Time series de-trend and df

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2 total messages Started by Len Makin Mon, 21 Dec 1992 01:39

#785

Author: Len Makin
Date: Mon, 21 Dec 1992 01:39

35 lines
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A Climate modeller colleague asked the following:
I am planning to remove trend from some GCM results (CO2 levels).
I suspect this affects degrees of freedom for t-tests, F-tests, etc.
Can  anyone help please?
I plan to remove the trend without changing the mean.
I suspect not changing mean implies that only one df is lost.

So if original data are 
y[1], y[2], ... , y[i], ... , y[N]
where i represents year, so data are equally spaced in time.

Detrended value for year i is:
y[i] - r(y,i) * ( sd(y) / sd(i) ) * ( i - mean(i) )

where r(y,i) is correlation between y & i
r(y,i) = ( sum(i*y[i]) - mean(i) * mean(y) ) / ( sd(i) * sd(y) )

Since i is simply 1, 2, ... , N
by algebra:
mean(i) = ( N + 1 ) / 2
sd(i) = sqrt( ( N * N - 1 ) / 12 )

In my case N = 20 for both control (normal CO2) & double CO2 samples.
I plan to detrend both samples before comparing 
(1) means using t-test 
(2) variances using F-test.

What is the effect of this detrending on degrees of freedom for these tests?
Are these tests meaningful?

Len.Makin@mel.dit.csiro.au               | Senior Computer Scientist    |
CSIRO Division of Information Technology | Supercomputing Support Group |
723 Swanston Street, Carlton             | Tel: +61 3 282 2622          |
VIC 3053, Australia                      | Fax: +61 3 282 2600          |

Re: Time series de-trend and df

#670

Author: R0264
Date: Tue, 22 Dec 1992 21:14

50 lines
2564 bytes

In article <1992Dec21.013932.24703@mel.dit.csiro.au>
len@mel.dit.csiro.au (Len Makin) writes:
 
>
>
>A Climate modeller colleague asked the following:
>I am planning to remove trend from some GCM results (CO2 levels).
>I suspect this affects degrees of freedom for t-tests, F-tests, etc.
>Can  anyone help please?
>I plan to remove the trend without changing the mean.
>I suspect not changing mean implies that only one df is lost.
>
>So if original data are
>y[1], y[2], ... , y[i], ... , y[N]
>where i represents year, so data are equally spaced in time.
>
>Detrended value for year i is:
>y[i] - r(y,i) * ( sd(y) / sd(i) ) * ( i - mean(i) )
>
>where r(y,i) is correlation between y & i
>r(y,i) = ( sum(i*y[i]) - mean(i) * mean(y) ) / ( sd(i) * sd(y) )
>
>Since i is simply 1, 2, ... , N
>by algebra:
>mean(i) = ( N + 1 ) / 2
>sd(i) = sqrt( ( N * N - 1 ) / 12 )
>
>In my case N = 20 for both control (normal CO2) & double CO2 samples.
>I plan to detrend both samples before comparing
>(1) means using t-test
>(2) variances using F-test.
>
>What is the effect of this detrending on degrees of freedom for these tests?
>Are these tests meaningful?
>
>Len.Makin@mel.dit.csiro.au               | Senior Computer Scientist    |
>CSIRO Division of Information Technology | Supercomputing Support Group |
>723 Swanston Street, Carlton             | Tel: +61 3 282 2622          |
>VIC 3053, Australia                      | Fax: +61 3 282 2600          |
>
See books on experimental design and analysis, under "Orthogonal
Polynomials".  Also, Robson, D.S. (1959), Biometrics,15,187-191, and
Emerson, P.L. (1968), Biometrics,24,695-701.
  The test of H0: "no linear trend" can be done with an F test with
1 df for numerator MS.  The detrended test for "between steps" would
be done with an F test with K-2 df for the numerater MS, because there
is 1 df subtracted for the grand mean, and another for the linear
trend component.  (by K I mean the number of time steps).  At least
I believe these principles carry over to the design you are talking
about.

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