估计季节周期的相位和幅度

Question

我有以下数据：

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')

这表明英国英格兰的1900年至2014年的月温度周期。

我想在this论文中概述的方法评估季节性温度循环的相位和幅度。具体来说，他们描述了给定12个月值（我们在这里），我们可以估算年度成分：

其中X（t）表示12个月的表面温度值，x（t + t0），t = 0.5，...，11.5，是每月降温的12个月值，其中因子为2是指正负频率。

然后季节周期的幅度和相位可以计算为

和

他们指出，每年的数据，他们使用傅里叶变换计算年度（每年一个周期）正弦分量，如上所示。

我对如何生成他们在这里展示的时间序列感到困惑。任何人都可以提供一些指导，说明我如何重现这些方法。注意，我也在matlab工作 - 如果有人对如何在该环境中实现这一点有任何建议。

以下是数据的子集。

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

从字面上看，Y的公式可以在MATLAB中表示为：

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

在不知道X的功能的情况下，我无法确定这确实可以被矢量化，或者您是否必须循环，这可以像：

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)

只需插入您想要的任何t0，循环上面的代码即可获得时间序列。