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This site is produced by Mike Haseler AKA Scottish Sceptic # About the "forecast"

If we examines records of temperature like that of the Central England Temperature series (shown right), we will find that climate varies naturally and that the variation present is what is known as "flicker noise" in reference to the similarity of the light intensity of a candle which tends to grow and diminish over time.

This kind of natural variation has the characteristic that long-term changes are larger than shorter-term and for typical flicker noise the relationship is that the size of the variation increases proportionally to the periodicity of the change. Or inverting, the size is equal to 1/f. This is why it is also referred to has 1/f type noise, although more accurately as 1/fn noise where m varies between 0 to 2.

The primary purpose of this site (so far) is to show what pure 1/f noise looks like and so it has been presented humorously on the front page as a "97% accurate" prediction. Indeed like all climate academics I am so confident in these predictions that if they aren't all entirely correct at the end of 100 years then I will gladly run a marathon naked, swim the Forth wearing a pink tutu and then climb Ben Nevis singing "I'm a teapot". 1/f climate "forecast"

## Technical details

There are 1024 data points in the plot but as the plot is only shown 500pixels wide there are around 4-500 discrete points. The 1/f noise is produced using 10 random number generators (giving integers -100 to 100) whereby the the first changes every 2nd point, the second every 4th, the third every 8th, etc. The random number generators are then summed and scaled to give a value from -2 to +2 - although the number seldom go above 1.

The decadal graph is produced by a rolling average equal to 0.05 x datan + .05 x previous. If α is 0.05, this equates to a moving average of (2/α)-1 points. If we take the graph as being a century, with around 400 points across, then the filter parameters used correspond roughly to a moving average filter of 10 years.

You should be able to spot the following typical characteristics of 1/f noise:

• Long term trends - because longer periods are larger, then changes taking longer than the sample period are largest and these will be present as trends.
• Short term cycles - 1/f noise is rich in longer periodic changes. As a result, particularly when shorter term changes are removed, the result can be to leave a series of short-term changes that appear to be some form of a cycle. Typically there will be 3-4 "peaks" and troughs in a row. There will not usually be less because if there are too few peaks and troughs we do not recognise them as a cycle. However because statistically it is becomes increasingly unlikely with more peaks and troughs that they will align so as to appear as a cycle, more than 4 is unusual.
• Steps - are apparent changes in "level". They occur because steps are rich in long-term periodicities. So they are relatively common in 1/f noise.
• Fractal noise - because the size of the variation increases with longer periods, this means that shorter sections tend to appear like longer sections but only scaled down. It is therefore common to find a short section that looks like the overall signal. So, if we take the global warming "signal" which warmed from 1910-1940, cooled and then warmed from 1970-2000 followed by a pause, it is usually possible to find a section in any of the signals that contain a similar /\/-
• Skewed probability distribution - unlike normal white noise, many of the probability distributions show a far from "bell shaped" curve. This is because the probility distribution of any point is Gaussian, but each one is skewed by a "memory" of what went before. So, if the previous values are high - the probability distribution for the next point will be pushed higher and conversely if it is lower. This means that if there is a run tending toward either extreme, we get a "tail" from the run and a final "bell" curve. Combined these appear as a lopsided curve. Probability distribution of "forecast"

## Probability distributions

I've plotted tow simple distributions of the values of the "forecast" split in equal buckets. The first shows the first half of the series of data points and the second is the second half.

This is the main purpose of the site! Which is to try to understand the statistics of 1/f type noise. In theory, the second distribution should be more likely to be offset higher or lower. This shows how this 1/f noise can appear to have "trends" when in fact all that is happening is natural variation.

However the bigger questions are these:

• When is a trend so large that it is no longer compatible with 1/f noise
• If we see this natural 1/f variation tending upwards, what then is the best estimate of the future trend? A rule of thumb says "the higher it is, the more likely it should decrease. That is certain true of white noise 1/f0. But red noise (random walk or 1/f2 is the opposite in that after an upward run, it is just as likely the next point will continue that upward run as that it will come down.

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