson")
print(correlation)## x y
## x 1.00 0.05
## y 0.05 1.00
##
## n= 192
##
##
## P
## x y
## x 0.529
## y 0.529From the output the y value is 0.529, the n value is 192. Use the table given in GY663 workshop to assess the correlation. Therefore go to row 200 at the 5% significant level and the value is 0.138. Therefore the correlation for Winter is significant as the value 0.529 is greater than 0.138.
Similarly, investigate if there is a signifcant correlation between Spring CET records and NAO.
correlation = rcorr(spring.ts$CET.spring.ts,spring.ts$NAO.ts, type="pearson")
print(correlation)## x y
## x 1.00 -0.09
## y -0.09 1.00
##
## n= 192
##
##
## P
## x y
## x 0.2027
## y 0.2027From the output the y value is 0.2027, the n value is 192. Use the table given in GY663 workshop to assess the correlation. Therefore go to row 200 at the 5% significant level and the value is 0.138. Therefore the correlation for Spring is significant as the value 0.2027 which is greater than 0.138.
Investigate the correlation between NAO and summer CET records.
correlation = rcorr(summer.ts$CET.summer.ts,summer.ts$NAO.ts, type="pearson")
print(correlation)## x y
## x 1.00 0.07
## y 0.07 1.00
##
## n= 192
##
##
## P
## x y
## x 0.363
## y 0.363From the output the y value is 0.363, the n value is 192. Use the table given in GY663 workshop to assess the correlation. Therefore go to row 200 at the 5% significant level and the value is 0.138. Therefore the correlation for Summer is significant as the y value is 0.363 which is greater than 0.138.
Lastly investigate if there is a significant correlation between Autumn CET records and NAO.
correlation = rcorr(autumn.ts$CET.autumn.ts,autumn.ts$NAO.ts, type="pearson")
print(correlation)## x y
## x 1.00 -0.07
## y -0.07 1.00
##
## n= 192
##
##
## P
## x y
## x 0.3434
## y 0.3434From the output the y value is 0.3434, the n value is 192. Use the table given in GY663 workshop to assess the correlation. Go to row 200 at the 5% significant level and the value is 0.138. Therefore the correlation for Autumn is significant as the value 0.3434 is greater than 0.138.
Calculate the cross spectral analysis between the two timeseries for each of the CET seasons.
Winter.DF <- crossSpectrum(winter.ts)
Winter.transferFunction <- Winter.DF$Pxy / Winter.DF$Pxx
Winter.transferAmp <- Mod(Winter.transferFunction)
Spring.DF <- crossSpectrum(spring.ts)
Spring.transferFunction <- Spring.DF$Pxy / Spring.DF$Pxx
Spring.transferAmp <- Mod(Spring.transferFunction)
Summer.DF <- crossSpectrum(summer.ts)
Summer.transferFunction <- Summer.DF$Pxy / Summer.DF$Pxx
Summer.transferAmp <- Mod(Summer.transferFunction)
Autumn.DF <- crossSpectrum(autumn.ts)
Autumn.transferFunction <- Autumn.DF$Pxy / Autumn.DF$Pxx
Autumn.transferAmp <- Mod(Autumn.transferFunction) Plot Winter cross spectral analysis.
plot(Winter.DF$freq,Winter.transferAmp,type='l',log='x', xlab="Frequency", main="Transfer Function Amplitude (Winter)") 
The highest peak from the cross spectral analysis, occurs at the frequency of 0.13.
Plot Spring cross spectral analysis.
plot(Spring.DF$freq,Spring.transferAmp,type='l',log='x', xlab="Frequency", main="Spring Transfer Function Amplitude") 
From the plot there are a number of high frequency peaks, however the two highest peaks occur at a relatively close frequency. They both occur around the frequency 0.400 and 0.425.
Plot Summer cross spectral analysis.
plot(Summer.DF$freq,Summer.transferAmp,type='l',log='x', xlab="Frequency", main="Summer Transfer Function Amplitude") 
From the output there appears to be one major peak in frequency. This occurs at a frequency of about 0.200.
Plot Autumn cross spectral analysis.
plot(Autumn.DF$freq,Autumn.transferAmp,type='l',log='x', xlab="Frequency", main="Autumn Transfer Function Amplitude") 
From the output there a number of significant peaks in frequency, however the highest peak occurs at around 0.140.
Cross Wavelet Power Spectrum
This method is used to assess how two datasets covary through time.
First calender age needs to be defined. The length is 192 as that matches the length of the NAO temperature data (1825 to 2016)
date <- seq(as.POSIXct("1825-01-30 00:00:00", format = "%Y"),by = "year",length.out = 192) Need to combine the new date column with the NAO and temperature data on a seasonal basis.This can be done using the cbind function.
winter.ts <- cbind(winter.ts,date)
spring.ts <- cbind(spring.ts,date)
summer.ts <- cbind(summer.ts,date)
autumn.ts <- cbind(autumn.ts,date)Conduct a wavelet coherency analysis for each of the seasons.
winter.wc <- analyze.coherency(winter.ts, c(1, 2), dt=1)## Smoothing the time series...
