In tests of the rigor of the Steffen/Wong statement that “not only is the OHC increasing, it is increasing faster“, we previously used a linear regression model including natural cycles. The question was raised about the confounding of an upward trend with part of the quadratic terms representing ‘acceleration’. This risk is increased by the short run of data (only 54 years) and also because the phase of the periodic terms is a free variable. The periodic is free because both sin() and cos() are used.

The phase can be bound easily by the simplification below. I introduce 1976 as a start date for the sin() periodic, the date of the Great Pacific Climate Shift, a widely recognized change in ocean and atmospheric phenomena. The code for obtaining the probability that the model is improved by a quadratic term is then:

` x<-time(SL)`

f<-(time(SL)-1976)*2*pi

m1<-lm(SL~x+sin(f/60))

m2<-lm(SL~x+sin(f/60)+I(x^2))

f<-ftest(m1,m2)

ptest<-1-pf(f[1],f[2],f[3])

The results below for existing data and data with an appended low value for 2009 show little change. The p values are 0.07 and 0.12 respectively, indicating no improvement in the model by adding a quadratic term. With the suggested alterations, we again find **no empirical justification for the Senator and her advisor’s statement**, when we include natural cycles in the model, i.e. for OHC to be increasing faster we have to exclude natural cycles as a possible explanation.

Previous result with free phase:

The change in OHC from peak to trough is 2*2.7 or 5.4 x10^{22} joules, while the linear increase in heat over the same period is 0.33*54 or 17.8X10^{22} joules. The coefficients of the linear regression suggest periodic oceanographic phenomena could account for 30% of the increase in OHC since 1958.

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David, the NODC corrected it’s OHC data today.

http://bobtisdale.blogspot.com/2009/10/nodcs-correction-to-ohc-0-700m-data.html

My apologies to Nick Stokes (who suggested caution) have been posted on the earlier thread.

Yes, I saw it. Does not affect these analyses though.

David, the NODC corrected it's OHC data today.http://bobtisdale.blogspot.com/2009/10/nodcs-co…My apologies to Nick Stokes (who suggested caution) have been posted on the earlier thread.

Yes, I saw it. Does not affect these analyses though.

What a surprise, another adjustment which goes in favour of AGW or lessons contrary evidence; still, the downturn is evident; and has there been any further developments about this;

http://wattsupwiththat.com/2009/06/02/anomalous-spike-in-ocean-heat-content/#more-8132

Is this an artifact of the transition in the measurement process? If so the idea that OHC is increasing at all is suspect let alone at an increasing rate.

What a surprise, another adjustment which goes in favour of AGW or lessons contrary evidence; still, the downturn is evident; and has there been any further developments about this;http://wattsupwiththat.com/2009/06/02/anomalous…Is this an artifact of the transition in the measurement process? If so the idea that OHC is increasing at all is suspect let alone at an increasing rate.

You compare “trend+cycle” to “trend+cycle+quadratic”. The latter will be penalized for having more parameters. I think you should compare the skill of these two models:

* trend + cycle

* quadratic

I suspect that the quality of their fits will be indistinguishable when you consider the level of noise in the OHC observations. I would argue that the quadratic is the simpler model because i think the cycle model has many more free parameters.

Wavelength, and phase are specified apriori, but i would still argue that they are a quasi-free parameters. If your hypothesis is “trend+PDO”, why dont you use a PDO index as predictor rather than an idealized cycle?

Note, I am not arguing against longish cycles in OHC.

Aslak, it is penalized, but only in the case where the number of data is few, due to the F test. In this case it probably would be, but that is what you want after all, as you don’t want to be overfitting.

Your choices probably are about the same, but refer to the discussion with Nick, a lot hands on the prior assumptions of how you think the data should be explained. I see the models in the RC post suggest an acceleration too, so there is another basis for a prior belief to take into account.

I am interested in the periodics as its a clean and transparent approach. I would never rely on just one type of model, and clearly someone should fit PDO, cSOI, etc to OHC.

Aslak, A more to the point answer to the issue of Occam’s razor that you raise, is what is the reality? We know that there are cycles, we know that there might be acceleration, the question is can the correct proportion be recovered and under what conditions? The number of parameters doesn’t seem like a basis for elimination of cycles, but more of a possible source of difficuly in estimating CIs.

