Re: Deseasonalization and detrending of Keeling curve


David Winsemius wrote:
"Pekka Jarvela" <pekkajarvela@xxxxxxxxx> wrote in
news:1169326747.707212.147920@xxxxxxxxxxxxxxxxxxxxxxxxxxxx:

CO2 concentrations in atmosphere measured at Mauna Loa in Hawaii are
known as Keeling curve, http://en.wikipedia.org/wiki/Keeling_curve You
can get the update date from
http://scrippsco2.ucsd.edu/data/in_situ_co2/mlo_in_situ_record.txt In
this page it is said that

"The "detrended" data is seasonally adjusted by removing a 4-harmonic
fit with a linear gain factor. The "fit" is based on a stiff spline
plus 4-harmonic functions with linear gain."

1. Is detrending fitting a line y = ax + b to data and then subtracting
this line from data?
2. What does "removing a 4-harmonic fit with a linear gain factor"
mean? Has this something to do with Fourier analysis?


http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1190&context=sio

"The number of harmonics refers to a portion of the fitting function which
involves sinusoidal terms with a fundamental period of one year plus higher
order Fourier components. Thus, 2 harmonics indicates that terms with periods
of 1 year and 6 months were fit, 4 harmonics indicates additional terms with
periods of 4 and 3 months."

See also
http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1110&context=sio


This data set can be adequately modeled as a ARIMA Model of the
follwing form.

Rather than assume a particular deterministiv form, the data
autocorrelative structure can be examined which yields Gaussian
Residuals while pointing to anaomalies that didn't followthe paradigm
....suggesting unusual events or readings..


MODEL STAGE: 888 25EST 1


MODEL STATISTICS AND EQUATION FOR THE CURRENT EQUATION (DETAILS
FOLLOW).


Estimation/Diagnostic Checking for Variable Y C02

: NEWLY IDENTIFIED VARIABLE X1 I~P00064 1964/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X2 I~P00160 1972/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X3 I~P00046 1962/ 10
PULSE
: NEWLY IDENTIFIED VARIABLE X4 I~P00509 2001/ 5
PULSE
: NEWLY IDENTIFIED VARIABLE X5 I~P00448 1996/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X6 I~P00266 1981/ 2
PULSE
: NEWLY IDENTIFIED VARIABLE X7 I~P00376 1990/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X8 I~P00506 2001/ 2
PULSE
: NEWLY IDENTIFIED VARIABLE X9 I~P00081 1965/ 9
PULSE
: NEWLY IDENTIFIED VARIABLE X10 I~P00079 1965/ 7
PULSE
: NEWLY IDENTIFIED VARIABLE X11 I~P00577 2007/ 1
PULSE
: NEWLY IDENTIFIED VARIABLE X12 I~P00350 1988/ 2
PULSE






Number of Residuals (R) =n 564

Number of Degrees of Freedom =n-m 549

Residual Mean =Sum R / n -.121718E-03

Sum of Squares =Sum R**2 66.4928

Variance var=SOS/(n) .117895

Adjusted Variance =SOS/(n-m) .121116

Standard Deviation =SQRT(Adj Var) .348018

Standard Error of the Mean =Standard Dev/ .148530E-01

Mean / its Standard Error =Mean/SEM -.819486E-02

Mean Absolute Deviation =Sum(ABS(R))/n .280075

AIC Value ( Uses var ) =nln +2m -1175.81

SBC Value ( Uses var ) =nln +m*lnn -1110.78

BIC Value ( Uses var ) =see Wei p153 -103.425

R Square = .999691

Durbin-Watson Statistic =[A-A(T-1)]**2/A**2 1.99054


D-W STATISTIC SUGGESTS NO SIGNIFICANT AUTOCORRELATION for lag1.


THE DURBIN-WATSON STATISTIC IS VALID ONLY FOR MODELS THAT HAVE A WHITE
NOISE
ERROR TERM AND NO LAGS OF THE Y SERIES. OTHERWISE IT IS INVALID.
IN THIS CASE THE TEST IS INVALID.




