The Drift Function Could Be Exactly Cubic

  The anomaly values used for fitting the drift function in the last blog were those of the lower ends of the sub-intervals. Using the centered x values for the sub-intervals and shifting the center of the anomaly to x = 0.0313 one finds that the coefficients are within 2% of a purely cubic equation.

The Drift Function For The Monthly Global Land Anomaly Appears To Be Stable

  To determine the drift function for the monthly global land anomaly the interval from -2 degrees to 2 degrees was divided up into 20 parts then the anomaly values were scanned to determine the intervals they fell in and the interval of the following anomaly. The result of the scan was a 20 x 20 matrix in which the columns represented the sub-interval, x, the first anomaly fell into and the rows, x', that of the next anomaly. Next an expected value was computed for each column to determine the most likely value of x' given that of x. Finally the expected change, Δx = x' - x, was computed for each value of x. The results are plotted below along with a least squares polynomial fit.

The anomaly data indicates that at higher anomaly values the anomaly is likely to decrease and at lower values it is likely to increase while remaining relatively stationary near the center. This indicates that the anomaly fluctuates about a stable point.

Stochastic Drift And The Central Limit Theorem

  I've been going through some books on stochastic processes such as Gillespie's Markov Processes and Kittel's Elementary Statistical Physics. The two characterizing functions of a continuous Markov process are the drift function and the diffusion function. These seem to be the analogs of what I have called "signal" and "noise" in the preceding blogs. It may be that the Earth's equilibrium temperatures are subject to drift over time which would be difficult to distinguish from global warming. If a drift is persistent it will have consequences over time but would not be as serious as a run-away greenhouse effect.

  Global warming does not appear to be a well studied as a stochastic process. Maxwell, Boltzmann and Gibbs introduced the concepts of statistical physics to the study of gases and Einstein's explanation of Brownian motion extended this to liquids. The Earth can be considered as a physical system in contact with an external heat source and it is normal for there to be fluctuations in temperature but I doubt that we need a Lyapunov control system to clamp the Earth's mean temperature to some arbitrarily chosen fixed value.

  The weighted sum of normal distribution functions that was found to be a good fit for the variation in the monthly global land anomaly may have a variation of the Lyapunov central limit theorem associated with it but it's definitely not a normal distribution.

NCDC September Monthly Climate Reports Out Today

  NOAA's National Climate Data Center regularly releases monthly climate reports for the US and the entire globe and the September 2013 reports are now available online.

Standard Deviation For A Sum Of Normal Distributions

  It can be shown using the mathematical tools found in textbooks on Statistical Mechanics that the standard deviation for a sum of m normal distributions whose individual standard deviations are σ_j is,

So, we don't need to calculate the expected value integral E(x²) with the weighted sum of the individual normal distributions if the individual stds are already known. One can also show that the mean for the summed distribution is the weighted sum of the means for the normal distributions.

Coherence in Statistical and Physical Processes

  We all know how variable the weather from time to time and from place to place and the global means reflect this to a smaller extent. Thermodynamics and Statistical Mechanics tell us that a large physical system can experience fluctuations from its equilibrium state. The Earth's weather also has seasonal variations due to the inclination of the Earth relative to its orbit and annual changes in its distance from the Sun. Cloudy weather can block the amount of sunlight that a location can receive throughout the year. The Earth's rotation produces a daily cycle of heating and cooling. Consequentially the Earth's weather is in a constant state of flux. There may be some coherence between one location and another nearby but there is no guarantee that this is so for large separations of time and space. There are statistical and physical processes for which the mean and variation of a distribution vary with time and there is no long term pattern to observations. We can check to see if there is a "coherence time" for the global anomalies and the test that we designed for predictions might be useful for this. I have put together a list of readings that might help to give another view on global warming along with two references on statistics.

Cyclostationary process

Stationary process

Thermal fluctuations

Gaussian noise

Ornstein-Uhlenbeck process

An Introduction to Scientific Research - Wilson

The Statistical Analysis of Experimental Data - Mandel

Using Statistics to Reject a Projection

  Instead of using an integer multiple of the standard deviation, s, for the limits of a test one can use an arbitrary multiple known as a z-value defined by z = [a-E(a)]/s where a is a limiting value for the difference of the anomaly from the smoothed anomaly or its fit. Using z = 2.0 for the monthly global land anomaly with the three harmonic fit we find that the probability for observing an outlier is q = 5.848% so out of the 1604 months we would expect about 94 outliers. The observed number for the anomaly data of 92 is close to the expected number which is reasonable since we chose the probability function to fit the data. For z =2.0 we would expect the observed number of outliers to range from 75 to 113.

  How can we decide whether or not it is safe to reject a projection when we have collected a number of observations after the prediction? In ten years one would have 120 months of data to work with and suppose for arguments sake we observe k_obs outliers. Using the binomial distribution we can compute the probability, P_k, of observing k outliers and also the probability, 1- P_k, of not observing that number. If the level of certainty is set at 95% and the probability of not observing k_obs outliers is greater than this value we can confidently reject the projection. We would have to pass the projection if k_obs was between 4 and 10 inclusively.