You compare “trend+cycle” to “trend+cycle+quadratic”. The latter will be penalized for having more parameters. I think you should compare the skill of these two models:* trend + cycle * quadraticI suspect that the quality of their fits will be indistinguishable when you consider the level of noise in the OHC observations. I would argue that the quadratic is the simpler model because i think the cycle model has many more free parameters. Wavelength, and phase are specified apriori, but i would still argue that they are a quasi-free parameters. If your hypothesis is “trend+PDO”, why dont you use a PDO index as predictor rather than an idealized cycle?Note, I am not arguing against longish cycles in OHC.

Aslak, it is penalized, but only in the case where the number of data isfew, due to the F test. In this case it probably would be, but that is whatyou want after all, as you don't want to be overfitting.Your choices probably are about the same, but refer to the discussion withNick, a lot hands on the prior assumptions of how you think the data shouldbe explained. I see the models in the RC post suggest an acceleration too,so there is another basis for a prior belief to take into account.I am interested in the periodics as its a clean and transparent approach. Iwould never rely on just one type of model, and clearly someone should fitPDO, cSOI, etc to OHC.

David,

This still won’t work. It’s better, because fixing the phase reduces the confounding a little. But not much.

To show this, I replaced the OHC data by a vector made up of just a parabola section plus white noise. No PDO there:

set.seed(199) # fix the seed to make it reproducible

x=1:52

C=x*x+50*x-400+500*rnorm(52) # artificial data

C<-ts(C,start=1958)

Then I did an extra fit stage at the start, fitting just a line:

l0<-lm(sw~x)l1 and l2 as before. Then I tested for the significance of first adding sin(f), then x^2.

ft<-ftest(l0,l1)ptest<-1-pf(ft[1],ft[2],ft[3])

print(ptest) # test for adding the sin

ft<-ftest(l1,l2)

ptest<-1-pf(ft[1],ft[2],ft[3])

print(ptest) # test for further adding quadratic

Adding sin(f) showed p=.0052. Highly significant,99.5%, even though there was no PDO there.

Then adding a quadratic (x^2) showed p=0.519, not even significant at 50%. Even though the underlying function

wasa quadratic.Same explanation – the sinusoid could fit the function well enough, and adding the quadratic made little improvement.

That is your point about not enough data to discriminate, you could do the same thing with sinusoidal synthetic data. Be a different story with more data though.

Another version is that if you want to do what you’re trying, and have the effect of each addition independently significant (or not), then the functions should be nearly orthogonal to avoid confounding. A sine with several periods is close to orthogonal to a smooth function like a quadratic. But one with a fractional period is not.

Yes thats another way to look at it, though another issue I am trying to get at is that such analysis should incorporate a universe of possible functions: linear, quadratic or periodic. To test only quadratic is to assume that response, so finding of significance is a circular argument.

Eg, the argument is made against natural cycles as a cause of recent AGW that it must be shown first that cycles are capable of causing the forcing seen in the last 50 years. However, there is a data history of strong climate cycles of around 60 years, so the onus should be to show the relative proportion of natural cycle vs linear or other type of forcing, even if they are not completely understood. My analysis shows that periodic is the greatest contribution to temperature increases of the last 50 years, and predicts a moderation of temperature increases (as AGW alarmists are now scrambling to explain).

That doesn’t mean it is all periodic, a small linear component persists of around 0.05C/decade, consistent with Lindzen, Spencer, Douglass etc.

David,

“My analysis shows that periodic is the greatest contribution to temperature increases of the last 50 years”Your analysis shows the same result for a parabolic increase plus noise. And there’s nothing to indicate that would moderate. Your inference doesn’t work.

Nick, You are thinking of OHC I think. Temperature is a longer record, though the analysis you suggest would be worth doing.

David,This still won't work. It's better, because fixing the phase reduces the confounding a little. But not much.To show this, I replaced the OHC data by a vector made up of just a parabola section plus white noise. No PDO there:

set.seed(199) # fix the seed to make it reproduciblex=1:52C=x*x+50*x-400+500*rnorm(52) # artificial dataC<-ts(C,start=1958)Then I did an extra fit stage at the start, fitting just a line:l0<-lm(sw~x)l1 and l2 as before. Then I tested for the significance of first adding sin(f), then x^2.ft<-ftest(l0,l1) ptest<-1-pf(ft[1],ft[2],ft[3]) print(ptest) # test for adding the sin ft<-ftest(l1,l2) ptest<-1-pf(ft[1],ft[2],ft[3]) print(ptest) # test for further adding quadraticAdding sin(f) showed p=.0052. Highly significant,99.5%, even though there was no PDO there.Then adding a quadratic (x^2) showed p=0.519, not even significant at 50%. Even though the underlying functionwasa quadratic.Same explanation – the sinusoid could fit the function well enough, and adding the quadratic made little improvement.That is your point about not enough data to discriminate, you could do the same thing with sinusoidal synthetic data. Be a different story with more data though.

