estimate phase and amplitude of a seasonal cycle

Question

I have the following data:

CET <- url("http://www.metoffice.gov.uk/hadobs/hadcet/cetml1659on.dat")
cet <- read.table(CET, sep = "", skip = 6, header = TRUE,
                  fill = TRUE, na.string = c(-99.99, -99.9))
names(cet) <- c(month.abb, "Annual")
cet <- cet[-nrow(cet), ]
rn <- as.numeric(rownames(cet))
Years <- rn[1]:rn[length(rn)]
annCET <- data.frame(Temperature = cet[, ncol(cet)],Year = Years)
cet <- cet[, -ncol(cet)]
cet <- stack(cet)[,2:1]
names(cet) <- c("Month","Temperature")
cet <- transform(cet, Year = (Year <- rep(Years, times = 12)),
                 nMonth = rep(1:12, each = length(Years)),
                 Date = as.Date(paste(Year, Month, "15", sep = "-"),format = "%Y-%b-%d"))
cet <- cet[with(cet, order(Date)), ]

idx <- cet$Year > 1900
cet <- cet[idx,]
cet <- cet[,c('Date','Temperature')]

plot(cet, type = 'l')

This demonstrates the monthly temperature cycle from 1900 to 2014 in England, UK.

I would like to evaluate the phase and amplitude of the seasonal cycle of temperature follwowing the methods outlined in this paper. Specifically, they describe that given 12 monthly values (as we have here) we can estimate the yearly component as:

where X(t) represents 12 monthly values of surface temperature, x(t+t0), t = 0.5,...,11.5, are 12 monthly values of the de-meaned monthly temperature, where the factor of two is to account for both positive and negative frequencies.

Then the amplitude and phase of the seasonal cycle can be calculated as

and

They specify, that each year of data, they calculate the yearly (one cycle per year) sinusoidal component using the Fourier transform, as the equation shown above.

I'm a bit stuck on how to generate the time series they demonstrate here. Can anyone please provide some guidance as to how I can reproduce these methods. Note, I also work in matlab - in case anyone has some suggestions as to how this would be achieved in that environment.

Here is a subset of the data.

