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Пишет Journal de Chaource (![[info]](http://lj.rossia.org/img/syndicated.gif){kind=small} lj_chaource)@ 2015-12-20 15:29:00 |
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The uncertainty in the average global warming
Previous posts - http://chaource.livejournal.com/135561.h
Suppose that the temperature T(t) has the form
T(t) = β t + R(t),
where R(t) is a stationary random process with zero mean and correlation function
C(x) = E [ R(t) R(t+x) ].
Also suppose that we only observe T(t) during the interval [ - D/2, D/2 ]. We can estimate β from these observations by the method of least squares: we approximate T(t) by a linear function,
T(t) = a + b t + error,
and we choose a and b such that the integral of square error over [ -D/2, D/2] is at the minimum.
The result is that a and b become random variables with standard deviation that we can compute by the same method as in my previous post. If R(t) is an oscillating function that makes n oscillations of amplitude M, the standard deviation of both a and b*D is of order M/n. The mean of "a" is zero, and the mean of "b" is β (i.e. these estimators are unbiased). The quantity b*D is the mean change in the temperature (i.e. the estimated amount of "climate change") during the observation interval.
I estimated the value of M/n to be of order 1 °C. So, for the Wikipedia-quoted data:
... a 15-year period starting in 1996 shows a rate of increase of 0.14 [0.03 to 0.24] °C per decade, but taking 15 years from 1997 the rate reduces to 0.07 [–0.02 to 0.18] °C per decade.
https://en.wikipedia.org/wiki/Global_war
we find that the total amount of climate change during the observation interval is of order 0.2 °C. This is below 1 sigma and not statistically significant.
In other words, this value could be 0.1 °C or -0.5 °C just as likely, due to random fluctuations, and so we cannot conclude that the temperature has increased or decreased during that interval.
My estimate is of course rough, but it can be made precise by a numerical calculation of the correlation function C(t). How do meteorologists estimate the uncertainty in their global warming?
Previous posts - http://chaource.livejournal.com/135561.h
Suppose that the temperature T(t) has the form
T(t) = β t + R(t),
where R(t) is a stationary random process with zero mean and correlation function
C(x) = E [ R(t) R(t+x) ].
Also suppose that we only observe T(t) during the interval [ - D/2, D/2 ]. We can estimate β from these observations by the method of least squares: we approximate T(t) by a linear function,
T(t) = a + b t + error,
and we choose a and b such that the integral of square error over [ -D/2, D/2] is at the minimum.
The result is that a and b become random variables with standard deviation that we can compute by the same method as in my previous post. If R(t) is an oscillating function that makes n oscillations of amplitude M, the standard deviation of both a and b*D is of order M/n. The mean of "a" is zero, and the mean of "b" is β (i.e. these estimators are unbiased). The quantity b*D is the mean change in the temperature (i.e. the estimated amount of "climate change") during the observation interval.
I estimated the value of M/n to be of order 1 °C. So, for the Wikipedia-quoted data:
... a 15-year period starting in 1996 shows a rate of increase of 0.14 [0.03 to 0.24] °C per decade, but taking 15 years from 1997 the rate reduces to 0.07 [–0.02 to 0.18] °C per decade.
https://en.wikipedia.org/wiki/Global_war
we find that the total amount of climate change during the observation interval is of order 0.2 °C. This is below 1 sigma and not statistically significant.
In other words, this value could be 0.1 °C or -0.5 °C just as likely, due to random fluctuations, and so we cannot conclude that the temperature has increased or decreased during that interval.
My estimate is of course rough, but it can be made precise by a numerical calculation of the correlation function C(t). How do meteorologists estimate the uncertainty in their global warming?