## Starting wavelet transformation and coherency computation...
## ... and simulations...
##
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## Class attributes are accessible through following names:
## series loess.span dt dj Wave.xy Angle sWave.xy sAngle Power.xy Power.xy.avg Power.xy.pval Power.xy.avg.pval Coherency Coherence Coherence.avg Coherence.pval Coherence.avg.pval Wave.x Wave.y Phase.x Phase.y Ampl.x Ampl.y Power.x Power.y Power.x.avg Power.y.avg Power.x.pval Power.y.pval Power.x.avg.pval Power.y.avg.pval sPower.x sPower.y Ridge.xy Ridge.co Ridge.x Ridge.y Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tzspring.wc <- analyze.coherency(spring.ts, c(1, 2), dt=1)## Smoothing the time series...
## Starting wavelet transformation and coherency computation...
## ... and simulations...
##
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## Class attributes are accessible through following names:
## series loess.span dt dj Wave.xy Angle sWave.xy sAngle Power.xy Power.xy.avg Power.xy.pval Power.xy.avg.pval Coherency Coherence Coherence.avg Coherence.pval Coherence.avg.pval Wave.x Wave.y Phase.x Phase.y Ampl.x Ampl.y Power.x Power.y Power.x.avg Power.y.avg Power.x.pval Power.y.pval Power.x.avg.pval Power.y.avg.pval sPower.x sPower.y Ridge.xy Ridge.co Ridge.x Ridge.y Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tzsummer.wc <- analyze.coherency(summer.ts, c(1, 2), dt=1)## Smoothing the time series...
## Starting wavelet transformation and coherency computation...
## ... and simulations...
##
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## Class attributes are accessible through following names:
## series loess.span dt dj Wave.xy Angle sWave.xy sAngle Power.xy Power.xy.avg Power.xy.pval Power.xy.avg.pval Coherency Coherence Coherence.avg Coherence.pval Coherence.avg.pval Wave.x Wave.y Phase.x Phase.y Ampl.x Ampl.y Power.x Power.y Power.x.avg Power.y.avg Power.x.pval Power.y.pval Power.x.avg.pval Power.y.avg.pval sPower.x sPower.y Ridge.xy Ridge.co Ridge.x Ridge.y Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tzautumn.wc <- analyze.coherency(autumn.ts, c(1, 2), dt=1)## Smoothing the time series...
## Starting wavelet transformation and coherency computation...
## ... and simulations...
##
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## Class attributes are accessible through following names:
## series loess.span dt dj Wave.xy Angle sWave.xy sAngle Power.xy Power.xy.avg Power.xy.pval Power.xy.avg.pval Coherency Coherence Coherence.avg Coherence.pval Coherence.avg.pval Wave.x Wave.y Phase.x Phase.y Ampl.x Ampl.y Power.x Power.y Power.x.avg Power.y.avg Power.x.pval Power.y.pval Power.x.avg.pval Power.y.avg.pval sPower.x sPower.y Ridge.xy Ridge.co Ridge.x Ridge.y Period Scale nc nr coi.1 coi.2 axis.1 axis.2 date.format date.tzPlot the wavelet coherency analysis for winter.
wc.image(winter.wc, main = "cross-wavelet power spectrum, x over y (Winter)",
legend.params = list(lab = "cross-wavelet power levels"),
periodlab = "period (years)", show.date = TRUE) 
According to the graph the significant cycles occur every eight to sixteen years. These cycles are relatively short accordind to the map.
Plot the wavelet coherency analysis for spring.
wc.image(spring.wc, main = "cross-wavelet power spectrum, x over y (Spring)",
legend.params = list(lab = "cross-wavelet power levels"),
periodlab = "period (years)", show.date = TRUE) 
According to the plot for spring the significant cycles occur every eight, sixteen to thirty four years.
Plot the wavelet coherency analysis for Summer.
wc.image(summer.wc, main = "cross-wavelet power spectrum, x over y (Summer)",
legend.params = list(lab = "cross-wavelet power levels"),
periodlab = "period (years)", show.date = TRUE) 
From the graph the significant cycles occur every eight years. The cycles are relatively short according to the graph.
Plot the wavelet coherency analysis for Autumn.
wc.image(autumn.wc, main = "cross-wavelet power spectrum, x over y (Autumn)",
legend.params = list(lab = "cross-wavelet power levels"),
periodlab = "period (years)", show.date = TRUE) 
According the the output, significant cycles occur every eight years for Autumn.
Use the following code to calculate the average cycles for winter CET records and NAO . The significant cycles are marked by red points.
wc.avg(winter.wc, siglvl = 0.05, sigcol = 'red',
periodlab = "period (years)", main = "Winter") 
From the graph it is clear that the significant cycles for Winter occur every eight to sixteen years, which is indicated by the red dots.
wc.avg(spring.wc, siglvl = 0.05, sigcol = 'red',
periodlab = "period (years)", main = "Spring") 
From the output it is clear that the significant cycles for Spring occur every eight years.
wc.avg(summer.wc, siglvl = 0.05, sigcol = 'red',
periodlab = "period (years)", main = "Summer") 
From the graph it is evident that the significant cycles for Summer occur every fourteen to eight years.
wc.avg(autumn.wc, siglvl = 0.05, sigcol = 'red',
periodlab = "period (years)", main = "Autumn") 
Lastly form the graph, significant cycles for Autumn occur every eight years.