Another version is that if you want to do what you're trying, and have the effect of each addition independently significant (or not), then the functions should be nearly orthogonal. A sine with several periods is close to orthogonal to a smooth function like a quadratic. But one with a fractional period is not.

Yes thats another way to look at it, though another issue I am trying to getat is that such analysis should incorporate a universe of possiblefunctions: linear, quadratic or periodic. To test only quadratic is toassume that response, so finding of significance is a circular argument.Eg, the argument is made against natural cycles as a cause of recent AGWthat it must be shown first that cycles are capable of causing the forcingseen in the last 50 years. However, there is a data history of strongclimate cycles of around 60 years, so the onus should be to show therelative proportion of natural cycle vs linear or other type of forcing,even if they are not completely understood. My analysis shows that periodicis the greatest contribution to temperature increases of the last 50 years,and predicts a moderation of temperature increases (as AGW alarmists are nowscrambling to explain).That doesn't mean it is all periodic, a small linear component persists ofaround 0.05C/decade, consistent with Lindzen, Spencer, Douglass etc.

David,

“My analysis shows that periodic is the greatest contribution to temperature increases of the last 50 years”Your analysis shows the same result for a parabolic increase plus noise. And there's nothing to indicate that would moderate. Your inference doesn't work.Nick, You are thinking of OHC I think. Temperature is a longer record,though the analysis you suggest would be worth doing.

The adjustment to NODC data [as usual the adjustment is up] still shows a down trend in recent OHC;

http://wattsupwiththat.com/2009/10/15/ocean-heat-content-cooling-gone-today-with-new-adjustment/

On top of that is the 2002-2003 ‘artifact’ which is responsible for over 1/2 of the OHC increase in the data range has not been addressed by NODC. And speaking of temperature SST is down;

http://wattsupwiththat.com/2009/10/15/ocean-heat-content-cooling-gone-today-with-new-adjustment/

And [thermosteric] sea-level increase is shot to bits;

http://www.ocean-sci-discuss.net/6/31/2009/osd-6-31-2009.html

Where’s the heat Nick? ERBE perhaps?

The adjustment to NODC data [as usual the adjustment is up] still shows a down trend in recent OHC;http://wattsupwiththat.com/2009/10/15/ocean-hea…On top of that is the 2002-2003 'artifact' which is responsible for over 1/2 of the OHC increase in the data range has not been addressed by NODC. And speaking of temperature SST is down;http://wattsupwiththat.com/2009/10/15/ocean-hea…And [thermosteric] sea-level increase is shot to bits;http://www.ocean-sci-discuss.net/6/31/2009/osd-…Where's the heat Nick? ERBE perhaps?

Aslak, A more to the point answer to the issue of Occam's razor that you raise, is what is the reality? We know that there are cycles, we know that there might be acceleration, the question is can the correct proportion be recovered and under what conditions? The number of parameters doesn't seem like a basis for elimination of cycles, but more of a possible source of difficuly in estimating CIs.

Nick, You are thinking of OHC I think. Temperature is a longer record, though the analysis you suggest would be worth doing.

The adjustment to NODC data [as usual the adjustment is up] still shows a down trend in recent OHC;http://wattsupwiththat.com/2009/10/15/ocean-hea…On top of that is the 2002-2003 'artifact' which is responsible for over 1/2 of the OHC increase in the data range has not been addressed by NODC. And speaking of temperature SST is down;http://wattsupwiththat.com/2009/10/15/ocean-hea…And [thermosteric] sea-level increase is shot to bits;http://www.ocean-sci-discuss.net/6/31/2009/osd-…Where's the heat Nick? ERBE perhaps?

Aslak, A more to the point answer to the issue of Occam's razor that you raise, is what is the reality? We know that there are cycles, we know that there might be acceleration, the question is can the correct proportion be recovered and under what conditions? The number of parameters doesn't seem like a basis for elimination of cycles, but more of a possible source of difficuly in estimating CIs.

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