Date    Temperature
1980-01-15  2.3
1980-02-15  5.7
1980-03-15  4.7
1980-04-15  8.8
1980-05-15  11.2
1980-06-15  13.8
1980-07-15  14.7
1980-08-15  15.9
1980-09-15  14.7
1980-10-15  9
1980-11-15  6.6
1980-12-15  5.6
1981-01-15  4.9
1981-02-15  3
1981-03-15  7.9
1981-04-15  7.8
1981-05-15  11.2
1981-06-15  13.2
1981-07-15  15.5
1981-08-15  16.2
1981-09-15  14.5
1981-10-15  8.6
1981-11-15  7.8
1981-12-15  0.3
1982-01-15  2.6
1982-02-15  4.8
1982-03-15  6.1
1982-04-15  8.6
1982-05-15  11.6
1982-06-15  15.5
1982-07-15  16.5
1982-08-15  15.7
1982-09-15  14.2
1982-10-15  10.1
1982-11-15  8
1982-12-15  4.4
1983-01-15  6.7
1983-02-15  1.7
1983-03-15  6.4
1983-04-15  6.8
1983-05-15  10.3
1983-06-15  14.4
1983-07-15  19.5
1983-08-15  17.3
1983-09-15  13.7
1983-10-15  10.5
1983-11-15  7.5
1983-12-15  5.6
1984-01-15  3.8
1984-02-15  3.3
1984-03-15  4.7
1984-04-15  8.1
1984-05-15  9.9
1984-06-15  14.5
1984-07-15  16.9
1984-08-15  17.6
1984-09-15  13.7
1984-10-15  11.1
1984-11-15  8
1984-12-15  5.2
1985-01-15  0.8
1985-02-15  2.1
1985-03-15  4.7
1985-04-15  8.3
1985-05-15  10.9
1985-06-15  12.7
1985-07-15  16.2
1985-08-15  14.6
1985-09-15  14.6
1985-10-15  11
1985-11-15  4.1
1985-12-15  6.3
1986-01-15  3.5
1986-02-15  -1.1
1986-03-15  4.9
1986-04-15  5.8
1986-05-15  11.1
1986-06-15  14.8
1986-07-15  15.9
1986-08-15  13.7
1986-09-15  11.3
1986-10-15  11
1986-11-15  7.8
1986-12-15  6.2
1987-01-15  0.8
1987-02-15  3.6
1987-03-15  4.1
1987-04-15  10.3
1987-05-15  10.1
1987-06-15  12.8
1987-07-15  15.9
1987-08-15  15.6
1987-09-15  13.6
1987-10-15  9.7
1987-11-15  6.5
1987-12-15  5.6
1988-01-15  5.3
1988-02-15  4.9
1988-03-15  6.4
1988-04-15  8.2
1988-05-15  11.9
1988-06-15  14.4
1988-07-15  14.7
1988-08-15  15.2
1988-09-15  13.2
1988-10-15  10.4
1988-11-15  5.2
1988-12-15  7.5
1989-01-15  6.1
1989-02-15  5.9
1989-03-15  7.5
1989-04-15  6.6
1989-05-15  13
1989-06-15  14.6
1989-07-15  18.2
1989-08-15  16.6
1989-09-15  14.7
1989-10-15  11.7
1989-11-15  6.2
1989-12-15  4.9
1990-01-15  6.5
1990-02-15  7.3
1990-03-15  8.3
1990-04-15  8
1990-05-15  12.6
1990-06-15  13.6
1990-07-15  16.9
1990-08-15  18
1990-09-15  13.2
1990-10-15  11.9
1990-11-15  6.9
1990-12-15  4.3
1991-01-15  3.3
1991-02-15  1.5
1991-03-15  7.9
1991-04-15  7.9
1991-05-15  10.8
1991-06-15  12.1
1991-07-15  17.3
1991-08-15  17.1
1991-09-15  14.7
1991-10-15  10.2
1991-11-15  6.8
1991-12-15  4.7
1992-01-15  3.7
1992-02-15  5.4
1992-03-15  7.5
1992-04-15  8.7
1992-05-15  13.6
1992-06-15  15.7
1992-07-15  16.2
1992-08-15  15.3
1992-09-15  13.4
1992-10-15  7.8
1992-11-15  7.4
1992-12-15  3.6
1993-01-15  5.9
1993-02-15  4.6
1993-03-15  6.7
1993-04-15  9.5
1993-05-15  11.4
1993-06-15  15
1993-07-15  15.2
1993-08-15  14.6
1993-09-15  12.4
1993-10-15  8.5
1993-11-15  4.6
1993-12-15  5.5
1994-01-15  5.3
1994-02-15  3.2
1994-03-15  7.7
1994-04-15  8.1
1994-05-15  10.7
1994-06-15  14.5
1994-07-15  18
1994-08-15  16
1994-09-15  12.7
1994-10-15  10.2
1994-11-15  10.1
1994-12-15  6.4
1995-01-15  4.8
1995-02-15  6.5
1995-03-15  5.6
1995-04-15  9.1
1995-05-15  11.6
1995-06-15  14.3
1995-07-15  18.6
1995-08-15  19.2
1995-09-15  13.7
1995-10-15  12.9
1995-11-15  7.7
1995-12-15  2.3
1996-01-15  4.3
1996-02-15  2.5
1996-03-15  4.5
1996-04-15  8.5
1996-05-15  9.1
1996-06-15  14.4
1996-07-15  16.5
1996-08-15  16.5
1996-09-15  13.6
1996-10-15  11.7
1996-11-15  5.9
1996-12-15  2.9
1997-01-15  2.5
1997-02-15  6.7
1997-03-15  8.4
1997-04-15  9
1997-05-15  11.5
1997-06-15  14.1
1997-07-15  16.7
1997-08-15  18.9
1997-09-15  14.2
1997-10-15  10.2
1997-11-15  8.4
1997-12-15  5.8
1998-01-15  5.2
1998-02-15  7.3
1998-03-15  7.9
1998-04-15  7.7
1998-05-15  13.1
1998-06-15  14.2
1998-07-15  15.5
1998-08-15  15.9
1998-09-15  14.9
1998-10-15  10.6
1998-11-15  6.2
1998-12-15  5.5
1999-01-15  5.5
1999-02-15  5.3