FORECASTING WITH FINAL MODEL



MODEL COMPONENT LAG COEFF STANDARD P
T
# (BOP) ERROR VALUE
VALUE


Differencing 12

1CONSTANT .120 .294E-01 .0001
4.07
2Autoregressive-Factor # 1 1 .916 .194E-01 .0000
47.23
3Moving Average-Factor # 2 1 .210 .481E-01 .0000
4.36


INPUT SERIES X1 I~P00064 1964/ 4 PULSE



Differencing 12

4Omega (input) -Factor # 3 0 -1.68 .198 .0000
-8.49


INPUT SERIES X2 I~P00160 1972/ 4 PULSE



Differencing 12

5Omega (input) -Factor # 4 0 .901 .197 .0000
4.57


INPUT SERIES X3 I~P00046 1962/ 10 PULSE



Differencing 12

6Omega (input) -Factor # 5 0 -.547 .198 .0058
-2.77


INPUT SERIES X4 I~P00509 2001/ 5 PULSE



Differencing 12

7Omega (input) -Factor # 6 0 .627 .197 .0015
3.18


INPUT SERIES X5 I~P00448 1996/ 4 PULSE



Differencing 12

8Omega (input) -Factor # 7 0 -.715 .198 .0003
-3.62


INPUT SERIES X6 I~P00266 1981/ 2 PULSE



Differencing 12

9Omega (input) -Factor # 8 0 .507 .197 .0104
2.57


INPUT SERIES X7 I~P00376 1990/ 4 PULSE



Differencing 12

10Omega (input) -Factor # 9 0 -.646 .197 .0011
-3.28


INPUT SERIES X8 I~P00506 2001/ 2 PULSE



Differencing 12

11Omega (input) -Factor # 10 0 .426 .198 .0317
2.15


INPUT SERIES X9 I~P00081 1965/ 9 PULSE



Differencing 12

12Omega (input) -Factor # 11 0 .554 .198 .0052
2.80


INPUT SERIES X 10 I~P00079 1965/ 7 PULSE



Differencing 12

13Omega (input) -Factor # 12 0 .554 .198 .0053
2.80


INPUT SERIES X 11 I~P00577 2007/ 1 PULSE



14Omega (input) -Factor # 13 0 -1.08 .344 .0018
-3.14


INPUT SERIES X 12 I~P00350 1988/ 2 PULSE



15Omega (input) -Factor # 14 0 .713 .278 .0107
2.56






MODEL STATISTICS AND EQUATION FOR THE CURRENT EQUATION (DETAILS
FOLLOW).


Estimation/Diagnostic Checking for Variable Y C02

: NEWLY IDENTIFIED VARIABLE X1 I~P00064 1964/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X2 I~P00160 1972/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X3 I~P00046 1962/ 10
PULSE
: NEWLY IDENTIFIED VARIABLE X4 I~P00509 2001/ 5
PULSE
: NEWLY IDENTIFIED VARIABLE X5 I~P00448 1996/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X6 I~P00266 1981/ 2
PULSE
: NEWLY IDENTIFIED VARIABLE X7 I~P00376 1990/ 4
PULSE
: NEWLY IDENTIFIED VARIABLE X8 I~P00506 2001/ 2
PULSE
: NEWLY IDENTIFIED VARIABLE X9 I~P00081 1965/ 9
PULSE
: NEWLY IDENTIFIED VARIABLE X10 I~P00079 1965/ 7
PULSE
: NEWLY IDENTIFIED VARIABLE X11 I~P00577 2007/ 1
PULSE
: NEWLY IDENTIFIED VARIABLE X12 I~P00350 1988/ 2
PULSE






Number of Residuals (R) =n 564

Number of Degrees of Freedom =n-m 549

Residual Mean =Sum R / n -.121718E-03

Sum of Squares =Sum R**2 66.4928

Variance var=SOS/(n) .117895

Adjusted Variance =SOS/(n-m) .121116

Standard Deviation =SQRT(Adj Var) .348018

Standard Error of the Mean =Standard Dev/ .148530E-01

Mean / its Standard Error =Mean/SEM -.819486E-02

Mean Absolute Deviation =Sum(ABS(R))/n .280075

AIC Value ( Uses var ) =nln +2m -1175.81

SBC Value ( Uses var ) =nln +m*lnn -1110.78

BIC Value ( Uses var ) =see Wei p153 -103.425

R Square = .999691

Durbin-Watson Statistic =[A-A(T-1)]**2/A**2 1.99054


D-W STATISTIC SUGGESTS NO SIGNIFICANT AUTOCORRELATION for lag1.

Hope this helps ..

Dave Reilly
Automatic Forecasting Systems
http://www.autobox.com

.

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