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and Its Predictability |
Some define the climate of a region in terms of its average weather over a number of years; i.e., the mean values of temperature, cloudiness, rainfall and so forth. This notion of climate is not adequate because the variability of weather is another essential aspect of the climate of a region. For example, Texas is noted for the variability of its weather.
The definition above could be extended to include not only mean values but also measures of variation such as ranges, extreme values and variances. However, what is being touched upon is that it is the probability distributions that define the climate of a region. These probability distributions of course depend upon the season and could be subject to other cycles and trends. They could also be subject to change due to exogenous circumstances such as volcanic eruptions.
The interesting question is whether these probability distributions of weather are completely general or whether they necessarily belong to a family of probability distributions and hence are describable by a limited set of parameters. Many but not all variables in nature have a so-called normal or bell-shaped distribution. Such distributions are completely specified by two parameters, the mean and variance. There is a central limit theorem which says that variables which are the sum of a large number of independent random variables will tend to have a normal distribution.
As an example of this matter consider the distribution of the changes in average global temperatures.
This is a reasonable approximation of a normal distribution. For more on the question of whether the distribution of changes in global average temperature is normal see Normality
Although the central limit theorem is quite general there are limits to its applicability. The independent random variables referred to in the previous sentence have to have finite variance.
There is a generalized central limit theorem which would suggest that the weather characteristics being the result of a large number of independent random variables would have a Levy-stable distribution. The normal distribution is one of those Levy-stable distributions. For an application of Levy-stable distributions to rainfall data see Rainfall in San Jose.
The Climate Data Center for the National Oceanic and Atmospheric Administration in Asheville, North Carolina has the best store of climate data. In the 1990's this organization, under the direction of Thomas R. Karl, began considering questions concerning possible shifts in the distributions of weather variables in connection with the debate on global warming. The results published suggested an increase in extreme weather events. It would be desirable if this organization separated itself from the polemics concerning global warming and climate change and focused on fundamental matters such as the nature of the distributions and how to rigorously test for changes.
In the early 1960's the meteorologist and mathematician Edward Lorenz was investigating nonlinear mathematical systems as analogues of weather forecasting models. By fortuitous chance he discovered the mathematical phenomenon of chaos.
A chaotic system is one that exhibits an extreme sensitivity to initial condiions. Even though the value for one time is deterministically given by the value at a previous time, say
the relationship can become so complex after a number of iterations that the later values as t→∞ appear to be random.
For example, consider the case of the logistics curve where f(x)=αx(1-x). For some values of α, x_{t} will go to an equilibrium point and remain there forever, as shown below:
For other values of α, x_{t} will approach a two period cycle; e.g.,
For still other values of α, x_{t} will approach a four period cycle; e.g.,
For yet other values of α, x_{t} will follow an infinite period cycle; i.e., one that never repeats. Such a solution is shown below:
The sequence appears to the eye to be chaotic and such that one value seems unrelated to the previous value. It is as though the values of x_{t} are random. However if x_{t+1} is plotted versus x_{t} the pattern is apparent:
The distribution of the values of x_{t} for the above chaotic case is as follows
For systems characterized by a deterministic transformation of the type
one thing of interest is equilibria; i.e., values of x such that x=f(x). There are some remarkable mathematical theorems, called fixed point theorems, having to do with the existence of such solutions. Repeating cycles can be included in such analysis by looking fixed points of iterations of the transformation. For example, two period cycles are fixed points of f(f(x)). (Equilibria are just special cases of a two period cycles.)
One can also look at how the distribution for x_{t} get transformed into the distribution for x_{t+1}. It is this perspective that is relevant in the matter of climate. A stable climate would be one that the distributions of weather variables at one point in time leads to the same distributions at later points in time. What is need is a fixed point theorem for distributions. Such a fixed point would constitute a stochastic equilibrium.
One approach to climate theory would be to look for probability density functions such the distribution of the output of a transformation is the same as the distribution of the inputs.
For illustration consider first the simple case when the transformation y=f(x) is monotonic. The monotonicity of the transformation guarantees that there exists an inverse function x=f^{-1}(y). If p(x) is the probablity density function of x then the probability density function of y, q(y), is given by
This means that q(y)=p(x)|f'(x)|, and thus q(y) is the same as f(x) if and only if f(x)=±1.
This applies only to monotonic transformations.
For the logistic function y=α[x−x²] and thus dy/dx=α[1−2x] and thus
If we are looking for the distribution of x such that y has the same distribution then p(x) must satisfy the functional equation
Functional equations are generally hard to solve. Let α=4.0. The distribution of the values of x for sample of size 10,000 looks like this:
The regularity of the result suggests that the reciprocal of the frequencies would be worth looking at. These are:
This suggests the solution is some function of x(1-x). If the solution were p(x)=β/[x(1-x)] where β is a parameter to be determined then the frequency f(x) multiplied times x(1-x) would be a constant over x. The plot of f(x)*x(1-x) is shown below:
This is not a constant over x. The fact that the plot has a parabolic shape suggests that a lower power of x(1-x) would give a plot more nearly constant. The plot of frequency multiplied times the square root of x(1-x) is closer to a constant.
Thus the solution appears to be
A numerical computation of the LHS and RHS of the functional equation
reveals that indeed p(x)=β/[x(1-x)]^{½} is a solution. The constant β is found so that integration of p(x) from 0 to 1 yields a value of 1.0.
The integral of β/[x(1-x)]^{½} is −sin^{-1}(1−2x) and thus over the interval [0,1] the integration is equal to π. Thus the probability distribution is
The graph of this function is shown below:
The above illustrates the existence of distributional solutions to a transformation. The formulation of the problem may be too stringent. Instead of requiring the output distribution to be exactly the same as the input distribution it appears that the process would have to be cyclic so that eventually the input distribution would reappear.
(To be continued.)
In statistical mechanics a system will, starting from a specific point, go through all of the possible configurations. Furthermore the average of characteristic of the system as it follows the time path from any initial starting point is the same and is equal to the average for probability distribution, usually called the ensemble, for the system.
This is what is assumed without any theoretical justification in climate model projections. That is to say, the projecters run their models and presume that some average of the projections will be the same as the average for the future distribution of the weather variables. This might be so, and then again it might not be so.
This question was investigated by Edward Lorenz before his death in 2008 at age 90. His work on chaos theory established why weather could not be accurately predicted beyond a week to ten days. Given this limit there is a serious question of whether future climate can be predicted. Lorenz' answer on the predictability of climate was a tentative affirmative. His paper is included in the published preceedings of a conference on the question of predictability, The Predictability of the Weather and Climate (2006) edited by Tim Palmer and Renate Hagedorn.
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