1999-03-15  7.4
1999-04-15  9.4
1999-05-15  12.9
1999-06-15  13.9
1999-07-15  17.7
1999-08-15  16.1
1999-09-15  15.6
1999-10-15  10.7
1999-11-15  7.9
1999-12-15  5
2000-01-15  4.9
2000-02-15  6.3
2000-03-15  7.6
2000-04-15  7.8
2000-05-15  12.1
2000-06-15  15.1
2000-07-15  15.5
2000-08-15  16.6
2000-09-15  14.7
2000-10-15  10.3
2000-11-15  7
2000-12-15  5.8
2001-01-15  3.2
2001-02-15  4.4
2001-03-15  5.2
2001-04-15  7.7
2001-05-15  12.6
2001-06-15  14.3
2001-07-15  17.2
2001-08-15  16.8
2001-09-15  13.4
2001-10-15  13.3
2001-11-15  7.5
2001-12-15  3.6
2002-01-15  5.5
2002-02-15  7
2002-03-15  7.6
2002-04-15  9.3
2002-05-15  11.8
2002-06-15  14.4
2002-07-15  16
2002-08-15  17
2002-09-15  14.4
2002-10-15  10.1
2002-11-15  8.5
2002-12-15  5.7
2003-01-15  4.5
2003-02-15  3.9
2003-03-15  7.5
2003-04-15  9.6
2003-05-15  12.1
2003-06-15  16.1
2003-07-15  17.6
2003-08-15  18.3
2003-09-15  14.3
2003-10-15  9.2
2003-11-15  8.1
2003-12-15  4.8
2004-01-15  5.2
2004-02-15  5.4
2004-03-15  6.5
2004-04-15  9.4
2004-05-15  12.1
2004-06-15  15.3
2004-07-15  15.8
2004-08-15  17.6
2004-09-15  14.9
2004-10-15  10.5
2004-11-15  7.7
2004-12-15  5.4
2005-01-15  6
2005-02-15  4.3
2005-03-15  7.2
2005-04-15  8.9
2005-05-15  11.4
2005-06-15  15.5
2005-07-15  16.9
2005-08-15  16.2
2005-09-15  15.2
2005-10-15  13.1
2005-11-15  6.2
2005-12-15  4.4
2006-01-15  4.3
2006-02-15  3.7
2006-03-15  4.9
2006-04-15  8.6
2006-05-15  12.3
2006-06-15  15.9
2006-07-15  19.7
2006-08-15  16.1
2006-09-15  16.8
2006-10-15  13
2006-11-15  8.1
2006-12-15  6.5
2007-01-15  7
2007-02-15  5.8
2007-03-15  7.2
2007-04-15  11.2
2007-05-15  11.9
2007-06-15  15.1
2007-07-15  15.2
2007-08-15  15.4
2007-09-15  13.8
2007-10-15  10.9
2007-11-15  7.3
2007-12-15  4.9
2008-01-15  6.6
2008-02-15  5.4
2008-03-15  6.1
2008-04-15  7.9
2008-05-15  13.4
2008-06-15  13.9
2008-07-15  16.2
2008-08-15  16.2
2008-09-15  13.5
2008-10-15  9.7
2008-11-15  7
2008-12-15  3.5
2009-01-15  3
2009-02-15  4.1
2009-03-15  7
2009-04-15  10
2009-05-15  12.1
2009-06-15  14.8
2009-07-15  16.1
2009-08-15  16.6
2009-09-15  14.2
2009-10-15  11.6
2009-11-15  8.7
2009-12-15  3.1
2010-01-15  1.4
2010-02-15  2.8
2010-03-15  6.1
2010-04-15  8.8
2010-05-15  10.7
2010-06-15  15.2
2010-07-15  17.1
2010-08-15  15.3
2010-09-15  13.8
2010-10-15  10.3
2010-11-15  5.2
2010-12-15  -0.7
2011-01-15  3.7
2011-02-15  6.4
2011-03-15  6.7
2011-04-15  11.8
2011-05-15  12.2
2011-06-15  13.8
2011-07-15  15.2
2011-08-15  15.4
2011-09-15  15.1
2011-10-15  12.6
2011-11-15  9.6
2011-12-15  6
2012-01-15  5.4
2012-02-15  3.8
2012-03-15  8.3
2012-04-15  7.2
2012-05-15  11.7
2012-06-15  13.5
2012-07-15  15.5
2012-08-15  16.6
2012-09-15  13
2012-10-15  9.7
2012-11-15  6.8
2012-12-15  4.8
2013-01-15  3.5
2013-02-15  3.2
2013-03-15  2.7
2013-04-15  7.5
2013-05-15  10.4
2013-06-15  13.6
2013-07-15  18.3
2013-08-15  16.9
2013-09-15  13.7
2013-10-15  12.5
2013-11-15  6.2
2013-12-15  6.3
2014-01-15  5.7
2014-02-15  6.2
2014-03-15  7.6
2014-04-15  10.2
2014-05-15  12.2
2014-06-15  15.1
2014-07-15  17.7
2014-08-15  14.9
2014-09-15  15.1
2014-10-15  12.5
2014-11-15  8.6
2014-12-15  5.2

Answer 1

Literally, the formula for Y can be represented in MATLAB as:

t=0.5:0.5:11.5; %//make sure the step size is indeed 0.5
Y = 1/6.*sum(exp(2*pi*i.*t/12).*X(t0-t); %// add the function for X
phi = atan2(imag(Y)/real(Y)); %// seasonal phase

without knowing the function for X I can't be sure this can indeed be vectorised, or whether you'd have to loop, which can be done like:

t=0.5:0.5:11.5; %//make sure the step size is indeed 0.5
Ytmp(numel(t),1)=0; %// initialise output
for ii = 1:numel(t)
    Ytmp(ii,1) = exp(2*pi*i.*t(ii)/12).*X(t0-t(ii));
end
Y = 1/6 * sum(Ytmp)

Just slot in any t0 you want, loop over the codes above and you have your time series.